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B.Rajgopal,ME08B011,19/7/2011.
INTERNSHIP PROGRAM
Transmission Error in Gears
Table of ContentsAcknowledgements 6
About the company 6
Objective 6
Problem statement 6
1.Transmission and gears 7
1)Introduction 7
2)Involute Profile 8
2.Noise in gear systems 9
1)Introduction 9
2)General ways of reducing noise 9
3)Reducing excitations 10
4)Sources of force variation 10
5)Perfect transmission 11
6)Gearbox internal responses 12
7)External responses 12
8)Overall path to noise 13
3.What is TE? 14
1)Introduction 14
2)What is TE? 15
3)Nomenclature 16
4)Path of contact 17
5)Pitch Point 19
6)Contact ratio 19
7)Elastic deflections 20
8)Tip Relief 21
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Table of Contents4.Transmission Error 22
1)Introduction 22
2)Effect of load on TE-Harrys maps 24
5.Static Transmission Error 25
1)Thin slice model 25
2)Geometric errors 27
3)Adjacent pitch errors 27
4)Helix corrections 28
5)Method of Approach 29
6.Algorithm of the Program 31
1)Input Parameters 31
2)Formulae involved 31
3)Code for evaluating TE 32
4)A simple code in Matlab 33
7.Actual gear system 36
1)Components of gear systems in a compressor 36
2)Schematic of gear train 36
3)Input Parameters 5
4)Gear data 37
i) Drive gear 37
ii) Driven gear 38
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Table of Contents8.Tooth stiffness 39
1)Parameters affecting C’ 39
2)Evaluation of tooth stiffness as per ISO standards 39
3)Determining tooth stiffness 40
4)Calculation 41
5)Profile shift factor 41
6)Theoretical stiffness C’th 42
7)Correction factor CM 43
8)Gear blank factor CR 43
9)Basic rack factor CB 44
10)Single tooth stiffness C’ 44
9.Misalignment 45
1)Drive train 45
2)Input shaft 46
3)Male rotor 47
Case1: only radial force acting 52
Case2: only moment acting 53
Case3: only gas loads acting 54
10. Case study 55
1)Varying base load 55
2)Varying contact ratio 56
3)Varying misalignment 57
4)Varying start of relief 58
Appendix- A59
# Program 1: 59
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# Program 2: 69
AcknowledgementsI express my sincere gratitude to Mr.Jayaselvan and
Mr.Raghu K Oruganty for their guidance and support
throughout my project.
I extend my thanks to Mr.James P Mathew, Mr.Sam P Joseph
and Mr.PoornachandraRao for their hospitality and help
throughout my internship in the company.
I hereby thank Prof. S.P. Venkatesan, Prof. Amitava Ghosh,
and Prof. Dhiman Chatterjee, Prof.Chandramouli for
providing me such a great opportunity to expose myself to
the working atmosphere of the esteemed ELGI Company.
Page | 4
About the companyELGI was established in 1960 as service station equipment and reciprocating compressor manufacturing company. Over the years ELGI has become a multi-product, multi-market company manufacturing technically superior products.
The ELGI product lines today, broadly comprise, Rotary Compressors, Reciprocating Compressors Centrifugal Compressors, Automotive Equipment, Diesel Engines and Manufacturing and engineering services.
ELGI endeavours to continuously improve processes, products and technology, with the objective of serving people better.
This in turn has made ELGI the undisputed leader in the products it manufactures with a strong reputation for quality and service.
Vision and Mission:
ELGI's vision and strategy has been defined and detailed in a mandate entitled Horizon 2015, which states that,
"ELGI aims to get a sizeable share of the world market for its compressors and automotive equipment business by 2015.
With Horizon 2015 ELGI strives to earn customer trust through the quality of our products and through our responsiveness and accountability to customers and partners.
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Objective To visit the esteemed ELGI company and observe
various processes of manufacturing compressors.
To get exposed to the working atmosphere of an
industry.
Enabling me to gain diverse experience academically
while at the same time enriching my technical skills in
the fields of Gears and Transmission and thereby
contributing effectively for the assigned project.
Problem statement
Study of Transmission error and its causes.
To predict the Transmission error for a given gear pair.
To the influence of various parameters on transmission
error.
Thereby, to propose a suitable rectification to reduce
Transmission error.
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1. Transmission and GearsIntroduction: For most of the motion transmission applications we use gears.
In the above figure, the angular motion of input shaft is transmitted through gears to the output shaft. To maintain the conjugate motion between the given gears, the tangential velocity of the point of contact of both gears must be same. And hence the angular velocities of the two gears will maintain constant ratio.
v1=v2
⇒ r1×ϖ1=r2×ϖ2
Where 1 refers input and 2 refers output.
The ratio
r2r1 is equal to the ratio
ϖ1ϖ2 and is called as the gear ratio.
Thus the input and output motions are related to each other and the relation depends on the gear ratio.
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Fig.1.1
Involute profile:The conjugate motion is also influenced by the profile of the tooth of the gear pair. Usually, to maintain conjugacy, we use involute profile for the gear teeth.
The involute profile can be visualised as the path followed by the tip of a straight line rolling on the cylinder. It can also be visualised as a string unwinding from the circle.
To produce this profile on a spur gear, a basic rack is used. For generating same profile on helical gears, the basic rack is inclined by an angle equal to the helix angle.
α t and α nare transverse and normal pressure angles respectively.
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Fig.1.2
Fig.1.3
2. Noise in gear systemsIntroduction:
To generate noise from gears the primary cause must be a force variation which generates a vibration (in the components).
It is then transmitted to the surrounding structure and when the vibration excites external panels, airborne noise is produced.
In most of the cases, the noise and vibrations will not cause mechanical failure of the transmission and does not affect the mechanical performance of the system.
It is perceived as a failure of quality by the consumers. Consumer expectations of noise quality are getting higher and higher.
General ways of reducing noise:Commonly used methods to reduce noise are:
1. Reducing the excitation at the gear teeth. Usually, for most of the systems, less input gives less output (noise) except for some non-linear systems.
2. Reducing the dynamic transmission of vibration from gear teeth to the sound radiating panels and out of the panels by inserting vibration insulators in the path or by altering the sound radiation properties of external panels.
3. Absorbing the noise after it has been generated or enclosing the whole system in a soundproof box.
4. Using anti-noise to cancel the noise in a particular position or limited number of positions or using cancellation methods to increase effectiveness of vibration isolators.
Of the above approaches, 3 and 4 are expensive and tend to be impracticable. Hence we focus more on 1 and 2 approaches from economic point of view.
