Sports have developed over time into an entire industry from the children who grow up playing sports to the professional players that fans love to watch. Because all sports involve movements of the human body, a bio-mechanical system, these movements may be modeled with mathematics and physics. Speciﬁcally, the sports of golf, tennis, and baseball can use the concepts of a double pendulum to maximize sport performance. The double pendulum model is applied to these sports by making the arm one pendulum and the club, racquet, or bat the second pendulum. A good understanding of the mechanics of a double pendulum and its connection to these sports can enhance the eﬀorts of players and aid them in their performance

Neal, Heston

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Bridges: A Journal of Student Research Issue 13 Article 5 2020 Using Mechanics of a Double Pendulum to Maximize Sport Performance Heston Neal Coastal Carolina University Follow this and additional works at: https://digitalcommons.coastal.edu/bridges  Part of the Engineering Science and Materials Commons, and the Mathematics Commons Recommended Citation Neal, Heston (2020) "Using Mechanics of a Double Pendulum to Maximize Sport Performance," Bridges: A Journal of Student Research: Vol. 13 : Iss. 13 , Article 5. Available at: https://digitalcommons.coastal.edu/bridges/vol13/iss13/5 This Article is brought to you for free and open access by the Journals and Peer-Reviewed Series at CCU Digital Commons. It has been accepted for inclusion in Bridges: A Journal of Student Research by an authorized editor of CCU Digital Commons. For more information, please contact commons@coastal.edu. Using Mechanics of a Double Pendulum to Maximize Sport PerformanceHeston NealEngineering Science DepartmentCoastal Carolina UniversityAdvisor: Prashant SansgiryABSTRACTSports have developed over time into an entire industry from the children who grow up playing sports to the professional players that fans love to watch. Because all sports involve movements of the human body, a bio-mechanical system, these movements may be modeled with mathematics and physics. Specifically, the sports of golf, tennis, and baseball can use the concepts of a double pendulum to maximize sport performance. The double pendulum model is applied to these sports by making the arm one pendulum and the club, racquet, or bat the second pendulum. A good understanding of the mechanics of a double pendulum and its connection to these sports can enhance the efforts of players and aid them in their performance.I. INTRODUCTIONA pendulum is defined as a small mass that is sus-pended from a wire or string that allows the mass toswing back and forth in a semi-circular motion [1]. Adouble pendulum is a similar system, but with two pen-dulums attached at a ”joint”. The double pendulum isa mathematical system that can be represented by twodifferential equations derived from Newton’s Second Lawof Motion, F = ma. Appendix A provides the derivationof a single pendulum to illustrate the origins of the dif-ferential equations representing a double pendulum. Avisual of a double pendulum can be seen in figure 1.FIG. 1. Visual Representation of a Double Pendulum [2]A. Introduction to Double Pendulum Model forGolfGolf was the original sport to be modeled by a doublependulum. For this model, it is assumed that the armsare one pendulum and the club is the other pendulumwith the wrist being the pivot point. It is assumed thatthe wrist ’hinge’ is restricted to prevent the club segmentfrom going too far behind the golfer’s head [3]. This is acombination of the tennis and baseball models.∗ Engineering Science Department Coastal Carolina University.B. Introduction to Double Pendulum Tennis ModelWe will next apply the general knowledge of a dou-ble pendulum motion to the sport of tennis. Here, theforearm is one pendulum and the racquet is the secondpendulum, with the joint or connection point being theperson’s wrist. Looking then at the physics of a tennisstroke, one can observe that gravity plays a minor rolebecause most of the forces in the system stem from themuscles in the arm and hand that swing in a horizon-tal or vertical plane [4]. Thus, analyzing this model willyield results that tennis players can use to potentiallymaximize their strokes and serves.C. Introduction to Double Pendulum BaseballModelA double pendulum can model two aspects of base-ball. The first instance is hitting the