Period pendulum

When looking at the snapshots in Figure A3.13.6 we see that the position of maximal probability oscillates back and forth along the stretching coordinate between the walls at = -20 and +25 pm, with an approximate period of 12 fs, which corresponds to the classical oscillation period r = 1 / v of a pendulum with... 

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

A vibration is a periodic motion or one that repeats itself after a certain interval of time. This time interval is referred to as the period of the vibration, T. A plot, or profile, of a vibration is shown in Figure 43.1, which shows the period, T, and the maximum displacement or amplitude, X - The inverse of the period, j, is called the frequency, f, of the vibration, which can be expressed in units of cycles per second (cps) or Hertz (Hz). A harmonic function is the simplest type of periodic motion and is shown in Figure 43.2, which is the harmonic function for the small oscillations of a simple pendulum. Such a relationship can be expressed by the equation ... 

There are several other comparable rheological experimental methods involving linear viscoelastic behavior. Among them are creep tests (constant stress), dynamic mechanical fatigue tests (forced periodic oscillation), and torsion pendulum tests (free oscillation). Viscoelastic data obtained from any of these techniques must be consistent data from the others. 

Relative rigidity = P02/P2 where P0 is period of pendulum of the control, and P is period of pendulum after irradiation. A value of 1 indicates no change less than 1 indicates softening. 

Assume that we have a pendulum (Fig. 6-14) provided with a piece of soft iron P placed coaxially with a coil C carrying an alternating current that is, the axis of the coil coincides with the longitudinal axis OP of the pendulum at rest. If the coil is excited, one finds that the pendulum in due course begins to oscillate, and th oscillations finally reach a stationary amplitude. It is important to note that between the period of oscillation of the pendulum and the period of the alternating current there exists no rational ratio, so that the question of the subharmonic effect is ruled out. 

If one makes this assumption, one can integrate the first equation of (6-185) assuming that the periodic motion of the pendulum exists with an unknown constant period Cl, obtaining... 

Peculiar particle velocity, 19 Pendulum problem, 382 Periodicity conditions, 377 Perturbed solution, 344 Pessimism-optimism rule, 316 Petermann, A., 723 Peterson, W., 212 Phase plane, 323 "Phase portrait, 336 Phase space, 13 Photons, 547... 

Thus, we have demonstrated that measuring the period of oscillations, T, the pendulum allows us to determine the field g. From Equation (3.28) we have... 

If T is the oscillation time of pendulum or the time the pendulum needs to swing from the angle to (half period ), then integration of Equation (3.38) gives... 

A and B are constants and they are independent of time. Bearing in mind that = X + iy, it is simple to find functions x t) and y t), which describe a motion of the pendulum on the earth s surface. In accordance with Equation (3.88) a solution is represented as a product of two functions. The first one characterizes a swinging of the pendulum with the angular velocity p, which depends only on the gravitational field and the length /, while the second is also a sinusoidal function and its period is defined by the frequency of the earth s rotation and the latitude of the point, (Foucault s pendulum). In order to understand the behavior of the pendulum at the beginning consider the simplest case when a rotation is absent, co — 0. Then, we have... 

Thus, in general the pendulum moves along an ellipse on the spherical surface with radius equal to the string length, /. This motion has a periodic character and the period is that of the swinging, T — Injco). In particular, if the initial conditions are... 

Therefore, the pendulum moves periodically along the arc, located in the plane XOZ, and this case was studied in detail in the previous section. Comparison of Equations (3.91 and 3.92) shows... 

In the case when the pendulum path is an ellipse the effect of the earth s rotation is to cause the ellipse to rotate with an angular velocity equal to (—co sin X). As we know, at the poles it is approximately 7.3 x 10 s. In accordance with Equation (3.94), for an observer located at the z-axis above the earth the rotation is clockwise in the northern hemisphere, (A>0), and counterclockwise in the southern hemisphere, (A<0). For example, at the North Pole the direction of the pendulum rotation and the earth are opposite to each other. During swinging the pendulum moves from the point a to the opposite point of the path b, which is shifted at some small distance because of the earth s rotation. Fig. 3.5c. Suppose that the radius of the circle is sq, then the displacement bd during a half period of swinging, Tjl, is... 

A striking example of the so formed class of kick-excited self-adaptive dynamical phenomena and systems is the model of a pendulum influenced by quasi-periodic short-term actions, as considered in papers (Damgov, 2004) - (Damgov and Trenchev, 1999). 

Damgov, V.N. and P.G. Georgiev Amplitude Spectrum of the Possible Pendulum Motions in the Field of an External. Nonlinear to the Coordinate Periodic Force. Dynamical Systems and Chaos, World Scientific, London, Vol. 2, P. 304 (1995)... 

In general, the 3D motion of the spherical pendulum is very complex, but for fixed initial angular displacements, values of the kinetic energy can be found (by trial and error) for which this motion is periodic. The approximation discussed above leads to the approximate description of the horizontal motion in terms of Mathieu functions, for which Flocquet analysis determines periodic solutions in terms of two integers k and n, which can be thought of as quantum numbers. 

So how is the spherical pendulum quantised The answer is that its motion is generally chaotic, except for discrete values of the initial projection speed, for which it is periodic. The precise details of this phenomenon are difficult to get a handle on, because, although the vertical motion is always described by periodic elliptic functions, the horizontal motion is described in terms of Lame functions, which are very difficult to study and for which periodicity is difficult to diagnose. 

Numerical simulations of the spherical pendulum for arbitrary values of K and W will usually reveal a very complicated, a periodic motion of the type shown in Fig. 2, but in some cases the motion is periodic. The theory can be found in Refs. [9,11], but is summarised here. Let r be any integer or simple fraction (such as 3/2, etc.). Then solutions of Mathieu s equation of the form... 

A regularly periodic motion characteristic of a pendulum, spring, or many chemical bonds. See Hooke s Law... 

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