Simulation and Analysis of Table Tennis Considering Spin Angular Speed Damping

Transcription

1 Simulation and Analsis o Table Tennis Considering Spin Angular Speed Damping Chen Jicheng, Wang Haojiao, Wang Xuejiao*, Lei Junwei 3, Pan Changpeng 4. Department o phsical education, Hainan Medical Universit, Hainan Province, China. First Ailiated Hospital o Hainan Medical College, Hainan Province, China 3. Department o control engineering, Nav aeronautical and astronautical universit, Yantai, China 4. Department o command, Nav aeronautical and astronautical universit, Yantai, China Abstract: It is ver complicated to analze the orce and moment as well as the kinematics modeling o table tennis ball in the process o ball hitting, especiall with high returning speed. In the course o light, the rotation speed o the ball is attenuated b the eect o air resistance. In this paper, the angular speed damping model is established or the irst time in consideration o the actors that the spin angular speed attenuates in the light process, and the estimation method o the deca time constant is proposed in this paper. Through the analsis o the orce and moment in the process o returning the ball, contact-table rebound, the kinematics and dnamics modeling o the whole process o ree-ling, contact-table rebound, hitting ball and second rebound ater across the net is established. The trajector o ling ball is simulated b means o computer simulation, considering the actors o dierent angles o swing, dierent strength o riction, as well as the dierent rotation generated b the dierent riction acting time on ball. The results can provide theoretical support or the training o table tennis plaers and the design o table tennis robot. Kewords: Simulation; table tennis; orce; movement; angular speed. Introduction The research on the modeling o moving bod has attracted more and more attentions. However, most o the current literatures ocus on high speed moving aircrat, rocket, supersonic missile and hpersonic vehicle. The main reason is that the modeling o the above-mentioned aircrat is complicated, and its research has high militar value and economic value. Because o the strong non-linearit o the orce and moment analsis o the low speed vehicles, the precise modeling and analsis are more complicated. Meanwhile the economic value o the low speed vehicle is not as good as that o the high speed vehicle, so the research on the low speed vehicle is less concerned. At present, the research on low speed aircrat is mainl ocused on airships, saucer-shaped aircrat, and low speed aeronautical devices. Thereore modeling o low speed ling object such as table tennis is not popular. Especiall in table tennis, the driving orce acting on the ball is mainl rom the plaer's hitting and the table s rebound, while the hitting time is ver short, which is similar to the shock response, so accurate modeling and simulation analsis are ver complex [-], and the research on the orce and moment o the ball return process is relativel rare. Thereore, this paper mainl carries on the research on the modeling and simulation o the orce and moment as well as the actors inlecting the trajector o the ball in the course o hitting process. The signiicance o this research is that it can be used or the uture design o intelligent table tennis robot, who can realize the intelligent hitting and compete with the super athlete in the same wa as the Alpha dog in the ield o Go, which will be a classic example o robot motion control. It is well known that the analsis o orce and moment as well as the kinematics modeling o table tennis ball in the process o ball hitting is ver complicated, especiall when the returning speed is high [-5]. There is not onl movement in horizontal, vertical and lateral directions in ree light, but also the rotation o the ball itsel. Though the ball is small, its analsis and modeling are as complex as the general low speed aircrat. At present, dierent approximation ormulas are put orward or rotating spheres with dierent speeds and Magnus orce generated b its rotation. The above calculation method can be applied to high speed sel-rotating shells, but also can be applied to the general speed o tennis, as well as the low speed o table tennis, onl the approximate aerodnamic lit coeicient or these three speed cases is dierent [6-]. However, the modeling method o current literature does not consider the damping o the rotation speed due to the eect o air resistance. The ball rotation speed damping characteristics are ver obvious especiall in the rain das or the air humidit is high [-3]. Thereore, this paper establishes the damping model under the air resistance to urther improve the modeling and analsis method o the table tennis light process.. Analsis o Force and moment, kinematics modeling during light F All the orces acting o the ball in the course o light includes gravit G, resistance d, buoanc Fa and Magnus F M. The orce diagram is illustrated in igure. F a V θ x F M F d G Figure Schematic diagram o the orce acting on a ball o ree light θ Where is angle between the speed direction o ling ball and the table. The orces are calculated as the ollowing ormulas. The gravit is calculated as ormula. Set the air densit as ρ, the buoanc is calculated as ormula. The resistance is calculated as ormula 3. G = mg () Journal o Residuals Science & Technolog, Vol. 3, No. 8, 6. 6 DEStech Publications, Inc. doi:.783/issn /3/8/

