Frequency Domain Analysis of Rattle in Gear Pairs and Clutches. Abstract. 1. Introduction

Transcription

1 The 00 International Congress and Exosition on Noise Control Engineering Dearborn, MI, USA. August 9-, 00 Frequency Doain Analysis of Rattle in Gear Pairs and Clutches T. C. Ki and R. Singh Acoustics and Dynaics Laboratory Deartent of M echanical Engineering and Center for Autootive Research The Ohio State University Colubus, Ohio , USA Abstract In order to truly understand rattle noise and vibration sources in vehicle drivetrains, a torsional syste odel with one or ore clearance non-linearities ust be develoed. Most rior analyses have focused on the deterination of tie doain resonses under haronic excitations using cubersoe nuerical tools. In this aer, we roose an alternative rediction schee that exaines the nonlinear frequency resonse characteristics. We resent a new sei-analytical schee, based on the ulti-ter haronic balance ethod (HBM). It is alied to the flywheel-clutch (with ulti-staged stiffnesses), clutch hub-sline (with backlash) and gear air (with backlash) sub-systes. Unlike the revious ethods, our ethod includes araetric continuation with self-adjustable frequency stes, and it is caable of efficiently finding both stable and unstable solutions in ulti-valued regies. Calculated results are successfully coared with exeriental data and nuerical siulations. This versatile ethod could be used to develo design criteria for a rattle-free syste.. Introduction Rattle noise sources arise fro gas and ulti-valued stiffness eleents in a wide variety of achinery, vehicle and industrial alications. In this article, we focus on the torsional vibrations of a tyical vehicle drivetrain that consists of any gear airs with backlashes and a clutch with two or ore stages in ters of its torque vs. relative angular dislaceent characteristics. Further, non-linear daing echaniss are often resent. To better understand this roble, consider the source-ath-receiver network of Figure where the sound ressure fro the rattle sources, at a given engine seed Ω, can be defined as follows. a ( Ω ) = ( Ω ) ( Ω ) θ&& j ( Ω ) = θ&& j( Ω ) k i,j,k a θ&& i j,k i, j,k θ && Here, is the ressure (at an interior or at the ass-by location), a is the structural acceleration of drivetrain housing, i denotes the ath fro housing vibration to location, i,j ()

2 Ω a i (a) I I (b) T e ( t ) (t) T e θ θ (t) T e I Clutch Gear Pair index j is the ath for torsional structure-borne noise, and θ & j is the torsional acceleration of the k th rattle source. In this analysis, ultile aths could be quantified using linear exeriental or coutational ethods. However, the knowledge of sources ust be obtained by using a non-linear differential equation solver. Note that here and θ & are defined at a given seed and thus one ust kee track of all eriodic and non-eriodic solutions. To illustrate the nature of this roble, exaine the two degree of freedo (DOF) torsional sub-syste of Figure that could be reduced to a single degree of freedo (SDOF) definite syste in ters of relative angular dislaceent ( t ). Figure 3 shows a generic for of the clearance tye non-linearity that induces rattle. Here the stiffness of the first stage is α with resect to the second stage of unity stiffness. The backlash roble is given when α = 0 and b to b regie will be equal to the aount of backlash. When α and b = 0, one find the tyical characteristics of a torque converter clutch in a vehicle with autoatic transission. Finally, the clutch corresonding to a anual transission is given by 0 < α <. Non-linear dynaics of a torsional syste with one or ore clearance non-linearities has been studied using digital [, ] and analog [3] siulation ethods. However, such analyses ehasize the deterination of resonse tie history at a given haronic excitation, and these require substantial coutational resources. The resulting tie history can be very colicated and ay include suer-haronic and sub-haronic resonances [-4], or even b f S ( ) Figure 3. Torque-dislaceent characteristics of clearance tye non-linearity. α b θ k Figure. Scheatic of a vehicle drivetrain with rattle sources. Kf (, & ) T D Kf (, & ) Figure. Torsional sub-syste with clearance nonlinearity. (a) DOF sei-definite syste; (b) SDOF definite syste. chaotic otions [-3]. Such a colex behavior revents an engineer fro efficiently judging the overall syste resonse. Moreover, one ust often run too any cases to find araeters that could avoid or reduce rattle. Therefore, fro the design oint of view, steady state frequency (seed) resonse functions of the governing syste are needed. Nonetheless, the concet of a frequency resonse function is ore colicated for the non-linear case, as it ust include sub- and suer-haronics, ulti-valued resonses of a single frequency excitation, and ju henoenon. One ust also consider the ossibility of non-eriodic and chaotic solutions [-3]. In this aer, we resent a new sei-analytical ethod that can efficiently calculate the relevant frequency and tie doain characteristics of a rattle source.

