Chapter 16
Computer Programs in Vibration
Analysis
16.1 Introduction
Based on the chapters dealt with, it is revealed that the linear vibration analysis of
a structure will involve one or more of the following types: free vibration analysis,
forced vibration analysis and random vibration analysis. The free vibration analysis
is essentially an eigenvalue problem. If the problem is to be solved on the basis of
matrix operations, then the mass matrix and the stiffness matrix of the structure are
to be formed. It may be achieved by using the direct stiffness method. Once they are
formed, the natural frequencies and the mode shapes may be determined by using
any one of the well-known techniques discussed in Chapter 6. In fact, the computer
program for the solution of the eigenvalue problem is readily available in textbooks
[1, 2] and journals [3–5].
16.2 Computer Program for Forced Vibration Analysis
The forced vibration analysis of structures can be carried out in two ways: the direct
integration method and the mode superposition method. As in free vibration analysis,
the mass matrix and the stiffness matrix of the structure are to be formed. Numerical
methods such as Newmark’s β method presented in Chapter 7 may be taken resort
to for the determination of the response. In order to obtain the response by the mode
superposition method, the first step lies in the free vibration analysis of the structure.
The natural frequencies and mode shapes thus determined become the input to the
problem. In this approach, coupled equations are reduced to n equations of single
degree of freedom system. The individual equation of normal mode can be solved by
using numerical techniques involving time integration. The outline of the program
on forced vibration by mode superposition method, after the free vibration analysis
is carried out and the results stored in a low-speed magnetic tape, is given in Table
16.1.
© 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_16
563
Computer Programs in Vibration
Analysis
16.1 Introduction
Based on the chapters dealt with, it is revealed that the linear vibration analysis of
a structure will involve one or more of the following types: free vibration analysis,
forced vibration analysis and random vibration analysis. The free vibration analysis
is essentially an eigenvalue problem. If the problem is to be solved on the basis of
matrix operations, then the mass matrix and the stiffness matrix of the structure are
to be formed. It may be achieved by using the direct stiffness method. Once they are
formed, the natural frequencies and the mode shapes may be determined by using
any one of the well-known techniques discussed in Chapter 6. In fact, the computer
program for the solution of the eigenvalue problem is readily available in textbooks
[1, 2] and journals [3–5].
16.2 Computer Program for Forced Vibration Analysis
The forced vibration analysis of structures can be carried out in two ways: the direct
integration method and the mode superposition method. As in free vibration analysis,
the mass matrix and the stiffness matrix of the structure are to be formed. Numerical
methods such as Newmark’s β method presented in Chapter 7 may be taken resort
to for the determination of the response. In order to obtain the response by the mode
superposition method, the first step lies in the free vibration analysis of the structure.
The natural frequencies and mode shapes thus determined become the input to the
problem. In this approach, coupled equations are reduced to n equations of single
degree of freedom system. The individual equation of normal mode can be solved by
using numerical techniques involving time integration. The outline of the program
on forced vibration by mode superposition method, after the free vibration analysis
is carried out and the results stored in a low-speed magnetic tape, is given in Table
16.1.
© 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_16
563