Analyzing SHO of a double pendulum

Transcription

1 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.1 Analyzing SHO of a double pendulum W.C. Beard, R. Paudel, B. Vest, T. Warn, and M.J. Madsen Department of Physics, Wabash College, Crawfordsville, IN (Dated: May 4, 2010) We measured the angle versus time by video capture for a physical double pendulum for small angles, and found that the period of the second normal mode did not agree with the Lagrangian model. We hope to improve our model to better approximate our setup in the future. Interests in non-linear motion have motivated a number of experiments. In an undergraduate laboratory, a double pendulum serves as a good apparatus to model a chaotic device [1]. We can anaylze the motion of the double pendulum to calculate the Lypanov exponent which is a key indicator of the chaotic motion[1, 2]. A double pendulum can be easily built in an undergraduate laboratory by attaching one metal rod to another [3]. The dynamics of such pendulum have been topic of interest for a number of years[1]. In the past, polarized photos of the pendulum were used to analyze the dynamics. In this project, we used a high speed camera[4] to analyze the motion of a double pendulum. [5]. The double pendulum built at Wabash Advanced Laboratory was used to study the motion. We modeled the upper arm and lower arm of the pendulum as solid blocks, and developed a Lagrangian model of the dynamics of the double pendulum. Following the analysis by Thornton and Marion[5], we find that the kinetic energy T of the two arms is composed of the motion of their cener of mass, (ẋ 2 i + ẏi 2 ) for the two arms i = 1, 2, and the rotational kinetic energy about the rotation axis as shown in Figure 1, 1/2I i θ2 i. The kinetic energy of the entire system is: T = 1 2 m 1( x y 2 1 ) I 2 1 1θ m 2( x y 2 2 ) I 2 2θ 2 (1) where m 1 and m 1 are the masses of the upper and lower arms of the pendulum respectively, I 1 and I 2 are the moments of inertia of the upper and lower arms (as shown in Figure 1). WCB and RP equally contributed to this paper.

2 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.2 We can find the moment of inertia, I 1 and I 2, modeling the two arms as solid blocks which is rotating about the pivot which is (lf/2 D/2) away from the center of mass y 10. The moment of inertia of the upper arm is I 1 = 1/6(m 1 (y w 2 1)) + m 1 (lf 1 /2 D 1 /2) 2 and that for the lower arm is I 2 = 1/6(m 2 (y w 2 2)) + m(lf 2 /2 D 2 /2) 2 The potential energy of the system is defined as the height of the center of mass of each arm above the equilibrium position. The offset potential V 0 is a constant that defines the potential energy in this frame. V = m 1 gy 1 + m 2 gy 2 + V 0 (2) We now shift the coordinate system from Cartesian coordinates x, y to a generalized coordinate system based on the two angles θ i defined as the angle of deviation of the center of mass of each arm from its equilibrium position about the rotation axis of the respective arm. From the geometry, and noting that the sign in each of these equations is negative as we are putting the origin at the top of the upper arm, we can see that, y1 = y 10 cos θ 1 (3) x1 = y 10 sin θ 1 (4) y2 = y 20 cos θ 2 + l 1 (cos θ 1 cos θ 2 ) (5) x2 = y 20 sin θ 2 + l 1 (sin θ 1 sin θ 2 ) (6) The Lagrangian of the system is given by L = T V. We then use the Euler-Lagrange equation [5] to get the two coupled differential equations, L d L θ i dt θ = 0 (7) i We can write the two equations for the two coordinates θ 1 and θ 2 as: g sin(θ1(t))(l1m2 + m1y10) θ1 (t) ( i1 + l1 2 m2 + m1y10 2) + l1m2(l1 y20) (8) θ2 (t) cos(θ1(t) θ2(t)) + l1m2(l1 y20)θ2 (t) 2 sin(θ1(t) θ2(t)) = 0 (9) m2 (l1 y20) ( g sin(θ2(t)) + l1θ1 (t) cos(θ1(t) θ2(t)) l1θ1 (t) 2 sin(θ1(t) θ2(t)) ) (10) θ2 (t) ( i2 + m2(l1 y20) 2) = 0 (11) We then look at the small-angle approximation to get, (I 2 + (y 20 l 1 ) 2 m 2 θ2 m 2 (l 1 y 20 )l 1 θ1 m 2 (l 1 y 20 )gθ 2 = 0. (12)

