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NASA Technical Memorandum 89901
Profile Modification to Minimize Spur Gear Dynamic Loading
Hsiang Hsi Lin Memphis State University Memphis, Tennessee
and
Dennis P. Townsend and Fred B. Oswald Lewis Research Center Clevekzandi Ohio ~ ~ ~~~~
Prepared for the Design Engineering Technical Conference sponsored by the American Society of Mechanical Engineers Orlando, Florida, September 24-28, 1988
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(NASA-TR-8990 1) F R O P I I E ? I C C l F I C A T I C l T C 187-2S918 L3blILIAZE SEUB G E E & . EY L A H l C L C A C l l r G ( N A S A ) 52 p Avail: &?IS HC e02/r;€ BO1 CSCL 131
Unclas G3137 0098898
PROFILE M O D I F I C A T I O N TO M I N I M I Z E SPUR GEAR DYNAMIC LOADING
Hsiang H s i L i n * Memphis State U n i v e r s i t y Memphi s, Tennessee 381 52
I .
I
I
Dennis P. Townsend and Fred 6. Oswald Nat iona l Aeronautlcs and Space Admin is t ra t ion
Lewis Research Center Cleveland, Ohio 44135
SUMMARY
An a n a l y t i c a l computer s lmulat ion program f o r dynamic modeling o f low- c o n t a c t - r a t i o spur gear systems i s presented. s t a t i c t ransmiss ion e r r o r and uses a f a s t Four ie r t rans form t o generate l t s frequency spectrum a t var ious t o o t h p r o f i l e mod i f i ca t ions .
- _. I - ine procedure computes the gear %
The dynamlc load ing response o f an unmodified ( p e r f e c t i n v o l u t e ) gear p a i r was compared w i t h t h a t of gears w i t h var ious p r o f i l e mod i f i ca t i ons . r e l a t i o n s were found between var ious p r o f i l e rnod i f i ca t lons and the r e s u l t i n g dynamic loads. t he gear ' s s t a t i c t ransmiss ion er ro r , gave a very good es t imat ion o f gear dynamic loading.
Cor-
An e f f e c t i v e e r r o r , obtained from frequency domain ana lys is o f
Design curves generated by dynamic s imu la t ion a t severa l p r o f i l e m o d l f i - ca t l ons are g iven f o r gear systems operated a t var ious app l ied loads. Optimum p r o f i l e mod i f i ca t i ons can then be determined from the design curves t o y i e l d a minimum dynamic e f f e c t f o r a gear system and t o p rov ide up-to-date knowledge f o r b e t t e r gear deslgn.
NOMENCLATURE
ampl i tude o f ith frequency component of gear p a l r s s t a t i c t ransmiss ion e r r o r , pm
damping values o f sha f ts and gear t o o t h mesh N-m-sec; N-sec
p i t c h e r ro r , pm
combined spacing e r r o r between succeeding t o o t h p a i r s a and b, pm
t ransmiss ion e r r o r
po la r mass moment o f i n e r t i a , m2-kg
NASA Lewis Research Center Summer Facul ty Fellow. *
K
Ln
Q ~ , , Q ~
Rb
T
Wn
wa , Wb
E
0
6
6
stiffness, N-mhad
normalized length of tooth profile modification zone
combined meshing compliances of the contacting tooth pairs a and b, pm/N
base radius, m
torque, N-m
total transmitted load, N/m
shared tooth loads for tooth pairs a and b, N/m
reference value of profile modification; minimum amount of tip relief recommended by Welbourn, pm
angular displacement, rad
angular velocity, rad/sec
angular accel erati on, rad/sec2
Subscripts:
f tooth contact frlction
9 meshing t o o t h pair
L 7 oad
a output torque
M motor
m lnput torque
sl shaft 1
s2 shaft 2
1 gear 1
2 gear 2
Superscripts:
a leadlng tooth pair
b lagging tooth pair
2
INTRODUCTION
Reducing the dynamic load ing and no ise of gear systems has been an impor- t a n t concern i n gear design. generated f rom gear ing i s b a s i c a l l y due t o gearbox v i b r a t i o n exc i ted by the dynamic load ( r e f s . 1 t o 9). This v i b r a t i o n i s t ransmi t ted through sha f t s and bear ings t o no i se - rad ia t i ng surfaces on t h e gearbox e x t e r i o r . M in imiz ing gear dynamic load ing w i l l reduce gear noise.
Many researchers have found t h a t t he no ise
The p r i n c i p a l source o f gear system v i b r a t i o n i s t he unsteady component of the r e l a t i v e angular motion o f meshing gear p a i r s . The s t a t i c t ransmiss ion e r r o r descr ibes t h i s displacement type o f v i b r a t o r y e x c i t a t i o n . The v a r i a t i o n of gear-pair meshing t o o t h s t i f f n e s s , which causes s t a t i c t ransmiss ion e r r o r ,
tee th . e r ro rs , and wear.
