Damping phase angle

Inertia foree + Damping foree + Spring foree + Impressed foree = 0 From the previous equation, the displaeement lags the impressed foree by the phase angle 6, and the spring foree aets opposite in direetion to... 

Figure 5-13b. Phase angle as a function of the frequency ratio for various amounts of viscous damping.

With damped vibration, the damping constant, c, is not equal to zero and the solution of the equation gets quite complex assuming the function, X =Xo sin(ft)/ — ). In this equation, cj) is the phase angle, or the number of degrees that the external force, Fo sin(ft)/), is ahead of the displacement, Xo sin(ft)/ — cj>). Using vector concepts, the... 

A technique for performing dynamic mechanical measurements in which the sample is oscillated mechanically at a fixed frequency. Storage modulus and damping are calculated from the applied strain and the resultant stress and shift in phase angle. 

The two constants, the amplitude A and the phase angle a, are determined by the initial conditions. It represents a damped oscillation. The formula for the frequency with damping, Eq. (10.9), indicates that the existence of damping... 

Offset value [f Amplitude [ Frequency (Hz) Time delay (sec) [0 Damping factor (1/sec) [(T Phase angle (degrees) jiT... 

We have a lot of flexibility since we can specify the Offset Value (DC Offset), Amplitude, FretJUeflCy, Time delay, Damping factor, and Phase angle. The dialog box below specifies a 1 kHz sine wave with a 1 V amplitude and 1 V offset ... 

The additional parameters add a time delay, exponential damping, and a phase angle to the waveform. Click the Apply button to view the waveform ... 

Two unknowns require two measurements. For a free oscillator these measurements are the resonant frequency and the damping. For a forced oscillator the favored combination is the amplitude ratio and phase angle over a range of applied frequencies. This combination is not available for the evaluation of coatings because of the requirement that one surface be free. The two measurements described in the example below are damping and phase angle. 

The reciprocal of the phase angle is called the quality factor Q and is often used in the description of electric circuits. The inverse of Q, i.e., Q, is often called the loss factor. The specific damping capacity of the SLS can, therefore, be expressed as AUIU=2tt8=2ttQ. Although these factors depend on frequency, they should be independent of the stress amplitude. In order to characterize the loss peak in Fig. 5.16, it is common practice to measure the peak width at 1/V2 of the maximum loss, which is approximately equal ioQ. ... 

Vibration forces applying a dynamic stress load to viscoelastic materials results in a phase shift by the phase angle 8 between stress a and elongation e. The tangent of 8 is called the mechanical loss factor d or mechanical damping. Damping is thus a measure of the heat produced by application of dynamic loads as a result of internal friction (dissipatiOTi) (Fig. 24). 

Spectra of this kind have basically been observed in some organic conductors (Le et al. 1993) and recently, apparently in HoNi2fi2C (sect. 7.1). Misfitting a spectrum of the type given by eq. (54) with an exponentially damped cosine oscillation (exp[-At] cos(fy,t + 0)) results in a phase angle 0 90° and a poor representation of the initial decay. These features are strong indicators that an incommensmate spin structure is present, which then is better represented by the Bessel function. A pertinent example is CeAl3 which will be discussed in sect. 9.3. 

FIGURE 11.17 Example of variations of the energies-per-entity of a damped oscillator as a function of the opposite of the phase angle, which is proportional to time. The curves of the capacitive, inductive, and conductive components of the effort are labeled C, L, and G, respectively. 

As for the temporal damped oscillator, these half-energies are related by two derivations (with + and - signs) with respect to an apparent phase angle f defined by... 

In Chapter 9, the wave fnnction of a spatiotemporal oscillator has been demonstrated as being the product of two individual wave functions, one for the temporal oscillator and the other for the spatial one. From this property, the phase angle (here an apparent one for taking into acconnt the damping) was expressed as a double contribution of the individual phase angles, according to the relationship... 

If the internal frictions are not dependent on the amplitude, then the plot of the Napierian logarithm of the amplitudes versus the number of cycles is linear with a slope equal to the logarithmic decrement. Moreover, during damping of a sound wave, the anelastic behavior leads to a lag between the stress and strain, and the phase angle, 8, between the two waves is then related to the Napierian logarithmic decrement by the simple equation ... 

Pig. 1. (a) When a sample is subjected to a sinusoidal oscillating stress, it responds in a similar strain wave, provided the material stays within its elastic limits. When the material responds to the applied wave perfectly elastically, an in-phase, storage, or elastic response is seen (b), while a viscous response gives an out-of-phase, loss, or viscous response (c). Viscoelastic materials fall in between these two extremes as shown in (d). For the real sample in (d), the phase angle S and the amplitude at peak k are the values used for the calculation of modulus, viscosity, damping, and other properties. 

Fig. 2. Schematic of intermittent contact mode SFM (a) cantilever vibrating freely in air vs (b) damped oscillation due to interactions with the surface (c) amplitude vs displacement dependence showing the steep decrease of the amplitude as function of distance (d) phase angle shift A< between excitation signal and cantilever response owing to interactions of the tip with the surface.

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