Usually initial development work has been done by development engineers on gear resonant frequencies or gear casing or sound radiating structures thus leaving only approach 1 as the prime target.
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Reducing excitations:The excitation is generally due to a force varying in either amplitude, direction or its point of application.
Variation of direction of contact force between gears can occur with unusual gear designs such as cycloidal and hypocycloidal gears. But with involute gears, the direction variation is only due to friction effects. This effect is small and can be neglected for normal industrial gears.
For involute gears of attainable accuracy it is the variation of the amplitude of contact force that gives dominant vibration excitation.
The inherent property of involute gives a constant force direction and constant gear ratio as well.
Sources of force variation: The source of force variation in involute gears is a variation in the
smoothness of the drive and is due to the combination of small variations of the form of the tooth from true involute.
Varying elastic deflection of the teeth.
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Fig.2.1
Perfect transmission:Any one involute should mate with another to give a constant velocity ratio while they are in contact. It is possible to have two gears of slightly different nominal pressure angle meshing satisfactorily since pressure angle is not a fundamental property of flank and depends on the centre distance at which the gears happen to be set.
The only relevant criteria are:
(a) Both gears must be (nearly) involutes.
(b) Before one pair of teeth finish their contact the next pair must bready to take over (contact ratio greater than 1.00).
Transverse contact ratio, ε α =path of contact/transverse base pitch.
Axial contact ratio,ε β =face width/axial pitch.
(c) The base pitches of both gears must be the same (except for tip relief) so that there is a smooth handover from one pair to the next. (The base pitch of a gear is the distance from one tooth’s flank to the next tooth's flank along the line of action and so tangential to the base circle.)
If the gears are perfectly involute, rigid and correctly spaced (no misalignment), there should not be any noise or vibration generated during the meshing.
In practice, for a variety of reasons, this does not occur and the idea of Transmission error came into existence.
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Fig.2.2
Gear box internal responses:The relative displacement between the teeth is generated by equal and opposite vibrating forces on the two gear teeth surfaces, moving them apart and deflecting them a sufficient distance to accommodate the T.E.
When we consider the internal responses of the gearbox, the input is the relative vibration between the gear teeth and the outputs (as far as noise is concerned) are the vibration forces transmitted through the bearings to the Gear case.
Axial forces cause vibrations if the end panels of gear cases are flat and are rather flexible. The resulting end panel vibrations are important if it is the gear case which is producing noise, but of little importance if it is vibration through the mounting feet that is the principal cause.
The assumption usually made is that, when modelling internal resonances and responses, the bearing housings can be taken as rigid. This is a reasonable idealisation as the bearing housing movements are typically less than 10% of the gear movements.
External responses:Masses are known and the stiffness can be calculated with reasonable precision, but there are problems with damping which cannot be predicted reliably. The path of the vibration from the bearing housings to the final radiating panels on either the gear case or external structure is usually complex. Fortunately, they can be tested experimentally.
First requirements is to establish whether it is the gear case itself which is the dominant noise source or, more commonly, whether the vibration is transmitted into the main structure to generate the noise.
Transmission to the structure is greatly affected by the isolators fitted between the gearbox and the structure. There is liable to be a large number of parallel paths for the vibration through the structure and an extremely large number of resonances which are so closely packed in frequency that they overlap. A statistical energy approach with the emphasis on energy transmission and losses over abroad frequency band can give a clear description.
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Overall path to noise:
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3. What is TE?Introduction:
θ1 and θ2 are the angles swept by the respective gears in a given time.
Ideally, θ1=θ2∗
r2r1 .
But, in reality, we find
θ2 '=θ2+δθ2 .
The relative variation in displacement between the gears acts via the system dynamic response to give a force variation and resulting vibration.
We define TE by imagining that the input gear is being driven at an absolutely steady angular velocity and we would then hope the output gear was rotating
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Pitch point
Fig.3.1
at a steady angular velocity. Any variation from this steady velocity gives a variation from the “correct position” of the output and this is the TE which will subsequently generate vibration.
More formally, T.E. is the difference between the angular position that the output shaft of arrive would occupy if the drive were perfect and the actual position of the output.
In practical terms, we take successive points of input and calculate where the output should be, and subtract it from measured output position to give TE.
Measurements are initially made in terms of angular displacement and hence the error initially appears in terms of radian. This angular deviation is then multiplied by the radius of the gear to give the transmission error. There is some uncertainty as to whether we should multiply by pitch circle diameter to get tangential movement at pitch circle radius or multiply by base circle radius to get movement along the pressure line.
For most noise purposes it is only the vibrating part of the T.E. that is important so any steady (elastic) deflections are ignored.
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Fig.3.2
Once we have obtained a decent understanding of what Transmission error is, we should understand various macro and micro geometry parameters of gears as they will also be influencing TE.
Base pitch is mostly in DIN standards. As per ISO standards, a parameter called module is used.
Module is the ratio of diameter of the pitch circle in millimetres to the number of teeth. The module of both the meshing gears has to be same for proper meshing to occur.
Some parameters of the basic rack profile used in generating the gears like rack addendum, pressure angle of rack etc. also influence the transmission error.
For helical gears, helix angle is an important parameter, as it reduces the stiffness at one end of the teeth and increases the buttressing on the other end.
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Fig.3.3
Path of contact:• The contact on a spur gear is a line that progresses from the root to the
tip on the driving gear and the tip to the root on the driven gear
Start of Active Profile (SAP) is the point where the contact between two gears starts. EAP is the End of Active Profile.
The contact takes place on a line tangent to the base circles. The limits of the contact are determined by the outside diameters (or the start of chamfer if applied).
The contact between the two mating gears usually happens between either one or two pairs of teeth. The range where the contact is only between one pair of tooth is called as Single Tooth Contact (STC) range.
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Fig.3.4
Fig.3.5
The top most point of the gear tooth in this region is called as HPSTC (Highest Point of Single Tooth Contact). Similarly the lowest point is called as LPSTC.
In the above figure, point 1 will be LPSTC for the drive gear. The same point becomes HPSTC for the driven gear.
The length of path of contact is from SAP to EAP.
Helical gears have a diagonal path of contact.
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Fig.3.6
Fig.3.7
Pitch point:The point of intersection of the common tangent to the base circles of both meshing gears and the line joining their centres is called as Pitch Point.
This point is in the range of Single Tooth Contact.
Contact ratio:
Page | 19
Fig.3.8
Fig.3.9
Elastic deflections:To achieve a smooth take-over, before one contact reaches the tip there must be another contact coming into action, one tooth space behind. For the theoretical ideal of a rigid gear the only requirement for a smooth take-over is that the base pitch, the distance between two successive teeth along the pressure line, should be exactly the same for both gears.