3 . New Solution Method Assue that the rattle source can be well reresented by a lued odel of Figures and 3. The excitation torque T e (t), fro an internal cobustion engine, fluctuates significantly between low (around the coression stage) and high (around the ignition stage) values. Therefore, decoose T e (t) into ean T and erturbation T (t) arts. The fundaental haronic Ω of T (t) deends on tye of the engine, nuber of cylinders and crankshaft configuration. Exress T e (t) via a Fourier series (with N haronics) as shown below where ϕ n is the hase angle at the n th haronic. T The ean N e ( t ) = T + Tn sin(nω t + n ) n= e t N e n cos( nω t + n ) n= ϕ or T ( t ) = T + T ϕ () T = T ( t) ter, under no ower condition, should be equal to the drag torque TD ( Ω, Λ ) generated within the transission where Λ is the gearbox teerature. After an alication of the initial conditions in ters of absolute angular dislaceents, θ & ( 0) = θ& (0) = 0, and θ (0) = 0, the relationshi between T D and T for Figure is written as T = D I T / I. Therefore, the couled non-linear second order differential equations for the sei-definite sub-syste of Figure a are reduced to a single equation for the equivalent SDOF syste of Figure b where = θ θ. N C K && + & + f (, & ) = F( t ) = F + Fn sin(nω t + ϕn ) (3) I I n= Here, C is the viscous daing coefficient, K is the linear stiffness corresonding to the second stage of the unity sloe in Figure a, I = II /( I + I) is the effective inertia, F = T / I + T / I is the effective external ean torque, and Fn = Tn / I is the effective external ulsating torque at the n th haronic. Next, we roose a new ulti-ter haronic balance ethod (HBM). It is essentially afor of the Galerkin s ethod based on the least squares fit in error reduction [5]. The residual or error r (t) is the ga between true inut and estiated solutions, and it should goes to zero for (t) that satisfies the non-linear differential equation. In tie doain analysis, the residual should be zero for all tie sans. This is called as the strong for since the satisfaction of tie san requires an extensive calculation. Fro equation (3), before noralization by the inertia ter I, the equation of otion becoes as below. The strong for residual is: r( t ) = T( t ) I & C& Kf (, & ) 0 t. (4) However, when the inut and resonse characteristics of a non-linear syste are eriodic, frequency doain analysis can be alied by introducing the Fourier transforation (I ) of both sides. The Fourier transforation of equation (4) yields the following weak for residual. I ( r( t )) = I( T( t )) II( & ) CI( & ) KI( f (, & )) (5) Let the Fourier coonent vector of the saled outut be a = I( ( t)), then the iniization of the residual in Newton-Rahson for is exressed as R R 0 + ( R / a ) a. Here, R 0 is the residual fro an initial guess, and the correction factor is defined as a = ( R / a) R. Finding a true a is an iterative rocess, and the Newton-Rahson

4 schee is used. The ter corrector and finding R / a is the Jacobian atrix J used in the Newton-Rahson c / a in J is the key of HBM. The ter c / a is strictly related to non-linearity in the sub-syste and defined as below. * c ( Γ f ) * f * f * f = = Γ = Γ = Γ Γ (6) a a a a The critical ter below is the instantaneous stiffness of non-linear function f ( ) at each saling oints. f f f f f = f = diag ( t0 ) ( t ) L ( tn ) ( tn ) (7) Additionally, an arc-length continuation schee [5] is ileented in HBM by introducing an indeendent scalar araeter ω, which is the excitation frequency over the rage of frequency-doain analysis. Refineent of the arc-length can be related to both nubers of iterations before convergence and changes in tangential angles of the frequency resonse function, and it greatly iroves the convergence of the algorith. With inial coutational cost, dynaic stability can also be analyzed in the frequency-doain by HBM. 3. Results and Discussion To deonstrate the validity of HBM, an exeriental syste [4] is odeled. Nuerical siulations (NS) based on a Runge-Kutta schee are also run and their redictions are coared with those fro HBM and easureents. As seen in Table, exeriental and siulation results atch exactly in eak-to-eak alitudes. After a close exaination of resonse curves in tie doain, it is found that both siulation schees yield the sae tie signatures as the exerient even in the non-linear stiffness transition regies. For the nuerical siulation, two runs, down and u frequency swees, are needed in order to achieve solution oints around the non-linear resonance eak. This is because the nuerical siulation tends to follow the stable results, and can only rovide a single value in a ultivalued solution regie; this results in the ju henoenon. This liitation can also be observed in non-linear exeriental result. However, our HBM with araetric continuation caability adjusts itself for better convergence using Newton-Rahson tye solutions, and calculates eriodic solutions in ulti-valued regies. A single run of HBM can rovide all results even in ulti-valued regies and the suer haronic eaks of (/n) Ω with the sae accuracy as nuerical siulation in less than /0 of calculation tie, as evident fro Table. Figure 4 shows a coarison between linear and non-linear frequency resonses as calculated by HBM using the oerating conditions of Table for the sub-syste of Figure and 3. Here, is the oerating (ean) oint around which dynaic erturbation takes lace. The non-linear frequency resonse is generated by lotting rs vs. Ω where Ω is the diensionless frequency (seed) and rs is the root-ean-square dislaceent. Corresonding to a haronic excitation at Ω, one could run the nuerical siulation and deterine the steady state resonse. Then we can calculate the rs value of ( t ) that ay include sub- or suer-haronics. The HBM yields this answer quickly unlike the nuerical siulation. The non-linear frequency resonse clearly deonstrates an alitude sensitivity leading to severe rattle noise. The fundaental haronic eak starts fro the no-iact case with

