WDS'5 Proceedings of Contributed Papers, Part III, 649 654, 25. ISBN 8-86732-59-2 MATFYZPRESS Linear Spring Oscillator with Two Different Types of Damping J. Bartoš Masaryk University, Faculty of Science, Department of Theoretical Physics and Astrophysics, Brno, Czech Republic Abstract. This paper presents a simple, cheap and easily accessible demonstration experiment in two modifications. A typical example of a physical system for which damping forces have to be involved into equations is a damped spring mechanical oscillator. Here we use two models of the damping force, being produced by Foucault currents and by the force of the drag in air. It turns out, maybe as a slightly surprising result, that the commonly used model of damping forces in mechanics - the air drag force linearly depending on velocity - is not realistic in some typical situations. Equations of motion are solved analyticaly, when possible, othervise they are solved numerically. The results of the demonstration experiments are compared with graphical outputs of numerical solutions. Introduction Exercices and problems in bachelor courses of general physics, especially in mechanics, are usually solved and demonstration experiments are usually interpreted assuming that forces originating from the motion of bodies in their enviroment (typically air drag forces) can be neglected. Standard formulations of physical problems and exercices involve such formulations as suppose that the air drag force is negligible, do not take the air drag force into accout, etc. This leads to a tendency of students to omit damping forces even in situations in which these forces play an essential role. For some mechanical systems a damping force depending linearly on the velocity of the moving body is taken into account. For such a case the equations of motion are often solvable analytically.unfortunately, the quadratic function of the velocity is more appropriate for describing this force, in many practical cases. Oscilations as a drag force study Experimental motivation We using a usual spring oscillator with a point-like mass. We can verify that the oscilations are practicaly undamped. The damping can be enhanced by adding various damping elements - for example an alluminium strip in a magnetic field connected to the oscillator (see Figure.1) or square cartoon plate (see Figure 4). A light-emitting diod is fastened to the mass of the oscillator. The photographs presented in Figure 1 and Figure 4 were obtained by a digital camera placed on a turning tripod. The camera was turned by hand for simplicity and cheapness of the experiment (we wanted to avoid the necessity of an additional equipment and give students themselves a possibility to perform the experiment). This caused some shaking of the photograph. On the other hand, as we show later, one can use a particular treatment of the data obtained from the photograph that eliminates these imperfections. Thus, the used experimental simplification does not limit the correctness of experimental results and corresponding conclusions. Technical parameters of the phorograph: equivalent of film sensitivity = ISO 4, diaphragm number = 5, time of exposition = 25 s (3 s, respectively), flash synchronized with the beginning of the exposition time. 649
Figure 1. Spring oscillator damped by Foucault currents Physical background On the left-hand side of Figure 1 we see linear sprig oscillator with light-emitting diod connected with alluminium strip that can move through magnetic field without contact with magnets. A metal strip with the conductivity σ moves (along z axis) with the velocity v in magnetic field with the flux density B perpendicular to v. It is well-known fact that the motion of the strip is damped by Foucault currents, the dependence is even the direct proportion on velocity (see e.g. discussion at the end of chapter 17-5 in [1]). F = Kσ B 2 v, σ being the conductivity and K is a constant given by the geometrical parametres. Spring oscillator with one degree of freedom damped with the force of the type F B = b v (where ż = v) has the folloving equation of motion (see e.g. chapter 15-8 in [3]): Equation (1) can be rewritten as: where γ = b 2m a ω = k m. Its solution with initial conditions used in the experiment is: m z = kz bż. (1) z + 2γż + ω 2 z =, (2) z = Ae γt cos ωt, (3) where ω = ω 2 γ2 Function (3) is given graphically in Figure 2 (for the presentation of Figures the code [4] was used). Experimental results and interpretation For verifying the linear model of damping force we use the photograph on Figure 1. 65
.1.5 z [m] -.5 -.1 5 1 15 2 25 Figure 2. A =.115 m, ω = 6 s 1, γ =.181 s 1..12.1.8 z [m].6.4.2 2 4 6 8 1 12 Figure 3. Fit function : z = Ae γt, A =.115m. The envelope function Ae γt in (3) with one free parameter γ is compared with minima and maxima of displacement as measured digitally from Figure 1 by the method of least squares (see Figure 3). We obtain γ = (.181 ±.5) s 1. An experiment with elecromagnetically damped oscillator was described also in [2]. Spring oscillator damped by air drag force Physical background On the left side of Figure 4 we can see a linear spring oscillator with a light-emitting diod and resistive square cartoon plate connected. In this experimental situation the drag force of the type F d = D v cannot be used. It is immediately evident that the curves in Figures 2 and 4 have a different behaviour. Let us estimate the quadratic (Newton) model of the damping force. Here we use motion along x axis to differentiate betwen the linear and the quadratic model. F d = Dv 2 v v = D v v, D = 1 CS. (4) 2 651