2 4 Fa = rgπr 3 Fd = Cdρ AV (3) C Where d is the drag coeicient obtained rom the wind tunnel experiment, A is the cross-sectional area o the ball given b the ormula 4. A= π r (4) The Magnus orce generated b the spinning o ball can be calculated as the ormula 5. Where CM FM = CMρ AV 3 is the lit coeicient, which can be calculated at low speed as the ormula 6, and is the rotating angular speed o the ball. C M =. gv / + (6) According to the above-mentioned stress analsis, suppose that the displacement o the ball along the x-axis is x, the speed is x, the acceleration rate is x, and the displacement along the -axis is, the speed is, the acceleration rate is, the ollowing dierential equation can be established as in 7. x = V x = V () (5) V x = x= ( FM sin θ Fd cos θ ) m V = = ( mg + Fa FM cos θ Fd sin θ ) m (7) 3. Modeling o rotation angular speed considering damping Considering the damping eect o the spin angular speed o the ball, the sign o the damping coeicient or the positive rotate speed and negative rotate speed is opposite, and the ollowing dierential equation 8 is established. () t = () t T (8) () = The initial condition is, and is the initial speed ater the rebound or the return hit. And T is the damping time constant, T 8s which can be selected as =, and the value can be approximatel estimated b experiments using high-speed camera to observe the speed attenuating time. I the initial rotation angular speed is a, ater the time period o t, the speed attenuates to b, then the ollowing equation 9 can be established to estimate the damping time constant as ollows. t T ae = b (9) Then we get the results. And then The damping time constant can be estimated as the ormula. t = ln T b a t = T ln b a t = ln a b The precision o the parameters can also be improved b multiple measurements. T () () () Journal o Residuals Science & Technolog, Vol. 3, No. 8, 6. 6 DEStech Publications, Inc. doi:.783/issn /3/8/

3 4. Analsis o the Force and moment or the ball hitting Assuming at the moment ball contact with racket, the normal orce is F, the acting time is F t t, the tangential orce is and its acting t > time is. Generall tangential orce acting time is longer than the normal orce acting time, that is F, due to the recover coeicients o the tangential orce and normal orce are dierent or the sponge; thereore the ball has a riction and rolling a distance in the rubber surace. In this paper, we ignore the change o the orce direction in the rolling o ball, and onl consider the average eect. At this point the orce diagram is as ollows in igure, and θ is the angle between the table and racket. F θ Figure Schematic diagram o orce acting on the ball First o all, establish the coordinate sstem with the contact point as the coordinate origin, and the normal as the F axis, the tangent as the axis. The initial speed o the vertical direction is not considered at irst, but considering the speed and angular velocit incremental caused b normal orce and tangential orce to table tennis. We can obtain the results as ormula 3 rom impulse theorem. FtF = m VF t = m V t r = I Where r is the radius o the ball, I is the moment o inertia,which is calculated b ormula: 3 I = mr V and V. F, are the speed increment o F-axis, speed increment o -axis and angular speed increment respectivel, considering the clockwise as the positive angular speed. And then establish the coordinate sstem according to the table tennis table, set the length o the table as x-axis, the height as the -axis, the intersection o middle line and the width as the point o origin, as illustrated in igure3. (3) Middle line o table o F θ θ x Figure 3 Coordinate sstem according to the table tennis table The above speeds are projected to the coordinate sstem established here. Then the ormula 4 can be obtained. V = V sinθ V x F V = V cosθ + V cosθ sinθ F (4) v Considering the initial vertical speed along the - axis is, the initial speed along the x-axis is v x, then speed o the ball ater the hitting is described as ormula 5. v = V + v v = V + v x x x Suppose the rotation angular speed is beore hitting, and takes counterclockwise as positive, and then the rotation angular speed ater hitting is given b ormula 6. (5) Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