ball with a bat.This case differs from the tennis model in that there aretwo hands on the bat instead of one, and so there aremore body segments to consider. As a result, the me-chanics of swinging a bat appears to be more complexmodeling problem. One way to simplify the analysis ofsuch mechanics is to view both arms acting together as asingle pendulum segment hinged to the bat at the wrist[5]. The bat would remain the second pendulum. Thisis a simpler way to look at the system of swinging a batto get a better understanding of what is happening me-chanically.The second aspect of baseball that can be modeledby a double pendulum is pitching. This system consistsof the arm being one pendulum, while the hand is thesecond pendulum. The wrist acts as the pivot or jointconnection [5].2II. ANALYZING DOUBLE PENDULUM SWINGEXPERIMENTAgain, the sports of tennis, baseball, and golf all havein common a swing of some sort. Whether that is swing-ing a racquet, club, or bat, the double pendulum is mod-eling this motion to get a desired result in the specificsport. This result can vary from maximizing the velocityof a tennis serve or stroke to maximizing the trajectoryof a baseball or golf ball. With this in mind, double pen-dulum experiments have in fact been conducted to try todetermine what a ”perfect swing” might look like [6].One paper described an experiment that constructed adouble pendulum out of two rods to mimic the swingingmotion done by athletes in the sports of tennis, golf, orbaseball. Essentially, the experiment used a video record-ing to break down the swinging motion of a double pen-dulum to determine a way to transfer all the kinetic en-ergy from the swing to the ball. The idea was to finda way to achieve the maximum transfer for each of thesubject sports [6]. This experiment defined a ”perfectswing” as the transfer of all initial potential energy tokinetic energy in the swing of the rod to all that kineticenergy being transferred into the ball [6].A. Conclusion of Swing ExperimentThe results of the preceding experiment presented inliterature revealed a couple of key characteristics of a”perfect swing” [6]. Firstly, the arm and the instrumentshould be lined up fairly closely. Secondly, the goal is tohave the angular momentum of the instrument reach itsmaximum value at the moment of impact with the ball[6].Based on the results of the swing experiment, it is im-portant to note what was discovered by these researchers[6]. Essentially, the experiment was using a self-madedouble pendulum to see how the motion could correlateto a tennis, baseball, or golf swing. This highlighted away that physics and mathematics could be correlated tosports enhancement. The experiment demonstrated thatanalyzing a double pendulum can help to determine thebest points of contact in a swing which yields the mostdesired results. This means that the experiment foundthe most ideal position to swing at a ball based on thesport. Therefore, this is showing how a double pendulumcan be used to model sports and maximize the results.III. DOUBLE PENDULUM MODELMAXIMIZING GOLF, TENNIS, AND BASEBALLThus far, models have been established to show thatgolf, tennis, and baseball, can be modeled by a doublependulum. How is that really useful though? Simplystating that these sports can be modeled by a doublependulum does not show why that concept is useful orallow for mathematics and physics to be correlated tothe world of sports. To better understand why this isimportant, the described models were tested to determineways of maximizing certain results.A. Double Pendulum Model Maximizing GolfSwingThe model of a double pendulum for a golf swing wasused to study the mechanics of a golf swing and con-tributed to the sport with a new optimal way to swing theclub. The model itself is fairly simple and looks specifi-cally at the downswing. The results were that the com-bination of the effect from inertia and centripetal forceacting on the club can create an efficient downswing ifthe arms are accelerating using the correct force [3]. Thismethod eliminates wrist torque which led to a concept ingolf known as the natural wrist release [3]. Eliminatingwrist torque is essentially helping golfers by allowing theclub to accelerate naturally towards the desired goal ofhitting the ball. This means that based on the model,researchers have experimented and found a way to makea golf swing more efficient, which in turn maximizes thesport of golf because