Chapter 11
Vibration of Ship and Aircraft as a Beam
11.1 Introduction
Aircrafts and ships are complex structures. They have curved surfaces with various
forms of intricate stiffening arrangements. For simplification of the analysis, they
are sometimes treated as a beam having varying rigidity. This no doubt facilitates the
analysis to a great extent, both for static and dynamic cases, but it introduces certain
special problems, particularly in vibration analysis. Unlike a building, a bridge or
a machine, the ship or the aircraft is unconstrained in space; that is, every point in
the structure may undergo displacement. This introduces rigid body displacements.
As a result of which, the stiffness matrix becomes singular. Various approaches have
been sought to obviate these difficulties. Some of them are presented in the following
[1–5].
11.2 Shift in Stiffness Matrix
Let us consider a multiple degrees of freedom system with lumped masses. The
equations of motion in free vibration are given by
p
2
[M]{φ} = [K ]{φ}
(11.1)
For an unconstrained structure, [K] is singular. Now to apply the general methods
to the solution of the eigenvalue problem as presented in Chap. 6 for this case, the
singularity of [K] is to be disturbed. This is done as follows :
Let us add α[M]{φ} to both sides of Eq. (11.1), where α is an arbitrary constant.
[K ]{φ} + α[M]{φ} = p
2
[M]{φ} + α[M]{φ}
or
([K ] + α[M]){φ} = ( p
2
+ α)[M]{φ}
(11.2)
© 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_11
425
Vibration of Ship and Aircraft as a Beam
11.1 Introduction
Aircrafts and ships are complex structures. They have curved surfaces with various
forms of intricate stiffening arrangements. For simplification of the analysis, they
are sometimes treated as a beam having varying rigidity. This no doubt facilitates the
analysis to a great extent, both for static and dynamic cases, but it introduces certain
special problems, particularly in vibration analysis. Unlike a building, a bridge or
a machine, the ship or the aircraft is unconstrained in space; that is, every point in
the structure may undergo displacement. This introduces rigid body displacements.
As a result of which, the stiffness matrix becomes singular. Various approaches have
been sought to obviate these difficulties. Some of them are presented in the following
[1–5].
11.2 Shift in Stiffness Matrix
Let us consider a multiple degrees of freedom system with lumped masses. The
equations of motion in free vibration are given by
p
2
[M]{φ} = [K ]{φ}
(11.1)
For an unconstrained structure, [K] is singular. Now to apply the general methods
to the solution of the eigenvalue problem as presented in Chap. 6 for this case, the
singularity of [K] is to be disturbed. This is done as follows :
Let us add α[M]{φ} to both sides of Eq. (11.1), where α is an arbitrary constant.
[K ]{φ} + α[M]{φ} = p
2
[M]{φ} + α[M]{φ}
or
([K ] + α[M]){φ} = ( p
2
+ α)[M]{φ}
(11.2)
© 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_11
425