! s pr!mar!!y due t n t h e per!nd!c a!ternat!on !n the nnmhers nf cnntact lng Secondary e f f e c t s i nc lude tooth p r o f i l e mod i f i ca t ions , machining
Modi fy ing the gear too th p r o f i l e has been found t o s i g n i f i c a n t l y a f f e c t t o o t h meshing s t i f f n e s s . Therefore, min imiz ing meshing s t i f f n e s s v a r i a t i o n t o achieve a smooth s t a t i c t ransmiss ion e r ro r has become a w ide ly used p r a c t i c e f o r reducing gear dynamic load. Much research has been done i n t h i s area, ye t t o the bes t o f t he authors ' knowledge there i s a l ack o f systemat ic work lead- i n g t o d e t a i l e d understanding o f how too th p r o f i l e m o d i f i c a t i o n a f f e c t s the dynamic response o f spur gear systems.
This paper presents an a n a l y t i c a l procedure and associated computer simu- l a t i o n t o sys temat ica l l y change the length o f t h e mod i f ied zone and the t o t a l amount o f p r o f i l e m o d i f i c a t i o n and t o study how t h i s a f f e c t s the s t a t i c t rans- miss ion e r r o r and dynamic load ing o f spur gears. A method i s presented f o r min imiz ing dynamic load ing through an opt imized p r o f i l e m o d i f i c a t i o n t o produce qu ie te r spur gears.
The dynamic load and transmission e r r o r f o r an i n v o l u t e spur gear p a i r and f o r var ious modi f ied gear p a i r s are presented i n t h e t ime domain (as e i t h e r degrees o f r o l l angle o r r o t a t i o n a l speed) and i n the frequency domain. The e f f e c t o f var ious p r o f i l e mod i f i ca t ions on gear dynamics i s discussed. The c h a r a c t e r i s t i c s o f dynamic loading and the Four ie r spectrum o f t he t o o t h p a i r s ' t ransmiss ion e r r o r a re compared. On the bas is o f t h i s comparison an e f f e c t i v e e r r o r , weighted from the frequency components o f s t a t i c t ransmission e r r o r , i s recommended as a c r i t e r i o n f o r optimum p r o f i l e m o d i f i c a t i o n t o minimize gear dynamic load. This procedure w i l l produce a gear se t opt imized f o r one p a r t i c u l a r design load.
For a gear system t h a t must operate over a range o f loads ( r a t h e r than a t a steady design load) , several curves are provided t h a t a l l o w the designer t o make i n t e l l i g e n t t radeo f f s t o produce a q u i e t gearbox.
THEORETICAL ANALYSIS
The dynamic model used f o r t he spur gear system was based on t h a t o f L i n and Huston ( r e f . 10). Other researchers ( r e f s . 4, 6, and 11) have used a s i m - i l a r system dynamic model approach. The t h e o r e t i c a l model, as shown i n f i g u r e 1, comprises th ree basic elements of a spur gear system, (1) the gears,
(2) the shafts, and (3) the connected masses. Given this model, the governing equations developed, using basic gear geometry and elementary vibration prin- ciples, may be expressed as follows:
J li + csl(hl - 6,) + K ( e - e,,) + c ( R 6 - ~ ~ ~ 6 , ) 1 1 sl 1 g bl 1
J 6 t Cs2(h2 - 6,) + KS1(e2 - el) + c ( R 6 - Rblhl) 2 2 g b2 2
Tf 2 t K [ R ( R 8 - R e ) ] = g b2 b2 2 bl 1
Similar procedures developed by Cornel1 (ref. 8) and Tavakoli (ref. 9) were used to determine the tooth spring stiffness by modeling the elastic behavior of the gear tooth. The range of tooth contact was divided into a sequence of contact positions.
The meshing analysis for static transmission error and load-sharing com- putation is simllar to that of Tavakoli and Houser (ref. 9). The load was assumed to be uniformly distributed along the tooth face width. Four equations were solved simultaneously to determine the load sharing and total transmission error of a low-contact-ratio (less than 2) mesh:
Wf + wi b = wn
where the subscript i the superscripts a and b represent the leading and lagging tooth pairs, respectively.
represents the contact point on the tooth profile and
The static transmission error has basic periodicities related to the gear tooth mesh frequency and the shaft rotational frequency. ponents attributable to elastic tooth deformations, to deviations of the tooth profile from the perfect involute profile, and to uniform lead or spacing errors. A Fourier spectrum analysis of the static transmission error wave
It consists of com-
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shows harmonic components t h a t occur a t i n t e g r a l m u l t i p l e s o f t he t o o t h meshing frequency ( r e f . 12). These components a re caused by t o o t h deformat ion and the d e v i a t i o n o f t o o t h surfaces f rom the p e r f e c t i n v o l u t e p r o f i l e . The lower har- monic f requencies occur a t t he i n t e g r a l m u l t i p l e s o f s h a f t r o t a t i n g f requencies and a re caused by t o o t h spacing errors . The equations o f mot ion i nc lude exc i - t a t i o n terms due t o t ransmiss ion errors . The c o n t r i b u t i o n o f each i n d i v i d u a l frequency component t o the dynamic loading response o f gear systems was inves- t i g a t e d i n t h i s study.
