4569846498

Modelling and analysis of suspension systems 207
We can now apply the triangle law of vector addition to equate the expression
for {V DG } 1 in equation (4.114) with {V D } 1 in equation (4.112) and
{V G } 1 in equation (4.109):
{V DG } 1 {V D } 1 {V G } 1 (4.115)
⎡ 314z
2164y⎤
⎢
⎥
⎢194z
2164x⎥
⎢
⎣
194y31
⎥
4x
⎦
(4.116)
Rearranging (4.116) yields the first three equations required to solve the
analysis:
Equation 1 3346f 3 216 4y 31 4z 0 (4.117)
Equation 2 2070f 2 1840f 3 216 4x 19 4z 0 (4.118)
Equation 3 63 250f 2 54 970f 3 31 4x 19 4y 0 (4.119)
We can now proceed to set up the next set of three equations working from
point H to point D and using the triangle law of vector addition:
{V DH } 1 {V D } 1 {V H } 1 (4.120)
Determine an expression for the velocity {V H } 1 at point H:
{V H } 1 {V HJ } 1 { 5 } 1 {R HJ } 1 (4.121)
⎡V
⎢
⎢
V
⎣⎢
V
⎤
⎥
⎥
mm/s (4.122)
⎦⎥
We already have an expression for {V D } 1 in equation (4.112) and can determine
an expression for the relative velocity {V DH } 1 of point D relative to
point H using
{V DH } 1 { 4 } 1 {R DH } 1 (4.123)
⎡V
⎢
⎢
V
⎣⎢
V
Hx
Hy
Hz
DHx
DHy
DHz
⎡3346 f3⎤
⎡ 0 ⎤
⎢
1840 f
⎥
3
⎢
2070 f
⎥
⎢ ⎥ ⎢ 2
⎥
mm/s
⎣⎢
54 970 f3⎦⎥
⎣⎢
63 250 f2
⎦⎥
⎤ ⎡ 0 5z
5y
⎤ ⎡ 0 ⎤ ⎡228 8
⎥ ⎢
⎥
⎥
⎢
⎢
⎥ ⎢
5z
0 5x⎥
228
⎢ ⎥ ⎢
8
5x
⎦⎥
⎢
⎥
⎣
5y
5x
0
⎦ ⎣⎢
8
⎦⎥
⎣⎢
228
5x
5z
5y
⎤ ⎡ 0 4z
4y
⎤ ⎡144
⎤ ⎡ 44 51
⎥ ⎢
⎥ ⎢ ⎥ ⎢
⎥
⎢ 4z
0 4x⎥
44
⎢ ⎥ ⎢ 144 51
⎦⎥
⎢
⎥
⎣
y x 0
⎦ ⎣⎢
51⎦⎥
⎢
4 4
⎣
144 44
4z
4y
4z
4x
4y
4x
⎤
⎥
⎥ mm/s
⎥
⎦
(4.124)
We can now apply the triangle law of vector addition to equate the expression
for {V DH } 1 in equation (4.124) with {V D } 1 in equation (4.112) and
{V H } 1 in equation (4.122):
{V DH } 1 {V D } 1 {V H } 1 (4.125)
⎡ 44 z 51
⎢
⎢ 144 z 51
⎢
⎣
144 y 44
4 4y
4 4x
4 4x
⎤ ⎡3346
f
⎥
⎢
⎥ ⎢
1840 f
⎥
⎦ ⎣⎢
54 970 f
3
3
3
⎤ ⎡228 z 8
⎥
⎢
⎥ ⎢
85x
⎦⎥
⎣⎢
2285x
5 5y
⎤
⎥
⎥
mm/s
⎦⎥
(4.126)