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 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 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 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: ; 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 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 . 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 † ˆ 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 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 is: damping c, and linear stiffness k. When the ball is mg not in contact with the ground, the equation of xà ˆ ; 5† motion, assuming no aerodynamic , 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 ˆ mg; 1† finite. where x is measured vertically up to the ball's with x ˆ 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 ˆ 0 after being dropped to the x 0† ˆ h0 and x_ 0† ˆ 0 for a ball released from rest time it first comes back to x ˆ 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† ˆ 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 ˆ mg 2†  exp ˆ 0 6† 2m k with the initial conditions of x 0† ˆ 0 and which in general has multiple solutions. x_ 0† ˆ v0 where v0 is the 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† ˆ 0 sin ! t ‡ cos ! t 2k! d k d   d v0 c mg   x t† ˆ exp t sin !d t c mg !d 2m k  exp t 3†   2m k  c  c ‰1 exp t cos !d t ‡ sin !d t : where the damped natural frequency, !d , is: 2m 2m!d 1 p (7) 2 !d ˆ 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 ˆ m 1 ‡ ; 14† v  c  ÁT  x t† ˆ 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 ˆ ln ": 15† ÁT  ÁT ˆ ; 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 ˆ 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† ! ˆ 1 ‡ ; 16† " ˆ 10† n ÁT  x_ 0† is the ratio of of separation to speed of and is a function of the contact time, ÁT, and the approach at the first bounce. The coefficient of coefficient of restitution, ". The damping ratio, : restitution can be shown to be a measure of the s !2 c lost in the collision. In a perfectly  ˆ 1 d ˆ p ; 2 , there is no loss and " ˆ 1. In an !n 2 km , 0 < " < 1, some kinetic energy is transformed into deformation of the material, can be found by substituting Equations (9) and , , and other forms of energy, and is (16) or Equations (14) and (15) giving: therefore unavailable. This loss is represented in "   #1=2 the model by damping, c. ln " ln " 2 The denominator of Equation (10) is simply the  ˆ 1 ‡ : 17†   speed of the ball prior to the first contact, v0, and the numerator is the rebound or post-impact speed The damping ratio depends solely on the coeffi- of the ball, v1. The latter can be found by differ- entiating Equation (7) and imposing the assump- cient of restitution, ". tion mg=k  jv0=!d j or alternatively differentiating Coefficient of restitution related to total time and Equation (8) directly to give: number of bounces cv  c  In the following, it is shown that the coefficient x_ t† ˆ 0 exp t sin ! t 2m! 2m d of restitution, ", can be related to the initial height d from which the ball is dropped, h , the total  c  0 v exp t cos ! t; 11† number of bounces of the ball, n, and the total 0 2m d time, Ttotal, that elapses from when the ball is and then substituting t ˆ ÁT with Equation (9) to dropped until it comes to rest. These parameters, give the rebound speed: indicated in the bounce history diagram of Fig. 3, can easily be determined in an experiment.   cÁT It is assumed that k, c, m, and " are constant v ˆ x_ ÁT† ˆ v exp : 12† 1 0 2m (independent of the velocity). With n and Ttotal assumed known, ÁT and ", and thus k and c, can From Equation (10), the coefficient of restitution be determined under the assumptions of constant can then be written simply as: ÁT and " for all bounces and negligible aero-   dynamic drag. cÁT The total time, T , is comprised of flight times " ˆ exp : 13† total 2m and contact times:

For fixed m and c, the shorter the contact time, Ttotal ˆ Tflight ‡ Tcontact: 18† 396 M. Nagurka and S. Huang

Equation (25) can pbe derived from i vi ˆ "vi1 ˆ " v0 and vi ˆ 2ghi. For the ball to come to rest: mg h ˆ "2nh  26† n 0 k where the upper limit is given by the equilibrium position of Equation (5). Substituting Equation (14) into the equality of Equation (26) and rearran- ging gives an expression for ÁT: v u "   # uh ln " 2 ÁT ˆ "nt 0 1 ‡ : 27† Fig. 3. Bounce history showing height versus time. g 