The gear t o o t h meshing process leads t o instantaneous load f l u c t u a t i o n s on t h e t e e t h even under constant loading cond i t ions . load f l u c t u a t i o n I s in f luenced by the damping e f f e c t o f t h e l u b r i c a n t and t h e p r o x i m i t y o f t he opera t ing frequencies t o the system na tu ra l f requencies. S t r u c t u r a l damping was no t considered.
The magnitude o f t he
To s i m p l i f y the analys is , t he dynamlc process was de f ined i n t h e r o t a t i n g p lane o f t he gear p a i r , and the d i f f e r e n t i a l equations o f mot ion were developed by us ing t h e t h e o r e t i c a l l i n e o f act ion. Damping due t o l u b r i c a t i o n , e tc . i s expressed as a constant damping fac to r t h a t i s t he r a t i o o f the damping c o e f f i - c i e n t t o the c r i t i c a l damping value. The damping f a c t o r used f o r t he t o o t h mesh was 0.10 (a t y p i c a l value f rom gear research l i t e r a t u r e ) .
For convenience, t he same amount and the same length o f p r o f i l e mod i f i ca- t i o n s were assumed t o be app l ied t o the t o o t h t i p o f both p i n i o n and gear. Since mod i fy ing the r o o t o f one member has the same e f f e c t as mod i fy ing the t i p o f t h e mat ing member, a l l mod i f i ca t i on was assumed t o be app l i ed a t the t o o t h t i p s . Ex t ra care must be taken i n modi fy ing the roo ts o f gear t e e t h because o f t he complex geometry, p a r t i c u l a r l y on gears w i t h small numbers o f tee th . I n some extreme cases w i t h low-contact - ra t io gears, r o o t m o d i f i c a t i o n can des t roy the e f f e c t s o f t i p mod i f i ca t ion , making i t p re fe rab le t o g i v e on ly t i p m o d i f i c a t i o n ( r e f . 5 ) .
The minimum amount o f conventional t i p r e l i e f was chosen as a re ference value i n t h i s study. This re ference value was designated A. According t o Welbourn ( r e f . 13), t he minimum t i p r e l i e f should be equal t o tw ice the maximum spacing e r r o r p lus the combined too th d e f l e c t i o n evaluated a t t he h ighes t p o i n t o f s ing le - too th contact (HPSTC).
The ana lys is was app l ied t o a sample s e t o f gears as s p e c i f i e d i n t a b l e I . These a r e i d e n t i c a l spur gears w i th s o l i d gear bodies and w i t h var ious l i n e a r p r o f i l e ( t i p r e l i e f ) mod i f i ca t ions . A t y p i c a l t o o t h p r o f i l e showing both the unmodif ied ( t r u e i n v o l u t e ) p r o f i l e and a mod i f ied p r o f i l e i s i l l u s t r a t e d i n f i g u r e 2(a). A sample p r o f j l e mod i f l ca t i on c h a r t t s shown i n f i g u r e 2(b), where t h e amount o f m o d i f i c a t i o n i s 1.00 A and the m o d i f i c a t i o n zone extends t o the HPSTC. This length o f mod i f i ca t i on f rom t o o t h t i p t o HPSTC i s designa- ted as t h e normalized length Ln. 1.00 Ln. c a l d i s tance ( p a r a l l e l t o t he t o o t h axis i n f i g . 2(a)) , i t i s a c t u a l l y de f ined i n terms o f t he gear r o l l angle.
I n t h i s case, the m o d i f i c a t i o n l eng th i s Note t h a t a l though the length o f m o d i f i c a t i o n i s shown as a v e r t i -
The optimum length o f t i p r e l i e f w i l l a l l o w load ing t o pass smoothly f rom one t o o t h t o the next . T i p r e l i e f should no t extend t o t h e p i t c h rad ius unless the contac t r a t i o I s a t l e a s t 2 ( r e f . 5 ) . To evaluate the e f f e c t o f t he length o f t i p r e l i e f , t he
The leng th required depends upon the contac t r a t i o .
5
modified zone was varied from zero to the pitch radius. Only linear tip relief was considered in this study. This means that the tip modification line (as in fig. 2(b)) i s straight.
The equations of motion were solved by a linearized Iterative procedure. The linearized equations were obtained by dividing the mesh period into n equal intervals. In the analysis, a constant input torque Tm was assumed. The output torque Ta was assumed to be fluctuating as a result of time- varying stiffness, friction, and damping in the gear mesh.
displacement were obtained by preloading the input shaft with the nominal torque carried by the system. from the nominal system operating speed.
To start the solution iteration process, initial values of the angular
Initial values of the angular speed were taken
The iterative procedure was as follows: the calculated values of angular displacement and angular speed after one period were compared with the assumed initial values. Unless the differences between them were smaller than preset tolerances, the procedure was repeated using the average of the initial and calculated values as new initial values.
