Science

Ebook

2009-02-05T17:15:00.001-08:00

Gear Design

2009-02-04T02:40:00.000-08:00

Design - DSGN 215, 221 Gear Design Home Page.

Introduction.
When mating gear teeth are designed to produce a constant angular velocity ratio during meshing they are said to have 'conjugate action'. To provide this an 'involute' type profile is almost universally used for tooth forms.

There are two modes that are important causes of gear failures. Bending stresses (leading to tooth breakage) which are a maximum at the tooth root and compressive stresses (leading to pitting) that are a maximum on the tooth face. Because tooth loading is cyclic, both of these mechanisms are of a fatigue nature. The design of gears needs to counter both of these potential failure modes. An important part of providing the resistance to the high contact stresses is to use gears of appropriate hardness. The lower the levels of impurities in a material, the better it is normally able to resist fatigue.

Link to page showing some gear nomenclature.

Quite a bit of information is available about gears on the following sites:
DR Gears site.
Quality Transmission Components site, go to QTC Technical Library.

Loads on Teeth

The tangential force on the teeth can be found from:

W_t = 60H/(3.14159.d.n) where:

W_t = transmitted load, N

H = power, W

d = gear diameter, m

n = speed, rev/min.

Bending Stresses - The Lewis Formula

Although this was published in 1893, it is still very widely used for assessing bending stresses when designing gears. The method involves moving the tangential force and applying it to the tooth tip and assuming the load is uniformly distributed accross the tooth width with the tooth acting as a simple cantilever of constant rectangular cross section, the beam depth being put equal to the thickness of the tooth root (t) and the beam width being put equal to the tooth, or gear, width (b_w).

The section modulus is I/c = b_wt²/6 so the bending stress is given by:

sigma_bending = M/(I/c) = 6W_tL/(b_wt²) eqn.1.

Assuming that the maximum bending stress is at point 'a'.

By similar triangles: or eqn.2

Rearranging eqn.1 gives:

Substitute the value for 'x' from eqn.2 and multiply the numerator and denominator by the circular pitch, 'p' gives:

let y=2x/3p then

This is the original Lewis equation and 'y' is called the Lewis form factor which may be determined graphically or by computation.

Engineers often now work with the 'diametral pitch', 'P', and or the 'module', 'm', which is 1/diametral pitch = 1/P

Then where Y = 2xP/3

Written in terms of the module:

sigma_bending = W_t/(b_wmY)

The Lewis form factor considers only static loading, it is dimensionless, independent of tooth size and is a only a function of tooth shape. It does not take into account the stress concentration that exists in the tooth fillet.

The Lewis formula is generally limited to pitch line velocities up to 7.6 m/s and based on tests (in the 19th Century) on cast iron gears with cast teeth, C G Barth suggested a modification involving a velocity factor, K_V.

In SI units this was K_V = 3.05/(3.05 + V)
For cut or milled teeth the Barth equation (in SI units) is often modified to: K_V = 6.1/(6.1 + v), giving

sigma_{bending, allowable} = W_t/b_wmYK_V

NB Recent Changes in Barth Equation
In about 2000 the AGMA (see below) re-defined the dynamic factor, K_V, as the inverse of that originally proposed by Barth, above. Consequently it is greater than 1 (and called K_V' here) and the expression for the allowable bending stress becomes:

sigma_{bending, allowable} = W_tK_V'/b_wmY

For a full depth tooth with a 20^o pressure angle, Y varies between 0.245 for a gear with 12 teeth to 0.471 for a gear with 300 teeth. (0.485 for a rack).

It is common for spur gears to be designed with a face width of between 3 and 5 times the circular pitch.

American Gear Manufacturers Association (AGMA) Code

Some key points from the AGMA approach to gear design are shown below.

AGMA have published graphs of allowable bending stresses and allowable surface contact stresses as a function of the Brinell hardness for some grades of through hardened steel.

Grade 2 steels, that have higher allowable stresses, are more closely specified than grade 1 steels.
This is commonly used and contains further refinements compared to the approach above, it also includes detailed guidance about materials.
A number of modifying factors are normally included in the AGMA code:

K_a = application factor - depends on the type of power source

K_s = size factor - increases above 1 for a module, m, of 6 mm or greater.

K_m = load distribution factor - depends mainly on face width.

K_v = dynamic factor - depending upon tooth accuracy, loads greater than the transmitted load may be generated.

Contact Stresses

These are determined by Hertzian contact stress analysis. The maximum pressure in the (rectangular) contact zone when two parallel cylinders are pressed together is given by:

where E' is the effective modulus of elasticity:

W' is the dimensionless load = w'/(E'/R_x)

w' is the load per unit width = normal load/gear width and

1/R_x = 2((1/pinion dia)+(1/gear dia))/sin(pressure angle)

The same modifying factors are again used with the contact stresses that were used with the bending stresses.

sigma_compressive = p_H(K_aK_s K_m/K_v)^0.5

Depending upon the heat treatment, the maximum bending stresses for hardened gears can be in the range of 200 - 360 MPa and the maximum compressive stress 500 to 1200 MPa.

Link to example gear tooth strength calculation, note file size: 214 kB.

Forces on Helical Gear Teeth
As the teeth on helical gears are inclined to the axis of the gear, the tooth force generates an axial or thrust load in addition to the radial force and the tangential force (which is the only one that does useful work). The tooth load can be resolved in the three directions as shown in this diagram.
The forces and resulting bending moments on the shaft carrying a helical gear are illustrated in this note.

Stresses in helical gear teeth
When calculating the stresses in helical gear teeth, the Lewis form factor for the 'virtual number of teeth' needs to be used rather than that for the actual number of teeth. This is because on looking along a tooth on a helical gear the apparent radius of the gear is greater than that of the gear blank (the cross section is an ellipse). The 'virtual number of teeth' is found by:

virtual number of teeth = actual number of teeth /(cosine of the helix angle)³

Supplementary note on Stresses in Gears
It should be noted that using the simplified version of these methods (including only K_v) on gears in automotive gear boxes gives high stresses, particularly for 1st gear. Gears for car gear boxes are probably manufactured to a high degree of accuracy, to keep noise levels low. The velocity correction factor, K_v, from the Barth equation, is over 100 years old and probably gives a conservative factor compared to that appropriate for modern high quality gears. Using all the AGMA factors - and noting that 'Y' the Lewis form factor is NOT used but the geometry factors J or I, (for bending and compressive stresses) are included, should give a more useful answer.
Note that 1st gear may be designed for a limited life as it so rarely operates under maximum load.

Link to example on helical gear tooth strength calculation, 85kB file size.

Torque Acting on a Gearbox
As the input and output torques associated with a gearbox are normally not the same, some 'holding' torque will be needed to prevent the gearbox rotating. This can be determined as shown in this diagram

Lubrication
During operation the teeth are sliding against one another, so to prevent wear lubrication is normally essential for heavily loaded gears. Even though a gearbox may have an efficiency of 97%, where considerable power is being transmitted, 3% loss as heat generated within the gearbox, may necessitate the provision of some type of forced cooling.

Acceleration of a Geared System

link to notes.

example on a geared hoist, 56 kB file size

More detailed information about gears and some helpful animations can be seen in the chapter on 'Gears' at Mechanical Engineering Department pages at the University of Western Australia.

Useful information (albeit in imperial units) can be found on the 'Boston Gear' web site: click here

Further Reading: Shigley and Mischke, chapters 13, 14 and 15.

Return to Module Introduction

David J Grieve, 24th November 2005.

Spur Gear Design

2009-02-04T02:36:00.000-08:00

The notes below relate to spur gears. Notes specific to helical gears are included on a separate page Helical Gears

Introduction

Gears are machine elements used to transmit rotary motion between two shafts, normally with a constant ratio. The pinion is the smallest gear and the larger gear is called the gear wheel.. A rack is a rectangular prism with gear teeth machined along one side- it is in effect a gear wheel with an infinite pitch circle diameter. In practice the action of gears in transmitting motion is a cam action each pair of mating teeth acting as cams. Gear design has evolved to such a level that throughout the motion of each contacting pair of teeth the velocity ratio of the gears is maintained fixed and the velocity ratio is still fixed as each subsequent pair of teeth come into contact. When the teeth action is such that the driving tooth moving at constant angular velocity produces a proportional constant velocity of the driven tooth the action is termed a conjugate action. The teeth shape universally selected for the gear teeth is the involute profile.

Consider one end of a piece of string is fastened to the OD of one cylinder and the other end of the string is fastened to the OD of another cylinder parallel to the first and both cylinders are rotated in the opposite directions to tension the string(see figure below). The point on the string midway between the cylinder P is marked. As the left hand cylinder rotates CCW the point moves towards this cylinder as it wraps on . The point moves away from the right hand cylinder as the string unwraps. The point traces the involute form of the gear teeth.

The lines normal to the point of contact of the gears always intersects the centre line joining the gear centres at one point called the pitch point. For each gear the circle passing through the pitch point is called the pitch circle. The gear ratio is proportional to the diameters of the two pitch circles. For metric gears (as adopted by most of the worlds nations) the gear proportions are based on the module.

m = (Pitch Circle Diameter(mm)) / (Number of teeth on gear).

In the USA the module is not used and instead the Diametric Pitch d _pis used

d _p = (Number of Teeth) / Diametrical Pitch (inches)

Profile of a standard 1mm module gear teeth for a gear with Infinite radius (Rack ).
Other module teeth profiles are directly proportion . e.g. 2mm module teeth are 2 x this profile

Many gears trains are very low power applications with an object of transmitting motion with minium torque e.g. watch and clock mechanisms, instruments, toys, music boxes etc. These applications do not require detailed strength calculations.

Standards

AGMA 2001-C95 or AGMA-2101-C95 Fundamental Rating factors and Calculation Methods for involute Spur Gear and Helical Gear Teeth
BS 436-4:1996, ISO 1328-1:1995..Spur and helical gears. Definitions and allowable values of deviations relevant to corresponding flanks of gear teeth
BS 436-5:1997, ISO 1328-2:1997..Spur and helical gears. Definitions and allowable values of deviations relevant to radial composite deviations and runout information
BS ISO 6336-1:1996 ..Calculation of load capacity of spur and helical gears. Basic principles, introduction and general influence factors
BS ISO 6336-2:1996..Calculation of load capacity of spur and helical gears. Calculation of surface durability (pitting)
BS ISO 6336-3:1996..Calculation of load capacity of spur and helical gears. Calculation of tooth bending strength
BS ISO 6336-5:2003..Calculation of load capacity of spur and helical gears. Strength and quality of materials

If it is necessary to design a gearbox from scratch the design process in selecting the gear size is not complicated - the various design formulea have all been developed over time and are available in the relevant standards. However significant effort, judgement and expertise is required in designing the whole system including the gears, shafts , bearings, gearbox, lubrication. For the same duty many different gear options are available for the type of gear , the materials and the quality. It is always preferable to procure gearboxes from specialised gearbox manufacturers

Terminology - spur gears

Diametral pitch (d _p )...... The number of teeth per one inch of pitch circle diameter.
Module. (m) ...... The length, in mm, of the pitch circle diameter per tooth.
Circular pitch (p)...... The distance between adjacent teeth measured along the are at the pitch circle diameter
Addendum ( h _a )...... The height of the tooth above the pitch circle diameter.
Centre distance (a)...... The distance between the axes of two gears in mesh.
Circular tooth thickness (ctt)...... The width of a tooth measured along the are at the pitch circle diameter.
Dedendum ( h _f )...... The depth of the tooth below the pitch circle diameter.
Outside diameter ( D _o )...... The outside diameter of the gear.
Base Circle diameter ( D _b ) ...... The diameter on which the involute teeth profile is based.
Pitch circle dia ( p ) ...... The diameter of the pitch circle.
Pitch point...... The point at which the pitch circle diameters of two gears in mesh coincide.
Pitch to back...... The distance on a rack between the pitch circle diameter line and the rear face of the rack.
Pressure angle ...... The angle between the tooth profile at the pitch circle diameter and a radial line passing through the same point.
Whole depth...... The total depth of the space between adjacent teeth.

Spur Gear Design

The spur gear is is simplest type of gear manufactured and is generally used for transmission of rotary motion between parallel shafts. The spur gear is the first choice option for gears except when high speeds, loads, and ratios direct towards other options. Other gear types may also be preferred to provide more silent low-vibration operation. A single spur gear is generally selected to have a ratio range of between 1:1 and 1:6 with a pitch line velocity up to 25 m/s. The spur gear has an operating efficiency of 98-99%. The pinion is made from a harder material than the wheel. A gear pair should be selected to have the highest number of teeth consistent with a suitable safety margin in strength and wear. The minimum number of teeth on a gear with a normal pressure angle of 20 desgrees is 18.

The preferred number of teeth are as follows

12 13 14 15 16 18 20 22 24 25 28 30 32 34 38 40 45 50 54 60
64 70 72 75 80 84 90 96 100 120 140 150 180 200 220 250

Materials used for gears

Mild steel is a poor material for gears as as it has poor resistance to surface loading. The carbon content for unhardened gears is generally 0.4%(min) with 0.55%(min) carbon for the pinions. Dissimilar materials should be used for the meshing gears - this particularly applies to alloy steels. Alloy steels have superior fatigue properties compared to carbon steels for comparable strengths. For extremely high gear loading case hardened steels are used the surface hardening method employed should be such to provide sufficient case depth for the final grinding process used.


Material	Notes	applications
Ferrous metals
Cast Iron	Low Cost easy to machine with high damping	Large moderate power, commercial gears
Cast Steels	Low cost, reasonable strength	Power gears with medium rating to commercial quality
Plain-Carbon Steels	Good machining, can be heat treated	Power gears with medium rating to commercial/medium quality
Alloy Steels	Heat Treatable to provide highest strength and durability	Highest power requirement. For precision and high precisiont
Stainless Steels (Aust)	Good corrosion resistance. Non-magnetic	Corrosion resistance with low power ratings. Up to precision quality
Stainless Steels (Mart)	Hardenable, Reasonable corrosion resistance, magnetic	Low to medium power ratings Up to high precision levels of quality
Non-Ferrous metals
Aluminium alloys	Light weight, non-corrosive and good machinability	Light duty instrument gears up to high precision quality
Brass alloys	Low cost, non-corrosive, excellent machinability	low cost commercial quality gears. Quality up to medium precision
Bronze alloys	Excellent machinability, low friction and good compatability with steel	For use with steel power gears. Quality up to high precision
Magnesium alloys	Light weight with poor corrosion resistance	Ligh weight low load gears. Quality up to medium precision
Nickel alloys	Low coefficient of thermal expansion. Poor machinability	Special gears for thermal applications to commercial quality
Titanium alloys	High strength, for low weight, good corrosion resistance	Special light weight high strength gears to medium precision
Di-cast alloys	Low cost with low precision and strength	High production, low quality gears to commercial quality
Sintered powder alloys	Low cost, low quality, moderate strength	High production, low quality to moderate commercial quality
Non metals
Acetal (Delrin	Wear resistant, low water absorbtion	Long life , low load bearings to commercial quality
Phenolic laminates	Low cost, low quality, moderate strength	High production, low quality to moderate commercial quality
Nylons	No lubrication, no lubricant, absorbs water	Long life at low loads to commercial quality
PTFE	Low friction and no lubrication	Special low friction gears to commercial quality

Equations for basic gear relationships

It is acceptable to marginally modify these relationships e.g to modify the addendum /dedendum to allow Centre Distance adjustments. Any changes modifications will affect the gear performance in good and bad ways...

Addendum	h _a = m = 0.3183 p
Base Circle diameter	D_b = d.cos α
Centre distance	a = ( d _g + d _p) / 2
Circular pitch	p = m.π
Circular tooth thickness	ctt = p/2
Dedendum	h _f = h - a = 1,25m = 0,3979 p
Module	m = d /n
Number of teeth	z = d / m
Outside diameter	D _o = (z + 2) x m
Pitch circle diameter	d = n . m ... (d _g = gear & d _p = pinion )
Whole depth(min)	h = 2.25 . m
Top land width(min)	t _o = 0,25 . m

Module (m)

The module is the ratio of the pitch diameter to the number of teeth. The unit of the module is milli-metres.Below is a diagram showing the relative size of teeth machined in a rack with module ranging from module values of 0,5 mm to 6 mm

The preferred module values are

0,5 0,8 1 1,25 1,5 2,5 3 4 5 6 8 10 12 16 20 25 32 40 50

Normal Pressure angle α

An important variable affecting the geometry of the gear teeth is the normal pressure angle. This is generally standardised at 20^o. Other pressure angles should be used only for special reasons and using considered judgment. The following changes result from increasing the pressure angle

Reduction in the danger of undercutting and interference
Reduction of slipping speeds
Increased loading capacity in contact, seizure and wear
Increased rigidity of the toothing
Increased noise and radial forces

Gears required to have low noise levels have pressure angles 15^o to17.5^o

Contact Ratio

The gear design is such that when in mesh the rotating gears have more than one gear in contact and transferring the torque for some of the time. This property is called the contact ratio. This is a ratio of the length of the line-of-action to the base pitch. The higher the contact ratio the more the load is shared between teeth. It is good practice to maintain a contact ratio of 1.2 or greater. Under no circumstances should the ratio drop below 1.1.

A contact ratio between 1 and 2 means that part of the time two pairs of teeth are in contact and during the remaining time one pair is in contact. A ratio between 2 and 3 means 2 or 3 pairs of teeth are always in contact. Such as high contact ratio generally is not obtained with external spur gears, but can be developed in the meshing of an internal and external spur gear pair or specially designed non-standard external spur gears.

                   (R_go² - R_gb² )^1/2 + (R_po² - R_pb² )^1/2  -  a sin α
contact ratio m =                                                                  
     p cos α

R _go = D _go / 2..Radius of Outside Dia of Gear
R _gb = D _gb / 2..Radius of Base Dia of Gear
R _po = D _po / 2..Radius of Outside Dia of Pinion
R _pb = D _pb / 2..Radius of Base Dia of Pinion
p = circular pitch.
a = ( d _g+ d _p )/2 = center distance.