3 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.3 Arm 1 Arm 2 m (g) ± ±.1 l fi (cm) ± ±.07 D (cm) 3.79 ± ±.01 w (cm) 3.18 ± ±.01 TABLE I: This table shows the measured mass m, length l fi and width w of both arms i of the double pendulum (see Figure 1). Though l 1, the length between the centers of the two tracking points on Arm 1, is used to calculate several k parameters, this quantity as well as the moments of inertia I 1 and I 2 were calculated from direct measurements, including l f and D. Following Rafat, we introduce new scaled variables to match those of the paper [2], We can then find the two normal modes of oscillation, k 1 = g(m 2 l 1 + m1y 10 ) (13) k 2 = I 1 + m 1 l m 1 y 2 10 (14) k 4 = m 2 (y 20 l 1 ) 2 + I 2 (15) k 5 = m 2 g(l 1 y 20 ) (16) k 6 = l 1 m 2 (l 1 y 20 ). (17) ω 2 ± = k 1k 4 + k 2 k 5 ± (k 1 k 4 k 2 k 5 ) 2 + 2k 1 k 5 k 2 6 2(k 2 k 4 k 2 6) We can thus find the time period of these modes, using pendulum measurements that are listed in Table I. Using Equations (12-16), we get the values for the k parameters and are then able to use Equation (17) to find the periods of the normal modes of oscillation, T and T +, all of which are shown in Table II. The predicted values of the periods of the first and second normal mode were T = ±.007 and T + = ±.002, respectively. We measured the normal mode frequency of the physical double-pendulum described above as a preliminary step towards verifying our data collection and analysis procedures. The double pendulum apparatus was constructed by Vest and Warn in the fall of 2009 [3], whose dimensions are shown in Figure 1. We captured the motion of the double pendulum with a high-speed camera in order to get θ i data as a function of time. We covered the background and the arms of the pendulum (18)

4 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.4 y y x D 1 w 1 x θ 1 (x 1, y 1 ) l 1 cosθ 1 y 20 l f1 l 1 (x 10, y 10 ) x 10 =0 θ 2 y 2 -l 1 cosθ 2 D 2 (x 2, y 2 ) y 20 -l 1 l f2 (x 20, y 20 ) x 20 =0 w 0 First normal mode Second normal mode FIG. 1: This figure shows the two arms of the double pendulum, and the dimensions that were used for calculating the normal mode periods of oscillation. These measured values are listed in Table I. Below the dimensional diagrams are representations of the two normal modes of oscillation. The periods T and T + of the first and second normal mode, respectively were calculated and compared to the experimental periods.

5 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.5 FIG. 2: This figure compares a recorded video frame before (above) and after (below) being processed with Photoshop. The contrast was increased, the image was converted to grayscale and a Gaussian blur was applied to the image. Before applying the changes, the tracking software had occasionally detected background noise as particles and sometimes counted a single spot as two, as the brightness withing the dots were nonuniform. The Gaussian blur made the dots more uniform, correcting this problem, and washed out most of the background noise. Quantity Calculated Value k k k k k T ±.007 T ±.002 TABLE II: This table shows the calculated k-parameters from Equations (12-16), as well as the calculated periods of the first (T ) and second (T + ) normal modes of oscillation.

6 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.6 with black felt-cloth, and then put three white reflective stickers at the end of each arm and at the point of rotation in order to make the movement more visible. The lights were turned off in the room, and the black background was used to minimize background light onto the camera, and felt was used for this background to prevent glare. A projector was used to emit white light onto the pendulum to maximize the amount of light reflected from the white stickers, and was placed 2.58 m away from the apparatus. A high speed camera was then set up next to the projector, 3 m away from the pendulum, so that the light from the projector would reflect off of the reflective tape nearly directly back to the camera, and not be scattered. It was our goal to move the projector and camera far enough back from the pendulum that the changes in the angle of oscillation would not put the light from the reflective dots at an excessively difficult angle for the camera to detect. We oscillated the pendulum at angles of less than 24 and took high speed video at 300 fps, as high speed movement of the pendulum has been known to produce video recording errors for frame rates under about 100 fps [3]. 3.5 m High Speed Camera Double Pendulum 0.5 m Projector 2.58 m FIG. 3: We set up the double pendulum and placed a high speed camera 3 m away. We placed a projector to emit white light onto the pendulum and made the room dark while taking the high speed video. We converted each frame of the video to a tiff file, and ran these tiffs through Photoshop batch operations to increase the contrast between the dots and the background and to Gaussian blur all of the dots, eliminating virtually all background light noise and distinctly separating each of the dots (see Fig. 2). We used the Autocontrast command in Photoshop,