a better swing increases trajectoryof the ball.B. Double Pendulum Maximizing TennisAs previously stated, the model for a double pendulumfor tennis correlates to one pendulum being the arm andthe second pendulum being the racquet with the wristthe joint [4]. Figure 2 illustrates this relationship[7].FIG. 2. Double Pendulum Model for Tennis [4]Once again, this model is effective in displaying thesystem but to utilize it, is to look at how this model canmaximize some aspect of the sport of tennis. The modelfor tennis can be maximized by looking at the best wayto collide with the ball at a given stroke for a desiredreturn[7]. The swing experiment of section 2 discussesthe transfer of energy to get the maximum output ofthe ball and the same can be discussed here. A way to3generate a better angular velocity is through the energytransfer of one pendulum to the next or from the arm tothe racquet[7]. To best achieve this, the angular velocityof the arm portion should be zero to allow the angularvelocity of the racquet to reach its maximum value [7].Another observation is that the speed of the reboundball can increase by 20 percent depending on the torque.Therefore, the sport of tennis can not only be modeledby a double pendulum but can be analyzed using saidpendulum to get results that can lead to an efficient wayto hit a tennis ball.C. Double Pendulum Model Maximization ofBaseballBaseball, in itself, is a unique sport because it is a verymechanical sport that can use different aspects of physicsto enhance the players motions for a better outcome. Forexample, pitchers focus on small mechanical adjustmentsto increase velocity. The introduction explained the me-chanics involved with a baseball swing and how a doublependulum can model this. It also introduced anotheraspect of baseball which can model a double pendulumthat is pitching. The double pendulum is useful in under-standing the mechanics of a baseball swing[5] but lookingat a way to maximize the sport, leads to a focus on thepitching model. Figure 3 shows the comparison of thedefinition of a double pendulum to the model applied toa pitching arm [8].FIG. 3. The figure on the left, a, shows the model of a base-ball pitch while the figure of the right, b, is a simple doublependulum model [8].A study was done to determine the best way to pro-duce maximum velocity by looking at how non-muscularforces like gravity forces generate, absorb, and transfermechanical energy [8]. The study was modeled by athree-dimensional double pendulum with a moving pivotmodel to use a multi-body power analysis [8]. This meansthat the a double pendulum model was derived and thenused in this study to analyze the dynamic behavior asso-ciated with maximizing a pitch. The focus of the studywas the assessment of the motion of the forearm andwrist during a pitch. The experiment associated withthis study consisted of using video footage to record acollegiate male baseball pitcher. The results yielded thatthe hand segment transferred mechanical energy from theinternal force which consisted primarily of the centrifu-gal force [8]. Another observation was that the muscletorque applied to the wrist joint didn’t increase the en-ergy but instead, absorbed the mechanical energy fromthe hand segment. Thus, the wrist is not able to createa ”high ball velocity”[8]. Therefore, this article used areal life pitcher and the double pendulum model to seethe points at which a pitcher could achieve a maximumvelocity by analyzing the transfer of mechanical energyin the system. This uses a double pendulum model tomaximize an aspect of a sport.IV. CONCLUSIONBaseball, golf, and tennis are three sports that have noreal player to player physical contact. Other sports havethe added unpredictable variables of one on one contactwithin each play. This makes them unique systems thatcan be modeled by other physics or mathematical meth-ods. For the purpose of this paper, they were all modeledto a double pendulum. The purpose of modeling themwas to find a way to use the physics of a double pen-dulum to improve the sport in some way. Golf used themodel to discover a back swing, and to better understandthe mechanics of the swing itself. Tennis used the modelto find an efficient way to hit the ball. Lastly, baseballused the model to understand the mechanics of a base-ball swing and to maximize the points at which a pitchercould achieve maximum velocity. Thus, a double pendu-lum is a simple mathematical