PRoCEEDINCS ITB Vol. t8, No. I, 1985 13

VIBRATION AND TOOTH STRAIN MEASUREMENT ON A PAIRMISALIGNED POWER TRANSMISSION HELICAL GEARS

By: Komang Bagiasna*, Kiyohiko Urnezawa** and Toshio Su;uki***

ABSTRACT

A pair of helical gears with the followin8 specifications: (l) the total contact ratio ofmorethan 2.0, and (2) the overlap ratio ofless tlun 1.0, has been investigated to determine ther€lationships of the gear sluf( alignntent errors with the vibration characteristics, and withthe tooth strains as well.

For this purpose an experimental set up has been made available to measure the accelerarronlevel, tooth root strains, and to carry out frequency analysis of the gear vibration signals.

It was found that at a cenitin level of gear siuft aiignrnent errors the characteristics ofgearvibration were strongly affected by errors. The results demonstrated that for the sameamount of error, the gear shaft deviation and gear shaft inclination applied at different sides,i.e., at gear leading side and at gear trailing side, produced different gear vibration character-irt ics as well as tooth strain values.

SARJ

Suatu penyelidikan dilakukan terhadap pasangan roda gigi miring yang rncmiliki spesifikasisebagai betikut : /ord I contact rotio lebih be sar dari 2,0 d an overlap rotie lebih kecil dari I ,0.Dalam penyeiidikrn ini dibuat suatu rc1 r?p percobaan untuk mengukur pcngaruh kewlaharrpemasangar poros roda gigi terhadap karakteristik getaran maupun r€gangan yang terjadipada roga gigi miring tersebut. Besaran yang diukur mencakup percepatan (akselerasi) getar-an, regangan pada akar gigt, dan dilakukan pula analisa frekuensi terhadap sinyal getarantersebut.

Data yang diperoleh menu jukkan bahwa tingkat kesalahan pemasangan yang tertentu mem-berikan pengaruh yang besar te(hadap karakteristik getaran maupun pola regangan pada rodagigi miring tersebut. Kesalahan deviasi poros maupun inklinasi poros yang sama besar, yangterjadi pada sisi depan dan pada sisi belakangan roda gigi, ternyata menghasilkan karakteris-tik getaran maupun harga regangan yang berbeda.

Guesr resenrcher, l \ ' lechanical l_ngjreerin! Deportmcnt. Ba.ndung lnsl i lule ofTcchnolDgy.Jl. Ganesha l0 Bandung. lndonesia.Professor, Research Lab. ot Precision Machinery and Elecrrcnics, Tokyo lnsdtute of TechnologyNagalsuta, Midori-ku Yokohama 227, Japan.Graduate Sludenl, Craduatc School ofCoordinated Sciencc, fokyo lnst i tute otTechnology

l 4 PROCEEDINCS ITB VoL 18, No. 1, 1985

l. Review of research on gear vibration by Prof. Umezawa, er 4r.

Helical gears area widely used for power transmission in vehicles and for appli-cation where smooth tooth meshing condition is required. It is known thatstiffness of the gear pair has a strong influence on vibration as well as on dyna-mic load which occured during power transmission.

At the Laboratory of Precision Machinery and Electronics of Tokyo lnstituteof Technology, Japan, intensive ongoning research works on gear vibrations,including gear noise prcblems have been conducted by Prof. Umezawa and hisgroup. Emphasis of the research were initially devoted in formulating thegoverning equations of gear tooth deflections and their solution. It was soonfollowed by research effods to determine tooth meshing conditions and toothdeflection along the contact lines on helical gears. As a result, formulas for ap-proxirnating del'lection of gear tooth and bending montent distribution of geartooth were developed. The validity of the approximations wefe subsequentlydetermined by comparing the calculated values with measured values obtainedfrom a rnodeled gear tooth (3,4,5). Measurements on actual gears were the nextresearch undertaking to analyse the tooth meshing and the behavior of thedriven gear under static lood and under lood transmission. Tooth profile modi-fications were implemented in a number of tests to reduce the level of gear vi-brations. Experimentations using a novel tooth flank modification method, asproposed by Prof. Umezawa, were conducted and the measured results were ingood agreement with the calculated values from the theoretical approximations( 6 , ? , 8 ) .

In real life situation, a gear assembly is seldom free from errors. These errorsnray be present due to manufacturing erors of the tooth and/or assemblingerrors of the gear. Several research actiyities were therefore carried out to inves-tigate the influence of gear errors on gear vibration. Of particular interest wasthe rotational vibration on spur geius caused by pressure angles as well as nor-nul pitch errors. In order to investigate this complex problem, a computerprogranr was developed to sinulate the gear vibration on spur gears. The resultsobtained by this simulation were validated by measured values from expe-riments. To examine the influence of Transverse Contact Ratio (TRCR) andoverlap Ratio (OLR) on the level of vibration, Prof. Umezawa proposed to ca-tegorizc the helical gears into three different classes ( I I ) :

I . Class I : (TRCR + OLR) less than 2.02. Class II : (TRCR + OLR) greater or equal to 2.0, but (OLR) less

tlian I .03. Class III : (OLR) greater than 1.0.

