Experimental Investigation of the Dynamics of a Wilberforce Pendulum

Friday 2 5 Lab 3 Experimental Investigation of the Dynamics of a Wilberforce Pendulum Barbara Hughey Lab Partner: Thalia Rubio 3/6/09 2.671 Measurement and Instrumentation Prof. Neville Hogan

Abstract The Wilberforce pendulum provides an important qualitative demonstration of coupled harmonic oscillators, but quantitative measurements are not typically performed. The present work examines the dynamics of a simple pendulum comprising a steel spring with attached mass using an ultrasonic motion detector. A series of measurements allowed determination of the coupling constant between the longitudinal and torsional modes of the spring-mass system. The motion depends on the moment of inertia, which could be varied by about 20%. The pendulum motion was recorded at six values of moment of inertia, and the frequencies of the two normal modes were determined with Fourier transform techniques. The coupling constant was measured to be (5.49 ± 0.38 10 5 for the 20 mm diameter steel spring used in the experiment. Correspondence of the measured mode frequency to that predicted by the model was qualitatively excellent, but exhibited a systematic shift to lower frequency that merits further study. 1 Introduction The Wilberforce pendulum is commonly used in introductory mechanics classes to demonstrate the interaction between two coupled harmonic oscillators, in this case, the longitudinal stretching and torsional twisting of a spiral spring attached to a mass. The pendulum is named after L.R. Wilberforce, a Demonstrator in Physics at the Cavendish Laboratory in Cambridge, England, in the late 19th and early 20th century. Wilberforce proposed the use of a loaded spiral spring to determine the Young s modulus of the spring material, but also recognized its potential to demonstrate the transfer of energy between two coupled modes of a harmonic oscillator [1]. It is the latter application that is familiar to generations of undergraduate physics and engineering students. A tuned Wilberforce pendulum provides an exciting demonstration, especially in a large lecture hall, because from far distances, the spring oscillations seem to periodically stop and start, apparently violating the law of conservation of energy. In reality, the energy has been transferred from longitudinal motion to torsional motion, but the latter can be difficult to observe unless one is close to the pendulum. The present study focused on a more quantitative analysis of the Wilberforce pendulum motion than is typically provided. In particular, the longitudinal motion of the pendulum bob was measured with an ultrasonic motion detector and the results were used to determine the coupling coefficient between the longitudinal and torsional modes. This parameter is difficult to measure by any other means, so the primary focus of this research was to determine if the coupling constant could accurately be measured by studying the dynamics of a Wilberforce pendulum. An additional objective of this study was to compare the observed mode frequencies with those predicted from the theory of two coupled harmonic oscillators. 1 Barbara Hughey 3/6/09

Fig. 1: Schematic diagram of typical Wilberforce pendulum 2 Coupled Harmonic Oscillators A schematic diagram of a typical Wilberforce pendulum is given in Fig. 1. A spiral spring with linear spring constant k in N/m and torsional spring constant δ in Nm is suspended from a fixed support. A pendulum bob with mass m in kg and moment of inertia I in kg m 2 is attached to the free end of the spring. A pair of nuts mounted on threaded rods extending from the pendulum bob can be used to adjust the moment of inertia. The coordinate system is defined so that the z direction is along the axis of the spring, and the Θ direction corresponds to rotation around the axis of symmetry of the system. Analysis of the equations of motion for a massless spring with no coupling between the translational and torsional motion yields the well-known equations for the natural frequency of the longitudinal oscillations, ω z in terms of k and m, and for the torsional natural frequency, ω θ in terms of δ and I: k ω z = m, and (1a) δ ω θ = I, (1b) where ω denotes frequency in rad/s. If the coupling between the translational and torsional motion is assumed to have the form ɛzθ/2, the coupled equations of motion can be solved to find the frequencies of the two normal modes of the system [2], ω1 2 = 1 2 ω2 θ + ω2 z + ω2 2 = 1 2 ω2 θ + ω2 z [ ( ) 2 ωθ 2 ɛ 2 ω2 z + mi [ ( ) 2 ωθ 2 ɛ 2 ω2 z + mi ] 1 2 ] 1 2, (2a) (2b) 2 Barbara Hughey 3/6/09

