Dynamic Model of a Badminton Stroke

Transcription

1 ISEA 28 CONFERENCE Dynamic Model of a Badminton Stroke M. Kwan* and J. Rasmussen Department of Mechanical Engineering, Aalborg University, 922 Aalborg East, Denmark Phone: / Fax: mak@ime.aau.dk Topics Abstract The purpose of this paper is to contribute to the understanding of dynamic racket behavior in badminton. The paper describes the development of a dynamic model of a badminton stroke based on experimental data. Motion capture and strain gage experiments are performed to clarify the movement and accelerations of a badminton smash stroke. Subsequent data processing reveals that due to the speed and acceleration of the movement, neither of the two methods provides the complete picture of the racket s dynamic behavior, but computational processing of the combination of experimental data together with physical modeling of the racket mechanics allows for an understanding of the basic mechanisms underlying racket movement and deformation. The paper concludes that... Keywords 1 Introduction Although badminton racket technology has advanced considerably, the relationship between design and response are still rather poorly understood. A review of the literature turns up very little on badminton rackets, whereas design of tennis rackets has been studied quite extensively (Brody, 1995; Cross, 1999; Lees, 23). Some tennis studies could be applicable to badminton as well as other racket sports in general, but given the distinct styles of play, the design goals of a tennis racket and a badminton racket are probably quite different. As the world s fastest racket sport, the game of badminton is very much about speed. With smashes reaching up to 332 km/h, the racket is subjected to significant dynamic effects. A dynamic model is therefore necessary to properly analyze racket performance. During the stroke, the racket experiences transverse deflection due to the high rotational and translational accelerations and it is likely that this deflection plays an important role in the transfer of momentum to the shuttle cock. Skilled players perceive significant differences between different high-quality rackets that superficially appear similar in terms of elasticity and mass properties. This indicates that small differences in the racket design have a high influence on the dynamics properties. An understanding of racket dynamics is therefore essential for further improvement of racket design. 2 Motion Capture Experiment *** JR: The section below should be elaborated more: Precisely how are the numbers processed? Where does the inaccuracy appear? What has been done in terms of filtering and such? Motion capture data were taken for ten trials of a smash stroke performed by an advanced player using a Qualisys system of eight high-speed cameras at the maximum frame rate of 24 Hz. Six markers were placed on the racket, two on the handle and four on the head, illustrated in Figure. A 12 mm spherical marker was fixed at the base, and reflective tape was used for the other five markers. Position data of the markers were passed through a low-pass Butterworth filter with a cut-off of 35 Hz and subsequently interpolated using a 6th order B-spline. An optimization-based method designed Paper No. -1- Copyright of ISEA 28

2 ISEA 28 CONFERENCE for over-determinate systems was then applied to determine the kinematics of the system (Andersen et al.). The racket was modeled as a flexible multi-body system with 7 dof, where the handle was considered a rigid segment and the revolute joint between the handle and shaft allowed transverse deflection. The results of the kinematic analysis are shown in Figures and. Figure 1: Illustration of racket anatomy and racket marker placement for motion capture. Due to the high accelerations involved and the limitations of the motion capture system used, the accuracy of marker positions may have been compromised. As seen in Figure, the error between measured and calculated marker positions are below 4 mm during most of the stroke. However, the errors increase up to nearly 2 cm around the time just before impact where accelerations are presumably the highest. The effects of the error can be seen in the calculated angular velocities and accelerations in Figure. Angular kinematics are given with respect to the local reference frame, defined in Figure. The angular accelerations are noisy and somewhat high, with peak accelerations around 15 rad/s 2. 2 p p 2 p 3 Error [cm] 1 p 4 p 5 p Figure 2: Error between calculated and measured marker positions during pre-impact part of stroke. Impact occurs just before t=.3 s. Paper No. -2- Copyright of ISEA 28