Presently the rL'search activities on gear include the investigations of gear noise

PROCEEDINGS ITB Vol. 18, No. I, 1985

problems, gear vibration induced by gear shaft misalignments, influence ofshaft length on gear vibration, and research on different types of gears.

2. The present work

Stiffness on tooth meshing has been reported to have a significant influence onthe nature of vibration on helical gears. Furthermore, improper alignments be-tween the shafts on a pair helical gear during power transmission affect thetooth meshing stiffness and the load distribution along the tooth facewidth. Todetermine the influence of gear shaft misalignments on vibration during powertransmission, a series of experiments have been conducted for three differentclasses of helical gears.

Simultaneous measurements were conducted on the vibration and the toothroot strains during tooth meshing. Tooth root strains, in particular, were mea-sured by using strain gages implanted at several tooth root fil lets. In addition,by incorporating a pair of spur gear in the experimentation, it was possible toperform a comparative study among the different classes of gears.

Helical gears used in the experiments were selected from the threc differentclasses, and designated as: Hl (from class I), H2 (from class II) and H3 (fromclass III). The spur gear was assigned as S.

In this partial report, only the experirnental results from helical gear H2 will bediscussed.

3. Line of action of helical gear (LAHG)

On helical gears, the gear tooth lns a certain helix angle and consequently theline of contact on a pair of helical gear teeth during tooth meshing is not pa-rallel to the tooth tip edge. The condition during tooth meshing between a pairof helical gear teeth, showing the geometrical arrangement as well as thc termi-nologies used in this report, is il lustrated in Figure l-

Tooth meshing starts at point a' and during this action, the line of contactmoves along bb',c',cc", d'd " on the plane of action. Tooth meshing is com-pleted when the line of contact reaches point f:

To analyse the line action of helical gcar (LAHG) during tooth meshing, itis necessary to specily that:

(a) LAHG is located in the middle of the tooth facewidth, and on the plane ofact ion.

(b) The origin of LAHG coincides with the pitch point of the helical gear.

l 5

l 6

Consequcnt ly , rhe LAHG of i lpo int a . and the meshing pointat point f.

PROCEEDINGS ITB VoL 18. No. L t985

pair of helical gear teeth starts meshing atmoves through point b, c, d, e and terminates

Base cyli.6er ol gc.r

E^d pointmc shia9

Lenqth ol the ti.Eol act ion

Lin. ol

Rane ol

contac t

actioh

Li^e ol act lonol hel ical gear

Tlp cyl inder ot 9€.r

Tlp cyl lnder ot pinion

Start point ol meshing

Erse cylrrdcr ol phion

Figure 1 The line of action of helical gear (LAHG in briefl

4. Experiment

4.I Instrumentation arul Experimental Set up

Tire arrangenrent of the mechanical components of tlte experimental set up isi l lust ratcd in FGure 2. The input shat l o f the tested gear pai rs ( l ) was con-nc 'c ted bya V-bc l t to a var iab le specd 55 KW induct ion motor(6) : andthe output sh l f t was coupled to an eddy current type dynamometer . Two p ie-zoelect r ic type accelerometers (ntanufacturer code: BBN 50[ ) were at tached at180 degree, opposite to each other, on each side of tlre gear blank. Each accele-

- g l / \ '

PROCEEDINGS ITB VoL 18. No. 1. 1985

@ T e s t 9 e a .

@ B a t t o e a r i n g ( 6 3 0 S )@ B a l t u e a r i n g ( 6 2 0 5 )

@ Dyn" -o- " t " | . ( eddycur ren t type)

@ I n d , J c t i o n m o t o / ( v a r i a b l e s p e e d )

t7

rometers had the following specifications : 5 mm in diameter, l0 mm long andweighted 2 grams. The sensitive direction of both accelerometer positions werearranged such that any one of the three different kinds of vibration, i.e, rota-tional (torsional), radial (transversal), and axial vibration coukl be measured.

0 . i v i ^ 9

0 r i v . ^

Figure 2 Set up of the mechanical components

The arrangement of electrical instrumentations is schematically shown in Fi-gure 3. The rotational spebd of the induction elcctric motor flM) was con-trolled either manually by using a speed controller (SC), or automatically bycombined use of speed controller and a swecp oscillator (SWO). The trans-

l 8 PROCEEDINGS ITB Vol. 18. No. 1. 1985

mitted torques were delivered by eddy current type dynamometer (ED) adjus-ted through torque controller (TC).

(RMS-va lues)l ' -

- - - - - --]

L -_ - - - - J

( Manua[ )

( Mechanica[Svc Io rn \

(A uto)

TS

Figure 3 Schematic arrangement of the electronical instrumentations

( FREO. Analysis)

qi

L - - _ _ _ - - J

Wave-form )

DA : dual differential amplif ierDC : digital counterDR : tape recorderDV : digital voltmeterFV : frequency to voltage converterlM : induction electric motorLPF : low pass fi l terMC : mini computerOS : oscil loscope

Additional anangement of strain gages forments at different tooth root fillets as wellmeters are shown in Figure 4.