(a) (b) Fig. 2: Calculated normal mode frequencies in Hz as a function of moment of inertia for coupling constant of a) 10 5 N and b) 4 10 5 N. The vertical line at I = 10 6 kg m 2 indicates the location of the resonance at which ω θ = ω z. The remaining model parameters are defined in the text. where the coupling coefficient ɛ has units N. The frequencies of the two normal modes are shown in Fig. 2 as a function of moment of inertia for two different values of coupling coefficient. The longitudinal spring constant, k, is assumed to be 0.1 N/m, the torsional spring constant, δ, is 10 5 Nm, and the pendulum mass, m is 10 gm. The coupling coefficient is a factor of four larger in Fig. 2a than in Fig. 2b. The frequency axis in Fig. 2 has been converted to Hz from rad/s. In the absence of coupling (ɛ = 0), the normal mode frequencies are given by (1). For the parameters of Fig. 2, the corresponding longitudinal frequency f z = ω z /2π = 0.503 Hz, corresponds to the horizontal portions of the curves shown in Fig. 2a. The angled portions of the curve correspond to motion primarily in the torsional mode, in which case the frequency is inversely proportional to the square root of I. The resonance condition between the two modes, ω θ = ω z, is indicated with a dotted blue line. In the absence of coupling, the two normal modes cross at this point and energy in one mode can not be transferred to the other mode. In other words, if the system initial condition is pure longitudinal oscillation, there will be no oscillation in the torsional mode, and vice versa. Increasing the coupling has the effect of creating repulsion between the two branches of the frequency curves, as shown by the increase in the semi-major axis of the resultant hyperbola evident in Fig. 2b. In this case, energy can be coupled between the two modes of the system so that even if the initial condition is purely longitudinal, torsional motion can and will occur. In order to use measurements of the frequency to determine the coupling constant, it is helpful to rearrange (2) (after dividing by 2π to convert from rad/s to Hz) in order to 3 Barbara Hughey 3/6/09

obtain the following equation. Definition of two new quantities, ( ) ( 2 δ f1 2 f 2 2 2 = 16π 4 Φ + ) ( 1 I [ ɛ 2 2δk 16π 4 m + k2 16π 4 m 2 ) 2 ] ( ) 1 L (3) ( f 2 1 f 2 2 ) 2 and (4a) J 1/I, allows (3) to be re-expressed in terms of quadratic fit constants C 0, C 1, C 2 as (4b) Φ = C 2 J 2 + C 1 J + C 0. (5) Comparison of (5) with (3) yields the coupling and spring constants in terms of the quadratic fit constants as δ = 4π 2 C 2, k = 4π 2 m C 0, and ɛ = 16π 4 mc 1 + 2δk, (6a) (6b) (6c) where δ and k in (6c) are determined from C 2 and C 0 via (6a) and (6b), respectively. Therefore, measurement of the two normal mode frequencies, f 1 and f 2 as a function of moment of inertia I allows determination of k, δ, and ɛ. Section A.1 on page 13 describes a MatLAB function written for this purpose 1. The present study focuses on the determination of coupling coefficient, ɛ. 3 Instrumented Wilberforce Pendulum Measurement of the dynamics of a Wilberforce pendulum may be accomplished with the apparatus shown in Fig. 3 on the next page. The pendulum itself comprises a ring stand supporting a spring onto which is mounted a cylindrical pendulum bob with moment of inertia adjustment arms. System specifications are given in Fig. 3 on the following page and Tab. 1 on the next page and 2. 3.1 Adjustment of Moment of Inertia The moment of inertia of the pendulum bob may be adjusted with nylon M5 nuts threaded on a pair of nylon rods extending 10.2 mm from the outside of the disk in diametrically 1 Added by Dayán Páez and not part of the original report draft. 4 Barbara Hughey 3/6/09

Steel spring Pendulum bob 1.1 m Adjustment nut Adjustment arm Vernier Motion Sensor, GO-MOT Computer with Vernier LoggerPro v3.6.1 Fig. 3: Apparatus used for the Wilberforce pendulum dynamics measurements Tab. 1: Steel spring parameters Parameter Dimension Units Outer diameter 20 mm Wire diameter 0.40 mm (AWG 26) Relaxed length 0.45 m Sprint constant 0.2 N/m 5 Barbara Hughey 3/6/09

