Compliance Effect Modeling based on Quasi-static Analysis for Real-time Multibody Vehicle Dynamics

1. 서 론

실시간 다물체 차량 해석을 위한 준정적법의 컴플라이언스 효과 모델링

정완희^† ․ 하경남^* ․ 김성수^**

Compliance Effect Modeling based on Quasi-static Analysis for Real-time Multibody Vehicle Dynamics

Wan Hee Jeong, Kyoung Nam Ha, Sung-Soo Kim

Key Words:

Compliance effect (컴플라이언스 효과), Quasi-static analysis method (준정적법) HILS 부분시스템 합성법

(Hardware-in-the Loop Simulation), Subsystem synthesis method( )

Abstract

Compliance effect consideration method for real-time multibody vehicle dynamics is proposed using quasi-static analysis. The multibody vehicle model without bush elements is used based on the subsystem synthesis method which provides real-time computation on the multibody vehicle model.

Reaction forces are computed in the suspension subsystem. According to deformation from the quasi-static analysis using reaction forces and bush stiffness, suspension hardpoint locations and suspension linkage orientation are changed. To validate the proposed method, quarter car simulations of McPherson strut and multilink suspension subsystems. Full car bump run simulations are also carried out comparing with the ADAMS vehicle model with bush elements. CPU times are also measured to see the real-time capabilities of the proposed method.

충남대학교 대학원 기계 기계설계 메카트로닉스 공학, ,

†

과 지능로봇시스템 전공, , BK21 메카트로닉스그룹 E-mail : [email protected]

TEL: (042) 821-7782 FAX : (042) 823-4919

충남대학교 대학원 기계 기계설계 메카트로닉스공학과

* , , ,

지능로봇 시스템 전공

충남대학교 메카트로닉스 공학과

**

일반 좌표 분할법 을 이용한 부분

2. (GCP)

시스템 차량 운동 방정식

Fig. 1 Topology of 4 subsystem

Mi

Pi

4 4

0 0 0

1 1

( _i) ( _i)

i= i=

+

∑

= +

∑

M



M Y

 

Q



P



) ˆ (

0 t Y

0( )

T T

qq q yq

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤ t

= −

⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎣ ⎦

q q

M Φ q P M

Φ 0 λ γ 0 Y

 

q

v u

) ˆ (

0 t

v T yq

u T yq v u T

vv vu

T uv uu

Y 0 M M γ P P λ

v u Φ 0

Φ

Φ M M

Φ M M

v u

v

u  





⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

−

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

0

*

*

*v Q Q Yˆ

M  

y q−

=

* 1 1

1 1

( ) ( )

( ) ( )

T T

vv vu vu

T uu

− −

− −

= − −

+

u v u v

u v u v

M M M Φ Φ Φ Φ M

Φ Φ M Φ Φ

* 1

1 1

( )

[ ( ) ]( )

T

q v u

T

vu u u

−

− −

= −

− −

u v

u v u

Q P Φ Φ P

M Φ Φ M Φ γ

T yqv yqu

y ( ¹ )

*

v

u

Φ

Φ M M

Q

= − ⁻

1 1

− −

=

_u

−

_{u v}

u Φ γ Φ Φ v

 

) (

)

( Φ

¹

P M u M v

λ

_u

 

uv uu u

T

− −

=

⁻

* 1

*

*_y^T( ) _y

yy

c M Q M Q

M = − ⁻

* 1

*

*

1 ) ( )

( _y^T _q

yqu y

c P M Φ γ Q M Q

P = − _u⁻ − ⁻

준정적 법을 이용한 컴플라이언스 3.

효과 모델링

조인트 반력 계산 3.1

r

i

f

_i,i-1

n

_i,i-1

n^뮃 i-1,i

Z^뮃_i-1,i Y^뮃_i-1,i

X^뮃_i-1,i f^뮃

i-1,i r_i

Z Y

X n^’_i+1,i

Z^’_i+1 Y^’

i+1 X^’_i+1 f^’_i+1,i Z^’_i

Y^’_i X^’_i O^’_i

O^’’_i,i+1 n^뮃_i,i+1 Z^뮃_i,i+1

Y^뮃_i,i+1

X^뮃_i,i+1 f^뮃_i,i+1 Z^c_i

Y^c_i X^c_i f^'_i,i-1 n^’_i,i-1

body i-1

body i

body i+1 C.G.