7 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.7 and set the Gaussian blur radius to 2.1 pixels. We input the series into ImageJ, a freeware image processing software package, about 900 frames at time, due to system memory constraints [6]. ImageJ would track the trajectories and store the coordinate data of the trajectories of the three reflected points pixels with respect to time, and output them into a text file. These raw coordinates were then extracted from the text files into a spreadsheet, which was then used to convert the relative position of the three reflective dots through time into angles θ i. Since the angle θ i can be given as θ i = tan(y i /x i ), we were able to get the angle data versus time by taking the inverse tangent of y i /x i for both of the angles. Once we had the θ i (t) data (shown in Figure 4), we applied a five-point smoothing function to eliminate any small errors in the digitization of the image files, interpolated the data, and then took the first time derivative of the interpolated data to plot the motion in phase space. We found the period of the motion by measuring the time difference between subsequent zero crossings of the interpolated data. While we did not analyze chaotic systems, we did measure the periods of smaller angle simple harmonic oscillation of the two normal modes, to compare the experimental results with our model. As can be seen in Figure 5, the periods of oscillation varied periodically, and the average period was measured to be T = ±.001 s for the first normal mode, and the second normal mode period was measured to be T + =.995 ±.062 s (95% CI). (The varying oscillations seen in Fig. 5 is likely due to damping that was stronger on one side of the pendulum, because of the large amount of felt material used to eliminate the glare.) This compares to the predicted values according to the model given in Table II of T + =.983 ±.002 and T = ±.007 s (95% CI). We were thus able to use video analysis to capture the motion of a double pendulum and measure the periods of the first and second normal modes to compare to a model based on the system s Lagrangian. We found that our data do not agree with our model. One of the reasons for this is because we used small angle approximation for our Lagrangian. The maximum amplitude in our data is 24 which is slightly off from the small angle approximation. In the future, we can take video with amplitude less than 15 which will then make it easier to use the small angle approximation. The other possibility is to extend the model for larger angle. Future possibilities for experimentation along this line would involve taking massive amounts of data of the double pendulum s chaotic motion at larger angles, and calculating Lyupanov exponents and constructing Poincaré sections from the data to

8 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.8 Θ1, Θ2 HradiansL Time HsL Θ1, Θ2 HradiansL Time HsL FIG. 4: These figures depict data of θ1 and θ2 versus time for the first and second normal modes, top to bottom respectively, with θ1 shown in gray points and θ2 shown in black points. quantify the system s chaos. [1] DeSerio, Robert. American Journal of Physics. 71, (2003). [2] Rafat, M. Z., Whealand, M. S., Bedding,T. R. American Journal of Physics. 77, 3 (2009). [3] Vest, Brad C, Warn, Thomas and Madsen, Martin. Wabash Journal of Physics. 381, 1-10

9 WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.9 Period T s n FIG. 5: This graph shows the period data for θ 1 for the small angle first normal mode simple harmonic oscillations of the double pendulum. The y axis shows the time of the periods, measured in seconds, and the x axis shows the numbering of which oscillation is represented. (2009). [4] L. Labous, A.D. Rosato, and R.N. Dave. Phys. Rev. E 56, 5717?5725 (1997) [5] Thornton, Stephen T., Marion, Jerry B. Classical Dynamics of particles and systems, Brook Cole. [6] ImageJ. [7] Poincaré, Henri. Sur le problme des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt. Acta Mathematica 13, (1890). [8] Levien, R. B., Tan, S. M.American Journal of Physics. 61, (193). [9] Shinbrot, Troy; Grebogi, Celso; Wisdom, Jack; Yorke, James A. American Journal of Physics. 60, (1992) 60, 6 (1992). [10] Hilborn, Robert C. American Journal of Physics. 72, (2004).