Damping amplitude

Fig. 1.11 Qualitative plots of solutions (normalized damping amplitude and phase shift) for a damped, driven oscillator as a function of the relative frequency ratio. In-phase vibrations correspond to a phase shift of zero, while out-of-phase means a phase shift of 180° (reradian). With increasing quality factor (decreasing damping) the resonance becomes sharper...

Fig. 1.15 (a) Theoretical amplitude curve as a function of sample—tip distance (d). The shadowed region corresponds to unstable behavior, (b) Corresponding force maximum as a function of the damped amplitude. Adapted from [15] with permission by Elsevier... 

The so-called SEDOR technique (spin-echo double resonance [9, 47, 48], see Fig. 5) allows the determination of internuclear distances, rjs, for isolated spin pairs consisting of a resonating J-spin and a non-resonating S-spin. Hahn s echo [49] of the resonating J-spins is observed with a time between the 7i/2-pulse and the 7T-pulse. A rr-pulse applied to the S-spins after a variable time ti (0characteristic damping of Hahn s echo of the resonating /-spins. The damped amplitude, of Hahn s echo under SEDOR conditions is determined by ... 

Hz. The free oscillation amplitude of the oscillating cantilever was typically around 50 nm, the setpoint amplitude (damped amplitude, when the tip was in intermittent contact) was slightly lower. In aU measmements, topographic (height mode) and viscoelastic (phase-mode) data were recorded simultaneously. It should be noted that semicrystaUine polymers are well suited for the... 

DAC DAC (distance amplitude control) up to 40 dB optimized for material with high sound damping 0.1 to 20 MHz (-3 dB)... 

The expression exp(-cxx) describes the reduction of the wave amplitude in absorbing materials. The damping coefficient a can be split into an absorption coefficient Oa and the scattering coefficient Oj. 

The damping coefficient a can be determined by measuring the exponential reduced wave amplitude p, at various points during propagation. 

In the fonner, the excitation amplitude to the lever (via the piezo) is kept constant, thus, if the lever experiences a damping close to the surface the actual oscillation amplitude falls. The latter involves compensatmg the excitation amplitude to keep the oscillation amplitude of the lever constant. This mode also readily provides a measure of the dissipation during the measurement [100]. 

All electrical equipment are designed for a specific BIL, as indicated in Tables 11.6, 13.2, 14.1, and 32.1(A) for motors, switchgears and bus systems respectively, and Tables 13.2 and 13.3 for the main power system (line clearances and insulators). If the actual severity of a prospective surge, i.e. its amplitude and/or rise time or both, is expected to be higher than these levels (higher amplitude and lower rise time) the same must be damped to a safe level, with the use of surge arresters, surge capacitors or both as discussed later. 

Figure 4-6 is an amplitude-speed eurve showing the loeation of the running speed to the eritieal speed, and the amplitude inerease near the eritieal speed. When the rotor amplifieation faetor, as measured at the vibration probe, is greater than or equal to 2.5, that frequeney is ealled eritieal and the eorresponding shaft rotational frequeney is ealled a eritieal speed. For the purposes of this standard, a eritieally damped system is one in whieh the amplifieation faetor is less than 2.5. 

In the absenee of damping the amplitude ratio beeomes infinite at tu =... 

The previous equation shows that when lu < uj ,8r is positive. Thus, when operating below the critical speed, the system rotates with the center of mass on the outside of the geometric center. Operating above the critical speed (lu > LUn), the shaft deflection 8r tends to infinity. Actually, this vibration is damped by outside forces. For very high speeds (lu >> LUn), the amplitude 8r equals —e, meaning that the disc rotates about its center of gravity. 

This speed becomes critical when the frequency of excitation is equal to one of the natural frequencies of the system. In forced vibration, the system is a function of the frequencies. These frequencies can also be multiples of rotor speed excited by frequencies other than the speed frequency such as blade passing frequencies, gear mesh frequencies, and other component frequencies. Figure 5-20 shows that for forced vibration, the critical frequency remains constant at any shaft speed. The critical speeds occur at one-half, one, and two times the rotor speed. The effect of damping in forced vibration reduces the amplitude, but it does not affect the frequency at which this phenomenon occurs. 

From Equation (17-4), one will find that the phase lag is a function of the relative rotating speed lu/lu and the damping factor (See Figure 17-1.) The force direction is not the same as the maximum amplitude. Thus, for maximum benefit, the correction weight must be applied opposite to the force direction. 

The damped (unbalance response) plots are not really very impressive, but they address the information every operator would like to see. Amplitude is plotted against speed at various stations such as at the probes (see... 

Figure 9-13). A response, of course, is a sudden rise in amplitude at a speed. The analysis is performed using a mathematical model that includes the effects of damping in the equations, making the model much more complex than any previous analysis. The University of Virginia carries out continuous research in which a consortium of users, vendors, and researchers provide funds, and data and interchange ideas to advance the science of rotor dynamics. Other organizations, such as the Bently Rotor Dynamics Research Corporation and Texas A M University, are also carrying on similar work on a continuing basis.

Damping in the system represents dissipation of vibratory energy that reduces the amplitudes in the system. 

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