In a previous article, we looked at the trampoline vibration mode of a pickleball paddle and attempted to model it like a tennis racquet, with a rigid outer frame and a compliant center. Since the calculated equivalent stiffness of the paddle was significantly greater than the measured static stiffness, we inferred that the dynamic loads from the paddle face might not react through the paddle edge, but instead through the paddle center of mass. In this respect, pickleball paddles might behave more like baseball bats than tennis racquets. While this does not affect paddle dynamics, it might indicate that current methods to measure, certify, or adjust paddle power based on the trampoline effect may inadequate. In this article, we will compare pickleball paddles with baseball bats to determine differences and commonalities in their dynamic response to impact loads.
Baseball Bat Dynamics
Conventional knowledge suggests that when striking a baseball, a solid wooden bat behaves like a diving board. That is, upon contact with the ball, the barrel of the bat bends backward, then at the end of contact, the barrel of the bat springs forward, propelling the ball back towards the field. In numerous studies, however, this scenario has been proven to be false. High speed photography confirms that when contact with the ball is made further down the barrel (Figure 1a), the bat bends backwards (away from the pitcher), and when contact is made closer to the handle (Figure 1b), the bat bends forward (towards the pitcher). Furthermore, when the ball rebounds from the bat, the bat is still bent.
Figure 1. Baseball Bat Bending at Contact
We can explain this in simple terms using principles of momentum exchange. When contact is made between the bat and the ball, the kinetic energy of the ball is converted into potential energy that is stored in bending of the bat and deformation (crushing) of the ball. Since the natural frequency of the bat bending mode is low (i.e., its period of oscillation is long), the bat does not have time to spring back before the ball has rebounded from the bat. Therefore, the potential energy in the bat is not converted back into kinetic energy of the ball. Similarly, since the ball still appears to be squashed when leaving the bat, the energy used to deform the ball is also not recovered. These losses reduce the apparent bat/ball coefficient of restitution (BBCoR).
The ”No Hands” Home Run
An interesting thing happened on May 27, 2012, when Todd Frazier of the Cincinnati Reds hit a “no hands” home run, where he essentially let go of the bat at the instant it made contact with the ball. Researchers have studied this extensively and determined that when the ball strikes the bat, a transverse acoustic wave is created that travels from the impact point to the handle. By the time this wave reflects back to the impact point on the bat, the ball has long since departed. From this, they concluded that “nothing on the handle end of the bat could possibly influence what happens to the ball: not the size or shape or weight or even the hands.” A player’s grip on the bat therefore has no influence on the ball-bat collision!
In our assessment of the “no hands” home run, we depart a bit from the baseball researchers. We contend that this was possible only because contact with the ball was made at the bat center of percussion, or “sweet spot”. This allowed the bat to follow its intended trajectory without rotation. For a right-handed hitter, if the ball is hit further down the barrel from the sweet spot, the bat would rotate clockwise (when viewed from above). Similarly, if the ball is hit closer to the handle, the bat would rotate counterclockwise. Energy must be expended to rotate the bat on impact, and therefore, this energy is not recovered by the ball on rebound. A batter will therefore hit the ball harder if contact is made at the bat sweet spot.
Hollow vs Solid Bats
Why are hollow aluminum and composite bats “hotter” (i.e., have a higher CoR) than wooden bats? According to the technical literature, hollow bats exhibit a “trampoline effect” caused by the so-called “hoop” vibration modes which do not occur in wooden bats. Figure 3 illustrates how the hoop vibration mode can provide the ball with greater velocity on rebound. In this example we see that the ball has lost contact with the bat before the hoop vibration dissipates. What enables this is that the hoop vibration mode occurs at a high enough frequency that the vibration energy is converted back into kinetic energy of the ball.
Figure 3. Bat Hoop Mode Illustration
Another simpler explanation may be related to the stiffness of the bats. Tests of hollow aluminum and composite bats confirm that they are stiffer than wooden bats. Therefore, on contact with the ball, less energy is stored in the bending deformation of the bat, which is not recoverable, and more energy is stored in the higher frequency bat hoop modes, which are recovered on rebound. Furthermore, because of its greater stiffness, the bat bending vibration modes may occur at a sufficiently high frequency to enable some of this energy to be converted back into kinetic energy of the ball. Simply stated, hollow aluminum and composite bats are “hotter” than wooden bats because they return more energy back to the ball on rebound.
Paddle Performance
How does this affect paddle performance? In previous articles, we attempted to characterize paddle performance based on static tests for face stiffness, throat stiffness, and core stiffness. Since we now suspect that paddles behave more like baseball bats than tennis racquets, the constraints (or boundary conditions) for these tests are not representative of how paddles would behave in real life. Specifically, with the load path going from the handle to the paddle center of mass (or center of percussion), the paddle is best modeled with free boundary conditions with the inertial reference at the paddle center of mass. This can only be assessed through the dynamic impulse tests we performed in previous articles.
When striking a ball, you are merely accelerating the paddle mass along a trajectory to contact the ball with a certain velocity at impact. This requires that you contact the ball at a certain location on the paddle face with the paddle face oriented in a certain direction. What are the desired contact locations on the paddle face?
For blocks and volleys at the kitchen line, the paddle face should be oriented towards the net and undergo pure translation with zero rotation while striking the ball. The optimum contact location will therefore be at the paddle center of mass, or cg location, where the force of the ball will cause a pure translation of the paddle. In a previous article, we showed how to determine the location of the paddle cg by balancing the paddle about a lateral axis on the paddle face.
For serves, groundstrokes and overhead smashes, the paddle is rotating about a pivot point which we determined in a previous article to coincide with the center of your palm. Therefore, striking the ball at the center of percussion (the so-called “sweet spot”) will result in zero translation at the pivot point, thereby maximizing energy that can be transmitted back to the ball. We calculated the sweet spot location for over one hundred paddles and have determined that it is on average about 2.50” towards the paddle tip from the paddle cg location.
Future Articles
In this article, we postulated that the dynamic loads from striking a ball are reacted through the paddle center of mass rather than through the edge of the paddle face. Therefore, a pickleball paddle behaves more like a baseball bat rather than a tennis racquet. When striking a ball, numerous paddle vibration modes are excited. These vibration modes can either dissipate energy from, or supply energy to the ball on return. In a future article we will look at the vibration modes of several pickleball paddles to illustrate how energy is transferred from the paddle to the ball and to develop criteria for classifying paddles based on their frequencies of their vibration modes.