RESULTS AND DISCUSSION
The foregoing analysis was applied to a typical set of low-contact-ratio spur gears whose specifications are given in table I. analytical work, the choice of gears used can be arbitrary. gears with solid gear bodies were selected for the study.
Since this is an Two identical
As a control case, the dynamic solution at design load was calculated for the sample gear with unmodified (true involute) tooth profile. transmission error and shared tooth load (fig. 3) were generated from the solutions of the simultaneous equations presented in the previous section.
Plots o f static
To investigate the effect of tooth profile modification, the amount of modification was varied from 0.25 to 1.50 of the reference value A in incre- ments of 0.25 A. At each amount of modification, the length of modification was also varied in fixed increments. If one of the tooth pairs in the double- contact region lost contact because of excessive profile modifications or tooth deflections, the meshing analysis equations were solved for the load and static transmission error of the tooth pair that maintained contact. how the static transmission error and the tooth load of a gear pair are affected by the change of the length of modification at a constant 1.25 A.
Figure 4 shows
Frequency analysis of static transmission error was performed by taking the fast Fourier transform ( F F T ) of its time wave. selected to avoid possible leakage error. body windup was neglected; only the tooth meshing frequency component and its harmonics, which are the major vibratory excitation source of gear dynamics, were included i n the analysis.
A periodic time signal was The dc component created by gear
The beneficial effects of profile modification can be seen by comparing figures 5 and 6. These figures show the static transmission error, the Fourier spectrum of the static transmission error, and the dynamic factor as a function
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o f speed ( a "speed sweepll) f o r unmodified gears ( f i g . 5) and gears mod i f ied w i t h 1.25 A l i n e a r t i p r e l i e f a long a m o d i f i c a t i o n l eng th o f 0.52 Ln ( f i g . 6) . The dynamic f a c t o r i s de f ined as the r a t i o o f the maximum dynamic t o o t h load t o the s t a t i c t o o t h load.
Since gear no ise and gear dynamic load ing a re o f t e n t y p i f i e d by s t rong components a t the too th mesh frequency and i t s f i r s t two m u l t i p l e s ( r e f . 9 ) , the r e l a t i o n between the gear dynamic f a c t o r and the s t a t i c t ransmiss ion e r r o r ' s f i r s t th ree harmonics o f the fundamental t o o t h mesh frequency was inves t iga ted . With no p r o f i l e mod i f i ca t ion , abrupt changes i n the t rans- miss ion e r r o r ( f i g . 5(a)) produced a very s t rong l i n e a t t he f i r s t harmonic o f the t o o t h meshing frequency A1 ( f l g . 5(b)) . With p r o f i l e mod i f i ca t i on , the changes i n t ransmission e r r o r occurred more smoothly ( f i g . 6 (a) ) . This resu l ted i n a much reduced f i r s t harmonic. a l though the t h i r d harmonic increased. The unmodified gears had a s t rong resonance a t about 11 000 rpm ( f i g . 5 (c ) ) . A t t h i s resonant speed the dynamic f a c t o r was about 2.2, which means t h a t t he maximum dynamic too th load dur ing contac t was 2.2 times the s t a t i c t o o t h load. The dynamic fac to r o f the modi f ied gears ( f i g . 6 ( c ) ) d i d no t exceed 1.5. This represents a reduct ion o f 32 percent i n t h e maximum dynamic load ing over t h a t f o r t he unmodified gears.
f a c t o r i n f i g u r e s 5 and 6 shows t h a t the gear dynamic f a c t o r seems t o be r e l a t e d p r i m a r i l y t o the magnitude o f the fundamental t o o t h mesh frequency.
Comparing the f i r s t t h ree peaks o f Four ie r spectrum w i t h the gear dynamic
The maximum gear dynamic f a c t o r was r e l a t e d t o t h e s t a t i c t ransmiss ion e r r o r ' s amplitudes o f t he f i r s t fou r harmonics o f the t o o t h mesh frequency, designated as lengths o f mod i f i ca t i on ( f i g . 7 ) . o f the maximum gear dynamic f a c t o r w i th respect t o the p r o f i l e m o d i f i c a t i o n length was most s i m i l a r t o t h a t o f AI. That means t h a t A1 should be weighted more than any other frequency component i n any r e l a t i o n between the gear dynamic f a c t o r and the Four ie r frequency components o f t he s t a t i c t ransmiss ion e r r o r .
AI, A2, A3,. and A4, f o r t he sample gear a t 1.00 8 and vary ing The t rend o f ampl i tude (shape o f the curve)
A suggested procedure f o r c a l c u l a t i n g an e f f e c t i v e t ransmiss ion e r r o r i s t o take t h e sum o f A1 w i t h the square r o o t o f t he sum o f the squares o f the f i r s t 12 Four ie r harmonic components o f t o o t h mesh frequency. magnitudes o f the harmonic components a f t e r the 12 th harmonic a re usua l l y smal l , t h e i r c o n t r i b u t i o n t o the v ib ra to ry e x c i t a t i o n o f gear dynamics i s n e g l i g i b l e .
