Modeling of impact dynamics of tennis ball with a flat surface

摘要

A two-mass model with a spring and a damper in the vertical direction, accounting for vertical translational motion and a torsional spring and a damper connecting the rotational motion of two masses is used to simulate the dynamics of a tennis ball as it comes into contact with a flat surface. The model is supposed to behave as a rigid body in the horizontal direction. The model is used to predict contact of the ball with the ground and applies from start of contact to end of contact. The springs and dampers for both the vertical and the rotational direction are linear. Differential equations of motion for the two-mass system are formulated in a plane. Two scenarios of contact are considered: Slip and no-slip. In the slip case, Coulomb??s law relates the tangential contact force acting on the outer mass with the normal contact force, whereas in the no-slip case, a kinematic constraint relates the horizontal coordinate of the center of mass of the system with the rotational coordinate of the outer mass. Incorporating these constraints in the differential equations of motion and applying initial conditions, the equations are solved for kinematics and kinetics of these two different scenarios by application of the methods for the solutions of second-order linear differential equations. Experimental data for incidence and rebound kinematics of the tennis ball with incidence zero spin, topspin and backspin is available. The incidence angles in the data range from 17 degrees up to 70 degrees. Simulations using the developed equations are performed and for some specific ratios of inner and outer mass and mass moments of inertia, along with the spring-damper coefficients, theoretical predictions for the kinematics of rebound agree well with the experimental data. In many cases of incidence, the simulations predict transition from sliding to rolling during the contact, which is in accordance with the results obtained from available experimental measurements conducted on tennis balls. Thus the two-mass model provides a satisfactory approximation of the tennis ball dynamics during contact.