Page 263 - Dynamics of Mechanical Systems

0593_C08_fm Page 244 Monday, May 6, 2002 2:45 PM

244 Dynamics of Mechanical Systems

applied to P, the acceleration of P in an inertial reference frame is related to F and m
through the expression:

F = m a (8.3.1)

As noted in the preceding section, an inertial reference frame (or a Newtonian reference
frame) is defined as a reference frame in which Newton’s laws are valid. This is a kind
of circular definition that has led dynamics theoreticians and philosophers to contemplate
and debate the existence of inertial or Newtonian reference frames. Intuitively, an inertial
reference frame is a reference frame that is at rest relative to the universe (or relative to
the “fixed stars”). Alternatively, an inertial reference frame is an axes system fixed in a
rigid body having infinite mass. For the study of most mechanical systems of practical
importance, the Earth may be considered to be an approximate inertial reference frame.
The analytical procedures of d’Alembert’s principle may be developed from Eq. (8.3.1)
by introducing the concept of an inertia force (see Section 6.9). Specifically, if a particle P
with mass m has an acceleration a in an inertial reference frame R then the inertia force
*
F on P in R is defined as:

D
*
F =−m a (8.3.2)
Observe that the negative sign in this definition means that the inertia force will always
be directed opposite to the acceleration. A familiar illustration of an inertia force is the
radial thrust of a small object attached to a string and spun in a circle. Another illustration
is the rearward thrust felt by an occupant of an automobile accelerating from rest.
By comparing Eqs. (8.3.1) and (8.3.2), the applied and inertia forces exerted on P are
seen to be related by the simple expression:

+
*
FF = 0 (8.3.3)
Equation (8.3.3) is an analytical expression of d’Alembert’s principle. Simply stated, the
sum of the applied and inertia forces on a particle is zero.
When d’Alembert’s principle is extended to a set of particles, or to rigid bodies, or to a
system of particles and rigid bodies, the principle may be stated simply: the combined
system of applied and inertia forces acting on a mechanical system is a zero system (see
Section 6.4). When sets of particles, rigid bodies, or systems are considered, interactive
forces, exerted between particles of the system on one another, cancel or “balance out”
due to the law of action and reaction (see Reference 8.31).
Applied forces, which are generally gravity, contact, or electromagnetic forces, are some-
times called active forces. In that context, inertia forces are at times called passive forces.
d’Alembert’s principle has analytical and computational advantages not enjoyed by
Newton’s laws. Specifically, with d’Alembert’s principle, dynamical systems may be stud-
ied as though they are static systems. This means, for example, that free-body diagrams
may be used to aid in the analysis. In such diagrams, inertia forces are simply included
along with the applied forces. We will illustrate the use of d’Alembert’s principle, with
the accompanying free-body diagrams, in the next several sections.