We have all experienced mis-hits where the ball either lands into the net or sails long over the opposite baseline. One key contributor to these mis-hits involves the rotation of the paddle face about the longitudinal (handle) axis. These rotations come about by not striking the ball at the center of the paddle, which causes a torque that twists the paddle either upwards or downwards. An upward rotation causes the ball to sail high or long, and a downward rotation causes the ball to hit the net or land short.
As we have seen in previous articles, the paddle/ball contact time is so short that it is impossible for players to correct the paddle orientation in mid-shot. We must therefore rely on physics that make the paddle less susceptible to mis-hits. A paddle’s twist weight provides inertia that resists these rotations, but how much do twist weights differ among the different paddles? Moreover, do these differences result in noticeable performance improvements? What are the differences between standard (square) paddles and elongated (rectangular) paddles? Let’s look at the science for answers to these questions.
What is Twist Weight?
We have discussed swing weight extensively in several articles (see for example, “Paddle Selection: Swing Weight”), but what about the twist weight? The twist weight is essentially the mass moment of inertia of the paddle about the longitudinal (handle) axis.
In previous articles, we calculated the mass moments of inertia (Imass) of paddles using the so-called “lumped mass” method. This method assumed that the masses of the paddle components (e.g., face sheets, cores, handles, etc.) could be converted into equivalent concentrated masses (mi) acting at offset distances (ri) from the paddle center of mass. That is,
While this method provides a good first-order approximation of the paddle mass moment of inertia for purposes of calculating the paddle swing weight, it lacks sufficient accuracy for calculating the paddle twist weight. A better method involves integration of each incremental paddle mass element (dm) over the entire paddle volume. That is,
It can be shown that for thin plates, the above integral equation can be solved in closed-form by knowing the area moment of inertia of the plate (Iarea), the material density (ρ), and the plate thickness (t):
Therefore, for a rectangular plate with a base length (b), height (h) and thickness (t), the mass moment of inertia can be calculated by:
Lumped Mass vs. Integral Calculation
As an illustration we will calculate the mass moment of inertia about the X-axis for an aluminum plate (ρ = 0.10 lb/in3) with a width (b) of 8”, height (h) of 10”, and thickness (t) of 0.25” using both the lumped mass and integral methods (Figure 1).
Using the lumped mass method, we would separate the plate into two concentrated masses located at the centroids of the upper and lower plates (Figure 1b). The calculated mass moment of inertia is therefore:
Imass = 2 * (8.0 * 5.0 * 0.25 * 0.1) * 2.52 = 12.50 lb-in2
Using the integral method,
Imass = 0.1 * 0.25 * 8.0 * 10.03 / 12 = 16.67 lb-in2
The lumped mass method therefore underpredicts the mass moment of inertia by nearly 25%! This is due to the fact that we are ignoring the areal inertia.
The problem with using the integral method for calculating the mass moment of inertia of a pickleball paddle is that there are no closed-form solutions for complex shapes. However, since CAD programs such as SketchUp are surface modelers, it is possible to use the advanced capabilities of this program to accurately calculate the swing weight and twist weight of paddles. Returning to our equation,
Imass = ρ t Iarea
We can use the SketchUp program to calculate the paddle area moment of inertia (Iarea) and convert this to its mass moments of inertia (Imass) by multiplying by the areal density of the paddle.
How Twist Weight Resists Rotation
According to Newton’s 2nd Law for Rotation, the torque (T) acting on a the paddle is equal to the product of the rotational inertia (I) multiplied by the angular acceleration (α). That is,
T = I α
From the above equation, it is apparent that for a given level of torque (T), a paddle with a greater rotational inertia (or twist weight) will provide a smaller angular acceleration (α).
The torque comes about by the force (F) of the ball striking paddle’s surface at a distance (d) from the paddle longitudinal axis as shown in Figure 2.
If you strike ball directly on center of the paddle (d=0), there will be no torque induced about the longitudinal axis. However, the farther off-axis you strike the ball, the greater the torque and thus the greater the angular error (θ).
Standard vs. Elongated Paddles
We used this updated integral methodology to re-calculate the swing weights and twist weights in our article “Paddle Technical Comparisons”. The average twist weight of more than forty paddles was found to be about 27 oz-in2, with a low of about 23 oz-in2 (ProKennex Pro Flight) and a high of about 32 oz-in2 (CRBN 2X).
Do the relatively small differences among the paddle twist weights result in noticeable differences in paddle performance? Conventional wisdom suggests that paddles with larger twist weights are more stable than paddles with smaller twist weights since the paddle inertia will resist rotation about the paddle longitudinal axis if the ball is hit off center. How does the paddle configuration affect its twist weight?
To illustrate the effects of twist weight, let’s look at the CRBN 1X and CRBN 2X paddles*. The composition of the paddles is virtually identical, except that the 1X has an elongated (rectangular) shape, whereas the 2X has a standard (square) shape that gives them different swing and twist weights as shown in Figure 3.
Let’s assume that for both paddles, we strike the ball 2” off the longitudinal axis (i.e., d = 2.0 in Figure 2). In a previous article we determined that a hard hit puts a force of 40 lbs on the ball over a duration of 4 milli-seconds (0.004 sec). The 40 lbs acting at 2.0” from the longitudinal axis provides a torque of 80 in-lb (9.0 N-m) to the paddle. Dividing the torque by the inertias of the CRBN 1X (I1) and CRBN 2X (I2) paddles, we get the following angular accelerations:
α1 = 18,256 radians/sec2 (for CRBN 1X)
α2 = 15,385 radians/sec2 (for CRBN 2X)
We can convert the angular accelerations (α) to angular rotations (θ) by integrating the accelerations twice with respect to the contact time (t=0.004 sec) and converting radians to degrees. That is,
θ1 = ½ α1 t2 = ½ (18,256 rad/s2) (0.004 s)2 = 0.146 rad = 8.4 degrees
θ2 = ½ α2 t2 = ½ (15,385 rad/s2) (0.004 s)2 = 0.123 rad = 7.1 degrees
The CRBN 1X paddle with the smaller twist weight will therefore rotate 1.3 degrees more than the CRBN 2X! If the ball is contacted below the longitudinal centerline, hitting the ball with the CRBN 1X paddle from the baseline will have a trajectory that is 6” lower than the CRBN 2X as the ball approaches the net. This could mean the difference between clearing the net and hitting the net on a low serve! Similarly, if the ball is contacted above the longitudinal centerline, hitting the ball with the CRBN 1X paddle will have a trajectory that is 12” higher than the CRBN 2X as the ball approaches the opposite baseline, which may cause it to sail past the baseline!
Conclusions
This article has confirmed that paddles with larger twist weights are less prone to unwanted rotation than those with smaller twist weights. In general, wider paddles are more forgiving than elongated paddles and can provide beginners to intermediate pickleball players with a couple of advantages: (1) there is more lateral area available to strike the ball, and (2) the larger twist weight will reduce unwanted rotation if the ball is hit off center. As players become more skilled at reliably hitting the ball at the center of their paddles, they should consider an elongated paddle for extra power and reach, at the expense of lower accuracy if the ball is hit off-center. In a future article, we will show you how to “tune” and adjust your paddle’s swing weight and twist weight by use of weighted tapes.
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