Chapter 8
Free Vibration Analysis of Continuous
Systems
8.1 Introduction
Structures analysed so far have been treated as discrete systems. Structures have been
idealised, and for convenience of computation, simpli-fying assumptions have been
introduced and as such results obtained can only be treated as approximate. But they
are, however, sufficiently accurate for most practical purposes. Increase of structural
discretisation will entail an increase of degrees of freedom, and this will improve the
accuracy of the results.
For all systems, the mass of the members is continuously distributed. As such,
specifying the displacement at every point in the system will require infinite number
of coordinates. The system in this case is assumed to have infinite degrees of freedom.
For such cases, the mass is inseparable from the elasticity of the system. Continuous
models of vibrating systems are indeed more realistic because structural properties
are distributed rather than concentrated at discrete points. The price however to be
paid for increased realism is the increased complexity.
The reason as to why the practical structures are reduced to discrete systems is
due to the fact that the analysis of continuous system is much more involved. The
argument is similar to that applied to the static analysis, in which for a somewhat
redundant structure, solutions obtained by classical methods are very tedious and
for such cases, matrix or the finite element analysis is preferred. However, there are
certain applications where modelling the system on the basis of distributed parameter
may be justified. In this chapter, the free vibration analysis of continuous systems
has been discussed. The equation of motion is a partial differential equation for
continuous systems. In the treatment of the entire chapter, it has been assumed that
the material is elastic, homogeneous and isotropic.
© 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_8
307
Free Vibration Analysis of Continuous
Systems
8.1 Introduction
Structures analysed so far have been treated as discrete systems. Structures have been
idealised, and for convenience of computation, simpli-fying assumptions have been
introduced and as such results obtained can only be treated as approximate. But they
are, however, sufficiently accurate for most practical purposes. Increase of structural
discretisation will entail an increase of degrees of freedom, and this will improve the
accuracy of the results.
For all systems, the mass of the members is continuously distributed. As such,
specifying the displacement at every point in the system will require infinite number
of coordinates. The system in this case is assumed to have infinite degrees of freedom.
For such cases, the mass is inseparable from the elasticity of the system. Continuous
models of vibrating systems are indeed more realistic because structural properties
are distributed rather than concentrated at discrete points. The price however to be
paid for increased realism is the increased complexity.
The reason as to why the practical structures are reduced to discrete systems is
due to the fact that the analysis of continuous system is much more involved. The
argument is similar to that applied to the static analysis, in which for a somewhat
redundant structure, solutions obtained by classical methods are very tedious and
for such cases, matrix or the finite element analysis is preferred. However, there are
certain applications where modelling the system on the basis of distributed parameter
may be justified. In this chapter, the free vibration analysis of continuous systems
has been discussed. The equation of motion is a partial differential equation for
continuous systems. In the treatment of the entire chapter, it has been assumed that
the material is elastic, homogeneous and isotropic.
© 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_8
307