PS : polver supplyRTSA : real t ime spectrum analyzerSA I strain amplif ierSC : speed controllerSR : s l ip r ingSwO : sweep oscillatorTC : torque converterTG : tested gearTS : triggering signalX-Y : x-y recorder

dynamic load and strain measure-as the positions of both accelero-

- - - t - - l TG iL ; : - J

- - - - - 1 |

I t v l I

TOOTH N0.23-=--_-_:t

^ ^ r ; l>. b. L4 WHEATSTONE

br id g e

TOOTH NO. l7

c,.: fil- v. L:,

Trailingsrde

LeadinQ

Figure 4 Strain gages and accelerometers arrangement

20 PROCEEDINGS ITB Vol. 18, No. 1, 1985

Data acquisition was conducted as follows. The acceleration signals neasuredby the accelerometers were sent to two low pass filters (LPF) and thet outprrtsignals were combined in a dual difl 'erential amplifiers (DA). Meanwhile strainsmeasured by the strain gages at tooth root wcre amplified by a strain amplifier(SA). These measured signals, i.e, accelerations and strains. were sent front thenrechanical systenr to the peripheral instruments through a slip ring (SR). Theoutput signals fronr the differential antplifiers and the strain amplifiers subse-qucntly became thc measured cxperimental data ready to be processed orstored lor later use.

4.2 Data processitrg

Tlrce kinds of data processing were pcrformed, narnely:

(a) RMS valucs of the v ibrat ion levelThe output of the differential anrplifier (DA) represcnting the vibration levcl(in tenns ot acceleration) was nreasured by a digital voltnleter (DV) and its di-gital output signals were then averaged by a nlini computer (MC). The averagedvalues (llMS-valucs) of the acceleration signals hence becante the assigned valueof thc Y-ordinate in the X-Y plot. At the &?ntc tinte, the varying rotationalspeed of the output shalt of thc tested gears (TC;), having a reduction ratio ofI .0, was tallied by a digital counter (DC). Using frequcncy to voltage convcrter(FV) the pulses wcrc convcrtcd into analog voltage signal and assigned as themoving val r res of the X- lbsc issa in the X-Y p lot .

(b) Frc<1uc ncy analys is dataA real tinre spectrunr analyzcr (RSTA) was used to process the output signal ofthe differential amplificr from the time domain to the frequency domain. Theresults werc plotted on an X-Y coordinate system with the X-axis rcpresentingthc lieqLrency and th(- Y-uxis as the vibration level.

(c) Wavc lorm dataData in the timc dornain were recorded by a tape recorder. These data includedthe acceleration signals from the differential amplifier (DA), the strain signalsfrom thc stra in anrp lifier and one-pulse p(-r rcvolution triggering signal ( TS ). Byretrieving the data frorn the tapc, data observations and analyses were accom-plished using plots obtained from a pen recorder.

4.3 Aligument of gear shaft

In addition to rotational speed and transmitted torquc tlut were sclected asmeasuring parunlcters. shaft alignrnents were also introduccd as another para-meter. Two kinds ol shaft misalignments, i.c. shaft deviation and shaft incli-nat io l l . were consic lcrcd in thc exper iment .

' l - i le tenninologies used in t i r is re-

PROCEEDINGS ITB Vol. 18. No. I. lg85

port to describc the alignment of the gcar shaft are in accordance with the ISOstandard.

ln the actual crperimental set up, the shaft misalignments were obtaiued by in-serting scveral slip gagcs either on the surfacc of the plate or on its side surface,as shown in Figure 5. Tire thickness of the slip gages choscn for the cxpcrin)ent wcre 0.2 mm and 0.4 mrn. The resulted sluft deviation and shaft incli-nation were mersured by two dial indicators. For tooth facewiclth of 10.0 rnm,the anglrlar crror and tlre alignrrel.|t error rcsrrited fron] two different slip gagethicknesses are shown in Table L

Table I Tested gear specification and its classification

2 l

H ] H2 H J S

Far€ v / id lh (6 i r \o 70 ') t o3 5

Pressure anq le ( ? 0 2AH e l i x a n a l e ( 30

) 0 t 0R e l e r e n c e d r a m e r e ( i f t 2 t 7 1200

- 0 t 7 2 0l d 0 t 6 :

0 4 5 0-q tr 8 5 ?t l 254 r65

9:2

Trans le r3e con lac i r . t io €s

Table 2 The values of shaft deviat ion and shaft incl ination

Slip'q.thickness

l m m )

Shaft Deviation Shatt Incl ination

0 (rad) F o r b = 2 0 m m 0 ltadl F o r b = 2 0 m m

o.z0.4

5 . 6 X 1 0 1

1 . 1 2 X 1 c t 3

1 1 .1 i rm 5.9 X 104

1 . 1 4 X 1 c r r

I 1 .9 l lm

22 .8 pm

'r)PROCEEDINGS ITB Vol. 18. No. l. 1985

d e v i a t i o n :

