ISSN : 974-7435 Volume 8 Issue 9 BTAIJ, 8(9), 13 [1173-118] Mathematical analysis o the table tennis hitting process based on dynamics and hydrodynamics Desheng Chang Department o Physical Education, Zhejiang University o science and technology, Zhejiang 313 (CHINA) Email: 1717748@qq.com ABSTRACT In this paper, it conducts mechanics analysis on the orce condition and motion state during the movement o table tennis ball, and uses the kinetic theory and luid mechanics theory to describe the mechanical eatures o the ball. Through the mechanical parameters proves the inluencing actors o the ball s movement states, obtains eective mathematical equations, and conirms the scientiic principles that the movement process ollows. In the attack, in order to receive better attack eect the ball must have high-speed movement and strong rotation; in the deending, you must irst determine the speed rotation and drop point o the ball, prepare preparedness to ight response time and improve deense capabilities. 13 Trade Science Inc. - INDIA KEYWORDS Bernoulli s principle; Table tennis; Kinetics; Hydrodynamics. INTRODUCTION The achievements o Chinese table tennis players in international competitions are not only inseparable with the leaps and bounds improvement o technique in peacetime training, but also closely related with the improvements o table tennis hitting technique. Predecessors through scientiic means and analysis correct method o made the hitting table tennis technology continue to improve, but rom another point o view we can also draw the appropriate conclusions, and can develop it urther. This article abandon previous research methods in these areas, takes the physical movement mechanics this natural science as a research tool, conducts a detailed introduction on the collision momentum o racket and table tennis, orce analysis, trajectory ormation during the movement o the ball, and positioning analysis on the alling point o the ball, rom the perspective o tennis biomechanics conducts simulation analysis o table tennis trajectory to eectively understand the relationship between orce and trajectory in table tennis movement. It promotes the world popularity o national sport sports, improves the training eiciency o table tennis, and makes the physical mechanics better serve the table tennis. THE MECHANICAL ANALYSIS O THE COLLISION PROCESS BETWEEN TABLE TENNIS AND RACKET This paper describes in detail the entire process o the collision between tennis ball and racket, the ball movement in the air, the collision between the tennis
1174 Mathematical analysis o the table tennis hitting process based on dynamics and hydrodynamics BTAIJ, 8(9) 13 ball and the table and ly out the table surace, and conducts the mechanical analysis. The role o racket on table tennis Racket hitting table tennis ball is a comple mechanical process, this article takes the eect o the racket to the ball as the unction process o two orce, the impact orce perpendicular to the racket surace and the riction tangential to the racket surace, and the initial velocity and the initial angular velocity is. Suppose the mass o the table tennis ball is m, the diameter is D, shown in igure 1, the racket is bending orward and has a â angle with the horizontal surace, shoot action time o the ball, tennis racket and speed o impact. Through these three equations the ollowing can be successully solved: I v m (4) I v (5) m DI 3I I md (6) e igure 1 : Analysis chart o racket initial parameter t The impulse o in time t is dt denoted by I, the impulse o in timeis denoted by. Go through the centroid o table tennis ball, is with tangent tennis ball, so riction produces a torque o D I I e (3) D to table tennis, the impulse moment o this moment in time t is t D dt, that is D I. Tennis rotation moment o inertia o its own center 3 D is I mr ( r ), the inal velocity in the direction e perpendicular