Harmonic displacement

Mathematically, the harmonic displacements of an array of coupled oscillators in any dimension may be described by the set of equations ... 

A Single Harmonic Displaced Oscillator Mode System... 

The moment of inertia of a ring of particles, mrki was used as the criterion for stability to define a closed orbit that combines circular motion with simple harmonic displacements. A more general discussion that substantiates the derivation is given by Goldstein [11]. The frequency of revolution is obtained in the form of a square root, defined by a set of integers,... 

Disturbance of the fluid sphere5 results in harmonic displacements, governed by Laplace s equation, which assumes a minimum average gradient of some... 

X-Ray results provide important information regarding molecular and atomic motions, through determination of the thermal factor (B), which gives a measure of the mean square (harmonic) displacement (u ) of an atom or group from its equilibrium position. The two are related by the Debye—Waller equation B = A highly mobile pro-... 

Lattice modes, or external modes, as well as internal modes produce harmonic displacements of the atoms. The internal modes result in local molecular deformation, whilst external modes are supposed to entrain the atoms when a molecule is rigidly displaced from its equilibrium position in the lattice. The time-dependent atomic position vector r can be expressed in terms of the internal displacement vector u nx, taken with respect to the molecular centre of mass, and the external displacement vector, Hext the displacement vectors have units of length. A, Eq. (A2.52). 

Both the column thickness and the column natural frequency are important in determining the effect of wind on column sway and the stresses induced. To determine the natural frequency of the column, Rayleigh s method, based on harmonic displacement for all elements of the column, is used. The natural vibration frequency formula has been derived as follows ... 

Applying harmonic voltage causes the film to undergo harmonic displacement. The normalized deflection variation with respect to electrical field change is depicted in Fig. 2.6. The displacement amplitude is dependent on both voltage... 

Basic aspects of the resonance technique may be illustrated by considering a linear visco-elastic medium between two parallel plates, one undergoing forced harmonic displacement, amplitude a (= cos cat), the other being fixed. As 0) is varied, resonances occur and a resonance bandwidth analysis yields the loss tangent, tan 5 [Whorlow, 1992 Ingard 1988]. 

Flexing of the cell corresponds to a harmonic displacement of the cell centre, i.e. S = S osin(wt). Thus the current generated by flexoelectricity is... 

The potential energy at each point s on the reaction path is taken to be the classical energy at that point plus the potential energy for harmonic displacements perpendicular to the path (6) ... 

For a dynamic strain arising from a harmonic displacement, the storage and loss moduli depends nonlinearly upon the strain amplitude A i as shown in Fig. 26 for a strain amplitude in the range AS S [0 1 0 6] and a frequency f= Itz co= 10 Hz... 

However, the restrictions on the dynamics posed by the inclusion of only quadratic terms in the expansion of the perpendicular vibrational motions, the fact that the reaction path itself may be poorly deflned in certain regions, bifurcations of the path should not be overlooked. In such cases it is possible to switch to the reaction surface or the reaction volume methods. These methods are more demanding from the point of view of the potential energy surface and also as far as the dynamical requirements are concerned. But they nevertheless reduce the problem considerably by demanding only the full information in a 2 or 3D space and the harmonic displacement of the spectator atoms connected to the reaction center. 

Each mode involves approximately harmonic displacements of the atoms from their equilibrium positions for each mode, i, all the atoms vibrate at a certain characteristic frequency, V/. The potential energy, V(r), of a harmonic oscillator is shown by the dashed line in Figure 1.1 as a function of the distance between the atoms, r. For any mode in which the atoms vibrate with simple harmonic motion (i.e., obeying Hooke s law), the vibrational energy states, can be described... 

Figure 4 illustrates the effect of the velocity exponent oc in terms of hysteresis loop /fvd(0 u(t) (Fig. 4b) and time history of the reaction force /fvd(0 (Fig- 4a). Assuming a harmonic displacement u(t) = sin(27t tIT) with period T = 0.5 s, it is shown that reducing the exponent a changes the hysteresis loop, from the ellipse... 

In Eq. 1 M is the damper displacement relative to the initial position of the damper and m is the damper velocity./o is the frictional force. C and n are the coefficients of the nonlinear dashpot. Suppose that the MR damper is subjected to a harmonic displacement motion... 

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