2010 vds-3d. Professor Kyongsu Yi. Vehicle Dynamics and Control Laboratory
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1 3D Dnamcs 2010 vds-3d Professor Kongsu Y 2010 VDCL Vehcle Dnamcs and Control Laborator 1
2 Knetcs of Rgd Bodes n Three Dmensons ma G Translatonal Dnamc Equaton of Rgd Bod (Newton Equaton F ma G Rotatonal Dnamc Equaton of Rgd Bod (Euler Equaton M G H G 2
3 Relaton between Postons n the Two Frame: Fxed Frame and Rotatng Frame unt vectors, ˆ, ˆj t ˆ( cos( t sn( t ˆ( j t sn( t cos( t Where, ĵ î t OXYZ Fxed Frame Ox Rotatng t Frame about OXYZ tme dervatve of unt vectors dˆ sn( cos( ˆ t t j dt dj ˆ cos( sn( ˆ t t dt Angular Veoct of Ox w.r.t. OXYZ tme dervatve of unt vectors (General Form Relaton between Postons n the Two Frame xˆ cos( t sn( t X ˆ j sn( t cos( t Y d dt ˆ ˆ 0 ˆ ˆ ˆ j ˆ 0 ˆ ˆ j j j ˆ ˆ ˆ ˆ k 0 k k k 3
4 Rate of Change of a partcle n the Two Frame: Fxed Frame and Rotatng Frame Poston Vector of a partcle, Q Q Q ˆ Q ˆ j Q kˆ x ĵj î Q Rate of Change of wth respect to Ox Q Q ˆ ˆ ˆ x Q jq k Ox Where, OXYZ Ox ˆk Fxed Frame Rotatng Frame about OXYZ Angular Veoct of Ox w.r.t. OXYZ Rate of Change of Q wth respect to OXYZ ˆ ˆ ˆ ˆ ˆ ˆ d dj dk Q Q OXYZ x Q jq kqx Q Q dt dt dt Q Ox Q Q Q Ox 4
5 Three-dmensonal Moton of a Partcle Relatve to a Rotatng Frame. V P' V P / F r Veloct Vector of Partcle P w.r.t. OXYZ V r r r Where, P OXYZ Ox V P V P' V P / F V V P/ F P' = absolute veloct vector of partcle P = veloct vector of pont P' of movng frame concdng wth P = veloct vector of P relatve to movng frame Where, OXYZ Ox Fxed Frame Rotatng Frame about OXYZ Angular Veoct of Ox w.r.t. OXYZ 5
6 Three-dmensonal Moton of a Partcle Relatve to a Rotatng Frame. Corols Acceleraton a c r 2 Ox V P/ F r r Ox Absolute Acceleraton of Partcle P w.r.t. OXYZ Where, Therefore, d a P VP r r Ox dt d dt d dt r r r Ox Ox Ox r rr OXYZ r r r Ox a r r r r r P Ox Ox Ox r r r 2 r Ox Ox a P/ F a a a P/ F P' c a P ' a c Where, a c = corols acceleraton 6
7 General Moton n Rgd Bod. Absolute Poston Vector of Poston P r r r P A P / A Absolute Veloct Vector of Poston P V r r r P P A P/ A V A V P/ A VA rp/ A r P / A r PA / constant 0 Absolute Acceleraton Vector of Poston P Where, OXYZ Ox Fxed Frame Rotatng Frame about OXYZ Angular Veoct of Ox w.r.t. OXYZ Angular Acceleraton of Ox w.r.t. OXYZ a PVPVAVP/ A a r ( r A PA / PA / 7
8 Rgd Bod Translatonal Dnamcs Translatonal Dnamc Equaton of Rgd Bod (Newton Equaton F m a G Acceleraton Vector of the Rgd Bod n the Global Frame ax v x wx vx a G a ( V Ax V Ax v w v a v w v v ( v w v w x v ( vxwvwx v ( vwxvxw ( v x v w v w ( v vxwvwx ( vvwxvxw 8
9 Rgd Bod Rotatonal Dnamcs v r Rotatonal Dnamc Equaton of Rgd d Bod (Euler Equaton M n H G Where, H r v m r r dm G 1 : Angular momentum of the Rgd bod wth respect to the frame of fxed orentaton. Angular momentum about x-axs 2 2 H w w x w xw dm x x x w dm w x dm w x dm x w I w I w I x x x x There, H x Iw x x Ixw Ixw HG H Ixwx Iw Iw H Ixwx Iw Iw 9
10 Rgd Bod Rotatonal Dnamcs Rotatonal Dnamc Equaton of Rgd Bod (Euler Equaton M H H G Angular Momentum (Smmetrc moment of nerta Hx Ixwx HG H Iw H Iw Ix Ix I 0 Rgd Bod Rotatonal Dnamcs H ( H H G G x G ( I x x xx wx Ixw I I I ww x I w Iw I ( Ix I ww x I w Iw I ( I Ix ww x 10