to the racket is v, the inal velocity along a direction parallel to the racket is v, the inal angular velocity is, according to the momentum conservation law and angular speed law in physics, we can obtain: I mv (1) I mv () igure : orce eploded view o the table tennis in and y direction The horizontal and vertical component v and v need to be decomposed into the horizontal and vertical directions. As shown in igure, and y respectively means the horizontal and vertical directions. Thus the ollowing relationship can be obtained: v I I v sin v cos sin cos (7) m m I I v y v sin v cos sin cos (8) m m By the ormula (7) (8) obtain the table tennis velocity ater the moment o collision with racket. Table Tennis movement trajectory Table tennis trajectory reers to the trajectory rom the table tennis leaving racket to just touching the table board, is the important actor to determine the touching point o the ball with the table board. Thereore, the analysis on the moving process is very important and is an important indicator o players to hit good shots. During the light table tennis mainly suers the vertically downward gravity, vertical upward buoyancy, also the resistance brought by air in the opposite direction o
BTAIJ, 8(9) 13 Desheng Chang 1175 movement. In the motion process o table tennis, due to the role o the racket, table tennis also rotates around its own ais, so we have to conduct detailed analysis on the rotation o the table tennis. or the table tennis trajectory analysis, we should assume that table tennis eercises in the plane oy, shown in igure 3. And the light speed o the ball is v, and the angle between the horizontal line is, the angular velocity o the ball rotates around itsel is, so the linear speed o rotation is, the ais o its rotation is perpendicular to plane oy. Table tennis ball s gravity is G mg, wherein, g is the commonly used 9.8m / s. We assume that the density o air is, so 1 3 buoyancy is buoyancy g D. Meanwhile, the resistance 6 that table tennis suers in the motion process is 1 resistance C Av, where isc the drag coeicient o table tennis, this resistance coeicient is related with the geometry o table tennis, D A 4 is the cross sectional area o table tennis. The rotation that table tennis has 3 in motion process is turn Cl D v, and where C is the l lit coeicient. d y 1 3 3 1 m mg g D C D v cos C Av sin L d (1) dt 6 These two equations are the orce analysis during the movement process o tennis in the air. Analyze the rotation o the ball in the movement process As shown in igure 4: apply a orce through centroid point O on the sphere, the sphere can only move along the direction parallel to the orce. As shown in igure 5: apply a orce deviate centroid point O on the sphere, the sphere can generate a parallel movement and produce rotation around itsel under the action o the. Its rotation eect is determined by the moment size o the on point O. rom the above analysis, giving the ball a orce not through the center o the sphere can make table tennis rotate up. igure 4 : Apply a orce through centroid point O on the sphere igure 3 : The orce analysis o table tennis lying in the air igure 3 is the orce analysis o table tennis lying in the air, the orce analysis directly determines the light path o table tennis; according to the stress analysis, the orce can be decomposed along and y direction, thus we can list equation epressions o motion as ollows. d 3 1 m C D v sin C Av cos L d (9) dt igure 5 : Apply a orce deviate centroid point O on the sphere Ball s rotation and riction orce rom the previous analysis, the key to make table tennis rotate up is that the orces acting on the ball is not passing through the center, and this orce is rom the riction o the ball on the racket. Shown in igure 6, when the racket strikes the ball, the racket have relative movement to the ball, this relative movement can produce riction.
