4569846498

Modelling and analysis of suspension systems 193
If a solution is assumed of the form
x i X i e t (4.76)
then
˙ẋ X e x
i
2 t
2
i
i
(4.77)
thus
m t 2 x t x t (k t k b ) x b (k b ) (4.78)
m b 2 x b x t (k b ) x b (k b ) (4.79)
which may be rearranged into the familiar eigenvalue problem:
2
⎡k k m
k
x
t b t b
⎤ t
⎢
0
(4.80)
2 ⎥ ⎡
⎣k k m
x
b b b ⎦ ⎣ ⎢
⎤
⎥
b ⎦
in which the determinant of the matrix can be used to find the eigensolution
when set to zero:
(k t k b m t 2 )(k b m b 2 ) k 2 b 0 (4.81)
(m t m b ) 4 (k b m t (k t k b )m b ) 2 k t k b k b 2 k b 2 0 (4.82)
which may be recognized as a quadratic in 2 and solved in the normal
manner:
2
b ± b 4ac
2a
2
(4.83)
a m t m b (4.84)
b k b m t (k t k b )m b (4.85)
c k t k b k b 2 k b 2 k t k b (4.86)
The calculated roots using this method are
2 : 56.3075
4920.87
: 7.50383i
70.1489i
frequencies: 1.19427 Hz
11.1645 Hz
which can be seen to agree exactly with the MSC.ADAMS model. However,
the differences between the exact method and the approximate method are
small – less than 0.15%. Thus the approximate method is a ‘good enough’
check for this system. This is generically true for quarter vehicle models,
where the second mode of vibration is typically an order of magnitude higher
than the first. However, for particularly stiff suspensions or compliant tyres
as may be used on circuit cars, the suitability of the approximate method
breaks down and therefore it should be used with some care.