Unfortunately, although gear teeth are short and stubby, they have elasticity and there are significant deflections. The deflection between two teeth is partly due to Hertzian contact deflections, which are non-linear, but mainly due to bulk tooth movement because the tooth acts as a rather short cantilever with a very complex stress distribution and some rotation occurs at the tooth root.
One more important observation is that the stiffness of the tooth is varying all over the contact line and is not constant thereby making the predictions even tougher. So, for these values we mainly depend on the finite element stressing software. But considering two conjugate teeth in mesh, this effect almost get nullified. Hence, considering constant tooth stiffness is also a reliable approximation.
Since the stresses are high at the tooth root, various manufacturers produce gears with different root shapes to affect strength. But the main variations arise from variation of pressure angle or undercutting and also to some extent from low tooth numbers.
Page | 20Fig.3.10
Tip Relief:Since there is deflection of the mating pair of teeth under load, it is not possible to have the next tip enter contact in the pure involute position because there would be sudden interference corresponding to the elastic deflection and the corner of the tooth tip would gouge into the mating surface.
Manufacturing errors can add up to this effect. Hence, it is necessary to relieve at the tip of the tooth to make sure that the corner does not dig in. correspondingly, the tooth tip is relieved in order to a give a gradual removal of force.
In addition a sharp corner plays havoc with the oil film locally as the oil squeezes out too easily allowing metal to metal contact and accelerated failure.
The amount of tip relief that should be given can be estimated by adding the worst case deflection, the possible pitch errors on each gear and the possible profile errors. Some extra tip relief might be needed if there is large temperature differential between two mating gears as one base pitch grows more than the other because of thermal expansion.
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Fig.3.11
4. Transmission error
Introduction:
The transmission error is a function of all the above stated parameters. The TE of unloaded spur gear pair with tip relief extending a third way down without root relief will look as shown in the figure below.
All distances along the profile are in terms of roll distance.
Roll distance =base radius X roll angle.
The horizontal lines represent pure involutes and the teeth are shown a little apart for clarity. The tip reliefs are linear in this case.
The combination of two teeth with perfect involutes in the centre is to give zero T.E. for this part of the mesh. Where there is tip relief it is irrelevant which gear has it as either gives a drop in the T.E. trace for the combination.
The combined effect of one pair of teeth meshing under no load would be to give a T.E. of the shape shown in second figure above. The same effect is obtained if only one of the teeth has both tip and root relief and the other have none.
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Fig.4.1
Putting several pairs of teeth in mesh in succession gives the effect shown in figure below. If there are no pitch or profile errors and no load applied so no elastic deflections, the central involute sections will be at the same level (of "zero" T.E.) and part way down the tip relief there will be a handover to the next contacting pair of teeth. One base pitch is then the distance from handover to handover. When we measure T.E. under no-load conditions we cannot see the parts shown dashed since handover to the next pair of teeth has occurred.
If there is base pitch error present, TE will raise in the section. It will also affect the handover process from one pair to the next as shown below.
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Fig.4.2
Fig.4.3
Effect of load on TE-Harrys maps:We wish to predict the T.E. under load as this is the excitation which will determine the vibration levels in operation. As soon as load is applied there are two regimes, one around the pitch point where only one pair of teeth are in contact and one near the handover points where there are two pairs in contact, sharing the load but not, in general, equally.
Since total load remains constant, assuming the stiffness of the teeth to be constant, combined deflection of two pairs in contact is equal to the deflection when only one pair is in contact. In particular, exactly at the changeover point, the loads and deflections of both the pairs are equal if there are no pitch errors and so the contact deflection should be half the value of deflection when only one pair is in contact.
The effect of handover process under varying loads was explained by Harris and hence the diagrams of effects of varying loads on TE are termed “Harris Maps”. At a particular "design load" the effects of tip relief are exactly cancelled by the elastic deflections (curve d) so there is no T.E.
Page | 24
Fig.4.4
Fig.4.5
5. Static Transmission ErrorIntroduction:If we already have a gearbox available and can run it slowly under design torque, without wrecking gears or bearings, then the reliable and the most straightforward approach is to measure the quasi-static T.E. This gives a very reliable answer with an accuracy of a fraction of micron.
Conversely, if an existing box is tested, it is an advantage to know what errors might have produced a given (undesirable) result.
Hence, coming up with a method of calculating static TE of given gearbox is important. A suitable model is to consider the thin slice assumption.
Thin slice model:The helical gear is assumed to act as if it were a pack of thin 2-dimensional slices with each slice of tooth behaving independently of its neighbours and deflecting solely due to the contact forces on that slice.
We should think of slice local to a tooth and assume that each slice is restrained normal to the tooth. But since assuming the tooth slice in transverse plane and is restrained axially will make negligible difference for helical gears of low helix angles, we consider the later.
Page | 25
Fig.5.1
The principal theoretical objection to the thin slice model is "buttressing" because we know that if we apply a load at one point only on a tooth the local deflections are less than the "thin slice" estimate because the neighbouring slices support or "buttress" the slice, due to shear stresses and to longitudinal bending stresses.At the end of a tooth the local stiffness is significantly lower because there is no support from the outboard end, as well as the effective modulus being lower due to axial expansion reducing Poisson's ratio effects.
The typical variation of applied load in the tooth slices is shown in the figure below.
However, reasonably accurate teeth do not have such sudden changes in the loading along the line of contact. Because of tooth relief, usually, the load should rise smoothly, reach a constant value, and again decrease smoothly.
Neighbouring slices have similar loads and deflections and hence the shear buttressing effects should be small and cancel out on either side except for the end slices. This practical effect will prove buttressing effect to be small and hence thin slice assumption is much more accurate than expected.
Page | 26
Fig.5.2
Geometric errors:Geometric errors are expressed in terms of various components measured separately by the gear testers. They are:
1. Pitch errors2. Profile errors 3. Lead errors.
Pitch error is the circumferential positional error of the surface of the gear tooth, best thought of as an angular misplacement.
Profile error is the deviation of the tooth surface from the actual involute measured in transverse or normal plane.
Lead error is the deviation, across the face of a gear, of an actual line on the tooth surface at a constant radius, from the ideal tooth helix at that radius.
Adjacent pitch errors:The mathematics of a series of random height (pitch) steps of equal length gives the result that there is no once-per-tooth or harmonics.
The result can be seen more straightforwardly if we integrate the N adjacent pitch errors since the integral of adjacent pitch is cumulative pitch which sums to zero round a full revolution of N teeth.
Adjacent pitch errors in good manufacturing are small and their contribution to steady noise at any given frequency is even smaller. Hence, we can afford to ignore their effect on noise.
