VI. BALL VIBRATIONS
An estimate of the losses due to vibrations induced in the
tennis ball was obtained by gluing a small (4mm34mm!
piezoelectric ceramic element, of thickness 0.3 mm, onto a
tennis ball and measuring the induced voltage by means of
light wires soldered onto the element. Results are shown in
Fig. 6 for a case where the ball was dropped from a height of
10 cm onto the 50 mm diam piezo. When the small piezo
element is located near the bottom of the ball, the force wave
form observed is similar to that observed with the large piezo element, but there is a delay of about 0.4 ms between the two
wave forms. The pulse decreases in amplitude and changes
shape as the location of the element is rotated away from the
bottom of the ball toward the top of the ball. The top of the
ball is only slightly effected by the compression and expansion
of the bottom of the ball, but there is a small-amplitude
oscillation at ;700 Hz. The oscillations are global in extent
and persist for about 2 ms after the ball rebounds. The 0.4 ms
delay observed between the large and small piezo signals is
roughly consistent with the fact that the initial impulse
propagates around the ball to give a period of oscillation of
;1.5 ms. The delay also coincides with the transition from a
high to a low stiffness state, indicating that the ball surface
may deform into a bending mode when the impulse propagates
to a point about 30° from the bottom of the ball. Since
the ball is hollow, it bends more easily than a solid ball, and
it is much easier to bend rubber than to compress it.
The amplitude of the oscillation shown in Fig. 6~c! is relatively
small when measured in terms of the displacement of
the ball surface. The induced voltage in the piezo is proportional
to the displacement of the surface, but it is also proportional
to the square of the frequency. Given that the
stored energy in the ball is proportional to the compression
squared and that the piezo output is proportional to the applied
force and hence to the second derivative of its displacement,
it is clear that the 700 Hz signal represents a relatively
small-amplitude, low-energy oscillation. An absolute value
for the energy stored in the oscillation was not obtained,
since the piezo was not calibrated and since it responds to
bending as well as to a force perpendicular to the surface.
Even at high impact speeds, ball vibrations do not store a
large amount of energy after the rebound. High-speed video
film of a ball impacting with concrete at 100 mph has recently
been obtained by the International Tennis Federation.
The film was recorded at 18 000 frames/s and shows the ball
oscillations clearly. Several frames from this video are
shown schematically in Fig. 7. The video image is consistent
with the results in Fig. 6 and shows that when the ball compresses
to about half its original diameter, the surface opposite
the contact surface oscillates with an amplitude of about
1 cm during the impact and at lower amplitude for several
ms after the ball rebounds.