Pendulum exercise

B. Pendulum exercise, after acute phase-active (Fig. 89-2)... 

FIG. 89-1 Pendulum exercise, acute phase—passive motion. 

Exercise. For a pendulum in a potential 1/(0) and subject to a constant torque x this equation is... 

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... 

The preceding analysis provides a powerful method for determining the diffusivities of species that produce an anelastic relaxation, such as the split-dumbbell interstitial point defects. A torsional pendulum can be used to find the frequency, u>p, corresponding to the Debye peak. The relaxation time is then calculated using the relation r = 1/ojp, and the diffusivity is obtained from the known relationships among the relaxation time, the jump frequency, and the diffusivity. For the split-dumbbell interstitials, the relaxation time is related to the jump frequency by Eq. 8.63, and the expression for the diffusivity (i.e., D = ra2/12), is derived in Exercise 8.6. Therefore, D = a2/18r. This method has been used to determine the diffusivities of a wide variety of interstitial species, particularly at low temperatures, where the jump frequency is low but still measurable through use of a torsion pendulum. A particularly important example is the determination of the diffusivity of C in b.c.c. Fe, which is taken up in Exercise 8.22. 

Solution. Using a torsion pendulum, find the anelastic relaxation time, r, by measuring the frequency of the Debye peak, cup, and applying the relation cupr = 1. Having r, the relationship between r and the C atom jump frequency F is found by using the procedure to find this relationship for the split-dumbbell interstitial point defects in Exercise 8.5. Assume the stress cycle shown in Fig. 8.16 and consider the anelastic relaxation that occurs just after the stress is removed. A C atom in a type 1 site can jump into two possible nearest-neighbor type 2 sites or two possible type 3 sites. Therefore,... 

In torsional braid analysis,a flexible, braided fiber, usually glass, is impregnated with the material to be studied. The impregnated fiber then becomes the torsion member in the pendulum. This type of device is useful for following the cure of a material that starts out as a liquid and cures to a solid (e.g., an epoxy). In analyzing the data from such a device, care must be exercised in separating interactions between the sample and supporting fiber. 

Point Oj on the line OC (Figure 2.10) at a distance L from the axis of rotation z is called the center of swing of the physical pendulum. It is noteworthy that if a pendulum is turned over and hung up on the horizontal axis passing through the point Oj the period of its oscillation does not change, point 0 being the new center of oscillation. We will leave the proof of this property as an exercise for the reader. 

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