Damped mechanical oscillator: Experiment and detailed energy analysis

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1 1 Damped mechanical oscillaor: Experimen and deailed energy analysis Tommaso Corridoni, DFA, Locarno, Swizerland Michele D Anna, Liceo canonale, Locarno, Swizerland Hans Fuchs, Zurich Universiy of Applied Sciences a Winerhur, Winerhur, Swizerland The following aricle appeared in The Physics Teacher 52, 88 (2014), and may be found a hp://dx.doi.org/ / Copyrigh (2015) AIP Publishing. This aricle may be downloaded for personal use only. Any oher use requires prior permission of he auhor and AIP Publishing. Inroducion The damped oscillaor is discussed in every high school exbook or inroducory physics course, and a large number of papers are devoed o i in physics didacics journals. Papers ypically focus on kinemaic and dynamic aspecs 1-5 and less ofen on energy. Among he laer, some are devoed o he peculiar decreasing behavior of energy characerized by ripples, 6,7 ha can be easily demonsraed by using a dynamic modeling approach. 8 In his noe we consider an oscillaor consising of a car running on a horizonal rack, wo springs, and a damping device creaed wih magnes and a meal plae aached o he car (Fig. 1). Using sensors and daa acquisiion sofware, 9 we measure kinemaic quaniies and hree forces: hose of he springs on he car and, separaely, he force beween magnes and he plae. A deailed analysis of he energy exchanges beween he car and he ineracing pars is obained. In paricular, we show ha only he energy exchanges wih he magnes are affeced by dissipaive processes while over a suiable ime inerval he ne energy exchanged beween car and springs equals zero. Experimenal se up and kinemaics The oscillaor is consruced from a car wih a meal plae on op and wo springs. The plae ineracs wih wo magnes aached o a force sensor (Fig. 1). Posiion and linear velociy are measured wih a roary moion sensor, while forces are obained wih he aid of hree separaed force sensors (for he ineracion of springs and car and for he ineracion of car and magnes). Figure 1 Kinemaic daa is aken by a roary moion sensor (RMS). The springs are fixed o wo force sensors (F 1 and F 2 ). The magnes responsible for he dissipaive ineracion are aached o a hird force sensor (F 3 ) allowing o deermine he horizonal value of he damping force on he car.

2 2 The mechanical characerisics of he oscillaor (addiional mass on he car; elasic consans of he springs) can be chosen quie freely, while he srengh of he magneic ineracion can be regulaed via he disance beween magnes and he plae. All quaniies necessary for he energy analysis, i.e., for energy ransfers, can be obained from measured forces and velociy. To perform he experimen, he car is displaced from he equilibrium posiion and hen released. Fig. 2 shows a ypical behavior of posiion vs. ime: he moion is damped. Figure 2 - Car posiion vs. ime, showing damped oscillaion. The iniial posiion of he oscillaor (0.110 m) is evidenced. Energy We consider now he energy aspecs of he oscillaor no only globally (inegraed) bu also by focusing on insananeous energy exchanges beween he car and he springs and beween he car and he magnes. In order o do his, we need o find he magniudes of he energy flows (power) ha characerize he ineracions. A firs insigh can be obained from a graphical represenaion of he velociy of he car and of he measured forces versus ime (Fig. 3 and Fig. 4). Figure 3 Force of one spring on he car (saring a abou 0.8 N) and car s velociy (saring a 0 m/s) vs. ime: he sensor F 1 in Fig. 1 les us measure he force of he spring (which is considered o be ideal) on he car. The force applied by he oher spring (no shown) is almos idenical. Since he velociy is zero a he urning poins and has a maximum when he car ransis hrough he equilibrium posiion, we undersand why measuremens show a shif of a quarer of a period beween he velociy and he spring force (in Fig. 3, only he velociy and he force on he car exered by he spring on he lef, measured by sensor F1, are displayed).

3 3 Figure 4 - Force of he magnes on he car and he car s velociy vs. ime. The velociy becomes negaive when he car is released. The force of he magnes on he car is proporional o he velociy bu wih opposie sign (Fig. 4). This agrees wih he assumpion ha he magneic force is of viscous ype. We now discuss he energy flows as funcions of ime. A firs, we consider he ineracion beween car and springs shown in Fig. 5. Figure 5 Energy exchange rae beween car and springs: posiive values mean an ingoing energy flow, i.e. he acion of he springs increases he energy of he car; a negaive value means an ouflow of energy from he car, i.e. he springs receive energy from he car. The rae of energy ransfer, i.e. he energy curren I E, is given by he insananeous relaion: (1) = [ + ] I ( ) F ( ) F ( ) v( ) Es s1 s2 where we separaely consider he conribuions of he wo springs used in he experimenal se up. The energy flow is equal o zero for wo saes of he sysem: when he car passes hrough he equilibrium posiion (he ne force of he springs is zero crossed poins in Fig. 5) or, alernaively, when he car invers is moion (when he velociy is zero cenered poins in Fig. 5). In order o undersand correcly he balance of energy, i is imporan o coherenly define he sign of an energy flow: posiive for incoming, negaive for ougoing. Fig. 5 is now easy o read: when he car goes away from he equilibrium posiion, an energy ransfer from he car o he springs akes place so ha he car s energy decreases; when he car moves back oward he equilibrium posiion, he energy flow reverses since he springs give back sored energy o he car.

