Measurement of badminton racket deflection during a stroke

Abstract

The compliance of a badminton racket is an important design consideration, which can be better understood by studying the deflection behaviour of the racket during a stroke. Deflection can be measured using direct methods, such as motion capture or high speed video, or by indirect methods, which then require a mathematical model in order to calculate the deflections from indirect measures. Indirect methods include strain gauges and accelerometers. Here, racket deflection is measured directly using motion capture and compared with deflections calculated from strain gauge data using a beam model. While the elastic behaviour is better calculated from strains than measured by motion capture, it is not possible to extract the whole motion of the racket from strain data. Motion capture is therefore also necessary to determine the rigid body velocity, in order to put the elastic velocity (as calculated from strains) in perspective.

Appendix: Calculation of strain–acceleration relationship

Motion capture data provide a measure of the accelerations which load the racket, and strains measure the deflections caused by these loads. Comparing the results from both methods is not straightforward, since strains unfortunately provide limited kinematic information. Thus, a loading term α _ε (t), composed of linear and angular accelerations, is developed, which can be extracted from strains using a beam model.

The racket is modelled as a uniform cantilever beam of length L, as shown in Fig. 10, subjected only to inertial loads which cause the racket to bend. The deflection of a beam can be represented analytically by the summation of the infinitely many natural modes of the beam [18]:

$$ w(x,t) = \sum\limits_{n = 1}^{\infty } {W_{n} (x)q_{n} (t)} $$

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where W _n (x) and q _n (t) represent the beam shape along x and the beam motion over time, for the nth mode of the beam. Assuming the first mode is dominant, such that n = 1 gives

Fig. 10

Beam model of racket

$$ w(x,t) = W_{1} (x)q_{1} (t) $$

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For fixed-free boundary conditions, the beam shape function W(x) is

$$ W_{n} (x) = \sin \,\beta_{n} x - \sin \,h\beta_{n} x - \alpha_{n} \left( {\cos \,\beta_{n} x - \cos \,h\beta_{n} x} \right) $$

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where

$$ \cos \,\beta_{n} L \cdot \cos \,h\beta_{n} L = - 1 $$

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$$ \alpha_{n} = {\frac{{\sin \,\beta_{n} L + \sin \,h\beta_{n} L}}{{\cos \,\beta_{n} L + \cos \,h\beta_{n} L}}} $$

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For n = 1, β _n L = 1.8751.

For a beam initially at rest, the dynamic response of the beam, q _n (t), subjected to the forcing function f(x, t) is given by

$$ q_{n} (t) = {\frac{1}{{\rho Ab\omega_{n} }}}\int\limits_{0}^{t} {Q_{n} (\tau )\sin \omega_{n} \left( {t - \tau } \right){\text{d}}\tau } $$

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where

$$ \omega_{n} = \beta_{n}^{2} \sqrt {{\text{EI}}/\rho A} $$

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$$ b = \int\limits_{0}^{L} {(W_{n} (x))^{2} {\text{d}}x} $$

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$$ Q_{n} (t) = \int\limits_{0}^{L} {f(x,t)W_{n} (x){\text{d}}x} $$

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Strain is proportional to the second derivative of the deflection with respect to x:

$$ \varepsilon (x,t)/R = w^{\prime\prime}(x,t) = w^{\prime\prime}(x)q_{n} (t) $$

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where R is the radius of the shaft where the strain gauge is placed, and

$$ W^{\prime\prime}_{n} (x) = - \beta_{n}^{2} (\sin \,\beta_{n} x + \sin \,\beta_{n} x - \alpha_{n} (\cos \,\beta_{n} x + \cos \,h\beta_{n} x)) $$

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Focusing on the transverse direction only, the acceleration of a point at position x along the beam is given by

$$ a(x,t) = a_{H} (t) + (x + h)\alpha_{T} (t) $$

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where a _H (t) is the translational acceleration of point H (at the base of the handle), α _T (t) is the rotational acceleration of the beam, h is the handle length, the distance from point H to point O.

For the forcing function f(x, t) = ρA a(x, t), the strain can be expressed explicitly as a convolution:

$$ \begin{aligned} \varepsilon (x_{sg} ,t) = & - RW^{\prime\prime}_{n} (x_{sg} ){\frac{{2(\alpha_{n} + \beta_{n} h)}}{{b\omega_{n} \beta_{n}^{2} }}}\int\limits_{0}^{t} {\alpha_{s} (t)\sin \omega_{n} (t - \tau ){\text{d}}\tau } \\ = & - RW^{\prime\prime}_{n} (x_{sg} )(\alpha_{\varepsilon } *g)(t) \\ \end{aligned} $$

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with

$$ \alpha_{\varepsilon } (t) = {\frac{{\beta_{n} }}{{\alpha_{n} + \beta_{n} h}}}a_{H} (t) + \alpha_{T} (t) $$

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$$ g(t) = - {\frac{{2(\alpha_{n} + \beta_{n} h)}}{{b\omega_{n} \beta_{n}^{2} }}}\sin \omega_{n} t $$

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Using Eq. 16, the ideal β _n can be solved for numerically, such that the measured strains ε(x _sg , t) and measured angular accelerations α _ε (t) satisfy the equation reasonably well. Since β _n L = 1.8751, the model beam length L can then easily be determined.

Deflections can also be calculated, as they are directly proportional to strains:

$$ w(x,t) = W_{n} (x)q_{n} (t) = W_{n} (x)\varepsilon (x_{sg} ,t)/RW^{\prime\prime}_{n} (x_{sg} ) $$

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The elastic velocity can then be calculated by taking the derivative

$$ \dot{w}(x,t) = W_{n} (x)\dot{q}_{n} (t) = W_{n} (x)\dot{\varepsilon }(x_{sg} ,t)/RW^{\prime\prime}_{n} (x_{sg} ) $$

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