Page | 27
Fig.5.3
Helix corrections:The standard methods of manufacture tend to give a profile which is consistent along the axial length of the teeth but the helix matching between two mounted gears is rarely "correct" along the tooth. In some cases there may be helix correction to allow for the pinion body bending and twisting under the imposed loads. More commonly, there is no attempt to correct exactly for distortion but there are helix corrections.
They are:
Tip relief (end relief). Crowning. Misalignment.
Hence, an analysis is needed for them.
Specifying the (consistent) profile is predominantly a question of specifying the tip reliefs on wheel and pinion. The more common linear relief starts abruptly from a point which is typically a roll distance about one third of a base pitch from the pitch point.
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Fig.5.4
Method of approach:An interference at this point due to elastic deflections under load and we start by arbitrarily assuming an amount of interference (cap in the program) at pitch point P.
Once the interference at P is "known" we can find the interference at all other points along the contact line by adding in the extra interference due to helical corrections or misalignment and subtracting any tip relief amounts. Summing the local slice interference times slice stiffness at each point gives the total force between the gears.
This force will not, at first, be the correct desired force but with a rough knowledge (or guess) of the overall contact stiffness we can correct the pitch point interference to get a better answer and carry on iterating until the total interference force is within a specified amount, perhaps 0.5%, of the applied force in the base pitch direction.
Helix corrections depend solely on x, the axial distance from the centre of the face width.
The interference between the gear flanks will be increased by b∗xwhere b is the relative (small) angle between the helices, due to manufacturing misalignments together with gear body movements due to support deflections and body distortions.
Crowning will reduce the interference by an amount crrel∗( x
0 .5 f )2
;
Where crrel is the amount of crowning relief at the ends and f is the face width.
Linear end relief also reduces interference by an amount
endrel∗(x−0 .5 ff ) , provided this is positive (or 0 if negative);
endrel is the amount of end relief and ff is the length of face width that has no end relief.
Page | 29
Tip relief corrections for a slice depend upon the distance (yppt) of the contact point from the pitch line.
Tip relief=bprlf∗ (|y|−positionofstartofrelief )
(0 .5Pb−positionofstartofrelief ) ;
Where bprlf is the relief at the ±0.5 Pb handover position (at zero load) and
Pb is the base pitch.
Page | 30
Fig.5.5
6. Algorithm of programINPUT PARAMETRES
Base load, Gear ratio, Module (m), Number of drive gear teeth (Z1),Helix angle (β), Pressure angle (α),
Contact ratio (εβ), Base pitch,Tip relief at the tip of the tooth,Start of tip relief (strlf),Web thickness (bs);Rim thickness (Sr);Basic rack dedundum (hfp);Pressure angle of basic rack profile (αp);Misalignment;Crowning;Formulae involved:
Number of driven gear teeth (Z2) =Z1
gearratio ;
Face width (b) =Contactratio*Basepitchtand ( β ) ;
Tip relief at 0.5 base pitch from pitch point (bprlf) =Tiprelief∗(0 .5∗basepitch−strlf )
(0 .5∗b−strlf )
Page | 31
CODE FOR EVALUATING TE:
x= b25
∗( ss−13 );
Where x is the distance from pitch point (varying from -12th slice to +12th slice when considering 25 tooth slices) and ss is the no. of tooth slice.
crown= x2∗gb2
4 ; Where g is the amount of crown given in m.
ccp=baseload∗contratib∗ttsht ;
ccp is the initial interference present (assumed).
yppt (contline ,:)=x*tan ( β )+(k−16 )∗basepitch16
+ (contline−3 )∗basepitch;
Yppt is the location of point of contact from pitch line.K is the iteration variable.TIP RELIEF:
relief (contline ,: )=bprlf∗( abs( yppt (contline ,:))−strlf )(0.5∗basepitch−strlf ) ;
Only positive values (actrel) considered.
int erf ( contline ,: )=ccp+misalignment∗xb
−actrel(contline ,: )−crown;
Only positive values of interference (totint) considered.
ffst=sum( tot int ) ; where ffst is the total interference along the face width.ff=ffst∗ttsht∗b
25 ; ff is the total contact force (due to interference);residualforce=ff−baseload ;Ifabs (residualforce )>0 .005∗baseload
ccp=ccp− residualforce( ttsht∗b) ;
elsete(1 , k )=ccp ;
Page | 32
stlddf= peak int*b∗contrati∗ttshtbaseload ; Where stlddf is the static load
distribution factor.peakint is the peak value of interference.
A simple Code in Matlab:% Program to estimate static transmission error% first enter known constants or may be entered by inputfacew=38.861e-3; %is divided into 25 slices of equal thickness
%base tangential load actingbaseload=input('Enter the base load ');
%%%tooth geometry bpitch=6.28e-3;contrati=1.6;misalig=11.942e-6;bprlf=25e-6;Strelief=0.256;tanbhelx=0.4559; %base helix angle of 24.51 degreesttsht=4.89e10; %tooth stiffness%%%
relst=strelief*bpitch;ss=(1:25);hor=ones(1,25);x=(facew/25)*(ss-13*hor);%distance from facewidth centrecrown=(x.*x)*8e-6/(facew*facew/4); %8 micron crown givenccp=baseload*contrati/facew/ttsht; % pitchpoint interferencete=zeros(1,32);for k=1:32; %complete tooth meshfor adj=1:15%loop to adjust contact forcefor contline=1:5; %considering 5 contact lines
yppt (contline ,:)=x*tan( β )+(k−16 )∗basepitch16
+ (contline−3 )∗basepitch
relief (contline ,: )=bprlf∗( abs( yppt (contline ,:))−strlf )(0.5∗basepitch−strlf )
Page | 33
posrel=(rlief(contline,:)>zeros(1,25));actrel(contline,:)=posrel.*rlief(contline,:);interf(contline,:)=ccp*hor+misalig*x/facew-actrel(contline,:); %local interference
% interference along the contact lineposint=interf (contline,:)>0; totint(contline,:)=interf(contline,:).*posint; %considering only positive values of interferenceenddisp(round((1e6*totint)'));pause %checking interference pattern%%%%plotting totintgrid on;a=1:25;b=1:5;surfl(a,b,interf);colormap hot;colorbar;%%%%
ffst=sum(sum(totint));%total interferenceff=ffst*ttsht*facew/25; %total interference forceresidf=ff-baseload; % residual force
%%%%%converegenceif abs(residf)>baseload*0.005;ccp=ccp-residf/(ttsht*facew);elsebreakend%%%%%end
if adj==15;disp('steady state force not reached');pauseendte(1,k)=ccp*1e6;intmax(1,k)=max(max(totint)); %maximum local interferenceend%iterate for next value of k
Page | 34
xx=1:32;peakint=max(intmax); %max interference during the cyclestlddf=peakint*facew*contrati*ttsht/baseload;disp('static load distribution factor');disp(stlddf);figureplot(xx,te);xlabel('Steps of 1/16th of a tooth mesh');tlabel('Transmission error in micron');
In the above program, first few lines are for entering the input values of the given gear pair. All the above given values are approximate values of each parameter just entered to give a rough idea. Nevertheless, one can change them anytime and input the values that belong to an actual gear pair.
