Chapter 4
Numerical Methods in Structural
Dynamics: Applied to SDF Systems
4.1 Introduction
The analysis of SDF systems requires the solution of the differential equation. For
simple cases of forcing functions, the solution of the equation can be obtained in
closed bound form. However, the dynamic loading obtained from the records for
many practical cases is not easily mathematically amenable and as such numerical
procedures are to be applied for obtaining the solution. In time domain analysis, the
numerical techniques can be classified as follows:
(1) Direct integration techniques
(2) Numerical evaluation of the Duhamel’s integral.
In direct integration techniques, the basic differential equation of a SDF system
is integrated using a step-by-step procedure [1–6]. The equations in this case are
not transformed to any other form. In the case of Duhamel’s integral, numerical
integration is performed over the integrand. The numerical procedures are much
more general than the rigorous approach, which are of limited usefulness due to
severe restrictions imposed on it by practical problems.
4.2 Direct Integration Techniques
In direct integration techniques, the time interval is divided into a discrete number
of steps t apart. The solution is attempted at each time step, i.e. the differential
equation is solved progressively from given or known initial conditions. Shorter the
time step, the more accurate is the solution. A few of the direct integration techniques
are given below.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_4
129
Numerical Methods in Structural
Dynamics: Applied to SDF Systems
4.1 Introduction
The analysis of SDF systems requires the solution of the differential equation. For
simple cases of forcing functions, the solution of the equation can be obtained in
closed bound form. However, the dynamic loading obtained from the records for
many practical cases is not easily mathematically amenable and as such numerical
procedures are to be applied for obtaining the solution. In time domain analysis, the
numerical techniques can be classified as follows:
(1) Direct integration techniques
(2) Numerical evaluation of the Duhamel’s integral.
In direct integration techniques, the basic differential equation of a SDF system
is integrated using a step-by-step procedure [1–6]. The equations in this case are
not transformed to any other form. In the case of Duhamel’s integral, numerical
integration is performed over the integrand. The numerical procedures are much
more general than the rigorous approach, which are of limited usefulness due to
severe restrictions imposed on it by practical problems.
4.2 Direct Integration Techniques
In direct integration techniques, the time interval is divided into a discrete number
of steps t apart. The solution is attempted at each time step, i.e. the differential
equation is solved progressively from given or known initial conditions. Shorter the
time step, the more accurate is the solution. A few of the direct integration techniques
are given below.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
M. Mukhopadhyay, Structural Dynamics,
https://doi.org/10.1007/978-3-030-69674-0_4
129