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12

Generalized Dynamics: Kane’s Equations

and Lagrange’s Equations

12.1 Introduction
In this chapter, we apply the procedures developed in Chapter 11 to obtain governing
dynamical equations of motion. We will consider two approaches: the use of Kane’s
equations and the use of Lagrange’s equations. We will discover that these approaches
are related. Indeed, we will use Kane’s equations to develop Lagrange’s equations. We
will illustrate and compare the use of Kane’s and Lagrange’s equations by obtaining
equations of motion for the various examples considered in Chapter 11. In addition, we
will also consider the triple-bar pendulum and the N-bar pendulum (an N-bar pendulum
may be considered as a model of a chain or cable).
Using either Kane’s equations or Lagrange’s equations to obtain equations of motion
has distinct advantages over using Newton’s laws, over the method of impulse and
momentum, and over the work–energy principle. These advantages include the automatic
elimination of nonworking internal constraint forces from the analysis. In addition, with
Kane’s equations and Lagrange’s equations, the exact same number of equations are
obtained as there are degrees of freedom. Moreover, these equations are obtained without
needing to make insightful or clever choices of summation directions or of reference points
to take moments about. Lagrange’s equations have the additional advantage of not requir-
ing computation of accelerations.
The advantages of Kane’s and Lagrange’s equations, however, do not come without
corresponding disadvantages. Indeed, if the constraint forces are eliminated, then they
are not determined and remain unknown in the analysis. On occasion, such forces may
be of interest, particularly in machine design. Also, with Lagrange’s equations we cannot
study nonholonomic systems (at least, not without a modification of the equations).
Nevertheless, on balance, the advantages of Kane’s and Lagrange’s equations outweigh
the disadvantages for a large class of systems of importance in machine dynamics.

12.2 Kane’s Equations
Kane’s equations provide an elegant formulation of the dynamical equations of motion.
Kane’s equations simply state that the sum of the generalized forces (both applied and
inertia forces), for each generalized coordinate, is zero. That is, for a mechanical system

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