I n c | | n a l

a . ISO s tanda rd f o r sha f ta I i gnmen t

D r i v e nGear -

0 r i v i ngGear

b . R e a l i z a t i o n o f i n p r o p e r s h a f ta l i g n m e n t

Figure 5 Shatt deviat ion and shaft incl ination

PROCEEDINGS ITB Vol. 18. No. I. 1985

5. Experimental resrlts of vibration measlrements on helical gears H-2

5.1 Influence of rotatianal peed on vibration level for different gear shaftalignments

In these experiments, three kinds of vibration level (in terms of RMS values)were measured, i.e:

(a) Rotational (torsional) vibration(b) Radial (tranwersal) vibration(c) Axial vibration

For the three different kinds of vibration level, measurements were carried outby varying the rotational speed in a continuous manner from about 600 RPMto 3400 RPM. At the same time, the RMS values of the accelemtion level weremeasured for the torque transmission of I 47 N.m ( l5kgf .ln).

The obtained results were categorized into two major findings, namely:

(a) The influence of gear shaft deviation on rotational, radial and axial vibra-tions, respectively.

(b) The influence of gear shaft inclination on rotational, radial and axial vibra-tions, respectively.

The results area presented in Figure 6 for gear shaft deviation, and in Figure 7for sear shaft inclination.

GEAR H-2 TOROUE 147N.mDeviat ion.

- - - - . 22 .3 um- . - . - . - . l l . l u m

0.0- - - - - - 1 1 . 9 u m" - l l . u u m

Jn/3

2000Speed rpm

Ilz kHz

23

r00

c: 5.!

o

ry\'-E.p---R

- _LS TS

a . R o l a t i o n a l ( t o r s i o n a l ) . . ' i b r a t l o n

24

Dev ia l ion :- - - - . ?? .3_ _ . _ . - , l t l

q0- - _ - _ l l . l u m

. . - 2 2 . 3 u m

Deviation.- __ - - . 22 .3_ _ . _ . _ . 1 1 . 1

0.0- - - - - - t l t u m

- 2 2 3 u m

GEAR H-2 TORQUE t {7Nm

:l .e,S=

PROCEEDINGS ITB Vot. 18, No. t, I9t5

GEAR H-? TOROUE t47|'lm

u,n trlq^ - F ] -

um 'g r l .= -L q t <

I

Ec

. : )u6

7.

lmHt

b . I i a C i a I ( t ! a n s v e r s a l ) v i D r a r r o n

II

E

EC9

v

I z kllzc . A x r a I v r b r a t l o n

Figure 6 Influence of rotational speed on vibration level for gear shalt deviation

0

PROCEEDI IGS ITB VoL 18.No. 1. 1985

l n c l i n a t i o n :- - - - . l l . o u m- - . - . 1 1 . 9 u m

0.0- - - - 1 1 . 9 u m' ' ' ' ' ' - 1 1 . 6 u m

I n c l i n a l i o n :- - - - . 2 2 . 8 u m- - - ' 1 1 9 u m

0.0- - - - 1 1 . 9 u m

- 2 2 . 8 u m

GEAR H-2 TOROUE 147 N m

0riven

25

100

c: f U

q/

oJ

.Tt

f h-.H.<"--r -zJ -Dr ,v i n9

,(

t.0l z k H z

a . R o t a l j . o n a l { t o r s i o n a l ) v r D r d c r o n

100GEAR H-2 TORQUE 147 fl.m

Driv en

iE-f.b F1 <'s-- -.1-T -

0r iv ing

E

Ec

o,

t z k H tb . F a d l d I ( t r a n s v e r s a l ) v i b r a t i o n

26 PROCEEDINGS nB Vol. I8,No. 1,1985

GEAR H-2 ToROUE 147 N.m2 5

E

E

c'a.0

!q)

I nc l i na t i on :- - - - . ?2 .8 um- - - , ] l . 9 um

00- . 5 U m

' ' - 22 .8 um

0r rven

t 0l z k H z

c . A x i a L v i b ) . a t . i o n

Figure 7 Influence ol rotational speed gn vibration for gear shaft inclination

Discussions

(l) In general, the gcar shaft devirtion gave significant influence on vibrationlevcl. On thc other hand, the shaft inclination error exerted influcnce onlyto a nrinor extent on the vibration level. Furthermore, errors at the gearlcading side yielded higher levcl of vibrltion as compared to the errors atthe gear trailing side.

(2) For both shaft misaliplnments, i.e. shaft dcviation and shaft incLination, thehighcst lcvel of vibration occurcd on rotational vibration. followed by ra-dial vibration, and the lowest on axial vibration. Henceforth, the axial vi-bration, on thc tested helical gcars H-2 was neglected in the subsequcnt in-vest rgatlo n,

(3) With respect to the vibration level of proper geirr shaft alignment, errorvalues of I l . l gm and l l .9 gnt (corresponding to s l ip gage th ickness of0.2 nrm) did not render much influence. However, error values of 22.3 prnand 22-8 pni (corresponding to the slip gage thickness of 0.4 mm) producedsignificant influence on the level of vibration at the gear leading side and atthe gear trailing side as well.