Spur gear Forces, torques, velocities & Powers

F = tooth force between contacting teeth (at angle pressure angle α to pitch line tangent. (N)
F _t = tangential component of tooth force (N)
F _s = Separating component of tooth force
α= Pressure angle
d ₁ = Pitch Circle Dia -driving gear (m)
d ₂ = Pitch Circle Dia -driven gear (m)
ω ₁ = Angular velocity of driver gear (Rads/s)
ω ₂ = Angular velocity of driven gear (Rads/s)
z ₁ = Number of teeth on driver gear
z ₂ = Number of teeth on driven gear
P = power transmitted (Watts)
M = torque (Nm)
η = efficiency

Tangential force on gears F _t = F cos α

Separating force on gears F _s = F _t tan α

Torque on driver gear T ₁ = F _t d ₁ / 2

Torque on driver gear T ₂ = F _t d ₂ / 2

Speed Ratio =ω ₁ / ω ₂ = d ₂ / d ₁ = z ₂ /z ₁

Input Power P ₁ = T₁ .ω ₁

Output Power P ₂ =η.T ₁ .ω ₂

Spur gear Strength and durability calculations

Designing spur gears is normally done in accordance with standards the two most popular series are listed under standards above:

The notes below relate to approximate methods for estimating gear strengths. The methods are really only useful for first approximations and/or selection of stock gears (ref links below). — Detailed design of spur and helical gears is best completed using the standards. Books are available providing the necessary guidance. Software is also available making the process very easy. A very reasonably priced and easy to use package is included in the links below (Mitcalc.com)

The determination of the capacity of gears to transfer the required torque for the desired operating life is completed by determining the strength of the gear teeth in bending and also the durability i.e of the teeth ( resistance to wearing/bearing/scuffing loads ) .. The equations below are based on methods used by Buckingham..

Bending

The basic bending stress for gear teeth is obtained by using the Lewis formula

σ = F_t / ( b_a. m. Y )

F _t = Tangential force on tooth
σ = Tooth Bending stress (MPa)
b _a = Face width (mm)
Y = Lewis Form Factor
m = Module (mm)

Note: The Lewis formula is often expressed as

σ = F_t / ( b_a. p. y )

Where y = Y/π and p = circular pitch

When a gear wheel is rotating the gear teeth come into contact with some degree of impact. To allow for this a velocity factor is introduced into the equation. This is given by the Barth equation for milled profile gears.

K _v = 6,1 / (6,1 +V )

V = the pitch line velocity = d.ω/2
Note: This factor is different for different gear conditions i.e K _v = ( 3.05 + V )/3.05 for cast iron, cast profile gears.

The Lewis formula is thus modified as follows

σ = K _v.F_t / ( b_a. m. Y )

Surface Durability

This calculation involves determining the contact stress between the gear teeth and uses the Herz Formula

σ _w = 2.F / ( π .b .l )

σ _w = largest surface pressure
F = force pressing the two cylinders (gears) together
l = length of the cylinders (gear)
b = halfwidth =

d ₁ ,d ₂ Are the diameters for the two contacting cylinders.
ν ₁, ν ₂ Poisson ratio for the two gear materials
E ₁ ,E ₂ Are the Young's Modulus Values for the two gears

To arrive at the formula used for gear calculations the following changes are made
F is replaced by F _t/ cos α
d is replaced by 2.r
l is replaced by W
The velocity factor K _v as described above is introduced.
Also an elastic constant Z _E is created

When the value of E used is in MPa then the units of C_p are √ MPa = KPa The resulting formula for the compressive stress developed is as shown below

The dynamic contact stress c_c developed by the transmitted torque must be less than the allowable contact stress S_e...

Note: Values for Allowable stress values S_e and Z_E for some materials are provided at Gear Table

r₁ = d₁ sin α /2
r₂ = d₂ sin α /2
Important Note: The above equations do not take into account the various factors which are integral to calculations completed using the relevant standards. These equations therefore yield results suitable for first estimate design purposes only...

Design Process To select gears from a stock gear catalogue or do a first approximation for a gear design select the gear material and obtain a safe working stress e.g Yield stress / Factor of Safety. /Safe fatigue stress

Determine the input speed, output speed, ratio, torque to be transmitted
Select materials for the gears (pinion is more highly loaded than gear)
Determine safe working stresses (uts /factor of safety or yield stress/factor of safety or Fatigue strength / Factor of safety )
Determine Allowable endurance Stress S_e
Select a module value and determine the resulting geometry of the gear
Use the lewis formula and the endurance formula to establish the resulting face width
If the gear proportions are reasonable then - proceed to more detailed evaluations
If the resulting face width is excessive - change the module or material or both and start again

The gear face width should be selected in the range 9-15 x module or for straight spur gears-up to 60% of the pinion diameter.

Internal Gears
Advantages:

Geometry ideal for epicyclic gear design
Allows compact design since the center distance is less than for external gears.
A high contact ratio is possible.
Good surface endurance due to a convex profile surface working against a concave surface.

Disadvantages:

Housing and bearing supports are more complicated, because the external gear nests within the internal gear.
Low ratios are unsuitable and in many cases impossible because of interferences.
Fabrication is limited to the shaper generating process, and usually special tooling is required.

Lewis form factor.
Table of lewis form factors for different tooth forms and pressure angles

No Teeth	Load Near Tip of Teeth								Load at Near Middle of Teeth
No Teeth	14 1/2 deg		20 deg FD		20 deg Stub		25 deg		14 1/2 deg		20 deg FD
	Y	y	Y	y	Y	y	Y	y	Y	y	Y	y
10	0,176	0,056	0,201	0,064	0,261	0,083	0,238	0,076
11	0,192	0,061	0,226	0,072	0,289	0,092	0,259	0,082
12	0,21	0,067	0,245	0,078	0,311	0,099	0,277	0,088	0,355	0,113	0,415	0,132
13	0,223	0,071	0,264	0,084	0,324	0,103	0,293	0,093	0,377	0,12	0,443	0,141
14	0,236	0,075	0,276	0,088	0,339	0,108	0,307	0,098	0,399	0,127	0,468	0,149
15	0,245	0,078	0,289	0,092	0,349	0,111	0,32	0,102	0,415	0,132	0,49	0,156
16	0,255	0,081	0,295	0,094	0,36	0,115	0,332	0,106	0,43	0,137	0,503	0,16
17	0,264	0,084	0,302	0,096	0,368	0,117	0,342	0,109	0,446	0,142	0,512	0,163
18	0,27	0,086	0,308	0,098	0,377	0,12	0,352	0,112	0,459	0,146	0,522	0,166
19	0,277	0,088	0,314	0,1	0,386	0,123	0,361	0,115	0,471	0,15	0,534	0,17
20	0,283	0,09	0,32	0,102	0,393	0,125	0,369	0,117	0,481	0,153	0,544	0,173
21	0,289	0,092	0,326	0,104	0,399	0,127	0,377	0,12	0,49	0,156	0,553	0,176
22	0,292	0,093	0,33	0,105	0,404	0,129	0,384	0,122	0,496	0,158	0,559	0,178
23	0,296	0,094	0,333	0,106	0,408	0,13	0,390	0,124	0,502	0,16	0,565	0,18
24	0,302	0,096	0,337	0,107	0,411	0,131	0,396	0,126	0,509	0,162	0,572	0,182
25	0,305	0,097	0,34	0,108	0,416	0,132	0,402	0,128	0,515	0,164	0,58	0,185
26	0,308	0,098	0,344	0,109	0,421	0,134	0,407	0,13	0,522	0,166	0,584	0,186
27	0,311	0,099	0,348	0,111	0,426	0,136	0,412	0,131	0,528	0,168	0,588	0,187
28	0,314	0,1	0,352	0,112	0,43	0,137	0,417	0,133	0,534	0,17	0,592	0,188
29	0,316	0,101	0,355	0,113	0,434	0,138	0,421	0,134	0,537	0,171	0,599	0,191
30	0,318	0,101	0,358	0,114	0,437	0,139	0,425	0,135	0,54	0,172	0,606	0,193
31	0,32	0,101	0,361	0,115	0,44	0,14	0,429	0,137	0,554	0,176	0,611	0,194
32	0,322	0,101	0,364	0,116	0,443	0,141	0,433	0,138	0,547	0,174	0,617	0,196
33	0,324	0,103	0,367	0,117	0,445	0,142	0,436	0,139	0,55	0,175	0,623	0,198
34	0,326	0,104	0,371	0,118	0,447	0,142	0,44	0,14	0,553	0,176	0,628	0,2
35	0,327	0,104	0,373	0,119	0,449	0,143	0,443	0,141	0,556	0,177	0,633	0,201
36	0,329	0,105	0,377	0,12	0,451	0,144	0,446	0,142	0,559	0,178	0,639	0,203
37	0,33	0,105	0,38	0,121	0,454	0,145	0,449	0,143	0,563	0,179	0,645	0,205
38	0,333	0,106	0,384	0,122	0,455	0,145	0,452	0,144	0,565	0,18	0,65	0,207
39	0,335	0,107	0,386	0,123	0,457	0,145	0,454	0,145	0,568	0,181	0,655	0,208
40	0,336	0,107	0,389	0,124	0,459	0,146	0,457	0,145	0,57	0,181	0,659	0,21
43	0,339	0,108	0,397	0,126	0,467	0,149	0,464	0,148	0,574	0,183	0,668	0,213
45	0,34	0,108	0,399	0,127	0,468	0,149	0,468	0,149	0,579	0,184	0,678	0,216
50	0,346	0,11	0,408	0,13	0,474	0,151	0,477	0,152	0,588	0,187	0,694	0,221
55	0,352	0,112	0,415	0,132	0,48	0,153	0,484	0,154	0,596	0,19	0,704	0,224
60	0,355	0,113	0,421	0,134	0,484	0,154	0,491	0,156	0,603	0,192	0,713	0,227
65	0,358	0,114	0,425	0,135	0,488	0,155	0,496	0,158	0,607	0,193	0,721	0,23
70	0,36	0,115	0,429	0,137	0,493	0,157	0,501	0,159	0,61	0,194	0,728	0,232
75	0,361	0,115	0,433	0,138	0,496	0,158	0,506	0,161	0,613	0,195	0,735	0,234
80	0,363	0,116	0,436	0,139	0,499	0,159	0,509	0,162	0,615	0,196	0,739	0,235
90	0,366	0,117	0,442	0,141	0,503	0,16	0,516	0,164	0,619	0,197	0,747	0,238
100	0,368	0,117	0,446	0,142	0,506	0,161	0,521	0,166	0,622	0,198	0,755	0,24
150	0,375	0,119	0,458	0,146	0,518	0,165	0,537	0,171	0,635	0,202	0,778	0,248
200	0,378	0,12	0,463	0,147	0,524	0,167	0,545	0,173	0,64	0,204	0,787	0,251
300	0,38	0,122	0,471	0,15	0,534	0,17	0,554	0,176	0,65	0,207	0,801	0,255
Rack	0,39	0,124	0,484	0,154	0,55	0,175	0,566	0,18	0,66	0,21	0,823	0,262

Gear

2009-02-04T02:25:00.000-08:00

Gear Design

Gears have been around for hundreds of years and are as old as almost any machinery ever invented by mankind. Gears were first used in various construction jobs, water raising devices and for weapons like catapults.

Nowadays gears are used on a daily basis and can be found in most people’s everyday life from clocks to cars rolling mills to marine engines. Gears are the most common means of transmitting power in mechanical engineering.

Gears are used in almost all mechanical devices and they do several important jobs, but most important, they provide a gear reduction. This is vital to ensure that even though there is enough power there is also enough torque(is a movement of force).

This site is a valuable resource about gear and gear design.

Bevel Gears

Bevel gears are useful when the direction of a shaft's rotation needs to be changed. They are usually mounted on shafts that are 90 degrees apart, but can be designed to work at other angles as well.

A good working example of a bevel gear is the mechanism used in a hand drill. As you turn the handle of the drill in a vertical direction, the bevel gears change the rotation of the chuck to a horizontal rotation. The bevel gear also works to increase the speed of the chuck so that its possible for the drill to work on a range of surfaces.

There are four types of bevel gears:

Straight Bevel Gears: These gears have a conical pitch surface and straight teeth tapering towards an apex.

Zero Bevel Gears: Are very similar to straight bevel gears except the teeth are curved.

Spiral Bevel Gears: The teeth are curved at an angle which then allows the contact to be gradual and smooth.

Hypoid Bevel Gears: These gears are similar to spiral bevel except that the pitch surfaces are hyperboloids

Helical Gears

Helical gears are so called because the angle of the teeth are inclined to the axis of the shafts in the form of a helix.

Helical gears are generally seen and described as high speed gears as they can take higher loads than equally sized spur gears. Also with a helical gear the two teeth start to engage and gradually increase as the gears rotate this gradual movement makes helical gears operate much more smoothly and quietly that spur gears. Its because of this design that helical gears are used in the majority of car transmissions.

Rack & Pinion Gears

Rack and pinion gears are used to convert rotation into linear motion or linear motion into rotation. The rack is the flat toothed part and the pinion is the gear. The diameter of the gear determines the speed that the rack moves as the pinion turns.

A perfect example of a rack and pinion gear system is the steering system on many cars. The driver turns the steering wheel which rotates the gear which then engages the rack so as the gear turns it slides the rack to the right or the left depending on which way the steering wheel is turned.

Spur Gears

Spur gears are the most common type of gear they have straight teeth and are mounted on parallel shafts. The main reason for the popularity of spur gears is their simplicity in design, easy manufacturer and maintenance. However due to their design spur gears create large stress on the gear teeth.

Spur gears are known as slow speed gears. Spur gears are seen as noisy due to their design so if noise is not a problem spur gears can be used at almost any speed. Spur gears are noisy because every time a gear tooth engages a tooth on the other gear, the teeth collide, and this impact makes a noise.

Spur gears can be found in applications like washing machines and electric screwdrivers but due to the noise you will never find them in your car.

Worm Gears

A worm gear is used when there is a requirement to reduce speed. It’s very common to see worm gears with reductions like 20:1 and as high as 300:1 or even greater depending on the situation.

A worm gear consists of a cylinder with a spiral groove mounted on a shaft, this is generally referred to as the worm shaft and a gear which is normally referred to as the worm wheel. The gear then meshes with the spiral groove on the cylinder and so when the cylinder rotates it causes the gear to rotate as well. So for each complete turn of the worm shaft the gear shaft advances only one tooth of the gear. So a gear with 20 teeth will see the speed reduced by a factor of 20:1.

The worm always drives the worm wheel around it is not reversible so the worm wheel can’t drive the worm to increase the speed. If it’s attempted the system will normally jam or lock.

Gear Manufacture

The materials that are used for gear manufacturing depend massively on the conditions that the gears will be operating under, conditions like wear and noise etc. Gear manufacturers use metallic or non-metallic materials.

The Metallic materials used in gear manufacture are normally available in cast iron, steel and bronze. Steel is used when there is a need for a high strength design. But cast iron is mainly used because of its good wearing properties, excellent machineability and the ease that complicated shapes can be created due to the casting method.Castings are created by pouring molten metal into a mould and once the metal as cooled it takes the shape of the mould.

The non metallic, materials used are generally wood, rawhide, compressed paper and synthetic resins like nylon. These types of materials are generally used when there is a need for the reduction in noise from the gears.

Gear Design

2009-02-04T02:17:00.000-08:00

Selection of Gear Type

(Reprinted from Handbook of Gears, Stock drive products)

Selection of Gear Materials

(Reprinted from Handbook of Gears, Stock drive products)

Formulae for gear forces

(Reprinted from Design Data, PSG Tech,1995)

Gear Force Diagram

Spur Gear Design

(Reprinted from Design Data, PSG Tech,1995)

DESIGN OF SPUR GEAR

Spur Gear Fundamental

1. Determine HorsePower based on Lewis Formula

Metalic Spur Gears :

W = SFY . 600 / (P . [600 + V] )

where W = Tooth Load, Lbs

S = Safe Material Stress (static)Lbs per Sq.in

F = Face Width, In.

Y = Tooth Form Factor

P = Diametral Pitch

D = Pitch Diameter

V = Pitch Line Velocity, Ft. per Min. = 0.262 . PD . RPM

For Non-Metalic Gears

W = S.F.Y. {(150 /[200 + V]) + 0.25} / P

Horse Power Rating (HP_L) = W . D. RPM / 126000

2. Calculate Design Horse Power

Design HP = HP_L * Service Load factor

3. Select the Gear / pinion with horse power capacity equal to or more than Design HP.

Ref :"Handbook of Gears" -Stock drive products

Table 1.15 Ratings for Steel Spur Gears

Helical Gear Design

(Reprinted from Design Data, PSG Tech,1995)

HELICAL GEAR DESIGN

Helical Gear Fundamental

1. Determine HorsePower based on Lewis Formula

Same as Spur Gear Design except the inclusion of helix angle

HP_Helical = HP_Spur * cos(ψ)

2. Calculate Design Horse Power

Same as Spur Gear

3. Select the Gear / pinion with horse power capacity equal to or more than Design HP

From "Handbook of Gears" -Stock drive products

Table 1.18 Ratings for Hardened Steel Helical Gears

SolidWork PLANAR JOINTS

2009-02-04T01:32:00.000-08:00

PLANAR JOINTS:

NOTE: if you are not familiar with the layout of SolidWorks, then click here to familiarize yourself with the layout. If you are unfamiliar with assemblies please see the assembly tutorial.

There are three types of Planar Joints: Pin Joint, Pin-in-Slot, and Sliding. Solidworks will allow us to study these joints in a way that a simple drawing or schematic would not allow. We will be able to actively move the joints and see the limitations of each joint type.

This tutorial uses files in the parts.zip package. Make sure you download and extract the files to your computer. The assemblies in this tutorial come from the "planar joints" directory. RED pins represent pins that are fixed from translating...each still allows rotation of the body connected to it.

The first Planar Joint is called a Pin Joint. It is one you are already familiar with since it can be seen in most mechanical systems.

It only permits two bodies to pivot relative to another.

As an example of a pin joint consider a scissors lift shown below. This mechanism, which serves to raise platform holding workers, has a series of links which are unfolded by several hydraulic cylinders. Each pair of links is connected by a pin joint which enables them to pivot with respect to each other:

To examine a simple pin joint in solidworks, follow these steps:

1.) Goto File->Open and select "pinjoint.SLDASM" from the planar joints folder

2.) using the rotate component button and the move component button see how the pin joint moves in space. Notice the limitations of this joint:

You can see that there are actually two pin joints in this assembly. The pin in a pin-joint could be fixed in position, or it can join two parts, both of which can move.

The second Planar Joint is called a Pin-in-Slot joint. A pin-in-slot joint allows the joined bodies to pivot with respect to each other and to translate with respect to each other in one direction. However translation in the perpendicular direction is restricted.

