© 2015 IOP Publishing Ltd. Trampolines can be found in many gardens and also in some playgrounds. They offer an easily accessible vertical motion that includes free fall. In this work, the motion on a trampoline is modelled by assuming a linear relation between force and deflection, giving harmonic oscillations for small amplitudes. An expression for the cycle-time is obtained in terms of maximum normalized force from the trampoline and the harmonic frequency. A simple expression is obtained for the ratio between air-time and harmonic period, and the maximum g-factor. The results are compared to experimental results, including accelerometer data showing 7g during bounces on a small trampoline in an amusement park play area. Similar results are obtained on a larger garden trampoline, and even larger accelerations have been measured for gymnastic trampolines

Eager, D

Pendrill, AM

Physics Education

Ann-Marie Pendrill

David Eager

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Free fall and harmonic oscillations: analyzing trampoline jumps

Pendrill, Ann-Marie

David, Eager

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Free fall and harmonic oscillations - analyzing trampoline jumpsPendrill, Ann-Marie; David, EagerPublished in:Physics EducationDOI:10.1088/0031-9120/50/1/642015Link to publicationCitation for published version (APA):Pendrill, A-M., & David, E. (2015). Free fall and harmonic oscillations - analyzing trampoline jumps. PhysicsEducation, (50), 64-70. https://doi.org/10.1088/0031-9120/50/1/64Total number of authors:2General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.Free fall and harmonic oscillations - analysingtrampoline jumpsAnn-Marie Pendrill1,2 and David Eager31) Department of Physics, University of Gothenburg, SE 412 96 Göteborg, Sweden2) National Resource Centre for Physics Education, Lund University, Box 118, SE221 00 Lund, Sweden3) Faculty of Engineering and Information Technology, University of TechnologySydney, AustraliaE-mail: Ann-Marie.Pendrill@physics.gu.se and David.Eager@uts.edu.auAbstract. Trampolines can be found in many gardens and also in some playgrounds.They offer an easily accessible vertical motion, including free fall. In this work, themotion on a trampoline is modelled by assuming a linear relation between force anddeflection, giving harmonic oscillations for small amplitudes. An expression for thecycle-time is obtained in terms of maximum normalised force from the trampolineand the harmonic frequency. A simple expression is obtained for the ratio betweenair-time and harmonic period, and the maximum g-factor. The results are comparedto experimental results, including accelerometer data showing 7g during bounces on asmall trampoline in an amusement park play area. Similar results are obtained in alarger garden trampoline and even stronger forces have been measured for gymnastictrampolines.(This is an author-created, un-copyedited version of an article published in PhysicsEducation 50 64-70 (2015). IOP Publishing Ltd is not responsible for any errors oromissions in this version of the manuscript or any version derived from it. The Versionof Record is available online at doi:10.1088/0031-9120/50/1/64.)1. IntroductionWhere do you experience the largest forces in an amusement park? Amusement parkstandards allow rides that make the rider feel up to 6 times heavier than normal,i.e. 6g, although many modern roller coasters stay well below 5g. As the Lisebergamusement park in Gothenburg [1] recently asked it’s Facebook followers to guess whichrides involves the largest forces, it was hardly surprising that the newer roller coastersand a free fall ride were the favourites. However, larger forces - up to 7g - were infact encountered in the small trampoline found in the children’s playground area atLiseberg (the ”Rabbit Land”). Similar data have also been obtained in a round domestictrampoline, such as the one shown in figure 1, where a spiral toy is used to provide avisual illustration of the forces on the user. (The same toy has been used e.g. to illustrateFree fall and harmonic oscillations - analysing trampoline jumps 2Figure 1. A spiral toy used as accelerometer during a trampoline jump. Note howthe spiral in the toy is contracted while the jumper is in the air, and expanded at thebottom of the jump. Also note how far down the trampoline is displaced in the lastphoto.forces in a swing [2]). When the feet are not in contact with the trampoline, the userexperiences free fall (0g), and the spiral in the toy is contracted. At the bottom of athe jump, when the trampoline bed experiences the largest displacement, a large forceis exerted on the user, as illustrated by the extension of the spiral in the toy. Below, wefirst analyse the motion during bounces in a trampoline, and then compare the resultsto data from