Duhamel Integrals

The response of a simple oscillator to a seismic event is given by the value of the spectral response (Fig. 15-6) if the earthquake is defined by its spectrum. Instead, when the earthquake is defined by the space-time history of the ground acceleration, its response can be calculated by the Duhamel integral ... 

Figure 15-12 is the same record applying the Duhamel Integral with the Microsoft Excel macro integraleduhamel.xls (prepared by the author and enclosed as the download file, DUHAMEL on the book s accompanying website). The Duhamel Integral has been calculated for 20 simple oscillator frequencies ranging from 0.1 Hz to 20 Hz. 

It is evident that these equations have the same form as the equation of motion of a simple oscillator with the substitution of the generalized coordinate y in place of x in the simple system. It is therefore evident that, once the estimate of p(x) has been made (even a tentative shape generally gives good results without the need of iterations), the coefficients of the equation can be calculated and the solution can be obtained by the methods valid for one degree of freedom systems (i.e. the Duhamel integral, response spectrum, etc.). 

In this equation, stands for the damping factor and v for the generalized displacement for the mode i. It is a measure for the response of the ith mode to the earthquake-induced excitation. is the so-called participation factor which constitutes a measure or unit for the participation of the Ith mode in the overall excursion during an external excitation. Usually, the equation is solved by means of the Duhamel integral, and the response of the Ith mode in relation to time t can be established from... 

The integral in Equation (4.3) is called a Duhamel integral, and it is a useful illustration of the consequences of the Boltzmann superposition principle to evaluate the response for a number of simple loading programmes. Recalling the development that leads to Equation (5.2) it can be seen that the Duhamel integral is most simply evaluated by treating it as the summation of a number of response terms. Consider two specific cases ... 

In the theory of Equation (10.19), four functions of stress go, gi, gi and Ua characterize the non-linearity and must be evaluated over the required stress range. Experimental regimes that involve periods of constant stress, during which the functions are constants, have proved useful for this purpose. For a single-step creep test at stress o applied at time t = 0, Equation (10.19) can be evaluated, noting that it contains a Duhamel integral like Equation (4.3), to give the result... 

Notice that in this case the first element of vector y(0, written in integral form, coincides with the well-known Duhamel integral. Starting by the integral form solution in state variables. 

The relative displacement response of a structure (Fig.2) resulting from a shock defined by the acceleration (it of the support is given by the Duhamel integral... 

Combining Equations 7 to 9 and describing the interaction of flame spread and rate of heat release by a superposition, Duhamel-type integral (19), a final form of the regression equation was given in (ii) as... 

That is, the polarization process represented by P,(t) will be established throughout a time evolution. To mathematically express this process, the following time convolution integral (normally named the Duhamel s integral) [32] is used ... 

Figure 1.55. Presentation of the primary field for determination of Duhamel s integral.

This integral is also called Duhamel s integral and it permits us to find the transient response for an arbitrary shape of a current excitation when the transient response of the medium for a step function excitation is already known. 

As was shown in Chapter 2, an arbitrary change of moment M t) with time can be presented with the help of a Duhamel s integral as a sum of successively turning on step functions h t — r) with magnitude M (r) (Fig. 9.1a) ... 

The above integration, however, is rather complicated, since Eq. (7-8) includes a number of parameters. There have been many trials to obtain analytical solutions for similar but simpler cases including the pioneering work of Rosen (1952), which used Duhamel s theorem to include intraparticle diffusion kinetics. 

Consider the coupled response displacement Y Ct) to the transient and random ground acceleration Ug(t) with zero mean. The simple input-output transfer relation in the time domain is given by the Duhamel s convolution integral... 

The Boltzman superposition principle (or integral) is applicable to stress analysis problems in two and three-dimensions where the stress or strain input varies with time, but first the approach will be introduced in this section only for one-dimensional or a uniaxial representation of the stress-strain (constitutive) relation. The superposition integral is also sometimes referred to as Duhamel s integral (see W.T. Thompson, Laplace Transforms, Prentice Hall, 1960). 

Doyle et al. [17], used Duhamel s superposition integral to numerically solve the solid phase diffusion, as described by Equation 25.36 The exact solution obtained for spherical particles provides a considerable improvement in computational speed however, it is limited to restrictive assumptions (e.g., perfectly spherical particles and constant diffusion coefficient in the solid phase, where the exact integral solution is possible). This approach still requires the numerical solution of the solid phase diffusion at each control volume. 

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. 

The integral with respect to the increments of the stochastic point process is the stochastic counterpart of the usual Duhamel convolution integral. 

Notice that the function Zo(tu, i) is the so-called evolutionary frequency response function of the oscillator (Li and Chen 2009). Remarkably, since the integrals (Eq. 34) are convolution integrals of Duhamel s type, they can be interpreted as the response, in terms of state variables, of the quiescent oscillator, at time t = 0, subjected to the deterministic complex function /(ffl, t) = exp(iffl0 fl((u, 0- By introducing the state variables, the evolutionary frequency response vector function can be defined as... 

The conventional filtered white-noise process f(t, k) is the stationary response of a linear time-invariant filter subjected to a white-noise process. White-noise w t) is a stationary random process in time that has a zero mean and a constant spectral density for all frequencies. The response of a linear filter to a white-noise process may be calculated by using the Duhamel convolution integral, and hence the general formulation of a filtered white-noise process can be written in the following form ... 

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