Page 157 - The Mechatronics Handbook

9.6 Rigid Body Multidimensional Dynamics

The modeling of bodies in mechanical systems presumes adoption of a “rigid body” that can involve
rotation as well as translation, and in this case the dynamic properties are more complex than those for
a point mass. In earlier sections of this chapter, a simple rigid body has already been introduced, and it
is especially useful for a large class of problems with rotation about a single fixed axis.
In the rigid body, the distance between any two elements of mass within a body is a constant. In some
cases, it is convenient to consider a continuous distribution of mass while in others a system of discrete
mass particles rigidly fixed together helps conceptualize the problem. In the latter, the rigid body prop-
erties can be found by summing over all the discrete particles, while in the continuous mass concept an
integral formulation is used. Either way, basic concepts can be formulated and relations derived for use
in rigid body dynamic analysis. Finally, the modeling in most engineering systems is restricted to classical
Newtonian mechanics, where the linear velocity–momentum relation holds (so energy and coenergy are
equal).

Kinematics of a Rigid Body

In this section, a brief overview is given of three-dimensional motion calculations for a rigid body. The
focus here is to present methods for analyzing rotation of a rigid body about a fixed axis and methods
for analyzing relative motion of a rigid body using translating and rotating axes. These concepts introduce
the basis for understanding more complex formulations. While vector descriptions (denoted using an
a
arrow over the symbol, ) are useful for understanding basic problems, more complex multibody systems
usually adopt a matrix formulation. The presentation here is brief and included for reference. A more
extensive discussion and examples can be found in introductory dynamics textbooks (e.g., [23]), where
a separate discussion is usually given on the special case of plane motion.
Rotation of a Body About a Fixed Point
Basic concepts are introduced here in relation to rotation of a rigid body about a fixed point. This basic
motion specifies that any point on the body lies on the surface of a sphere with a radius centered at the
fixed point. The body can be said to have spherical motion.
Euler’s Theorem. Euler’s theorem states that any displacement of a body in spherical motion can be
expressed as a rotation about a line that passes through the center of the spherical motion. This axis can
be referred to as the orientational axis of rotation [26]. For example, two rotations about different axes
passing through a fixed point of rotation are equivalent to a single resultant rotation about an axis passing
through that point.
Finite Rotations. If the rotations used in Euler’s theorem are finite, the order of application is impor-
tant because finite rotations do not obey the law of vector addition.
Infinitesimal Rotations. Infinitesimally small rotations can be added vectorially in any manner, and
these are generally considered when defining rigid body motions.
θ
Angular Velocity. A body subjected to rotation d about a fixed point will have an angular velocity
θ
θ
w defined by the time derivative d /dt, in a direction collinear with d . If the body is subjected to two
component angular motions that define w 1 and w 2 , then the body has a resultant angular velocity, w =
w 1 + w 2 .
Angular Acceleration. A body’s angular acceleration is found from the time derivative of the angular
α
ω
velocity, = , and in general the acceleration is not collinear with velocity.
ω
ω
v
r
Motion of Points in the Body. Given , the velocity of a point on the body is = × , where r
is a position vector to the point as measured relative to the fixed point of rotation. The acceleration of
ω
ω
α
a point on the body is then, = α × + × ( × ). r
r
Relating Vector Time Derivatives in Coordinate Systems
It is often the case that we need to determine the time rate of change of a vector such as A in Fig. 9.29
relative to different coordinate systems. Specifically, it may be easier to determine A in x a , y a , z a , but we
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