1176 Mathematical analysis o the table tennis hitting process based on dynamics and hydrodynamics BTAIJ, 8(9) 13 Shown in igure 6, the moment o hitting the ball, the athletes pull the racket upward with orce; the ball suers the two orces; by this time is through the center o the sphere, and thereore does not produce torque. The time the ball light orward under the eect o, is tangent to the ball, generating the eect that the ball counterclockwise rotates, which is topspin ball in the table tennis movement process. Similarly, i pulling the racket in dierent directions at the racket hitting instant, it will produce dierent direction o the ball and riction tangent to the ball. thus creating downward lateral pressure on the ball. The light path o the ball is low, and topspin ball is just the opposite. Sidespin ball will appear lateral pressure; the role o this lateral pressure is to make the light direction o the ball side rotation, similar to the banana kick in ootball. (igure 9) trajectory is the parabolic trajectory o the no spiral ball, while trajectorya is the S-type trajectory o the strong side spiral ball. igure 6 : Ball s rotation and riction orce In act, in table tennis movement process: cut, chop, twist, pull, belt, lit and other technique actions reer to make the racket and the ball produce side relative movement when the racket and the ball contact instant, so that the ball suers the eect o the lateral riction and produces rotation. Bernoulli s Principle and curveball and rotation ball In a variety o table tennis light trajectories, oten there will be many curve balls whose trajectories are not in the same plane, that is, table tennis light is not in a plane, similar to banana kick in the ootball. So, why the ball will appear dierent and various arcs, air makes trouble mainly. To solve this problem we must understand the Bernoulli principle. In igure 7, blow between the reedom o the two pendulous white piece o paper, so that the air occurs low, we will ind two papers will be attracted to each other; according to the Bernoulli principle, the intensity o pressure is small when the luid low rate is big, and the intensity o pressure is big when the luid low rate is small, so that two pieces o paper will receive lateral pressure 1 and and be attracted to each other. In the orward process o table tennis, due to the rotation o the ball there will have a similar situation as shown in igure 8. Rotation ball study: the low rate o air above the ball is small relative to the ball and the low rate o air below the ball is big relative to the ball, igure7 : The Bernoulli principle phenomenon o two white papers igure 8 : The Bernoulli principle phenomenon o rotation igure 9 : The Bernoulli principle movement trajectory o the rotation ball THE MECHANICAL ANALYSIS ON THE HITTING OUT BALL Analysis on topspin ball Topspin ball lies in the air and rotates orward along the horizontal ais (igure 1); Due to viscosity, the ball will rotate along with the surrounding air. When the ball lights orward, the air circulation above the ball is in contrast with air resistance in the head-on ace, the air low rate o this region is small; the air circulation
BTAIJ, 8(9) 13 Desheng Chang 1177 below the ball is in accordance with air resistance in the head-on ace, the air low rate o this region is larger. According to Bernoulli s theorem, the intensity o pressure is small when the luid low rate is big, and the intensity o pressure is big when the luid low rate is small, so the upper edge o the ball has a strong pressure, the lower edge o the ball has a small pressure, the dierential pressure is generated along the upper edge and the lower edge and the direction is downward; As the sphere circulation o the upper and lower edges is perpendicular to the direction o the air circulation, it does not cause the pressure dierence. According to the parallelogram law, the joint orce (P) acting on the ball points to the ront below o the ball. rom the above analysis, topspin ball has a downward trend during the light, the light arc is relatively steep and the light distance is shorter than the non-rotating ball. the air low o the upper edge and lower edge o the ball does not change, and no pressure dierence is generated; the air circulation o the ront edge and back edge o the ball is perpendicular to the air low direction, the pressure dierence is also not generated; The air circulation in the ball s let is in the opposite direction o air low, and the air circulation in the ball s right is in the same direction o air low. The let and right side o the ball generates a pressure dierence, and the direction is right; the joint orce (P) acting on the ball points to the rear right o the ball, thereore, let sidespin ball has the right rotation trend in the air. While the light curvature o right side spin ball is in contrast with the let spin ball and has the