Chapter 12
Finite Element Method in Vibration
Analysis
12.1 Introduction to the Finite Element Method
In the finite element method, the continuum is divided into a finite number of meshes
by imaginary lines. For one-dimensional continuum, two adjacent elements placed
side-by-side will meet at a point. Two dimensional continuum will have adjacent
elements meeting at a common edge. Strictly speaking, the continuity requirement
along the edge should be satisfied, but for the sake of simplicity, it is assumed
that the elements are connected only at the nodal points and the continuum has
to be continuous through those points. Total structure obtained as an assembly of
elements is then analysed. In the finite element analysis, therefore, the continuum
is divided into a finite number of elements, having finite dimensions and reducing
the continuum having infinite degrees of freedom to finite degrees of unknowns. The
term finite element was coined by Clough [1].
The finite element method has many advantages. The method can be efficiently
applied to cater to the irregular geometry. Material anisotropy and nonhomogeneity
can be dealt without much difficulty. The method takes into account any type of
boundary condition in an efficient manner. Of the various approaches prevalent in
the analysis by the finite element method, one based on the assumed displacement
pattern within the element is widely used. It is perhaps the simplest among the finite
element techniques and has been extensively used and applied successfully to a large
variety of stress analysis problems [1–5].
12.2 Torsional Vibration of Shafts
A shaft element of Fig. 12.1 has two nodes at its two ends. The unknown displacements at each end are the angles of twist φ 1 and φ 2 . The displacement function, which
is the angle of twist in the present case is given by
© 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_12
443
Finite Element Method in Vibration
Analysis
12.1 Introduction to the Finite Element Method
In the finite element method, the continuum is divided into a finite number of meshes
by imaginary lines. For one-dimensional continuum, two adjacent elements placed
side-by-side will meet at a point. Two dimensional continuum will have adjacent
elements meeting at a common edge. Strictly speaking, the continuity requirement
along the edge should be satisfied, but for the sake of simplicity, it is assumed
that the elements are connected only at the nodal points and the continuum has
to be continuous through those points. Total structure obtained as an assembly of
elements is then analysed. In the finite element analysis, therefore, the continuum
is divided into a finite number of elements, having finite dimensions and reducing
the continuum having infinite degrees of freedom to finite degrees of unknowns. The
term finite element was coined by Clough [1].
The finite element method has many advantages. The method can be efficiently
applied to cater to the irregular geometry. Material anisotropy and nonhomogeneity
can be dealt without much difficulty. The method takes into account any type of
boundary condition in an efficient manner. Of the various approaches prevalent in
the analysis by the finite element method, one based on the assumed displacement
pattern within the element is widely used. It is perhaps the simplest among the finite
element techniques and has been extensively used and applied successfully to a large
variety of stress analysis problems [1–5].
12.2 Torsional Vibration of Shafts
A shaft element of Fig. 12.1 has two nodes at its two ends. The unknown displacements at each end are the angles of twist φ 1 and φ 2 . The displacement function, which
is the angle of twist in the present case is given by
© 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_12
443