3 ISEA 28 CONFERENCE α [rad/s 2 ] ω [rad/s] Figure 3: Angular velocities [rad/s] and accelerations [rad/s 2 ] of the racket about its axial x-axis (blue), y-axis (green), and z-axis (red) during pre-impact part of stroke. Impact occurs just before t=.3 s. The resulting load on the racket was then computed, despite the questionable accuracy of the kinematics derived above. The load is proportional to the acceleration normal to the racket, containing contributions from both linear and angular accelerations. The normal acceleration generated at the tip of the racket is shown in Figure. At the tip of the racket, the normal acceleration is dominated by the rotational component, implying that the angular accelerations contribute much more to the speed of the racket tip than the linear accelerations. The acceleration is especially noisy during backswing, from t= to.2 s, followed by an increase (in the negative direction) as the racket accelerates forward. As the racket continues forward, unloading of the racket begins (t=.25 s) as the normal acceleration decreases back to zero Linear Rotational Total Acceleration [m/s 2 ] Figure 4: Contributions to total normal acceleration of racket tip (red) from translational accelerations (blue) and rotational effects (green) during pre-impact part of stroke. Impact occurs just before t=.3 s. Paper No. -3- Copyright of ISEA 28

4 ISEA 28 CONFERENCE 3 Strain Gage Experiment The motion capture experiment revealed that the high accelerations in the motion combined with the spatial and temporal resolution limitations of the system makes it difficult to obtain reliable values for the accelerations, and thus for the dynamic forces in the system. An alternative is to measure a more direct representation of the forces in the system, namely the strain in racket shaft. Two strain gages were placed on the end of the shaft 8 mm from the top cap in a half-bridge configuration. Ten trials of a smash stroke were performed by the same player and the same racket from the motion capture trials. Through a series of simple bending tests for a clamped racket with a concentrated end load, a linear relationship between the strains and the measured deflection at the tip of the racket was determined. Using this relationship between strain and end deflection for a concentrated load, the end deflection assuming a triangular distributed load was calculated as a function of the strain, shown in Figure Deflection at Tip [m] Strain [µ m/m] Figure 5: Linear relationship between measured strain at shaft base and measured deflection at tip for concentrated load (blue). Calculated tip deflection for an increasing triangular distributed load (red). Applying the linear relationship to the smash stroke data gives the tip deflection curves seen in Figure. Over a stroke, the maximum strain of around 8 µm/m corresponds to a tip deflection of about 5 cm. Paper No. -4- Copyright of ISEA 28

5 ISEA 28 CONFERENCE.6.4 Deflection at Tip [m] Figure 6: Deflection at racket tip over one full stroke. Impact occurs around t=.46 s. During the stroke, the racket is first swung backward and then forward before hitting the shuttle. The backswing generally takes under.2 s, and the forward stroke lasts around.1 s. In Figure, a negative deflection of the racket can be seen (around t=[.14,.34] s) during the backswing (positive normal acceleration), followed by a much greater positive deflection during the forward stroke (negative normal acceleration) until impact around t=.46 s. Figure 7: Illustration of racket deflection during backswing (left) and forward swing (right) before impact. For the maximum benefit from the elastic deformation of the racket, impact should occur when the deflection of the racket returns to zero, where the tip velocity will be highest. However, the elastic deformation will still be advantageous as long as the racket tip is moving forward (towards the shuttle), indicated by a negative deflection slope in this case, so there is a narrow window of about.1 s where slight deviations in the timing of the impact can be tolerated. As perhaps a more reliable source than the motion capture data, the strain gauge data was also used to derive racket kinematic. The racket was assumed to experience rotation only, due to an applied torque at the handle. The magnitude of the torque τ was computed from the strain ɛ, given by τ = EIɛ/r where EI is the flexural rigidity of the shaft and r is the shaft radius. The angular acceleration of the racket due to the torque was determined from α = τ/(i cm + md 2 ) where I c m is the racket moment of inertia, m is the total mass, and d is the distance from the strain gauge to the racket center of mass. The resulting normal acceleration of the racket tip is given in Figure. Paper No. -5- Copyright of ISEA 28