5 Ω =.00, then reduces to Ω = when it gets into single-sided iacts, and finally saturates to Ω = for double-sided iacts. In addition, the non-linear frequency resonse exhibits several active suer-haronic resonses below Ω = 0. 40, and a subharonic resonse above Ω =. 0. The linear frequency resonse curve can not obviously redict these henoena and their alitudes are lower than those fro non-linear analysis. For exale, there would at least a 0 db difference in sound ressures generated by a rattle source between the linear and non-linear siulations, around 0. 6 < Ω < Table : Coarison of siulation and exerient Exerient NS HBM θ (rad/s ), Coutation Tie (T) - 3,640 s 350 s starts fro the nd stage Table : Oerating condition of the sub-syste of Figures and 3 Iact Case K =. 0, I =. 0, C = 0. 05, α = 0. 5 F = 0., F = , β = , b = 0 stiffness Double-sided HBM Stable HBM Unstable Linear HBM (n=) HBM (n=) NS U NS Down rs A B A B Ω Figure 4. Coarison between linear and nonlinear frequency resonses Ω Figure 5. Coarison between nuerical siulation and HBM HBM (n=) NS HBM (n=) NS Figure 6. Tie history of quasi-eriodic resonse ( A at Ω = ) Cycles of Ω Cycles of Ω Figure 7. Tie history of resonse with a subharonic ( B at Ω =. 4). Furtherore, the stability indicator in our HBM shows the ossible aearances of chaotic or quasi-eriodic ( A ), and sub-haronic ( B ) regies. As noted in Figures 4 and 5, there is a sall regie ( A ) where the HBM and nuerical siulation results do not atch well. Fro the nuerical siulation, this is a quasi-eriodic regie as shown in Figure 6. Since

6 HBM is assued to have only eriodic solutions, chaotic and quasi-eriodic or aeriodic resonses could result in unstable solutions. As seen in Figure 6, the eriod is doubled in the nuerical siulation when coared to the eriodic HBM solution. Yet, another instability occurs in the sub-haronic regie ( B ). Even though sub-haronic resonses are eriodic, they are unstable in ters of the assued Ω solution. Re-running HBM with nω can siulate the nth sub-haronic resonances. The HBM assues the eriodic solution but with a frequency Ω. It results in a sub-haronic eak at Ω =. 4, as evident fro the tie doain signature in Figure 7. Nuerical siulation and HBM results atch exactly since sub-haronic resonse is still eriodic. The ean oerating oint igrates back and forth fro the second stage of the stiffness to the first stage (Figure 3) during the frequency swee. An aearance by the sub-haronic resonance deends uon the Ω searation between the riary haronic and ean oerating oint crossover oint. When the searation in Ω is greater than the riary haronic resonance frequency ( Ω = in Figure 4), then a subharonic resonance eak would aear around Ω = Conclusion The HBM can be used to generate new insights about the rattle sources and thus develo useful design guidelines. For exale, when designing a gear air or a clutch, resonse as of Figure 5 and the like can be used to avoid ossible vibro-iacts. In the case of a clutch, the range given by 0. 0 < α < and 0. 3 < Ω < 3. 0 should be avoided. This is because suer- and sub-haronics as well as quasi-eriodic resonses, as shown in Figure 5, are found there. The resonse as need to be constructed according to each α value, given certain ean and alternating torque conditions. Our sei-analytical ethod (HBM) with araetric continuation and stability analysis can effectively redict the non-linear frequency (seed) resonse of a torsional rattle syste even in the resence quasi-eriodic, chaotic and sub-haronic regies. Furtherore, the resonse as develoed by the HBM rovide iortant clues and suggestions that could lead to the avoidance or reduction of vibroiacts. Finally, ongoing research work is exaining non-linear daing echaniss. Acknowledgeents This roject has been suorted by the Gear Rattle Industrial Consortiu (Eaton R & D, Sicer Clutch, CRF Fiat, Luk, SAAB and Volvo Trucks over the eriod), and DailerChrysler Challenge Fund since 000. Personal discussions with T. E. Rook in develoing the HBM code are gratefully acknowledged. References. R. SINGH, H. XIE and R. J. COMPARIN 989 Journal of Sound and Vibration 3, Analysis of an autootive neutral gear rattle.. R. J. COMPARIN and R. SINGH 989 Journal of Sound and Vibration 34(), Nonlinear frequency resonse characteristics of an iact air. 3. R. J. COMPARIN and R. SINGH 990 Journal of Sound and Vibration 4, 0-4. Frequency resonse of ulti-degree of freedo syste with clearances. 4. C. PADMANABHAN, R. C. BARLOW, T. E. ROOK, and R. SINGH 995 Journal of Mechanical Design 7, Coutational issues associated with gear rattle analysis. 5. G. VON GROLL and D. J. EWINS 00 Journal of Sound and Vibration 4(), The Haronic Balance Method with Arc-Length Continuation in Rotor/Stator Contact Probles.