4 = + (6) 5. Analsis o the orce and moment in the process o hitting The table tennis ball will rebound ater the irst touch o the table. Assuming that the component o the speed ater the irst bounce V V along the x-axis is xt, the component along the -axis is t, the rotation angular speed beore the rebound is t, and the rotation angular speed ater the bounce is t. Suppose the orce acting on the ball during the bounce is normal orce Nt and a riction orce t. Since the table can be regarded as a rigid bod, it is assumed that the acting time o the above-mentioned two orces is the same. Assuming T that the acting time is t, and the rebound process is ver short, thus the impact on the ball speed b gravit, buoanc, air resistance and Magnus orce can be ignored in this process. There are two situations concerning the orce analsis during the rebound process, and the irst case is that the rotation speed o the ball is weakened, and then the absolute speed at rebound point is given as 7. Va = Vxt t r < (7) The ball lies relativel backward to the table, and the direction o the riction orce is positive along the x-axis direction, as shown in Figure 4, the ling speed along the x-axis is enhanced but the ball spinning is weakened ater rebound. t θ x N t t V Figure 4 Schematic diagram o the orce on ball when the rotating speed is weakened The second case is that the spinning speed is enhanced ater the rebound, and the absolute speed at rebound point is given b ormula 8. Va = Vxt t r > (8) The ball lies relativel orward to the table, and the direction o the riction orce is negative along the x-axis direction, the ling speed along the x-axis is decreased but the ball spinning is enhanced ater rebound. According to the above stress analsis, assuming the riction coeicient is µ, and then we get ormula 9. The impulse can be described as ormula. S N = N µ = t S = dt = N µ dt t t t (9) Ndt () Obviousl we get ormula. N Analsis o the irst case o rebound, the ollowing results can be derived rom the theorem o impulse, as given in ormula. S = µ S mv mv = S xt xt mv mv = S t t N I I = Sr t t Suppose the vertical speed recover coeicient is Γ, and then we get ormula 3. t t The solution o the above equation is given as the ollows in 4. V V = ΓV = ΓV t t V = V µ V ( +Γ) xt xt t 3µ t = t + Vt ( +Γ) r () () (3) (4) Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

5 Analsis o the second case o rebound, the ollowing results can be derived rom the theorem o impulse, as given in ormula 5. The solution o the above equation is given as the ormulas 6. V mv mv = S xt xt mv mv = S t t N I I = Sr t t = ΓV t t V = V + µ V ( +Γ) xt xt t 3µ t = t Vt ( +Γ) r (5) (6) 6. Simulation analsis o the trajector o the ball Through the above analsis o the light process, the ollowing is the simulation o the trajector o the ball ater athlete's batting. It is T = 8s assumed that the attenuating time constant is, the starting position o the ball awa rom the table edge is.5m. As shown in Figure 5, the normal orce is.39n, and the tangential orce is.979n in the moment o the ball back, the speed in the vertical direction is.878 m/s ater the ball back, the orward ling speed is -9. m/s, and the angle between the racket and the table is 45 degrees. The speed increment in tangential direction is m/s, and m/s in normal direction produced b batting motion. The detailed trajector is shown in Figure 5.. Speed o Vs, ater return the ball : Speed o Vb,ater return the ball: Angle o the pat when return the ball,sita:45 Time constant o damping o rotation speed:8 Speed o Vx, ater return the ball : Speed o V,ater return the ball:.878 Force Fb, normal orce :.39 Force F,tangential orce: Figure 5 Simulation trajector o the table tennis ball It can be seen rom Figure 5, the table tennis ball rebounds twice ater serving, and the parameter design o batting the ball is shown in Figure 5, the ball is just over the net when it is batted back. In the processes o three times rebound and the ball is batted, the rotation speed is shown in Figure 6, which is continuousl damping when the ball is back. 6 4 gama Figure 6 Simulation curve o the ball's rotation speed Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

6 T = s Set the damping time, the trajector o the ball is shown in Figure Speed o Vs, ater return the ball :.34 Speed o Vb,ater return the ball: Angle o the pat when return the ball,sita:45 Time constant o damping o rotation speed: Speed o Vx, ater return the ball :-9.97 Speed o V,ater return the ball:.878 Force Fb, normal orce :.39 Force F,tangential orce: Figure 7 Simulation curve o the trajector o the ball At this case, the ball's rotational speed decreases sharpl, as shown in igure gama The height o the ball is shown in Figure Figure 8 Simulation curve o the ball's rotational speed Figure 9 Simulation curve o the height o the ball The orward speed curve o the ball is shown in Figure below,and the vertical speed curve is shown in Figure below. Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

7 5 Vx Figure Simulation curve o orward speed 3 V Figure Simulation curve o vertical speed It can be seen rom Figure that the vertical speed is still increases rom.4 m/s to.8 m/s when the ball is back, and the changes o the vertical speed is more signiicant when the ball is rebounded. The angle changes curve o the speed is shown in Figure. 3 sita Figure Simulation curve o the angle changes o speed Then consider increasing the normal orce and tangential orce when batting the ball, making the ball track to the edge o the table, and test the accurac o the athletes to control the orce demanding the ball over the net without out o bounds. Firstl, maintain the tangential orce and batting angle unchanged, increasing the normal orce o the impact rom.39 N to.58 N, and the trajector has changed as shown in Figure 3. Obviousl, the back ball touches the net. The main reason is that orward speed increases, meanwhile the height is too low. So, reduce the normal orce, and increase the height o the ball, and the trajector is shown in Figure 4. Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