6 ISEA 28 CONFERENCE 4 2 Acceleration [m/s 2 ] Figure 8: Estimated normal acceleration of racket tip during pre-impact part of stroke, calculated from strain measurements. Impact occurs around t=.35 s. Comparing with the normal acceleration derived from motion capture data in Figure, the shape of the curves during the forward swing phase are quite similar, with the peak acceleration near 8 m/s 2. 4 Computational model *** JR: Possibly a shortened version of the Lagrange model leading to the conclusion that the model does not capture the correct dynamics, possibly due to lack of damping and incorrect boundary conditions. The simplified racket model is composed of two uniform beam elements, representing the shaft and the head of the racket. Each element has uniform mass per unit length, ρa, and uniform flexural rigidity, EI. The uniform beam assumption is a good approximation for the shaft, but is a rather gross simplification of the racket head. The handle is assumed to be completely rigid. The origin of the body-fixed coordinate system xyz of the racket, located at the base of the handle, can be expressed in the ground coordinate system XY Z by the vector r, as seen in Figure. The rotation of the racket coordinate system is related to global by the transformation matrix T, such that xyz = T XY Z. Figure 9: Beam model (one element). Paper No. -6- Copyright of ISEA 28

7 ISEA 28 CONFERENCE In addition to the six degrees of freedom from rigid body motion, the racket is allowed to bend in the transverse direction. The finite beam element of length L is represented by two nodes, one at each end. The displacement vector q contains the linear and rotational displacements at each node, q(t) = [v 1 θ 1 v 2 θ 2 ] T, and the displacements along x between the nodes are described by shaping functions, N(x) = [N 1 N 2 N 3 N 4 ]. The deflection of the beam w is then given by w(x, t) = N(x)q(t). Dynamic equations describing the flexible motion of the racket due to both inertial and centrifugal effects are derived using Hamilton s principle (e.g. Yang, 24). Applying Lagrangian mechanics, the kinetic energy of a beam element is given by T = 1 L 2 ρa r r dx (1) where r is the position vector (in local coordinates) of a point P on the racket from the global origin. Since the racket experiences only tranverse deflection, the position vector of point P from the local origin is p = R + x, w,, where R is the length of the rigid handle. Then r = T r + p (2) r = T r + ω p + p (3) where ω contains the angular velocities of the racket about its body-fixed axes. The potential energy consists of elastic energy and energy from centrifugal force. Neglecting axial deformation, the potential energy is given by V = 1 L 2 EI (w ) 2 dx + 1 f c (x) (w ) 2 dx (4) 2 where f c (x) is the centrifugal force (per unit length) that arises from rotation, given by L L L f c (x) = ρa a c (ζ) dζ = ρa x x r i dζ (5) where i is a unit vector along the x-axis of the racket. The centrifugal acceleration a c does not include deflection terms, such that r = T r + ω p (6) where p = R + x,, and r = T r + p represent the undeformed position vectors of point P from the local and global origin, respectively. Applying Lagrange s equations, L q d dt ( ) L = (7) q where the Lagrangian L = T V, the non-linear equations of motion take the form: [m] { q} + ([k] [k 1 ] + [k 2 ] + [k 3 ]) {q} = [c] T + [c 1 ] T (8) where m and k are the mass and stiffness matrices of the beam element. Dynamic stiffening is represented by the stiffness matrices k 1, k 2, and k 3, and the forcing functions due to rotational and translational accelerations are given by the vectors c and c 1, respectively. The equations were solved using a fourth order Runge-Kutta integration. Mass, geometric, and stiffness properties for the shaft and head segments were determined experimentally, shown in Table 1. Using these values, a natural frequency of 14.9 Hz was calculated for the beam model in the clamped condition, which corresponded well with the racket s natural frequency of 15.1 Hz. Paper No. -7- Copyright of ISEA 28

