Displacement response

Ground displacement response spectrum in terms of spectral displacement versus natural period of oscillations (Figure 14.14). 

Figure 14.14 Displacement response spectra of the Koyna (India), Chamoli (India) and EL Centro (USA) earthquakes...

The equation of motion given above may be solved for maximum displacement response using Figure B.2. Transformation factors K, K, and Xlm are provided for a variety of structural elements in References 7, 73, 75, 92, and 93. Solutions in terms of maximum displacement response of the nonlinear SDOF model to transient loads are also provided by these references in graphical form or in the form of empirical equations. 

Actuators Pressure Stroke (Displacement) Response Time Reliability... 

Relation Between the Displacement Response Function and the Time-Dependent Diffusion Coefficient... 

The displacement response function x t, f) characterizes the average displacement (x(t) — x(to)) at time t, due to a unit impulse of force taking place at a previous time t < t. Is is easily deduced from the equation of motion (61), in which one adds to the random force F(t) a nonrandom force proportional to 5 (t — t ). One thus gets... 

Adding to the Langevin force in Eq. (73) a nonrandom force proportional to 8(t — t1), one gets the expression of the displacement response function ... 

Exactly like its classical analog Eq. (94), Eq. (125) allows one to express the displacement response function in terms of the time-dependent diffusion coefficient. However, contrary to the classical case in which Xxx( 0 is directly proportional to D(t — t ), in the quantum formulation Xxv( 0 is a convolution product, for the value t — t of the argument, of the functions D(t ) and logcoth(7i fi /2 h). Inverting the convolution equation (125) yields an expression for D(t) in terms of the dissipative part of the displacement response function ... 

Since the displacement response function does not depend on the bath temperature [Eq. (115) or (116)], Eq. (126) displays the above quoted property that, at any fixed time t,D(t) is a monotonic increasing function of the temperature. In the infinitely short memory limit, taking into account the corresponding expression (79) of %xx, one gets from Eq. (126)... 

For a particle evolving in a thermal bath, we focused our interest on the particle displacement, a dynamic variable which does not equilibrate with the bath, even at large times. As far as this variable is concerned, the equilibrium FDT does not hold. We showed how one can instead write a modified FDT relating the displacement response and correlation functions, provided that one introduces an effective temperature, associated with this dynamical variable. Except in the classical limit, the effective temperature is not simply proportional to the bath temperature, so that the FDT violation cannot be reduced to a simple rescaling of the latter. In the classical limit and at large times, the fluctuation-dissipation ratio T/Teff, which is equal to 1 /2 for standard Brownian motion, is a self-similar function of the ratio of the observation time to the waiting time when the diffusion is anomalous. 

Fig. 11 Force versus displacement response of T-Peel specimens bonded with XD4600 adhesive and tested at 5 mm/min, 23 °C.

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