Resonance and damping

Both resonance and damping can cause some confusion and the explanations of the underlying physics can become muddled in a viva situation. Although the deeper mathematics of the topic are complex, a basic understanding of the underlying principles is all the examiners will want to see. 

The condition in which an object or system is subjected to an oscillating force having a frequency close to its own natural frequency. 

The frequency of oscillation that an object or system will adopt freely when set in motion or supplied with energy (hertz, Hz). 

We have all felt resonance when we hear the sound of a lorry s engine begin to make the window pane vibrate. The natural frequency of the window is having energy supplied to it by the sound waves emanating from the lorry. The principle is best represented diagrammatically. 

The curve shows the amplitude of oscillation of an object or system as the frequency of the input oscillation is steadily increased. Start by drawing a normal sine wave whose wavelength decreases as the input frequency increases. Demonstrate a particular frequency at which the amplitude rises to a peak. By no means does this have to occur at a high frequency it depends on what the natural frequency of the system is. Label the peak amplitude frequency as the resonant frequency. Make sure that, after the peak, the amplitude dies away again towards the baseline. 

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The aim of this appendix is to show that the g ACF (304) involving Davydov coupling, Fermi resonances, and damping, may be viewed, after some simplifications, as formally equivalent to that used by Marechal [83] in his peeling-off approach of Fermi resonances. 

A favourite application is the substitution of materials for boxes in sound generation because the material is optimal for suppressing unwanted resonances and damping oscillations. 

Here, the whole shear cell is located on a vibrating plate, mounted on vertical leaf springs, see Fig. 3. On the shear base and the shear ring, there are located one piezoelectric accelerometer each for measuring the vibration acceleration a and a, respectively. The arrangement whole shear cell vibrated can be used for investigations on the resonance and damping characteristics of cohesive powders. 

The displacements of the system at resonance will be a function of the magnitude of the driving or excitation source and damping. 

As illustrated in Figure 44.42, a resonance peak represents a large amount of energy. This energy is the result of both the amplitude of the peak and the broad area under the peak. This combination of high peak amplitude and broad-based energy content is typical of most resonance problems. The damping system associated with a resonance frequency is indicated by the sharpness or width of the response curve, ci) , when measured at the half-power point. i MAX is the maximum resonance and Rmax/V is the half-power point for a typical resonance-response curve. 

In this section we shall give the connections between the nonadiabatic and damped treatments of Fermi resonances [53,73] within the strong anharmonic coupling framework and the former theory of Witkowski and Wojcik [74] which is adiabatic and undamped, involving implicitly the exchange approximation (approximation later defined in Section IV.C). 

Dynamic mechanical testers apply a small sinusoidal stress or strain to a small sample of the polymer to be examined and measure resonant frequency and damping versus temperature and forced frequency. Instrument software computes dynamic storage modulus (G ), dynamic loss modulus (G") and tan delta or damping factor. Measurements over a wide range of frequency and temperature provide a fingerprint of the polymer with sensitivity highly superior to DSC. 

Low-frequency acquisition of the curves corresponds to a non-inertial regime wherein the mass of the cantilever does not play any role and the system can be treated as two springs in series. The in-phase and out-of-phase mechanical response of the cantilever in FMM-SFM was interpreted in terms of stiffness and damping properties of the sample, respectively [125,126]. This interpretation works rather good for compliant materials, but can be problematic for stiff samples. Assuming low damping, the cantilever response (Eqs. 9 and 10) below the resonance frequency (O0 for the case of is given by... 

Here, [G))cnnl (l)] is the ACF of the g part of the system involving Fermi resonances and quantum indirect damping, whereas [G -1 (f)]M are the ACFs corresponding to the u parts involving neither Fermi resonance nor indirect damping that are given by Eq. (276). 

Eq.(297). For simplicity, we are here limit to the special case of two Fermi resonances and the situation where the indirect damping is missing and where the direct damping of the first excited state of the high frequency mode is the same as those of the first harmonics of the two bending modes. Then, the Hamiltonian (297) reduces to... 

The ACF involving Davydov coupling, without Fermi resonance and without direct and indirect dampings. 

P-Dav( )] without Fermi resonance and indirect damping. The SD involving Davydov coupling and direct and indirect dampings, but without Fermi resonance. 

Atomic processes are very fast, so that intrinsic properties obey equilibrium statistics. An intermediate regime is characterized by typical magnetostatic and anisotropy energies per atom, about 0.1 meV, which correspond to times of order t0 0.1 ns. Examples are ferromagnetic resonance and related precession and damping phenomena. When energy barriers are involved, thermal excitations lead to a relatively slow relaxation governed by the Boltzmann-Arrhenius law [99, 133-137]... 

Equation (1) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. Even for linear polarization, the oscillation of the induced dipole moment may be damped (by material resonances) and thereby phase-shifted with respect to the oscillation of the external electric field. The usual way of expressing this phase shift is by considering the relationship between the Fourier components of the induced effect (oscillation of the induced dipole) and the stimulus (the electric field), with the damping and phase shift conveniently expressed by treating the terms involved as complex. Thus, the linear polarizability can be written as... 

Fast detection of the electro-acoustic impedance is a condition for successful kinetic studies. Soares [64] introduced a circuit to measure both resonant frequency and damping resistance R, though not as fast as simple active oscillator methods mainly used for resonant frequency measurement. Most active circuits operate in the series frequency a>s although some oscillators are designed to operate in the parallel frequency wp , which is slightly higher and very susceptible to the value of Ca. 

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