8 ISEA 28 CONFERENCE Table 1: Material properties of beam segments. Mass Length ρa EI Segment [kg] [m] [kg/m] [N-m 2 ] Shaft Head Using kinematic inputs from the motion capture data, the beam model produces the deformation curve seen in Figure. In the clamped case, the predicted deflections are somewhat high, but the frequency of the dynamic response is also too high. One possible explanation is the clamped boundary condition, which may not be very representative of the handheld case (Cross; Brody). To loosen up the boundary conditions, the end of the racket was changed from clamped to pinned with a torsional spring of stiffness k t. A very stiff spring (k t >1e6 N-m/rad) approaches the clamped condition, and lowering the stiffness leads to a reduction in the response frequency. Figure shows the elastic deflection at the tip, over one stroke with k t =5 N-m/rad..1.8 Clamped Pinned spring Deflection at Tip [m] Figure 1: Deflection at beam tip during pre-impact part of stroke for clamped condition (blue) and pinned with spring k t =5 N-m/rad (green), using motion capture data. Impact occurs just before t=.3 s. Figure shows the deflections of the racket tip using kinematic inputs from the strain gauge data, for the clamped condition and the pinned case with a torsional spring of stiffness k t =5 N-m/rad. In both Figures and, the deflections of the racket tip are still rather high, when compared to the values found from strain gauge measurements, shown in Figure. This could be caused by an overestimation of the accelerations from the experimental data and/or a lack of damping in the model. Paper No. -8- Copyright of ISEA 28

9 ISEA 28 CONFERENCE.2.15 Clamped Pinned spring Deflection at Tip [m] Figure 11: Deflection at beam tip during pre-impact part of stroke for clamped condition (blue) and pinned with spring k t =5 N-m/rad (green), using motion capture data. Impact occurs around t=.35 s. 5 Conclusions 1. Our current mocap technology is only partially useful. Is it possible that state-of-the-art technology will be enough? What frame rate and spatial resolution will be sufficient, based on gaps in our mocap results? 2. The strain gage experiment reveals surprisingly large elastic deflections. Is this reasonable? 3. What can we say about the timing of the stroke? How sensitive is elastic advantage to timing? Motion capture could be better with spherical markers drilled into racket to improve spatial resolution and higher frame rate to improve temporal resolution and avoid disappearing markers. (how high is necessary?) The elastic energy of the deflected racket is advantageous as long as impact occurs while the racket is continuing to rotate forward. The ideal timing is to wait long enough for the racket to recover from the backwards deflection until it is vertical again, although ±.5 s is the window of opportunity where the elastic energy would still be beneficial. what is this frequency dependent on, the stroke (kinematics) or the racket (material)? References [A1] Andersen, M.S. [B1] Brody, H. How would a physicist design a tennis racket? In Physics Today, 48(3):26-31, [C1] Cross, R. Impact of a ball with a bat or racket. In American Journal of Physics, 67:692-72, [L1] Lees, A. Science and the major racket sports: a review. 21:77-732, 23. In Journal of Sports Sciences, Paper No. -9- Copyright of ISEA 28

10 ISEA 28 CONFERENCE [YJ1] Yang J.B., Jiang L.L., and Chen D.C.H. Dynamic modelling and control of a rotating Euler-Bernoulli beam. In Journal of Sound and Vibration, 274: , 24. Churchyard. The dynamic behavior of a rotating flexible beam based on finite beam elements has been well documented, e.g. Yang. Montagny et al. developed such a model and used a badminton racket as a case study, which appears to be the only study related to badminton racket dynamics. However, these rotating beam models are generally limited to planar motion and rotation about a fixed point. Here the racket is allowed to translate and rotate in 3D space. Appropriate boundary conditions to describe the handheld racket are also discussed. Here the racket is represented as a two-segment Bernoulli-Euler beam, and the focus is on understanding what dynamic effects take place over the stroke before impact, i.e. deformation of the racket. *** Paper No. -1- Copyright of ISEA 28