Because the
Ae = A1 t [E A:] 1'2
i =1
This re fe rence value i s termed " e f f e c t i v e e r ro r , " s ince i t comprises the f r e - quency components of the s t a t i c t ransmission e r r o r . The e f f e c t o f p r o f i l e m o d i f i c a t i o n length on e f f e c t i v e e r r o r (curve i n f i g . 7 ( b ) ) co r re la tes w e l l w i t h the gear dynamic f a c t o r curve ( f i g . 7 (a ) ) . gear t o o t h p r o f i l e appears t o be an exce l len t i n d i c a t o r f o r t he gear dynamic f a c t o r .
A, The e f f e c t i v e e r r o r o f a
It may ampl i fy t he pena l ty of llmistuned'l p r o f i l e m o d i f i c a t i o n due t o
7
t he weighted e f f e c t o f t he A1 component. Therefore, t h e e f f e c t i v e e r r o r can be a s e n s i t i v e device f o r op t im iz ing gear t o o t h p r o f i l e mod i f i ca t i on . I t can be used f o r tun ing the l eng th and amount o f p r o f i l e m o d i f i c a t i o n i n order t o minimize poss ib le dynamic e x c i t a t i o n and thus lower t h e gear dynamic loading. The less the e f f e c t i v e e r r o r , the smal ler t he gear dynamic loading. I n addi- t i o n , gear system dynamic f a c t o r s can be determined w i thou t going through the time-consuming i t e r a t i o n procedure t o so lve the d i f f e r e n t i a l equations o f motion. the s t a t i c t ransmission e r r o r w i l l g i v e a good est imate o f t h e gear dynamic f a c t o r . by vary ing gear too th p r o f i l e s and eva lua t ing the r e s u l t a n t e f f e c t i v e e r r o r o f the meshing too th p a i r s .
A simple c a l c u l a t i o n o f t he Four ie r spectrum and e f f e c t i v e e r r o r o f
B e t t e r gear dynamic design can be achieved w i t h l ess t ime and e f f o r t
From f i gu re 8(a) i t i s apparent t h a t the l eng th o f m o d i f l c a t i o n should be decreased for gears w i t h a g rea ter amount o f m o d i f i c a t i o n t o achieve the minimum dynamic e f f e c t . mod i f i ca t i on . For 1.25 A, t he minimum dynamic f a c t o r was obtained w i t h a 20-percent length reduc t ion f rom t h a t f o r 1.00 A. For gears w i t h a g rea ter amount o f p r o f i l e mod i f i ca t i on , t he v a r i a t i o n o f t h e gear dynamic f a c t o r w i t h respect t o length o f m o d i f i c a t i o n was more s e n s i t i v e than f o r gears w i t h a smal ler amount o f p r o f i l e mod i f i ca t i on . The v a r i a t i o n o f the gear dynamic f a c t o r w i t h length o f p r o f i l e m o d i f i c a t i o n was very s i m i l a r t o the e f f e c t i v e e r r o r curve f o r both gears ( f i g . 8) .
The optimum decrease depends on the amount o f p r o f i l e
When t h e too th p r o f i l e m o d i f i c a t i o n amount i s l ess than the minimum t i p r e l i e f dynamic e f f e c t ( f i g . 9 ) . As i n f i g u r e 8, t he optimum leng th o f mod i f ied t o o t h p r o f i l e depends on the prescr ibed amount o f p r o f i l e mod i f i ca t i on . imate ly Ln = 1.09 was optimum f o r 0.75 A, Ln = 1.21 was optimum f o r 0.50 A, and Ln = 1.28 was optimum f o r 0.25 A. For gears w i t h a smal ler amount o f pro- f i l e mod i f i ca t ion , the length o f m o d i f i c a t i o n has a l ess s i g n i f i c a n t e f f e c t on the gear dynamic f a c t o r than f o r gears w i t h a g rea ter amount o f p r o f i l e mod i f i ca t i on .
A, t h e length o f p r o f i l e m o d i f i c a t i o n should be increased t o minimize
Here approx-
F igure 10, which shows the e f f e c t o f l eng th o f p r o f i l e m o d i f i c a t i o n on the gear dynamic f a c t o r a t var ious amounts o f mod i f i ca t i on , can be used as a design c h a r t t o determine the optimum m o d i f i c a t i o n l eng th f o r minimum dynamic e f f e c t . As an example, consider a gear w i t h 1.00 A (minimum amount o f t i p r e l i e f ) , operat ing a t a load smal ler than design load such t h a t i t i s equiva- l e n t t o operat ing-a long the curve represented by 1.50 A ( p o i n t C i n The optimum length o f p r o f i l e m o d i f i c a t i o n i n t h i s case should be 0 ins tead o f 1.00 Ln. I f the gear mentioned above were operated a t a loads equiva lent t o opera t ing between the 1.00 A curve and the 1.50 the optimum length o f m o d i f i c a t i o n should be 0.75 Ln, corresponding i n f i g u r e 10, t h e i n t e r s e c t i o n o f the 1.00 A curve and the 1.50 A choice o f p o i n t B, or p o i n t C, o r any p o i n t o ther than A, would y i e des i rab le higher dynamic f a c t o r s under t h i s range of loads.
f i g . 10).
range o f A curve, t o p o i n t A curve. The d l e s s
68 Ln
The envelope o f minimum dynamic f a c t o r s achievable f o r gears w i t h the
This envelope i s more s e n s i t i v e t o l eng th of m o d i f i c a t i o n f o r gears prescr ibed amount o f p r o f i l e mod i f i ca t i on i s shown as dashed l i n e s i n f i gu res 9 and 10. w i t h a smal ler amount o f p r o f i l e m o d i f i c a t i o n than f o r gears w i t h a g rea ter amount o f p r o f i l e mod i f i ca t i on .
