Normal coordinates in a system of coupled oscillators and influence of the masses of the springs
Alvaro Suarez, Daniel Baccino, Martin Monteiro, Arturo C. Marti
NNormal coordinates in a system of coupledoscillators and influence of the masses of the springs ´Alvaro Su´arez , Daniel Baccino , Mart´ın Monteiro , Arturo C.Mart´ı IPA-ANEP, Montevideo, Uruguay Universidad ORT Uruguay Instituto de F´ısica, Universidad de la Rep´ublica, UruguayE-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
E-mail: [email protected]
Abstract.
Experimental analysis of the motion in a system of two coupled oscillatorswith arbitrary initial conditions was performed and the normal coordinates wereobtained directly. The system consisted of two gliders moving on an air track, joinedtogether by a spring and joined by two other springs to the fixed ends. From thepositions of the center of mass and the relative distance, acquired by analysis of thedigital video of the experiment, normal coordinates were obtained, and by a non linearfit the normal frequencies were also obtained. It is shown that although the masses ofthe springs are relatively small compared to that of the gliders, it is necessary to takethem into consideration to improve the agreement with the experimental results. Thisexperimental-theoretical proposal is targeted to an undergraduate laboratory. a r X i v : . [ phy s i c s . e d - ph ] J u l ormal coordinates in a system of coupled oscillators
1. Statement of the problem
In mechanical systems with several degrees of freedom, in general terms it is not possibleto obtain complete solutions of the equations of motion. A notable exception to thisrule is the dynamics of conservative systems slightly separated from a point of stableequilibrium. In this case, the general solutions known as small oscillations are givenby simple periodic solutions or normal modes characterized by their frequency, knownas normal frequency. Knowledge of these normal modes allows the description ingeneral terms of the dynamics of the system. Notably, if the initial conditions aresuitably chosen, it is possible to excite the normal modes one by one and, settingaside perturbations, the system will continue to oscillate in the same normal mode.This selection of initial conditions (excepting some systems with symmetries) is nottrivial a priori, and can only be made after having solved the differential equationsof motion. In general terms, given arbitrary initial conditions, the motion will be alinear combination of all the normal modes of oscillation. However, there is a methodfor uncoupling the equations of motion and expressing them in terms of new variablescalled normal coordinates which behave as simple uncoupled oscillators. The generalprocess for obtaining these normal coordinates is, from the mathematical point of view,very straightforward but it often lacks physical meaning. Analysis of normal modes alsoplays a very important role in Solid State Physics where atoms in a crystal lattice aremodelled as a set of masses joined by springs oscillating at preestablished frequencies.The normal modes of the crystal lattice – phonons – allow explanation of the electricaland thermodynamic properties of materials.In university courses, the study of coupled oscillations usually begins in courseson Waves or Classical Mechanics with systems having two degrees of freedom. Usuallythe equations of motion are obtained, the normal frequencies are calculated and thenormal modes of oscillation are incorporated as a tool to describe the dynamics ofthe oscillators [1, 2]. Then the concept of normal coordinates is introduced as amathematical contrivance to uncouple the system of differential equations of motionand it is shown that they give rise to equations analogous to those of simple harmonicmotion with the frequency of the normal mode. Unlike normal modes of oscillation,where different experiments are usually done to excite each particular mode, the useof these normal coordinates is treated as secondary, without much interest from thepractical point of view.In more advanced courses of Analytical Mechanics, the study of oscillating systemsis taken up again, but from the perspective of Lagrange formalism. It is shown in generalthat for systems with n degrees of freedom, motion can be described using a set of thesame number of normal coordinates, each oscillating with a defined normal frequency.Again, as in the basic courses, the absence of experiments to study specifically thedynamics of normal coordinates could induce students to hold the false idea that suchcoordinates are only a mathematical contrivance.Several proposals for experiments for analysis of normal modes and normal ormal coordinates in a system of coupled oscillators
2. Normal modes in coupled systems
We consider a system of two coupled masses joined together and to the fixed ends withsprings as indicated in Figure 1. Our hypothesis is that the masses of both gliders M and M are equal to M and the three elastic constants k , k and k are all equal,and equal to k . The gliders can move in the direction indicated and friction betweenthe gliders and the surface is negligible. We call the coordinates of each glider withrespect to the equilibrium position x and x , and the distance between the blocks inthe equilibrium position d . The equations of motion can be found easily according to ormal coordinates in a system of coupled oscillators M d x dt = − kx + kx , (1) M d x dt = kx − kx . (2)We can find the normal coordinates directly, without using analytical mechanicstools, by adding and subtracting the above equations, obtaining directly an uncoupledsystem of equations, for the coordinates q S = x + x and q A = x − x : M d q S dt = − kq S , (3) M d q A dt = − kq A . (4)where q S and q A are the normal coordinates of the system, each having simple harmonicmotion with frequencies w S = (cid:113) k/M and w A = (cid:113) k/M , independently of thecharacteristics of the motion of each of the bodies that make up the system.We can observe that, as the position of the center of mass measured from theequilibrium position of glider 1 is and the position of glider 2 with respect to 1 is x / ,the normal coordinates q S and q A are given by q S = 2 x cm − d (5) q A = x / − d (6)So we see that, given the change over time of the center of mass and the relativeposition of one glider with respect to the other, we can easily find the change overtime of each normal coordinate, whose oscillation frequencies w S and w A correspond tothe frequencies of the normal modes of symmetric and antisymmetric oscillation in thesystem of two gliders, Eqs. 1-2.
