Damped resonance response

Tani discusses the dynamics of displacive ferroelectrics and obtains a damped-resonant response. 

FIGURE 14.5 Pressure step response computed from the model (tq = 2, Tsm = 5, and G = 10). The step was produced by altering the initial condition for pressure at t = 0. Note that pressure and resistance are 180° out of phase. A damped resonant response is evident with a resonant frequency of 0.1 Hz. 

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. 

Examples of the Applications of Damping. As was pointed out earlier, structural damping can control "free" responses, i.e., (a) resonant response and spatial... 

As an example of principle number 4 above, consider a locally excited large panel on which resonant bending waves account for most of the vibratory response. However, assume that these resonant waves have a wavelength shorter than that of free waves in the surrounding air. In such a case the resonant waves are poorly coupled to the air, and radiate very little sound. What radiation there is can be dominated by non-resonant forced motion around the drive point (and at other discontinuities). As a result, applied damping can reduce the resonant response, but not the forced motion and the radiation of sound. 

The dielectric response shown in Fig. 43a resembles a highly damped resonance curve with the peak frequency v m located in the THz region. The s// L(v) spectra in Fig. 43b are compared for y =2.8 (curve 1) andy L = 2 (curve 2), for which the lifetimes t are respectively 0.089 and 0.14 ps. For greater y L (i.e., for shorter lifetime t l) the absorption curves become shallower. The frequency vj m of the loss-peak intensity, marked in Fig. 43b by the vertical lines, agrees with the estimate (193) only approximately. It is worthwhile to emphasize the following ... 

The discussion in this section is applicable to any polar solid (i.e., one that has a permanent dipole) in which relaxation of the permanent dipoles occurs. In principle this approach could be applicable to piezoelectric and ferroelectric solids at temperatures above their transition temperature as well (see next chapter). However, the same response, as shown in Fig. 14.11, can also occur as a result of heavily damped resonance. This is easily seen in Fig. 14.7a as /. which is a measure of the damping, increases, the resultant resonance curves become flatter. Experimentally it is not easy to distinguish between the two phenomena. 

The loss tangent determines such macroscopic physical properties as the damping of free vibrations, the attenuation of propagated waves, and the frequency width of a resonance response. It can often be more conveniently measured than any other viscoelastic function, by observations of these phenomena, and is of considerable practical interest. It is less susceptible of direct theoretical interpretation than the other functions, however. 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic 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 temis + ) correspond to tlie corrections to the hamionic... 

We now want to study the consequences of such a model with respect to the optical properties of a composite medium. For such a purpose, we will consider the phenomenological Lorentz-Drude model, based on the classical dispersion theory, in order to describe qualitatively the various components [20]. Therefore, a Drude term defined by the plasma frequency and scattering rate, will describe the optical response of the bulk metal or will define the intrinsic metallic properties (i.e., Zm((a) in Eq.(6)) of the small particles, while a harmonic Lorentz oscillator, defined by the resonance frequency, the damping and the mode strength parameters, will describe the insulating host (i.e., /((0) in Eq.(6)). 

We can design a controller by specifying an upper limit on the value of Mpco. The smaller the system damping ratio C the larger the value of Mpco is, and the more overshoot, or underdamping we expect in the time domain response. Needless to say, excessive resonance is undesirable. 

In situations where absorption of the incident radiation by the transducing gas is troublesome a piezoelectric transducer (made from barium titanate, for example) can be attached to the sample (or sample cuvette in the case of liquids) to detect the thermal wave generated in the sample by the modulated light (8,9). The low frequency, critically damped thermal wave bends the sample and transducer thus producing the piezoelectric response. The piezoelectric transducer will also respond to a sound wave in the solid or liquid but only efficiently at a resonant frequency of the transducer typically of the order of 10 to 100 KHz (see Figure 4). Thus neither in the case of microphonic nor piezoelectric detection is the PA effect strictly an acoustic phenomenon but rather a thermal diffusion phenomenon, and the term "photoacoustic" is a now well established misnomer. 

A continuation of the preceeding diaphragm integration indicates a seemingly resonant condition after several cycles of the applied wall reactions. This result has little effect on the first response peaks and disappears with the application of a reasonable amount of damping. 

Note In centrifugal pumps, the typical damped response to unbalance does not show a peak in displacement at resonance large enough to assess the amplification factor. With this limitation, assessment of the damped response to unbalance is restricted to comparing rotor displacement to the available clearance. 

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... 

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