Duhamel s integral

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 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). 

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 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... 

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. 

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... 

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. 

---