Along each line of contact (line 16) the distance (yppt) of each xslice contact from the pitch line is the sum of the base helix effect, themovement due to the 32 steps and the movement due to the change from onecontact line to the next.
The transmission error predicted for a misalignment of around 12 micron:
Page | 35
7. Actual gear systemComponents of gear system in a compressor:
Gear housing Motor Input shaft Drive gear Driven gear Male rotor Female rotor Bearings
Page | 36
Fig.6.1
Schematic of a gear train:
We neglect the female rotor part as its contribution towards TE is very less compared to male rotor.
Gear data:To calculate the transmission error, one must calculate the tooth stiffness of the given gear pair and misalignment in the gear train.
Drive gear:
Page | 37
Fig.7.1
All the symbols are as per the ELGI standards.
Driven gear:
Page | 38
Fig.7.2
8. Tooth stiffness
Page | 39
Fig.7.3
Tooth stiffness represents the requisite load over 1m face width, directed along the line of action to produce the deformation amounting 1m of deviation in the pair of teeth in contact. Single tooth stiffness of a gear is denoted by C’. Tooth stiffness also depends on various parameters.
Parameters affecting C’: Tooth data Blank design Specific load normal to tooth flank Shaft-hub connection Mesh misalignment Modulus of elasticity of the material of gear
Evaluation of tooth stiffness: (as per ISO standards)There are two methods of calculating tooth stiffness.
1. The tooth stiffness is determined by comprehensive analysis including all influences. This can be done by making direct measurements on the gear pair of interest. Values based on the theory of elasticitycan be calculated or determined by finite element methods.
2. This method is based on studies of the elastic behaviour of solid disc spur gears.
With the help of a series expansion, a sample expression was derived for cylindrical gears conjugate to astandard basic rack profile according to ISO 53.
1. This was based on an assumed specificloading of Ft/b = 300 N/mm.
2. Using this method, theoretical single stiffnesses, c th, are′ obtained.
3. Differences between these theoretical results and the results of measurements are adjusted by means of a correction factor, CM, and extension section to adjust for low specific loading.4. Additional correction factors, determined by measurement and
theoretical means, allow this method to be applied to gears
Page | 40
consisting of rims and webs (factor CR), similar to gears conjugate to other basic rack profiles (factor CB) and helical gears (factor cos β).
5. By superposition of the single stiffness of all tooth pairs
simultaneously in contact, an expression for the calculation of C γ
was developed. Its accuracy was verified by measurement results.
Determining C’ by method 2:
Subject to the conditions and assumptions described. C and ′ C γ as determined by Method 2 are, in general, sufficiently accurate for the calculation of the dynamic factor and face load factors as well as for the determination of profile and helix modifications for gears in accordance with the following:
a) External gears;
b) Any basic rack profile;
c) Spur and helical gears with β <= 45°;
d) Steel/steel gear pairs;
e) Any design of gear blank;
f) shaft-hub fitting spreads the transfer of torque evenly around the
circumference (pinion integral with shaft, interference fit or splinted fitting);
g) Specific load (Ft KA) / b W 100 N/mm.
Page | 41
Calculation:Firstly, the given helical gear has to be imagined as a virtual spur gear.
The number of teeth on the virtual spur gear pair,
Zn1=Z1
(cos β )3 ;
Zn2=Z2
(cos β )3 ; Where Z1 and Z2 are the number of teeth in helical pair.
Profile shift factor:Profile shift factors X1 and X2 influence the theoretical tooth stiffness C’th.
The distance between the reference line of the basic rack and the reference diameter of the gear during cutting is called as radial displacement of the tool (Δabs).
X1=Δabs1m
X2=Δabs2m Where X1 and X2 are profile shift coefficients.
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Fig.8.1
Theoretical stiffness C’th:C′th is appropriate to solid disc gears and to the standard basic rack tooth profile. C′th for a helicalgear is the theoretical single stiffness relevant to the appropriate virtual spur gear.
C ' th=1q ' ;
Where q’ is the minimum value for the flexibility of a pair of teeth.
q '=C1+C2Z n1
+C3Zn 2
+C4 X1+C5X1Zn 1
+C6X2+C7X2Zn2
+C8X12+C9X22 ;
The values of coefficients as per ISO standards are:
C1 C2 C3 C4 C5 C6 C7 C8 C9
0.04723 0.15551 0.25791 -0.00635 -0.11654 -0.00193 -0.24188 0.00529 0.00182
For the given drive and driven gear,
Considering there is no profile shift, i.e. X1=X2=0,
And substituting all the above coefficients,
The minimum flexibility turns out to be
q’=0.056
and hence, theoretical stiffness for the given pair of gears,
C’th=17.704.
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Correction factor: Correction factor CM accounts for the difference between measured value and theoretical value of stiffness for solid disc gears.
CM=0.8
Gear blank factor:Gear blank factor CR accounts for flexibility of gear rims and webs.
For solid disc gears, CR=1
CR=1+ln (bs
b)5exp(sR 5m)
Boundary conditions:
When bs
b < 0.2 substitute bs
b = 0.2
When bs
b > 1.2 substitute bs
b = 1.2
When sR
m< 1 substitute sR
m= 1
For the given drive gear,
b=38.86 mm,
bs=13.461 mm,
sR=9.398 mm,
m=2.
Hence, CR=0.917
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Basic rack factor:CB accounts for the deviations of the actual basic rack profile of the gear, from the standard basic rack profile.
Basic rack factor(CB) =1+0.5*(1.25-
hfp
m )*(1-0.02*(20-αp));
If the wheel and pinion have different basic rack profiles, CB has to be
calculated for both of them and should be averaged.
Considering the basic rack profile
Dedendum, hfp=1.2m,
Pressure angle of rack, αp=20 degrees,
We have CB=1.025
Single tooth stiffness:
Single tooth stiffness C’ depends on all above mentioned factors as well as theoretical tooth stiffness. It also depends on the helix angle of the gears.