11 Major Course Contents Part 1: Lateral Vehcle Dnamcs 11VehcleDnamcModel Planar Model 1.3 Tre Models 1.4 Bccle Model Bank angle/crosswnd 1.5 Understeer/oversteer 1.6 Dnamc model nterms of error wrt road 1.7 lane keepng model 1.8 Lateral stablt Control Part 2: Longtudnal Vehcle Dnamcs 1 Longtudnal Dnamc Model 2 Engne model 3 Transmsson 4 Brake Part 3: Vehcle Control Sstems
12 Part.1 Lateral Vehcle Dnamcs 1. Vehcle Dnamc Model 2. Planar Model 3. Tre Models 4. Bccle Model 5. Understeer/oversteer 6. Dnamc model n terms of error w.r.t. road 7. lane keepng model 8. Vehcle Stablt Control 12
13 1. Vehcle Dnamc Model Vehcle State - Roll: -Ptch: -Yaw: - X: x -Y: - Z: Rear Vew Left Vew l f lr Top Vew 13
14 1. Vehcle Dnamc Model Translatonal Dnamc Equaton of Vehcle (Newton Equaton F ma G Rotatonal Dnamc Equaton of Vehcle (Euler Equaton M G H G obal Y Glo 14
15 1. Vehcle Dnamc Model Translatonal Dnamc Equaton of Vehcle (Newton Equaton F m a G ax vx vx ( vx v v ag a ( V CG.. VCG.. v v ( v vx v a ( v v v v vx Rotatonal Dnamc Equaton of Vehcle (Euler Equaton M G H G I x ( x x Ix Ix I I I H G I ( H G x HG I I I ( Ix I I I I I ( IIx 15
16 2. 3DOF Planar Moton Model 16
17 2. 3DOF Planar Moton Model Assumpton of 3DOF Vehcle Planar Moton Model T T 1 Ignore Roll, Ptch Moton ( Ignore Suspenson Dnamcs ( F constant v 0 Translatonal Dnamc Equaton of Vehcle (Newton Equaton t F m a G ax vx 0 vx ( vx v a G a ( V CG.. VCG.. v 0 v ( v vx a Rotatonal Dnamc Equaton of Vehcle (Euler Equaton M G H G Ixx H G I ( H G x H G I I I I 17
18 2. 3DOF Planar Moton Model -x-axs Moton Dnamc Equaton F ma m( v v x x x 2 Ftx cos( f Ft sn( f Ftx3 Ftx4 1 f F F t1 F t2 - -axs Moton Dnamc Equaton F ma m( v v x 2 F tx sn( f F t cos( f F t 3 F t 4 1 F Ftx1 v tx2 x - aw-axs Moton Dnamc Equaton M H I 2 4 lf [ Ftx sn( f Ft cos( f ] lr Ft 1 3 twftx 1Ftx2 cos( f twftx3ftx4 F t3 F t4 twft1ft2sn( f F tx3 F tx4 4 Where, f Front Steerng Angle v 2t w l f l r 18
19 3. Tre Model 3.1 Pacejka Tre Model Slp Angle Lateral Tre Model Slp Rato Longtudnal Tre Model Combned Tre Model Self Algnng Moment 3.2 Dugoff s Tre Model 19
20 3. Tre Model Ftx Ft M t Longtudnal Tre Force Lateral Tre Force Self Algnng Moment Slp Angle Wheel Angular Speed M t F tx F t
21 3. Tre Model Tre Deformaton Longtudnal Tre Force F tx x x Ft Lateral Tre Force F t Ft Ftx Ft xx F t < Longtudnall tre deformaton > < Laterall tre deformaton > REF: Rea N. Jaar, Vehcle Dnamcs: Theor and Applcaton, pp101 ~ 105, Sprnger, 2008 F t
22 3. Tre Model Longtudnall Tre Deformaton Longtudnal Tre Force F tx x Ft r V t Longtudnall Tre Deformaton x r V t xr Vt REF: Rajesh Rajaman, Vehcle Dnamcs and Control, pp391 ~ 394, Sprnger, 2006
23 3. Tre Model Laterall Tre Deformaton Shear Stress Dstrbuton < Bottom vew of a laterall deflected and turnng tre > Lateral Tre Force and Self Algnng Moment x Drec cton of Wheel Travel Lateral Tre Force F t da Ftx F t M t Self Algnng Moment M t Ft ax ax Pneumatc Tral
24 3. Tre Model 3.1 Pacejka Tre Model Slp Angle Lateral Tre Model Slp Rato Longtudnal Tre Model Combned Tre Model Self Algnng Moment 3.2 Dugoff s Tre Model 24
25 3.1.1 Slp Angle The angle between the orentaton of the tre and the orentaton of the Wheel Where, Tre Slp Angle at - th Wheel Steerng Angle at - th Wheel tx t Angle between V and V at -th Wheel 1 f V t V tx tan V V t tx v l v l tan( tan( f f 1 2 vx tw vx tw v lr v lr tan( 3 tan( 4 v t v t x w x w 25
