On the Integration Schemes of Retrieving Impulse Response Functions from Transfer Functions

The numerical inverse Laplace transformation (NILM) makes use of numerical integration. Generally, a high-order scheme of numerical integration renders high accuracy. However, surprisingly, this is not true for the NILM to the transfer function. Numerical examples show that the performance of higher-order schemes is no better than that of the trapezoidal scheme. In particular, the solutions from high-order scheme deviate from the exact one markedly over the rear portion of the period of interest. The underlying essence is examined. The deviation can be reduced by decreasing the frequency-sampling interval. 1. Introduction Some linear dynamic systems are described at first in the frequency domain (FD) via the frequency response function (FRF) or transfer function . For example, the characteristic of an unbounded media is relatively conveniently described in FD, as are the cases of soil-structure interaction and crack analysis [1–6]. Another salient example is where the transfer function is modified directly in FD to match some special material properties, as is the case of a hysteretic damping model [7–12]. The media property of attenuating wave propagation is also easily described via the FD expression [13]. The unitary impulse response function (UIRF) is the inverse Fourier transform of the FRF as or the inverse Laplace transform Here the real number is the convergence abscissa, that is, all the poles lie on the left side of . Though and are equivalent for describing a linear time invariant system, sometimes, is preferred, such as when inspecting the system causality, or computing in time domain [3, 7, 8, 12, 14]. For simple cases, the transform from to can be carried out analytically [15–17]. But from an engineering point of view, a numerical approach is recommended, especially when the closed-form solutions do not exist, for example, in the case of an ideal hysteretic model [8–11]. The numerical inverse Laplace transform (NILT) appears in many engineering problems [18, 19], and lots of algorithms have been constructed. Novel approaches are still developing, for example, Wang’s approach based on the wavelet [20]. The first issue in implementing the NILT is the infinite integral bound of (1.2). This is addressed by choosing a large enough to accommodate the essential part of , and ignoring contribution beyond the , that is, Provided that attenuates very fast as , and is large enough, then the approximation (1.3) is acceptable. Otherwise, if attenuates slowly, then, even if is very small for , we cannot ignore the truncating error arbitrarily, because