Figure 4..25.2.15.1.5 y [m] -.5 -.1 -.15 -.2 5 1 15 2 25 3 Figure 5. A =.22 m, ω = 4.9 s 1, β =.38 s 1. The equation of motion is then: ẍ + βẋ ẋ + ω 2 x =, (5) where β = D/m and ω is again the frequency of the corresponding undamped oscillator. Numerical solution of equation (5) in the time interval 3 s is shown in Figure 5. We can see the close similarity with the experimental curve on Figure 4. Let us now verify this similarity by exact considerations. First of all let us note that the period of the solution in Figure 5, given by time intervals between every two neighbour maxima (or minima, or every other equilibrum positions) is constant to a good approximation. We use the measured heights of experimental maxima and minima only to eliminate the imperfections of the experimental graph caused by turning the camera by hand. Comparing the calculated and experimental results we proceed in two steps: a) By an approximate calculation based on energy considerations we obtain the descending amplitude of oscillations as a funcion of time. Then we estimate the experimental data (heights of maxima and minima) by this trial function. 652
b) We look for the best fit of heights of maxima and minima obtained from the numerical solution of (5) with the experimental data. Let us discuss the method a) in more detail. Consider time intervals [t n, t n +t ] between an arbitrarily chosen n-th maximum equilibrium displacement x n of our damped oscillator and the immediately following minimum. (The graph in Figure 5 shows that the length of these time intervals is approximately constant within the studied range [, 3] s.) The intervals are short enough for describing the real damped motion by undamped oscillations with the frequency ω = π/t and the amplitude A = x n, the displacement and velocity being x(t n + t) = Acos ωt, v(t n + t) = Aω sin ωt, t [, π ω ]. The energy change to time ratio of damped oscillations in the mentioned time interval can be calculated as the work of the damping force related to t, i.e. de dt = ω π mβa3 ω 3. = 1 t mβv(t) v(t) dx(t) = t π/ω sin 3 ωtdt = 4 3π mβa3 ω 3. On the other hand, for the corresponding undamped oscillations we have E = 1 2 mω2 A 2 de dt = mω2 A da dt. Comparing the last two expressions for de dt we obtain The solution of this equation is K is an integration constant. A = da dt = 4βω 3π A2. 1 ct + K, Experimental results and interpretation Case a c = 4βω 3π. (6) The fit of formula (6) with experimental maxima or minima (see Figure 6), gives the coefficient of drag resistance C, in equation (4) for Newton drag force. Our value is C = 1.3 which corresponds with the expected value for the cartoon plate C = 1. Moreover we obtain the coefficient β =.38 s 1. Case b In the method b) we compare the experimental data with those obtained by numerical solution of the equation of motion (5) and maxima or minima measured from Figure 4. A qualitative comparison of the numerical solution of (5) for given β and ω with the real trajectory of our oscillator immediately shows a strong disagreement. The experimental frequency ω corresponds to much smaller ω than calculated from k/m. Since the elastic constant k of the spring is given, this discrepancy is undoubtedly caused by greater effective oscillating mass then m. Suppose that ω 2 = k m + m, 653
.25.2.15 y [m].1.5 1 2 3 4 5 6 7 8 9 1 Figure 6. Fit function: x = 1 ct+k, c = 4βω 3π. where m is the real mass of the oscillator, i.e. the mass of the weight together with the damping plate, and m is an additional mass which can be interpreted as the contribution of the air dragged by the damping plate. We can determine this mass by requiring the best fit of data {[t n, x n ]} obtained for n-th maximum from the numerical solution of (5) with free parameters β and ω and the corresponding experimental data with t n = nt. Conclusion From the results of the experiments presented it is evident that we cannot reduce the discussion problems with damping forces to the linear dependence on velocity only. Every experimental situations needs a careful analysis and explanation. References [1] R. P. Feynman, R. B. Leighton, M. Sands: The Feynman Lectures on Physics, Volume II.Addison- Wesley Publishing Company, Reading,Massachusetts; Menlo Park, California; London; Amsterdam; Don Mills, Ontario; Sydney, 1977. [2] L. McCarthy: On the electromagneticaly damped mechanical harmonic oscillator, Am. J. Physics 64(7), July 1996 [3] D. Halliday, R. Resnick, J. Walker: Fundamentals of physics - 7th edition. Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, Singapore, 24. [4] http://www.octave.org 654