C '=CM∗CB∗CR∗C 'th*cos (β )
For helix angle, β=24.51 degrees,
C’=1.33¿10 10 .
Page | 45
9. MisalignmentDrive train:Misalignment is another very important parameter that has significant impact on the value of Transmission Error.
For calculation of misalignment, a clear idea of drive train is needed.
In the above drive train, we should calculate the misalignment. Misalignment due to female rotor is very little and hence can be neglected.
Page | 46
Drive train
Fig.9.1
Input shaft:Section between B-B and A-A is of our concern as the bearings are located at that position and the deflection is considered to be in between the bearings.
The given gear train operates generally at 18 kW power and 1460 rpm.
So, the corresponding moment acting on the input shaft is 117.79 N-m.
Shaft present between the bearings,
Page | 47
Female rotor (zoomed)
Fig.9.2
Fig.9.3
Hence, effective diameter, D=
(20 .5×50 )+(34×41.3 )(20.5+34 ) =44.572 mm.
To find effective deflection, we assume a shaft of effective diameter throughout and find the deflections of it when moment and force are acting separately and add them.
Operating power P=18 kW.
N=1460 rpm.
Torque on the gear, T=
Pω =117.79 Nm.
Tangential force, Ft=
Tradius =
117.79×2173 .648
×103
Ft=1356.652 N.
Axial force, Fa=Ft×tan β
=618.53N.
Radial force, Fr=Ft×tan α
cos β =542.68N
Page | 48
Fig.9.4
1. When only Fr is acting:
Deflection at the point of application of force is
P=Fr=542.68N
L=54.5 mm
b=37.5 mm
E=200GPa (steel)
I=πD4
64
x=a=17 mm
Where, D is the effective diameter.
D=44.572 mm
Substituting all the values in the above equation,
We have deflection
δ 1=0.0506 μm
Page | 49
Fig.9.5
2. When only moment is acting:
M=Fa×radius
=53.7 Nm.
Deflection for the above mentioned figure,
At the point of application of force
M0=M=53.7 Nm
a=37.5 mm
b=17 mm
l= 54.5 mm
E=200 GPa
I=πD4
64
Where, D=44.572 mm.
Substituting all the above values,
We get the deflection,
δ2=0 .078 μm
Hence total deflection at gear location, δ12=0.128μm.
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Fig.9.6
Male rotor:
Similarly analysis is done and the misalignment of the male rotor is also done.
The only difference being that the male rotor part is overhung.
The male rotor has a hollow cylindrical part.
Hence, whole of the diameter will not contribute to the effective diameter.
So, the moment of inertia of only the hollow cylindrical part is calculated and then is compared to that of a solid cylinder to get the diameter of it.
This computed diameter of the solid cylinder is used for calculating effective diameter of male rotor.
Ihollowcylinder=3.38¿10−6 m4.
And from this,
Page | 51
Fig.9.7
The apparent diameter of the solid cylinder =91.1 mm.
The composite mean diameter of the male rotor is computed in a similar length weighted mean.
Effective diameter, D=69.208 mm.
Fg is the gas load acting on the rotor.
Fg=1353N
For both the gears, the tangential load acting is same.
Hence, Ft=1356.652N
Fa=618.53N
Fr=542.68N.
This problem again is solved by splitting it into 3 different cases and then summing up the misalignment in each of them to get the overall misalignment.
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Fig.9.8
Case1: only Fr is acting.
For the above figure,
Slope of the shaft can be given by the equation:
θ=Wba
a3
3EI1a+Wb 2
2EI
Where,
θ is the slope of the line.
In the above case,
W=Fr=542.68N
b=56.7 mm
a=197.2 mm
E=200 GPa (steel)
I=πD4
64 where D is the effective diameter.
D=69.208 mm.
Substituting all the values in the above equation,
Page | 53
Fig.9.10
We have, θ=9 .87×10−6
degree.
Hence, deflection, δ 1=b tan (θ )=bθ=0.559μ.
Case2: when only moment is acting
Assuming that the deflection due to moment is present only in the overhung region,
Deflection for the above figure is
δ=−M 0 x
2
2 EI
Where X is the distance from rigid support,
Comparing with male rotor,
M0=Fa×radius
=618 .53×54 . 952×10−3
2
=16.99 N-m.
x=56.7 mm
E= 200 GPa
Page | 54
Fig.9.11
I=1 .125×10−6
m4.
Substituting in the above equation,
δ2=0.121μm.
Case3: only gas loads, Fg is acting.
We neglect the deflection of the overhung part of the shaft due to gas loads as
we assume the bearings to be rigid supports.
Hence, we need not compute the deflection due to gas loads as we are only
concerned with the deflection at the gear location (which will give
misalignment).
Therefore,
Deflection at the position of the gear
δ=δ1+δ2
=0.68μm.
The sum of deflections in input shaft and male rotor at the gear location gives
the overall misalignment in gear train in microns.
Misalignment=0.128+0.68
=0.808μm.
Page | 55
10. Case StudyPrediction of TE is simple once we know mesh misalignment and tooth stiffness along with other gear parameters.
Once prediction of TE is done, the influence of various parameters on TE is observed.
Varying baseload:
As we increase baseload, the value of TE is also increasing. All systems may not follow same trend. In some systems, with load, TE fluctuations decrease to a minimum value and then rise again and for other systems, vice-versa. The load at which TE fluctuations are minimum is called as design load. We want our gear system to be operating at the design load for better results.
Page | 56
Fig.10.1
Varying contact ratio:
As it is evident in the above figure, TE fluctuations decrease with increase in contact ratio.
As we increase the contact ratio, the number of teeth pairs that share the load increases. And so, the contact deflections are less. As a result, TE also decreases.
However, for integral contact ratio, various misalignment effects come into play and the TE will shoot up. Hence, integral values of contact ratio are not preferred.
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Fig.10.2
Varying misalignment:
In the above figure, though the mean value of TE is decreasing with increase in
misalignment, the peak to peak value is increasing.
One important thing to note is that it is the fluctuation of TE which causes
noise and we are not bothered about its mean value.
Hence, we try to reduce misalignment as much as possible. Sometimes,
misalignment between the gears is induced deliberately for better outputs.
Page | 58
Fig.10.3
Varying start of relief:Varying the start of relief diameter also influences the TE.
In the above figure, TE becomes better as we increase the start of relief diameter, i.e. giving a short tip relief.
It is observed that a short tip relief is better to operate at low loads. Similarly, gears with long relief have low TE at heavier loads.
