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M 17E “Pohl's Wheel (Linear and Nonlinear Oscillations)”

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M 17E “Pohl's Wheel (Linear and Nonlinear Oscillations)” Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum M 17e “Pohl’s wheel (linear and nonlinear oscillations)” Tasks 1. Determine the frequency fd, the damping constant and the logarithmic decrement of the rotating pendulum after Pohl at eight different attenuations (currents IW of the eddy current brake). Plot the logarithmic decrement as a function of the square of the current IW and determine the straight line parameters via linear regression. 2. Experimentally, determine the directional torque D and the moment of inertia J of the rotating pendulum. 3. Record the resonance curve of the rotating pendulum at one of the given attenuations. Determine the resonance frequency and the damping constant and compare the values to those obtained in task 1. 4. Experimentally, try to observe for the nonlinear pendulum the first bifurcation and (qualitatively) the transition to chaotic motion. To this end measure the amplitudes close to the resonance of the forced oscillation for various values of the eddy current brake current IW (to be decreased in small steps) and plot them graphically. Additional task For the driven pendulum, simulate the first bifurcation and the transition to chaotic oscillations by changing the attenuation. A computer algebra program is available for download in the download area of the Undergraduate Physics Lab website. Literature Physics, M. Alonso, E. F. Finn, Chap. 10.10 The Physics of Vibrations and Waves, H. J. Pain, Wiley 1968, Chap. 3 Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Mechanik, 2.3 Java Applet about mechanical resonance: http://www.walter-fendt.de/ph14d/cpendula.htm Accessories Pohl’s wheel (ELWE company), CASSY-interface (Leybold Didactic), PC Keywords for preparation - torques on a rotating pendulum, moment of inertia of a cylinder - rotating pendulum, homogeneous and inhomogeneous linear differential equations with constant coefficients, eigenfrequency, damping constant, logarithmic decrement - methods for determining attenuation (logarithmic decrement, resonance curve) - principle of operation of an eddy current brake - principles of nonlinear dynamics, transition into chaotic states, bifurcations Remarks In this experiment free, damped and forced oscillations of a rotating pendulum after Pohl are studied. The rotating pendulum has an eigenfrequency, depending on the directional torque of the spring and its moment of inertia. By increasing the damping, in case of free oscillations one can observe a stronger attenuation of the amplitude as a function of time as well as a decrease of the eigenfrequency. In case of a periodically driven pendulum the resonance can be realized by matching external frequency and eigenfrequency. In case of resonance the maximum amplitude is a function of the damping. Furthermore the pendulum can be provided with an additional mass leading to an imbalance. This will cause an additional restoring torque and the relation between deflection angle of the pendulum and restoring torque becomes nonlinear. This results in a qualitatively different oscillation behaviour of the pendulum, allowing for different amplitude states in between which the pendulum deflection fluctuates in response to the initial conditions as well as small perturbations. The movement of the pendulum appears to be random. The emergence of two different amplitude states is called bifurcation. A further decrease of the attenuation will lead to bifurcations of higher order. Accordingly, there are various possibilities for the dynamic evolution of such a system (random or chaotic behaviour). The pendulum after Pohl used in the experiment (Fig.4) consists of a ring-shaped copper disk with a homogeneous mass distribution, attached to a rotation axis through the center of mass. The rest position is enforced by the spiral spring and the deflection can be read off an angle scale. For controlling the damping of the oscillation the current in the inductor of the eddy current brake can be adjusted. Additionally the pendulum can be actuated on by a regulated stepping motor (that is connected to the disk via an extender wheel and a rod) with different frequencies. The oscillation recorder and further electronic components allow for a contactless measurement of the copper disk’s motion. Voltages proportional to the measured signals, i.e. the time dependence of both angular deflection and angular velocity, are generated and fed to an ADC (Analog-to-Digital Converting) interface for further processing in the computer. Basic knowledge Linear systems – rotating pendulum without external torque In the simplest case, i.e. without external drive, the system can be modelled in the following way: if the pendulum is deflected by an angle and released, it will perform a damped oscillation around the equilibrium angle 0. This oscillation is called damped natural oscillation (eigenoscillation) of the system. The equation of motion of the system can be derived from the equilibrium of torques MMMTFD , (1) with MJT torque of inertia MDF restoring torque of the spring M D damping torque (moment of inertia J, directional torque of the spring D, damping coefficient ). Inserting into the equation above leads to the differential equation of the damped natural oscillation of the pendulum: JD 0 (2a) 2 or 20 0 . (2b) 2 22 (/2J , 0 DJ/ , d0 ) Equation (2b) is a second order, homogeneous, linear differential equation. In the solution of such a differential equation three different cases have to be considered: 2 22 1. 