Transactions of the JSME (in Japanese)
:
2
*1
Automobile plane-motion model including roll motion: Second report
(Taking into consideration phase difference of cornering forces between front and rear
wheels)
Hideki SAKAI
*1 *1 Kindai Univ. Dept. of Robotics Engineering1 Umenobe, Takaya, Higashihiroshima-shi, Hiroshima 739-2116, Japan
Abstract
The roll motion of the vehicle has an effect on the yaw natural frequency. Because the yaw natural frequency formula considering this effect is the solution to a quartic equation, the expression of the formula is thought to be complex and incomprehensible. Accordingly, a seemingly comprehensible approximate formula was suggested in the former paper. In this approximation process, it was assumed that the cornering forces of the front and rear wheels are generated simultaneously. On the other hand, a later report have indicated that the yaw resonance when the vehicle drives at a certain speed has a cornering forces phase difference of 90 degrees between the front and rear wheels. Therefore this paper formulates the yaw natural frequency and yaw damping ratio assuming the phase difference to be 90 degrees. As results of that, qualitatively, the design variables that dominate the characteristic equation are appropriately included in the yaw natural frequency and yaw damping ratio formulas, and quantitatively, the approximation error is reduced. Consequently, these formulas are believed to be more appropriate than the previously proposed formulas. These formulas indicate that, when the load distribution ratio of the front wheels becomes larger, the yaw natural frequency decreases and the yaw damping ratio increases. In addition, this paper also indicates the scope of these approximate formulas quantitatively.
Key words : Automobile, Roll, Natural frequency, Damping ratio, Roll stiffness, Roll arm length
1999 1985 4 /2[rad] 90[deg] 2015a *1 739-2116 1
E-mail of corresponding author: [email protected]
Received 15 July 2016
/2[rad] 1 2) 3 4 5) 6 7 0 8 2 0 o o-x-y-z 1 x y z
Fig.1 The coordinate system
(A) Top view (B) Rear view
Fig.2 Vehicle model
x y z o f r lr lf r 2Fr 2Ff 2Kr V m,Iz C.G x y o roll axis Kx Cx m, Ix z y roll axis ground line
1 0 2 Cx Ff Fr g H h Ix Iz Kf Kr Kx l lf lr m r V f r 1 Kx 2 y z x 2010 r f F F r mV( ) 2 2 (1) r r f f zr l F lF I 2 2 (2) ) ( 2 ) ' ( x f r x x C K mhg hF F I (3) 2Ff 2Fr ) ( 2 2Ff Kf f (4) r r r K F 2 2 (5) f r
V h r V lf f (6) V h r V lr r (7) (6) (7) 3 2 mhg K Kx 'x (8) g 2001 mg l l K C r f f 2 * (9) mg l l K C f r r 2 * (10) C*f C*r 2Kf 2Kr C*f C*r (1) m g g 2015b Cf Cr C*f C*r g C m l l K C f r f f * 2 (11) g C m l l K C r f r r * 2 (12) Cf Cr Cf 100 Cr 200[m/s 2 ] Cf Cr A 1999 r f C C l A 1 1 1 (13)