8
Since the characteristic of effective error at varying modification lengths gives a good indication of the gear dynamic factor of a gear pair, an example is shown in figure 11. example with the tooth profile modified at 1.00 A for one gear and 1.25 A for the mating gear. Ln = 0.65 for both members. The relative position of the effective error curves in figure 11(a) should indicate a corresponding position for the dynamic factor curves in fig- ure ll(b). For this particular case the gear dynamic factor was predicted to be approximately 1.4. the gear system equations of motion (fig. ll(c)) was found to be 1.39, indeed close to the value predicted from the effective error in figure ll(b).
The sample gear in table I was used for this
The length of modification was
The maximum gear dynamic factor calculated by solving
CONCLUSIONS
An analysis and a computer program were developed to investigate the effect of linear profile modifications on the dynamic loading response of a spur gear system. tooth mesh frequency components of transmission error was also studied. Applying the program to a pair of identical low-contact-ratio spur gears revealed the following:
The relation between the gear tooth dynamic factor and the
1. The dynamic characteristics of a spur gear system are affected significantly by tooth profile modifications.
2. The dynamic (load) factor can be simulated analytically by the effective error, which is calculated from the frequency components of a gear pair's static transmission error.
3. The effective error is a good indicator for tuning the length and amount of profile modification to reduce gear dynamic loading.
4. If gears are to be operated at less than the design load, the length of the modification zone should be reduced. Conversely, if gears are to be operated at greater than the design load, the length of modification should be increased.
5. An increase in the applied load (or a decrease in the total amount of tip relief) reduces the sensitivity of the gears to changes in the length of profile modifications.
6. The dynamic tooth loads on gears that must operate over a range of loads can be minimized by using profile modifications optimized according to the procedures outlined i n this work.
The results obtained herein should be useful for predicting the vibra- tion excitation of spur gear systems and for modifying tooth profiles for improved gear dynamic performance.
To fully understand and best utilize gear tooth profile modification, it is recommended that the analysis be extended to nonlinear profile modifica- tions. Experimental tests should be performed to verify the analytical results.
9
REFERENCES
1. Harris, S.L., 1958, "Dynamic Loads on the Teeth of Spur Gears," Proceedings of the Institute of Mechanical Engineers, Vol. 172, No. 2, pp. 87-112.
2. Selreg, A. and Houser, D.R., 1970, IIEvaluation of Dynamic Factors for Spur and Helical Gears," Journal of Engineering for Industry, Vol. 92, No. 2, pp. 504-515.
3. Terauchi, Y., Nadano, H. , and Nohara, M. , 1982, "On the Effect of the Tooth Profile Modification on the Dynamic Load and the Sound Level of the Spur Gear," JSME Bulletin, Vol . 25, No. 207, pp. 1474-1481.
4. Kubo, A. and Kiyono, S., 1980, "Vibrational Excitation of Cylindrical Involute Gears Due to Tooth Form Error," JSME Bulletin, Vol. 23, No. 183, pp. 1536-1543.
5. Smith, J.D., 1983, Gears and Their Vibration: A Basic Approach to Under- standing Gear Noise, Marcel Dekker, New York.
6. Kasuba, R. and Evans, J., 1981, "An Extended Model for Determining Dynamic Loads in Spur Gearing,Il Journal of Mechanical Design, Vol. 103, No. 2, pp. 398-409.
7. Cornell, R.W. and Westervelt, W.W., 1978, "Dynamic Tooth Loads and Stress- ing for High Contact Ratio Spur Gears," Journal of Mechanical Design, Vol. 100, NO. 1 , pp. 69-76.
8. Cornell, R., 1981, "Compliance and Stress Sensitivity of Spur Gear Teeth," Journal o f Mechanical Design, Vol. 103, No. 2, pp. 447-459.
9. Tavakoli, M.S. and Houser, D . R . , 1986, llOptimum Profile Modifications for the Minimization of Static Transmtssion Errors of Spur Gears," Journal of Mechanisms, Transmissions. and Automation in Design, Vol. 108, No. 1 , pp. 86-95.
10. Lln, H.H. and Huston, R.L., 1986, "Dynamic Loading on Parallel Shaft Gears. 'I NASA CR-179473.
1 1 . Pintz, A., Kasuba, R., Frater, J.L., and August, R., 1983, "Dynamic Effects of Internal Spur Gear Drives," NASA CR-3692.