the trampoline in Liseberg’s Rabbit land.2. From harmonic oscillator to free fallFor a simplified analysis, we neglect energy losses, both in contact with the trampolineand in the air. We consider a person with mass m jumping on a trampoline. As longas the person is in contact with the trampoline, it exerts an upwards force, given byF (z) = −kz, where k is the ”spring constant” of the trampoline, and z is the verticaldisplacement from the rest position of the trampoline (positive is upwards). Whenthe person stands still, the force from the trampoline exactly counteracts gravity, i.e.kz = −mg. For small oscillations around this equilibrium position, starting at thelowest point at time t = 0 the motion can be described by z(t) = −mg/k − A cosωt,where ω2 = k/m. We can thus writez(t) = −g/ω2 − A cosωtVelocity and acceleration are obtained by taking derivatives, givingv(t) = Aω sinωtFree fall and harmonic oscillations - analysing trampoline jumps 3Figure 2. Theoretical values for the elevation, speed and normalised force during ajump reaching 3g. The green line represents acceleration, the blue line velocity, andthe red line displacement. The data are presented using dimensionless quantities.a(t) = Aω2 cosωtThe period for small oscillations is given by T0 = 2π/ω. As long as Ag/ω2 the userremains in contact with the trampoline bed and the user’s motion is sinusoidal onthe z-axis. For larger amplitudes, the separation of the user from the trampoline bedcommences he force increases beyond 2g. As the force increases the user experienceslonger and longer periods of 0g (air-time).Like children on a swing, the trampoline user is continually converting energy toreach incrementally higher with each oscillation. While the feet are in contact with thetrampoline, the users can add energy to the system, by raising the centre of mass whenthe forces are largest, e.g. by stretching the legs or swinging the arms. While the feetare in contact with the trampoline, small changes in the body position allows the userto lift the centre of mass and thus increase the maximum potential energy on successivejumps. The details of the movements will not be considered here, but examples can befound e.g. in movie recordings from the 2012 Olympic games [3]. The potential energyis then converted to kinetic energy as the user falls and this kinetic energy in convertedinto to spring energy as the feet make contact with the trampoline and the user stretchesthe trampoline bed in a downward direction. As the user falls from greater and greaterheights the downward displacement of the trampoline bed becomes larger, causing largerforces from the trampoline on the user. During the contact time, the force from thetrampoline makes the speed rapidly decrease to zero at the bottom dead centre, andthen increasing again until the user passes the equilibrium position.Free fall and harmonic oscillations - analysing trampoline jumps 4Figure 3. Theoretical values for the displacement, velocity and normalised forceduring a jump reaching 3g, 5g and 7g. The green line represents acceleration, the blueline velocity, and the red line displacement.Since the whole body accelerates, every part of the body experiences strong forcesthat are analogous to the forces of a stronger gravitational field. These forces arecommonly referred to as g-forces. The trampoline doesn’t just provide coordinationand exercise for the users heart, leg muscles, core strength, long-bones and drain thepoisons from the lymphatic system, the elevated gravitational force is applied to eachand every cell of the user’s body. Each cell is cycled from 0g to more than 5g each timethe trampoline user cycles from weightless to a gravitational field 5 times that of earth.This cycling ‘exercises’ each and every cell of the user’s body.If the trampoline user experiences Ng, the maximum upward force from thetrampoline can be written as Nmg. This corresponds to a maximum downwarddeflection zN = −Nmg/k, and an amplitude A = (N − 1)mg/k = (N − 1)gω2.Let us now consider in more detail a jump, starting at the lowest point. The motionduring the first part of the jump when the jumper is in contact with the trampoline canstill be written as,z(t) = −g/ω2 − A cosωt = −g(1 + (N − 1) cosωt)ω2However, since the trampoline does not pull the jumper downwards, this expressionholds only as long as z(t) < 0, i. e. for ωt < arccos(−1/(N − 1)). After this time, thedisplacement becomes positive and only gravity acts on the trampoline user, who willbe in free fall until touching the trampoline again. The user then leaves the trampolineFree fall and harmonic oscillations - analysing trampoline jumps 5Figure 4. Dependence of the bouncing period for different values of the maximumforce, given as a ratio to the harmonic oscillator period, T0, for amplitudes smallerthan mg/k (blue). The