Iz m 1 m l l I r f z (14) (2) Iz m 1999 Iz/(lflrm)=0.85 1.05[ ] (14) (1) (12) (14) 0 r 0 0 1 1 2 2 3 2 2 2 2 s K m h V C V C s K m h V C l l V C l l V lC l C l C s V C C s s K C s K I x r f x r f f r r f r r f x x x x (15) s (15) 1( ) 0 2 r 2015c (3) 2Ff=2Fr=0 (15) { } 0 (14) 2 2013a (1) (2) (4) (7) (11) (12) (14) =0 h=0 (15) 2 2 h2m/Kx h 2 m/Kx (6) (7) 3 2 2 (15) h=0,Cx=0,Ix=0 2 5 0 1 2 2 V lC l C l C s V C C s f r r f r (16) (16) n0 0 2 0 1 V lC l C l Cr f r n (17) V C Cf r n 2 0 0 (18)
n0 0 (7) (6) (7) 0 1997 5 ay ay (1) (2) r f y F F ma 2 2 (19) r r f fF lF l 2 2 0 (20) 2Ff 2Fr y r f ma l l F 2 (21) y f r ma l l F 2 (22) (21) (22) (3) 0 y x a K hm (23) y y x a K m h h y 2 (24) y 2 2kf
2 2 2 2 h K l l a K m h ma l l y F k x r y x y r f f (25) 1 k*f f 2Ff 2001 s V k K K s s F f f f f f * 2 2 1 2 ) ( ) ( 2 (26) f= f -(26) 2k*f (25) 2kf 2Kf (11) s V K mC h K s s F x f f f f 2 1 2 ) ( ) ( 2 (27) r s V K mC h K s s F x r r r r 2 1 2 ) ( ) ( 2 (28) r= r (27) (28) (4) (5) -2Kf -2Kr 0 (1) (2) s V K mC h V r l K s V K mC h V r l K s r mV x r r r x f f f 2 2 1 2 1 2 ) ( (29) s V K mC h V r l K l s V K mC h V r l K l rs I x r r r r x f f f f z 2 2 1 2 1 2 (30) (1) (2) s 1 (29) (30) s 3 (29) (30) s 1 (27) (28)
s=0 1 1997 (27) (28) f x f f f K s V K mC h s s F 2 1 2 ) ( ) ( 2 (31) r x r r r s K V K mC h s s F 2 1 2 ) ( ) ( 2 (32) (4) (5) -2Kf -2Kr 0 (1) (2) V r l s V K mC h K V r l s V K mC h K s r mV r x r r f x f f 2 2 1 2 1 2 ) ( (33) V r l s V K mC h K l V r l s V K mC h K l mrs l l r x r r r f x f f f r f 2 2 1 2 1 2 (34) s 1 (34) (14) (33) (34) previous previous
previous previous previous
2 previous previous h2m/Kx 0 1 0 2 2 2 1 n x r f previous K m h V C V C (35) x r r r f f f n previous previous K m h V C l C V C V C l C V C 2 2 2 2 2 0 0 2 1 (36) (35) (36) ( ) rad/s2 h2m/Kx 1/(rad/s 2 ) (35) (36) (35) (36) n0 Cf 2015a =0 h=0 n0 Cf (17)
(37) V V =0 r f lC V 0 (38) V=V =0 n0 =0 (38),(17) l Cr f n 0 (39) 2Ff 2Fr 1 1 2 2 2 s V l s C l l l F F r r f f r (40) 1998 V=V =0 s= =0 2Ff 2Fr j mg l l F mg l l F r f f r 2 2 (41) j 2Fr 2Ff /2[rad] 90[deg] V=V =0 V =0 f=0 ( 2013b) (24) 2Ff 2Fr V =0 2Fr 2Ff /2[rad] 90[deg] V= V =0 V= V =0 9 V 2Fr 2Ff /2[rad] 90[deg] 2Ff 2Fr 2Ff f 2Fr r f r 2.1 5) f 2Ff 0 1 2 V lCr
r 2Fr 5 5.3 2Ff 2Fr 2Ff yf 9 (3) h F Kx f 2 f 0 (42) 2Ff (21) f y x r f a K hm l l (43) y x r f f a K m h l l h y 2 (44) (24) lr/l <1 2Ff 2Fr /2[rad] 2Ff yf (44) 2 2kf 2 2 2 ' 2 h K a K mh l l ma l l y F k x y x r y r f f f (45) kf kf l/lr >1 2 f (26) 2k*f (27) s V K mC h l l K s s F x f r f f f 2 1 2 ) ( ) ( 2 (46) (27) lr/l <1 (28)
