Question 1.4.6: Consider the compound pendulum of Figure 1.19 pinned to rota......

A compound pendulum results from a simple pendulum configuration (Examples 1.1.1 and 1.4.2) if there is a significant mass distribution along its length. In the figure, G is the center of mass, O is the pivot point, and θ(t) is the angular displacement of the centerline of the pendulum of mass m and moment of inertia J measured about the z-axis at point O. Point C is the center of percussion, which is defined as the distance q_{0} along the centerline such that a simple pendulum (a massless rod pivoted at zero with mass m at its tip, as in Example 1.4.2) of radius q_{0} has the same period. Hence

q_{0} = \frac{J}{mr}

where r is the distance from the pivot point to the center of mass. Note that the pivot point O and the center of percussion C can be interchanged to produce a pendulum with the same frequency. The radius of gyration, k_{0}, is the radius of a ring that has the same moment of inertia as the rigid body does. The radius of gyration and center of percussion are related by

q_{0}r = k²_{0}

Consider the equation of motion of the compound pendulum. Taking moments about its pivot point O yields

\sum{M_{0} } = J \ddot{θ}(t) = -m\mathtt{g}r \sin  θ(t)

For small θ(t) this nonlinear equation becomes (sin ~ θ)

J \ddot{θ}(t)  +  m\mathtt{g}r θ(t) = 0

The natural frequency of oscillation becomes

w_{n} = \sqrt{\frac{m\mathtt{g}r}{J}}

This frequency can be expressed in terms of the center of percussion as

w_{n} = \sqrt{\frac{\mathtt{g}}{q_{0}}}

which is just the frequency of a simple pendulum of length q_{0}. This can be seen by examining the forces acting on the simple (massless rod) pendulum of Examples 1.1.1, 1.4.2, and Figure 1.20(a) or recalling the result obtained in these examples.

Summing moments about O yields

ml² \ddot{θ} = -m\mathtt{g}l \sin  θ

or after approximating sin θ with θ,

\ddot{θ}  +  \frac{\mathtt{g}}{l}  θ = 0

This yields the simple pendulum frequency of w_{n} = \sqrt{\mathtt{g}/l}, which is equivalent to that obtained previously for the compound pendulum using l = q_{0}.

Next, consider the uniformly shaped compound pendulum of Figure 1.20(b) of length l. Here it is desired to calculate the center of percussion and radius of gyration.

The mass moment of inertia about point O is J, so that summing moments about O yields

J \ddot{θ} = -m\mathtt{g} \frac{1}{2}  \sin  θ

since the mass is assumed to be evenly distributed and the center of mass is at r = l/2. The moment of inertia for a slender rod about O is J = \frac{1}{3} ml²; hence, the equation of motion is

\frac{ml²}{3}  \ddot{θ} + m\mathtt{g}  \frac{1}{2}  θ = 0

where sin θ has again been approximated by θ, assuming small motion. This becomes

\ddot{θ}  +  \frac{3}{2}  \frac{\mathtt{g}}{l}  θ = 0

so that the natural frequency is

w_{n} = \sqrt{\frac{3}{2}  \frac{\mathtt{g}}{l}}

The center of percussion becomes

q_{0} = \frac{J}{mr} = \frac{2}{3}l

and the radius of gyration becomes

k_{0} = \sqrt{q_{0}r} = \frac{1}{\sqrt{3}}

These positions are marked on Figure 1.20(b).

The center of percussion and pivot point play a significant role in designing an automobile. The center of percussion is the point on an object where it may be struck (impacted) producing forces that cancel, causing no motion at the point of support. The axle of the front wheels of an automobile is considered as the pivot point of a compound pendulum parallel to the road. If the back wheels hit a bump, the frequency of oscillation of the center of percussion will annoy passengers. Hence automobiles are designed such that the center of percussion falls over the axle and suspension system, away from passengers.

The concept of center of percussion is used in many swinging, or pendulum-like, situations. This notion is sometimes used to define the “sweet spot” in a tennis racket or baseball bat and defines where the ball should be hit. If the hammer is shaped so that the impact point is at the center of percussion (i.e., the hammer’s head), then ideally no force is felt if it is held at the “end” of the pendulum.