Damping linear

The dynamic equilibrium of damped, linear elastic, SDOF system illustrated in Figure 6.3 is expressed mathematically as follows. 

If e = 0, the system (19)—120) describes a damped linear oscillator governed by the equation... 

In some fields, especially in acoustics, special names are given to the number 1 when expressing physical quantities defined in terms of the logarithm of a ratio. For a damped linear oscillation the amplitude of a quantity as a function of time is given by... 

To expose the source of the difficulties, we begin with a practice problem that can be solved exactly. Consider the weakly damped linear oscillator... 

Caughey, T. K. and O Kelly, M. E. J. Classical normal modes in damped linear dynamic systems. Journal of Applied Mechanics (ASME), 32(12) (1965), 583-588. 

The loss factor rj for the damped linear SDOF is defined by... 

The general matrix equation of motion for a damped linear structure is given by... 

Thus, the approximation applicable to slightly damped linear oscillators with intermediate eigenfrequencies exqjJied by a broad band process having a narrow-band output renders the PSD of the modal driving forces in the form... 

The broadening Fj is proportional to the probability of the excited state k) decaying into any of the other states, and it is related to the lifetime of the excited state as r. = l/Fj . For Fjt = 0, the lifetime is infinite and O Eq. 5.14 is recovered from O Eq. 5.20. Unfortunately, it is not possible to account for the finite lifetime of each individual excited state in approximate theories based on the response equations (O Eq. 5.4). We would be forced to use a sum-over-states expression, which is computationally intractable. Moreover, the lifetimes caimot be adequately determined within a semiclassical radiation theory as employed here and a fully quantized description of the electromagnetic field is required. In addition, aU decay mechanisms would have to be taken into account, for example, radiative decay, thermal excitations, and collision-induced transitions. Damped response theory for approximate electronic wave functions is therefore based on two simplifying assumptions (1) all broadening parameters are assumed to be identical, Fi = F2 = = r, and (2) the value of F is treated as an empirical parameter. With a single empirical broadening parameter, the response equations take the same form as in O Eq. 5.4 with the substitution to to+iTjl, and the damped linear response function can be calculated from first-order wave function parameters, which are now inherently complex. For absorption spectra, this leads to a Lorentzian line-shape function which is identical for all transitions. 

The damped linear response function computed from the real part of Eq. 5 JO with the indicated values of the broadening F (corresponding to 0,250, 500,1,000, and 2,000 cm ). Electric dipole transition moments and excitation energies were obtained using CCSD for the hydrogen molecule with the aug-cc-pVDZ basis set... 

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... 

Adhikari S, Wagner N (2004) Direct time-domain integration method for exponentially damped linear systems. Comput Struct 82 2453-2461 Alehashem SMS, Keyhani A, Pourmohammad H (2008) Behavior and performance of structures equipped with ADAS and TADAS dampers (a comparison with conventional structures). 14th World Conference on Earthquake Engineering, 12-17 Oct 2008, Beijing Bayat M, Abdollahzadeh G (2011) Analysis of the steel braced frames equipped with ADAS devices under the far field records. Latin Am J Solids Struct 8 163-181... 

Campbell KW (1981) Near-source attenuation of peak horizontal acceleration. Bull Seismol Soc Am 71 2039-2070 Campbell KW, Bozorgnia Y (2008) NGA ground motion model for the geometric mean horizontal component of PGA, PGV, PGD and 5 % damped linear elastic response spectra for periods ranging fiom 0.01 to 10 s. Earthq Spectra 24 139-171... 

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... 

Due to the recent focus in sustainable earthquake-resistant design, there is a renewed interest to consider general nonviscous and nonpropor-tionally damped linear dynamic systems as new... 

Caughey TK, O Kelly MEJ (1965) Classical normal modes in damped linear dynamic systems. Trans ASME J Appl Mech 32 583-588 McTavish DJ, Hughes PC (1993) Modeling of linear viscoelastic space structures. Tran ASME J Vib Acoust 115 103-110... 

Adhikari S (2001) Classical normal modes in non-viscously damped linear systems. AIAA J 39(5) 978-980... 

Adhikari S (2002) Dynamics of non-viscously damped linear systems. ASCE J Eng Mech 128(3) 328-339... 

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