The damped driven pendulum and applications Presentation by, Bhargava Kanchibotla, Department of Physics, Texas Tech University.
Overview The Dynamics of a simple pendulum. Dynamic steady state. Cases of dynamic steady state. Applications Conclusions
The Dynamics of Simple Pendulum The angular acceleration of the pendulum is produced by a gravitational torque mgr sinφ Corresponding to the equation of motion mr 2 2 d φ + mgrsinφ = 2 dt 0
Continued. Small angular displacement linearizes the problem by making the torque proportional to the displacement and the motion is simple harmonic with characteristic frequency ω 0 = g R
Continued.. If a torque is applied to a stationary pendulum, it swings through an angle φ and the restoring force restores the pendulum as equilibrium position N = mgr sinφ
Continued. Greater the torque, the larger the angle φ and there is a critical torque The critical torque assumes a value 90 degrees. If N exceeds the critical value, then the applied torque becomes larger then the restoring torque N c
Continued. Assuming the damping force, the equation motion of the pendulum is given by the following equation adding the restoring and the damping torque as 2 2 d φ dφ N = mr + η + 2 dt dt mgrsinφ
Dynamic Steady State When a constant torque is applied to the pendulum at rest, there will be a initial transient behavior that eventually settles down to a dynamic steady state after the transients die out. There are several cases of this dynamic steady state.
Static Steady state 1. For low applied torque, N N C there is a steady state N = N C in which all the time derivatives vanish after the initial oscillations have dies out. sin φ
Dynamics steady state For under damped motion, with a constant applied torque, we have the following equation as below torque = N mgrsinφ = mr The torque has specific values at four particular angles 2 2 d φ 2 dt
Continued If the applied torque exceeds the critical torque, the motion will be continuously accelerated rotation and the pendulum increases its energy as the time goes. The angular speed also increases, but with fluctuations that repeat every cycle as indicated in the figure
Continued.. When damping is present with ω << c and N > N C the angular speed increases untill the damping term approaches the applied torque.the acceleration fluctuates around an average that is zero and the pendulum undergoes a quasi static motion ω 0
Quasi Static motion This has the following equation and the solutions are as follows φ φ ω sin 1 + = dt d N N C C N C forn < = 0 ω c C C N forn N N > = 1 2 ω ω C C C N forn N N >> = ω ω
Quasi Static Motion The solutions of the quasi static motion can be plotted as shown in the figure
Quasi Static Motion The cyclic variations in angular speed for points A and B are in this plot. At Point A, the applied torque has a value and the net torque varies between 0.2N C and 2.2 N C N = 1. 2N C
Continued. For point B, we have N = 2 N C so the net torque varies between N C and 3N C producing more regular variations in angular speed.
Continued For a negligible damping case η 0and ω c >> ω 0 We have the following solutions for all valuses of N ω = 0 forn N C N ω = C for N N ω 0 C
Continued These solutions are plotted as a graph.it is seen that the system exhibits hyteresis When the torque is increased for N < N C the pendulum stabilized at the angle (pi) satisfying the relation N = sin φ N C
Continued. When N reaches the critical torque the angular speed jumps to the value ω C and then rises linearly with further increase in N For decreasing torque, ω remains proportional to N all the way to the origin as shown.
Continued. For the particular case ω we have the C = 2ω 0 following plot. For increasing torques there is the usual initial rise in then at zero frequency until the critical value N C is reached.
Continued. For decreasing torques. There is a hyteresis with the zero average frequency reached at a torque ' which is less than N. N C C
Application The damped driven pendulum equation has a particularly important applications in solid state physics When 2superconductorsin close proximity with a thin layer of insulating material between them, the arrangement constitutes a Joseph son junction,
Continued. Current exceeding the value is accompanied by the presence of a voltage, and the plots of current I versus voltage V for the junction exhibit hysteresis.the Joseph son junction satisfies the same differential equation as the damped oscillator with the current playing the role of the torque, the voltage playing the role of the average angular speed, the capacitance acting like a moment of inertia and the electrical conductance serving as the viscosity.
Conclusion The damped driven pendulum is a very important system that has a very significant application in the field of solid state physics. Also the system is very important to be understood as it has a lot of physics involved in