(4) Cune peaks, which for some cases corresponding to fn/2 and fn/3, shiftedtoward lower rotatiorlal speed (lower tooth meshing frequency) for both

PROCEEDINGS ITB Vol. 18. Na 1. 1985

type of rnisalignments. These shifts were clearly observed for curves witherror values resulted from slip gage thickness of 0.4 mrn.

(5) For error values corresponding to the gage thickness of 0.4 mm, the radialvibration level caused by gear shalt deviation was, generally, higher than theone produced by gear shaft inclination.

5-2 Inffuence of transmitted torque on rotational vibration for different shaftawnments

As was discussed earlier slip gage thickness of 0.4 mm produced values at thegear leading side as well as at the gear trailing side which caused rotational vi-bration to occur at high vibration lejrel. Therefore, the following investigationwas conducted on rotational vibration prirndrily to explore the behavior of tor-que transmision and the level of vibration due to improper gear shaft alignment.

On a graph of acceleration level (RMS values) versus rotational speed, thevalues of acceleration level at different rotational speeds, i.e. : 800, 1000, 1460,2000, 2150, and 2300 RPM were plotted for a vaiation values of torquetransmiss ion values of 49.0,73.5,98.0, 122.5, and 141.0 N-m. These rotat io-nal speeds were selected because they produced vibrations beyond the reso-nance frequencies but stil l in the normal range of operating speed.For data processing, the maxinrum transmitted torque was specified to be147.0 N-m, corresponding to the tangential ransmitted force of 2800 N (285.7kgf) on the pitch circle.

The computed results of gear shaft deviation and gear shaft inclination are pre-sented in Figure 8 and 9, respectively. These plots are in good ageement withthe compiled result of Figure 10, presented as the composite curvcs of differenttorque transmission values.

27

(a) SHAFT O€vtaTtON

AT tEaO lNG SrO€ : + 22 .1 um

/ .9 735 t 22 .5 t L7

TOROUE (N m)

19 735 98 122 .5 117

IORO(E (N m)

(b) tlo C€vlATtON

€- {,0a{ 3 052 7 09

E^ 4 0

I

z - _o z u;

1 y'l-1).;7+], :a --,1 -,"

- !

p|;tr3 o

28 PROCEEDINGS ITB Vol 18, No. 1, 1985

{c) SHAFT O€VIAIION

a T T R a t L t N G S | D E : - 2 2 . j u mE

E

z9

(L

-

E

E

z9

E

Ezo

(!

(l

E

zo

L O

30

20

t0

0t9 735 9B 1225 u7

TOROUE (N m)

Fig{.rre 8 Influence ot transmitted torque on rotational vibralion for gear shatt deviation

( a ) SHAFT INCLINATION

A T I € ^ O I N G S l O f + 2 2 . B u n

49 735 98 1225 r47.IOROUE (N m)

1.0

30

?o

t0

0

(b) t\O ll.lCLlNATlOtl

4 0

l0

20

t0

0

LO

30

20

r0

0r.9 l3-5 98 ]22.5 1L',7

TOROTJ€ (N m)

(c) SHAF I INCLINAIIoN

A T T R A I L I N G S I D E ] - 2 2 . 8 u m

49 73.5 98 r22 5 14?TOROUE (N m)

Figure 9 Influence of transmitted torque on rotational vibrational Jor geAr shaft deviation

PROCEEDINGS ITB Vol. I8. No. I , i 9B5

Discussions

Based on the results shown indiscgssed, namely :

the previous llgures, the foltowing cases can be

l n t3

2000Speed rpm

29

(l) In general, the curves suggest an increasing level of rotarjonal vibration. ata constaltt rotational speed, when the transmitted torque is increased.

(2) Wi th h igher rotat ional speeds i .e . : 1460,2000,2150, and 2300 RpM, theincrease of torque transmission yielded higher level of vibration on eearshaft deviltion (corresponding to the slip gage thickness of 0.4 mm) t-hanon gear shaf t inc l inat io n.

(3) Likewisc, for both gear shaft misalignments, i.e.: gear shafl deviation andgcar shaft incLination, errors at gear leading side produced higher level ofvibration than the one caused by errors at the gear trailing side.

(4) Thc composite graphs presented in Figure 10, shown that the curve peaks,which correspond to half and onethird of the looth meshing resonance fre-quency, shift toward highcr rotational speed (tooth meshing frequency).These tooth nrcshing rcvtnance frequencies wcre determined by frequencyanalysis of the vibration signals. In addition, gear shaft deviation ofJ2.3 pn dLre to irnproper sluft alignnrent caused the level of vibration toincreasc if the transmittcd torque valucs were increased. These conditionswere clcarly observcd for rotational speed of g00 RpM to 3400 RpM.

r00 Torque ( N m) G E A R H - 2

/ o

/ J . 5

9B12? 5

C

c: ) U.!

