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THB11 9/19/03 7:33 PM Page 317

CAM SYSTEM MODELING 317

idealized elements: point masses or lumped inertias, massless springs that provide force
as some function of displacement, and massless dampers that produce force as a function
of some velocity in the system. Once the properties of the individual components have
been identified, the individual masses, springs, and dampers can be lumped together to
produce simple models of the complete system. This process of lumping in many cases
requires making judgments about which elements of the system are dominant and which
can be ignored. For instance, elements with very little compliance can often be assumed
rigid and elements with very small mass can often be assumed massless in order to sim-
plify the modeling process.
The chapter begins by presenting some fundamental concepts from vibrations and
system dynamics that prove useful in model reduction and assessment. The next three sec-
tions treat each of the basic modeling elements—mass or inertia elements, spring elements,
and dampers—individually and discuss the rules for obtaining equivalent elements when
multiple elements are lumped or combined. The final section of the chapter applies all of
the concepts of modeling, simplification, and evaluation to produce a simple model of an
automotive valve-gear system.

11.2 DYNAMIC SYSTEM MODELING
AND REDUCTION

This section introduces a few concepts from the fields of vibrations and system dynamics
that provide useful vocabulary and intuition for developing and reducing models of
dynamic systems. Obviously, it is impossible to cover the entire field of vibrations in
a few pages, so only the aspects most relevant to later discussions in this chapter are
included. For a more complete treatment of these concepts and the field of vibrations in
general, useful resources are Ungar (1985) and the texts by De Silva (2000) and Meirovich
(2001). Although not discussed in this chapter, the concepts of lumped parameter model-
ing and model reduction applied here to mechanical systems can also be applied to systems
that include electrical, thermal, and fluidic elements. More general treatments of this mate-
rial that include all of these energy domains can be found in Cannon (1967), Layton (1998),
and Karnopp et al. (2000).

11.2.1 Natural Frequencies and Modes of Vibration
One concept that proves useful in discussing any system modeled as a combination of
masses, springs, and dampers is the natural frequency associated with a given mode of
vibration. Figure 11.1 illustrates the simplest such system, consisting of a single mass and
a single spring with the displacement x measured from the equilibrium position of the
spring. Invoking Newton’s second law
F = ˙˙ (11.1)
mx
and assuming that the spring provides a force proportional to its displacement from
equilibrium,
F =- kx (11.2)

a force balance of this system gives
mx ˙˙ =- kx (11.3)