8 Speed o Vs, ater return the ball :.34 Speed o Vb,ater return the ball:9.58 Angle o the pat when return the ball,sita:45 Time constant o damping o rotation speed: Speed o Vx, ater return the ball :-9.79 Speed o V,ater return the ball:.3646 Force Fb, normal orce :.587 Force F,tangential orce: Figure 3 Simulation curve o the trajector o the ball Speed o Vs, ater return the ball :.34 Speed o Vb,ater return the ball:7.6 Angle o the pat when return the ball,sita:45 Time constant o damping o rotation speed: Speed o Vx, ater return the ball : Speed o V,ater return the ball:.756 Force Fb, normal orce :.58 Force F,tangential orce: Figure 4 Simulation curve o the trajector o the ball As shown in Figure 4, when orward impact orce is reduced to.54 N, the height o the ball increases, and the drop point is close to the bottom line. Mainl the hitting is not so good because the orward speed is too slow and the height is too high. Tr to increase the riction between the ball and racket, which makes the ball close to the bottom line and the height lower. Secondl, maintaining the normal orce and batting angle unchanged, increase the tangential orce to enhance the riction, and make the drop point close to the bottom line. As shown in Figure 5, increase the tangential riction orce to 3.9 N, the drop point is close to the bottom line o the table Speed o Vs, ater return the ball :.956 Speed o Vb,ater return the ball: Angle o the pat when return the ball,sita:45 Time constant o damping o rotation speed: Speed o Vx, ater return the ball :-.3 Speed o V,ater return the ball:.699 Force Fb, normal orce :.39 Force F,tangential orce: Figure 5 Simulation curve o the trajector o the ball Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

9 Finall, maintaining the batting angle unchanged, increase the tangential orce and normal orce simultaneousl to improve the riction and speed o the ball. As shown in Figure 5, the tangential riction increases to 4.7 N, the normal orce increases to 3.9 N, the back trajector is relativel lat and the rebound speed is relativel higher, as seen in Figure 7, and this is a good batting Speed o Vs, ater return the ball :5.993 Speed o Vb,ater return the ball:.956 Angle o the pat when return the ball,sita:45 Time constant o damping o rotation speed: Speed o Vx, ater return the ball : Speed o V,ater return the ball:.39 Force Fb, normal orce :3.98 Force F,tangential orce: Figure 6 Simulation curve o the trajector o the ball 5 Vx Figure 7 Simulation curve o orward speed The above is the simulation results with the starting position o the ball awa rom the table edge.6 meter and the batting angle is ixed at 45 degrees. The impact o dierent riction orce and batting orce on the trajector o the ball as well as the qualit o batting is ver complicated. I take it considered, such as position and angle o batting the ball, table tennis skills and tactical choices become more complicated. 7. Conclusion Based on the analsis o the normal and tangential riction orces acting on the ball, and the analsis o the rebound orce as well as the aerodnamic analsis o the ree-ling process, a set o analtical method or analzing the trajector o the table tennis ball is established b means o modeling o rotation angular speed damping and estimating o damping constant time, and the trajector is simulated b computer. The detailed simulation results show the motion trajector o the ball under dierent combinations o riction orce and normal orce. The results also show that the modeling method proposed in this paper can accuratel simulate the movement o the ball and its correlation with the hitting action. Considering the dierent selection o the intercept batting position and batting angle, obviousl the simulation analsis o the above situation is ver complicated. Based on the theoretical analsis, this paper onl simulates the trajector o the ball with ixed batting position and ixed batting angle. More abundant simulation and analsis work will be carried out in the uture to urther improve the theoretical research. Acknowledgements This work is supported b the 6 Scientiic Research Project o Hainan Provincial Department o Education (HnK6-5). Reerences [] Z. Sun, G. X. Yu, M. Gou, Aerodnamic principles o table tennis loop and numerical analsis o its ling route. Chin. Spor. Science. Vol. 4, No. 8, 69-7(8) [] Y. Q. Zhou, Z. N. Ye, Z. H. Wu, Calculation and analsis o air resistance in ball games. Phs. Engi. Vol., 55 () [3] J. Fang, Research on table tennis collision using the simulation model o the computer. Jour. Tianjin Inst. Phs. Edu. Vol. 3, No. 8, (3) [4] X. S. He, Theoretical mechanics: Advanced dnamics. Northwestern Poltechnical Universit Press. Xian (3) [5] F. G. Jiang, X. C. Li, Q. Y. Xu, Establishment and Simulation o the kinematics model o table tennis. Jour. Quu Norm. Univ. Vol., No. 34, 4 (8) [6] Hinds William C, Aerosol Technolog: properties, behavior and measurement o airborne particles. John wile & Sons Inc., New York (98) Journal o Residuals Science & Technolog, Vol. 3, No. 8, DEStech Publications, Inc. doi:.783/issn /3/8/

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