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316 CAM DESIGN HANDBOOK

11.1 INTRODUCTION

“All models are wrong. Some models are useful.”—George Box
“Everything should be made as simple as possible, but not simpler.”—Albert Einstein

Engineering modeling is the art of reducing a physical system to a mathematical
description in order to analyze system performance. Unlike many familiar problems in
mathematics and engineering, the problem of producing a mathematical model for a given
system does not possess a unique solution. Many models of various degrees of complex-
ity and detail can be derived for any physical system, and the selection of the right model
for a given purpose can at times seem a bit overwhelming. The concepts of usefulness and
simplicity outlined in the quotes above prove to be helpful guides when faced with such
a challenge. As the quotes above suggest, the goal is rarely to create the most detailed or
realistic model conceivable, but instead to produce a representation that captures the details
of interest in a simple, concise manner. Transforming a system from physical reality to a
set of equations invariably involves some simplification in the description and some uncer-
tainty in the measurement or calculation of model parameters. Since it is therefore impos-
sible to reproduce the physical system perfectly in the mathematical world, the proper
evaluation of a model’s quality is really an evaluation of its usefulness and not its detail.
The operative question thus becomes: can the model explain the behavior it was devel-
oped to explain to the degree of accuracy desired?
Given a choice among models that pass this test of usefulness, there are often advan-
tages to using the simplest possible representation. These advantages stem not from com-
putational efficiency (after all, many commercially available computer codes allow the
engineer to build multibody dynamic models of cam-follower systems that are more
complex than those discussed here), but rather from the engineering insight provided by
simple models. Often, simple models enable the engineer to develop analytical results that
save time in the design process. Instead of running a battery of simulations on a complex
model to see how the frequency of an oscillation in the system varies with the length of
one element, a simple model could provide an algebraic relationship indicating that the
frequency changes with the square root of the length. Simple models also enable the engi-
neer to apply intuition from very simple systems (for instance, the idealized mass-spring-
damper) to more complex systems (such as an automotive valve train). The danger is
that the model will be too simple and some relevant behavior will be ignored. In the design
of engineering systems, therefore, different models with varying complexity each have a
role to play. Complex models offer the comfort of additional completeness and a
closer match to experiment while simple models provide intuition and a means of sepa-
rating primary and secondary effects. Knowing which model to use when is indeed a chal-
lenge, but one that can be greatly reduced with the help of a few concepts discussed in later
sections.
This chapter provides the basic tools necessary to produce dynamic models of cam-
follower systems (and many other mechanical systems), simplify the models through some
physical arguments, and roughly assess whether or not such models are useful or too
simple. The models developed may be used to analyze dynamic phenomena (Rothbart,
1956) such as cam hop (where the cam and follower actually separate), the dynamic forces
that occur in each component, and the noise generated by the system while in operation.
The particular modeling approach taken is known as “lumped parameter” modeling since
the physical characteristics of components are combined, or lumped, to produce simpler
representations. All components of the cam-follower system possess some mass or inertia,
exhibit some compliance or spring-like behavior, and dissipate some energy from the
system. For modeling purposes, it is convenient to separate these effects into a series of