s V K mC h l l K s s F x r f r r r 2 1 2 ) ( ) ( 2 (47) (28) lf/l <1 (46) (47) (46) (47) s=0 1 f x f r f f K s V K mC h l l s s F 2 1 2 ) ( ) ( 2 (48) r x r f r r s K V K mC h l l s s F 2 1 2 ) ( ) ( 2 (49) (4) (5) -2Kf -2Kr 0 (1) (2) V r l s V K mC h l l K V r l s V K mC h l l K s r mV r x r f r f x f r f 2 2 1 2 1 2 ) ( (50) V r l s V K mC h l l K l V r l s V K mC h l l K l mrs l l r x r f r r f x f r f f r f 2 2 1 2 1 2 (51) n n n 2 n h 2 m/Kx 0 1 0 2 2 2 1 n x r f f r n K m h V C l l V C l l (52) x r r r f f f f r n n K m h V C l C V C l l V C l C V C l l 2 2 2 2 2 0 0 2 1 (53) (35) (36) (Cf/V) n lr/l (Cr/V) n lf/l n=1,2 (52) (53) ( ) (rad/s) 2 h2m/Kx 1/(rad/s) 2
(52) (53) (52) (53) n Cf/V Cr/V h 2 m/Kx lr/l lf/l 4 4 (15) (52) (53) lr/l lf/l (35) (36) (35) (36) (52) (53) (52) (53) (52) (53) 3 3 H=0.55[m] (2.0[deg/4.98(m/s2)]=0.00701[rad/(m/s2)]) h=0.45[m] Ix Ix H 2 m 2012 1999 Cf=100 Cr=200[m/s 2 ] m 3 (15) previous e 13<V<25[m/s] previous n e (52) (53) (35) (36) 4 5 r n n (50) (51) previous previous (33) (34) (50) (51) (33) (34) (50) (51) (33) (34)
(A) Natural frequency (B) Damping ratio
Fig. 3 Yawing natural frequency and damping ratio as a function of forward speed ( lr/l=0.6, l=2.5[m], Cf=100, Cr=200[(m/s 2
)/rad], m=1500[kg], h=0.45[m] , H=0.55[m], Ix=H2m, Cx=2000[Nm/(rad/s)], Kx=96.3[kNm/rad])
V [m/s]
Exact solution (Planar motion) n previous n0 0 5 10 15 20 25 0 10 20 30 40 50
Exact solution (Planar motion)
previous 0 V [m/s] 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50
Fig. 4 Yaw velocity response in frequency domain ( lr/l=0.6, l=2.5[m], Cf=100, Cr=200[(m/s 2 )/rad], m=1500[kg], h=0.45[m] , H=0.55[m], Ix=H 2 m, Cx=2000[Nm/(rad/s)], Kx=96.3[kNm/rad] , V=30[m/s])
(A) Steering input (B) Yaw velocity response Fig. 5 Yaw velocity response in time domain ( lr/l=0.6, l=2.5[m], Cf=100, Cr=200[(m/s
2 )/rad], m=1500[kg], h=0.45[m] , H=0.55[m], Ix=H 2 m, Cx=2000[Nm/(rad/s)], Kx=96.3[kNm/rad] , V=30[m/s]) Cf Cr,Kx, Ix,Iz e e xC.G/ n0 e/ n0 2015 e xC.G x x G xC I K . (54)
Exact solution (Planar motion) Eq.(50) and (51) Eq.(33) and (34) 2D.O.F model 2 4 6 2 4 6 8 - /2 - /4 -3 /4 -frequency [rad/s] 8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0.1 0 time [s] 0.2 0.4 0.6 0.8 1 1.2 1.4 0.1 0.2 0.3 0.4 0.5 time [s]
Exact solution (Planar motion) Eq.(50) and (51)
Eq.(33) and (34) 2D.O.F model
xC.G/ n0 e/ n0 Kx e Cf Cr,Ix,Iz Kx n/ n0 e/ n0 previous/ n0 6 A Kx previous/0 e/0 / 0 6(B) e 6(B) e/0 xC.G/ n0 1.5 / 0 V (52) xC.G/ n0<1.5 (52) (53) xC.G/ n0<1.5 xC.G/ n0>1.5 xC.G/ n0 3 V=30[m/s] xC.G/ n0=1.85
(A) Yawing natural frequency (B) Yawing damping ratio Fig.6 Prediction accuracy as a function of natural frequencies ratio ( lr/l=0.6, l=2.5[m], Cf=100, Cr=200[(m/s