CJ'=

,j

r0kHz

a . C e a r s h a f L d e v r a t i o n a t t h eg e a r l e a d r n g s i d e : + 2 2 . 3 u m

30

Torque ( N.m)

PROCEEDINGS ITB Vol- l,8, No. l, 1985

G E A R H - 2

r.973.598122.51 t 7

lz ktsz

" ;:: : ; : : l i ' i ; ' : i : : :n-"!,1!"u'

Figu.e l0 Influenc€ of rotational spe€d on rotational vibration for different values oftransmitted torque

6. Experimental results of strain nreasurement on helical gears H-2

6.I Tooth root strain along the line of action for different strain gage positions

In order to know the behavior of tooth root strajn along the line of action(LAHG) during tooth meshing of a pair of heLical gear teeth, the stmin valueson ten d i f ferent posi t ions a long LAHG, i .e . : 0 . 0 .25, 0.5, 0 .75, 1.0, 1 .25, I .5 .1.15.2.0, and 2-25 Ptn (where Ptn is the transverse normal pitch) were evalu-ated and plotted. The evaluations were carried out on the strain wave formdata. at a constant transmitted torque of 196 N-m (20.0 kgf.m). This torquevalue corresponded to the tangential transmitted force of 3733.3 N (380.95kg| on the pitch circle. The plotted result are showr in Figurcs 1l and l2 forgear sluft deviation and gear shat't inclination, respectively.

Similar data have been evaluated for transmitted torque of98 N.m ( 10.0 kgf.m),and the obtained results show similar strain conditions except the strain ampli-tudes were smaller than those ones obtained for the transmitted torque of196.0 N.m.

E

c:

qr

PROCaEDINCS nB Vd. I8,No. 1,1985

Discrusions( I ) Gear slnft deviation and gear shaft inclination

a. Figures I I and 12 showed that the maximum tooth root strains causedby shaft deviation is higher than those ones caused by shaft inclination.When the gear shaft deviation at the gear trailing side is 22.3 tm thehighest strain value is 880 pe, corresponding to a stress value of l8l .3MPa or 18.5 kgf/mm2. But when the gear shaft inclination at the gearleading side is 22.8 pn, the maximum strain is 500 trre , corresponding toa stress value of 103.0 MPa or 10.5 kgf/mm2. Both shaft alignmenterrors were obtained from slip gages thickness of 0.4 mm.

b. For both gear shaft misalignments, errors at the gear leading sidedeated an opposite strain phenomenon at the position of strain gage I .However, it did not occur for the same error at the gear trailing side;except for shaft irclination of 22.8 pm.

c. Errors at the gear leading side increased the strain at the position ofstrain gage 2, but it suppressed the strain at strain gage 4. On the otherhand, errors at the gear trailing side generated an opposite condition.

d. Errors distorted the tooth root strain distribution of a proper gear shaftalignment; however, the maximum tooth root strain still occuredaround 1.0-1.25 ftn.

(2) Errors at the gear trailing side caused a maximum tooth root strain valueswhich were higher than those ones generated by ermrs at the gear leadingside, as slrcwn in Figure I l.

3 l

z

Ftn

o

zEo

a . I € .d i ng s i de + l l , l un b . L e a d i n g s i d e + 2 2 , f u r

PROCEEDINGS ITB Vol. 18. No. I. 1985

500

Pl n

r 000

- 4 0

b oooU

'o 400

z I '00

(t-a ^

z - v w

a(Yt-u l -

r0 r .5

LAI IG

Pt n

_20

c . N o e r ro r d . T r a i I i n g s i d e - 2 : , . 1 t r r o

o

zafiF

GEAR H-2 TOROUE 196 N.m

S T R A I N G A G E :

oo-o I Leading sideL--L-i 7

.{-a 3L--*---^ 4 Trailing side

Figure 11 Tooth root strains along the l ineof action at different strain qage positions.Gear shaft deviation

Pl n

e . T r a i l r n g s i d e - L i , I u n r

(3) Gear shaft inclinationWith respect to the tooth root strain of proper gear shaft alignment, incli-nation eror did not affect the tooth root strain severely, except to straingage l. Maximum tooth root strains occured around l' 0 to 1.25 Ptn; andtlreb values wcre slightly lower than those ones of the proper gear sha ftalignment.

PROCEEDINCS ITR Yol. 18, No. I , lq85

z 2O0acrul ^

500

700

J J

500

100Io

(rF

200

a - L e a d r n g s i d e + I l , 9 u r n n ! s i i e + 2 2 , 3 u m

ln

l n

0.5 l0

LAI iG

5 r.0

LAHG

.25 Pt n

9

=rrt-a

z

o:F

r0 r .5

LAHG

25PI N

_20

500

400

200

-20

t0 .

