Moderate-to-large damping

When the relaxation is not overdamped we need to consider the full Kramers equation (14.41) or, using Eqs (14.42) and (14.43), Eq. (14.44) forf. In contrast to Eq. (14.45) that describes the overdamped limit in terms of the stochastic position variable x, we now need to consider two stochastic variables, x and v, and their probability distribution. The solution of this more difficult problem is facilitated by invoking another simplification procedure, based on the observation that if the 

Furthermore, for high barrier, these boundary conditions are satisfied already quite close to the barrier on both sides. In the relevant close neighborhood of the barrier we expand the potential up to quadratic terms and neglect higher-order terms. 

Using this together with a steady-state condition (dP/dt = df /dt = 0) in Eq. (14.44) leads to 

Note that with the simplified potential (14.61) our problem becomes mathematically similarto that of a harmonic oscillator, albeit with a negative force constant. Because of its linear character we may anticipate that a linear transformation on the variables X and V can lead to a separation of variables. With this in mind we follow Kramers by making the ansatz that Eq. (14.62) may be satisfied by a function f of one linear combination of x and v, that is, we seek a solution of the form 

Such solution may indeed be found see Appendix 14A for technical details. The function/(v,x) is found in the form 

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Seismic Actions Due to Near-Fauit Ground Motion, Fig. 11 Normalized 5% damped equal-ductility pseudovelocity response spectra of elastic-perfectly plastic SDOF systems (a) all earthquakes (Mw 5.6-7.6), (b) moderate earthquakes (Mw 5.6-6.3), (c) moderate-to-large earthquakes (Mw 6.4-6.7), and (d) large earthquakes... 

The ensemble of the normalized elastic response spectra illustrated in the first panel of Fig. 11 can be utilized to derive normalized elastic design spectra for moderate, moderate-to-large, and large earthquakes, as well as for the entire set of seismic events considered by Mavroeidis et al. (2004). The solid and dashed lines in the top panel of Fig. 11 represent the mean and mean-plus-one-standard-deviation 5% damped normalized elastic response spectra. These average elastic response spectra can be used to derive normalized elastic design spectra for two different nonexceedance probability levels. 

Between the most and least difficult elements lies a broad spectrum of moderately difficult processes. Although most of these processes are dynamically complex, their behavior can be modeled, to a large extent, by a combination of dead time plus single capacity. The proportional band required to critically damp a single-capacity process is zero. For a dead-time process. It Is Infinite. It would appear, then, that the proportional band requirement Is related to the dead time in a process, divided by Its time constant. Any proportional band, hence any process, would fit somewhere In this spectrum of processes. A discussion of multicapacity processes In Chap. 2 will reaffirm this point. 

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