TORSIONAL VIBRATION SUPPRESSION IN AUTOMATIC TRANSMISSION POWERTRAIN USING CENTRIFUGAL PEN- DULUM VIBRATION ABSORBER

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1 TORSIONAL VIBRATION SUPPRESSION IN AUTOMATIC TRANSMISSION POWERTRAIN USING CENTRIFUGAL PEN- DULUM VIBRATION ABSORBER Takashi Nakae and Takahiro Ryu Oita University, Faculty of Engineering, Deartment of Mechanical and Energy Systems Engineering, 7 Dannoharu, Oita-shi, Oita , Jaan tnakae@oita-u.ac.j Kenichiro Matsuzaki Kagoshima University, Graduate School of Science and Engineering, Korimoto 1-1-4, Kagoshima-shi, Kagoshima 89-65, Jaan There are numerous roblems related to vibration in the owertrain that revent imrovements in fuel economy in vehicles. The rimary vibration source is engine forced vibration. Engine torque fluctuations create engine seed fluctuations as a result of combustion. In the latest trend in engine technology, diesel engines and higher-ower engines are widely used. These engines contribute to strong torsional vibrations in owertrain systems. A torque converter is an element that transfers torque from the engine to the gear train in the automatic transmission of an automobile. With the goal of imroving fuel efficiency, a lock-u clutch system in the torque converter has been develoed that locks the inut and outut sides directly from low engine rotation seeds. Therefore, the torsional vibration in owertrain systems is increased from low engine rotation seeds. In order to address this roblem, a new vibration suression method which is centrifugal endulum absorber must be designed so as to effectively absorb the torsional vibration over a wide range of engine rotation seeds. However, a new vibration roblem occurs when the centrifugal endulum absorber is attached to a owertrain system. In the resent aer, a centrifugal endulum vibration absorber is alied in order to effectively absorb the torsional vibration. The generation mechanism and countermeasures for this unusual vibration are investigated. The centrifugal endulum vibration absorber has a characteristic that its natural frequency changes in roortion to the engine rotation seed. Keywords: Vibration of a rotating body, Centrifugal endulum vibration absorber, Forced vibration, Torsional vibration, Automatic transmission owertrain 1. Introduction Diesel engines, higher-ower engines, and low cylinder engines are widely used in the latest engine technologies. These engines contribute to strong torsional vibration in owertrain systems. The requirement for imroved fuel efficiency has led to the develoment of a lock-u clutch system for low engine rotation seeds. Since engine torque fluctuations are transmitted directly to the tires through the torque converter, the gear train, and the drive shaft in the lock-u condition, the desire to increase ride comfort imrovement is increasing. A centrifugal endulum vibration absorber, the natural frequency of which changes in roortion to the engine rotation seed, is also under develoment, and several studies have investigated its use to suress torsional vibration [1]-[6]. Ishida clarified exerimentally and analytically the dynamic behavior of a centrifugal endulum absorber considering the nonlinear characteristics of the absorber in order to absorb the torsional vibration of a simle rotating body model []. The results showed that the natural frequency of the 1

2 ICSV4, London, 3-7 July 17 centrifugal endulum absorber changes in roortion to the engine rotation seed, and the antiresonance frequency changes with the engine rotation seed so as to effectively absorb the torsional vibration over a wide range of engine rotation seeds. Pfabe analytically investigated the otimum design of the centrifugal endulum absorber in terms of mass reduction and minimizing the sace occuied by the centrifugal endulum absorber [3]. However, a new vibration roblem occurs when the centrifugal endulum absorber is attached to a owertrain system. Figure 1 shows the simulated results