3. Experimental set-up
The experimental system is composed of two gliders moving on an air track and threesprings, one joining the gliders together and the other two joining each of the glidersto the ends of the track, arranged a in linear fashion as shown in Figure 1. The airtrack minimizes the friction between the gliders and the track. The air track and thegliders (SF-9214), the set of 3 springs (ME-9830) and the air source (SF-9216) wereprovided by PASCO. The masses of the gliders, measured on electronic scales, were M = M = 0 . k = k = k = 3 . ormal coordinates in a system of coupled oscillators Figure 1.
Diagram of the experimental set-up composed of two gliders joined bysprings to each other and to the fixed ends. To minimize friction the gliders move onan air track. gliders is recorded by a digital camera. In this case the camera built into a SamsungGalaxy S10e smartphone was used, fixed to a support so that its optical axis was ata right angle to the track. The digital video obtained was analysed using Tracker freesoftware [15]. This software is commonly used to record the motion of point masses indifferent situations, for example the bob of a pendulum [16], the trajectory of a modelcar [17], or more complicated systems [18] by recording the coordinates in the laboratoryframe of reference in each video frame. Other capabilities of the software allow workingwith a set of particle systems and obtaining their centers of mass, as well as studyingthe relative motions between different particles of the set [19, 20]. In this work we usethese capabilities to determine the center of mass of a system consisting of two coupledoscillators, as well as the coordinates of each oscillator from the point of view of theframe of reference of the other.
4. Experimental results and discussion
We determined the positions of the gliders, x ( t ) and x ( t ) , by means of the automatictracking provided by the Tracker software for arbitrary initial conditions. Analysis ofthe video recording of the experiment gave the changes over time shown in Fig. 2. Ascan be seen, the motion of the gliders was complex and was not simple harmonic motion.Afterwards, given the positions of the gliders as a function of time, using the Trackersoftware we determined the changes over time of the position coordinates of the centerof mass of the system and of the motion of glider 2 with respect to glider 1. Figure 3displays two screenshots of the Tracker software. Top panel illustrates the adquisitionof the temporal evolution of the center of mass, x cm and q S , while the bottom panel thatof x / and q A . The normal coordinates q S and q A were obtained, thanks to the Tracker,using Eqs. 5-6. It is worth noting that, as expected, x cm and x / , proportional to the q S and q A respectively, follow a sinoudail evolution with the corresponding frequenciesof the normal coordinates. It is remarkable that Tracker allows to fit the sinousoidalcurves, and easily obtain the behavior or the normal coordinates. To recapitulate, ormal coordinates in a system of coupled oscillators Figure 2.
Temporal evolutions of the coordinates of the gliders, x ( t ) (blue) and x ( t ) (red), for arbitrary initial conditions. Figure 3.
Two Tracker screenshots. The top panel shows the positions of the centerof mass and the symmetric coordinate q S . The botom panel displays the relative x / and the antisymmetric q A coordinates (relative to the glider 1). ormal coordinates in a system of coupled oscillators Figure 4.
