Chapter 6
Free Vibration of Multiple Degrees
of Freedom System
6.1 Introduction
After getting a taste of two degrees of freedom system, we now proceed to a more
general treatment of the problem. In the previous chapter, it is revealed that the two
degrees of freedom system are more involved in computation than the single degree
of freedom system. When more number of masses are considered, the mathematical
formulation becomes much more complicated.
The theory of multiple degrees of freedom system (MDF system) is important, because they are applicable not only to systems which may be classified as
discrete, but also to the numerical solution of the continuous systems converted into
discrete systems, by lumping the mass at significant points. In many cases, a structure represented by its model of a single degree of freedom does not describe it
adequately. A truer representation of such cases will be the consideration of multiple
degrees of freedom system of the structure. Many structures such as framed structures
have essentially lumped masses, since the mass of the columns is often negligible
compared to floors and the system can be treated as multiple degrees of freedom
system. The free vibration analysis of the multiple degrees of freedom system is
presented in this chapter.
6.2 Equations of Motion of MDF Systems
A frame structure with rigid floors and multiple discs mounted on a shaft are shown
in Fig. 6.1. Both these systems can be reduced to equivalent spring–mass-damper
system shown in Fig. 6.2a.
The freebody diagrams of the three masses are shown in Fig. 6.2b. Applying
D’Alembert’s principle, the equations of motion are written as follows:
© 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_6
183
Free Vibration of Multiple Degrees
of Freedom System
6.1 Introduction
After getting a taste of two degrees of freedom system, we now proceed to a more
general treatment of the problem. In the previous chapter, it is revealed that the two
degrees of freedom system are more involved in computation than the single degree
of freedom system. When more number of masses are considered, the mathematical
formulation becomes much more complicated.
The theory of multiple degrees of freedom system (MDF system) is important, because they are applicable not only to systems which may be classified as
discrete, but also to the numerical solution of the continuous systems converted into
discrete systems, by lumping the mass at significant points. In many cases, a structure represented by its model of a single degree of freedom does not describe it
adequately. A truer representation of such cases will be the consideration of multiple
degrees of freedom system of the structure. Many structures such as framed structures
have essentially lumped masses, since the mass of the columns is often negligible
compared to floors and the system can be treated as multiple degrees of freedom
system. The free vibration analysis of the multiple degrees of freedom system is
presented in this chapter.
6.2 Equations of Motion of MDF Systems
A frame structure with rigid floors and multiple discs mounted on a shaft are shown
in Fig. 6.1. Both these systems can be reduced to equivalent spring–mass-damper
system shown in Fig. 6.2a.
The freebody diagrams of the three masses are shown in Fig. 6.2b. Applying
D’Alembert’s principle, the equations of motion are written as follows:
© 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_6
183