Page | 59
Fig.10.4
Fig.10.5
Appendix-AProgram1:A program in Matlab using GUI for the prediction of TE (“init1.m”)
close all;clear;clc;
figure( ...'Name','Transmission Error ', ...'NumberTitle','off','MenuBar','none')set(gcf,'Position',[0 30 1450 750])set(gcf,'Units','normalized')
%Title
h=uicontrol(...'Style','text', ...'Fontsize',20,...'Units','normalized', ...'BackgroundColor',[0 0.5 1], ...'Position',[0.045 0.92 0.91 0.05], ...'HorizontalAlignment','center',...'String',' Transmission Error');
% ********* Main frame ***************************
h=uicontrol( ...'Style','frame', ...'Units','normalized', ...'Position',[.045 .39 .3 .465], ...'BackgroundColor',[0.50 0.50 0.50]);% Then the text label INPUT
h=uicontrol( ...'Style','text', ...'Fontsize',16,...'Units','normalized', ...'BackgroundColor',[0 0.5 1], ...'Position',[0.09 0.86 0.2 0.05], ...'String','Drive');
h=uicontrol( ...'Style','frame', ...'Units','normalized', ...'Position',[.045 .09 .3 .23], ...'BackgroundColor',[0.50 0.5 0.5]);
Page | 60
% Then the text label INPUT
h=uicontrol( ...'Style','text', ...'Fontsize',16,...'Units','normalized', ...'BackgroundColor',[0 0.5 1], ...'Position',[0.09 0.33 0.2 0.05], ...'String','Additional Parameters');
x1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','profile shift factor, X1',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.25 0.195 0.045]);
adc1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','0',...'Units','normalized', ...'Position',[0.25 0.25 0.085 0.045]);
x2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','profile shift factor, X2',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.2 0.195 0.045]);
adc2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','0',...'Units','normalized', ...'Position',[0.25 0.2 0.085 0.045]);
slice_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','no. of teeth slices',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.15 0.195 0.045]);
slice=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','25',...'Units','normalized', ...'Position',[0.25 0.15 0.085 0.045]);
Page | 61
conv1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','convergence in percentage',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.1 0.195 0.045]);
conv1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','0.5',...'Units','normalized', ...'Position',[0.25 0.1 0.085 0.045]);
h=uicontrol( ...'Style','frame', ...'Units','normalized', ...'Position',[.35 .39 .3 .465], ...'BackgroundColor',[0.50 0.50 0.50]);
% Then the text label INPUT
h=uicontrol( ...'Style','text', ...'Fontsize',16,...'Units','normalized', ...'BackgroundColor',[0 0.5 1], ...'Position',[0.4 0.86 0.2 0.05], ...'String','Common');
h=uicontrol( ...'Style','frame', ...'Units','normalized', ...'Position',[.66 .39 .3 .465], ...'BackgroundColor',[0.50 0.50 0.50]);
% Then the text label INPUT
h=uicontrol( ...'Style','text', ...'Fontsize',16,...'Units','normalized', ...'BackgroundColor',[0 0.5 1], ...'Position',[0.7 0.86 0.2 0.05], ...'String','Driven');
Page | 62
txt_title=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','Transmitted power in KW',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.8 0.195 0.045]);
base=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','18',...'Units','normalized', ...'Position',[0.56 0.8 0.085 0.045]);
text_title=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','contact ratio',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.75 0.195 0.045]);
c=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','1.6',...'Units','normalized', ...'Position',[0.56 0.75 0.085 0.045]);
txt_title=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','helix angle in degrees',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.70 0.195 0.045]);
he=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','24.51',...'Units','normalized', ...'Position',[0.56 0.70 0.085 0.045]);
teeth1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','no. of teeth on drive gear',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.8 0.195 0.045]);
Page | 63
t1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','79',...'Units','normalized', ...'Position',[0.25 0.8 0.085 0.045]);
teeth2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','no. of teeth on driven gear',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.67 0.8 0.195 0.045]);
t2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','25',...'Units','normalized', ...'Position',[0.87 0.8 0.085 0.045]);
presang_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','pressure angle in degrees',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.6 0.195 0.045]);
pa=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','20',...'Units','normalized', ...'Position',[0.56 0.6 0.085 0.045]);
gearatio_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','gear ratio',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.55 0.195 0.045]);
ge=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','3.16',...'Units','normalized', ...'Position',[0.56 0.55 0.085 0.045]);
Page | 64
module_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','normal module in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.5 0.195 0.045]);mo=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','2',...'Units','normalized', ...'Position',[0.56 0.5 0.085 0.045]);
relief1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','relief at the tip of the tooth od drive gear in micron',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.75 0.195 0.045]);
tip1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','15',...'Units','normalized', ...'Position',[0.25 0.75 0.085 0.045]);
relief2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','relief at the tip of the tooth of driven gear in micron',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.67 0.75 0.195 0.045]);
tip2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','18',...'Units','normalized', ...'Position',[0.87 0.75 0.085 0.045]);
strlief1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','start of relief diameter of the drive gear in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...
Page | 65
'Position',[0.05 0.7 0.195 0.045]);
st1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','176.873',...'Units','normalized', ...'Position',[0.25 0.7 0.085 0.045]);
strlief2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','start of relief diameter of the driven gear in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.67 0.7 0.195 0.045]);
st2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','58.364',...'Units','normalized', ...'Position',[0.87 0.7 0.085 0.045]);
web1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','web thickness of drive gear in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.65 0.195 0.045]);
web1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','13.461',...'Units','normalized', ...'Position',[0.25 0.65 0.085 0.045]);
web2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','web thickness of driven gear in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.67 0.65 0.195 0.045]);
web2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','0',...'Units','normalized', ...'Position',[0.87 0.65 0.085 0.045]);
Page | 66
rim1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','rim thickness of drive gear in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.6 0.195 0.045]);
rim1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','9.398',...'Units','normalized', ...'Position',[0.25 0.6 0.085 0.045]);
rim2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','rim thickness of driven gear in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.67 0.6 0.195 0.045]);
rim2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','0',...'Units','normalized', ...'Position',[0.87 0.6 0.085 0.045]);
dedendum_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','basic rack dedendum in mm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.65 0.195 0.045]);
ded=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','2.4',...'Units','normalized', ...'Position',[0.56 0.65 0.085 0.045]);
pangle_rack_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','pressure angle of basic rack profile in degree',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.4 0.195 0.045]);
Page | 67
phap=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','20',...'Units','normalized', ...'Position',[0.56 0.4 0.085 0.045]);
crown1_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','crown of the drive gear in micron',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.55 0.195 0.045]);
crow1=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','0',...'Units','normalized', ...'Position',[0.25 0.55 0.085 0.045]);
crown2_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','crown of the driven gear in micron',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.67 0.55 0.195 0.045]);
crow2=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','14',...'Units','normalized', ...'Position',[0.87 0.55 0.085 0.045]);
misalignment_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','misalignment in micron',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.359 0.45 0.195 0.045]);
mis=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','4',...'Units','normalized', ...'Position',[0.56 0.45 0.085 0.045]);
Page | 68
omega_txt=uicontrol(gcf,...'Style','Text','BackgroundColor','white',...'String','angular velocity of the drive gear in rpm',...'Units','normalized', ...'HorizontalAlignment','Left',...'Position',[0.05 0.5 0.195 0.045]);
om=uicontrol(gcf,...'Style','Edit','BackgroundColor','yellow',...'String','1460',...'Units','normalized', ...'Position',[0.25 0.5 0.085 0.045]);
frame_as = uicontrol(gcf,...'Style','frame','BackgroundColor','green',...'Units','Normalized',...'Position', [0.52 0.08 0.1 0.06]);
% Calls code.m file for initialisation
pbstart1 = uicontrol(gcf,'Style','push',...'BackgroundColor','white','FontWeight','bold',...'Units','Normalized','Position',[0.53 0.09 0.08 0.04],...'String','RUN','Callback','code');
f1=figure;
&&&&&&&&&&&&&&&&The above program, when run returns the following window.