As an example of a pin-in-slot joint, consider the motorized door opener shown. The end of one member has a pin with a roller, which rolls in a slot in the door:

To examine a simple pin-in-slot joint in solidworks, follow these steps:

1.) Goto File->Open and select "pininslot.SLDASM" from the planar joints folder

2.) using the rotate component button and the move component button see how the pin-in-slot joint moves in space. Notice the limitations of this joint:

You can see that the link with a slot and a hole is pinned at its hole to some fixed body which is not shown, but that a second link is connected to the slot with a pin. The pin-in-slot joint is that connecting the two links.

ALSO: Notice that you can only translate the link with two holes, not rotate it. This is a limitation of Solidworks. A work around to this problem is to "fix" the link in space by right clicking on link3slide in the Feature Manager Design Tree and click "Fix":

You may get a message that the assembly cannot be solved with this mate. However, you will find that it probably works. Now the link with two holes can be both translated and rotated. You can return to the initial state in which the slotted link floats by right clicking on link3slide in the Feature Manager Design Tree and selecting "Float":

The third Planar Joint is called a sliding joint. A sliding joint prevents two bodies from rotating with respect to one other and permits the bodies to translate with respect to one another only in a single direction.

As an example of a sliding joint, consider the mechanism for adjusting the position of the back to the exercise machine. The black sleeve can only slide on the white member. Notice that the sleeve is locked into position by the spring loaded pin (with the black handle) which engages one of the holes in the white member. But when this pin is retracted, the sleeve can slide. Notice that another link is pinned to the sleeve:

To examine a simple pin-in-slot joint in solidworks, follow these steps:

1.) Goto File->Open and select "slidingjoint.SLDASM" from the planar joints folder

2.) using the move component button see how the sliding joint moves in space. Notice the limitations of this joint:

You can see that a link with a slot and a hole is pinned at its hole to some fixed body which is not shown. A second member with a square peg engages the slot. While the link with the slot can pivot about its pin joint, the second member can only slide in one direction relative to the slotted link. The sliding joint is that connecting the two links.

Using Planar Joints to Form Mechanisms:

NOTE: if you are not familiar with the layout of SolidWorks, then click here to familiarize yourself with the layout. If you are unfamiliar with assemblies please see the assembly tutorial.

By connecting members of various shapes and sizes with planar joints, the motion (input) of one body brings about the desired (output) motion of another body. Two common input motions are: rotation of a shaft (by a motor) and translation of a body (by actuating a hydraulic or pneumatic cylinder). These are also two commonly desired output motions: pivoting a body about a point and translating a body along a line.

However, the input body often cannot be attached directly to the output body. Therefore, a mechanism converts the input motion to the output motion.

To illustrate this effect, we show three mechanisms which accomplish the same purpose: pivoting a member about a point. The member could be a door pivoting about its hinge:

This tutorial uses files in the parts.zip package. Make sure you download and extract the files to your computer. The assemblies in this tutorial come from the "pivot" directory. RED pins represent pins that are fixed from translating...each still allows rotation of the body connected to it.

Method 1 (Pivot1.SLDASM)

The pivoting member (top) is acted upon by a link (middle), which is in turn driven by a second link (bottom). The bottom link is pivoted by a motor. The motor is not shown, but the shaft of the motor (shown here as a fixed pin) would engage the link causing it to turn. Besides the hinge of the top green member (which is like the hinge of a door), this method involves two pin joints. The mechanism at work here is called a four bar linkage. The "fourth" link joins the two red dots. In the four bar linkage, one link always joins the two fixed pins.

Method 2 (Pivot2.SLDASM)

The pivoting member (top) has a slot in which a pin (blue) slides. The pin is connected to an L-shaped member. The L-shaped member would be pivoted by a motor just as the bottom link above is pivoted by a motor. Besides the hinge of the top member, this method involves a pin in slot joint.

Method 3 (Pivot3.SLDASM)

The pivoting member (top) is acted upon by the piston of a hydraulic (or pneumatic) cylinder. The hydraulic cylinder case is connected to a fixed pin about which it can pivot. The piston is moved back and forth in the case by the flow of compressed fluid or air. As the cylinder extends or contracts, the top member pivots about its pin. Besides the hinge of the top member, this method involves a sliding joint and two pin joints. (The piston and case are connected by a sliding joint.) The mechanism at work here is an inverted version of the crank and slider mechanism.

MATING:

NOTE: if you are not familiar with the layout of SolidWorks, then click here to familiarize yourself with the layout

SolidWorks has a simple, yet powerful mating feature. It is used for joining parts in an assembly and simulating how they fit together and move together. The picture of the engine above shows an intricate assembly. This tutorial will cover the most basic mates that we will use to simulate simple mechanical systems. To use mates we will be working with more than one part and therefore must be in the assembly mode of SolidWorks. To enter this mode follow these steps:

Start SolidWorks and goto file->new
Double click the "assembly" icon:

you will notice a new toolbar appearing to the left of the screen. This is the only noticeable difference between part mode, and assembly mode:

This tutorial uses part files from the parts.zip archive. Make sure you download and extract the files to your computer before continuing...

Concentric Mate:

The most common mate is called a concentric mate, and as the name implies, it is a mate between two concentric features. Any time you want a pinned connection or a piston cylinder type connection you will use a concentric mate. We will mate a pin to a link in the following example:

First you will add the pin to the assembly. Goto Insert->Component->From File

Find the "Pins" folder and double click "pin2inch.SLDPRT"

Click anywhere on the screen to place the part near to where you want it.

NOTE: In solidworks assembly mode ,the FIRST part you insert is automatically fixed in space. This means it can not rotate or translate. Every other part you add is "floated" in space which means it can rotate and translate. For more information about fixing and floating parts please see the Planar Joints Tutorial.

To add the link follow the same steps for adding the pin. Add "link1.SLDPRT" from the "Links" folder
Your screen should now look similar to this:

Select the Mate icon from the assembly toolbar and mate options will appear. Select the inside face of the hole on the link and the outside face (circumference) of the pin. Watch this animation for clarification:

Your assembly should now look similar to this:

To check that the mate worked try moving the link around the screen using the move component button . The link will remain concentric with the pin, although it can move parallel to the pin (and even off it).

To further restrict our pin we will want to do a second mate:

Coincident Mate:

A coincident mate, like the name sounds, is a mate between two features that you want to coincide with each other. Generally we use it for making two planes parallel and coincident. In this tutorial we will use it to mount the link onto the pin so that it cannot fall off of it. When we do this, the link will spin around the pin, but it will not be able to slide up and down the pin. Follow these steps to achieve this mate:

Click the mate button on the assembly toolbar
Select the top face of the link and the top face of the pin. If you can't easily select the surfaces, use the zoom and move buttons to navigate around the object until you can clearly see the features. Watch the following animation for clarification:

[NOTE: you can also use the middle mouse scroll button to zoom and rotate]

Your assembly should now look similar to this:

To check that the mate worked try moving the link around the screen using the "Move Component" button . The movement of the link movement should be restricted to be concentric with the pin and parallel with the top of the pin.

you are now ready to try mating exercise 1

Mating Exercise 1:

Try to create the following assembly for more practice:

Troubleshooting:

it is always a good idea to position the part close to its final mated position before you set up the mate. You can move the part using the move component button .
If you mess up badly, you can always edit->undo. Likewise, if you want to go back to a certain point, you can use the undo list to see your undo options. If this does not work, Solidworks keeps a record of all mates in the Feature Manager Design Tree under 'MateGroup#' If you expand this list, you can manually delete any mates you have made by selecting the mate and then hitting Delete on the keyboard. Likewise, you can right click and select delete:

MODIFYING PARTS:

NOTE: if you are not familiar with the layout of SolidWorks, then click here to familiarize yourself with the layout

This tutorial will use "link3slide.SLDPRT" from the parts.zip file. You can use these techniques on any part though. The steps in this tutorial are similar to the steps in the dimensioning tutorial. The more you understand solidworks the more you realize that even the most complex parts are made and changed in a similar fashion to the steps in these tutorials.

1.) Use file->open and browse to the links folder. Select "link3slide.SLDPRT" and click open:

2.) You can only add features in sketch mode. To enter sketch mode you must first decide where you want to add the feature. You can add a feature to any plane. You can usually find the sketch under an "extrude" in the feature manager design tree. You will have to click the to reveal the sketch:

3.) Right click on the sketch and select ‘edit sketch.’ You will notice that the rest of the part disappears or becomes transparent. Do not worry if some of the features of the part become transparent or disappear. They have NOT been deleted. They have simply been removed to simplify the screen and highlight the sketch you are currently working with. See the advanced dimensioning tutorial for an example of "disappearing" parts:

4.) You will now be in sketch mode and can use all the buttons on the right of the screen (picture rotated to save space):

5.) If your view of the part is skewed, click the front view on the standard view menu to rotate the sketch (view->toolbars->standard view):

6.) Your window should now look like this:

7.) To add another hole to the link select "draw circle" from the sketch toolbar:

8.) The cursor changes and you can now draw a circle on the link:

9.) to edit the radius of the circle either follow the steps in the dimensioning tutorial or modify the radius in the feature manager design tree:

10.) Exit sketch mode by clicking the purple arrow in the top, right hand corner:

11.) The new hole will be cut out of the link and your link should now look like this:

12.) To remove the hole you re-enter edit sketch mode. Do this by right clicking on the sketch and selecting edit sketch:

13.) Select the hole you just created by clicking on it with your mouse and hit the delete key on your keyboard. The hole will disappear. Exit sketch mode by clicking on the purple arrow:

You are now ready to try Modification Exercise 1

If you want to modify any extrusion follow this example:

1.) Use file->open and browse to the links folder. Select "link3slide.SLDPRT" and click open:

2.) Right click on "Base-Extrude" in the Feature Manager Design tree and click "Edit Definition":

3.) You will be presented with extrusion options. In this example change the Depth dimension from 1.00in to 3.00in and click the green check mark:

4.) The link will now be three times thicker:

You are now ready to try Modification Exercise 2

Modification Exercise 1:

for more practice try adding a square hole to the same link:

Modification Exercise 2:

starting with "linkwithsquareboss.SLDPRT" stretch the boss to 2 inches:

NOTE: make sure you are familiar with the dimensioning tutorial

Troubleshooting:

● If you mess up badly, you can always edit->undo. Likewise, if you want to go back to a certain point, you can use the undo list to see your undo options

CREATING A SIMPLE PART FROM SCRATCH:

The purpose of this tutorial is to create a simple bracket from scratch. Every new part in solidworks begins with a sketch. If you notice this bracket is just a 2d shape that has been extruded to create a 3d object. So to create this part we will create a 2d sketch and extrude it.

1.) Select 'File' -> 'New'' and double click 'Part':

2.) On the main toolbar click 'Sketch' to enter sketch mode. Then click the 'Sketch' button to the right to enter sketch mode:

A set of planes will now appear. Select the plane labeled Front to start your sketch on that plane:

3.) Draw a rectangle starting at the origin. [click once on the origin and a second time at any arbitrary point in the 1st quadrant] :

4.) Click 'Smart Dimension' on the main toolbar and then change the height to .08m and the length to .32m:

For help on dimensioning click here

5.) Now select the 'Circle Tool' and draw a circle similar to the one in the picture below:

6.) Click the circumference of the circle and enter the following values into the panel on the right side of the screen:

These numbers correspond to the absolute location of the circle's midpoint, and the radius of the circle. Namely, the midpoint of the circle is located at (0.25,0.04) and its radius is 0.03m.

7.) Click the little arrow next to the Features button on the main toolbar to bring up the following screen. Then click 'Extruded Boss/Base':

8.) In the pane that appears to the left, enter 0.01m in the box and click the green arrow:

9.) Your model will now look like this:

10.) To create the boss on the surface, you need to start a new sketch on the face of the model. To do this, select the face of the model and click the 'Sketch' button on the main toolbar:

11.) Change to a front view of the model and draw a rectangle using the Rectangle button:

12.) Dimension the rectangle using the 'Smart Dimension' button to be 0.06 m square:

13.) To locate the rectangle in the proper place, use the 'Smart Dimension' tool and select the left edge of the boss, and the left end of the goldish rectangle underneath. Enter 0.01m as the distance:

14.) Repeat this for the gap between the top edge of the boss, and the top edge of the goldish rectangle:

15.) Click the little arrow next to the Features button on the main toolbar to bring up the following screen. Then click 'Extruded Boss/Base':

16.) Enter 0.02m in the D1 box, click the green check mark, and you will get the following model:

CREATING A TRUSS STRUCTURE FROM SCRATCH:

The purpose of this tutorial is to create a truss-like bracket from scratch. Every new part in solidworks begins with a sketch. If you notice this bracket is just a 2d shape that has been extruded to create a 3d object. So to create this part we will create a 2d sketch and extrude it.

before you begin this tutorial make sure you have the 'sketch' , 'sketch tools' , and 'features' toolbars open. Make sure the base units are Meters. Optionally, you may want to turn on the drawing grid. Click here if you don't know how to add these options.

1.) Select 'File' -> 'New'' and double click 'Part':

2.) On the sketch toolbar click 'Sketch' to enter sketch mode. Then click the 'Rectangle' tool:

3.) Draw a rectangle starting at the origin. [click on the origin and drag the mouse to another point and click again] :

4.) Dimension the rectangle using the 'Dimension Tool' and by clicking on two edge's. Change the height to 0.08m and the length to 0.32m:

For more help on dimensioning click here

5.) Select the 'Line Tool' and draw a triangle on the part:

6.) In order to place the triangle exactly where you want it and make it the proper size you must use the 'Dimension Tool.' First you should line adjust the spacing around the triangle. In this example we will change the spacing to 0.01m:

For more help on dimensioning click here

7.) Next you should adjust the length of the triangle's base. Notice the height is already defined. In this example we will the base and height are 0.06m:

8.) Draw and dimension the upper triangle in the same way as the lower triangle. Notice the vertical and horizontal spacing between the two triangles is set to 0.005m:

9.) The sharp corners in the triangles will lead to very high stress concentrations. Fillets are used to lower the stresses in the corners. Use a fillet radius of 0.003m on each corner. To add a fillet, click the 'Fillet Tool,' enter the fillet radius, then click the two lines that create the corner you want to fillet:

10.) Instead of drawing each triangle again, you should use the 'Linear sketch and repeat tool' to add more triangles along the length of the bracket. Using this tool is simple. First select the 2 triangles you want to repeat. Then click the button, choose the number of copies you want, and the spacing between the copies. It will dynamically preview any changes you make. In this example we have 4 triangles, with a spacing of 0.08m.:

7.) Click the purple arrow to exit sketch mode.

8.) Click the extrude button on the 'Features' toolbar and enter .01m in the d1 text box:

If you want to import this file into ANSYS for analysis, you must save it in the IGES file format. For instructions on how to do this, click here.

Before importing a SolidWorks part into ANSYS, while still in SolidWorks you must export the part in the IGES file format. Click here if you have not done that yet.

MODEL:

START ANSYS AND IMPORT FILE:

Open ANSYS: on cmu cluster machines its under math & stats:

From the file menu select Import>IGES... and select the following options:

Locate your IGS file using the browse button. When you find the file it should look something like this:

Your screen wll now look like this (notice the axis directions):

MATERIAL PROPERTIES:

In order for ansys to do a Fine Element Analysis [FEA] we need to specify what kind of element we want to use. For this 3d solid we will be using a 10 node tetrahedron shaped element[solid187]. To set this click on Preprocessor>Element Type>Add/Edit/Delete:

Click 'Add' on the next window(Elements Type) and then find 'Tet 10 node' under 'Structural Mass' -> 'Solids':

click OK and then click Close on the 'Elements Type' window

Now we must specify what type of material this solid is. The problem specifies the bracket is to be made of aluminum. To enter material data click on Preprocessor>Material Props>Material Models:

Select Structural>Elastic>Isotropic and enter the following numbers into the window that appears:

EX refers to Young Modulus, which is 7e10 Pa for aluminum

PRXY refers to poisson ratio, which is .33 for aluminum

click OK, then close the Define Material Model Behavior' window

MESHING:

To mesh the volume into individual elements, go to Proprocessor>Meshing>MeshTool:

The MeshTool window will pop-up on the right side of your screen. Select the following options and click 'Mesh':

Select the bracket and click ok:

NOTE: If ANSYS gives you an error about going over the maximum number of elements, you must adjust the Smart Size slider to a number higher than 6. The reason for this error is because the educational version limits the maximum number of elements.

Your mesh should look something like this:

While not necessary to solve the problem, for more accurate results it is often a good idea to add more elements in certain areas of interest. This is called Refining the Mesh and can be done by selecting Preprocessor>Meshing>Modify Mesh>Refine at>Areas:

We are concerned with stresses in the holes so add more elements around the holes by selecting the areas that makes up the 2 holes (4 areas in all).

in the next window select 3 and click ok:

Your mesh will now look like this. Notice there are approximatly three times the elements around the circles:

this picture better illustrates the new elements:

BOUNDARY CONDITIONS:

Now we have to apply the loadings to our meshed volume. This problem has 1 mounted area and 1 force.

Before we apply forces you should familiarize yourself with the pan/zoom/rotate tool. You can find it under PlotCtrls:

It has the basic engineering views as well as buttons for rotating about each axis, and buttons for zooming in and zooming out:

Once you are familiar with this tool, you can continue on to apply forces

Since the left side of the bracket is mounted it will not displace in any direction. To set this choose Preprocessor>Loads>Define Loads>Apply>Structural>Displacement>On Areas, select the mounted face and click OK (NOTE: you will have to rotate your view so you can easily select this area):

on the window that pops-up select All DOF and enter a Displacement of 0...the click OK:

To apply the force select Define Loads>Structural>Force/Moment>On Nodes:

You will apply the forces at the end of the bracket, at the two corners. This will make the loading symmetric about the center plane of the bracket. Select the two nodes near the end of the bar (NOTE: select 'iso' on the 'pan/zoom/rotate' tool and zoom in a little to easily select these nodes):

A window will appear asking for the force data. To have a net force of 1200 N, you will need to apply 600 N to each of the nodes you have chosen. Enter the values shown below: Notice a downward force is -600 because the positive y direction points up.