LAHG

25Pl n

-2

c . l l o e r t o r _ d . T r r l l r n g s r d e - i 2 , u u n r

1

GEAR H-2 TOROUE t96 N.m

S T R A I N G A G E :

(>€ lLeading s ided-r}-,-d 2

a-a-a 3

,-*-- L 4 Trail ing sjde

Figure l2 Tooth root strains along the l ineof action al different strain gage positions.Gear shaft inclinatione . T r a 1 l i n g s i d e I 1 , 3 u t r

34 PR0CEEDINGS ITB YoL 18, No. I , 1985

6.2 Tooth root strains during tooth meshing at different struin gage posilions

During tooth meshing, tooth root strains at four different strain gage positions

were examined for a set of constant values of Ptn. The tooth root strains along

the tooth facewidth for Ptn values of 0.25 (at the start oftooth meshing), 1.5,

1.0 (<iuring tooth meshing), I .5 and 2.0 (at the end of tooth meshing) were ob-

served at different times. These observations described the load distribution as

well.

The results are shown in Figures 13 and 14 for gear shaft deviation and gear

shaft inclination, respectively. The evaluation was carried out for the trans-

mitted torque of 196 N-m (20.0 kef-m).

6 06 0

: . 0=-

T O O r s R O O r 5 1 R , u N

A 1 O I F F E R ! N I S I R A I N 6 A 6 E

P O S I I I O N O U R I N G T O O I 1 1 M E S H I N G

Sarr I 0 tv l ^ r lo r C?, !s .0 .nh

r00r r MtSr r16 Pogro !

cF. - -a o r t P ,n . . rh . ! ^H6d - - - 4 , 0 5 0Lo ro0

. ^ IO0

ro iou l r96ro t 10r0rh ,

t 0

! 5

a .

0

5 ] A A I N 6 A G :

Figule 13 Tooth root stralns

Gear shaft deviation.

(see iabte 2 )difJerent strain gage positions during tooth meshing'

PROCEEDINGS ITB Vol. |8, l tb.I , Ia8) J ]

6 0 0 1

5 ] R A I N C 4 G E

- ? 0

(see iable 2 )

Figure 14 Tooth root strain at dif ferent strain qage posit ion during tooth meshing.

Gear shaJt incl ination

DiscLtsslons

( l ) The gear shaf t dev iat ion of 1 l . l gm and 21.3 pm at the gear )eading s ideyielded tooth root strains similar to thosc which occured with proper gearshaf t a l ignment . The maxi rnum st ra in of 55 ge ( the st ress of I I3 .3 MPaor I L6 kgf/mmr ) was observed at strain gage 2 fbr Ptrr value of 1.0. On thcother hand, erors at the gear trailing side yielded maximum tooth rootst ra in of E50 ge ( the st ress of 175.1 MPa or I L6 kgf l rnm2 at s t rx in gage 4for Ptn va lues of I and L5.

(2) The relationship of tooth root strains and gcar shall inclination showedsimilar trend as the tooth train versus gear siratl deviation relationshil-), e\-cept the st ra in.va lucs were smal l r . - r . T i re maximunt s t ra in of i25 l1€ ( the

stress of 25.8 MPa or 2.(r kgf/mm'z)was observed for Ptn valtte of 1.0 ats t ra in grge 2.

. 0 0

. '00

0

6 0 0

: 4 o c

?c 2 0 (

' 0 ? h m

.// -/,

\, /! slRAlfr

c . N 0 E F R O F

n" / /

*c " !

PROCEEDINGS ITR Vo l. 18. No. 1. 1985

6.3 Maximum tooth root strain at different st/ain gage positions

Evaluation of the recorded strain data (wave form measurements of the strain)was also carried out to show the relationship between the maximum tooth rootstrains at four different strain gage positions with the two gear shaft alignmenteffois. Both gear shaft alignment erors were evaluated either at the gear trail-ing side or at the gear leading sidc. The evaluation was made for the transmittedtorque of I 47 N-m ( I 5.0 kgf-m) and the obtained results are shown in Figure 15-

GEAR :H_2 TORQUE:147N.M(15KgTM)STRAIN GAGE POSITION:

1231'0.07 0.41 0.59 0.84 Ptn

I - - f L5+0.4nrm Cl -_oNO ERROR A- . TS-0.4mm

t-} - { l LS '02nrm A- ' { TS-0.2mm: GEAR Lt f "0 lNC Sl l l t TS : GEAR TRAILING SIDI

600

/ r00

200

0/ . 1

E T D A I N I | / : A T : t rJ I t \ - ' \ r r t \ - / f 1 \ J L

(o

><UJ

z

(Fa

A. SHAFT DEVIATION

Ptn=l

PROCEEDINGS ITR Vol. l8. No. 1.1985

600

400

200

n1

Figure 15 Maximum toothshaft alignment.

b.SHAFT INCLINATION

q T t n - |

STRAIN GAGEroot strain at different strain gage positions and different gear

(oIO

XUJ

z

K.F(r)

Discussions

(l) From the strain measurentents, it was observed that a pair of helical gears

H-2 were sensitive to gear shaft misalignment, particularly to the gear shaft

tleviation at the gear trailing side. On the other hand, they are less sensitivc

to gear shaft deviation at the gear lcading side. For the transmitted torque

of 147 N-nr, corresponding to the tangential transmitted force of 2800 N(285.7 kgtl on the pitch circle, tlle maxintum strain value rvas 575 4e (the

s t r c s s o f | 1 8 . 5 M P a o r 1 3 . I k g f / n r n r l t .