Fig. 2 A free body diagram of multibody system

부싱 특성을 고려한 하드 포인트 변경 3.2

Fig. 3 ADAMS bushing model characteristics

"

"

T

i-1,i i-1 i-1,i i,i-1

' T

i-1,i i-1 i-1,i i,i-1

f = (A C ) (-f ) n = (A C ) (-n )

"

i-1,i

f

radial

f

f

Axis

"

"

[( ) ( ) 0]

[0 0 ( ) ]

T

x y

T z

=

=

" "

radial i-1,i i-1,i

"

Axis i-1,i

f f f

f f

₍₁₆₎

"

"

/ / / /

x

y

radial radial radial

Axis Axis Axis

x

y y

f K

f K

K K

θ θ

δ δ θ θ

=

=

=

=

"

i-1,i x

"

i-1,i

(n )

(n )

⁽¹⁷⁾

x, y

θ θ

−1 Δ

i ,i

C

'

_ "

'

_ "

_ _

[ ]

[ ]

x radial x radial

y radial y radial

T radial x radial y Axis

T

x y z

f

f

δ δ

δ δ

δ δ δ

δ δ δ

⎛ ⎞

⎜ ⎟

= ⋅⎜⎝ ⎟⎠

⎛ ⎞

⎜ ⎟

= ⋅⎜⎝ ⎟⎠

=

=

radial

radial

i

f

f

δ

(18)

1

1 0 0 cos 0 sin

0 cos sin 0 1 0

0 sin cos sin 0 cos

y y

x x

x x y y

θ θ

θ θ

θ θ θ θ

−

⎡ ⎤

⎡ ⎤

⎢ ⎥

⎢ ⎥

=⎢ − ⎥⎢ ⎥

⎢ ⎥

⎢ ⎥ −

⎣ ⎦ ⎣ ⎦

Δ i ,i

C

(19)

δ

i

1

1 1 1

T

−

− − −

⋅

⋅

* '

i,i+1 i,i+1 i ,i i

* Δ

i ,i i ,i i ,i

s = s + C δ

C = C C

Start Read Input

suspension (FR) suspension

(RL)

suspension (RR)

suspension (FL)

Chassis acceleration

Acceleration (RL)

Acceleration (RR)

Acceleration (FL)

Acceleration (FR)

Compute joint reaction force

Update Hard point Numerical

integrator

ˆ

i , i

Y q

i i

i i

K , L B , D

1 1 1

− − ⋅ −

* '

i,i +1 i,i+ 1 i

* Δ

i,i i, i i,i

s = s + δ

C = C C

ˆi, i-1 i iˆi ∑ⁿ l ll

l= i+1

R = -L + K Y + K B q

* i,i+1

s

* i,i+1

C

t +Δt

Y

Fig. 4 Data flow diagram of hard point changing

process

차량 모델링 및 시뮬레이션 4.

차량 현가장치 모델링 4.1

Fig. 5

McPherson strut suspension

Fig. 6 Multi-link suspension

차량 현가 시뮬레이션 4.2 1/4

Fig. 7 1/4 car wheel travel simulation result

전 차량 모델의 범프 통과 시뮬레이션 결과 4.3

비교

Fig. 8 Full car simulation result (chassis body

vertical motion)

실시간 성 검증 5.

Table 1 CPU time comparison

6. 결 론

후 기

참고문헌

(1) S.-S. Kim, "A Subsystem Synthesis Method for Efficient Vehicle Multibody Dynamics," Multibody

System Dynamics, Vol 7, pp. 189-207, 2002.

(2) J. H. Lee, “An Efficient Vehicle Dynamics Analysis Method Including Kinematic and Compliance Characteristics of Suspensions", Master’s

Thesis, Kook Min Univ., 1998.

(3) S.S. Kim, J. Y. You, “An Efficient Constraint Force Computation in Multibody Systems", JSME

international journal. Series C, Mechanical systems, machine elements and manufacturing,v.46 no.2,, 2003.

Method

Total simulation Time (sec)

Average simulation time(sec/step size)

Total CPU time to Real

time (%)

Ratio

kinematics 1.5622 0.000195 19.5 1

compliance 1.6241 0.000203 20.3 1.041