Page | 69
Default values for each parameter are pre entered in the above code in the format of strings. Variation in any parameter can be done in the “yellow” coloured edit boxes. If no variation is done, the program assigns default values to the respective parameter.
But the values, both default as well as entered, are considered as strings by the above program.
To convert them into numbers and operate on them, we need to write one more code. The push button “run” present in the bottom of the window calls back the program “code.m” when clicked on it.
Program2: “code.m”A code to convert the input strings into numbers and get TE
&&&&&&&&
%Converting input strings into numbers
contrati=str2num(get(c,'string')); %#ok<*ST2NM>helixangle=str2num(get(he,'string'));z1=str2num(get(t1,'string'));pangle=str2num(get(pa,'string'));gearratio=str2num(get(ge,'string'));m=str2num(get(mo,'string'))*10^(-3);
Page | 70
Fig.10
tiprelief1=str2num(get(tip1,'string'))*10^(-6);tiprelief2=str2num(get(tip2,'string'))*10^(-6);std1=str2num(get(st1,'string'))*10^(-3);std2=str2num(get(st2,'string'))*10^(-3);bs1=str2num(get(web1,'string'))*10^(-3);bs2=str2num(get(web2,'string'))*10^(-3);sr1=str2num(get(rim1,'string'))*10^(-3);sr2=str2num(get(rim2,'string'))*10^(-3);hfp=str2num(get(ded,'string'))*10^(-3);alphap=str2num(get(phap,'string'));crown1=str2num(get(crow1,'string'))*10^(-6);crown2=str2num(get(crow2,'string'))*10^(-6);misalig=str2num(get(mis,'string'))*10^(-6);power=str2num(get(base,'string'))*10^(3);omega=str2num(get(om,'string'))*2*pi/60;x1=str2num(get(adc1,'string'));x2=str2num(get(adc2,'string'));sl=str2num(get(slice,'string'));conv=str2num(get(conv1,'string'));bpitch=pi*m;%%%%%the values of inputs%%%%z2=z1/gearratio;b=contrati*bpitch/tand(helixangle);strlf1=(std1-z1*m/cosd(helixangle));strlf2=(std2-z2*m/cosd(helixangle));%%%%%%%%%%%%%%%%%%%%%%%%%
bprlf1=tiprelief1*(0.5*bpitch-strlf1)/(0.5*b-strlf1);bprlf2=tiprelief2*(0.5*bpitch-strlf2)/(0.5*b-strlf2);
baseload=2*power/(z1*m/cosd(helixangle)*omega);
%calculating tooth stiffness
zn1=z1/(cosd(helixangle))^3;zn2=z2/(cosd(helixangle))^3;
q=0.04723+(0.15551/zn1)+(0.25791/zn2)-(0.00635*x1)-(0.11654*x1/zn1)-(0.00193*x2)-(0.24188*x2/zn2)+(0.00529*x1^2)+0.00182*x2^2;cth=1/q;cm=0.8;%%the values of inputs for tooth stiffness calculation%%
cr=1+log(bs1/b)/(5*exp(sr1/(5*m)));
%%%%%%%%%%%%%%%%%%%%%%
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cb=1+0.5*(1.25-hfp/m)*(1-0.02*(20-alphap));tts=cth*cr*cm*cb*cosd(helixangle);ttsht=tts*10^9;
relst1=strlf1;relst2=strlf2;ss=(1:sl);hor=ones(1,sl);x=(b/sl)*(ss-(sl+1)/2*hor);crown=(x.*x)*crown1/(b*b/4)+(x.*x)*crown2/(b*b/4);ccp=baseload*contrati/b/ttsht;te=zeros(1,32);for k=1:32;for adj=1:15for contline=1:5;
yppt(contline,:)=x*tand(helixangle)+hor*(k-16)*bpitch/16+hor*(contline-3)*bpitch;
if yppt<=0;rlief(contline,:)=bprlf1*(abs(yppt(contline,:))-relst1*hor)/((0.5*bpitch-strlf1));elserlief(contline,:)=bprlf2*(abs(yppt(contline,:))-relst2*hor)/((0.5*bpitch-strlf2));end;
posrel=(rlief(contline,:)>zeros(1,sl));actrel(contline,:)=posrel.*rlief(contline,:);interf(contline,:)=ccp*hor+misalig*x/b-actrel(contline,:)-crown;posint=interf(contline,:)>0;totint(contline,:)=interf(contline,:).*posint;endffst=sum(sum(totint));ff=ffst*ttsht*b/sl;residf=ff-baseload;if abs(residf)>baseload*conv/100;ccp=ccp-residf/(ttsht*b);elsebreakendend
if adj==15;disp('steady state force not reached');pauseendte(1,k)=ccp*1e6;intmax(1,k)=max(max(totint));
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enda=1:sl;asd=1:5;figure;
grid on;surfl(a,asd,interf);colormap hot;colorbar;xx=1:32;peakint=max(intmax);stlddf=peakint*b*contrati*ttsht/baseload;disp('static load distribution factor');disp(stlddf);figure(f1);plot(xx,te);hold on;
&&&&&&
3. Run init1.m4. Vary parameters (if needed) from the default.5. Click on “Run” button when you are done with changing parameters.6. You will obtain a plot of TE and tooth slices.
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Plot obtained for default values enteredY-axis: TE in micronX-axis: 1/16th of a tooth mesh.
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Fig.11.1