Tab. 2: Pendulum bob parameters Item Dimensions Mass (g) Disk 25.4 mm OD 40 mm L 9.0 Arm M5 threaded rod 10.2 mm L 0.2 ( 2) Nut standard M5 nut 0.3 ( 2) I disc = 1 2 M discr 2 R L l ( I nuts = 2 13 M rod L 2 + M rod R 2) I nuts = 2M nut (l + R) 2 Fig. 4: Diagram of the pendulum bob defining the contribution of the disk, rods, and adjustable nuts to the total moment of inertia opposed directions, as shown in Fig. 4. The minimum moment of inertia can be obtained with both nuts at the edge of the disc, as shown in Fig. 4, which also defines the contributions of the disc and the threaded rods to the moment of inertia. The computed moment of inertia for the pendulum bob with both nuts at their minimum radius (just outside the disc) is 9.01 10 7 kg m 2. With both nuts in the extended position, the computed moment of inertia is 11.2 10 7 kg m 2. The location of each nut was measured with Mitutoyo Digimatic calipers (Model CD-S6 CT) with a resolution of 0.01 mm. The mass of each component was measured with an OHaus Scout Pro SP202 digital balance with 10 g resolution. The uncertainty due to instrument resolution contributed less than 1% uncertainty to the computed moment of inertia. The pendulum motion was measured for six positions of the adjustment nuts starting at the minimum distance and increasing in steps of about 2 mm to a distance of 9.3 mm. The corresponding range of moment of inertia was 9.05 to 10.9 10 7 kg m 2. 3.2 Detection of Pendulum Motion The Vernier Instruments motion sensor includes a gold foil transducer that produces short bursts of ultrasonic waves [3]. These waves are reflected off nearby objects and detected by the sensor. The delay time is used with the speed of sound in air to determine the distance to the nearest object. The instrument includes a temperature sensor to compensate for the dependence of sound speed on temperature. The ultrasonic waves are emitted into an expanding cone with angle 15 20 degrees, so care must be taken that other objects do not fall within this cone and affect the measurements. The Go!Motion sensor has a range of 0.15 to 6 m with a resolution of 1 mm, more than adequate for the present experiment. Data acquisition rates up to 30 Hz are easily achieved using this sensor. Higher sampling rates could be used provided the environment was relatively noise-free, but 6 Barbara Hughey 3/6/09

were not needed for this experiment, as discussed in the next section. 3.3 Determination of Normal Mode Frequencies The oscillation frequency for the coupled pendulum will be approximately equal to the longitudinal resonance frequency given in (1a). For the current pendulum, this value corresponds to about 0.5 Hz. This relatively low frequency is advantageous for visual observation of the pendulum oscillations, but requires relatively long measurement times in order to obtain sufficient frequency resolution to resolve the two normal mode frequencies. Frequency spectra were obtained using the MathCAD fast Fourier transform function, FFT. The FFT routine requires that the number of data points be equal to a power of two, chosen to be 2 12 = 4096 for the present work. A sampling rate of 15 Hz was used for all measurements, resulting in a total sample time of 273 sec. The frequency resolution is the inverse of the total sample time, or f = 3.66 mhz. The Nyquist frequency was 7.5 Hz, a factor of 10 greater than the oscillation frequency, and thus aliasing was not of concern. The Fourier spectrum was plotted in MathCAD and the Trace feature of the plot was used to determine the two peak frequencies for each data set. 4 Results and Discussion The linear oscillation of the pendulum bob was measured with the motion sensor for six values of moment of inertia between 9.05 and 10.9 10 7 kg m 2. Fourier analysis of the resulting position vs. time data sets was used to determine the frequencies of the two coupled modes. Analysis of the dependence of mode frequency on moment of inertia was used to determine the torsional and translational spring constants as well as the coupling constant. The measured linear spring constant was compared to the value provided in the spring specifications. Specifications were not provided for the torsional spring constant or the coupling constant. Finally, the measured mode frequencies were compared to those predicted from the model with the experimentally determined values of the spring constants and coupling coefficient. 4.1 Measurement of Oscillation Frequency Representative data of the distance from the rest point of the pendulum bob as a function of time are shown in Fig. 5 on the following page. These data were acquired with I = 9.05 10 7 kg m 2. The left plot (Fig. 5a on the next page) shows the long-time behavior, and the right plot (Fig. 5b on the following page) is an expanded view of the region near the first transition point. The sample rate was 15 Hz, and the motion was sampled for 273 sec, or a total of 4096 points. The large amplitude variations caused by beating of the two modes are readily apparent in Fig. 5 on the next page. The left plot (Fig. 5a on the following page) shows a slight exponential decay in the oscillations due to energy loss, presumably in the spring. The behavior near the transition between linear and torsional motion is shown in the expanded 7 Barbara Hughey 3/6/09

