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9

Principles of Impulse and Momentum

9.1 Introduction
Impact is a common phenomenon in machine dynamics. Collisions occur repeatedly
between machine elements such as gear teeth, cams and followers, clutch plates, brake
pads, chain links, and gripper jaws. The principles of impulse and momentum are ideally
suited for the study of such collisions. These collisions produce impact phenomena where
large forces occur over a short time interval.
As with d’Alembert’s principle, the principles of impulse and momentum may be
developed directly from Newton’s laws. For impact phenomena, these principles of
impulse and momentum are more convenient to use than either Newton’s laws or
d’Alembert’s principle in that accelerations do not need to be computed. In this chapter,
we will examine these principles and their applications.
From a strictly theoretical perspective, impact phenomena and the resulting contact
stresses and deformations are difficult topics. From a global perspective, however, the
overall behavior of colliding particles or bodies is relatively easy to study using the
principles of impulse and momentum, and the results agree relatively well with observed
phenomena. Therefore, our emphasis will be placed upon understanding these principles
and their underlying assumptions from a global perspective.

9.2 Impulse

When bodies collide the impact time is usually very short — one tenth of a second or less.
The forces, however, may be quite large. Such forces are conveniently represented by
*
impulses. Specifically, if a force F is applied between bodies over a time t , the linear
impulse of F is defined as:

t *
D
I = ∫ Fdt (9.2.1)
O

Typically, the magnitude of the impact force will have the form shown in Figure 9.2.1.
It is often triangular. The magnitude of the impulse is then the average of the impact force
magnitude multiplied by the impact time t . The direction of the impulse is the same as
*
that of the force. The dimensions of an impulse are thus force–time.

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