Substituting Equations (23), (24), and (27) into The sum of the flight times, Tflight, is: (18) gives: Xn s T ˆ 1 T ‡ T ; 19† flight 2 0 i h T ˆ 0 iˆ1 total g where the time to the first impact is 1 T , as 2 s 3 2 0     indicated in Fig. 3. Noting that the flight time p 1 ‡ " 2"n ln " 2  4 2 ‡ n"n 1 ‡ 5: for the ith bounce (i  1) can be written as: 1 "  T ˆ "T 20† i i1 (28) since v ˆ "v and v ˆ 1 gT , the sum of the flight i i1 i 2 i Equation (28) can be viewed as a single equation times for n bounces can be calculated using for unknown " in terms of h0, n, and Ttotal. Equation (19) as: Alternatively, it can be written in nondimensional 1 n1 form: Tflight ˆ T0 ‡ T1 1 ‡ " ‡ . . . ‡ " † 2 s     p   1 "n1 n 2 1 1 ‡ " 2" 2 n ln " ˆ T0 ‡ T0"  ˆ ‡ n" 1 ‡ ; 2 1 " 1 " 2  or 29†   n where 1 1 ‡ " 2" Tflight ˆ T0 : 21† 2 1 " T  ˆ total 30† 1 Since the time to the first impact: 2 T0 s is the total time divided by the time to the first 1 2h0 T0 ˆ ; 22† impact. Similarly, the contact time of Equation 2 g (27) can be presented in nondimensional form: s where h is the initial height of the ball, the total p   0 2 ln " 2 flight time for n bounces can alternatively be Á ˆ "n 1 ‡ ; 31† expressed as: 2  s   where 2h 1 ‡ " 2"n T ˆ 0 : 23† flight g 1 " ÁT Á ˆ ; 32† 1 T0 In Equation (18) the sum of the contact times, 2 Tcontact, can be written: is the contact time divided by the time to the first impact. Tcontact ˆ nÁT; 24† assuming the contact time, ÁT, at each bounce is Approximate relationships identical. It is possible to develop an expression for It is possible to develop simplified approximate ÁT in terms of ", n, and h0. For the i-th bounce, relationships for the case of j ln "†=j  1, which is the height the ball can reach is: representative of a ball with significant bounce, such as a ping-pong ball. For example, a ratio of j ln "†=j 2i hi ˆ " h0: 25† of 0.1 or smaller corresponds to 0:73 < " < 1. A Mass-Spring-Damper Model of a Bouncing Ball 397

For this case (for which ‰ ln "†=Š2 is 0.01 or where, from Equation (39), the coefficient of smaller), the stiffness, k, given by Equation (14) restitution, ", can be written: can be approximated as:  1   2 "  : 42† k  m : 33†  ‡ 1 ÁT Alternatively, it could be expressed as an approxi- mation of Equation (16): NUMERICAL AND EXPERIMENTAL  STUDIES !n  ; 34† ÁT A ping-pong ball (new Harvard, one-star) was which itself is an approximation of Equation (9). dropped from rest from a measured initial height Thus, approximately the contact time at a single of 30.5 cm onto a laboratory bench with a butcher- bounce is simply  times the inverse of the block top. The acoustic signals accompanying the undamped natural frequency. The contact time ball-table impacts were recorded using a micro- can also be approximated from Equation (27) as: phone attached to the sound card of a PC. The s method follows the procedure described in [5]. h From the temporal history of the bounce ÁT  "n 0 ; 35† of successive impacts, the total number of bounces g was determined to be n ˆ 64 and the total bounce time was determined to be Ttotal ˆ 6:2 s. The mass or, in nondimensional form, from Equation (31) of the ball used in the experiment was measured to as: be m ˆ 2:70 g. p The sequence of steps to determine the model 2 Á  "n: 36† parameters is as follows: 2 1. With known initial height h0, number of From Equation (17), the damping ratio, , for the bounces, n, and total time Ttotal, determine the case of higher values of " is: coefficient of restitution, ", from Equation (28). 2. With known h and n and calculated ", deter- ln " 0   37† mine the contact time, ÁT, from Equation (27).  3. With known m and calculated " and ÁT, providing a simple direct connection between the determine the stiffness, k, and damping, c, damping ratio and the coefficient of restitution. from Equations (14) and (15), respectively. For larger n, Equation (28) can be simplified to: s  Predicted coefficient of restitution 2h 1 ‡ " The coefficient of restitution, ", can be found T  0 ; 38† total g 1 " from Equation (28) given h0, n, and Ttotal. The relationship is shown in Fig. 4 for the case of which expresses T as the product of a factor h0 ˆ 30:5 cm and indicates that the total time total does not depend significantly on the number of depending only on "p(inparenthe ses) and the time bounces, especially for a large number of bounces to the first impact, 2h0=g, or in nondimensional form: where the asymptotic values are given by the approximation of Equation (38). 1 ‡ " Figure 4 provides a means to identify by inspec-   : 39† 1 " tion the coefficient of restitution. For Ttotal ˆ 6:2 s and n ˆ 64, the coefficient of restitution is Equations (38) and (39), which neglect the contact " ˆ 0:92. This value is also predicted from the times, do not depend on n, and can be rearranged approximate Equation (38), or equivalently from to find a simple equation for " in terms of Ttotal for Equation (42) with  ˆ 24:9 from Equations (22) a given h0. and (30). (Note that the time to the first impact is Finally, it is possible to arrive at expressions for 0.249 s from Equation (22).) the stiffness/mass and damping/mass ratios in terms of the coefficient of restitution and number of bounces. From Equations (14) and (27), Predicted contact time With the coefficient of restitution determined, k g the contact time, ÁT, at a single bounce can be ˆ "2n; 40† m h0 found from Equation (27) or the approximation of Equation (35). Equation (27) is shown graphically and from Equations (15) and (35): in Fig. 5, from which ÁT can be determined by r inspection given the values of n and ". In the c 2 g  "n ln " 41† experiment, for n ˆ 64 and " ˆ 0:92, the predicted m  h0 contact time ÁT ˆ 3:2 ms. 398 M. Nagurka and S. Huang

