Methods of Solution of the Equation of Motion

  • Gian Paolo Cimellaro
  • Sebastiano Marasco
Part of the Geotechnical, Geological and Earthquake Engineering book series (GGEE, volume 45)


The chapter analyzes different methods of solution of the equation of motion. The equation of motion for a forced SDOF system can be solved in closed form if the external excitation can be expressed as a harmonic function (analytical solution). Moreover, the dynamic response of a system subjected to a generic excitation can be evaluated using other approaches based on the decomposition of the irregular external force (Fourier series or Duhamel integral application). In these cases, the solution is achieved by the superposition property, so that they can be applied for a linear system. Clearly, this represents a limit to of the dynamic response of a real system in which the applied excitation causes irreversible deformation. In order to bypass the intrinsic limit of the previously proposed solution approaches, numerical methods are used. All the following examples and considerations are related to a damped SDOF system subjected to an external excitation.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Gian Paolo Cimellaro
    • 1
  • Sebastiano Marasco
    • 1
  1. 1.Department of Structural, Geotechnical and Building Engineering (DISEG)Politecnico di TorinoTorinoItaly

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