Your bracket should now look like this:

SOLVE:

To run the FEA on the bracket select Solution>Solve>Current LS:

and your bracket will look like this:

POSTPROCESSING:

To see the deformed shape of the bracket select General Postproc>Plot Results>Deformed Shape click OK on the window that appears and your bracket should look something like this:

To get various results for the bracket select General Postproc>Plot Results>Contour Plot>Nodal Solu:

To see the stress in the X direction select the following:

use the 'Pan/Zoom/Rotate' tool to see the 3d stresses. Notice MX and MN locate maximum and minimum stresses

or stress in the Y direction:

Introduction Solidwork

2009-02-04T01:14:00.000-08:00

Place your mouse over the image to find out about different toolbars in SolidWorks

WORKING WITH PARTS IN SOLIDWORKS:

NOTE: if you are not familiar with the layout of SolidWorks, then click here to familiarize yourself with the layout

SolidWorks allows you to view 3d parts from all different angles. There are infinite ways to view an object, but engineers usually only concern themselves with 4 main views. You can see these views in the above picture of the green link. The main views are Side, Front, Top, and Isometric.

For this tutorial you can open up any part in Solidworks. The tutorial will use the green link above to demonstrate, but any 3d part will work just as well. This link can be found in the parts.zip archive

Start SolidWorks and goto file>open
Find your part and click open.

Viewing the Part:

You can access the standard engineering views by clicking the Standard Views button on the main toolbar:

Each small box on the "Standard View" menu corresponds to a view of the object. The best way to understand what each view means is to click each view and see what happens.

Below is a front, back and isometric view of the link:

You can select a non-shaded view using the "View" toolbar: and by clicking on the non-shaded box(with hidden lines) . There is also an option for non-shaded without hidden lines, and non-shaded with solid hidden lines:

What if we want to view the indent in more detail? The standard views do not give a good view of it. Sometimes the only way to get a better look of a part or feature is to control the view manually. Use the rotate and zoom tools to get a better view of the indent. NOTE: if you have a mouse with a scroll wheel, you can rotate and zoom without these buttons. Push down on the wheel to rotate, and spin the wheel to zoom in and out:

Extracting Dimensions From the Part:

The simplest method of extracting the dimensions of a part is to make the dimensions visible in the default view:
Right click where it says annotations on the feature manager design tree, and click where it says 'Show Feature Dimensions':

If there is a check mark next to 'Show Feature Dimensions,' all the defined dimensions will appear.
Dimensions are easier to see if the part is viewed as shaded. Sometimes rotating the part will make some of the dimensions more readable.
Another method of extracting the dimensions of a part is to use the Measure Tool:
Go to the main file menu and select Tools>Measure. The cursor will change to a ruler and a dialog box will appear with the title 'Measure.' Switch to Isometric View and then click the circular hole. The Measure Tool window will display all the properties of this hole:

Likewise, you can use the measure tool to measure the distance between any 2 lines. For example, the distance between the top and bottom of the link from the side is 1.50 inches (shading turned off for clarity):

Another method to extract dimensions of the link is to use the sketch mode. This will be most useful later on when we are interested in changing dimensions:
Goto the feature manager design tree and locate 'Base-Extrude.' Click the to reveal the sketch.

Right click on 'Sketch1' and select ‘edit sketch.’

The view will change and you will now see a 2d drawing with all the dimensions:

To exit sketch mode click the purple arrow in the top, right corner, or right click in the drawing area and select 'exit sketch':

Solving Problems with Solidworks:

Checklist for creating mechanisms:

1.) [ ] If your assembly has pins that should be located in specific locations.....Place them using the move component tool and FIX them in place

[ ] Concentric Mate your links to the fixed pins

[ ] Face mate the end of the pins to the side of the links

2.) [ ] If your assembly has objects that must be located in specific locations.....Place them using the move component tool and FIX them in place.

Measure Tool: The Measure Tool is used to get the distance between points, the angle between lines, the displacement of parts in an assembly, and anything else you would use a ruler or protractor for in real life. To open the Measure Tool you can select it from: Tools->Measure...

Or you can select the measure tool using the Tools Button on the main toolbar:

Using the tool is very simple. You can select points or lines and it gives you information regarding the two selected entities. Pay particular attention to delta x, y, and z values. It allows you to determine distance between points independently from the global origin:

To measure the angle between 2 parts you must select 2 intersecting lines or edges on those parts. Look at the following picture for clarification(the 2 red edges were selected):

To change the units for display, click the Options... button and use the drop down menu to select the units you want:

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Move Component: It is often important to be able to move a part in an assembly by a specified displacement or angle or to a specific xyz coordinate. The move component tool can be used for this if you understand how to properly use the tool. It is located on the main toolbar:

The Move Component tool has the following options (they are explained in detail below):

1.) Free Drag option is the default and it allows you to drag the selected part wherever you want. It is useful for positioning parts for mates or for moving assemblies. It is not useful for accurate movement.

2.) Along Assembly XYZ is an inaccurate method of moving a part in exclusively the x y or z direction.

3.)Move to XYZ position: This is useful if you want to move a point on a part to a specific xyz coordinate. Consider the following link, which is located somewhere random in space.

To move the link to the origin, first select a point on the link. (Notice that the origin is below the selected point and to the right at the location 0,0,0)

Next click the Move Component button on the assembly toolbar and select 'To XYZ Position.'

Enter (0,0,0) as the coordinates and hit apply:

Now the the point on the link is at the origin of the assembly:

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4.) Move by delta XYZ: This is useful when you want to move a part a specific distance relative to its current location. You can choose to move by any combination of x, y and z distances as long as the part is not fixed or restrained by a mate. In this example, move a pin in a slot by 10mm to the right. Note that since the pin is mated to the inside of the slot you cannot move it in the y or z directions.

The first step is to select a point on the pin, or the pin itself:

Next click the Move Component button on the assembly toolbar and select 'By delta XYZ.'

Change the value in deltaX to 10mm and hit apply:

Now the pin has moved 10mm to the right:

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Rotate Component: It is often important to be able to rotate a component in an assembly by a specific angle about the x y or z axis. The rotate component tool can be used for this if you understand how to properly use the tool. It is located on the main toolbar:

In this example the green link will be rotated around the pin by 45 degrees in the positive z direction. To do this, click the Rotate Component button. In the window that appears to the left of the assembly, use the drop down list to select 'By Delta XYZ'

Enter 45 into the Z text box and click apply:

The link will rotate 45 degrees and stop:

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Changing Units:

Select 'Tools' -> 'Options.' On the 'Document Properties' window select 'Units' and change them to whatever you want:

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Exporting to ANSYS: ANSYS uses a different file format from SolidWorks but it can still read SolidWorks parts as long as you first convert them to the IGES[Initial Graphics Exchange Specification] format.

1.) After saving the file in SolidWorks as the usual .sldprt file, and while that file is still open in SolidWorks, select 'File' -> 'Save As...' and change 'Save as Type' to 'IGES File (*.igs)':

2.) Click Options in the 'Save As' window and change 'Surface Representation' to 'ANSYS':

3.) click 'OK' and then 'Save'

click here to import model into ANSYS

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Printing: To print what you see on the screen you have to change a setting in page setup. 'File' -> 'Page Setup'.

Otherwise the printout will be the actual size of the part you are working with. In some cases this is larger than a piece of paper.

The Chemical Equation

2009-01-30T03:57:00.001-08:00

The Chemical Equation

Chemical transformations take place according to strict 'rules' because of the nature of molecular structure. Chemical reactions are really the rearrangement of the connectivity between the atoms in the reagenmts to produce products. The Ancients saw that chemical reactions obeyed the law of multiple proportions very precisely and correctly concluded that this was significant. Because the creation or destruction of nuclei is very unlikely without a nuclear reactor, the same number of atoms of each type (element) must exist both before and after any chemical transformation. This means that molecules must react in simple, whole-number ratios (i.e. the law of multiple proportions), as can be seen by the following example:

Even though we could try to write down the chemical reaction as

X₁ * CH₄  +  X₂ * O₂  =  X₃ * CO₂  +  X₄ * H₂O

but this equation is not useful because it is not 'balanced'. Not balanced means it does not have the correct (or determined) stoichiometric coefficients. The process of 'balancing' a chemical equation is simply determining a set of coefficients {X_i}, which represent the proportions of whole molecules in the reaction that balance the number of atoms of each element on both sides of the equal sign. Note: an equal sign, as well as single and double headed arrows, is sometimes used to separate reactants and products in a chemical equation. Nomatter what is used as a separator, a chemical reaction must be written as an equation, which means that the same number of each type of atom exist each side. (Sometimes unscrupulous chemistry intructors provide unbalanced chemical equations (which are really not equations at all) in problems given to students, but they argue that these are the cards that life deals us. Nonetheless, the equations always must be checked before proceeding further with the use of said equation.) How do you balance a chemical equation?

Identify all the different elements in the chemical equation.
Count the number of atoms on both sides of the equal sign of the element that appears in the fewest different molecules in the equation. Make sure to use the correct molecular formula (structure) in this arithmetic
Adjust the Stoichiometric Coefficents of the species that contain this element to balance the count of this element in the equation.
Repeat the last two steps for each element identified in step 1. You must maintain the ratio of coefficients of every molecule that has been balanced for a previous element.
When all the elements have been balanced, multiply the coefficients in the entire equation by any number you wish. This is usually done to obtain the smallest whole number (integer) coefficients, {X_i}

For the above equation:

Carbon:   X₁ = X₃Hydrogen: X₄ = (2)*X₁Oxygen:   X₂ = X₃ + (1/2)*X₄

If one chooses X₁ =1, then X₃ = 1, then X₄ = 2, then X₂ = 2, and the equation is balanced. Practice some more equation balancing on your own.

Counting Atoms by Weight

Hardware store owners solved a problem long ago with a method that can be applied to almost everything, including chemistry. People need nails. Store owners have nails, but want money for them. People need lots of nails, but counting lots of nails so that you can charge for them is a drag, because they are small and pointy. Shopkeepers decided on an easier method than counting each individual nail, they simply sold nails by weight. But how many nails do you get when you buy a pound of nails? It obviously depends on the weight of an individual nail. Even now, the weight of an individual nail is still found on in Hardware Stores; It is listed as Penny Weight. A 10 penny nail weighs 1/2 oz (1 penny = 1/20 oz). So, how many 10 penny nails are in 1 lb of nails? answer.

Remember that molecules are so tiny, that instead of counting individual particles, we may want to count a "chemist's dozen", or mole, of molecules instead. So, three possible ways of specifying the 'amount' of a chemical are possible, by mass (by weight) / by mole / by molecular count. These are obviously interconvertible measures:

Isotopes and the 'Average' Mass of an Atom

In order to count molecules, we need to weigh them, because they are too small to count individually. Molecules also come in different sizes, just like nails, but there are actually an infinite number of possible molecular weights! We are saved by the fact that all molecules are made up of atoms, and there are only about 100 of these, so all we need to know is the atomic weights of the elements.

We can measure the charge to mass ration of ions in a mass spectrometer, shown below.

When we do this, we find out that the atoms of a given element do not all weigh exactly the same, but come is varieties with the same number of protons but different numbers of neutrons called isotopes. (Note: the mass of the electron lost upon ionization in this measurement is negligible, and known, and since the absolute charge of the ions is also known, the mass spectrometer may yield the exact mass of each elemental atom) Since we usually need to count billions and billions of atoms, and we cannot normally tell the difference between the isotopes since they are identical chemically, we use the average atomic weights of the elements in mole calculations and in the determination of average molecular weights. For example, for every 100,000 Hydrogen (Z=1) atoms(ions) we count in our mass spectrometer, 99,985 atoms weigh 1.007825 grams/mole and 15 weigh 2.0140 grams/mole. The minor isotope of hydrogen is so important it is called deuterium, but it is not a different element. The presence of naturally occuring deuterium makes the average atomic mass of the element with Z=1 as follows:

Atomic Mass of Hydrogen = (0.99985)*(1.007825) + (0.000015)*(2.0140) = 1.00797 g/mol

The natural abundance of the isotopes of Neon are as follows:

Abundance(²⁰Ne) = 90.92  %
Abundance(²¹Ne) =  0.257 %
Abundance(²²Ne) =  8.82  %

and the atomic masses of these isotopes are:

Atomic Mass(²⁰Ne) = 19.99244 g/mol
Atomic Mass(²¹Ne) = 20.99395 g/mol
Atomic Mass(²²Ne) = 21.99138 g/mol

What is the average atomic weight of Neon from these data? answer

Calculating Molecular Weight
Once we know the average weight (mass) of all the elements (these are usually listed in the Periodic Table), we can calculate the molar mass of all the molecules just by knowing their molecular formula. For example, the molecular weight of Methane, CH₄, is simply

MW{Methane} =
(1 carbon atom per molecule)*(12.011 grams per mole for carbon atoms)
+ (4 atoms of hydrogen per molecule)*(1.00797 grams per mole for hydrogen atoms)
= 16.0429 grams per mole methane

To calculate the isotopically averaged molecular mass of a substance (the weight in grams of one mole of molecules of the compound) simply use the average atomic weight (in g/mol) of each of the elements in the molecular formula multiplied by the number of times each element appears in each molecule.

Limiting Reagents

Consider the reacton of Hydrogen gas and Oxygen gas to for water. Lets assume for a moment we can 'see' each of the reacting molecules. The reaction might look like this:

Why weren't all of the reactants consumed in the reaction? Because the initial mixture was not in the proper stoichiometric proportions! 'Proper stoichiometric proportions' means in thre proportions that appear in the balanced chemical reaction. (What is the balanced chemical equation for the transformation? answer.) Whenever the reaction mixture does not contain stoichiometric proportions, one of the reagents is said to be limiting; When it runs out, the reaction must stop. In the case above, the hydrogen gas was the limiting reagent. How do you know which one of the reagents is limiting? You calculate the yield of the products from the amount of each one of the reactants, and the one that produces the least products is limiting.

Here is a little simulation that you might enjoy showing the effect of non-stoichiometric reactant concentration ratios on reaction yield for a couple of simple transformations.

Aerodynamic Design of a Windmill

2009-01-30T03:53:00.000-08:00

Aerodynamic Design of a Windmill

Introduction

The airfoils MH 102 to MH 110 were part of the design of an optimum windmill. The windmill itself as well as the airfoils have been designed using direct inverse design methods.

Today, the design of horizontal axis windmills can be performed with good results, using inverse design methods, based on the minimum induced loss windmill, as defined by Glauert and Prandtl during the 1920s. For the analysis under off-design conditions, simple blade element methods and more complex vortex lattice methods lead to quite accurate results. The final blade geometry can be tailored exactly to the desired main operating range by using a suitable inverse design method for the airfoil sections.

For a small horizontal axis wind turbine, the main project data were proposed by the manufacturer, based on market studies. The basic parameters of the windmill were a constant speed generator which could be switched between two velocities of rotation and that the blade angle should be constant and not adjustable. A gearbox between power generator and the rotor of the windmill permitted the decoupling of aerodynamics and the power generator.

To avoid destruction of the windmill due to over speed caused by high wind speeds, it was decided to direct the aerodynamic layout towards a stall regulated machine. (such a windmill has no variable pitch blades to control the mill at higher wind speeds; instead, the airfoils are designed to stall sharply when the operating limits are exceeded, thus limiting the power output).

After some preliminary design studies, the parameters for the design, as listed below, were selected.

Table 1: Definition of the Design Point
Technical Data
diameter	16 m
	gear 1	gear 2
rotor speed	80 1/min	120 1/min
at wind speed	6.0 m/s	9.0 m/s
tip Mach number	0.2	0.3
power loading	450 W/m²
max. power output	90 kW
useful wind speed range	5
stall controlled, fixed pitch, no adjustable blade angle

Regarding low manufacturing costs and competitive energy costs, the number of blades was limited to two. To achieve high yearly power on times, with short low power and non operating times, a specific power loading of 450 W/m² had been selected. A lower loading would increase the full power times, but was not possible due to a diameter restriction.

On the one hand, simple momentum theory correctly predicts maximum efficiency to occur at maximum diameter, but on the other hand the tip Mach number is directly proportional to the diameter for a fixed velocity of rotation. The tip Mach number was limited to 0.3, as a compromise between aerodynamics and noise constraints. Together with the available gearboxes, this lead to local Reynolds numbers of more than 500`000, which is sufficient to achieve high L/D ratios, which are essential for good performance.

Blade Design

The geometry of the blades is determined by the task to transform as much energy as possible from the incoming air flow into mechanical, respectively electric power. Thus the aerodynamic design of the windmill should fulfill the minimum induced loss principle. The basic aerodynamic design of the windmill was based on Glauerts optimum windmill theory, which was embedded into the framework of an existing general blade element code. This code uses two dimensional airfoil polars, which gives very good results for attached flow conditions. For cross checks, additional analysis runs were performed, using a vortex lattice code. During the preliminary design of the blades, the operating conditions for the local airfoil sections were defined in terms of Reynolds and Mach numbers as well as lift coefficient range. These conditions were used for the design of new airfoils, which were then used in the windmill design method to find the optimum blade shapes. Later additions to the code make it possible to account for the boundary layer of the ground by performing several analysis at different azimuthal blade positions.

Airfoil Design

Besides the number of blades, the planform and the power loading, the airfoil sections are of utmost importance for the performance of a windmill. Here maximum L/D ratios are desired to maximize efficiency, taking into account, that the surface of a wind turbine will not be perfectly smooth during the whole life span of a windmill - the airfoils should have good L over D values with rough surfaces too.
For the special case of a fixed pitch windmill, the characteristics of the airfoils are controlling the power versus wind speed performance curve. To avoid over speed conditions, the maximum power of the windmill has to be strictly limited, which can be achieved by a specially designed family of airfoils. These airfoils feature a distinct, but not necessarily hard primary stall, which leads to a limitation of the maximum power of the windmill. To avoid noise and structural problems, the airfoils have been designed to have a soft post stall plateau, followed by a soft secondary stall.

The five airfoils MH 102 to MH 110 make up the new family, which shows no dramatic sensitivity with respect to surface roughness throughout the operating range.

