5.3 Work and Energy Methods for Rigid Bodies

In many situations, applying the equations of motion to predict the dynamical response of a rigid body is inconvenient. An alternate work-energy principle, akin to the one encountered in the case of particle dynamics is especially handy when describing motion resulting from the cumulative effects of forces acting through distances. This is simpler still when the active forces are conservative; permitting velocity changes to be determined simply by analyzing energy states at the beginning and end of the motion interval.

Figure 5.3.1: The work-energy principle permits analyses of the effects of forces acting over intervals on the motion of a rigid body.

An analogous work and energy principle applicable to rigid bodies follows simply by applying the version encountered previously to each of the particles comprising a rigid body. In particular, for a rigid body subjected to a system of forces and couples,

where U is the work done by all forces and couples that act on the body, and T is the total kinetic energy of the body. As we already know how to determine the work done by translating forces, all that remains to be done before we can put this principle to use, is to determine expressions for the work done by moments, and the kinetic energy possessed by rigid bodies.
 

5.3.1 Work Done and Power Expended by Forces and Couples

Figure 5.3.2: A force F applied at point A of a body as it moves from state 1 to state 2.

The expression for the work done by an individual force acting on a rigid body during an interval of interest is the same as that seen in the case of particles previously,

where the integral is performed along the path traced by the point of application of the force as it moves from  to  during the interval. Further, the power expended by this force is simply,

where  is the velocity of the point of application of the force.

Figure 5.3.3: Work done by a couple  as the body moves from state 1 to state 2.

To determine the work done by a couple acting on the rigid body, consider the effect of the couple  shown. During an interval , the body rotates through , with line  moving to position . As before, consider the motion in two parts, first a translation to intermediate position  and then a rotation through  about . Observe that the work done by the forces during the translation cancels so that the work done is simple  due to the rotational component of the motion. During a finite rotation of the body from  to  then,

where the work done is positive if the couple acts in the same sense as the angle change. The power developed by the couple at a given instant is simply the rate at which it is doing work,

where  is the angular velocity of the body. If the senses of M and  are the same, the power is positive and energy is supplied to the body.

Using these expressions, the total work done on a rigid body by a system of forces and couples is simply the sum of the contributions due to individual forces and couples,

Observe that as individual forces are typically associated with distinct points of application, the work done by each force must be computed independently, by integration along the path traced by the point of application of the force.
 

5.3.2 Kinetic Energy of a Rigid Body

As before, to derive expressions for the total kinetic energy of a rigid body, we will first determine an expression for the total kinetic energy of a system of particles. This will then be specialized to rigid bodies by introducing rigid body kinematics.

Figure 5.3.4: A rigid body as a system particles with center of mass C.

Consider a rigid body comprised of particles  moving with velocity . The total kinetic energy of the system is then,

Next we decompose particle velocities relative to the center of mass,

where  is the velocity of the particle as seen by a translating observer at C. Substituting, we obtain,

Proceeding to tinker with the final term, observe by definition of the center of mass,

Therefore, the final term reduces to,

Yielding an expression for the total kinetic energy of a system of particles,

All that remains is to introduce the kinematics of rigid body motion , yielding

where  is the moment of inertia about the center of mass. Thus, the kinetic energy of a rigid body in plane motion is comprised of separate contributions to the total kinetic energy resulting from the translational velocity  of the mass center and the angular velocity  of the body about the mass center. Further, the form of the two terms reflects relationships between translational and rotational motion referred to previously: both terms are multiples of the squares of velocities with the appropriate inertia:
 

Observe that when translating rigid bodies () are considered, the expression for the kinetic energy reduces to the familiar form,

which is hardly surprising as all particles of the body translate with the center of mass.

Figure 5.3.5: A rigid body rotating about fixed axis O.

In the case of fixed axis rotation, on the other hand, as  where  is the distance from the axis of rotation to the center of mass, we have the reduced form,

where  is the mass moment of inertia about O.
 

5.3.3 Work-Energy Principle

Figure 5.3.6: A rigid body moving from state 1 to state 2 under the action of external forces and moments.

Consolidating our results, we equate the total work done on a rigid body during an interval of interest to the change in its kinetic energy during the interval,

where,

or, for fixed axis rotation,

Observe that as in the case of particles, work contributions arising from conservative forces can be determined using the appropriate potential energy.

The total power developed at an instant by a system of forces and moments is then,

where  is the velocity of the point of application of each active force.

Again, as problems encountered are often complex, we recommend the following procedure:

• Kinematics: Identify the class of motion, and solve for any linear and angular velocities that can be determined from the kinematical information provided. If the motion is constrained determine relationships between the linear and angular quantities.
• Free body diagram: Draw a free body diagram and determine the active forces.
• Work-energy principle: Compute the work done by all active forces and couples acting on the body. Observe that the work computation is very simple if the system considered is conservative.