(a) Full data set (b) Detailed view of data near first transition point Fig. 5: Measurement of pendulum bob oscillation for I = 9.05 10 7 kg m 2 and initial displacement of 0.04 m. Data acquisition parameters are specified in the text. view of Fig. 5b. A small amount of noise is apparent in the expanded view. We hypothesize that these fluctuations were due to a slight side-to-side component of the pendulum bob motion, which could be verified by recording the motion with a video camera and analyzing the video to obtain the transverse position as a function of time. Data similar to those shown in Fig. 5 were recorded for 5 additional values of moment of inertia between 9.37 and 10.9 10 7 kg m 2. The sampling parameters were maintained at the same values (15 Hz sample rate for 237 sec) and the initial displacement of the pendulum was held constant at 40 mm. Results near resonance (I = 9.85 10 7 kg m 2 are shown in Fig. 6 on the next page for comparison. Notice the much slower beat frequency caused by the smaller difference in frequency between the two modes. The Fourier transform of each data set was used to determine the mode frequencies as a function of moment of inertia. The Fourier spectra corresponding to the data of Fig. 5, 6 are shown in Fig. 7 on the next page. The left plot in Fig. 7a on the following page shows the Fourier spectrum of the data set displayed in Fig. 5. The two frequency peaks are well resolved. The right plot (Fig. 7b on the following page) is the Fourier spectrum of the data set displayed in Fig. 6 on the next page, showing that near resonance there are two overlapping, but still resolvable, frequency peaks. The total sample time of 237 sec corresponds to a Fourier spectrum resolution of 3.7 mhz. The measured frequencies as a function of moment of inertia for all six data sets are shown in Fig. 8 on page 10. The vertical error bars show the Fourier transform resolution of ±1.8 mhz. As can be seen in Fig. 8 on page 10, the frequency pairs lie along a hyperbola, as described in Section 2 on page 2. In the next section we describe how the spring constants and coupling constant were obtained from the data shown in Fig. 8 on page 10. 8 Barbara Hughey 3/6/09

(a) Full data set (b) Detailed view Fig. 6: Measurement of pendulum bob oscillation near resonance (I = 9.85 10 7 kg m 2 ). Note the larger time scale compared to Fig. 5b on the previous page. (a) I = 9.05 10 7 kg m 2, showing well-separated peaks at 747.3 and 710.6 mhz (b) Near resonance, I = 9.85 10 7 kg m 2, showing resolvable peaks at 721.6 and 710.6 mhz Fig. 7: Fourier spectra of measured oscillations for two different values of moment of inertia. The Fourier spectrum resolution is 3.7 mhz. 9 Barbara Hughey 3/6/09

Fig. 8: Measured oscillation frequency as a function of moment of inertia. The upper and lower frequency for each data set are indicated in red and blue, respectively. Fig. 9: Quadratic fit to Φ vs. J. The fit coefficients are listed in Tab. 3 on the next page, along with the derived values for k, δ, ɛ. 4.2 Determination of Coupling and Spring Constants As discussed in Section 2 on page 2, the measured frequency as a function of moment of inertia can be used to determine the spring constants k and δ for the linear and torsional motion, as well as the coupling constant, ɛ. Shown in Fig. 9 are the values of Φ [(6a)] obtained from the measured oscillation frequency as a function of the inverse of the moment of inertia, J [(6b)]. The parameters obtained from a linear least-square fit of a quadratic function to the data are given in Tab. 3 on the following page with 95% confidence intervals. These values are used to find the coupling and spring constants, also listed in Tab. 3 on the next page with associated 95% confidence uncertainty. The longitudinal spring constant obtained from the measurements of 0.202 ± 0.14 N/m agrees with the specified value of 0.2 N/m. We were not provided specifications on δ or ɛ, so could not compare these measurements with expected values. In fact, as demonstrated, this experiment has allowed us to determine the coupling coefficient with 7% uncertainty. 10 Barbara Hughey 3/6/09

