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This equation says that the change in the angular momentum about the center of mass is equal to the
total moment of all the forces about the center of mass. The equation can also be applied about a fixed
point of rotation, which is not necessarily the center of mass, as in the example of the compound
pendulum given below.
Equations (7.4) and (7.9) are known as the Newton–Euler equations of motion. Without constraints,
they represent six coupled second order differential equations for the position of the center of mass and
for the angular orientation of the rigid body.

Multibody Dynamics

In a serial link robot arm, as shown in Fig. 7.2, we have a set of connected rigid bodies. Each body is
subject to both applied and constraint forces and moments. The dynamical equations of motion involve
the solution of the Newton–Euler equations for each rigid link subject to the geometric or kinematics
a
constraints between each of the bodies as in (7.2). The forces on each body will have applied terms F ,
c
from actuators or external mechanical sources, and internal constraint forces F . When friction is absent,
the work done by these constraint forces is zero. This property can be used to write equations of motion
in terms of scalar energy functions, known as Lagrange’s equations (see below).
Whatever the method used to derive the equation of motions, the dynamical equations of motion for
multibody systems in terms of generalized coordinates {q k (t)} have the form

∑ m ij q ˙˙ j + ∑ ∑ m ijk q ˙ jq ˙ k = Q i (7.10)

The first term on the left involves a generalized symmetric mass matrix m ij = m ji . The second term
includes Coriolis and centripetal acceleration. The right-hand side includes all the force and control
terms. This equation has a quadratic nonlinearity in the generalized velocities. These quadratic terms
usually drop out for rigid body problems with a single axis of rotation. However, the nonlinear inertia
terms generally appear in problems with simultaneous rotation about two or three axes as in multi-link
robot arms (Fig. 7.2), gyroscope problems, and slewing momentum wheels in satellites.
In modern dynamic simulation software, called multibody codes, these equations are automatically
derived and integrated once the user specifies the geometry, forces, and controls. Some of these codes
are called ADAMS, DADS, Working Model, and NEWEUL. However, the designer must use caution as
these codes are sometimes poor at modeling friction and impacts between bodies.

7.5 Simple Dynamic Models

Two simple examples of the application of the angular momentum law are now given. The first is for
rigid body rotation about a single axis and the second has two axes of rotation.

Compound Pendulum

When a body is constrained to a single rotary degree of freedom and is acted on by the force of gravity
as in Fig. 7.5, the equation of motion takes the form, where θ is the angle from the vertical,

IJ (– m 1 L 1 – m 2 L 2 )g sin q = Tt() (7.11)

2 2
where T(t) is the applied torque, I = m 1 L 1 + m 2 L 2 is the moment of inertia (properly called the second
moment of mass). The above equation is nonlinear in the sine function of the angle. In the case of small
motions about θ = 0, the equation becomes a linear differential equation and one can look for solutions
of the form θ = A cosωt, when T(t) = 0. For this case the pendulum exhibits sinusoidal motion with

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