Damped pendulum

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)... 

Damped pendulum) Find and classify the fixed points of d + bd + sin0 = O for all b>0, and plot the phase portraits for the qualitatively different cases. 

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). 

Figure 7-13. Damped pendulum method for measurement of friction. A view of the moving specimen in the cradle of the fixed specimens. After Y. Tamai [13].

The main features of SF, such as excitation intensity dependence, emission pulse shortening, and time delay, can be described within a simplified semiclassical approach, which uses Maxwell-Bloch equations while neglecting the dipole-dipole interaction [113,114]. It was shown by Bonifacio and Lugatio [113] that in a mean field approximation the system of noninteracting emitters is described by the damped pendulum equations with two driving terms, as given below ... 

This involves the determination of the damping of the oscillations of a torsion pendulum, disk, or ring such as illustrated in Fig. IV-8. Gaines [1] gives the equation... 

Free- Vibration Methods. Free-vibration instmments subject a specimen to a displacement and allow it to vibrate freely. The oscillations are monitored for frequency and damping characteristics as they disappear. The displacement is repeated again and again as the specimen is heated or cooled. The results are used to calculate storage and loss modulus data. The torsional pendulum and torsional braid analy2er (TBA) are examples of free-vibration instmments. 

Fig. 42. Torsion pendulum and typical damped sine wave output. P is the period of the motion and M2 are successive ampHtudes (241).

Since singular points are identified with the positions of equilibria, the significance of the three principal singular points is very simple, namely the node characterizes an aperiodically damped motion, the focus, an oscillatory damped motion, and the saddle point, an essentially unstable motion occurring, for instance, in the neighborhood of the upper (unstable) equilibrium position of the pendulum. 

One such case arises in the theory of clocks. As is known, a dock is a mechanism consisting of two parts a torsional pendulum with a small damping, and an escapement mechanism replenishing the energy lost by damping in the torsional pendulum. 

The left-hand side of the second equation of (6-186) is the pendulum equation (J being the moment of inertia, D, the coefficient of damping and C, the coefficient of the restoring moment). 

There are other, less commonly used, methods for measuring hardness. One is an impact method in which an indenter is dropped from a known height onto a specimen, and either the size of the indentation, or the coefficient of restitution, is measured. Another is the pendulum method in which a rocking pendulum is applied to a specimen surface. The damping of the pendulum s oscillations is a measure of the hardness. Still another is Moh s scratch method in which the ability of one specimen to scratch another is observed. These methods are described in various books (McColm, 1990), but only the... 

Figure 5 Schematic diagram of a torsion pendulum and a typical damped oscillation curve. Modified from L. E. Nielsen,...

Note 3 A damping curve is usually obtained using a torsion pendulum, involving the measurement of decrease in the axial, torsional displacement of a specimen of uniform cross-section of known shape, with the torsional displacement initiated using a torsion bar of known moment of inertia. 

Figure 5.80 Torsion pendulum for the determination of shear modulus aud damping as functions of temperature at frequencies around 1 Hz. Reprinted, by permission, from N. G. McCrum, C. P. Buckley, and C. B. BucknaU, Principles of Polymer Engineering, 2nd ed., p. 133. Copyright 1997 by Oxford University Press.

While there are many methods for measuring modulus and damping, one of the simplest involves using a torsion pendulum (23, 24, 25) as illustrated in Figure 11. 

We may now express the damping term of the torsion pendulum experiment in terms of the dissipation factor by the simple equation (an approximation which holds for most cases) ... 

We begin with an innocuous case. Consider a pendulum suspended in air and consequently subject to damping accompanied by a Langevin force. This force is, of course, the same as the one in equation (1.1) for the Brownian particle, because the collisions of the air molecules are the same. They depend on the instantaneous value of V, but they are insensitive to the fact that there is a mechanical force acting on the particle as well. Hence for small amplitudes the motion is governed by the linear equation (1.10). For larger amplitudes the equation becomes nonlinear ... 

Exercise. A pendulum obeying the equation Mx = — sin x is suspended in air, which causes damping and fluctuations. Show that it obeys the bivariate nonlinear Fokker-Planck equation, or Kramers equation,... 

AW/W can be measured with a torsion pendulum, in which a specimen in the form of a wire containing the point defects is made the active element and strained periodically in torsion as in Fig. 8.18. If the pendulum is put into free torsional oscillation, its amplitude will slowly decay (damp out), due to the dissipation of energy. As shown in Exercise 8.20, the maximum potential energy (the elastic energy, W) stored in the pendulum is proportional to the square of the amplitude of its oscillation, A. The amplitude of the oscillations therefore decreases according to... 

Before considering particular test methods, it is useful to survey the principles and terms used in dynamic testing. There are basically two classes of dynamic motion, free vibration in which the test piece is set into oscillation and the amplitude allowed to decay due to damping in the system, and forced vibration in which the oscillation is maintained by external means. These are illustrated in Figure 9.1 together with a subdivision of forced vibration in which the test piece is subjected to a series of half-cycles. The two classes could be sub-divided in a number of ways, for example forced vibration machines may operate at resonance or away from resonance. Wave propagation (e.g. ultrasonics) is a form of forced vibration method and rebound resilience is a simple unforced method consisting of one half-cycle. The most common type of free vibration apparatus is the torsion pendulum. 

Surface viscosity has an important influence on the deformation of films and can also provide information about structure. Gaines [14] describes various methods of measuring this quantity. The damped torsion pendulum as developed by Langmuir and Schaefer [65] is probably the best device for making such measurements. Recent measurements of this type have been made by Buhaenko et al. [66]. Malcolm [67, 68] and Daniel and Hart [69] have carried out experiments which illustrate the important influence which viscosity has on the study of isotherms. 

Dynamic-Mechanical Measurement. This is a very sensitive tool and has been used intensively by Nielsen (17) and by Takayanagi (18). When the damping curves from a torsion pendulum test are obtained for the parent components and for the polyblend and die results are compared, a compatible polyblend will show a damping maximum between those of the parent polymers whereas the incompatible polyblend gives two damping maxima at temperatures corresponding to those of the parent components. Dynamic mechanical measurement can also give information on the moduli of the parent polymer and the polyblend. It can be shear modulus or tensile modulus. If the modulus-temperature curve of a polyblend locates between those of the two parent polymers, the polyblend is compatible. If the modulus-temperature curve shows multiple transitions, the polyblend is incompatible. 

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