system but when appliedto the world of sports, it can yield helpful results.Appendix A: Derivation of a Single PendulumA way of understanding the differential equations of adouble pendulum is to first analyze a single pendulumand how it is derived. The first step is to look at a freebody diagram and see what forces are acting on the jointwhere the mass is hung. Thus, figure 4 represents thefree body diagram of a single pendulum[1].FIG. 4. Free body diagram of a single pendulum [1]Thus, from the free body diagram an equation can beformed using Newton’s second law, F = ma. To find4the total force, F , the individual forces on the free bodydiagram must be added as follows,F = maT cos θj − T sin θi−mgj = maNow, we have just found the left side of the equation,the sum of the forces. To find the right side, we will lookat what the acceleration will be and multiply that by themass. In physics, acceleration is the second derivative ofthe position and velocity is the first derivative of positionwhich is found using kinematics. Kinematics is defined asthe motion of objects without looking at forces. For this,trigonometry is used to find the initial positions in the xand y direction. Thus, the position of the pendulum is,s(t) = L sin θi− cos θjv(t) = θ′L cos θi+ θ′L sin θja(t) = L[θ′′ cos θi− θ′2 sin θi+ θ′′ sin θj + θ′2 cos θj].The above equations are simply showing the derivativeof the position to get velocity and then the derivativeof velocity to get the acceleration. Therefore, to finishNewton’s second law, F = ma, the mass just needs to bemultiplied by the acceleration found above and then setequal to the sum of the forces.T cos θj − T sin θi−mgj= m(L[θ′′ cos θi− θ′2 sin θi+ θ′′ sin θj + θ′2 cos θj])To simplify that equation, the terms are separated intotwo equations by the unit vectors. Thus,i : −T sin θ = mL(θ′′ cos θ − θ′2 sin θ) (A1)j : T cos θ −mg = mL(θ′′ sin θ + θ′2 cos θ). (A2)Now, we will eliminate the T terms by multiplying equa-tion A1 by cos θ and equation A2 by sin θ to get−T sin θ cos θ = mL(θ′′ cos2 θ − θ′2 sin θ cos θ) (A3)T sin θ cos θ −mg sin θ = mL(θ′′ sin2 θ + θ′2 cos θ sin θ)(A4)Using algebraic manipulation and simplification weget,θ′′ = − gLsin θ (A5)The above is the second order differential equation for asingle pendulum.A double pendulum follows the same method but in-stead of one differential equation it is derived into twosecond order differential equations because there are fourinitial equations from the free body diagram and theforces involved. The free body diagram for a double pen-dulum is displayed in figure 5.FIG. 5. Free body diagram of a Double Pendulum[9]Following the same steps as above, the differentialequations depicting a double pendulum are as follows,θ′′1 =−g(2m1 +m2) sin θ1 −m2g sin (θ1 − 2θ2) − 2 sin (θ1 − θ2)m2(θ′22 L2 + θ′21 cos (θ1 − θ2))L1(2m1 +m2 −m2 cos (2θ1 − 2θ2))θ′′2 =2 sin (θ1 − θ2)(θ′21 L1(m1 +m2) + g(m1 +m2) cos θ1 + θ′22 L2m2 cos (θ1 − θ2)L2(2m1 +m2 −m2 cos(2θ1 − 2θ2))[1] R. H. Paul Peter Urone, College physics (OpenStax, Hous-ton, 2012) Chap. 16.[2] Double pendulum dynamics, .[3] E. W. S. R. O. Nils Betzler, Stuart Monk and G. Sha,From the double pendulum model to full-body simulation:evolution of golf swing modeling, (2008).[4] R. Cross, a double pendulum model of tennis strokes,American Journal of Physics (2011).[5] R. Cross, Mechanics of swinging a bat, (2008).[6] R. Cross, A double pendulum swing experiment: In searchof a better bat, American Journal of Physics (2004).[7] S.-H. Youn, Double pendulum model for a tennis strokeincluding a collision process, (2015).[8] H. O. Tsutomu Jinji, Ken Ohta, Multi-body power analy-sis of the baseball pitching based on a double pendulum,9th Conference of the International Sports EngineeringAssociation (ISEA) (2012).[9] R. Allain, The serious physics behind a double pendulumfidget spinner, (2017).Heston Neal double majored in Mathematics and Engineering Science at CCU.  Her research into the double pendulum began as a robotics project in Math 490 before she combined her interest in sports and experience as a soccer player into the article printed here. Heston achieved President’s List honors during her undergraduate career and graduated in May 2020.   

Using Mechanics of a Double Pendulum to Maximize Sport Performance

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Using Mechanics of a Double Pendulum to Maximize Sport Performance

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