4 4 One may wonder now if i is possible o recover experimenally he simple and well known resul for an ideal spring: for any properly chosen period beween wo consecuive passages of he car hrough he equilibrium posiion, for insance a ime A and B (crossed poins in Fig. 5), he energy balance for he (ideal) springs mus resul in a value of zero: B B (2) Es [ s1 s2 ] A I ( ) d = F ( ) + F ( ) v( ) d = 0 A The graphical inegraing ool of he sofware used allows us o es his requiremen: wihin he experimenal accuracy, he value of he seleced area in Fig. 5 equals zero (a correc calibraion of he force sensors and an adequae choice of he sampling frequency are crucial for his resul). A very imporan consequence of his propery is ha for each half period of he oscillaion he ime inerval of he moion away from he equilibrium posiion mus be shorer han ha of he corresponding moion oward he equilibrium posiion, while he oal ime, corresponding o half a period, remains consan (Fig. 5). Only in he ideal fricionless case would he away and oward moions be of he same duraion. The asymmery of hese wo ime inervals depends only on srengh and ype of he magneic force. In Fig. 6 he absolue values of he energy exchanges in he away and he oward moion for some half periods, obained by numerical inegraion, are compared. Exchanged energy (10-3 J) away 1oward 2away 2oward 3away 3oward 4away 4oward Figure 6 Energy exchanges beween wo consecuive ransis rough he equilibrium posiion: noe ha away and oward exchanges cancel for proper consecuive inervals while he magniude of he exchange decreases from half-period o half-period. In order o complee our discussion, we will now consider he energy exchange as a resul of a dissipaive process. In Fig. 7, he energy flow associaed wih he ineracion beween he magnes and he plae fixed on he car is shown. This quaniy is given by he insananeous relaion: (3) I ( ) = F ( ) v( ) Em m Since he magneic force and he velociy have opposie signs (Fig. 4), heir produc is always nonposiive, aking he value zero only in he posiions where he direcion of moion is reversed and he velociy is zero (cenered poins, Fig. 7). Daa shows ha his is indeed a purely dissipaive ineracion.

5 5 Figure 7 Energy exchange due o he dissipaive magneic force: he energy flow is always ougoing so ha he value of he rae a which energy is exchanged is always nonposiive (zero a he urning poins). The inegral over he whole process gives he oal amoun of energy dissipaed in he ineracion beween he magnes and he plae. One may wonder again if i is possible o recover anoher simple resul: he energy dissipaed should be given by he value of he energy iniially sored in he springs: (4) 0? 1 I d = K A 2 2 Em( ) sys ini Wih he help of he graphical inegraing ool, i is possible o deermine he value of he energy leaving he sysem, i.e., he energy dissipaed. The resuling amoun can be compared o he energy iniially sored in he springs which is calculaed from he measured iniial posiion (Fig. 2). Concree resuls show a difference ha depends upon he srengh of he magneic damping. This resul indicaes ha here are oher sources of dissipaion such as air, wheels, ec. In he example shown here, wih relaively low damping, elasic spring consans of 6.8 N/m and 7.0 N/m respecively, and an iniial ampliude of 11.0 cm, we obain J for he oal iniial energy sored in he springs, whereas he amoun of dissipaed energy equals J. Summary In his noe, we limied ourselves o a careful experimen and he presenaion of he main feaures of he differen energy flows in a damped mechanical oscillaor. We were able o show how an experimen can clarify he differing roles played by he spring forces and he dissipaive force(s) in he exchange of energy. This sudy can be exended boh experimenally and wih he help of dynamical compuer modeling. By performing addiional experimens wihou magneic damping, we can recognize he role of differen forms of mechanical ineracion in our daa. A dynamical model, on he oher hand, helps wih a more complee quaniaive analysis ha will allow us o disinguish beween differen forms of damping (sliding and/or viscous). This combined experimenal and compuer modeling approach will be discussed in a forhcoming paper. References: 1. Barra C., Srobel G. L., Sliding fricion and he harmonic oscillaor, Am. J. Phys , (May 1981).

6 6 2. Campbell K., Eason D., Bad Vibes: The Damped Oscillaor, Phys. Teach (May 1992). 3. Marchewka A., Abbo D.S., Beichner R. J., Oscillaor damped by a consan-magniude fricion force, Am. J. Phys , (April 2004). 4. Molina M. I., Exponenial Versus Linear Ampliude Decay in Damped Oscillaors, Phys. Teach (November 2004). 5. Kamela M., An Oscillaing Sysem wih Sliding Fricion, Phys. Teach (February 2007). 6. Karlow E. A., Ripples in he energy of a damped oscillaor, Am. J. Phys , (July 1994). 7. Basano L., Oonello P., Palesini V., Ripples in he energy of a damped oscillaor: The experimenal poin of view, Am. J. Phys , (Ocober 1996). 8. D Anna M., Lubini P., Marhl M., Grubelnik V., The energy of a damped oscillaor, Proceedings GIREP-EPEC Conference, Froniers of Physics Educaion, Opaja (2007). 9. In our experimen he daa were colleced using PASCO s Capsone sofware, he PS-2120 Roary moion sensor and hree PS-2189 High resoluion force sensors (hp://pasco.com).