(2) For proper gear shaft alignment, il was obsened that the strain values were

not much affected by gear shaft inclination, exccpt wllen the gear shaft

deviation at the gear leading side reached 22.8 pe.

7 Conclusions

An expcrimental set up, deviced for geu vibration and gear strain measurement,

has been successfull bLrilt to lncasure the RMS values of gear acccleratiolr. to

carry out frequency analysis of gear vibration signals on a pair of helical gears.

This set up was also capable of reading signals of : ( I ) gear vibration' (2) tooth

root strains, and (3) one-revolution triggering pulses in the time domain. Mea-

surements were conducted on a pair of helical gears with the folowing specifi-

38 PROCEEDINGS ITB Vol. 18. No. 1. 1985

cations, i.e., the total contact ratio of over 2.0 and the overlap ratio of less than1 . 0 .

It.was observed, among the three modes of misaligned helical gear vibrations,the highest vibration level occured by rotational (torsional) vibration.

Gear shaft alignment errors (gear shaft deviation or gear shaft inclination) atthe gcar lcading side has sgnificant influence on the gear vibration characteris-tics, i.e., RMS values of the acceleration and tooth meshing frequency. This isespecially evident for error value of 22 pm obtainable by the slip gage thicknessof 0.4 mm.

In addition, if the valuc of transmitted torque was raised, the level of the gearrotational vibration increased accordingly.

Gcar shaft deviation or gear shaft inclination at the gear trailing side stronglyinfluenced the tooth root strains on these helical gears.Finally, the calculatcd data should be helpful in constructing other theoreticalmodels of vibration of misaligned helical gears.

8 Acknowledgement

One of the authors who was a guest researcher of Prof. Umezawa at the Labora-tory of Precision Machinery and Electronics, T.l.T., would like to acknowledgethe members of this rcsearch group. Sincere thanks to Mr. H. Houjoh for hiskind help, discussions and lriendship. This acknowledgement is also conveyedto tire Japan Society for Promotion of Science (JSPS) for its financial assis-tancc during the author's sojourn in Japan. Special thanks to the reading co-mittee of the ITB Proceedings for their corrections and efforts so that theirublication of this paper becomes possible.

9 References

l . Umezawa, K. , J . lsh ikawa. and K. Hayashi , 'Def lect ion due to a concen-trated load on a cantilever thick plate of finite length for gears', Bull. JSME,V o l . 1 2 . N o . 5 3 ( 1 9 6 9 ) . 1 2 . 0 4 l 2 I I

2. Umezawa, K., 'Deflections and moments due to a concentrated load on arack shapcd cantilever plate with finite width for gears',Bull. JSME,Yol.1 5 , N o . 7 9 ( 1 9 7 2 ) , 1 1 6 , 1 3 0

3. Umezawa. K.,'The meslring test on helical gears under load transmission(lst report, The airproximate formula for deflection ofgeartooth)', Bul/.JSME,Yol . 15, No. 90 (1972) ,1632-1639

PROCEEDINGS ITB Vol. t8,tljo. L 1985

Umezawa, K.,'The meshing test on helical gears under load transmission(2nd report, The approximate formula for bending moment distributionof sear tooth) ' , -Bul l . JSUE, Vol . 16,No.92(1973) ,407-413

Umezawa, K. and J.Ishikawa, 'Deflection due to contact between gearteeth wi th f inete width ' . r r l l l . JSME,Yol . 16, No. 97 ( l9?3) , 1085 1093

Umezawa, K., 'The meshing test on helical gear under load transmission(3rd report, The static behaviours of driven gear)', Bull. JSME, Yol. 17,N o . I 1 2 , ( 1 9 7 4 ) , 1 3 4 8 1 3 5 5

Umezawa, K. and J. Ishikawa, 'On cylindrical gear without statical beha-viours of driven gear under any load', Bull. JSME, VoL 18, No. 122(197 5\.8'1s -904

Sato, T., K. Umezawa, and J. Ishikawa, 'Effects of contact ratio and pro-file corection on gear rotational', Bu . JSME, Vol. 26, No. 221 (1983).2 0 1 0 - 2 0 1 6

Umezawa, K., T. Sato, and J. Ishikawa, 'Simulation on rotational vibra-t ion of spur gears ' , B u l l . J SM E, Vol . 27, No. 223 (1984r , I 02- I 09

Umezawa, K., T. Sato, and K. Khono, 'lnfluence of gcar erors on rota-tional vibration of power transmission spur gear',.Bull. JSME, Vol. 27, No.225 (1984\

Umezawa, K., 'Vibration of power transmission helical gear with narrowfacewidth', ASME, 04-Det-l 09

39

5 .

6 .

7 .

8 .

9 .

10 .

l t .