26 3.1.2 Lateral Tre Model x Lateral Tre Force at the -th Wheel F D C B S 1 t sn( tan ( v Where, E E S B S 1 (1 ( h tan ( ( h B 5200 Ft Ft 5200 B 0.22 C D F F t t E 1.6 S 0 S 0 h v M t F t < Lateral Tre Force > Normal Tre Force at the -th Wheel ml ml F K r r g F K r r g r r t1 t1 1o 1 t2 t 2 2o 2 2 lf lr 2 lf lr ml f ml f Ft3 Kt3r3o r3 g Ft 4 Kt 4r4o r4 g 2 l l 2 l l f r f r Where, r Orgnal Radus of the Tre r Effectve Rollng Radus of the - th Wheel K o t Tre Stffness at - th Wheel 26
27 3.1.2 Lateral Tre Model Slp Angle versus Lateral Tre Force Curve 27
28 3.1.3 Slp Rato Durng Brakng Durng Tracton r V cos( cos( t V t r Vt cos( r Where, Angular Veloct of the - th Wheel r Tre Radus of the - th Wheel Tre Slp Angle at - th Wheel V t cos( r Sde Vew f V t V t V tx V1 ( v l ( v t 2 2 t1 f x w V ( v l ( v t 2 2 t2 f x w V ( v l ( v t 2 2 t3 r x w V ( v l ( v t 2 2 t4 r x w Top Vew 28
29 3.1.4 Longtudnal Tre Model Longtudnal Tre Force at the -th Wheel Where, F D C B S 1 tx x sn( x tan ( xx vx E E S B S x 1 x (1 x( hx tan ( x( hx Bx 0 Durng Tracton ( Ft 1940 Ft 1940 Ft 1940 Bx 22 Cx 1.35 Dx E 3.6 S 0 S 0 x hx vx 0 Durng Brakng ( Ft 1940 Ft 1940 Ft 1940 Bx 22 Cx 1.35 Dx E 0.1 S 0 S 0 x hx vx 29
30 3.1.4 Longtudnal Tre Model Slp Rato versus Longtudnal Tre Force Curve 30
31 3.1.5 Combned Tre Model Pacejka Tre Model 1 Longtudnal Tre Model: F (, tx Ftx 0 Ft 2 Lateral Tre Model: F F 0 (, F t t t There s no correcton between Longtudnal and Lateral Tre Model Normaled Slp m * 1 Normaled Slp Rato: * 2 Normaled Slp Angle: Correcton Factor: m ( ( * * 2 * 2 Where, f ( 0 m 0.1 elsewhere Ft 650 m Combned Tre Model based on Pacejka Tre Model F F (, F * tx * * tx0 m t * * Ft F * t0( m, Ft
32 3.1.5 Combned Tre Model Combned Tre Force Longtudnal Tre Force Lateral aea Tre Force oce F F (, F F F (, F * * * * tx * tx0 m t t * t0 m t Longtudn nal force [N] alpha=0[deg] alpha=5[deg] alpha=10[deg] alpha=15[deg] alpha=20[deg] Slp rato Lateral fo orce [N] Slp rato=0 Slp rato=0.005 Slp rato=0.01 Slp rato=0.05 Slp rato= Slp angle [deg]
33 3.1.6 Self Algnng Moment Self Algnng Moment at the -th Wheel Where, M D C B S 1 t sn( tan ( v E 1 (1 E( Sh tan ( B( Sh B x Drecton of Wheel Travel B C D C Ft exp F t F D F F t t t Ft M t BCD B C < Self Algnng Moment > D g E F F 4.04 S t t h 0 S 0 v
34 3.1.6 Self Algnng Moment Slp Angle versus Self Algnng Moment Curve F = 2500 N F = 3500 N F = 4500 N Self Algnng Mo oment [Nm ] Slp Angle [deg]
35 3.2 Dugoff s Tre Model Longtudnal Tre Force Ftx Cx f( 1 1 Lateral Tre Force tan( Ft C f ( 1 F t Ft 2 tan( Where, 1 2 C C 2 x F tx f (2 f ( 1 1 f ( 1 C C x Longtudnal Tre Stffness Lateral Tre Stffness Tre/Road Frcton Coeffcent 35
36 3.2 Dugoff s Tre Model Longtudnal Tre Force Lateral Tre Force Ftx C tan( x f( Ft C f( 1 1 Where, 0 Where, F = 2500 N F = 3500 N F = 4500 N 3000 F = 2500 N F = 3500 N F = 4500 N [N] Longtudn nal Tre Force Lateral Tre Force [N] Slp Rato Slp Angle [deg] 36
37 3.2 Dugoff s Tre Model Longtudnal Tre Force Lateral Tre Force Ftx C tan( x f( Ft C f( 1 1 F N Where, Ft 4500N Where, 4500 t nal Tre Force [N] Slp Angle = 0 deg Slp Angle = 3 deg Slp Angle = 6 deg Slp Angle = 9 deg l Tre Force [N N] Slp Rato = 0 Slp Rato = 0.04 Slp Rato = 0.08 Slp Rato = 0.12 Longtud Lateral Slp Rato Slp Angle [deg] 37
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