0 : overdamped case (3) 22 2. 0 : aperiodic limiting case (4) 22 3. 0 : oscillatory case (5) 0 denotes the natural angular frequency (angular eigenfrequency of the undamped oscillator) and d the free angular resonance frequency of the damped oscillator. The natural angular frequency 0 does not depend on the oscillation amplitude. This result is an important characteristic of harmonic oscillators, as they are described by linear equations of motion. As a solution for the weakly damped case one obtains an exponentially decreasing oscillation that is 1 characterized by the relaxation time and the free angular resonance frequency fdd /2 . The free resonance frequency of an attenuated oscillator is always smaller than the eigenfrequency of the corresponding undamped system. The damping constant , respectively the logarithmic decrement can be easily obtained from the temporal decrease of the oscillation amplitude (t) : ()t Td ln (period Tdd 2/). (6) ()tT d An illustrative representation of the oscillation behaviour can be found in phase space ( Diagram). For a point particle in three-dimensional space the phase space is defined as the set of six-tuples made up by the three spatial and three momentum coordinates. In case of Pohl’s wheel, space and momentum coordinates are one-dimensional quantities. The phase space is then reduced to two dimensions. In a two-dimensional graph the angular velocity ()t is plotted against the angular deflection()t of the pendulum, which yields a time independent geometrical curve, also called trajectory. In case of a weakly damped harmonic oscillator the trajectory is a spiral approaching the zero of the coordinate system (which is a so-called attractor). Rotating pendulum with external torque Attaching an additional, external, periodic torque M0 sin( t) to the pendulum, using an actuator, one obtains a forced oscillation. After a certain settling time the frequency of the pendulum is equal to that of the external drive. The equation of motion is J D M0 sin( t ). (7) This is a second order inhomogeneous, linear differential equation. A general solution of Eq. (7) is a superposition of the general solution of the corresponding homogeneous differential equation (2a) and a particular solution of the inhomogeneous equation. A particular solution can be found, using the ansatz p (t ) A ( )sin( t ) (8) For the amplitude one obtains MMJ/ A() 00. (9) 22 2222 2 22 22 J (00 ) ( ) 4 denotes the driving frequency and 0 DJ/ the eigenfrequency of the undamped system. The phase shift is given by 2 tan ( ) 2 2 2 2 (9a) J()()00 The general solution to Eq. (7) is found by adding this solution to the one of Eq. (2a): 3 t (t ) A ( )sin( t ) Ce cos( d ) . (10) One can see that the oscillations of the free and the driven system superpose. After a settling time the second term is exponentially damped away. The pendulum is then oscillating with the driving frequency of the stepping motor, albeit with a phase shift with respect to the drive. In the case of a forced oscillation the trajectory in phase space is an ellipse that does not change after the settling time. The amplitude of the forced oscillation reaches a maximum M M A() 0 (11) R 2 2 2 2 2 2J J (0 RR ) 4 d 22 at the resonance angular frequency R 0 2 . For weak damping the resonance angular frequency is approximated by D . R 0 J The other limiting values of Eq. (9) are MM00 A(0) 2 ; A( ) 0 . JD0 Plotting the amplitude A versus the angular frequency yields the resonance curve, see Fig. 1. The curve is not symmetric with respect to the resonance angular frequency. As full width at half maximum (FWHM) the difference between the two angular frequencies 1 and 2 is defined at which the resonance amplitude decreases to the value AAA(12 ) ( ) ( R ) / 2 ; at this point the power has decreased to half of the maximum value. An important relation between the FWHM of the resonance curve and the damping constant is 2 , respectively 2. (12) Eq. (12) describes the uncertainty between frequency and life time of a damped linear oscillator. Strong damping leads to a short ‘life time’ of the oscillation and results in broad resonance curves. Narrow resonance curves correspond to systems with a longer ‘life time’ and thus a small damping. A() / A( ) R 1.0 Fig. 1 Resonance curve 0.8 1/2 0.6 0.4 0.2 0.0 4 0.0 0.5 1.0 1.5 2.0 / R The damping describes the dissipation of energy that is fed into the system by the actuator.
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