Fig. 4. Total time, Ttotal , as a function of number of bounces, n, and coefficient of restitution, " (marked on curve), from Equation (28) for a drop height, h0, of 30.5 cm.

Predicted stiffness and damping For the case of the ping-pong ball dropped from Given values for the coefficient of restitution and an initial height of 30.5 cm and with ÁT deter- contact time, the linear stiffness and viscous damp- mined to be 3.2 ms, k=m ˆ 9:6  105 s2. For ing of an equivalent mass-spring-damper model of m ˆ 2:7 g, the stiffness k ˆ 2:6 N/mm (or kPa). the ball can be determined. An expression for the For " ˆ 0:92 and ÁT ˆ 3:2 ms, c=m ˆ 50 s1, stiffness is given in Equation (14) and in simplified and for m ˆ 2:7 g the damping coefficient approximate form in Equation (33). Figure 6 c ˆ 0:135 N-s/m. graphically depicts these relationships in terms of k=m as a function of " for different values of ÁT. Predicted natural frequency and damping ratio The approximate equation (33) provides a highly From Equation (16) or the approximation from accurate prediction of k from Equation (14), a rearrangement of Equation (34) it is possible to showing only slight deviation at smaller values of find the natural frequency, !n. For ÁT ˆ 3:2 ms ". The damping coefficient c, given by Equation and " ˆ 0:92, !n ˆ 980 rad/s or 155 Hz. From (15), is plotted in Fig. 7 in terms of c=m as a function Equation (17) or the approximation of Equation of " and ÁT, showing clear dependence on both. (37) it is possible to predict the damping ratio, .

Fig. 5. Contact time, ÁT, at a single bounce as a function of number of bounces, n, and coefficient of restitution, " (marked on curve), from Equation (27), and from approximation from Equation (35), for a drop height, h0, of 30.5 cm. A Mass-Spring-Damper Model of a Bouncing Ball 399

Fig. 6. Stiffness divided by mass, k=m, as a function of coefficient of restitution, ", and contact time, ÁT, from Equation (14) and for approximation from Equation (33).

For " ˆ 0:92,  ˆ 0:026, indicating a very lightly mass-spring-damper model precludes accounting underdamped system. for nonlinear stiffness and effects in the impact. For example, the model cannot represent Hertzian force-displacement characteristics (e.g., DISCUSSION force related to displacement to the 2/3 power) nor Coulomb-type damping and hysteretic losses The vertical bounce history of a ball, dropped at the interface that are presumed to act. from rest, provides clues that can be used to In addition to fixed model parameters for the ball, determine an equivalent mechanical model of the it is assumed that at all impacts there is a constant ball. In the formulation, the ball is modeled as coefficient of restitution, ". A consequence of having constant and linear mass, stiffness, and assuming that the parameters m, k, c and " are damping properties (m, k, and c, respectively), constant is that contact time, ÁT, at each bounce which are related to key features of the bounce is constant. A more advanced model, incorpo- behavior (namely, h0, n, and Ttotal). This fixed rating nonlinear elements and/or a changing