Temporal evolution of the normal coordinates q S (blue circles) and q A (redcircles) and curves obtained by non linear fitting to sinusoidal functions (solid lines). the temporal evolutions of the coordinates were determined and fitted to sinusoidalfunctions, as shown in Fig. 4. From the parameters of the fitted curves, we obtain thenormal frequencies ω S and ω A , with their margin of uncertainty: ω S = 3 . rad/s (7) ω A = 6 . rad/s. (8)We will refer to these frequencies as having been obtained by the normal coordinatesmethod.For a deeper study of the dynamics of the system, we calculated the normalfrequencies by the traditional method which involves setting the initial conditions so thatthe gliders oscillate in the normal modes, first in symmetric and then in antisymmetricmotion. We call this the normal modes method. As with the previous method, by meansof video analysis we obtained the changes over time of the coordinates (not shown here)and fitted the results to sinusoidal functions. The frequencies obtained were ω S = 3 . rad/s (9) ω A = 6 . rad/s. (10)which were slightly different from the values obtained by the normal coordinates methodbut always within the uncertainty margin. ormal coordinates in a system of coupled oscillators
5. Discussion
In this section we discuss the experimental results and interpret them in the light of theresults of the theoretical model. In the model presented in section 2, comprising glidersand ideal springs without mass, the results for the normal frequencies are ω S = 4 . ω A = 6 . M and the spring of mass m behave as a single body of mass2 M + m . This new body is joined on both sides to identical springs, each having elasticconstant k and mass m . Thus both springs behave as a single spring with effectiveelastic constant k eff = 2 k and mass 2 m . Using the approximation that 1 / M eff = 2 M + m + 2 m/
3. Then the frequency of symmetricoscillation is: ω S = (cid:115) kM + 5 m/ . (11)We can carry out a similar analysis for the normal mode of antisymmetricoscillation. In this case the midpoint of the central spring is a fixed point, so thesystem can be divided into two halves. Each half is composed of a spring with elasticconstant k and mass m joined to a block of mass M , which in turn is joined to anotherspring which is half as long as the original spring, so that it has elastic constant 2 k andmass m/
2. This system then has an effective elastic constant k eff = k + 2 k and effectivemass M eff = M + m/
2. Finally we obtain the frequency of asymmetric oscillation: ω A = (cid:115) kM + m/ . (12)Now we can compare the experimental results obtained by the methods describedabove, with the results of the theoretical models which depend on the spring constantsand the masses of the gliders, in the case which assumes the springs have no mass, as ormal coordinates in a system of coupled oscillators – Normal coordinates Theoretical model Theoretical model Normal modesmethod (massless springs) (springs with mass) method ω S ω A Table 1.
Comparison between experimental results and models.
We can see that there is good agreement between the normal frequencies obtainedby all four procedures. The small discrepancy between the experimental results may bedue to the fact that it is not possible to excite only a single mode and inevitably a mixtureoccurs with energy transfer from one mode to the other, alternately. Comparing theexperimental results with the theoretical model there is also very satisfactory agreement,especially when the correction for the mass of the springs is taken into account.
6. Conclusion
In this work we developed an experimental method which allows direct visualization ofthe changes over time of the normal coordinates of a system comprising two coupledoscillators with arbitrary initial conditions and studied the influence of the masses ofthe springs. Due to the video analysis capabilities of Tracker software it was easy tofind the changes over time of the center of mass of the system and of the motion of oneglider with respect to the other and afterwards of the normal coordinates. Finally weobserved that the changes of the normal coordinates followed simple harmonic motionand we measured the frequencies by means of non linear fitting. We compared theresults obtained by exciting each of the normal modes separately, and compared thesewith the predictions of theoretical models with and without correction for the mass ofthe springs. Agreement between the experimental results and the predictions of thetheoretical models was very good in all cases. However, taking into account the massof the springs improved considerably the agreement.It should be noted that the ease and speed of data processing makes the activitypresented here suitable not only for undergraduate laboratory courses, but also fordirect analysis of the video of the experiment in theoretical classes, using for exampleactive methodologies such as Interactive Lecture Demonstrations [24] and emphasizingactivities with sequences like POE (Predict – Observe – Explain). Finally, we stress thatthe most significant contribution of this work is the possibility of showing the changesover time of the normal coordinates in advanced courses of mechanics and waves andallowing students to visualize their dynamics, showing that these coordinates are notjust a mere mathematical contrivance for solving systems of differential equations. ormal coordinates in a system of coupled oscillators Acknowledgments
We acknowledge financial support from grant FSED 3 2019 1 157320 (ANII-CFE,Uruguay) and CSIC Grupos I+D (UdelaR, Uruguay).
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