Forced harmonic oscillator

The j-th harmonic bath mode is characterized by the mass mj, coordinate Xj, momentum pxj and frequency coj. The exact equation of motion for each of the bath oscillators is mjxj + mj(0 Xj = Cj q and has the form of a forced harmonic oscillator equation of motion, ft may be solved in terms of the time dependence of the reaction coordinate and the initial value of the oscillator coordinate and momentum. This solution is then placed into the exact equation of motion for the reaction coordinate and after an integration by parts, one obtains a GLE whose... 

The behaviour of dielectrics in alternating electric fields may be treated in the framework of forced harmonic oscillation. The displacement is then given by... 

A schematic view of the nanomechanical GMR device to be considered is presented in Fig. 1. Two fully spin-polarized magnets with fully spin-polarized electrons serve as source and drain electrodes in a tunneling device. In this paper we will consider the situation when the electrodes have exactly opposite polarization. A mechanically movable quantum dot (described by a time-dependent displacement x(t)), where a single energy level is available for electrons, performs forced harmonic oscillations with period T = 2-k/uj between the leads. The external magnetic field is perpendicular to the orientation of the magnetization in both leads. 

One of most popular techniques for dynamic mechanical analysis is the torsion pendulum method. In a modification of this method designed to follow curing processes, a torsion bar is manufactured from a braid of fibers impregnated with the composition to be studied this is the so-called torsional braid analysis (TBA) method.61 62,148 The forced harmonic oscillation method has been also used and has proven to be valuable. This method employs various types of rheogoniometers and vibroreometers,1 9,150 which measure the absolute values of the viscoelastic properties of the system under study these properties can be measured at any stage of the process. The use of computers further contributes to improvements in dynamic mechanical analysis methods for rheokinetic measurements. As will be seen below, new possibilities are opened up by applying computer methods to results of dynamic measurements. 

Then Equations (6) become the equations of motion for a forced harmonic oscillator, letting us solve exactly ... 

Substituting back the solution of C in the Euler Lagrange equations for X we obtain the equation for a forced harmonic oscillator. 

You might worry that (2) is not general enough because it doesn t include any explicit time dependence. How do we deal with time-dependent or nonautonomous equations like the forced harmonic oscillator mx + bx + kx = F cos t In this case too there s an easy trick that allows us to rewrite the system in the form (2). We let x, = x and Xj = X as before but now we introduce x, = t. Then x, = 1 and so the equivalent system is... 

The virtue of this change of variables is that it allows us to visualize a phase space with trajectories frozen in it. Otherwise, if we allowed explicit time dependence, the vectors and the trajectories would always be wiggling—this would ruin the geometric picture we re trying to build. A more physical motivation is that the state of the forced harmonic oscillator is truly three-dimensional we need to know three numbers, x, x, and t, to predict the future, given the present. So a three-dimensional phase space is natural. 

The cost, however, is that some of our terminology i s nontraditional. For example, the forced harmonic oscillator would traditionally be regarded as a second-order linear equation, whereas we will regard it as a third-order nonlinear system, since (3) is nonlinear, thanks to the cosine term. As we ll see later in the book, forced oscillators have many of the properties associated with nonlinear systems, and so there are genuine conceptual advantages to our choice of language. 

The viscoelastic parameters are generally measured by dynamic oscillatory measurements. Apparatus of three different configurations can be used cone and plate, parallel plates, or concentric cylinders. In the case of cone and plate geometry, the test material is contained between a cone and a plate with the angle between cone and plate being small (<4°). The bottom member undergoes forced harmonic oscillations about its axis and this motion is transmitted through the test material to the top member, the motion of which is constrained by a torsion bar. The relevant measurements are the amplitude ratio of the motions of the two members and the associated phase lag. From this information it is relatively simple to determine G and G". 

The Forced Harmonic Oscillator Inhomogeneous Linear Differential Equations... 

EXAMPLE 8.5 Let us assume that the external force on a forced harmonic oscillator is... 

Figure 8.4 The position of a forced harmonic oscillator as a function of time for the case a = lA(o.

A forced harmonic oscillator with a circular frequency a> = 6.283 s (frequency V = 1.000s ) is exposed to an external force Fq sin (at) with circular frequency a = 7.540 s such that in the solution of Eq. (8.59) becomes... 

Obtain the solution for Eq. (8.55) and (8.56) for the forced harmonic oscillator using Laplace transforms. 

It is a matter of some algebra, using the known solution for forced harmonic oscillators (cf. Ref. 12) to show that the oscillator energy at time t is given in terms of the initial conditions of the bath as... 

The equation of a damped, forced harmonic oscillator is given by ... 

If the loading of the indentor varies periodically with time, we would expect the response of the half-space to reflect this periodicity after a long time, when transient effects have died away. This situation is quite analogous to that of the forced harmonic oscillator subject to frictional resistance. The contact interval will, in particular, vary with the same period but in a manner that is not simply related to the load since from (3.10.17), for example, we see that the relation between the two quantities is not linear. We have therefore... 

The amplitudes of forced harmonic oscillations at the frequencies Vj = 400 Hz and V2 = 600 Hz are equal to each other. Determine the resonance frequency Neglect... 

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