2 )/rad], m=1500[kg], h=0.45[m], H=0.55[m], Ix=h 2 m, Cx=2000[Nm/(rad/s)] , V=30[m/s]) lr/l (52) (53) lr/l 3 lr/l e e 7 8 lr/l 0.353 0.673 2016 0.3 0.7 lr/l 7 lr/l e (52) (13) Cr>Cf Cr lf/l lr/l (52) lr/l lf/l V V 7 n lr/l (53) lf/l=l-lr/l lr/l x r r f f x r f r n K m h C lC V C lC V V K m h V C V C l l l 2 3 2 3 2 4 2 2 2 1 1 3 2 ) / ( (55) 0 V V V xC.G/ n0 [-] 0 0.5 1 1.5 2 0 1 2 3 4 e/ n0 n/ n0 previous/ n0 xC.G/ n0 [-] 0 0.5 1 1.5 2 0 1 2 3 4 e/0 / 0 previous/ 0
2 2 3 3 3 r f f r n C C C C l V (56) V n lr/l lr/l n V> V (53) lf/l lr/l lr/l n n V V lr/l (56) 3 V V =17.2[m/s] lr/l (52) (53) V 12.7[m/s]
Fig. 7 Yawing natural frequency as a function of the ratio of normal load distribution of front axle (l=2.5[m], Cf=100,
Cr=200[(m/s 2
)/rad], m=1500[kg], h=0.45[m] , H=0.55[m], Ix=H 2
m, Cx=2000[Nm/(rad/s)], Kx=96.3[kNm/rad])
Fig. 8 Yawing damping ratio as a function of the ratio of normal load distribution of front axle (l=2.5[m], Cf=100, Cr=200[(m/s 2 )/rad], m=1500[kg], h=0.45[m] , H=0.55[m], Ix=H 2 m, Cx=2000[Nm/(rad/s)], Kx=96.3[kNm/rad]) 5.1 n Cf/V Cr/V h 2 m/Kx lr/l lf/l 4 xR.A m h I K x x A xR. 2 (57) 20015c (54)
The ratio of normal load distribution of front axle lr/l [-]
V=15[m/s] V=20[m/s] V=30[m/s] V=40[m/s] 1 1.1 1.2 1.3 0.3 0.4 0.5 0.6 0.7 V=50[m/s]
The ratio of normal load distribution of front axle lr/l [-]
0.8 0.9 1 1.1 0.3 0.4 0.5 0.6 0.7 V=15[m/s] V=20[m/s] V=30[m/s] V=40[m/s] V=50[m/s]
2 . 2 . 2 1 1 G xC A xR x K m h (58) xR.A xC.G h 2 m/Kx 1 2 3 4 1.5 5) ( ) -- No.91-10 (2010) pp.13-18 Vol.39 No.3(1985) pp.275-285 (2001), pp.167-168 , C , Vol.65, No.633(1999), pp.1960-1965 = Vol.29 No.2(1998) pp.133-138 , C , Vol.79, No.801(2013a), pp.1681-1692 Vol.44 No.2(2013b) pp.441-448 2015 DVD(2015a) G1800301.
, , Vol.81, No.824(2015b), DOI 10.1299/transjsme.14-00663
Vol.46 No.2(2015c) pp.385-391
, , Vol.82, No.839(2016), DOI: 10.1299/transjsme.16-00019. , C , Vol.63, No.608(1997), pp.1179-1183 , C , Vol.65, No.633(1999), pp.1954-1959 Vol.45 No.3(2012) pp.709-716 Vol.46 No.5(2015) pp.911-917 References
Fujioka, T. and Yamamoto, M., Equations of motion of a vehicle, including roll motion by use of unsprang-mass coordinate systems -in order to release so-called Segel's trap of accompanying roll motion-, Proceedings of 2010 JSAE Annual congress(Spring)(2010), pp.13-18 (in Japanese).
Irie, N. and Shibahata, Y., Improvement of vehicle controllability and stability by rear suspension, Journal of Society of Automotive Engineers of Japan, Vol.39, No.3(1985), pp.275-285 (in Japanese).
Kageyama, K. and Kageyama, I., Jidousha rikigaku (2001), pp.167-168, Rikohtosho (in Japanese).
