Classical damping

Finally, we can recover the classical damping constant from Eq. (A3.30) by writing... 

Now, we see that the initial uncertainty of the packet I a) keeps getting smaller with the progression of time and becomes negligible as t — °°. Also, the evolution of the wave packet continually proceeds toward the motion of a classic damped oscillator with the progression of time. 

Another point of interest is the close similarity between the lineshapes associated with the quantum damped two-level system, Eq. (9.40), and the classical damped harmonic oscillator. We will return to this issue in Section 9.3. 

Figure P14.8 also shows the error norm, ej, versus the number of Ritz vectors (from Problem 14.7). The error is smaller when Ritz vectors are used, because they are derived from the force distribution. Ritz vectors are useful for dynamic analysis of large systems with classical damping, since the vibration properties of the system can be obtained by solving, a smaller eigenvalue problem of order 7, instead of original eigenvalue problem of size N. It must be noted that the resulting frequencies and mode shapes are approximations to the...

In Sect. 2.8 we saw that the mean lifetime r, of a molecular level E/, which decays exponentially by spontaneous emission, is related to the Einstein coefficient Ai by Ti = 1/A/. Replacing the classical damping constant y by the spontaneous transition probability A/, we can use the classical formulas (3.9-3.11) as a correct description of the frequency distribution of spontaneous emission and its linewidth. The natural halfwidth of a spectral line spontaneously emitted from the level Ei is, according to (3.11),... 

The methods presented in this article make use of the FFT (Fast Fourier transform) of measured acceleration data on a selected frequency band around the modes of interest. The modes are assumed to be classically damped. 

As for the free vibrations, the motion of an n-dof classically damped structure can be thought as a linear combination of normal modes cj)j, each of them vibrating with the associated circular frequency coj and damping Indeed, using the modal coordinates q(t) as defined in Eq. 6 and Eq. 11 and left multiplying by 4>, we obtain n uncoupled differential equation of motion in modal coordinates ... 

Vibrations of Classically Damped MDOF Structures in the Frequency Domain... 

Papagiannopoulos GA, Beskos DE (2006) On a modal damping identification model of building structures. Arch AppI Mech 76 443-463 Papagiannopoulos GA, Beskos DE (2009) On a modal damping identification modal for non-classically damped linear building stmctures subjected to earthquakes. Soil Dyn Earthquake Eng 29 583-589... 

The latter system of two algebraic equations is then to be solved simultaneously for ao and ai. A further generalization of classical damping can... 

The first more simplistic approach disregards all the coupled contributions of Eq. 31b by essentially zeroing in the row vector C all elements apart from C, i.e., C = [0...Ci...0]. Subsequently, the remaining analysis becomes identical to the above described for the classically damped case. 

Starting from the classically damped modal equation Eq. 22d, one can rewrite it as... 

After bringing the structural system to its modal description equivalent, the solutions pursued whether in terms of modal displacements or in terms of modal accelerations and velocities were always expressed in the time domain. Considering the case of the classically damped system with periodic loading and focusing on the probably most significant part of the response, the forced or else for this case steady state, one may suggest some alternatives to Eqs. 17 and 25. The reason is that the Duhamel s integral that provide the steady-state time response involves the convolution operation between the applied load and the unit-impulse response function. This term tends to perplex calculations. 

For classically damped structures the modal damping matrix H is a diagonal matrix listing the... 

In this study a unitary approach to evaluate the spectral characteristics of the structural response, to perform the reliability assessment, of classically damped linear systems subjected to stationary or nonstationary mono-/multi-correlated zero-mean Gaussian excitations, is described. 

Borino G, Muscolino G (1986) Mode-superposition methods in dynamic analysis of classically and non-classically damped linear systems. Earthq Eng Struct Dyn 14 705-717... 

In Sect.2.7 we saw that the mean lifetime x. of a molecular level E., which decays exponentially by spontaneous emission, is related to the Einstein coefficient by x. = 1/A. Replacing the classical damping constant... 

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