green line gives the time in contact with the trampoline. Alsoshown (red) is the ratio between air-time and cycle-time.with a speedvt = Aω sin(arccos−1N − 1) = (N − 1)mgk√k/m√1− 1(N − 1)2which can be rewritten asvt = g√N2 − 2N/ωThis velocity also gives the maximum height during the jump: h = v2/2g. Lighterjumpers have a higher ω =√k/m and will reach a lower velocity for the sameacceleration at the bottom of the jump. Similarly, a stiffer trampoline will give lesserspeed for the same maximum acceleration. Neglegting energy losses, the jumper willland again after a timetair = 2√N2 − 2N/ω = T0√N2 − 2N/πFigure 2 shows the displacement, speed and forces during different parts of thebounce for a maximum force of 3g. Figure 3 shows the elevation, speed and force duringthe motion for a maximum force of 3mg, 5mg and 7mg. The time in contact with thetrampoline during a full cycle is given bytc = 2 arccos(−1/(N − 1))/ω = T0 arccos(−1/(N − 1))/πFor high bounces (large N values), the contact time during a cycle approaches T0/2.Free fall and harmonic oscillations - analysing trampoline jumps 6Using the analysis above we find en expression for the total time for one period onthe trampoline:T = tc + tair = T0arccos(−1/(N − 1)) +√N2 − 2NπFigure 4 shows how the ratio T/T0 for the bounces depends on the maximum force. Theratio of airtime to cycle-time q = tair/T grows with for high bounces and is also shownin figure 4.Figure 5. Accelerometer data for bounces on the 1.3m circular trampoline at Liseberg(sampling frequency 25 Hz). In addition, data for a gymnastic trampoline (samplingfrequency 100 Hz) has been included, exceeding 9g, which clearly exhibit the dampingand how cycle-time increases with the g-factor. The longer period for around 7g in thesecond graph also shows that the ratio k/m was smaller for that case.3. Experimental resultsAs preparation for Liseberg’s question about the largest g-force, it was decided toperform a measurement also on the small trampoline (1.3 m diameter) in the children’splay area. The data were collected using the Wireless Dynamic Sensor from Vernier[5] placed in a data vest. Using the accompanying program Logger Pro [6], the datacould also be synchronised with a movie recording of the jump, as shown in the videoabstract accompanying this paper. Figure 5 shows the force acting on a trampoline userduring bouncing, both in the small trampoline at Liseberg and in a larger gymnasticstrampoline.It is worth noting that, in spite of the name, an accelerometer does not measureacceleration, but instead the vector f = (a− g)/|g|. Thus, for free fall, when a = g,Free fall and harmonic oscillations - analysing trampoline jumps 7Figure 6. Details of the accelerometer data for the trampoline at Liseberg shown infigure 5. The red curve shows the magnitude of the vector sum of the components,whereas the blue and green curves show the z and x components. The negative x inthe beginning of the contact time indicates the user leaned somewhat forward, whereasthe negative z values after the contact time may have been obtained by pulling theknees up. Data were collected at 0.04 s intervals (25 Hz).it becomes zero. As expected, the accelerometer data are essentially zero g when thejumper is in the air, (apart from user movement while air-bourne during the jump). Thestrongest force acts at the lowest point, where the trampoline mat is most extended.The surprising result was that the largest g-force in the amusement park are to befound in the children’s area: around 7g. To compare the results with the theoreticalanalysis above, the resonance frequency for small oscillations is needed. Thus, a fewsmall bounces were performed, where the feet remained in contact with the trampoline.Figure 6 shows in more detail a few of the high bounces, as well as the final smallbounces. The ratio between these two periods should be directly related to the g-force,as seen in figure 4.4. Comparison between theory and experimentFrom figure 6 we can extract the cycle-times for small oscillations and for the highestbounces. We find that the ratio between these two cycle-times corresponds to amaximum force of about 6.2mg. In figure 7 the theoretical graph is superimposed on theexperimental data. The accelerometer data, however, shows more than 7g, which canpossibly be accounted for by the bouncer giving an extra push at the bottom. VernierFree fall and harmonic oscillations - analysing trampoline jumps 8does not specify the quality of the data from the WDSS sensor beyond 6g [5], but therelatively smooth data shown in figure 7 does not indicate any obvious problem.The relation between bounce period and force shown in figure 4 can be used toanalyse e.g. the elephant bouncing in [7], where the video-analysis gave a periodT ≈ 2.2 s, with a contact time less than 0.3 s. For large accelerations, the