let rotation trend. igure 11 : Bernoulli mechanical analysis o the let-spin ball The dropping on table analysis o the spin ball igure 1 : Bernoulli mechanical analysis o spin ball Analysis on the under-spin ball Under-spin ball is on the contrary, it lies in the air and rotates backward along the horizontal ais (igure 1); the air circulation above the ball is in accordance with the air low direction, the air circulation below the ball is in contrast with the air low direction. Thereore, the intensity o pressure o the ball s upper edge is small, the intensity o pressure o the ball s lower edge is big, pressure dierence is generated between the upper and lower edges and the direction is upward; The ront edge and back edge o the ball have no pressure dierence, the joint orce (P) acting on the ball points to the top rear o the ball. Thereore, under-spin ball has an upward trend during the light, the light arc is relatively big and the lying speed is more and more slow. Analysis on the let and right spin ball Let spin ball in light rotates let along the approimate vertical line ais (igure 11, top view). During light When the spin ball drops on the table, in addition to the interaction orce in the vertical direction between the ball and the table (reaction orce), the rotation orce o the ball in the horizontal direction also acts on the table, the rotational orce can acts on the table as rictional orce. Thus, the table gives the ball an equal and opposite reaction orce, i.e. the reaction o the rotational orce. igure 1 : Bernoulli mechanical analysis when the rotation ball drops on the table board 1) The dropping o table board analysis o the upperspin ball: When the topspin ball drops on the table, the rotation orce o the ball acts on the table rearward in the horizontal direction, and the reaction o the rotation orce is orward (igure 1). This orce
1178 Mathematical analysis o the table tennis hitting process based on dynamics and hydrodynamics BTAIJ, 8(9) 13 gives the ball a orward speed U, so the relected velocity o the ball is no longer V, but V. And it is resultant velocity V with U. The relection angle o topspin ball signiicantly is greater than the incidence angle (â> á), but the speed o the ball ater bouncing up rom the table is ast, the impulsion is big; it remains rotation state in advancing process. ) The dropping o table board analysis o the underspin ball: When the under-spin ball drops on the table, opposite to the topspin ball, the rotational orce o the ball is orward along the horizontal direction acting on table board, the rotation reaction orce o table-board applied to the ball is backward (igure 1). This orce gives the ball to increase a backward speed õ, so the relection speed o the ball is V. And it is resultant velocity V with õ. Ater the ball drops on the table, the relection angle is smaller than the angle o incidence (â <á), the orward speed o the ball ater a rebound is weakened and the impulsion is reduced, the ball becomes higher ater dropping on the table. I the ball s under-spin is very strong, the orward orce o the ball itsel is very small, and the backward reaction orce that tables gives the ball is larger than the orward orce, then the ball will appear rebound ater dropping on the table. 3) Dropping table-board analysis o side spin ball: When side spin ball drops on the table, because the ball is rotating approimate to the vertical ais, the ball does not change its orce on the table-board ater being dropped because o let and right side spin. Thereore, ater the side-spin ball drops o the table its light arc continues turning according to the original direction. lat block and touching the racket analysis o the spin ball When the topspin ball touches the racket, the continuing rotation ball will apply a riction rom up to down on a racket, and the racket will give the ball a reaction orce rom the bottom up (igure 13), this orce increases the angle that the ball bounce upwards. So ater the lat block and touching racket the topspin ball will rebound upward and easy to be out o bounds. The under-spin ball is opposite to the upper-spin ball; when the ball hits the racket ace, the reaction orce direction o rotation is downward; the eect o this orce is to decrease the angle o the ball rebound. So the under-spin ball ater lat block and touching racket can easily o the net. igure 13 : The mechanical analysis o the spin ball ater the lat block and touching racket Based on the above analysis, when lat blocking (catching the ball) the let-spin ball, the ball s bounce direction is right, and the ball is easily out o bounds to the right; when lat blocking the right-spin ball, the ball s bounce direction is let, and the ball is easily out o bounds to the let. THE ORCE WHEN THE BALL CONTACTS WITH THE TABLE-BOARD Ball receives riction o the table when the ball touching the table board, the size o the