4569846498

208 Multibody Systems Approach to Vehicle Dynamics
Rearranging (4.126) yields the next three equations required to solve the
analysis:
Equation 4 3346f 3 51 4y 44 4z 8 5y 228 5z 0 (4.127)
Equation 5 1840f 3 51 4x 144 4z 8 5x 0 (4.128)
Equation 6 54 970f 3 44 4x 144 4y 228 5x 0 (4.129)
We can now proceed to set up the next set of three equations working from
point G to point P and using the triangle law of vector addition:
{V PG } 1 {V P } 1 {V G } 1 (4.130)
Determine an expression for the relative velocity {V PG } 1 of point P relative
to point G:
{V PG } 1 { 4 } 1 {R PG } 1 (4.131)
⎡V
⎢
⎢
V
⎣⎢
V
PGx
PGy
PGz
⎤
⎥
⎥ mm/s
⎥
⎦
(4.132)
We already have an expression for {V G } 1 in equation (4.109) and we can
define the vector {V P } 1 in terms of the known vertical velocity component
V Pz and the unknown components V Px and V Py :
{ V }
P
⎤ ⎡ 0
⎥ ⎢
⎥
⎢
⎦⎥
⎢
⎣
1
⎡ VPx
⎤
⎢
V
⎥
⎢ Py mm/s
⎥
⎣⎢
3366⎦⎥
4z
4y
4z
4x
0
4y
4x
(4.133)
We can now apply the triangle law of vector addition to equate the expression
for {V PG } 1 in equation (4.132) with {V P } 1 in equation (4.133) and
{V G } 1 in equation (4.109):
{V PG } 1 {V P } 1 {V G } 1 (4.134)
⎡5 84z
1764y⎤
⎡ VPx
⎤ ⎡ 0
⎢
⎥
7 4 1764
⎢
z
x V
⎥
⎢
⎢
⎥ Py 2070 f
⎢ ⎥ ⎢
⎢
⎣
7 4y
58
⎥
4x
⎦ ⎣⎢
3366⎦⎥
⎣⎢
63 250 f
0
⎤ ⎡ 7
⎤ ⎡58 176
⎥ ⎢ ⎥ ⎢
⎥
58
⎢ ⎥ ⎢ 7 176
⎥
⎦ ⎣⎢
176⎦⎥
⎢
⎣
7 4y
584x
(4.135)
Rearranging (4.135) yields the next set of three equations required to solve
the analysis:
Equation 7 176 4y 58 4z V Px 0 (4.136a)
Equation 8 2070f 2 176 4x 7 4z V Py 0 (4.136b)
Equation 9 63 250f 2 58 4x 7 4y 3366 (4.136c)
This leaves us with nine equations and 10 unknowns. The last equation is
obtained by constraining the rotation of the tie rod (Body 5) to prevent spin
2
2
⎤
⎥
⎥
mm/s
⎦⎥
4z
4y
4z
4x