Because the preliminary windmill design and analysis defined ranges for the lift coefficient and the Reynolds number, Epplers inverse design method was ideally suited for the design of these airfoil sections.

r/R	c/R	r	c	ß	Xd	Airfoil
				1)	2)	3)
[-]	[-]	[mm]	[mm]	[°]	[mm]
0.000	0.0000	0.0	0.1	84.500	.	.
0.040	0.1162	320.0	930.0	38.050	.	.
0.080	0.1094	640.0	875.1	25.844	271	MH 102
0.120	0.0988	960.0	790.4	17.883	.	.
0.160	0.0917	1280.0	733.8	12.686	.	.
0.200	0.0890	1600.0	711.9	9.077	214	MH 102
0.240	0.0842	1920.0	673.3	6.678	.	.
0.280	0.0814	2240.0	651.4	4.901	.	.
0.320	0.0783	2560.0	626.6	3.540	.	.
0.360	0.0745	2880.0	595.9	2.471	.	.
0.400	0.0717	3200.0	573.3	1.612	134	MH 104
0.440	0.0689	3520.0	551.4	1.110	.	.
0.480	0.0668	3840.0	534.8	0.725	.	.
0.520	0.0653	4160.0	522.6	0.435	.	.
0.560	0.0643	4480.0	514.1	0.219	.	.
0.600	0.0635	4800.0	508.3	0.061	115	MH 106
0.640	0.0606	5120.0	485.0	-0.207	.	.
0.680	0.0579	5440.0	463.4	-0.436	.	.
0.720	0.0553	5760.0	442.7	-0.631	.	.
0.760	0.0528	6080.0	422.1	-0.794	.	.
0.800	0.0499	6400.0	399.4	-0.938	89	MH 108
0.840	0.0468	6720.0	374.7	-1.061	.	.
0.880	0.0429	7040.0	342.9	-1.158	.	.
0.920	0.0371	7360.0	297.1	-1.247	.	.
0.960	0.0280	7680.0	223.9	-1.316	46	MH 110
1.000	0.0000	8000.0	50.0	-1.377	.	.

Table 2: Blade Geometry

blade-angle defined by plane of rotation and x-axis of airfoil-section
Xd - position of airfoils should form a straight line to minimize torsional Loads
airfoil-orientation is inverse (y-axis pointing with the wind)

Performance

The figure below shows the power delivered by the windmill at the two velocity of rotation settings. The lower 80 rpm value is used to start up the windmill and for wind speeds up to 8 m/s, whereas the 120 rpm setting is used for wind speeds between 8 and 14 m/s. The design power of 90 kW is reached at a wind speed of 13 m/s. For comparison, the chart also contains the power curve of a windmill using conventional airfoil sections, which result in a undesired power output at higher wind speeds.

Performance chart of the optimum stall controlled windmill.

Wind Scales

Beaufort Wind Scale
Beaufort Number or Force	Wind Speed			Description	Effects Land / Sea
Beaufort Number or Force	mph	km/hr	knots	Description	Effects Land / Sea
0	<1	<1	<1	Calm	Still, calm air, smoke will rise vertically. Water is mirror-like.
1	1-3 mph	1-5 kph	1-3 knots	Light Air	Rising smoke drifts, wind vane is inactive. Small ripples appear on water surface.
2	4-7 mph	6-11 kph	4-6 knots	Light Breeze	Leaves rustle, can feel wind on your face, wind vanes begin to move. Small wavelets develop, crests are glassy.
3	8-12 mph	12-19 kph	7-10 knots	Gentle Breeze	Leaves and small twigs move, light weight flags extend. Large wavelets, crests start to break, some whitecaps.
4	13-18 mph	20-28 kph	11-16 knots	Moderate Breeze	Small branches move, raises dust, leaves and paper. Small waves develop, becoming longer, whitecaps.
5	19-24 mph	29-38 kph	17-21 knots	Fresh Breeze	Small trees sway. White crested wavelets (whitecaps) form, some spray.
6	25-31 mph	39-49 kph	22-27 knots	Strong Breeze	Large tree branches move, telephone wires begin to "whistle", umbrellas are difficult to keep under control. Larger waves form, whitecaps prevalent, spray.
7	32-38 mph	50-61 kph	28-33 knots	Moderate or Near Gale	Large trees sway, becoming difficult to walk. Larger waves develop, white foam from breaking waves begins to be blown.
8	39-46 mph	62-74 kph	34-40 knots	Gale or Fresh Gale	Twigs and small branches are broken from trees, walking is difficult. Moderately large waves with blown foam.
9	47-54 mph	75-88 kph	41-47 knots	Strong Gale	Slight damage occurs to buildings, shingles are blown off of roofs. High waves (6 meters), rolling seas, dense foam, Blowing spray reduces visibility.
10	55-63 mph	89-102 kph	48-55 knots	Whole Gale or Storm	Trees are broken or uprooted, building damage is considerable. Large waves (6-9 meters), overhanging crests, sea becomes white with foam, heavy rolling, reduced visibility.
11	64-72 mph	103-117 kph	56-63 knots	Violent Storm	Extensive widespread damage. Large waves (9-14 meters), white foam, visibility further reduced.
12	73+ mph	118+ kph	64+ knots	Hurricane	Extreme destruction, devastation. Large waves over 14 meters, air filled with foam, sea white with foam and driving spray, little visibility.

Saffir-Simpson Hurricane Scale
Category	Wind Strength - Pressure	Effects
1	65 to 83 knots 74 to 95 mph 119 to 153 kph > 980 mb	Storm surge generally 4-5 ft above normal. No real damage to building structures. Damage primarily to unanchored mobile homes, shrubbery, and trees. Some damage to poorly constructed signs. Also, some coastal road flooding and minor pier damage. Hurricanes Allison of 1995 and Danny of 1997 were Category One hurricanes at peak intensity.
2	84 to 95 knots 96 to 110 mph 154 to 177 kph 980 - 965 mb	Storm surge generally 6-8 feet above normal. Some roofing material, door, and window damage of buildings. Considerable damage to shrubbery and trees with some trees blown down. Considerable damage to mobile homes, poorly constructed signs, and piers. Coastal and low-lying escape routes flood 2-4 hours before arrival of the hurricane center. Small craft in unprotected anchorages break moorings. Hurricane Bertha of 1996 was a Category Two hurricane when it hit the North Carolina coast, while Hurricane Marilyn of 1995 was a Category Two Hurricane when it passed through the Virgin Islands.
3	96 to 113 knots 111 to 130 mph 178 to 209 kph 964 - 945 mb	Storm surge generally 9-12 ft above normal. Some structural damage to small residences and utility buildings with a minor amount of curtainwall failures. Damage to shrubbery and trees with foliage blown off trees and large tress blown down. Mobile homes and poorly constructed signs are destroyed. Low-lying escape routes are cut by rising water 3-5 hours before arrival of the hurricane center. Flooding near the coast destroys smaller structures with larger structures damaged by battering of floating debris. Terrain continuously lower than 5 ft above mean sea level may be flooded inland 8 miles (13 km) or more. Evacuation of low-lying residences with several blocks of the shoreline may be required. Hurricanes Roxanne of 1995 and Fran of 1996 were Category Three hurricanes at landfall on the Yucatan Peninsula of Mexico and in North Carolina, respectively.
4	114 to 134 knots 131 to 155 mph 210 to 249 kph 944- 920 mb	Storm surge generally 13-18 ft above normal. More extensive curtainwall failures with some complete roof structure failures on small residences. Shrubs, trees, and all signs are blown down. Complete destruction of mobile homes. Extensive damage to doors and windows. Low-lying escape routes may be cut by rising water 3-5 hours before arrival of the hurricane center. Major damage to lower floors of structures near the shore. Terrain lower than 10 ft above sea level may be flooded requiring massive evacuation of residential areas as far inland as 6 miles (10 km). Hurricane Luis of 1995 was a Category Four hurricane while moving over the Leeward Islands. Hurricanes Felix and Opal of 1995 also reached Category Four status at peak intensity.
5	135+ knots 155+ mph 249+ kph <>	Storm surge generally greater than 18 ft above normal. Complete roof failure on many residences and industrial buildings. Some complete building failures with small utility buildings blown over or away. All shrubs, trees, and signs blown down. Complete destruction of mobile homes. Severe and extensive window and door damage. Low-lying escape routes are cut by rising water 3-5 hours before arrival of the hurricane center. Major damage to lower floors of all structures located less than 15 ft above sea level and within 500 yards of the shoreline. Massive evacuation of residential areas on low ground within 5-10 miles (8-16 km) of the shoreline may be required. There were no Category Five hurricanes in 1995, 1996, or 1997. Hurricane Gilbert of 1988 was a Category Five hurricane at peak intensity and is the strongest Atlantic tropical cyclone of record.

The effects described in the Saffir-Simpson scale are from the
National Hurricane Center

Dvorak Current Intensity Chart
The Dvorak technique is a method using enhanced Infrared and/or visible satellite imagery to quantitatively estimate the intensity of a tropical system. CI -- Current Intensity MWS -- Mean Wind Speed MSLP -- Mean Sea Level Atmospheric Pressure in Millibars
CI Number	MWS (Knots)	MSLP (Atlantic)	MSLP (Pacific)	Saffir-Simpson Category (Approximate)
1	25 Knots
1.5	25 Knots
2	30 Knots	1009 mb	1000 mb
2.5	35 Knots	1005 mb	997 mb
3	45 Knots	1000 mb	991 mb
3.5	55 Knots	994 mb	984 mb
4	65 Knots	987 mb	976 mb	1 (64-83 KTS)
4.5	77 Knots	979 mb	966 mb	1 (64-83 KTS); 2 (84-96 KTS)
5	90 Knots	970 mb	954 mb	2 (84-96 KTS); 3 (97-113 KTS)
5.5	102 Knots	960 mb	941 mb	3 (97-113 KTS)
6	115 Knots	948 mb	927 mb	4 (114-135 KTS)
6.5	127 Knots	935 mb	914 mb	4 (114-135 KTS)
7	140 Knots	921 mb	898 mb	5 (136+ KTS)
7.5	155 Knots	906 mb	879 mb	5 (136+ KTS)
8	170 Knots	890 mb	858 mb	5 (136+ KTS)

Dvorak Chart from NOAA Satellite Services Division

Wind Warnings
Category	Day Flags / Night Lights	Description
Small Craft Advisory	Red over white lights	Forecast winds of 18 to 33 knots (21 to 38 mph). Small Craft Advisories may also be issued for hazardous sea conditions or lower wind speeds that may affect small craft operations.
Gale Warning	White over red lights	Forecast winds of 34 to 47 knots (39 to 54 mph)
Storm Warning	Red over red lights	Forecast winds of 48 knots (55 mph) or greater
Tropical Storm Warning	Red over red lights	Forecast winds of 48 to 63 knots (55 to 73 mph) associated with a tropical storm
Hurricane Warning	Red over white over red	Forecast winds of 64 knots (74 mph) or higher associated with a hurricane

Fujita Tornado Scale
F-Scale / Intensity Phrase	Wind Strength / Frequency	Description of Damage
F0 Gale tornado	40-72 mph 35-62 knots 64-116 kph 29%	Minimal Damage - Some damage to chimneys, TV antennas, roof shingles and windows. Breaks branches off trees, pushes over shallow-rooted trees, damages sign boards.
F1 Moderate tornado	73-112 mph 63-97 knots 117-180 kph 40%	Moderate Damage - Automobiles overturned, carports destroyed, trees uprooted, peels surface off roofs, mobile homes pushed off foundations or overturned, moving autos pushed off the roads.
F2 Significant tornado	113-157 mph 98-136 knots 181-253 kph 24%	Major Damage - Roofs torn off frame homes, sheds and outbuildings are demolished, mobile homes overturned or destroyed, boxcars pushed over; large trees snapped or uprooted, light object missiles generated.
F3 Severe tornado	158-206 mph 137-179 knots 254-332 kph 6%	Severe Damage - Exterior walls and roofs blown off well-built houses, metal buildings collapsed or are severely damaged, trains overturned, forests and farmland flattened, heavy cars lifted off the ground and thrown.
F4 Devastating tornado	207-260 mph 180-226 knots 333-419 kph 2%	Devastating Damage - Few walls, if any, standing in well-built houses, structures with weak foundations blown off some distance, large steel and concrete missiles thrown far distances, cars thrown.
F5 Incredible tornado	261-318 mph 227-276 knots 420-512 kph less than 1%	Incredible Damage - Homes leveled with all debris removed, strong frame houses lifted off foundations and carried considerable distances to disintegrate. Schools, motels, and other larger structures have considerable damage with exterior walls and roofs gone, steel re-inforced concrete structures badly damaged. Automobile sized missiles fly through the air in excess of 100 meters, trees debarked.
F6 Inconceivable tornado	319-379 mph 277-329 knots 513-610 kph less than 1%	These winds are very unlikely. The small area of damage they might produce would probably not be recognizable along with the mess produced by F4 and F5 wind that would surround the F6 winds. Missiles, such as cars and refrigerators would do serious secondary damage that could not be directly identified as F6 damage. If this level is ever achieved, evidence for it might only be found in some manner of ground swirl pattern, for it may never be identifiable through engineering studies

Weather Map Wind Symbols

1 knot = 1 nautical mile per hour = 6076 feet per hour = 1.15078 mph
1 mph = 1 mile per hour = 5280 feet per hour = 0.86898 knots per hour

Electrical Machinery

These are the ways electrical energy and mechanical energy are transformed into each other by rotating machines

Rotating electrical machines -- generators and motors -- are devices that transform mechanical power into electrical power, and vice-versa. Electrical power from a central power station can be transmitted and subdivided very efficiently and conveniently. The operation of electrical machines is explained by four general principles, that will be briefly presented below. These principles are not difficult to understand, and illuminate most of the reasons for the stages in the historical development of electrical power, and especially of electric railways. This page discusses motors in general, but the specific application to electric locomotives is emphasized. Electricity is the medium that carries power from the prime mover, whether at a central power station or on the locomotive, to its point of application at the rail, and allows it to be controlled conveniently.

Power is rate of doing work. One horsepower means lifting 550 pounds by one foot in one second. Mechanical power is force times speed. One watt is a current of one ampere (A) flowing in a potential difference or voltage of one volt (V). Electrical power is current (in amperes, A) times voltage (in volts, V). 746W is equivalent to 1 hp. A medium-sized electric locomotive might have a rating of 2000kW, or 2680 hp. At 85% efficiency, and a voltage of 15kV, 157A is drawn from the overhead contact wire. Torque is the rotational equivalent of force, often useful in speaking of motors. It is force times perpendicular distance, and power is torque times rotational speed in radians per second.

The first principle is that an electrical current causes a magnetic field which surrounds it like a whirlpool, and that this field, which is not material but rather a region of influence on other currents and magnets, is guided and greatly strengthened (by more than a thousand times) by passing through iron. When the current reverses in direction, so does the magnetic field. Currents deep in the earth cause its magnetic field, and the energy to drive them comes from either the rotation of the earth or the flow of heat within the earth. The field acts on the compass needle, which is a magnet. This principle can be called "electromagnet action."

The second is that an electrical current in a magnetic field (produced by some other currents) experiences a force perpendicular to both the direction of the current and the direction of the magnetic field, and reverses if either of these reverses in direction. The force is proportional to the current and to the strength of the magnetic field. This principle can be called "motor action."

The third is that an electrical conductor, such as a copper wire, moving in a magnetic field has an electrical current induced in it. This is expressed by the creation of an electromotive force or voltage, which causes current to flow just as the voltage of a battery does. The effect is maximum when the wire, the motion, and the magnetic field are all mutually perpendicular. This principle can be called "generator action."

And the fourth principle is that a changing magnetic field causes a voltage in any circuit through which it passes. The change can be caused by changing the current producing the magnetic field, or by moving the sources of the magnetic field. This principle can be called "transformer action."

A rotating electrical machine consists of a field and an armature that rotate with respect to each other. The armature is the part of the machine in which the energy conversion takes place. The field provides the magnetic field to aid this process. In DC machines, the field is stationary (the stator) and the armature rotates within it (the rotor), because the rotation is necessary to switch the armature connections by means of the commutator, but it is only the relative motion that counts. In an alternator, the armature is stationary and the field rotates. The field consists of an iron core to carry the magnetic field, and a winding to excite the magnetic field by the current passing through it (first principle). The magnetic field is a passive but essential component in the operation of the machine. Like the field, the armature also consists of iron to complete the magnetic circuit, and is separated by a short air gap from the iron of the field. It is important that the air gap be as small as possible and remain uniform as the armature rotates.

The armature also has windings. In a generator, these conductors are moved in the magnetic field producing a voltage (generator action). If a circuit is completed and current flows in these windings, a force is produced resisting the rotation of the armature (motor action) so that the driving machinery experiences a mechanical resistance and does work, which is being transformed into electrical energy. In a motor, these conductors are supplied with an electrical current, so that a force acts on them in the magnetic field (second principle), and this force can do external work. When the armature rotates while exerting the force, work is done, but a voltage is also produced opposing the applied voltage, resisting the flow of current in the armature (third principle), implying a change of electrical work into mechanical work.

This opposing voltage generated when the armature of a motor turns is called counter-electromotive force. It might seem that it resists current flow through the motor, and of course it does, but it is really the essential factor in turning electrical into mechanical energy. Only the current that is driven into a counter-emf appears as mechanical work at the motor shaft; all else is wasted, the energy going into heat instead of mechanical work. Early inventors of electric motors did not realize this, and tried simply to get as much current into the motor as possible, which only burned the motor up without producing any mechanical effect.

Current is supplied to the armature through sliding contacts formed by graphite blocks (called brushes because originally brushes of phosphor bronze wire were used instead) pressing against copper rings. It is usually necessary to change the connections of the armature windings as they rotate with respect to the magnetic field, and this can conveniently be done by making the copper rings in segments. The result is the rotary switch called the commutator. These days, semiconductor switches can be used for this purpose in small motors, eliminating the commutator, but the principle is the same. The commutator and brushes are the only parts of a machine that normally require maintenance, except for the bearings and other mechanical elements. If it is not necessary to switch the current, as in AC machines, the moving contacts are called slip rings.

Contemplate now the complete chain of energy flow from the prime mover, a steam or internal combustion engine, to the point where the mechanical power is finally applied. The transformation at each end must take place with a smooth mutual reaction, based on the second and third principles. This was not properly understood until the 1880's, so that practical electric transmission of power was delayed until that time. Power that is not delivered to the load is lost as heat in electrical resistance, which is equivalent to mechanical friction. Heat is produced in the generator, transmission lines, and motor, and limits the amount of power that can be handled.

Electrical motors were invented early, in the 1830's, as soon as the magnetic effects of electrical currents and the magnetic properties of iron became known. The motors of Christie and Pixii are typical of these, which used the repulsion and attraction between electromagnetic poles switched by a commutator. Small motors of this kind are still made for classroom demonstration. Attempts in the 1840's to make these motors more powerful and larger failed completely, because the magnetic forces do not scale proportionally to distance, and the significance of counter-emf was not known. The motors of Davenport and of Long in the United States are examples of these unsuccessful attempts to scale up classroom demonstrations to practical size.

More success was encountered in making generators, usually by moving permanent magnets (thereby creating a moving magnetic field) with respect to coils of wire wound around iron cores, to generate alternating currents for supplying arc lights and direct currents for electrolysis tanks (transformer action). These generators all ran quite hot because of their lack of efficiency, but supplied the greater currents required for these applications more cheaply than chemical batteries. This industry evolved into the electrical power industries of later years.