12. Mark. W.D.. 1978. "Analysis of the Vibratory Excitation of Gear Systems: Basic Theory," Journal of the Acoustical Soiiety of America, Vol.-63, No. 5, pp. 1409-1430.
13. Welbourn, D . B . , 1979, "Fundamental Knowledge of Gear Noise - A Survey," Noise and Vibration of Engines and Transmissions, Mechanical Engineering Publications, London, pp. 9-14.
10
TABLE I. - GEAR DATA
n n
U U. SHAFT 2
Gear tooth . . . . . . . . . Standard full-depth tooth Module (diametral pitch), mn . . . . . . . . , 3.18 (8) Pressure angle, deg . . . . . . . . . . . . . . . . 20 Face width, mn (in.) . . . , . . . . . . . . 25.4 (1.0) Design load, N/m (lb/in.) . . . . . . . 350 OOO (2000) Theoretical contact ratio . . . . . . . . . . . . 1.64
Nt!!!!!!Er c f teeth . . . . , . . . . . . . . . . . . . -a LO
LOAD
T M ~ M A-
(( ( GEAR 1
/ - GEAR 2
Ks1 Kg k52
JM J1 J2 JL
cs1 c g cs2
FIG. 1 COMPUTER RODEL OF SPUR GEAR SYSTEM
11
ii' 'L
TRUE INVOLUTE TOOTH PROF I LE 7
30
20
10
0 .
~ I O U N T OF PROFILE
MODIFICATION, Ln \
MODIFICATION, A
MODIFIED PROFILE
TOOTH CONTACT, LPSTC
(A> GEAR TOOTH WITH MODIFIED TOOTH PROFILE
- T I P
I -
-
LENGTH OF PROFILE \\ MODIFICATION, Ln \
~ I O U N T OF PROFILE * \ / /
MODIFICATION, A
T MODIFIED PROFILE j,' I LHIGHEST CONTACT. SINGLE-TOOTH POINT HPSTC OF [ \ LPITCH POINT
LLOWEST POINT OF SINGLE- TOOTH CONTACT, LPSTC
(A> GEAR TOOTH WITH MODIFIED TOOTH PROFILE
5 20 “Ir L
I I I I I (A) STATIC TRANSMISSION ERROR FOR ONE MESH CYCLE
4x105 r
d 4 I !- 0 0 !-
ROLL ANGLE. DEG
(B) SHARED TOOTH LOAD FOR ONE MESH CYCLE
FIG. 3 SPUR GEAR PAIR UNDER DESIGN LOAD, NO PROFILE NOD IF ICATION
13
n
NORMALIZED LENGTH OF
TOOTH PROFILE MOD IF I C A T I O N
ZONE, Ln
/-\I. 30
(A) STATIC TRANSMISSION ERROR FOR ONE MESH CYCLE
15 20 25 30 35 ROLL ANGLE, DEG
(B) SHARED TOOTH LOAD FOR ONE MESH CYCLE
F I G . 4 EFFECT OF VARYING LENGTH OF PROFILE M O D I F I - CATION ZONE AT CONSTANT AMOUNT OF PROFILE MODIFICA- T ION, 1.25A
x 1 .
10 20 30 40 50 ANGLE OF ROTATION, DEG
(A) STATIC TRANSMISSION ERROR
A2
i A3
0 2 4 6 8 10 12 HARMONICS OF TOOTH MESH FREQUENCY
(B) FREQUENCY SPECTRUM OF STATIC TRANSMISSION ERROR 2 . 5 1
ROTATING SPEED, RPM
(C) DYNAMIC FACTOR AS FUNCTION OF ROTATING SPEED
FIG. 5 SPUR GEAR PAIR UNDER DESIGN LOAD - NO PROFILE MOD IF ICAT I ON
15
I 50
1.5
1.0
.5
ANGLE OF ROTATION, DEG
(A) STATIC TRANSMISSION ERROR
- A 2
-
A 1 -
L 1
I A4 A 1 1 A12
A a A A
HARMONICS OF TOOTH MESH FREQUENCY
(B) FREQUENCY SPECTRUM OF STATIC TRANSMISSION ERROR
/
1 5 i l O3
.5
eo;’ -- 3 6 9 12
ROTATING SPEED, RPM
(C) DYNAMIC FACTOR AS FUNCTION OF ROTATING SPEED
FIG. 6 SPUR GEAR PAIR UNDER DESIGN LOAD - PROFILE MODIFICATION: AMOUNT. 1.25 A; LENGTH, 0.52 Ln
16
2.5 - 8 HIGHEST POINT OF I-
SI NGLE-TOOTH
(A) GEAR TOOTH DYNAMIC FACTOR
n 1 ----
I I
I 2 1 31. I
0 ' t / I
LENGTH OF PROFILE MODIFICATION, L,
(B) TRANSMISSION ERROR FREQUENCY COMPONENTS
F I G . 7 EFFECT OF VARYING LENGTH OF PROFILE MODIF ICATION ZONE AT CONSTANT AMOUNT OF PROFILE NODIFICATION. 1.00 A
AND EFFECT I VE ERRORS
17
AMOUNT OF PROFILE MOD IF ICATION
1.00 A 1.25 A
- --- HIGHEST POINT OF SI NGLE-TOOTH CONTACT
CT
LL
(A) GEAR TOOTH DYNAMIC FACTOR
4
W 2 5 2 W LL LL W
LENGTH OF PROFILE MODIFICATION, Ln
(B) EFFECTIVE ERROR FIG. 8 EFFECT OF VARYING LENGTH OF PROFILE MODIFICATION ZONE AT TWO AMOUNTS OF MODIFI- CAT I ON
18
AMOUNT OF PROFILE MODIFICATION
e 0 I-
L L s
2.0 : L
> n Z i r . z 3
1.0‘
- 0.25 A .50 A .75 A
--- --- 2.5 r
’
e 0 I- V
L L 2.0 : L >- n
3 1.5 c(
3 1.0
e- I HIGHEST POINT OF
.