contact timeis close to T0/2, giving T0 ≈ 0.6 s (although the period seems close to 1 s for the fewelephant bounces shown in the movie before the elephant leaves the trampoline.) Thisratio would correspond to a maximum force of over 11mg. Rhett Alain [7] concludedfor other reasons that the movie was a fake.The relation shown in figure 4 can also be applied e.g. to the investigations ofdifferent trampoline spring systems by Eager et al [4]. In that work, it was found thatthe average ratio of air-time to cycle-time varied between 0.58 and 0.61 for the threetrampolines tested. The largest ratio corresponded to approximately 5g, slightly largerthat expected from the graph.The analysis leading to the graph in figure 4 is based on the assumption that thedeflection on the trampoline is proportional to the force. However, for deflections thatare large compared to the trampoline dimensions, the linear approximation is less valid[8]. This may account for the small differences found in the ratio obtained from Eageret al [4] and in the experimental data shown in figure 7, which were collected for jumpson the 1.3 m diameter trampoline in the amusement park.Figure 7. Comparison between theoretical results and accelerometer data. Samplingrate 25 Hz..Free fall and harmonic oscillations - analysing trampoline jumps 95. DiscussionMotion on a trampoline combines two common text-book examples: The harmonicoscillator and free fall. Whereas the free fall is independent of mass, the harmonicoscillator frequency is not. The analysis of the motion opens possibilities for a discussionof what variables influences the motion, and what variables are best suited for presentingthe results. Different trampoline stiffnesses and user masses can be combined into asingle variable, ω =√(k/m) or T0 = 2π/ω. The time dependence of displacement,speed and acceleration can then be expressed in terms of ω or T0, combined with thefactor N characterising the maximum force from the trampoline. The maximum g-forceon the user can be obtained from the ratio of airtime during the bounce. Trampolinesare commonly available and more use should be made of them in physics and scienceteaching. Studying motion upon the small trampoline at Liseberg will be included asone of the possible activities during future physics days. The trampoline motion issuitable for modelling, as well as successive refinement of the model, using e.g. videoanalysis and accelerometer data.AcknowledgementsWe would like to express our appreciation to Michael Nilsson at Liseberg who suggestedthe measurements in the trampoline at Liseberg and who volunteered to jump with theequipment. AMP would also like to thank her daughter for jumping with the spiral toyin figure 1.References[1] Liseberg http://www.liseberg.se and https://www.facebook.com/liseberg.se[2] Pendrill A-M and Williams G 2005 Swings and Slides Phys. Educ. 40 527[3] Women’s Trampoline Qualification - Gymnastics — London 2012 Olympicshttps://www.youtube.com/watch?v=TtEt Zngnxc[4] Eager D, Chapman C and Bondoc K 2012 Characterisation of trampoline bounce using acceleration7th Australasian Congress on Applied Mechanics, ACAM[5] Vernier, Wireless Dynamic Sensor System, http://www.vernier.com/products/sensors/wdss/[6] Vernier, Logger Pro, http://www.vernier.com/products/software/lp/[7] Rhett A 2009 Elephants can’t jump. Really. http://scienceblogs.com/dotphysics/2009/07/27/elephants-cant-jump-really/[8] http://physics.stackexchange.com/questions/1467/how-far-does-a-trampoline-vertically-deform-based-on-the-mass-of-the-object

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Free fall and harmonic oscillations - analyzing trampoline jumps

Trampolines can be found in many gardens and also in some playgrounds. They offer an easily accessible vertical motion that includes free fall. In this work, the motion on a trampoline is modelled by assuming a linear relation between force and deflection, giving harmonic oscillations for small amplitudes. An expression for the cycle-time is obtained in terms of maximum normalized force from the trampoline and the harmonic frequency. A simple expression is obtained for the ratio between air-time and harmonic period, and the maximum g-factor. The results are compared to experimental results, including accelerometer data showing 7g during bounces on a small trampoline in an amusement park play area. Similar results are obtained on a larger garden trampoline, and even larger accelerations have been measured for gymnastic trampolines

Pendrill, Ann-Marie,

Eager, D.

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Free fall and harmonic oscillations: Analyzing trampoline jumps

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Free fall and harmonic oscillations: Analyzing trampoline jumps

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