Siemens and Gramme solved the problem of efficiency in the late 1870's by introducing magnetic circuits that did not change as the armature rotated, so that the electrical reactions were smooth and constant. Siemens' first machine (a generator) of 1866 is shown at the right, and a Gramme dynamo, which could also serve as a motor if the brushes were repositioned, is shown at the left. These machines had smooth armatures with conductors on their surfaces. It was still thought that the conductors actually had to be immersed in the magnetic field to produce forces. Soon it was discovered that if the conductors were put into slots in the armature surface, the same result was obtained. This was far superior mechanically, and also made a smaller air gap possible.

The first long-distance transmission of electrical power took place in 1886 over the 8 km between Kriegstetten and Solothurn in Switzerland. Two Gramme machines in series were used as generators, and two similar machines in series at the other end were used as motors. The line voltage was 2000V, and the wire 6mm in diameter (1/4"). The shaft-to-shaft efficiency was 75%, and the installation remained in service until 1908.

Edison's famous Z-type dynamos (as direct-current generators are often called) appeared in 1879 to supply his carbon-filament incandescent lamps. These had long fields on the mistaken assumption that this gave a more powerful magnet (like a longer lever), showing how little magnetic circuits were understood at the time. This arrangement allowed a great deal of magnetic leakage between the long arms, and made the flux distribution in the armature nonuniform. Hopkinson, an engineer with Edison's British company, rationalized the field geometry, making a very good generator of the modern type a few years later. The field was symmetrical with respect to the armature, and short. A closely related type, the Manchester dynamo, is shown below. Compare its compact and short magnetic circuit with that of the Edison Z. Note the brush holder and the brass commutator on the armature. This is a two-pole machine, because the field has one N pole and one S pole.

One thing that may worry you if you examine an electrical machine closely also worried early designers. They put the wires on the surface of the armature where they would actually be in the magnetic field and experience motor or generator action, in the way we have explained it here by our principles. However, wires are now always placed in slots cut in the armature iron, allowing the air gap to be made smaller and the magnetic circuit much more efficient. The overall result is the same as if the wires were actually in the magnetic field, but the mechanism is slightly different. Now the armature current in the motor magnetizes the armature iron, and the interaction of this magnet with the field poles provides the force. In a generator, the field magnetizes the armature iron, and this field moves past the conductors as the armature rotates, with an effect like a transformer. Siemens, I believe, was the one who first saw this and the great improvement it could make in electrical machines.

The ways that windings of wire are arranged in modern machines are shown at the right. The windings are either around the pole pieces, or placed in slots on the surface. The part that rotates is called the rotor, and the part that remains at rest is called the stator. Both are of a magnetic core iron alloy, and are laminated if they are subject to alternating magnetic fields, to reduce eddy-current losses. DC machines typically have a salient-pole field on the stator, with the field windings on the pole pieces, and a non-salient pole winding on the armature, forming the rotor. The magnetic field of the stator is constant, while the field in the armature alternates. Therefore, the armature is laminated. The actions of salient and non-salient pole windings are equivalent. A non-salient pole winding can be arranged to give any desired spatial distribution of magnetic field. The typical salient-pole winding of a DC machine provides field-free regions between the poles that aids commutation, since switching can be done while the armature conductors are in this region and not generating any emf. In both salient and non-salient pole machines, the windings are firmly held mechanically.

The windings of motors and generators can be connected in one of two basic fashions. If the field windings and the armature windings are in series, they are called series-connected. In this case, the field windings are of heavy-gauge wire to carry the main motor current. The field becomes stronger as the armature current increases, leading to a very great force at low speeds. If the field and armature are in parallel, they are called shunt-connected. The field winding consists of rather fine wire. If the voltage applied to the motor is constant, then the field strength is also constant. If a generator is rotated at constant speed, then the output voltage is independent of the load. There are intermediate cases where the field has both series and shunt windings, and such machines are called compound.

Most direct-current power-station generators are mainly shunt-connected, and most traction motors mainly series-connected, as you might expect from the requirements of the two services: constant voltage in the first case, high starting torque in the second. Rotating machines can be made for voltages up to about 2000V, the restrictions being insulation and flashover at the commutator.

It is not easy to change DC voltages. One way to do this was to use a dynamotor, which had a normal field winding, but dual armature windings and two commutators. One winding was supplied at the input voltage and drove the dynamotor by motor action. The other winding supplied the output voltage. This can really be considered a kind of AC transformer. The input commutator creates AC from DC, and the output commutator changes the new AC voltage to DC. In World War II, when radios required a plate supply of, say 300 V, dynamotors were used to obtain this voltage from 6 V battery supply.

The speed of a direct-current motor is determined by both the field strength and the load. If there is no load, the speed is such that the voltage produced in accordance with the third principle exactly balances the applied voltage, and the armature current is zero. As the load is increased, the speed decreases to allow current to be drawn so the necessary electrical power can be converted. When the motor stalls, it is exerting its maximum force. Therefore, the speed of a shunt motor, or one in which the field is produced by a permanent magnet, is determined by the applied voltage, and can be adjusted finely.

If voltage is applied to a series motor without a load, the motor speeds up. As it does so, the field current decreases so the motor must speed up some more to generate the same back voltage. This keeps up until the motor flies apart. The loss of load on a series motor is a serious thing, and must be guarded against. When a loaded series motor is rotating with the maximum voltage applied to it, the current just produces the required amount of force with the existing field strength. If the field is weakened by reducing the current in it (by putting a resistance in series with it, for example) the motor must speed up to compensate. This is one method of speed control for direct-current motors.

A large direct-current motor must not be started by applying the full voltage across it while it is at rest, especially a series motor. The heavy current and field will create a great jolt that may damage the motor and its mechanical connections. A starting resistance is used to limit the initial current to only the amount necessary to put the motor into rotation. As it speeds up, the starting resistance can be removed in steps. In normal operation, the starting resistance should be removed, since it represents a significant loss of power. For further speed control when more than one motor is used, as on a streetcar or locomotive, the motors can be connected in series to start, and in parallel to run . In each case, the field can be weakened to give a higher speed. With four motors, series, series-parallel, and parallel connections, with field weakening, gives six speed levels that can be designed for service requirements. This could give, for example, speeds of 10, 15, 20, 30, 40, and 60 mph with starting resistance switched out.

A direct-current motor can be reversed by reversing the direction of the current in either the field or the armature. There is more to the story, however. The small demonstration locomotive of Werner Siemens of 1879 reversed by means of gears, and some authors have implied that it was not yet known how to reverse a DC motor, which is absurd. The problem lay in the brushes and commutator. The brushes must be placed so that their switching action takes place at the moment when the current in the windings being switched is zero. As the load on the motor increases, the armature current increases, and the magnetic field it produces causes the total magnetic field to change so that the brushes must be shifted to a new point of zero current. If this is not done, there is sparking at the commutator, which is rather destructive. The brushes on Siemens' locomotive were set at an average position for the load, so that if the motor were reversed, the brushes would have been at an improper position, and sparking would result. Therefore, he used reversing gears and the motor continued to rotate in the same direction.

A better way out of this difficulty was soon found. Small poles, called commutating poles, were placed between the main field poles. These windings are in series with the armature, and proportioned so that they cancel the varying field of the armature. The optimum brush position then becomes independent of motor load or direction of rotation.

Semiconductors have made possible a continuous control of voltage to the traction motors. This is called chopper control, and gives direct-current locomotives the same fine speed control as alternating-current locomotives. Using semiconductors, direct-current electrification is possible without any rotating machines, and all the advantages of both alternating and direct currents can be exploited.

If a commutator is not used on a generator, merely slip rings to connect with its windings without any switching, its output voltage alternates smoothly from positive to negative, which can be made to be a pretty good sine curve by using non-salient-pole windings. It is easier to generate such an alternating current by rotating the field inside a stationary armature, so that the main current does not have to flow through the sliding contacts. The sliding contacts, which are called slip rings, need handle only the lower voltage and current of the field windings. High voltages can be generated because it is much easier to insulate the stator, and because these voltages do not have to be handled by the slip rings. This is such an advantage that practically all electricity is generated as alternating current, and the generators are called alternators. These advantages were even reflected in automobiles, which previously used 6V dynamos, but now all use 12V alternators. Typically, the stators are non-salient-pole, while the rotating fields have salient poles.

Alternating current has the compelling advantage that its voltage can be changed easily and efficiently by a transformer, which is a closed iron core surrounded by two windings (first and fourth principles). The ratio of the voltages in the two windings is the same as the ratio of the number of turns, and the ratio of the currents inversely, so that the power remains the same. Since there are no mechanical parts, the efficiency of transformers is very high, and maintenance very low. Alternating current is transformed to higher voltage and smaller current for transmission, and back to lower voltages for use. Transformers with taps can be used to obtain a series of voltages if desired. In fact, an almost continuous voltage variation without loss is possible.

If direct current is required for motors, alternating current can be converted to direct current in four ways. First, a motor-generator set can be used, running at constant speed. A better solution is the rotary converter, essentially a DC generator rotated by AC supplied to it. These were the only practical ways at first for large power requirements, especially for electric railways. Later, rectifiers based on electrical discharges, notably the mercury-arc rectifier, made conversion possible without rotating machines, and with currents appropriate for locomotives. Both of these methods have now been completely superseded by solid-state (silicon) rectifiers, which are trouble-free and easily controlled. Single-phase 50/60 Hz alternating current can be supplied at high voltage, reduced in voltage by a transformer, rectified by solid state rectifiers, and applied to direct-current traction motors, making a very serviceable and economical locomotive that is today's standard.

The Ward-Leonard system of speed control for a direct current motor used a generator driven by a motor supplied from any kind of current, alternating, three-phase, or direct. By control of the generator and motor fields, control of the motor over a wide range of speeds was possible. This system was used in a few locomotives to permit the use of direct-current traction motors with single-phase alternating current supply. The problem was that three machines of equal capacity were required in place of one, a very costly alternative. Semiconductor rectifiers or chopper control have completely replaced the Ward-Leonard system.

At first sight, it would seem easy to make alternating current motors. If the polarity of the supply to a DC motor is reversed, the motor continues to turn in the same direction. If AC is supplied, the torque might be pulsating, but at least it would always be in the correct direction. Unfortunately, this is not true (except at unrealistically low frequencies) because of the existence of reactance that causes phase differences and inductive kicks. The magnetic fields and the currents get out of register, torques are small, and there are sparks everywhere. However, with the series motor there is some hope, because the same current that creates the field also passes through the armature, so they must stay in phase. In fact, series motors can be designed for AC, and they have been brought to a high state of excellence, although best at 16 2/3 or 25 Hz.

Most AC motors, however, depend on a rotating field produced in the stator, or stationary part of the motor. This is a magnetic field produced by windings through which alternating currents are passed that seems to rotate around the stator with time. The number of poles P is the number of north and south poles around the circumference, and is determined by how the windings are placed on the stator. The rate of rotation of the field is given by N = 120 f/P rpm, where f is the frequency of the alternating current. A four-pole field rotates at 1800 rpm, for example, with 60 Hz current. This is called the synchronous speed. The easiest way to produce such a field is to use an AC supply that consists of several voltages with a constant phase difference between them, called a polyphase supply. One could use two voltages at 90° phase difference, and let each voltage supply alternate coils in a four-pole machine. Better, however, to use three voltages at 120°, called three-phase, for then the power flows as evenly as with DC. These voltages are generated by separate windings on the alternator, and are supplied over three wires. These stators have no salient poles, but have a smooth surface with the windings embedded in slots. Three-phase current is normally used for transmission and distribution of electrical power, since it is the most efficient means in terms of the copper required and transmission losses.

Suppose we have such a stator with a rotating magnetic field. For the armature, we use a drum armature as for a DC motor, but simply short-circuit the windings. Then, when the motor is not rotating, we have what is very much like a transformer with a shorted secondary winding. It is excited at the frequency of the stator currents. Unlike a transformer secondary, this secondary can rotate in response to the forces exerted on it (its windings are in a magnetic field). When it does so, it tends to follow the rotating field more and more closely. When it is rotating at exactly the same speed as the rotating field, the apparent frequency to the rotor is zero, and no currents are induced at all. If it rotates slightly slower, or slips, the frequency increases, and more currents are generated. The slip is the difference in rotating speed divided by the synchronous speed, expressed as a percentage. The forces on these currents provide the torque exerted by the motor. The torque increases about proportionately to the slip. At some point, however, the reactance of the armature windings come into play not only to limit the currents, but to move them out of phase with the magnetic field. So the torque levels off, and passes a maximum value called the breakdown torque. The motor now slows down more and more rapidly, the torque more and more out of phase. When the rotor comes to rest, the starting torque is produced, and the motor will not move until the required torque is less than this amount.

This kind of motor is called an induction motor, which has no DC analog. It rotates a bit more slowly than the speed of the rotating stator field, which is called the synchronous speed. Large polyphase motors may have a slip of only 1% or so at normal load, so the induction motor is essentially a constant-speed motor. Its speed can only be greatly changed by changing the frequency of the supply. The starting torque is considerable, but at the cost of rather high currents out of phase with the voltage. The lagging power factor of an induction motor can be a problem. Nevertheless, the lack of a commutator and brushes, and that it has a reasonable starting torque, make the induction motor the most commonly used AC motor. A typical motor has a rotor composed of parallel thick copper or aluminum conductors connected to a ring of the same material at the ends. This is called a "squirrel cage" rotor.

Squirrel-cage rotors, because of their low impedance, run with small slip and give high torque at speed. However, their starting torque is poor. If your sink garbage disposal is on the same circuit with some lights, note the lights dim briefly when you switch it on. This is the result of the high reactive current drawn by the static induction motor, which can be 4 times the normal load current. Fuses must be slow-blow to let this pass before deciding that there is some problem. There is usually a circuit breaker on the disposal that opens if the motor is stalled. You might also note the limited starting torque when the motor fails to start if it gets something in its teeth. By adding resistance to the rotor, better starting torque can be obtained, at the cost of poorer running at speed. Ways have been devised to cut out the resistance after starting, but these involve a wound rotor with brushes, or complicated double windings, and have not been worth the bother.

Small single-phase induction motors can be made that create the extra phase that is necessary for starting the motor internally. Examples are the split-phase, capacitor start and shaded-pole motors used on motors of a few hp and less. Split-phase motors use a separate winding of higher inductance to produce a phase shift; capacitor motors use a capacitor for the same effect, and the capacitor can be switched out after the motor has started and has come up to speed. A shaded-pole motor has a copper ring around one side of the poles to retard the flux there, so the field appears to rotate. These single-phase motors can operate on 50/60 Hz, but are not suitable for traction motors because of their poor starting characteristics, and their low efficiency.

An unusual type of single-phase AC motor has non-salient-pole stator and rotor. The stator is supplied by the single-phase AC current, while the rotor is very similar to a DC armature, with commutator and brushes. The brushes are short-circuited, and the armature current is created by induction from the stator. If the brushes connect turns that are in field-free regions, any armature current would be in phase to produce a torque. However, with this brush position, there is no current, since the emf's cancel. If the brushes connect turns that are under the poles, the induced armature currents are a maximum, but the forces exerted on them cancel. At some intermediate brush position, there is both current and a net force. The motor can be reversed by moving the brushes past a pole to an equivalent position. Such a machine is called a repulsion motor, and it acts somewhat like a series motor. Although it gives good torque, commutation at the brushes can be difficult, with sparking. The theory of the repulsion motor is rather difficult.

A non-salient-pole stator with a rotating field can be used with a salient-pole rotor to make an AC motor. This is, in fact, exactly like an alternator except that the mechanical power is output, not input. The rotor is fed DC by slip rings to magnetize it. As the stator field rotates (electrically), it will carry the rotor poles around with it at the synchronous speed (mechanically). For this reason, this type of motor is called a synchronous motor. A synchronous motor is essentially an alternator run backwards. Rotor current must be supplied, because there is no relative motion to induce it as in an induction motor. The magnetic field in the rotor is constant, and corresponds to the constant flux in a transformer. However, its alternating variation as seen by the stator windings is now produced by its motion.

A few poles of a synchronous motor are shown in the diagram on the left. Stator poles are shown as salient for clarity, though the stator is usually non-salient-pole. The rotor and stator poles remain in fixed relative positions. The rotor poles are carried along by the rotation, while the stator poles move as they are excited by the polyphase supply. When there is no torque exerted by the rotor, conditions are as in (a). When the motor supplies a torque, the rotor lags a bit behind, as shown in (b). The lines of force suggest the pull on the rotor, since there is a tension along the magnetic field. There is now a phase difference between rotor and stator, inducing a voltage that reduces the back emf presented to the supply, and the current to the motor stator, in phase with the voltage, must increase to bring the back emf up to the supply voltage, exactly as in a transformer. It is rather difficult to see this except by a detailed analysis involving phasor diagrams, but the force and energy relations can be verified as for a DC motor. Energy supplied against a back emf is transformed to an equal amount of mechanical enerby.

The motor torque depends on the angle the rotor lags behind the rotating stator field. If the torque required is too high, the motor may drop out of synchronization, and then come to rest. Such motors can supply considerable power, but run at a fixed speed determined by the current frequency, and cannot start against a load. They may be started by an auxilary motor, by acting as an induction motor, or by other means, before load is applied. They are, therefore, unsuitable as traction motors, but can be the motor in a motor-generator set. They can drive a three-phase alternator in this arrangement, which can feed three-phase traction motors, or, of course, can drive a direct-current generator, as in a Ward-Leonard control. The synchronous motor requires DC excitation, and has slip rings to maintain, so it is not as commonly used as the induction motor. If the DC field excitation is changed, the synchronous motor can be made to draw reactive, wattless quadrature current and act as a capacitor or an inductor. Only for a certain level of excitation is the current in phase with the applied voltage. Synchronous motors without a mechanical load can be used to correct power factor in a transmission system.

One type of small synchronous motor is used to drive clocks. Tesla chose 60 Hz as the power frequency looking toward this application. More useful for this purpose are subsynchronous motors, that run at a fraction of the synchronous speed, making the gearing easier. These motors can have a toothed rotor, or a rotor consisting of copper rods, and will lock in at a subsynchronous speed, though they are fundamentally two-pole motors operating at 3600 rpm. Such motors cannot supply much torque.

