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320 CAM DESIGN HANDBOOK

T
and the vector x = [x s x t] . As with the single degree of freedom system, solutions of the
form
x t () = Acos (w n t + u ) f (11.10)
exist to these equations for some constant modal vector u which represents the shape of
the system motion. Solving this equation of motion for the exact natural frequencies of
this system gives:
2
) ]
Km + ( K + K M) [ Km + ( K + K MK K Mm
-
4
2
w = s s t ± s s t s t . (11.11)
n
2 Mm 2 Mm
There are thus two natural frequencies for the two degrees of freedom system, each asso-
ciated with its own modal vector u. The exact shape of the system’s motion at a particu-
lar natural frequency (known as a mode shape) can be determined analytically by solving
T
for the vector u. In this case, solving the equation above for u = [u s u t] gives:
u t = K -w 2 n M . (11.12)
s
u s K s
Like the natural frequencies, the mode shapes are properties of the system. The exact
amplitude of the motion requires knowledge of the initial conditions, but the ratio of the
motions associated with each mass remains fixed.
To put some physical intuition behind this example, consider specific parameter values
of M = 890lbm, m = 100lbm, K s = 97lbf/in, and K t = 1140lbf/in which are reasonable
numbers for passenger car suspensions. Equation (11.11) indicates that for this choice of
parameters, free vibrations will occur at frequencies of about 1Hz and 11Hz. The mode
shape corresponding to the low frequency mode is given by u t/u s = 0.079 and the mode
shape corresponding to the high frequency mode is u t/u s =-110. This indicates that the
low frequency mode consists mainly of motion of the larger mass while the high frequency
mode consists mainly of motion of the smaller mass. The negative sign indicates that the
two motions are 180° out of phase with each other. Physically, these mathematical results
describe two modes of vibration: the sprung mass bouncing on the suspension spring with
the tire remaining relatively rigid (the body mode), and the tire bouncing between the road
and the vehicle body while the body remains approximately stationary (the tire hop mode).
The effects of vibrations at these modes are familiar to most drivers or passengers. The
1Hz mode represents the slow body bounce that can occur with particular spacings of
pavement joints while the body mode produces the “slapping” sound as the tire bounces
on a rough stretch of pavement.
Unlike the single mass-spring-damper system, the two-mass system possesses two
degrees of freedom and two natural frequencies. Loosely speaking, whenever a system
consists of lumped masses and springs, each mass that is connected to other masses or the
environment through a spring generates one degree of freedom and one natural frequency
(for a more formal treatment of the number of variables needed to characterize a general
dynamic system see Layton, 1998 or Karnopp, 2000). While the presence of damping
alters the system somewhat in that oscillations die out instead of continuing indefinitely,
it does not change the basic number of degrees of freedom or natural frequencies. Damping
will slightly change the exact numerical value of the natural frequencies relative to cal-
culations made in the undamped case, but not the underlying intuition.
As demonstrated above, natural frequencies and mode shapes are concepts that can be
formulated in a rigorous mathematical framework. From the standpoint of modeling,
however, the qualitative use of these concepts in judging the complexity and sufficiency of
a given model is often as important (or more important) than exact quantitative calculations.