riction orce is related with primary inertial orce o the ball and the riction coeicient between the ball and the table board, the relevance with the value o X is small (unless X is very large). The direction o riction orce is perpendicular to the ront and back direction o the ball and opposite to the speed direction o the ball relatively to the table. Ater touching the table the ball bounces, the relection angle is greater than the incidence angle, shown in igure 14. According to collision theory, the relationship between the incident angle A and the relection angle B beore and ater the ball touches the table board: tan B tan A k Where, k is the coeicient o restitution, less than 1 (which may be determined eperimentally). It is thus clear that, there is always B> A, i.e. the moving route o ball bouncing ater touching the table is below the line
BTAIJ, 8(9) 13 Desheng Chang 1179 that the ball comes; or loop drive ball, because the ball rotates intensely even ater touching the platorm, the movement route o ball bouncing ater touching the table needs to plus the action on the ball, which continues to downward bend and side bend. Suppose the speed o the coming ball o the table is v, v and the angle between v and the vertical line is á; the instantaneous the coming ball drops on the table, the angular velocity o the ball around the horizontal ais O is, ater the collision with the table, due to the instantaneous riction eect, the horizontal velocity o the ball at the contact point is, recovery coeicient o the ball is K, the radius o the ball is r, the mass o the ball is m. By the collision centroid motion theorem: mu mv s (11) Wherein s is the impulse acting on the ball, u is the sphere center speed when the ball bounces. The projection o the ormula (11) into ais is: mu mv s (1) s Take sphere center O as the centroid, by the impulsive moment theorem: J ( s r (13) c J is the Moment o inertia o the ball, and the ball c is hollow, with J c ( / 3 ) mr ; as the horizontal velocity o the touching point ater the collision is equal to, so: r u (14) Substitute the ormula (13) into ormula (14) and eliminate s by the ormula (1) to obtain: igure 14 : Motion state diagram when the ball drops on the table board mr v J cr 1 ( 3v sin r ) (15) J mr 5 u c Ater the collision the ball s movement in the vertical direction is related to the recovery actor, thereore: u y Kv Kv cos (16) y So the bounce angle when the ball rebounded rom the table-board is: u 1 r tan ( 3 tan ) u 5K v cos (17) y CONCLUSIONS In Table tennis, when the athlete is on the oensive, to receive a better oensive eect the ball must have high-speed movement and strong rotation. The main way is increasing the hitting power o the pat on the ball to increase a greater acceleration o the ball, making the ball produce a greater speed in an instant. In deense, you must irst determine the speed rotation and placement o the ball, prepare preparedness to ight or the reaction time and improve deense capabilities. This paper conducted detailed mechanics analysis and speciic physics calculation on the various stages o the table tennis movement process, obtained the detailed calculation ormula o table tennis motion law, made the tennis ans have a deeper understanding and understanding o physics knowledge; at the same time, also provided some theoretical oundation or table tennis players and tennis enthusiasts to better grasp the technology, made the mechanics theoretical knowledge in physics be more closely applied to real lie. REERENCES [1] Bing Zhang, Yan eng; The Special Quality Evaluation o the Triple Jump and the Dierential Equation Model o Long Jump Mechanics Based on Gray Correlation Analysis. International Journal o Applied Mathematics and Statistics, 4(1), 136-143 (13). [] Cai Cui; Application o Mathematical Model or Simulation o 1-Meter Race. International Journal o Applied Mathematics and Statistics, 4(1), 39-316 (13). [3] ang Jie; The studies on Table Tennis Collision using computer simulation.journal o Tianjin Institute o Physical Education, 18(3), 47-5 (3). [4] Haibin Wang, Shuye Yang; An Analysis o Hurdle
118 Mathematical analysis o the table tennis hitting process based on dynamics and hydrodynamics BTAIJ, 8(9) 13 Perormance Prediction Based On Mechanical Analysis and Gray Prediction Model. International Journal o Applied Mathematics and Statistics, 39(9), 43-5 (13). [5] Hongwei Yang; Evaluation Model o Physical itness o Young Tennis Athletes Based On AHP- TOPSIS Comprehensive Evaluation. International Journal o Applied Mathematics and Statistics, 39(9), 188-195 (13). [6] Sun Zai, Yu Guang-in, Guo Mei; The aerodynamics principle and light trajectory simulation o loop table tennis ball. China Sport Science, (4), 69-75 (8). [7] Zhou Yu-qing, Ye Zhao-ning, Hao Zong-han; The calculation and analysis o air resistance in ball movement. Physics and Engineering, (1), 55-6 ().