A synchronous converter can convert AC to DC with high efficiency. This is a machine much like a DC generator, but with a polyphase non-salient-pole field. Polyphase, usually six-phase, power is supplied to the stator, while the rotor rotates at the synchronous speed. DC can now be taken from the commutator and brushes. The armature current is the difference of the AC and DC currents, so it is rather small and this helps the efficiency of the machine. The DC side of the machine can be used to bring it up to synchronous speed quite conveniently, if DC is available from batteries or some other source. Synchronous converters were much used before solid-state rectifiers became available.

Quite recently, semiconductors have made possible the creation of three-phase power of variable frequency, so that it becomes applicable to traction motors. Electrical power generated by an alternator, or supplied by a contact wire, is first converted to direct current, then to variable-frequency three-phase for supplying induction traction motors in a locomotive. In this way, commutators and other troublesome sliding contacts are completely eliminated, reducing maintenance costs. This has occurred even though DC has distinct advantages for traction.

Improvements in motors since the turn of the century, making them smaller, lighter, and more efficient for the same output, have been due to three main factors. The first was rational design of the magnetic circuit, to make the best use of the iron. Next was improved magnetic material, especially silicon iron with low losses, that meant less iron could be used and motors could be more compact. Finally came better insulating materials that required less room and could stand higher temperatures. These benefits extended to all other electrical machinery as well, including generating apparatus and transformers.

A supply in the range 600-700V DC was common for street railways, rapid transit, and interurban electric railways, in spite of the heavy currents required. Remember that power is voltage times current, so doubling the voltage halves the current for the same power, and the losses are proportional to the square of the current, so a decrease in current is very desirable. A voltage much above this cannot safely be used for third rails and where work must be done around energized equipment. With normal precautions, such voltages are not fatal and require actual contact for danger. 1500V was initially popular for main line electrification with an overhead contact wire. Somewhat higher voltages, up to 3000V, were later used to reduce the traction currents as far as possible, although 3000V generally requires traction motors to be permanently in series. 11, 15, and 20kV are used for alternating-current electric railways using an elevated contact wire. These are the highest voltages that can be safely used taking normal clearances into account. Personnel must be kept well out of the way of such voltages, which, unlike the lower voltages just mentioned, can reach out and be fatal at considerable distances. Transmission may be at 100kV and over, where the conductors are specially insulated on high towers.

Although we have seen a bewildering variety of rotating electrical machines, there is really only one fundamental principle at work. These machines are transformers between electrical and mechanical energy, just as the usual AC transformer transforms between electrical energy at different voltages. In fact, the ordinary transformer was also called a static transformer. These machines are dynamic transformers. On the mechanical side, the energy is transmitted by rotation. Torque times angle is work, torque times angular velocity is power. Every machine has a magnetic circuit, in which certain fluxes are established that are analogous to the flux in a static transformer. This magnetic field assists in the energy transfer, but does not receive or give energy itself. The magnetic flux links conductors, and can exert forces on them if they carry currents, and can induce emf's in them if the flux changes. The forces determine the mechanical power, while the emf's determine the electrical power. There are other energy effects, such as I²R loss in the conductors and the eddy currents, iron losses due to hysteresis, mechanical friction and windage, and the alternating flows of reactive volt-amperes. However, the induced emf's and the forces are always such as to represent an ideal conversion between electrical (emf times current) and mechanical (force times distance) energy, accompanied by these losses, which can be minimized, but are unavoidable.

When the conductors are on the surface of the rotor, actually in the magnetic field of the air gap, it is easy to see how the emf is induced, and how, if current flows, the force times distance equals the emf times current, as we pointed out at the beginning of this paper. It is not so easy to see this if the windings are on the stator, while the force is on the rotor, and if the conductors are buried in the iron, or wrapped around the poles. Nevertheless, a careful analysis would show in every case exactly the same relations that are so evident with conductors in the air gap.

To generate electrical energy, we move conductors (armature) in a magnetic field (field) at rest, or else move a magnetic field (rotor) relative to conductors at rest (stator). In either case we get a periodically reversing emf that can be made sinusoidal by careful design. If generated in the rotor, the motion can be used to switch the connections so that the output current is unidirectional, and more or less constant. Mechanical energy always enters by the rotor, but electrical energy can be taken either from the rotor or the stator. Whenever any current is drawn, the mechanical side feels the effect as an increased drag. The energy received from the mechanical side is always greater than the electrical energy delivered.

To generate mechanical energy, we can place movable conductors (armature) in a steady magnetic field (field). Torque is produced when we drive a current through the conductors, and when the armature moves, we feel an electrical opposition to supplying the current. Or, we can wind the stator to produce a rotating magnetic field, and have this field drag along conductors on a rotor. Current is driven through the rotor conductors either by induction, or by an external source. In either case, when mechanical energy is drawn, there is an electrical effect amounting to an increase of the current driven into a back emf. The energy delivered to the mechanical side is always less than the electrical energy supplied.

The only general and satisfactory way to understand the forces on the rotor of an electrical machine is by considering the magnetic field over an imaginary surface surrounding the rotor. Electric fields play no role in the forces in electrical machinery. From the magnetic field, the Maxwell stress tensor can be found, and from it the forces on the rotor, by integrating the shearing stresses over the surface. In the usual DC motor, if no armature current exists, the magnetic field passes symmetrically through the armature, and there is no net torque on it. When armature current flows, it creates a transverse component of the magnetic field, so the total field is "twisted." This field has a tangential component in the air gap that is responsible for the torque on the armature. The picture of current-carrying wires in a magnetic field is of little help in understanding an actual motor.

Rotary motion is ideal for most applications. On the mechanical side, it allows the use of an excellent and efficient mechanical transformer, gearing. Bearings provide a convenient support for rotating shafts. Turbines provide rotary power for generators, and they are efficient prime movers when used at constant speed and constant power. Just as there are reciprocating engines, one can conceive of reciprocating electrical machines. However, one notices that reciprocating motion usually has to be transformed to rotary motion for applications, aside from such things as driving a reciprocating pump. Even here, the centrifugal pump, with rotary motion, is used where possible. Reciprocating electrical machines have been tried, but were very unsatisfactory, and there is no reason to resurrect them. Motors with a linear stator would also seem quite impractical mechanically, if not electrically. The only application would be to traction, but here they are a solution without a problem. Professor E. Laithwaite of Imperial College promoted linear motors vigorously, and there are still efforts in this direction, usually with magnetic levitation as well. The best use of this idea would be if the power were supplied to the stator, with a passive "cursor," but this would be hopelessly uneconomic for practical transport, though feasible for very short distances. There is no essential difference in operation between these linear motors and the rotary ones.

Fans and Wind Power

2009-01-30T03:52:00.000-08:00

Fans and Wind Power

Introduction

The wind, like water, is a natural source of power. With water, we use its attraction by gravity as it descends from the mountains to the sea. With wind, we use the kinetic energy of the air. The energy of either fluid per unit volume can be expressed as the sum of three terms, ρgz + p + ρV²/2, corresponding to elevation, pressure and velocity. One form can be converted into another quite freely, but the sum must remain constant, or even decrease due to dissipative forces like friction. It is convenient to divide each term by ρg, the weight per unit volume or specific weight, so that they have the dimensions of length. Then, energy is called head of the fluid in question.

The most significant difference between air and water in regard to energy is that they are very different in density. Water is about 800 times as heavy as air, volume for volume. This means that energy is much more tightly packed in water, so that water machines can be conveniently small. Air machines, on the other hand, must handle large volumes of air, and so are large and cumbersome. Air weighs about 0.075 pcf (70°F, 29.92 inHg) or 1.293 x 10^-3 g/cm³ (0°C, 760 mmHg). Air is also compressible, approximately obeying the ideal gas law pV = nRT, where R is the molar gas constant, 8.31441 x 10⁷ J/K-gmol or 1545.33 ft-lb/R-lbmol. K = °C + 273.15 and R = °F + 491.7. Water, on the other hand, is incompressible in normal situations. Air is about 10 times more viscous than water, its kinetic viscosity being about 0.15 cm²/s. The discussion here applies directly to air, but the behavior of water is very similar.

The most popular unit of pressure for air machines in U.S. engineering has been the inch of water gauge. An inch of water is 1.867 mmHg, or 0.036 psi, or 1.489 mb (1489 dyne/cm²). Gauge pressure is pressure above atmospheric, which is considered to act at all points equally, so that it vanishes from equations. To find pressure in inches of water, multiply air head in feet by 0.1442, or air head in metres by 0.473.

Flow velocity in air or water is often measured with a Pitot Tube, as shown in the figure at the left. The impact tube faces directly into the flow, while the static tube is perpendicular to the flow. Both functions can be combined in a single unit. From Bernoulli's equation, p_t = p_s + ρV²/2. In U.S. engineering units, ρ = w/g, where w is pcf and g is 32.1725 ft/s². Therefore, V is proportional to the square root of the pressure difference Δp. For air, if V is in fps and p is in inches of water, then V = 66.8√Δp, assuming air weighs 0.075 pcf (in Denver, it is closer to 0.060 pcf). For water, the coefficient is 2.32. In practice, the pitot tube must be traversed to measure the velocity in annular areas of equal volume of a circular duct. It is often assumed that the average velocity V is 0.83 of the velocity at the centre of the duct, but this is approximate.

Henri Pitot immersed a bent glass tube in the Seine to determine the velocity of its water in 1730. The Pitot tube has been used in hydraulics since then. Perhaps its greatest common application is as an airspeed indicator for aircraft.

Let's estimate how much wind power is available to us. The average wind in Denver is 7.5 mph or 11 fps. The corresponding velocity head is (11)²/(2)(32.2) = 1.879 ft of air. Multiplying by the specific weight, 0.075 pcf, the energy is 0.1409 ft-lb/ft³. Assume a windmill of 50 ft diameter. The area will be 1964 ft², and so 21,600 ft³ of air will be gathered per second, or 3040 ft-lb/s. Since 1 hp is 550 ft-lb/s, this is 5.5 hp. The maximum efficiency of the windmill will be no greater than 65%, the efficiency of a propeller fan, so the useful output of our 50 ft windmill will be about 3.6 hp or 2.7 kWh. To capture greater power, we need either a higher wind velocity or a larger windmill. If the windmill is 150 ft in diameter, 3 times larger, and the wind speed is four times larger at 30 mph, then the power output would be multiplied by 576 to 2073 hp or 1555 kWh. This illustrates at least two important things: wind power is not very dense, and requires large machines to capture it; and it varies rapidly with the force of the wind (as V³). The past month (July 2003) in Denver has been practically windless, and would generate very little power.

Water mills come to us from antiquity, but windmills do not. Wind has been used as the motive power for boats for millennia. This application uses only the pressure of the wind on a sail, and the techniques for controlling it were highly developed. It does not require machinery, except for the minor machinery for handling rigging. In this article, we consider only the use of the wind in producing mechanical power. The basic theory of turbines, of which fans and windmills are examples, is given in Turbines.

The Fan

Fans are used to move air (or other gases) in large volume at low gauge pressures. A windmill is a fan in reverse. A fan consists of a wheel or impeller, and a housing. Sometimes the housing is absent, and we have just the impeller, as in an aircraft propeller. The two principal types of fans are the axial-flow and the centrifugal. We will talk mainly of propeller-type axial flow fans here, but the general principles will apply to both types. The ducting and other appurtenances associated with a fan are called the system, which may be absent when a fan is used in the free air just to generate a breeze. If the system is at the output of a fan, the fan is called a blower, while if the system is at the input, the fan is an exhauster. The moving part of the fan is the impeller or wheel, and the stationary part the housing. A propeller fan may have a housing as simple as a circular aperture, called the shroud, which nonetheless makes the fan more efficient. At the other limit, the fan may be enclosed in a duct and work against static pressure.

An arrangement for a fan test is shown at the right. The flow resistance of the duct can be varied at input or output. If A is the area of the duct, then Q = VA, where V is the flow velocity. If the input and output are completely unrestricted, then the pressure difference is zero and the flow is a maximum. If the duct is blocked at each end so that the flow is zero, the pressure difference Δp will be a maximum. The most important variables for a fan are its discharge at outlet in cfm or m³/s, which together with the area of the fan gives the output velocity V, and the total pressure difference Δp. For propeller fans, this pressure difference is in the range 0.5 to 1.5 inches of water. The velocity pressure at the output is ρV²/2, and the static pressure at the output is the total pressure less the velocity pressure. The output power is the total power in the output, W = (p_s + ρV²/2)Q. The efficiency is this power divided by the input power W', e = W/W'.

There may be an egg crate straightener about 6 duct diameters from the input. This is a lattice of square passages of side 0.075 to 0.15 of the duct diameter, and three times as long as the length of a side. This and other details are mentioned in the standard specifications for fan tests of the Association of Heating and Ventilating Engineers, or the ASME, which are excellent sources of information about fans.

Fan characteristics as determined by a test in a duct are shown at the left. These are just the general shapes of the curves, not the results for any particular fan. At zero flow the fan maintains a pressure difference of p_max. At free discharge, the fan produces a flow of Q_max. The pressure curves give the results for intermediate cases. p_s is the static pressure, as would be measured by a manometer with an opening in the wall of the duct. p_t is the total pressure, as would be measured by a Pitot tube. The difference between them is p_v, the kinetic energy measured in pressure units. The input power and the efficiency are shown as percentages at the right. The power curve is not quite a straight line. Most problems involving fans in ducts can be solved with the use of the characteristic curves. For example, the head loss due to friction in the system is a parabola open upwards. The intersection of this curve with the p_t curve will give the flow Q under those conditions.

Imagine the fan blades as elements of a helix, moving the air like an Archimedean screw. In one turn, let the helix advance a distance L, called the pitch of the fan. If θ is the inclination of a blade, then L = πD tanθ, or L/D = π tanθ. The amount of air moved in one revolution will then be (πD²/4)L, and if the impeller makes n rev/min, then the discharge Q = (πD³/4)(L/D)(n) cfm. A certain propeller fan mentioned in a handbook has the characteristics n = 1150 rpm, D = 2 ft, Q = 5000 cfm. For this fan, A = 3.142 ft², and V = 26.52 fps. From these figures, L/D = 0.69, or θ = 12.4°. This is a quite reasonable figure. The actual pitch of the blades is larger, since there is back flow or slip. For a slip of 50%, the blade inclination would be 24°. Slip is usually less than this, around 40% to 30%. In any case, the effective L/D is probably more or less typical for the fan design.

If D is held constant, and n is varied, we see that Q is proportional to n. Since V is proportional to Q, and Δp is proportional to V² (we presume that Δp is just the velocity pressure at the output in free discharge), it follows that Δp is proportional to n². Finally, the power output is proportional to Q and to V², so W is proportional to n³. These relations for varying n and constant D are called "Fan Law No. 1."

The tip velocity of a fan blade is a good reference velocity for the other velocities involved. This is V' = ωD/2 = nD/19.10 fps. If we vary D, but keep the tip velocity constant, then n varies inversely with D. In these conditions, Q will be proportional to D², assuming L/D constant. Since the area is also proportional to D², the velocity will be constant, as well as Δp. The power output W will be proportional to the discharge times the square of the constant V, so it will be proportional to D². These relations are "Fan Law No. 2." Vector summation of the blade velocity at any radial position and the speed of approach of the air will give the angle of attack on the blade. Blades are usually twisted (greater pitch for smaller radius) to equalize the effect over the effective area. Small blades may be flat and thin, but large blades should have a rounded leading edqe and a feathered trailing edge. An airfoil section is often used for propeller blades. For an aircraft propeller, the blade angle is from 10° to 28° at 0.75R. The tip velocity should not exceed the local speed of sound to avoid the creation of shock waves.

We can think of other conditions as well. For example, let us vary D, but now keep the angular velocity constant, so that the tip velocity is proportional to D. In this case, Q varies as the cube of D, Δp as the square of D, and W as the fifth power. This can be combined with Fan Law No. 1 to show that when D and n are both varied independently, Q ≈ D³n, Δp ≈ D²n² and W ≈ D⁵n³.

The frictional loss from air flow in a duct can be estimated by the pipe flow equations. If L is the length of a circular duct of diameter D, then the head loss is h' = 0.015(L/D)(V²/2g), where I have chosen what seems to be a reasonable value for the constant, usually written 4f and the one found in the Moody Chart and other references. For other duct shapes, replace D by 4R, where R is the hydraulic radius (area/perimeter). For air, the "wetted perimeter" is just the perimeter, of course. A square duct of side a has R = a/4.

Torque is the rate of change of angular momentum, just as force is the rate of change of linear momentum. When a fluid exerts a torque on a turbine runner, the reaction is a change in angular momentum of the fluid. The air that leaves a fan is rotating, the reaction to the torque that turns the impeller. Fluid is given angular momentum by the guide vanes which, ideally, is destroyed by the torque exerted on the runner. With some machines, however, the water at the exit may still have considerable angular momentum, and the energy in this motion is energy that does not appear at the shaft. Where velocity in the exit fluid is part of the desired output (as with a fan), vanes to straighten out the flow help to recover some of the energy that would otherwise be lost.

Fans for use under low pressure differences generally have a small hub, with the blades occupying most of the cross-sectional area. As the pressure differential increases, it becomes more efficient to concentrate the blade area near the periphery of the impeller. The hub then becomes larger, and the blades are stubby vanes on its surface. This is seen at the forward end of a jet engine, where the fan forms the compressor that efficiently decelerates the air relative to the engine, raising its pressure. Energy is added by burning fuel in the compressed air. Its velocity increases as it returns to atmospheric pressure, forming a jet the reaction to whose momentum provides the thrust. The exhaust drives a turbine that extracts some energy to operate the compressor.

The centrifugal fan is a very simple device. It consists of an impeller with blades that can be simply radial, though there are certain benefits to curved blades. It is fed from the centre, and the output is taken from a scroll case on the outside. The air is simply whirled around and centrifugal force causes the pressure to rise on the outside. It is a curious machine in that, unlike most power machines, it cannot be run in reverse to produce a torque. There are, of course gas turbines that can produce work efficiently, but they are very different from a centrifugal fan run in reverse.

Slip-Stream Analysis

An aircraft propeller is a fan with free discharge, whose purpose is to add velocity to the air it encounters. A jet engine, or even a rocket engine, has an identical function, so will be included in the following discussion. The reaction to the momentum added to the air is the thrust of the propeller, that pulls the aircraft through the air. This propels the aircraft although it has no connection with the ground, a somewhat marvellous phenomenon. The familiar blast of air behind a rotating propeller is called the slip-stream. When a single-engine aircraft is moving through the air, the velocity of the air behind the propeller is greater than the speed of the aircraft, and so produces greater parasitic drag than would otherwise be expected. Propellers on the wing, or behind the fuselage, do not produce this added drag.

