In several articles (see for example “Paddle Dynamics“), we alluded to the idea that the amount of power in a pickleball paddle is maximized with lower-stiffness paddles. This is due to the fact that when you strike the ball, the force of the ball causes the paddle face to deform like a trampoline and the paddle throat to bend like a diving board. The paddle stores the kinetic energy of the ball in the form of potential energy resulting in the deformation of the paddle face and throat much like the compression of a spring. When the paddle springs back during your stroke, the stored potential energy is returned to the ball in the form of kinetic energy that propels the ball forward with a higher velocity on your return.
It would therefore stand to reason that players who want the maximum power potential of a paddle will select the softest paddle available. However, if softer paddles have more power, why are the manufacturers making their paddles out of stiffer materials, such as carbon fiber and Kevlar? What are the limitations if your paddle is too soft or too stiff? Is there an ideal paddle stiffness that suits your ability and style of play that maximizes power and control? Does paddle stiffness alone determine its power potential? To answer these questions, we need to start from the “basics” of the Impulse-Momentum equation.
Impulse and Momentum
In several previous articles (see for example, “Paddle Weight and Momentum“) , we showed that for paddles and balls of identical mass (m), the return velocity of the ball (Δv) is proportional to the force applied to the ball (F) over the contact time (Δt). That is,
If you are trying to select among paddles of similar swing weight, the maximum amount of force (F) and the swing velocity (v) that you can develop will be about the same, and the mass (m) of the ball and paddle are constant. So the only way to increase the return velocity of the ball is to increase the paddle/ball contact time. But how can you do that? The contact time is affected by numerous factors, including the incoming ball speed, the paddle swing weight and velocity, and ball stiffness. As we will see in this article, the contact time is also intrinsic property of the paddle, governed by its effective mass and stiffness.
We can model the ball-paddle contact as a half-sine wave function as shown in Figure 1 below. Here, the paddle deformation starts at zero prior to contact with the ball, reaches a maximum amplitude, then reduces back to zero when the ball is released. The paddle then continues to vibrate after the ball is released. Assuming that this impulse load will excite the fundamental paddle vibration frequency, we can calculate the excitation frequency by assuming that the contact time is equal to one-half the period of the excitation frequency.
In numerous articles (see for example, “Paddle Weight and Momentum“), we determined that the contact time needed to return the ball at maximum velocity must be about 4 milliseconds. Knowing from Figure 1 that the contact time (Δt) is equal to one half the wave period (T), we can calculate the excitation frequency (fexc) as follows:
The key to developing maximum power and ball return velocity is to tune the paddle natural frequency (fn) to the excitation frequency (fexc). Paddles with natural frequencies that are lower than the excitation frequency will release the ball too late (or too slowly) and paddles with natural frequencies that are higher than the excitation frequency will release the ball too soon (or too fast). In either case, the amount of energy provided to the ball will not be optimum. When the paddle natural frequency matches the excitation frequency, the power potential is maximized. In the next section we will look at how we can tune the paddle natural frequency to the excitation frequency.
Paddle Ideal Stiffness
As we described in previous articles (see for example “Paddle Dynamics“), the paddle and ball behave like a mass-spring system, where the combined (or effective) mass of the ball and paddle oscillate on the combined (or effective) stiffness of the paddle face and throat (Figure 2).
We can simplify this dynamical system to an effective mass (meff) and an effective stiffness (keff) that oscillate at a natural frequency (fn) according to the well known equation:
As described above, when the paddle natural frequency (fn) matches the paddle/ball excitation frequency, the paddle power potential is maximized. If the effective mass of the paddles and balls are about the same, we can maximize the paddle power by tuning the paddle effective stiffness (keff) to an ideal stiffness (kideal). That is,
How do we find the effective mass? The effective mass (meff) must include the entire mass of the ball (0.9 oz) plus only a portion of the paddle mass. The average paddle mass was found to be about 8 oz. We must include the mass of the paddle from the contact point to the top of the paddle since this mass is not subjected to bending and is therefore “dead weight”. Our analysis found that the average amount of “dead weight” (Figure 2) above the paddle cG is about 55% of the total paddle mass (or 4.4 oz for an 8 oz paddle). The effective mass (meff) must therefore be 5.3 oz (8.57E-04 sl). Using this mass in the above equation, we find that the paddle ideal stiffness (kideal) for an “average” paddle must be about 530 lb/in.
Paddle Natural Frequencies
In our “Paddle Technical Data” table, we used our three-point bending apparatus to measure the face and bending stiffnesses of several different paddles. We combined the paddle face stiffness (kf) and the paddle throat stiffness (kt) to obtain the paddle effective stiffness (keff) according to the springs in series equation:
As shown in our “Paddle Technical Data” table, the average effective stiffness for the non-EVA paddles is about 460 lb/in with a range of about 230 lb/in for the softest paddle and over 620 lb/in for the stiffest paddle. EVA paddles, on the other hand, have an effective stiffness in the 160 lb/in range. What does this mean?
In general, we cannot classify paddles based on stiffness alone, because the paddle natural frequency is based on the ratio of its stiffness to its mass. Therefore, a lighter paddle with a lower stiffness can have as much power as a heavier paddle with a higher stiffness, especially in light of the fact that you might be able to swing a lighter paddle faster than a heavier one. We therefore report the paddle natural frequency along with the paddle stiffness in our “Paddle Technical Data” table. The key here is for the paddle to have a natural frequency that matches the excitation frequency of 125 Hz, which corresponds to a contact time of 4 milliseconds. This criteria allows us to scientifically classify paddles as “hybrid”, “control”, or “power” paddles, as follows:
Paddles with natural frequencies in the range of 118 Hz to 125 Hz will have an ideal stiffness-to-mass ratio and a contact time of about 4.0 to 4.25 milliseconds. Paddles with natural frequencies greater than 125 Hz will have a contact times that are less than 4.0 milliseconds, and cannot develop ball return velocities that approach the theoretical maximum velocity of 40 MPH (see “How Fast is a Pickleball Serve?“) but may be highly accurate. Paddles with natural frequencies less than 118 Hz will have a lot of power, but may be difficult to control and may be prone to mis-hits.
Hybrid Paddles will have natural frequencies in the range of 118 Hz to 125 Hz. These paddles will have an ideal stiffness-to-mass ratio and a contact time of about 4.0 to 4.25 milliseconds. These paddles might be suitable or most players, as they will have good all-around performance for power and control shots. Paddles with natural frequencies closer to 118 Hz would be classified as “power-hybrid” paddles, and those with natural frequencies closer to 125 Hz would be classified as “control-hybrid” paddles.
Control Paddles will have natural frequencies that are greater than 125 Hz will have a contact times that are less than 4.0 milliseconds. As a result, these paddles cannot develop ball return velocities that approach the theoretical maximum velocity of 40 MPH (see “How Fast is a Pickleball Serve?“). These paddles might be suitable for players who do not rely on power and speed, but instead make most of their points based on accuracy and finesse. The performance of these paddles will be highly repeatable and predictable.
Power Paddles will have natural frequencies less than 118 Hz with contact times that can be greater than 4.25 milliseconds. These paddles can develop a lot of power and ball speed, but may be difficult to control since the ball launch angle will be dependent on the amount of force applied to the ball. These paddles might be suitable for experienced players who can control their swing speed and can apply topspin to the ball to keep it in bounds. They also might be suitable for players with lower arm strength and swing velocity who need a springier paddle to develop ball velocity.
Future Articles
In future articles we will use the information in our “Paddle Technical Data” table to perform scientifically-based reviews of several different paddles.
