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CAM SYSTEM MODELING 321

For simple systems like this suspension model and many cam-follower systems, the vibra-
tional modes can often be analyzed intuitively and this information used to simplify models.

11.2.2 Model Sufficiency and Model Reduction
Natural frequencies provide some insight into the applicability or sufficiency of the model.
For mechanical systems, if the system is forced at the natural frequency, resonance (the
maximum amplification of the forcing input) will occur. Intuitively, since the system
naturally tends to vibrate at certain frequencies and in certain modes, motion at these
frequencies is, in a sense, easier to initiate and maintain. Resonance is an important feature
to capture in a mathematical model since it is not uncommon for the ratio of system motion
to input magnitude to increase by an order of magnitude or more at resonance if the system
is lightly damped. In a cam follower system, therefore, if the cam rotates at speeds up to
3000rpm (or 50Hz), the dynamic model of the follower system must, as a bare minimum,
include all natural frequencies up to 50Hz to capture the basic motion of the follower with
reasonable fidelity. Similarly, if experiments show a resonance at 1000Hz but the highest
natural frequency predicted by the model is 100Hz, the model requires additional refine-
ment. It is “too simple” to predict the observed behavior.
Continuing with the example of the suspension in Fig. 11.2, this two-mass model is
sufficient for simulating the vehicle response to road excitations in the range of around 0
to 15Hz. If a test driver complains about road vibrations being amplified around 50 or 60
Hz, the model is too simple, since the mass responsible for this vibration has been lumped
into either the body or the tire in the two-mass model. Conversely, if the goal of the mod-
eling process is to capture the fundamental response of the vehicle to vibrations in the
range of 1Hz, information about the resonance at 11Hz is extraneous and the model is
more complex than needed. Of course the more complex model could still be used. Instead
of retaining the additional complexity in the model, however, it is often beneficial to look
for a simpler model that retains the important characteristics of the original model. This
process is known as model reduction. In this case, the objective of the reduction is to create
a model with a single natural frequency that predicts the resonance associated with the
low-frequency mode (the sprung mass bouncing on the tire). The task then is to choose
an equivalent mass and stiffness for a simplified model with a single natural frequency
that resembles Fig. 11.3a. It cannot be emphasized enough that since any model will be
an approximation of the real system, there is no unique or correct way to simplify. Sim-
plification is again part of the art of engineering modeling.
The idea of natural frequencies can provide some guidance, however. Since natural fre-
quencies are associated with masses and stiffnesses (more formally, with mass and stiff-
ness matrices) and the goal is to remove the higher natural frequencies, the model can be
simplified by ignoring masses or replacing certain spring elements with rigid links (thus
assuming infinite stiffness). These assumptions remove degrees of freedom from the
system and reduce the number of natural frequencies and the complexity of the resulting
equations of motion. It is also possible, in the manner suggested by Chen (1982), to lump
masses and stiffnesses together without explicitly assuming any of these to be zero or infi-
nite. Three possible ways to apply these concepts to the suspension system are:

Approach 1: Assume Infinite Tire Stiffness. Assume that the stiffer spring (the tire)
is rigid and use the mass of the car and the stiffness of the suspension spring to determine
natural frequency. In this model, the mass of the tire and the degree of freedom associ-
ated with the tire are neglected since the rigid spring prevents the tire from moving rela-
tive to the road. This simplification is illustrated in Fig. 11.3b. The natural frequency in
this case is given by