Pendulum analogy

We will phrase our discussion in terms of the Josephson junction, but will mention the pendulum analog whenever it seems helpful. 

The nonlinear study of bifurcations of the elastic equilibrium of a straight bar involves, in a way, a change of the physical point of view, mostly due to the mathematical difficulties related to the direct approach of the Bernoulli-Euler (B.-E.) equation. This aspect gave rise to various models describing the same phenomenon, such as Kirchhoff s pendulum analogy [1], as well as to different methods of calculus, such as Thompson s potential energy method [2], [3]. In this paper, we use the linear equivalence method (LEM) to a B.-E. type model, thus deducing an approach for the critical and postcritical behavior of the cantilever bar. 

Figure 4.9 Mechanical pendulum analogy to describe electrode reaction and equilibrium.

Region I Activation Polarization 131 Pendulum analogy beyond equilibrium... 

Figure 4.10 Schematic of pendulum analogy of small net anodic current.

Figure 4.11 Schematic of pendulum analogy of larger net anodic current and energy input.

For simplicity, take the specific case where ki = k2 = k. Write the matrix of force constants analogous to matrix (5-29). Diagonalize this matrix. What are the roots Discuss the motion of the double pendulum in contrast to two coupled, tethered masses (Fig. 5-1). 

Suppose that the same pendulum moves about the axis passing through the point S. Then it is eharaeterized by a new reduced length I and, by analogy with Equation (3.57), we have... 

A few words should be said about the difference between resonance and molecular vibrations. Although vibrations take place, they are oscillations about an equilibrium position determined by the structure of the resonance hybrid, and they should not be confused with the resonance among the contributing forms. The molecule does not resonate or vibrate" from one canonical structure to another. In this sense the term resonance is unfortunate because it has caused unnecessary confusion by invoking a picture of vibration. The term arises from a mathematical analogy between the molecule and the classical phenomenon of resonance between coupled pendulums, or other mechanical systems. 

Pn is sometimes said to represent the coupling of q with term resonance integral has similar roots (Coulson C. A., Valence, Oxford University Press, Oxford, 2nd edn, p. 79). 

How can the result of unique steady state be consistent with the observed oscillation in Figure 5.9 The answer is that the steady state, which mathematically exists, is physically impossible since it is unstable. By unstable, we mean that no matter how close the system comes to the unstable steady state, the dynamics leads the system away from the steady state rather than to it. This is analogous to the situation of a simple pendulum, which has an unstable steady state when the weight is suspended at exactly at 180° from its resting position. (Stability analysis, which is an important topic in model analysis and in differential equations in general, is discussed in detail in a number of texts, including [146].)... 

Fig. 2. Automated torsion pendulum schematic. An analog electrical signal results from using a light beam passing through a pair of polarizers, one of which oscillates with the pendulum. The penduluin is aligned for linear response and initiated by a computer that also processes the damped waves to provide the elastic modulus and mechanical damping data, which are plotted vs. temperature or time...

This mechanical analog has often proved useful in visualizing the dynamics of Josephson junctions. Sullivan and Zimmerman (1971) actually constructed such a mechanical analog, and measured the average rotation rate of the pendulum as a function of the applied torque this is the analog of the physically important I-V curve (current-voltage curve) for the Josephson junction. 

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). 

The analogy to the waterwheel breaks down at still higher Rayleigh numbers, when turbulence develops and the convective motion becomes complex in space as well as time (Drazin and Reid 1981, Berge et al. 1984, Manneville 1990). In contrast, the waterwheel settles into a pendulum-like pattern of reversals, turning once to the left, then back to the right, and so on indefinitely (see Example 9.5.2). 

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