Let's analyze the action of the propeller, using the conservation of energy, momentum and mass. As usual, we will be able to make considerable progress. In the figure, the dotted line marks out a cylindrical region containing the air influenced by the propeller, the slip stream. At the left, velocities relative to the aircraft are shown, while at the right are the absolute velocities. V is the speed of the aircraft, and the relative velocity of approach of the air. V' is the speed of the air behind the propeller. Since V' > V for positive thrust, the area of the slip stream is smaller in the wake of the propeller than in front of it. The absolute velocity in the slip stream is ΔV = V' - V. It extends from a diameter of 0.2D on the axis to 0.8D - 0.9D.

Let us assume that the propeller produces a pressure difference Δp that is turned into a velocity difference in a short distance. The thrust can now be expressed in two ways, T = (πD²/4)Δp = QρΔV. The energy equation in the relative motion gives V²/2g + Δp/ρg = V'²/2g. Solving for Δp, we get Δp = ρΔV(V + ΔV/2). Then, using the relation between Δp and ΔV given by the thrust formulas, we find Q = (πD²/4)(V + ΔV/2). Using this in ΔV = T/Qρ, we get a quadratic equation for ΔV, with the solution ΔV = V[√(1 - K) - 1], where K = 8T/πD²V². This formula relates ΔV to the thrust, propeller diameter and aircraft velocity.

It is seen that the aircraft leaves behind an energy ΔV²/2g in the slip-stream. This energy soon mixes with the other air in turbulence. This loss is a necessary part of the propulsion, since if ΔV = 0 there is no thrust. The propulsive efficiency η of the propeller is the ratio of the total useful ouput, TV = QρVΔV, to the input energy, which will be the sum of the useful work and the energy left in the slip-stream. The result is η = 1/(1 + ΔV/2V) = 2/(1 + V'/V). V' cannot be less than V, of course, so η <>

Propeller and jet propulsion can be compared on the basis of thrust and propulsive efficiency. The thrust is T = ρQ(V' - V), so it depends jointly on the area of the slipstream and the velocity difference. If V = 200 mph, for example, V' should not greatly exceed 200 mph. If V' = 300 mph, then η will be 0.8, a reasonable figure. For an adequate thrust, this means a large flow Q, since the velocity difference will only be 100 mph. This can be achieved by a large propeller diameter D. A jet engine, however, has a much smaller area and would not be able to provide the required Q with the given V'. On the other hand, if V = 500 mph, then V' = 750 mph would give the same propulsion efficiency of 0.8 and a velocity difference of 250 mph. Now the required T can be obtained with the dimensions of a jet engine, since the higher velocity increases both factors contributing to T.

Thrust can also be expressed as T = ηP/V, where P is the power supplied to the propeller. If P is in hp and V is in mph, the thrust in pounds is T = 375ηP/V. The tip velocity of the propeller is V" = ωD/2 = πnD, where n is rps. A design ratio often used with propellers is N = V/nD, where V is in fps, D is in ft, and n is in rps. This dimensionless ratio is also N = πV/V". The tip velocity should be kept well below the speed of sound to avoid the ceation of shock waves. In practice, N is usually between 0.8 and 1.1, which implies V"/V = 3 to 4. If propellers are provided with multiple blades, the same flow Q can be obtained at a slower speed. If there are M blades, then we have V = MV"/3, taking the smaller end of the range. The speed of sound is about 340 m/s or 760 mph, so if V" is restricted to 50% of the speed of sound, V = 127M mph. For a two-bladed propeller, this means a limit of 264 mph, for a three-bladed propeller, 381 mph, and for a four-bladed propeller, 485 mph. This seems to agree with practice. The maximum velocity of flow over the propeller is greater than the tip speed.

In all of this, we have neglected the rotation of the propeller, and the vortex motion in the wake. The propeller gives angular momentum to the air in the slip stream, and the propeller tip sheds vortices in a helix. This added motion will reduce the propulsion efficiency, but does not play a large role.

If the efficiency is calculated in the relative motion, the maximum efficiency available is only 50%, since an amount of energy equal to the useful energy is contributed to the wake. In this coordinate system, the propeller is not moving, and so cannot produce useful work by means of the thrust. The energy that in the absolute system is useful work here goes into the wake. In fact, if you make an energy balance in the relative system, you will find that the energy contains two terms, one of which is exactly the thrust energy in the absolute system, and the other is the energy left in the wake. Of course, in this system both appear in the wake.

The aircraft is also supported in the air by the reaction to an air jet, in this case air forced downward by the wings. A helicopter even uses a fan for this purpose, and its analysis is the same as we have just presented. All this is possible because air is actually pretty heavy, with each cubic metre weighing about a kilogram. It seems insubstantial to us, but air has considerable inertia.

The Windmill

Wind as a land power source does not have as long a history as water as a land power source. Its low energy density and its unreliability are sufficient reason for this. Horizontal windmills were known in Persia, perhaps by the 8th century BCE, and the idea was carried to the Far East by prisoners of Genghis Khan, where it was considerably developed in China to drive irrigation machinery. These mills have nothing to do with the European windmill. The first documentation of windmills in Europe dates from 1185 (or 1105), with only the name mentioned, though by 1300 they were becoming common in Northern Europe. They spread to other places, where they developed many local peculiarities during their adaptation. The machinery was obviously developed from that of the Roman water mill, and the availability of millwrights was necessary to its creation. Their major use was always to turn the heavy millstones for grinding grain into flour, but before electricity was available they found many other applications, such as land drainage, pumping, sawing and ore crushing.

The European windmill consisted of sails, usually four in number, attached to sail stocks that rotated a stout horizontal windshaft. There was a large brake wheel on the windshaft with cogs driving a wallower, usually a lantern gear, that rotated the main shaft. In early mills, the main shaft rotated the upper millstone, or runner, which rested on the bedstone. Medieval mills had symmetrical sails, with the stock in the centre supporting transverse sail bars, usually braced on their outer edges with hemlaths. The canvas sail strips were threaded above and below alternate sail bars and tied tightly. The later common sail was entirely on the trailing edge of the sail, and the canvas was laid on top of it and tied down. Cords allowed the sails to be reefed as necessary, depending on the wind. Setting the sails was an arduous and difficult job. Eventually, canvas was replaced by rotating slats controlled by springs. Later, the slats could even be adjusted while the sails were in motion. Previously, the mill had to be stopped for this to be done. The leading edges of the stocks were later given fairings for smoother air flow, which made a considerable improvement. Sail spans of 50 to 70 feet were common. The windmill was built entirely of timber, connected by mortises and trenails (wooden pegs), and used as little costly wrought iron as possible. Since the wood was not exposed to water, as in the case of a water mill, it did not suffer from rot.

The basic machinery of wooden wind and water mills is shown at the right. The similarity is obvious. The millstones that turn grain into flour were about 4 ft in diameter, and rotated at 120-125 rpm. The preparation of millstones by grooving was an art. They had to be very carefully adjusted to just not touch, kept apart by the grain that was being ground. Moving stones could not be allowed to run "dry." Good stone for millstones was rare. In England, the best was a hard sandstone from the Pennines known as "millstone grit." A stone from the Rhine was superior to this, as was the "French burr" made from volcanic rock. The water mill drove the runner from below, by means of a spindle whose mace head turned the iron rynd let into the runner. The windmill drove from above, as shown. A wooden vat surrounded the millstones (not shown). On top of it, the horse supported the hopper from which grain was discharged into the eye of the runner. To drive more than one set of millstones, a "head and tail" arrangment could be used, with two crown wheels on the wind- or watershaft. A large spur gear could be mounted on the vertical shaft, running several sets of stones around its periphery. Metal gearing allowed even more complicated ways to distribute the power, and to attach accessories such as sack lifts.

The sails had to be directed into the wind by moving the windshaft. The first mills were post mills, small houses called bucks that contained the machinery and rotated on a stout post at the centre. The floor of the buck was supported by the crown tree, which rotated on a pintle in an enclosure called the roundhouse, when it was not open to the air. The tail pole extended from the rear of the buck, and was moved to point the sails into the wind and tied to one of a ring of stakes. The simplicity of this kind of mill meant that it never entirely disappeared, especially for small mills.

Another method was to mount the windshaft in a rotating cap. Only the cap rotated, while the power came down the main shaft at any cap position, since it was at the centre. In the tower mill, which arrived in the 15th century, the timber cap was mounted on a masonry or brick tower, usually round. The mills of La Mancha that excited Don Quixote were of this type, whitewashed and with slowly rotating sails. About this time, the horizontal windshaft was replaced by a windshaft that was inclined upwards slightly. This gave a much better stress distribution in the cap, and also more clearance below. The gearing was no problem, since all that was necessary was to incline the cogs on the crown wheel. The smock mill is so-called because its flaring tower, which could be of shingled or weatherboarded timber and octagonal in cross-section, looked like a rural smock. As these mills became taller to reach stronger winds, it was necessary to build a gallery around the tower so the miller could attend to the sails, and also arrange the tail pole to adjust the orientation. Dutch mills had a characteristic way of bracing the tail pole from a cross-piece on the cap by outrigger-like struts. Later, some mills were turned by a worm gear or winch inside the mill. The reader is probably familiar with the appearance of these mills, many of which have been preserved (except, alas, in the United States, where I believe they are all long gone).

When a windmill was not in operation, the sails were placed in an X position to equalize forces on the stocks. When the mill was ready to begin operation, the sails would be moved vertically one by one so the miller could reach them to set the sails, as long as this was necessary. The brake wheel held the sails immovable by the action of a weight. To allow the sails to move, the brake was "pulled off." In case of emergency, the sails could be stopped quickly by releasing the rope.

It was not only necessary to face the mill into the wind just so the sails would be turned efficiently, but also for important stability reasons. The mill was built to resist force from the front, and was in danger if tail-winded by a strong gust or thunderstorm. If the wind rose suddenly, and the brake could not hold the windshaft, friction rapidly set the mill on fire, especially if the miller ran out of grain to put between the millstones. Unusually strong winds are dangerous to windmills today, a hazard water mills do not face.

In 1745 Edmund Lee invented the fantail, or fly, a small windmill with an axis perpendicular to the windshaft. If the wind had a component from the side, the rotation of this wheel drove machinery that rotated the cap accordingly. In this way the mill was automatically kept facing the wind. The fantail was widely adopted in England and on the Continent, except, curiously, for the Netherlands, where the braced tail pole was retained.

A famous and excellent kind of windmill that can still be commonly seen, though less than formerly, is the American wind pump, simply called a "windmill" in the United States. It has an annular sail, which is very strong and durable, composed of many radial vanes. A tail vane keeps the sail faced into the wind. this vane is hinged so that it can be latched parallel to the sail when the mill is not intended to work. A cranked windshaft moves the vertical pump rod up and down to operate the pump in the well beneath it directly. The machinery is mounted at the top of a tower made from angle iron in the better machines, of wood in the lesser. This mill pumps water for cattle in isolated locations, and will work unattended, pumping whenever there is sufficient wind from any direction. Large mills of this type even provided locomotive water for the Union Pacific (as a photograph shows) at certain locations where the installation of a steam engine was not warranted. There could be a device that folded the tail if the wind exceeded 30 mph, or even speed governors. One example of a small mill had a 6' wheel and a 19' redwood tower. Among manufacturers were the Fairbury Windmill Co. of Fairbury, Nebraska and the Chicago Aermotor Co. A Fairbury windmill with an 8' wheel and 33' tower, restored by Bill Alexander, is shown at the left. Today, electricity has taken over most similar tasks once performed by the wind. Even the provision of small amounts of electricity for battery charging is now usually done with solar cells. However, windmills are made with geared heads for driving generators. Because of the variation in speed, the control of output voltage must be carefully considered.

An 8' wheel has an area of 50.2 ft². The maximum operating wind velocity is 30 mph, or 44 fps, which gives 2.25 ft-lb/ft³. The total power available in the wind intercepted is then 4976 ft-lb/s or about 9 hp. At an efficiency of 50%, this means that a maximum of 4.5 hp is available. With an average wind of 15 mph or so, about 0.56 hp should be available, which can still pump a lot of water. The rapid variation of output with wind speed is one of the difficulties in applying wind power. Windmills are most useful for winds of Beaufort Force 4 to Force 6, or 15 to 30 mph. Over this range, their power varies by a factor of 8. Weaker winds will not provide sufficient power, while stronger winds may be damaging, and require that either the vanes be feathered or the wheel turned parallel to the wind.

The horizontal windmill was mentioned above in connection with its early appearance in Persia. On a small scale, horizontal windmills are still common, in devices like the cup anemometer. The drag coefficient of a hemisphere or cone presenting its convex side to the wind is less than when it presents its concave side, so if two or more such cups are mounted on an axis, they rotate in the wind. A similar device is the ventilating stack with a rotating, S-shaped vane on top. The wind-operated prayer wheel of Central Asia seems to have been a device of this kind, possibly suggesting the Persian windmills. There was no gearing inside a Persian mill. It is a very long way between prayer wheels and wind-operated toys to a mill that can turn heavy millstones. Larger examples were in towers with walls and openings that act in the same was as jets, acting on a runner inside with fixed vanes. These devices are not efficient, but have the advantage that they do not have to be turned into the wind, working equally well with wind from any direction. Note that they are actually impulse turbines, while the European windmill is closer to being a reaction turbine.

Wind is getting its own back, however, in wind farms to generate electricity. A good example is the North Hoyle wind farm 7 km off the North Wales coast between Prestatyn and Rhyl. 30 turbines will be installed, with a total nominal capacity of 60 MW. The hub of windshaft is 67 m above the sea, and the sails are 80 m in diameter, sweeping out 5027 m². The windshaft is inclined by 6°, and the blades are coned by 2°. Each of the three glass-fibre reinforced epoxy blades is 39 m long, with a chord 3.52 m at the root and 0.48 m at the tip, and a twist of 13°. The pitch of the blades is regulated hydraulically. A blade can rotate 95° in all, and is full feathered for stopping. The windshaft is yawed by a ring gear and pinions, just like a cap mill, but electrically driven. The controls keep the windshaft rotating at a nominal 18.1 rpm, provided there is sufficient wind. The nacelle at the top of the post is also glass-fibre reinforced epoxy. It contains the gearbox and the alternator. The gearbox is planetary and helical, giving a fixed ratio of 1:92.6 for 50 Hz operation and 1:111.1 for 60 Hz operation. The alternator is a 4-pole induction generator with 690V output, and can generate either 50 Hz or 60 Hz. An induction generator is like an induction motor run in reverse. The rated speed of the rotor is 1680 or 2016 rpm. With slip, the stator field rotates at either 1500 or 1800 rpm, producing 50 Hz or 60 Hz power. The controls keep the windshaft rotating at a nominal 18.1 rpm, provided there is sufficient wind.

Unlike synchronous alternators, induction generators cannot supply reactive power, so connecting them to the grid involves control difficulties. Also, rated power can be produced only about 20% to 30% of the time, which calls for some kind of power storage. Even pumped hydroelectric has been suggested, but it is an expensive solution.

The average wind at the site is given as 10 m/s, which is a brisk 48 mph. At this speed, wind contains 61.25 J/m³. The flow is 50,270 m²/s, so the total power in the intercepted wind is about 3 MW. This would make the efficiency of the unit 67%, which seems a bit high. The wind speed for producing 2 MW is not given, so perhaps 2 MW is an optimistic estimate of the actual power. The turbine will stop at 25 m/s, and restart when the wind falls to 20 m/s. At 20 m/s, the power of the wind will be 24 MW, so getting 2 MW should be no problem, but the efficiency will drop to 8%. Efficiency is rather unimportant, since wind costs nothing. The efficiency could be improved somewhat by increasing the number of blades to 4 or even 6, but this would also increase the force on the structure in a high wind. Wind turbine blades are probably limited to 2 or 3 for this reason. At the North Hoyle site, the 50-year 10 minute gust is 46 m/s, and the 50-year 2-second gust is 60.3 m/s. This 60 m/s gust is 134 mph, well in the hurricane range, and something rightly to be feared.

Since the output of this wind farm is about that of a good hydroelectric turbine and alternator (80,000 hp), the economics of the enterprise are, it would seem, doubtful. Wind power burns no fuel, however, and is a hazard only to birds and ships. At sea, it is at least out of sight. Together with the higher winds over the ocean, this is an excellent reason for offshore wind farms. Modern windmills are actually very much like the older ones, but differences in materials and how the turbines are used makes them look different. The old mills were built almost entirely of wood, which determines the appearance of the structures and machinery, while new ones are metal and plastic. The sails of a modern windmill are aerofoils, and can be feathered automatically to control the torque and speed. In the older mills, this was done by reefing the sails, or later by controlling rotating slats. Fairings on the leading edges of the old sails brought them closer to aerofoils, and the angle of attack could sometimes be varied as well. The sails and windshaft of a new windmill are mounted on top of a post, just as in a post mill, with the orientation controlled by electric motors instead of a tail pole. Mechanical power does not have to be transmitted to the ground for grinding grain, but is used close to the sails by putting the alternator directly on the windshaft (with gearing). Thinner sails can be used because of the greater angular velocity; in the older mills, speed had to be kept low for several reasons. The same necessity of intercepting a large area to gather sufficient power is common to both. Modern windmills are distinctly less attractive than the old ones, verging on visual pollution of the landscape when used en masse on account of the low power density, but they are no worse than advertising, electric transmission wires and four-lane highways.

References

J. Reynolds, Windmills and Watermills (New York: Praeger, 1970). Very well illustrated; covers all kinds of historic mills.

S. Strandh, A History of the Machine (New York: A&W Publishers, 1979). pp. 108-111. Well-illustrated. The Chinese horizontal mill and the Mediterranean jib-rigged sails are shown.

R. L. Daugherty and J. B. Franzini, Fluid Mechanics, 6th ed. (New York: McGraw-Hill, 1965). pp. 162-165.

J. K. Salisbury, ed., Kent's Mechanical Engineer's Handbook, 12th ed. (New York: John Wiley & Sons, 1950). Power Volume, pp. 1-57 to 1-96; 15-18 to 15-22; 15-38 to 15-40.

The site Windmill World has many links to windpump information, most of them broken and some of the nasty American variety that leaves webturds. Be sure not to enable cookies when using this link. There are still a few honest sites and good pictures, if little technical information.

An excellent wind power website is National Wind Power.

Peter Fairley, Steady as she Blows, IEEE Spectrum, August 2003, pp. 35-39.

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