*
-
POINT
- 0
I
I S I NGLE-TOOTH
I
1.2 1.0 .8 .6 .4 .2 0 LENGTH OF PROF I LE MOD I F I CAT I ON, L n
FIG. 9 DESIGN CHART FOR OPTIMAL LENGTH OF PROFILE MODIFICATION ZONE AT LESS THAN 1.00 A (EQUIVALENT TO OVERLOAD CONDITION AT 1.00 A )
AMOUNT OF PROFILE MODIF I CAT I ON
1.00 A 1.25 A 1.50 A
--- --- 2.5 p
1.2 1.0 .8 .6 .4 .2 0 LENGTH OF PROFILE MODIFICATION, L”
FIG. 10 DESIGN CHART FOR OPTIMAL LENGTH OF PROFILE MODIFICATION ZONE AT GREATER THAN 1.00 A (EQUIVA- LENT TO UNDERLOAD CONDITION AT 1.00 A )
19
5r AMOUNT OF PROFILE MODIFICATION
( A > CALCULATED EFFECTIVE ERROR AT Ln = 0 .65
~ 1 . 2 5 A = 2.5 r e DYNAMIC FACTOR
1 . 2 1.0 .8 .6 .4 . 2 0 LENGTH OF PROFILE MODIFICATION, Ln
(B> PREDICTION OF GEAR TOOTH DYNAMIC FACTOR AT Ln = 0 .65
.-
e o --- 0
ROTATING SPEED, RPM
(C) CALCULATED DYNAMIC FACTOR FROM EQUATIONS OF MOTION FIG. 11 EXAMPLE OF 1 .OO A ON ONE GEAR AND
1 - 2 5 A ON MATING GEAR
20
NASA 1. Report No.
NASA TM-89901
Report Documentation Page 2. Government Accession No.
7. Key Words (Suggested by Author(s))
Spur gears; Dynamic load; Profile modifications; Equations of motion transmission error; Frequency analysis
Profile Modification to Minimize Spur Gear Dynamic Load i ng
18. Distribution Statement
Unclassified - Unlimited Subject Category 37
I
7. Author(s)
3. Security Classif. (of this report) Unclassified
Hsiang Hsi Lin, Dennis P. Townsend, and Fred B. Oswald
21. No of pages 21 j z z pr'T+02 1 20. Security Classif. (of this page)
Unclassif led
9. Performing Organization Name and Address
Nat 1 onal Aeronaut1 cs and Space Admi n 1 st rat 1 on Lewis Research Center Cleveland, Ohio 44135-3191
2. Sponsoring Agency Name and Address
National Aeronautics and Space Administration Washington, D.C. 20546-0001
5. Supplementary Notes
3. Recipient's Catalog No.
5. Report Date
6. Performing Organization Code
8. Performing Organization Repon No.
E-3597 10. Work Unit No.
505-63-51 11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
Prepared for the Design Engineering Technical Conference, sponsored by the American Society of Mechanical Engineers, Orlando, Florida, September 24-28, 1988. Hsiang Hsi Lin, Summer Faculty Fellow, presently at Memphis State University, Memphis, Tennessee 38152; Dennis P. Townsend and Fred B. Oswald, NASA Lewis Research Center.
6. Abstract
An analytical computer simulation program for dynamic modeling of low-contact- ratio spur gear systems is presented. The procedure computes the static trans- mission err'or of the gears operating under load and uses a fast Fourier transform to generate the frequency spectrum of the static transmlssion error at various tooth profile modifications. The dynamic loading response of an unmodified (per- fect Involute) gear pair was compared with that of gears with various profile modifications. Correlations were found between various profile modifications and the resulting dynamic loads. An effect-ive error, obtained from frequency domain analysis of the static transmission error of the gears, gave a very good indication of the optimum profile modification to reduce gear dynamic loading. Design curves generated by dynamic sirnulation at various profile modifications are given for gear systems operated at various loads. tions can be determined from these design curves for improved gear design.
Optimum profile modifica-