for an actual automobile. The figure shows the frequency resonse curve for the torque fluctuations of one drive shaft when the centrifugal endulum absorber is attached to a torque converter. A large eak aears at an engine seed of aroximately 1,5 rm. An unusual vibration at aroximately 1,5 rm was also confirmed in a traveling test using an actual automobile to which a centrifugal endulum absorber was attached. In the resent study, the generation mechanism and countermeasures for this unusual vibration are investigated. We analyzed the reduction of engine torsional vibration using a centrifugal endulum vibration absorber. The model considered herein includes the engine, the torque converter, the gear train, and the drive shaft. We focused on the otimum design of the ratio between the natural frequency of the centrifugal endulum vibration absorber and the engine rotation seed, as well as the mass of the centrifugal endulum vibration absorber for suressing the torsional vibration.. Theoretical analysis Engine seed rm Figure 1: Frequency resonse curve for actual automobile simulation..1 Analytical model Figure shows a model of the gear train based on the actual automatic transmission of an FF vehicle. This model includes the engine (four-cylinder), T/C rimary, T/C middle, T/C secondary, the gear train, the differential gear, and the centrifugal endulum absorber. The system is modeled by a five-degree-of-freedom system. Drive shafts are modeled by rotational srings. The wheels are assumed to be fixed ends. In Fig. (a), J1,, J6 are the moments of inertia of the comonents, K1, K, K3, and K4 are the rotational sring constants, 1,, 3 and 4 are the angular dislacements of each element, C1, C, C3, and C4 are the daming coefficients of each daming element, Fd is the dynamic torque from the engine, and is the inut angular frequency of the dynamic torque from engine forced vibration. The inertia of the engine J1 and the inertia of the T/C rimary J are locked directly by a lock-u clutch. Moreover, r is the total gear ratio of the gear train. As shown in Fig. (a), a centrifugal endulum absorber is attached to T/C secondary 3 that rotates at angular velocity which is also the engine seed. As shown in Fig. (b),, m, r, and R are the angular dislacement, mass, length of the centrifugal endulum absorber, and the osition at which the centrifugal endulum ICSV4, London, 3-7 July 17

3 ICSV4, London, 3-7 July 17 T/C secondary Differential gear y F d cos t K 1 K K 3 K 4 r J 1 J J J 3 4 J 5 J 6 R Engine C 1 C C 3 r C 4 T/C T/C rimary middle Gear train Drive shaft Centrifugal endulum absorber t+ x (a) Model of the gear train with centrifugal endulum absorber Figure : Analytical model. (b) Centrifugal endulum absorber absorber is attached, resectively. From the rotating system of coordinates shown in Fig. (b), the coordinates of the osition of the center of mass of the centrifugal endulum absorber are as follows: x R cos( t 3) r cos( t 3 ), (1) y R sin( t 3) r sin( t 3 ). () The velocities of the centrifugal endulum absorber are given by the following equations: v R )sin( t ) r ( )sin( t ), (3) v x y R ( )cos( t ) r ( )cos( t ). (4) ( The kinetic energy of the system is given by the following equation: T ( J1 J)( 1) J3( ) ( J4 J )( 3) m( vx vy ) J5( 4) J6( 4) where is the rotation seed ratio of the gear train ( = 1/r). The otential energy of the system is given by the following equation: U K1 ( 1 ) K( 3) K3( 3 4) K4( 4). (6) Lagrange s equations of motion relating to 1 through 4 and, taking system daming into consideration are given by Eq. (7) through Eq. (11): ( J J ) C ( ) K ( ) F cost. (7) J C ( ) C ( ) K ( ) K ( ). (8) ( J4 J mr mr mrrcos ) 3 ( mr mrr cos ) mrr ( 3 )sin C( 3 ) C3( 3 4) K( 3 ) K3( 3 4) ) C ( ) C K ( ) K. (1) ( J 5 J d (5) (9) ICSV4, London, 3-7 July 17 3