Fig. 7. Damping coefficient divided by mass, c=m, as a function of coefficient of restitution, ", and contact time, ÁT, from Equation (15). 400 M. Nagurka and S. Huang coefficient of restitution, would be needed to clearly related topics. The analysis is limited by account for a contact time that decreases with several questionable assumptions, such as fixed, each bounce. linear properties of the ball, constant coefficient of The total time from when the ball is dropped restitution, constant contact time, and no aerody- until when it comes to rest is comprised of flight namic drag. These assumptions and others, such as times and contact times. Although the sum of the vertical motion only of a non-rotating ball, can contact times (Tcontact) for all bounces is a small each be addressed in more detail with discussion of fraction of the sum of the flight times (Tflight), it is their significance, and more advanced models can included in the model. (Here, this fraction is be developed. The analysis is suited to the level of approximately 3.5%.) The analysis neglects aero- an undergraduate engineering student. dynamic effects, which occur in reality during the flight times. By not accounting for aerodynamic drag of the ball during flight, the approach gives a CLOSING higher coefficient of restitution than otherwise would be predicted. This paper examines relationships linking linear Several observations can be made: mechanical model parameters, namely the mass, stiffness, and damping of a bouncing ball, with . the larger the contact time ÁT, the smaller the classical mechanical impact parameters, i.e., the stiffness k and the damping c; coefficient of restitution and the time of contact . the larger the coefficient of restitution ", the between a ball and a surface. Under the assump- smaller the damping c; tion of no aerodynamic drag and constant coeffi- . the coefficient of restitution " does not strongly cient of restitution for all bounces, the stiffness/ influence the stiffness k; mass and damping/mass ratios, or alternatively the . the larger the coefficient of restitution ", the natural frequency and damping ratio, can be larger the total time; expressed explicitly in terms of the coefficient of . the number of bounces n (for n > 0) does not restitution and time of contact. Ball properties are strongly influence the total time. shown to be related to parameters easily found in Finally, as an aside, it is noted that some people, an experiment, namely, the height from which the including the sight-impaired, can gauge reasonably ball is dropped, the number of bounces, and the accurately the drop height of a ping-pong ball total bounce time. Also considered is the special based on listening to the bounce time history. case of bouncing with higher values of coeffi- This ability may be a subconscious synthesis of cient of restitution for which simple approximate knowing the total time and number of bounces. expressions can be derived for model parameters. The results of an experimental test are used to Pedagogy provide predictions of the equivalent stiffness and In the interest of motivating students to stretch damping, natural frequency and damping ratio, beyond the rather rigid formalism of classical and coefficient of restitution for a bouncing ping- presentations, this paper bridges two distinct but pong ball.

REFERENCES

1. L. Flansburg and K. Hudnut, Dynamic Solutions for Linear Elastic Collisions, American J. Physics, 47, 1979, pp. 911±914. 2. E. Falcon, C. Laroche, S. Fauve and C. Costa, Behavior of one inelastic ball bouncing repeatedly off the ground, The European Physical Journal, 3, 1998, pp. 45±57. 3. L. Pauchard and S. Rica, Contact and compression of elastic spherical shells: the physics of a ping- pong ball, Philosophical Magazine B, 78(2), 1998, pp. 225±233. 4. C. Cross, The Bounce of a Ball, American Journal of Physics, 67(3), 1999, pp. 222±227. 5. M. L. Nagurka, A simple dynamics experiment based on acoustic emission, Mechatronics, 12(2), 2002, pp. 229±239. 6. M. L. Nagurka, Aerodynamic effects in a dropped ping-pong ball experiment, Int. J. Eng. Educ., 19(4), 2003, pp. 623±630. 7. C. Wu, L. Li and C. Thornton, Rebound behavior of spheres for plastic impacts, Int. J. Eng. Educ., 28, 2003, pp. 929±946. 8. E. Kanso and P. Papadopoulos, Dynamics of pseudo-rigid ball impact on rigid foundation, Int. J. Non-Linear Mechanics, 39, 2004, pp. 299±309.

Mark Nagurka is an Associate Professor of Mechanical and Biomedical Engineering at Marquette University (Milwaukee, WI). He earned a Ph.D. in Mechanical Engineering from M.I.T. (Cambridge, MA), taught at Carnegie Mellon University (Pittsburgh, PA), and was a Senior Research Engineer at the Carnegie Mellon Research Institute (Pittsburgh, PA). Dr. Nagurka is a registered Professional Engineer in Wisconsin and Pennsylvania, a Fellow of the American Society of Mechanical Engineers (ASME), and a former Fulbright A Mass-Spring-Damper Model of a Bouncing Ball 401

Scholar. His research interests include mechatronics, automation, control system design, human/machine interaction, and vehicle dynamics.

Shuguang Huang is a Research Assistant Professor in the Department of Mechanical and Industrial Engineering at Marquette University (Milwaukee WI). He received a BS degree in Mechanics and an MS degree in Theoretical Mechanics from Peking University (Beijing, China), earned a Ph.D. degree in Mechanical Engineering from Marquette University, and taught in the Department of Engineering Mechanics at Tsinghua University (Beijing, China). Dr. Huang is a member of IEEE. His general interests include classical mechanics, control and design of robotic systems.