Harmonic motion damped

The return to equilibrium of a polarized region is quite different in the Debye and Lorentz models. Suppose that a material composed of Lorentz oscillators is electrically polarized and the static electric field is suddenly removed. The charges equilibrate by executing damped harmonic motion about their equilibrium positions. This can be seen by setting the right side of (9.3) equal to zero and solving the homogeneous differential equation with the initial conditions x = x0 and x = 0 at t = 0 the result is the damped harmonic oscillator equation ... 

The motion of a protein on its PES can be described as anharmonic motions near local minima (i.e. conformations), with rare hops between conformations. While the system executes this motion, we can record, for example, the distance Q(t) between two residues. If the Fourier transform of Q(t) is relatively peaked, then the distance between these residues varied in time like a damped harmonic motion. The quantity Q(t) is not an oscillator with energy levels, that is embedded in the enzyme, rather it is an internal distance of, for example, residues that participate in the equilibrium fluctuations of the enzyme. 

The behavior of the rider along the path BC can be analyzed exactly. The equation is that of damped harmonic motion with a constant resisting force added ... 

To account for the observed (slight) motion of tissue during isovolumic relaxation, we have modeled isovolumic relaxation kinematically. In accordance with Newton s law, inertial, recoil, and resistive terms are required. By changing variables from displacement (x) to pressure (P) in accordance with LaPlace s law, the expression for damped, harmonic motion applies. Lumped parameters account for elastic recoil (Ei) and viscosity (1/lt), and we include inertia as well [9] ... 

In an early study of lysozyme ([McCammon et al. 1976]), the two domains of this protein were assumed to be rigid, and the hinge-bending motion in the presence of solvent was described by the Langevin equation for a damped harmonic oscillator. The angular displacement 0 from the equilibrium position is thus governed by... 

In a damped forced vibration system such as the one shown in Figure 43.14, the motion of the mass M has two parts (1) the damped free vibration at the damped natural frequency and (2) the steady-state harmonic motions at the forcing frequency. The damped natural frequency component decays quickly, but the steady state harmonic associated with the external force remains as long as the energy force is present. 

The usefulness of spectral densities in nonequilibrium statistical mechanics, spectroscopy, and quantum mechanics is indicated in Section I. In Section II we discuss a number of known properties of spectral densities, which follow from only the form of their definitions, the equations of motion, and equilibrium properties of the system of interest. These properties, particularly the moments of spectral density, do not require an actual solution to the equations of motion, in order to be evaluated. Section III introduces methods which allow one to determine optimum error bounds for certain well-defined averages over spectral densities using only the equilibrium properties discussed in Section II. These averages have certain physical interpretations, such as the response to a damped harmonic perturbation, and the second-order perturbation energy. Finally, Section IV discusses extrapolation methods for estimating spectral densities themselves, from the equilibrium properties, combined with qualitative estimates of the way the spectral densities fall off at high frequencies. 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

Although in general the calculation of 8hm for modulation is difficult, a simplistic understanding can be obtained through consideration of the flow pattern above an oscillating infinite plate [13]. The velocity profile u(y,t) parallel to the wall, at a distance y above the wall, is a lagged, damped simple-harmonic motion ... 

The interaction of a light wave and electrons in atoms in a solid was first analysed by H. A. Lorentz using a classical model of a damped harmonic oscillator subject to a force determined by the local electric field in the medium, see Equation (2.28). Since an atom is small compared with the wavelength of the radiation, the electric field can be regarded as constant across the atom, when the equation of motion becomes ... 

In Eq. (4-31), the first three terms describe a simple damped harmonic oscillator the first term is due to molecular accelerations, the second is due to viscous drag, and the third is due to the restoring force. Qq is the oscillator frequency, which is of order 10 sec", and p is a viscous damping coefficient. The crucial term producing the dynamic glass transition is, of course, the fourth term, which has the form of a memory integral, in which molecular motions produce a delayed response. The kernel m(t — t ) is determined self-consistently by the time-dependent structure. One simple choice relating m(s) to the structure is ... 

For the case of no damping, 7 = 0, this describes a harmonic motion ii)(I,) = — Q2 w(t). The damping can be made to simulate a continuous spectrum of the second mode, a faked reservoir. 

Under conditions of small damping, the elastic (storage) modulus (G ) equals the torsional modulus of the rod (G) and thus the equation for simple harmonic motion may be rewritten ... 

The torsional potential of mean force (Fig. 24) and the correlation function for the torsional motions of the Tyr-21 ring in BPTI suggest that the time dependence of A can be described by the Langevin equation for a damped harmonic oscillator (see Chapt. IV.C and D). 

To obtain the standard form of Onsager s theory [37,38], we next linearize the thermodynamic forces in eqs. (A. 15) and (A.28). This linearization reduces these equations to coupled damped harmonic oscillator equations of motion. 

Eq. (A.43) is identical to the equations of motion for a set of n - p macroscopic coupled damped harmonic oscillators each of unit mass with the set having coordinate vector y(f), dynamical matrix and friction matrix P ... 

We now discuss a damped harmonic oscillator, which is a harmonic oscillator that is subject to an additional force that is proportional to the velocity, such as a frictional force due to fairly slow motion of an object through a fluid. 

This expression shows that if the detuning Ara is negative (i.e. red detuned from resonance), then the cooling force will oppose the motion and be proportional to the atomic velocity. The one-dimensional motion of the atom, subject to an opposing force proportional to its velocity, is described by a damped harmonic oscillator. The Doppler damping or friction coefficient is the proportionality factor. 

Equation (Cl.4.42) expresses the equation of motion of a damped harmonic oscillator with mass m. 

Smith J, Cusack S, Tidor B, Karplus M Inelastic neutron scattering analysis of low-frequency motions in proteins Harmonic and damped harmonic models of bovine pancreatic trypsin inhibitor. J. Chem. Phys. 1990,93 2974—2991. 

The oscillation of a real spring will eventually be damped out, in the absence of external driving forces. A reasonable approximation for damping is a force retarding the motion, proportional to the instantaneous velocity f = —bv= —bdx/dt, where b is called the damping constant. The differential equation for a damped harmonic oscillator can be written as... 

After the fading away of initial disturbances the motion of the rotational pendulum in vacuum always can be described by a simple damped harmonic oscillation. Eor initial conditions a(0) = oq, a(0) = 0, we have for the angular amplitude... 

Fig. 6.12. Computed Mossbauer spectra of damped harmonically bound particles in Brownian motion as a function of a, the ratio between the harmonic and the frictional force constants. (Nowik et al., 1983.)...

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