4 ICSV4, London, 3-7 July 17 ( mr mrrcos ) mr 3 c mrr( 3) sin where c in Eq. (11) is the daming coefficient of the centrifugal endulum absorber. Linearizing Eqs. (9) and (11), we obtain the following: ( J4 J mr mr mrr ) 3 ( mr mrr) C( 3 ) C3( 3 4) K( 3 ) K3( 3 4) (11) (1) ( mr mrr) mr c mrr. (13) 3 Here, the equation of motion of undamed free vibration of only the centrifugal endulum absorber, its natural frequencyn and its order n are given as follows: mr mrr, n n, R n. (14) r The angular frequency of the dynamic torque from the engine forced vibration is twice the rotation seed of the four-cylinder engine (. In order to suress the vibration comonent the order n must be set to (n = ). If c is set to and n is set to in Eq. (13), is not vibration solution, and the dynamic amlitude of is, because 3 is in Eq. (13), then the torque fluctuation is in the linear analysis. Therefore, c must be considered in the linear analysis. The Runge-Kutta-Gill method is used for numerical calculation in the linear and nonlinear analyses. As standard arameters of the centrifugal endulum absorber, m, r, R, n, c, and Fd are set to 1.59 kg,.8 m,.83 m,.,.1 Nms/rad, and 6 Nm, resectively. 3. Results of numerical calculations 3.1 Natural frequencies and natural modes Table 1 lists the natural frequencies and natural modes. The centrifugal endulum absorber is not attached. In Table 1, the natural frequency for the second mode is 3.85 Hz. In the second mode,, 3, and 4 vibrate out of hase with resect to 1. Table 1: Natural frequencies and natural modes 1st nd 3rd 4th fn (Hz) θ θ θ θ Torque fluctuations for one drive shaft without centrifugal endulum absorber This section describes numerical calculations for a system without a centrifugal endulum absorber. In this forced vibration analysis, we focused on the torque fluctuation for one drive shaft Td, which is given as follows: T d C 4 4 K 4 4. (15) 4 ICSV4, London, 3-7 July 17

5 ICSV4, London, 3-7 July C 1 =3Nms/rad C 1 =3Nms/rad Figure 3: Torque fluctuation for one drive shaft without centrifugal endulum absorber. Figure 3 shows the frequency resonse curves for the drive shaft. The abscissa indicates the engine rotation seed, and the ordinate indicates half the eak-to-eak amlitude of the torque fluctuation for one drive shaft Td. Here, the daming coefficient C1 is set to 3 and 3 Nms/rad. The large eak around 1, rm in Fig. 3 is the main resonance for the second mode when C1 is set to 3 Nms/rad. When C1 is set to 3 Nms/rad, the large eak around 1,4 rm is also the main resonance for the second mode. In the condition of C1 = 3 Nms/rad, it is confirmed that the damed natural frequency of the third mode is suressed by the over daming for C Relationshi between engine rotation seed and natural frequency In order to clarify the cause of the unusual vibration that occurs when the centrifugal endulum absorber is attached to a torque converter, the relationshi between the engine rotation seed and the natural frequency was investigated by alying eigenvalue analysis, as shown in Fig. 4. The abscissa indicates the engine rotation seed, and the ordinate indicates the undamed and damed natural frequency of a five-degree-of-freedom system in Fig. 4(a) and 4(b) (C1 = 3 Nms/rad), resectively. The red line indicates the inut forced vibration frequency determined by the engine rotation seed. It can be seen that the natural frequency increases with engine seed and intersects the forced vibration frequency at aroximately 1,5 rm in Fig. 4(a). The main resonance will occur in this range of engine seed. The range of engine seed that the damed natural frequencies intersect the forced vibration frequency becomes wide by daming C1 in Fig. 4(b). 1 1 Natural frequency Hz Damed natural frequency Hz (a) Undamed natural frequency (b) C 1 = 3 Nms/rad Figure 4: Natural frequency of five-degree-of-freedom system. 3.4 Torque fluctuation for one drive shaft with centrifugal endulum absorber Figures 5(a) and 5(b) show the frequency resonse for the torque fluctuations for one drive shaft ICSV4, London, 3-7 July 17 5