Kitahama, K. and Sakai, H., Identification method of automobile response to steering input using normalized cornering stiffness, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.65, No. 633(1999), pp.1960-1965 (in Japanese).
Sakai, H., Theoretical study on skid tendency to steering input -Extension of the force and moment method into frequency domain-, Transactions of Society of Automotive Engineers of Japan, Vol.29, No.2 (1998), pp.133 138 (in Japanese). Sakai, H., A physical interpretation of lead time constant of yaw angular velocity in planar motion of automobile,
Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.79, No. 801(2013a), pp.1681-1692 (in Japanese).
Sakai, H., Stability under force control and its index, Transactions of Society of Automotive Engineers of Japan, Vol.44, No.2 (2013b), pp.441 448 (in Japanese).
Sakai, H., Inner model of automobile (Theoretical consideration to yaw resonance phenomenon), Proceedings of Mechanical Engineering Congress, 2015 Japan, DVD (2015a), G1800301 (in Japanese).
Sakai, H., A consideration to natural frequencies under force control, Transactions of the JSME (in Japanese), Vol.81, No. 824(2015b), DOI 10.1299/transjsme.14-00663.
Sakai, H., A Study on change of roll natural frequency, Transactions of Society of Automotive Engineers of Japan, Vol.46, No.2 (2015c), pp.385 391 (in Japanese).
Sakai, H., Fundamental study of cornering limit characteristics during braking and driving, Transactions of the JSME (in Japanese), Vol.82, No. 839(2016), DOI: 10.1299/transjsme.16-00019.
Sakai, H. and Satoh, Y., Automobile model taking into account of a tire dynamic property, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.63, No. 608(1997), pp.1179-1183 (in Japanese).
Sakai, H. and Satoh, Y., Automobile plane-motion model including roll motion, Transactions of the Japan Society of Mechanical Engineers, Series C, Vol.65, No. 633(1999), pp.1954-1959 (in Japanese).
Sakai, H. and Yamamoto, Y., Damping control to enhance transient cornering feel - Control to synchronize pitching with rolling utilized Kalman filter -, Transactions of Society of Automotive Engineers of Japan, Vol.43, No.3 (2012), pp.709 716 (in Japanese).
Yamamoto, M. and Fujioka, T., Effects of vehicle parameters and their interactions on coupled dynamics in roll and plane motions, Transactions of Society of Automotive Engineers of Japan, Vol.46, No.5 (2015), pp.911-917 (in Japanese).
![Fig. 3 Yawing natural frequency and damping ratio as a function of forward speed ( l r /l=0.6, l=2.5[m], C f =100, C r =200[(m/s 2 )/rad], m=1500[kg], h=0.45[m] , H=0.55[m], I x =H 2 m, C x =2000[Nm/(rad/s)], K x =96.3[kNm/rad])](https://thumb-ap.123doks.com/thumbv2/123deta/9929244.1386882/12.892.128.782.676.914/yawing-natural-frequency-damping-ratio-function-forward-speed.webp)
![Fig. 4 Yaw velocity response in frequency domain ( l r /l=0.6, l=2.5[m], C f =100, C r =200[(m/s 2 )/rad], m=1500[kg], h=0.45[m] , H=0.55[m], I x =H 2 m, C x =2000[Nm/(rad/s)], K x =96.3[kNm/rad] , V=30[m/s])](https://thumb-ap.123doks.com/thumbv2/123deta/9929244.1386882/13.892.275.621.133.486/fig-yaw-velocity-response-frequency-domain-rad-nm.webp)
![Fig. 8 Yawing damping ratio as a function of the ratio of normal load distribution of front axle (l=2.5[m], C f =100, C r =200[(m/s 2 )/rad], m=1500[kg], h=0.45[m] , H=0.55[m], I x =H 2 m, C x =2000[Nm/(rad/s)], K x =96.3[kNm/rad]) 5.1 n C f /V C r /V h 2](https://thumb-ap.123doks.com/thumbv2/123deta/9929244.1386882/15.892.274.631.620.830/fig-yawing-damping-ratio-function-ratio-normal-distribution.webp)
