In a previous article, we evaluated the paddle / ball coefficient of restitution test (PBCoR) using principles of conservation of momentum and energy. This evaluation found that after accounting for the elastic deformation of the ball and the rigid-body motions of the ball and paddle, only 2.3% of the kinetic energy remains for deformation of the paddle! With such a small amount of energy going into the paddle, it might be difficult to differentiate one paddle from another due to potentially large variances in ball deformation and/or ball/paddle rigid-body motion. Is there a better way to measure the PBCoR (or paddle power or reactivity)?
We first need to determine the paddle design characteristics that affect its PBCoR. How much of a paddle’s reactivity is affected by its shape, weight, thickness, materials, or stiffness? If you are a paddle manufacturer, what paddle parameters can you “tweak” to gain the maximum PBCoR without exceeding the USAP allowable limits? Similarly, if you are a consumer, what should you look for in a paddle to maximize power? Is there an easier way to predict the paddle reactivity without having to run the PBCoR test?
To answer these questions, we looked at two paddles, the Ronbus Ripple v1, which was de-listed by USAP for failing the USAP PBCoR test, and the modified Ronbus Ripple v2, which passed with a PBCoR of 0.43. Through this evaluation, we developed a dynamic analysis technique that is independent of the ball, that may indicate a paddle’s reactivity, power, or PBCoR. To understand why this technique works, we first need to look at the technical basis for CoR in the sport of baseball.
Baseball / Bat Coefficient of Restitution (BBCoR)
Baseball has traditionally used wooden bats, but these bats were prone to breakage. Starting in the 1960’s, technological advancements in materials and manufacturing allowed bats to be made from more durable aluminum and composites. These bats, however, were deemed “too hot”, enabling batters to hit the ball harder, faster, and farther than they could with conventional wooden bats. Why do these bats produce more power than wooden bats? The answer can be found through an examination of bat dynamics.
Hollow aluminum or composite baseball bats contain so-called “hoop vibration modes” that are not present in wooden bats. These hoop modes make the bat behave like a trampoline, which helps to rebound the ball at a higher velocity than an equivalent wooden bat. Furthermore, it is believed that the hoop modes allow the bat to return proportionately more energy towards the rebound velocity of the ball, since the bat has greater collision efficiency than the ball. The physics behind the trampoline effect in baseball bats is well documented in a 2004 article by Nathan, Russell and Smith.
In an earlier (2000) article, Prof. Nathan developed an analytical model of the dynamics of a baseball / bat collision, where he found that vibrations of the bat reduce the effective mass of the bat, which tends to reduce the rebound speed of the ball. In his analysis, Nathan found that the so-called nodal locations of the bat (i.e., locations with minimum vibration amplitude) returned the most energy to the ball. But wait a second! This runs counter to conventional knowledge that the trampoline-like vibrations of the bat contribute to ball velocity! Nathan further clarified that his 2000 paper pertained to lower frequency bending vibrations of wooden bats that dissipate energy and reduce CoR. His 2004 paper, on the other hand, pertained to hollow aluminum and composite bats that have higher frequency hoop vibration modes that return energy to the ball, and thereby increase the CoR.
Aluminum and composite bats are not allowed in professional baseball, but they are allowed at the collegiate, high school, and little league levels. To limit the reactivity of these bats at the college level, the NCAA requires that the bats be certified not to exceed a pre-determined maximum baseball / bat coefficient of restitution (BBCoR). A form of this test has since been adopted by USAP for evaluation of pickleball paddles (PBCoR).
Do pickleball paddles exhibit the same dynamic behavior as baseball bats? One might make the case that the paddle lower frequency diving board modes are similar to the baseball bat bending modes, and the higher frequency paddle trampoline modes might be similar to the baseball bat hoop modes. What about nodal frequencies and locations? Do pickleball paddles have them, and if so, do they behave similar to baseball bats? To answer these questions, we need to perform dynamic tests of pickleball paddles.
Paddle / Ball Coefficient of Restitution (PBCoR)
Ronbus Pickleball provided us with two paddles for testing — their original Ripple paddle (v1) and their Ripple Version 2 (v2) paddle (Figure 1). The Ripple paddles use the Ronbus-patented Fiber-Infused-Rebounding-Elastomer (FIRE) technology which is supposed to enhance the trampoline effect. Unfortunately, the Ripple v1 paddle was de-listed by the USAP for exceeding the allowable limit of 44 in their PBCoR test. Ronbus then re-designed the paddle and created the Ripple v2 paddle which was re-tested to a PBCoR of 43. We analyzed and tested these paddles to determine what measurable characteristics might account for the differences in PBCoR.
The mass and stiffness properties of both paddles were analyzed and shown in Table 1. This analysis shows that the mass properties of both paddles are within reasonable limits. An examination of the stiffness properties, however, shows that the Ripple v1 paddle has a significantly lower stiffness than either the average paddle or the v2 paddle. Are the low stiffness properties of the Ripple v1 paddle sufficient to reduce its PBCoR below the USAP acceptable limits? Is it possible to determine this without re-running the PBCoR test?