6 ICSV4, London, 3-7 July c =Nms/rad c =.1Nms/rad 1 8 c =Nms/rad c =.1Nms/rad (a) Linear analysis (b) Nonlinear analysis Figure 5: Torque fluctuations for one drive shaft with centrifugal endulum absorber. T d when the centrifugal endulum absorber is attached, obtained through linear and nonlinear analyses, resectively. Here, the daming coefficient C1 is set to 3 Nms/rad. From Fig. 5(a), the vibration amlitude of the main resonance of the second mode in Fig. 3 was reduced for a wide range of the engine seed by the centrifugal endulum absorber. However, the large amlitude area around 1,5 rm in Fig. 5(a) was confirmed. The main resonance occurred in this range of engine seed because the damed natural frequency intersects for a wide range of the engine seed around 1,5 rm, as shown in Fig. 4(b). Furthermore, the large eak at around 1,5 rm in the nonlinear analysis results shown in Fig. 5(b) is larger than that in the linear analysis results shown in Fig. 5(a) because of the nonlinearity of the centrifugal endulum absorber. In the condition of c = Nms/rad, the nonlinearity at around 1,5 rm in Fig. 5(b) is larger than that in the condition of c =.1 Nms/rad. This is the generation mechanism for the unusual vibration caused by the centrifugal endulum absorber, as shown in Fig Effect of mass of centrifugal endulum absorber The effect of the mass of the centrifugal endulum absorber on the unusual vibration was investigated through nonlinear analysis. Figure 6 shows the frequency resonse curves for the torque fluctuations for one drive shaft Td for the cases in which the mass of the centrifugal endulum absorber m is.795, 1.59 (standard), and 3.18 kg. Here, the daming coefficient C1 is set to 3 Nms/rad. As can be seen, the unusual vibration at around 1,5 rm is suressed as the mass of the centrifugal endulum absorber increases. This is because the restoring torque of the centrifugal endulum absorber, which is the circumferential force of the centrifugal force, increases as the mass of the centrifugal endulum absorber increases. Figure 7 shows the relationshi between the mass of the centrifugal endulum absorber and the vibration amlitude of the angular dislacement of the centrifugal endulum absorber. The abscissa indicates the engine rotation seed, and the ordinate indicates half the eak-to-eak amlitude of the angular dislacement of the centrifugal endulum absorber obtained in the same calculation for Fig. 6. As shown in Fig. 7, the angular dislacement of the centrifugal endulum absorber decreased as the mass of the centrifugal endulum absorber increased for a range of the engine seed around, rm. Figure 8 shows the relationshi between the ratio of the natural frequency of the centrifugal endulum absorber to the forced vibration frequency and the angular dislacement of the centrifugal endulum absorber for an engine seed of 1,5 rm and an order n of.. The restoring force of the centrifugal endulum absorber has the characteristic of a soft sring, whereby the natural frequency decreases with increasing angular dislacement. Therefore, the decrease in the natural frequency from the order n =. can be small to decrease the angular dislacement of the centrifugal endulum absorber. The unusual vibration around 1,5 rm was suressed, as shown in Fig. 6, as the mass of the centrifugal endulum absorber increased to decrease the angular dislacement. 6 ICSV4, London, 3-7 July 17

7 ICSV4, London, 3-7 July =.795kg =1.59kg =3.18kg Amlitude of endulum rad =.795kg =1.59kg =3.18kg Figure 6: Torque fluctuations of one drive shaft with centrifugal endulum absorber Figure 7: Amlitude of angular dislacement of centrifugal endulum absorber 1.5 Amlitude of endulum rad Natural frequency ratio Figure 8: Relationshi between natural frequency ratio and angular dislacement of centrifugal endulum absorber. 