4569846498

206 Multibody Systems Approach to Vehicle Dynamics
Thus this analysis can proceed if we can develop 10 equations to solve the
10 unknowns:
f 2 , f 3 , x 4 , y 4 , z 4 , x 5 , y 5 , z 5 , V Px , V Py
Working through the problem it will be seen that a strategy can be
developed using the triangle law of vector addition to generate sets of equations.
As a starting point we will develop a set of three equations using
{V DG } 1 {V D } 1 {V G } 1 (4.106)
In this form the equation (4.106) does not introduce any of the 10
unknowns listed above. It will therefore be necessary to initially define
{V DG } 1 , {V D } 1 and {V G } 1 in terms of the angular velocity vectors that contain
unknowns requiring solution. The first step in the analysis can therefore
proceed as follows.
Determine an expression for the velocity {V G } 1 at point G:
{V G } 1 {V GE } 1 { 2 } 1 {R GE } 1 (4.107)
⎡230⎤
{ 2} 1 f2{ R } 1f
⎢
EF 2 0
⎥
rad/s
⎢ ⎥
⎣⎢
0 ⎦⎥
(4.108)
⎡V
⎢
⎢
V
⎣⎢
V
Gx
Gy
Gz
⎤
⎥
⎥
f
⎦⎥
⎡0 0 0 ⎤ ⎡115
⎤ ⎡ 0 ⎤
⎢
0 0 230
⎥ ⎢
275
⎥
⎢
f
⎥
⎢
⎥ ⎢ ⎥ ⎢
2070 ⎥
mm/s
⎣⎢
0 230 0 ⎦⎥
⎣⎢
9
⎦⎥
⎣⎢
63 250 f2
⎦⎥
2 2
(4.109)
Determine an expression for the velocity {V D } 1 at point D:
{V D } 1 {V DA } 1 { 3 } 1 {R DA } 1 (4.110)
{ } { }
f R f
3 1 3 AB 1 3
⎡230⎤
⎢
0
⎥
rad/s
⎢ ⎥
⎣⎢
14 ⎦⎥
(4.111)
⎡V
⎢
⎢
V
⎣⎢
V
Dx
Dy
Dz
⎤
⎥
⎥
f
⎦⎥
3
⎡ 0 14 0 ⎤ ⎡115
⎤ ⎡3346
f3
⎤
⎢
14 0 230
⎥ ⎢
239
⎥
⎢
f
⎥
⎢
⎥ ⎢ ⎥ ⎢
1840 3
⎥
mm/s
⎣⎢
0 230 0 ⎦⎥
⎣⎢
15
⎦⎥
⎣⎢
54 970 f3
⎦⎥
(4.112)
Determine an expression for the relative velocity {V DG } 1 of point D relative
to point G:
{V DG } 1 { 4 } 1 {R DG } 1 (4.113)
⎡V
⎢
⎢
V
⎣⎢
V
DGx
DGy
DGz
⎤
⎥
⎥
⎦⎥
⎡ 0
⎢
⎢
⎢
⎣
4z
4y
4z
4x
0
4y
4x
0
⎤ ⎡19⎤
⎡ 31 216
⎥ ⎢ ⎥ ⎢
⎥ 31
⎢ ⎥ ⎢19 216
⎥
⎦ ⎣⎢
216 ⎦⎥
⎢
⎣
19 31
4z
4y
4z
4x
4y
4x
⎤
⎥
⎥ mm/s
⎥
⎦
(4.114)