Table 1. Mass & Stiffness Properties of Ronbus Ripple Paddles
Paddle Dynamic Testing
In previous articles, we estimated the natural frequencies of the paddles based on the assumption that the paddles were constrained at the handle. This over-emphasized the diving board mode, resulting in lower paddle natural frequencies. According to Nathan, the shock wave created in the collision between a ball and bat is of sufficiently short duration that the wave does not have a chance to propagate into the handle by the time the collision has ended. In other words, a player’s grip on the bat does not affect how the bat responds! This means that we can treat the bat as essentially unconstrained and freely suspended in space during the collision. Another key observation is that the ball does not have time to spring back to its pre-collision configuration before the collision event ends. The collision time is therefore shorter than the contact time. We now believe that Nathan’s observations are applicable (to some extent) for a pickleball paddle’s interaction with a ball.
In our dynamic tests, we assume that the ball is contacting the geometric center of the paddle face (Figure 2). Since our accelerometer weighs only 0.25 oz, we provided an extra 0.75 oz to account for the weight of a 1 oz ball. We then performed an impulse modal test by striking the paddle at various locations with a modal impulse hammer to extract the paddle vibration modes.
This test provides the frequency response functions (FRFs) between each impact location and the ball (accelerometer) location. These FRFs define the input/output relationships between the acceleration of the ball due to impact at various locations on the paddle. The FRFs of particular interest are those where the impact and response locations coincide, known as the driving point transfer functions (DPXFs), which give us insight into the dynamics of the paddle striking the ball.
On a linear plot (Figure 3), we see that the peaks in the driving point frequency response functions of the paddles occur at about 394 Hz and 494 Hz for the Ripple v1 paddle, and about 438 Hz and 687 Hz for the Ripple v2 paddle. As shown in Figure 4, the vibration modes at 385 Hz and 438 Hz correspond to the paddle diving board modes, whereas the vibration modes at 492 Hz and 687 Hz correspond to the paddle trampoline modes (Figure 5). These vibration modes are virtually identical for the v1 and v2 paddles, except that they occur at different frequencies.
Figure 4. Diving Board Modes
Figure 5. Trampoline Modes
Now that we know the vibration modes, how do we find the so-called nodal points described in Nathan’s 2000 article? Nathan does this analytically by using modal superposition, however, this information is readily available to us since we are directly measuring the paddle impulse response. By looking at the transfer function data on a log-log plot, the nodal vibration frequencies are readily apparent at 222 Hz and 271 Hz (Figure 6).
An examination of these two vibration modes (Figure 7) verifies that the centers of the paddle (the ball striking location) are stationary. We hypothesize that if the period of the nodal frequency coincides with the period of the collision time, the PBCoR will be maximized.
Figure 7. Nodal Vibration Modes
Paddle Reactivity and PBCoR
In this article, we examined two paddles, the Ronbus Ripple v1 which exceeded the USAP allowable PBCoR, and the Ronbus Ripple v2 which was certified to an acceptable PBCoR of 43. This examination indicated the following:
- The mass properties of the v1 and v2 paddles do not vary significantly from the average of about 80 paddles tested. Paddle mass properties would therefore not be expected to significantly affect the PBCoR.
- The static stiffness of the v1 paddle was significantly lower than the v2 paddle and the average paddle by as much as 40%. A reduced paddle static stiffness may therefore contribute to a higher PBCoR. Further work should be done to identify the specific stiffness component(s) that affects PBCoR (i.e., paddle throat, face, or core stiffness). Once known, we can establish allowable limits for these stiffnesses so as not to exceed PBCoR.
- The paddle modal dynamic tests showed that it is possible to differentiate between the v1 and v2 paddles based on their driving point transfer functions and vibration mode shapes. These differences are manifested in upward shifts of the frequencies for the diving board and trampoline vibration modes for the v2 paddle when compared to the v1 paddle. Further work should be done to establish minimum frequencies of these vibration modes to remain within acceptable PBCoR limits.
- The paddle driving point transfer functions identified the frequency of the paddle nodal point when the ball strikes the center of the paddle. Assuming that the collision time between the paddle and ball is one-half of the contact time (0.004 sec), we hypothesize that paddles must have a minimum nodal frequency of 200–250 Hz to satisfy PBCoR requirements.
- Pickleball Science has tested over 24 paddles (so far) and have found that paddles that will not pass the PBCoR test will consistently have nodal frequencies less than about 200-250 Hz, and trampoline vibration frequencies less than 500 Hz. We will verify whether this is true for all paddles and discuss these test results in a future article.
- For most paddles, the amplification factor for the diving board mode is very low in comparison to the amplification factor for the trampoline mode. This would indicate that the diving board mode for these paddles does not contribute to the PBCoR.
The paddle modal dynamic test can be used by paddle manufacturers during the paddle development stage to quickly assess whether a paddle will pass or fail the PBCoR test prior to expensive and time-consuming certification testing at the USAP lab. These tests can also identify how the paddle’s dynamic characteristics can be adjusted to maximize paddle power while ensuring that the paddles will not exceed USAP PBCoR limits. The dynamic test apparatus can be configured as a “pass/fail test” for use at tournaments to identify paddles that might have been within PBCoR limits when new but now exceed PBCoR limits because they are broken in.
We are continuing our tests of several paddles, fine-tuning the test methodology, and refining the paddle reactivity criteria. In our next article, we will review the test results of several paddles to determine how they correlate with the PBCoR test and rank them according to paddle power or reactivity. We are actively seeking paddles to test and are particularly interested in paddles that do not pass the PBCoR test and paddles that are known to degrade with usage and eventually exceed PBCoR limits.
