Driven pendulum

Condensed-matter physics (Josephson junction, charge-density waves) Mechanics (Overdamped pendulum driven by a constant torque)... 

We now consider a simple mechanical example of a nonuniform oscillator an overdamped pendulum driven by a constant torque. Let 0 denote the angle between the pendulum and the downward vertical, and suppose that d increases counterclockwise (Figure 4.4.1). 

Pendulum driven by constant torque) The equation 0-t-sin0 = y describes the dynamics of an undamped pendulum driven by a constant torque, or an undamped Josephson junction driven by a constant bias current. 

This section deals with a physical problem in which both homoclinic and infinite-period bifurcations arise. The problem was introduced back in Sections 4.4 and 4.6. At that time we were studying the dynamics of a damped pendulum driven by a constant torque, or equivalently, its high-tech analog, a superconducting Josephson junction driven by a constant current. Because we weren t ready for two-dimensional systems, we reduced both problems to vector fields on the circle by looking at the heavily overdamped limit of negligible mass (for the pendulum) or negligible capacitance (for the Josephson junction). 

Improved electric clock (Alexander Bain) With John Barwise, Bain develops an electric clock with a pendulum driven hy electric impulses to regulate the clock s accuracy. 

The average prices of the batch centrifuge are shown in Fig. 18-154. All the models include the drive motor and control. In Fig. 18-154, the inverting filter, horizontal peeler, and the advanced vertical peeler are the premium baskets especially used for specialty chemicals and pharmaceuticals. Control versatility with the use of programmable logic control (PLC), automation, and cake-heel removal are the key features which are responsible for the higher price. The underdriven, top-driven, and pendulum baskets are less expensive with fewer features. 

The inhomogeneously AC driven, damped pendulum system can be described by the following equation ... 

Damgov, V.N. Quantized Oscillations and Irregular Behaviour of Inhomoge-neously Driven, Damped Pendulum. Dynamical Systems and Chaos. World Scientific, London, Vol. 2, P. 558 (1995)... 

An instrument for comparative determinations of the performance of different explosives. A mortar, provided with a borehole, into which a snugly fitting solid steel projectile has been inserted, is suspended at the end of a 10 It long pendulum rod. Ten grams of the explosive to be tested are detonated in the combustion chamber. The projectile is driven out of the mortar by the fumes, and the recoil of the mortar is a measure of the energy of the projectile the magnitude determined is the deflection of the pendulum. This deflection, which is also known as... 

It is universal for a large class of period doubling scenarios. Physical examples of this route to chaos include the driven pendulum (Baker and Gollub (1990)) and ion traps (Blumel (1995b)). 

A clue to the analytical understanding of these numerical results comes from the fact that the ampUtude of the transients for hi a appears to be random. This could be accounted for by a theory based on ampUtude randomness in an externally driven pendulum system. ... 

Recently, Douarche et al verified the transient ES FR and steady state ES FR for a harmonic oscillator (a brass pendulum in a water-glycerol solution, that is driven out of equilibrium by an applied torque). They also developed a steady state relation for a system with a sinusoidal forcing, and showed that the convergence time was considerably longer in this case. 

In Section 8.5 we used a Poincare map to prove the existence of a periodic orbit for the driven pendulum and Josephson junction. Now we discuss Poincare maps more generally. 

Consider the driven pendulum + atj) + sintl) = I. By numerical computation of the phase portrait, verify that if a is fixed and sufficiently small, the system s stable limit cycle is destroyed in a homoclinic bifurcation as / decreases. Show that if a is toolarge, the bifurcation is an infinite-period bifurcation instead. 

As tools for analyzing differential equations. We have already encountered maps in this role. For instance, Poincare maps allowed us to prove the existence of a periodic solution for the driven pendulum and Josephson junction (Section 8.5), and to analyze the stability of periodic solutions in general (Section 8.7). The Lorenz map (Section 9.4) provided strong evidence that the Lorenz attractor is truly strange, and is not just a long-period limit cycle. 

Chaos in the damped driven pendulum) Consider the forced pendulum 0-l-fe0-l-sin0 - Fcosr, with b = 0.22, F = U (Grebogi et al. 1987). 

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