Harmonic oscillator forced, resonance

For a weakly perturbed, harmonic damped driven oscillator, the resonance is shifted on the frequency scale depending on the sign of the interaction forces (Fig. 1.13). The availability of analytical expressions facilitates the applications of the weakly perturbed harmonic oscillator models for AM-AFM. Such harmonic models may be useful to illustrate the concepts used in AM-AFM well enough however, in most practical imaging cases, they do not describe the experiments [15]. 

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. 

In the context considered here, a resonance is a near match of frequency between two coupled oscillations. Such a resonance will produce energy transfer from one of the oscillators to the other. A nonlinear resonance is a resonance arising from the nonlinearity of the restoring force in one or both of the oscillators, or in other words, due to the anharmonicity of one or both of the oscillators. For a harmonic oscillator, of course, the frequency of oscillation is independent of the energy or amplitude of the oscillation. Molecular vibrational modes, however, are both anharmonic, particularly at energies sufficient for unimolecular reaction, and the energy dependence of the oscillator frequency is critical to mode-mode energy transfer. 

The resonant frequency (v) of any harmonic oscillator is proportional to (fc/A/) where M is the effective mass of the system and / is a restoring force constant for the oscillator. 

The Fourier component of interaction force F(t) on the transition frequency (2-184) characterizes the level of resonance. Matrix elements for harmonic oscillators m y n) are non-zero only for one-quantum transitions, n = m . The W relaxation probability of the one-quantum exchange as a function of translational temperature To can be found by averaging the probability over Maxwelhan distribution ... 

Figure 3.30. Amplitude and phase versus frequency for an idealized cantilever with Oi = nm Q = 10, and (Oo = 150 kHz. Solid lines denote response for the cantilever driven at resonance according to a damped driven harmonic oscillator model. In (A), dashed lines represent the effect of an attractive force gradient—reduction in amplitude and decrease in phase. In (B), dashed lines represent the effect of a repulsive...

The magiiitude of the displacement caused by an oscillating electric field can be obtained by treating the system as a damped harmonic oscillator in which the restoring force is characterized by u>o, the resonant frequency of the system. 

The first resonance peak is fitted after subtracting background noise. Since thermal excitation acts as a white noise driving force, the power spectrum should follow the response function of a simple harmonic oscillator ... 

N/cm = 200 N/m. The body can move along a horizontal pivot without friction. The other end of the spring is fixed (refer to Figure 1.22 and Section 2.7). The oscillations occur in viscous media. An external harmonic variable force operates on the body F(t) = F coscot where Fq is the force amphmde value (Fg = 3 N) and co is its angular frequency. For this system, define the resonant frequency and resonant amphtude Make calculations for two values of the resistance coefficients Tj = 0.5 kg/sec and rj = 5 kg/sec. 

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... 

When an electromagnetic wave interacts with resonators, the effect of quantization of all possible stationary stable oscillating amplitudes arises without the requirement of any specifically organized conditions (like the inhomogeneous action of external harmonic force). 

The most simple, but general, model to describe the interaction of optical radiation with solids is a classical model, due to Lorentz, in which it is assumed that the valence electrons are bound to specific atoms in the solid by harmonic forces. These harmonic forces are the Coulomb forces that tend to restore the valence electrons into specific orbits around the atomic nuclei. Therefore, the solid is considered as a collection of atomic oscillators, each one with its characteristic natural frequency. We presume that if we excite one of these atomic oscillators with its natural frequency (the resonance frequency), a resonant process will be produced. From the quantum viewpoint, these frequencies correspond to those needed to produce valence band to conduction band transitions. In the first approach we consider only a unique resonant frequency, >o in other words, the solid consists of a collection of equivalent atomic oscillators. In this approach, coq would correspond to the gap frequency. 

Simple harmonic motion, such as the (undamped by frictional forces) sinusoidal oscillation of a weight suspended by a spring can also be thought of in terms of the projection of a vector traveling in a circular path. This is something you should have covered in your elementary mechanics classes, of course, but we will reexamine it here, first because it is important in infrared spectroscopy, and second because it provides some illumination concerning resonance. 

Here E(t) denotes the applied optical field, and -e and m represent, respectively, the electronic charge and mass. The (angular) frequency coq defines the resonance of the harmonic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the terms + ) correspond to the corrections to the harmonic... 

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