Tab. 3: Quadratic fit coefficients and corresponding values for k, δ, and ɛ. Quadratic Fit Coefficients Wilberforce Pendulum Parameters Name Value Units Name Value Units C 0 0.262 ± 0.037 sec 4 k 0.202 ± 0.014 N/m C 1 ( 5.2 ± 0.73) 10 7 m 2 kg s 4 δ (2.02 ± 0.14) 10 5 Nm C 2 (2.61 ± 0.36 10 13 m 4 kg 2 s 4 ɛ (5.49 ± 0.38 10 5 N Fig. 10: Comparison of measured normal mode frequencies with those obtained from the coupled harmonic oscillator equations with experimentally determined values for the spring and coupling constants, k, δ, and ɛ. Points with error bars correspond to the measurements and the solid lines represent the calculated values from (2). 4.3 Comparison of Measured and Predicted Frequency Now that we have determined the spring and coupling constants, we can compare the predicted mode frequencies from (2) with the measured frequencies. This comparison is shown in Fig. 10. Evident in Fig. 10 is a systematic shift to lower frequency that results in a significant discrepancy between the model and the data, especially for values of moment of inertia below resonance. Independent measurement of the torsional spring constant, δ, might help illuminate the source of this discrepancy. 5 Conclusions A compact apparatus was used to investigate the dynamics of a Wilberforce pendulum. The frequencies of the two coupled modes were measured for six values of moment of inertia starting below resonance and ending above resonance. We measured a longitudinal spring constant of 0.202 ± 0.14 N/m which agrees with the specified value of 0.2 N/m. 11 Barbara Hughey 3/6/09

The torsional spring constant was measured to be (2.02 ± 0.14) 10 5 Nm. The parameter of most interest, the coupling constant, was found to be (5.49 ± 0.38) 10 5 N, demonstrating that a relatively simple series of measurements of the dynamics of a Wilberforce pendulum can be used to accurately determine the longitudinal-torsional coupling constant of a spring. A comparison of measured to predicted mode frequencies for the pendulum showed excellent qualitative agreement, but indicated a small but significant systematic shift of the data to lower frequency. Further research is needed to identify the source of this discrepancy. The relative ease of performing measurements on a Wilberforce pendulum suggests some interesting extensions of this research. Adding a measurement of the rotational motion of the pendulum bob would be useful not only for providing an independent measure of the rotational spring constant, but, more interestingly, would allow simultaneous independent measurement of the kinetic energy in the longitudinal and torsional modes. This would allow the transfer of energy between the two modes to be studied as a function of time. It may even be possible to use these techniques to identify the energy loss mechanism responsible for the decrease in oscillation amplitude with time. Acknowledgments The author would like to thank Prof. Thomas Peacock of the Department of Mechanical Engineering at the Massachusetts Institute of Technology for suggesting the Wilberforce pendulum, Prof. Stephen Crandall for the loan of the pendulum setup, and Prof. Neville Hogan for insightful comments. References [1] LR Wilberforce. On the Vibrations of a Loaded Sprial Spring. Philosophical Magazine 5th Series. 38 (1894). pp. 386 392. [2] E Debowska, S Jakubowicz, and Z Mazur. Computer Visualization of the beating of a Wilberforce Pendulum. Eur. J. Phys. 20 (1999). pp. 89 95. [3] Go!Motion User Manual (Order Code GO-MOT). Vernier Software and Technology, available at http://www2.vernier.com/booklets/go-mot.pdf. Accessed 9/18/2008. 12 Barbara Hughey 3/6/09

A MatLAB code The following MatLAB scripts were written as part of this exercise. A.1 dke.m 1 %% Returns delta, k, and epsilon for a Wilberforce Pendulum with 2 %% frequencies f1, f2, mass m, and moment of inertia I. 3 %% 4 %% [delta, k, epsilon] = dke(f1, f2, m, I) 5 %% 6 %% Fits quadratic to square of squared-differences of frequencies in 7 %% order to determine the parameters 8 9 function [dt, k, ep] = dke(f1, f2, m, II) 10 Phi = (f1.^2 - f2.^2).^2; 11 JJ = 1/II; 12 13 %% Fit quadratic to Phi 14 coeffs = polyfit(jj * ones(size(phi)), Phi, 2); 15 C0 = coeffs[1]; C1 = coeffs[2]; C2 = coeffs[3]; 16 17 dt = 4 * pi^2 * sqrt(c2); 18 k = 4 * pi^2 * m * sqrt(c0); 19 ep = sqrt(16 * pi^4 * m * C1 + 2 * dt * k); 13 Barbara Hughey 3/6/09