## 2 Materials and Methods
The vibration signals for the 91 (17%) rackets characterised previously at the brand were obtained using a wireless accelerometer (MetaWear CPro, Mbientlab Inc) sampling at 800 Hz, as described in [28]. A check using a racket model (Ti.S6, HEAD) located within both the brand and museum confirmed that the two accelerometers gave the same result for natural frequency (176 Hz). The recorded signals for all rackets were post-processed in MATLAB(r) to identify the natural frequency and damping ratio. The script performed a Fourier transform of the signal with the MATLAB(r) function _fft_, and the first peak in the power spectrum (f\({}_{p}\)) was assigned to the natural frequency of the racket. Similar to [28], the wired accelerometer was calibrated using a shaker excited at frequencies of 100, 150 and 200 Hz. Each calibration test was repeated three times, with the frequency from the accelerometer always matching the excitation value. Damping ratio was calculated with the half-power bandwidth method (Figure 2), as commonly used for estimating damping in multi-DOF systems [32]. The method first involves identifying the two half-power frequencies (f\({}_{1}\) and f\({}_{2}\)), which are located either side of the peak frequency were the amplitude (A\({}_{1}\)) is equal to the amplitude at the peak (A\({}_{p}\)) divided by \(\sqrt{2}\) (i.e., \(A_{1}\approx 0.7A_{p}\)). The half-power damping ratio can then be obtained using Equation (3),  
\[\zeta=\frac{f_{2}-f_{1}}{2\ f_{p}} \tag{3}\]  
where \(f_{2}-f_{1}\) is the frequency range. The number of data points in the region of the power spectrum that corresponds to the natural frequency (\(\pm\) 10 Hz of f\({}_{p}\)) was increased using a _spline_ function in MATLAB(r), to improve the frequency resolution (0.02 Hz) and the accuracy of the damping estimation, as shown in Figure 2. When applying the half-power bandwidth method, for a given natural frequency, damping ratio will increase with the frequency range (wider and less pronounced peak), while for a given frequency range, damping ratio will increase as natural frequency decreases. Each racket was tested twice, with the mean natural frequency (to 1 Hz) and damping ratio reported. The uncertainty in the calculation of damping ratio was dependent on both the damping ratio and natural frequency of the racket, as outlined in Equation (4),  
\[\zeta_{uncertainty}=\left(\frac{0.2}{f_{p}}\right)\times(0.1+\zeta) \tag{4}\]  
For example, the uncertainty in damping ratio for a wooden racket with a natural frequency of 100 Hz and damping ratio 1% would be 2.2% of the calculated value. Four hundred and ninety-four (94%) rackets had an uncertainty in damping ratio below 5%, 28 (4%) rackets had an uncertainty between 5 and 10%, and 3 (\(<\)1%) rackets had an uncertainty between 10 and 15%. The highest uncertainty in damping ratio was associated with metal rackets, because they have both low frequency and low damping. The resistance of a racket to bending can be increased using stiffer materials or larger cross sections. Fibre-polymer composites allow for rackets with large hollow cross sections, combining a high second moment of area with low mass. Measuring the cross section of hollow rackets was not possible without cutting them or taking them to a three-dimensional imaging device, such as a computed tomography scanner, so the ratio of the frame depth to thickness was used as a proxy for the second moment of area. The natural frequency of a freely suspended racket is often used as an analogue to its stiffness [33], although it is also affected by mass [34]. To reduce the dimensionality of the number of measured variables, a principal component analysis (PCA) [35] was conducted for all racket frame measurements (racket, head (int.) and grip length, racket width (int. ), frame thickness and depth, frequency and damping, mass, CoM location, and transverse and polar MOI). The three principal components that accounted for the most variance in the measured variables were then further analysed.