3.6 Effect of order of centrifugal endulum absorber In this section, the effects of the order of the centrifugal endulum absorber on the unusual vibration were investigated. Figures 9(a) and 9(b) show the frequency resonse for the torque fluctuations for one drive shaft Td when the dynamic torque Fd is set to 6 and 1 Nm, resectively, and the order n of the centrifugal endulum absorber is set to 1.9,. (standard), and.1 by changing the osition at which the centrifugal endulum absorber is attached R in Eq. (14). Here, the daming coefficient C1 is set to 3 Nms/rad. As shown in Fig. 9(a), the centrifugal endulum absorber with n =.1 is the most effective at suressing the unusual vibration around 1,5 rm when the dynamic torque Fd is set to 6 Nm. However the centrifugal endulum absorber with n =. is the most effective when Fd is set to 1 Nm, as shown in Fig. 9(b). The angular dislacement of the centrifugal endulum absorber is large enough to decrease its natural frequency when the dynamic inut torque F d is 6 Nm. Therefore, the order n must be set to a slightly higher value n = n=1.9 n=. n= n=1.9 n=. n= (a) F d = 6 Nm (b) F d = 1 Nm Figure 9: Torque fluctuations for one drive shaft with centrifugal endulum absorber. ICSV4, London, 3-7 July 17 7

8 ICSV4, London, 3-7 July 17.1 in advance when taking into consideration the decrease in the natural frequency when Fd is 6 Nm. The decrease in the natural frequency is small when Fd is 1 Nm. Therefore, the centrifugal endulum absorber with n =. is the most effective, as shown in Fig. 9(b). 4. Conclusions The resent aer focused on the theoretically otimum value for the ratio between the natural frequency of the centrifugal endulum vibration absorber and the engine rotation seed, as well as the mass of the centrifugal endulum vibration absorber, for suressing torsional vibration in an automatic transmission owertrain. The generation mechanism and countermeasures for the unusual vibration caused by the centrifugal endulum absorber were also investigated. The following are the main conclusions of the resent study: (1) The main resonance occurred at an engine seed of around 1,5 rm because the natural frequency intersects the engine seed at around 1,5 rm due to the attachment of the centrifugal endulum absorber. Furthermore, the resonance increased because of the nonlinearity of the centrifugal endulum absorber. This is the generation mechanism for the unusual vibration caused by the centrifugal endulum absorber. () The unusual vibration around 1,5 rm was suressed as the mass of the centrifugal endulum absorber increased because the restoring torque of the centrifugal endulum absorber increases and its angular dislacement decreases. (3) There exist otimum values for the order n of the centrifugal endulum absorber that suress the unusual vibration, and these values change with the dynamic inut torque from the engine because the restoring force of the centrifugal endulum absorber has the characteristic of a soft sring, whereby the natural frequency decreases with increasing angular dislacement. ACKNOWLEDGEMENT The resent study was suorted in art by JSPS KAKENHI Grant Number 15K5866. REFERENCES 1 J.P. Den Hartog, Tuned Pendulums as Torsional Vibration Eliminators, Stehen Timoshenko 6th Anniversary Volume, (1938). Ishida, Y., Inoue, T., Kagawa, T., Ueda, M. Nonlinear Analysis of a Torsional Vibration of a Rotor with Centrifugal Pendulum Vibration Absorbers and Its Suression, Transaction of the Jaan Society of Mechanical Engineers, Series C, Volume 71, No.78, August, (5). 3 Pfrabe, M., Woernle, C. Reduction of Periodic Torsional Vibration using Centrifugal Pendulum Vibration Absorbers, Proceedings in Alied Mathematics and Mechanics, Volume 9, December, (9). 4 Yamaura, H., Ono, K., Toyota, K. Otimal Tuning Method for a Swing Reduction of Gondola Lift by Pendulum Tye Dynamic Absorber, Transaction of the Jaan Society of Mechanical Engineers(in Jaanese), Series C, Volume 7, No.7, May, (1993). 5 Mayet, J., Ulbrich, H. Tautochronic Centrifugal Pendulum Absorbers General Design and Analysis, Journal of Sound and Vibration, Volume 333, Setember, (13). 6 Swank, M., Lindemann, P. Dynamic Absorbers for Modern Powertrains, SAE Technical Paer, May, (11). 8 ICSV4, London, 3-7 July 17