Marquette University e-Publications@Marquette Mechanical Engineering Faculty Research and Mechanical Engineering, Department of Publications 1-1-2006 A Mass-Spring-Damper Model of a Bouncing Ball Mark L. Nagurka Marquette University, [email protected] Shuguang Huang Marquette University, [email protected] Published version. International Journal of Engineering Education, Vol. 22, No. 2 (2006): 393-401. Permalink. © 2006 Tempus Publications. Used with permission. Int. J. Engng Ed. Vol. 22, No. 2, pp. 393±401, 2006 0949-149X/91 $3.00+0.00 Printed in Great Britain. # 2006 TEMPUS Publications. A Mass-Spring-Damper Model of a Bouncing Ball* MARK NAGURKA and SHUGUANG HUANG Department of Mechanical and Industrial Engineering, Marquette University, Milwaukee, WI 53201, USA. E-mail: [email protected], [email protected] The mechanical properties of a vertically dropped ball, represented by an equivalent mass-spring- damper model, are shown to be related to impact parameters. In particular, the paper develops relationships connecting the mass, stiffness and damping of a linear ball model to the coefficient of restitution and the contact time of the ball with the surface during one bounce. The paper also shows that the ball model parameters are functions of quantities readily determined in an experiment: (i) the height from which the ball is dropped from rest, (ii) the number of bounces, and (iii) the time elapsing between dropping the ball and the ball coming to rest. For a ball with significant bounce, approximate expressions are derived for the model parameters as well as for the natural frequency and damping ratio. Results from numerical and experimental studies of a bouncing ping-pong ball are presented. Keywords: mechanics; dynamics; mass-spring damper model; coefficient of restitution INTRODUCTION Motivation A challenging and meaningful part of engineer- THE BOUNCING BEHAVIOR of a dropped ball ing education is the development of system models is a classic problem studied in the literature [1±8] that reflect significant features of physical beha- and treated in textbooks of physics and dynamics vior. In many courses, classical models are that address the subject of impact. Many of these presented, at times with limited motivation. books also present, but in a separate section, the Many undergraduate engineering textbooks pres- concept of mass, stiffness, and damping as the ent the same material in the same sequence, further three elemental properties of a mechanical stifling the excitement and creativity of developing dynamic system. The behavior of such a system models based on physical insight and simplifying is represented by models comprised of series and assumptions. This paper is an attempt to take a parallel arrangements of masses, springs, and fresh look at two topics that are well established dampers, generally taken as linear elements. and related but not connected in the typical However, the books and literature do not connect presentation. The hope is that the student is the mechanical `primitives' of mass, stiffness and afforded an opportunity to learn more about damping to impact-related properties such as the both topics by building the connection. coefficient of restitution, a measure of rebound The physical properties of a real ball can be behavior in a collision. represented by its mass-spring-damping character- This paper develops this connection for a parti- istics. These characteristics are introduced in cular system, namely, a bouncing ball represented courses such as system dynamics and fundamen- by a linear mass-spring-damper model. It is shown tals of vibrations. In these courses as well as that the properties of the ball model can be related classical dynamics courses, the topic of impact of to the coefficient of restitution and the bounce bodies is often presented, but the connection contact time. Furthermore, assuming constant between the physics of collisions and the equiva- properties, it is shown that the coefficient of lent mechanical properties is generally missed. The restitution can be related to quantities easily natural question is whether one can determine the obtained in an experiment, namely, the height linear stiffness and viscous damping of a bounced from which the ball is dropped from rest, the ball based on its impact behavior, essentially total number of bounces, and the total bounce replacing the ball by an equivalent mass-spring- time. For a ball with significant bounce, such as a damper system. The development presented here is ping-pong ball, approximate expressions for the based on a linear model that only partially properties are derived. The analytical findings are captures the true physics. However, the intent is used to predict model properties of a ping-pong to prompt engineering