Chapter 7
Forced Vibration Analysis of Multiple
Degrees of Freedom System
7.1 Introduction
Most of the structures represented as multiple degrees of freedom system are
subjected to dynamic loads, for which they are to be analysed. The equations that
result for these systems are all coupled. The equations of motion are transformed
for facilitating the analysis. The method which is adopted for this purpose is known
as the mode superposition method. There is yet another method developed for this
purpose, which is termed as the mode-acceleration method. The mode-acceleration
method may be treated as an extension of mode superposition method; the main
difference between them lies in obtaining the final response of the system. The
mode-acceleration method gives better convergence characteristics. The equations
of motion for a MDF system may also be directly evaluated. For many problems,
particularly when external loading is complicated, the direct integration is performed
numerically. Different methods for the forced vibration analysis of MDF systems are
presented in this chapter.
7.2 Mode Superposition Method for the Determination
of Response of Undamped MDF System
The equations of motion for an undamped MDF system are given by
[M]{ ¨
x} + [K ]{x} = {F(t)}
(7.1)
where [M], [K] and {F(t)} are the mass, stiffness and the loading matrix. {x} and { ¨
x}
are the displacement and acceleration matrices. For a n-degree of freedom system,
Eq. (7.1) possesses n unknowns and they are to be obtained from simultaneous
solution of n equations.
© 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_7
261
Forced Vibration Analysis of Multiple
Degrees of Freedom System
7.1 Introduction
Most of the structures represented as multiple degrees of freedom system are
subjected to dynamic loads, for which they are to be analysed. The equations that
result for these systems are all coupled. The equations of motion are transformed
for facilitating the analysis. The method which is adopted for this purpose is known
as the mode superposition method. There is yet another method developed for this
purpose, which is termed as the mode-acceleration method. The mode-acceleration
method may be treated as an extension of mode superposition method; the main
difference between them lies in obtaining the final response of the system. The
mode-acceleration method gives better convergence characteristics. The equations
of motion for a MDF system may also be directly evaluated. For many problems,
particularly when external loading is complicated, the direct integration is performed
numerically. Different methods for the forced vibration analysis of MDF systems are
presented in this chapter.
7.2 Mode Superposition Method for the Determination
of Response of Undamped MDF System
The equations of motion for an undamped MDF system are given by
[M]{ ¨
x} + [K ]{x} = {F(t)}
(7.1)
where [M], [K] and {F(t)} are the mass, stiffness and the loading matrix. {x} and { ¨
x}
are the displacement and acceleration matrices. For a n-degree of freedom system,
Eq. (7.1) possesses n unknowns and they are to be obtained from simultaneous
solution of n equations.
© 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_7
261