students to synthesize ball. material and better appreciate and question assumptions inherent in the classical derivations, with the ultimate aim being an enhanced under- * Accepted 30 September 2005. standing of fundamentals. This paper does not 393 394 M. Nagurka and S. Huang present a definitive advanced model, but rather the and exp e , i.e., the base of the natural simplest model for determining the equivalent logarithm raised to the power . mechanical properties of the ball based on para- Equation (3) gives the motion of the ball during meters readily available experimentally. contact with the ground and applies only when x t 0. Bounce behavior, involving deformation, restitution, and then rebound, requires an under- MASS-SPRING-DAMPER MODEL damped solution for which !d > 0 or 4km c2 > 0. The `steady' or rest solution, The behavior of a vertically dropped ball is applying after the bounces have died out, can be studied using the model illustrated in Fig. 1, obtained by setting t ! 1 in Equation (3). The where the ball is represented by its mass m, viscous equilibrium position is: damping c, and linear stiffness k. When the ball is mg not in contact with the ground, the equation of xà ; 5 motion, assuming no aerodynamic drag, can be k written simply as: and when jxj jxÃj there will be no further bounces. It follows that the number of bounces is mx mg; 1 finite. where x is measured vertically up to the ball's center of mass with x 0 corresponding to initial Contact time at first bounce contact, i.e., when the ball just contacts the ground The contact time, ÁT, at the first bounce, shown with no deformation. The initial conditions are in exaggerated view in Fig. 2, is the time from when the ball reaches x 0 after being dropped to the x 0 h0 and x_ 0 0 for a ball released from rest time it first comes back to x 0. Mathematically, from height h0. The solution of this simple problem appears in physics and mechanics textbooks, lead- the contact time is the first finite solution of the ing to the classical results of vertical projectile equation x ÁT 0, i.e., from Equation (3) it is motion. the minimum non-zero solution of: When the ball is in contact with the ground, cg 2kv0 mg deformation and restitution occur. The equation sin !d ÁT cos !d ÁT of motion is then: 2k!d k cÁT mg mx cx_ kx mg 2  exp 0 6 2m k with the initial conditions of x 0 0 and which in general has multiple solutions. x_ 0 v0 where v0 is the velocity of the ball just prior to contact with the ground. Integrating Equation (6) is difficult to solve analytically. A Equation (2) gives: solution can be found numerically, or alternatively an approximate solution can be obtained by first cg 2kv mg writing Equation (3) in the rearranged form: x t 0 sin ! t cos ! t 2k! d k d d v0 c mg x t exp t sin !d t c mg !d 2m k  exp t 3 2m k c c Â1 exp t cos !d t sin !d t : where the damped natural frequency, !d , is: 2m 2m!d 1 p (7) 2 !d 4km c 4 2m The maximum magnitude of the first term on the right-hand side, v0=!d , is the dynamic deformation due to the impact for the incoming velocity v0; the Fig. 1. A mass-spring-damper model of a ball showing impact Fig. 2. Height versus time showing exaggerated view at first phases at the first bounce. bounce. A Mass-Spring-Damper Model of a Bouncing Ball 395 maximum magnitude of the second term, mg=k, is ÁT, the larger the coefficient of restitution, ", and the static deformation due to the weight. Assuming vice-versa. mg=k jv0=!d j, which is reasonable for a bounc- By manipulating Equations (4), (9), and (13), the ing ball such as a ping-pong ball, the second term stiffness, k, can be expressed as: on the right-hand side in Equation (7) can be " # neglected and x t can be approximated as: 2 ln " 2 k m 1 ; 14 v c ÁT x t 0 exp t sin ! t: 8 ! 2m d d and by rearranging Equation (13) the viscous The contact time, ÁT, can be found as the mini- damping, c, can be written as: mum nonzero solution of Equation (8) set equal to 2m zero giving: c ln ": 15 ÁT ÁT ; 9 ! From Equations (14) and (15), a shorter contact d time, ÁT, corresponds to both a higher stiffness, k, where !d is specified by Equation (4). Equation (9) and damping, c. In addition, as " increases, there is represents an approximate solution for the contact a negligible change in k and a reduction in c. time at the first bounce. Natural frequency and damping ratio at first Stiffness and damping at first bounce bounce p The ball stiffness, k, and damping, c, properties The undamped natural frequency, !n k=m, can be related to the contact time, ÁT, and the can be written from Equation (14) as: coefficient of restitution, ", where: s ln " 2 x_ ÁT ! 1 ; 16 " 10 n ÁT x_ 0 is the ratio of speed of separation to speed of and is a function of the contact time, ÁT, and the approach at the first bounce.