Chapter 14
Nonlinear Vibration
14.1 Introduction
In all the preceding chapters, we have discussed different linear systems. All the
differential equations that have been formed and dealt with so far, whether a single
equation in the case of SDF system or a set of coupled equations in the case of
MDF system, are linear. But there are certain vibration phenomena, which cannot be
predicted by the linear theory. The vibration of a string, the belt friction drive, the
shimming of automobile wheels and the inclusion of nonlinear damping are some of
the examples of nonlinear vibration [1–5].
The principle of superposition does not hold good for nonlinear systems. If the
excitation to a system is trebled, it does not mean that the response will also be trebled.
Thus, Duhamel’s integral cannot be applied to determine the response of these SDF
systems. Another characteristic of nonlinear system is that, it can have more than
one equilibrium position, whereas a linear system can have only one equilibrium
position.
Nonlinear problems are classified into two categories: geometric nonlinearity
and material nonlinearity. So far we have dealt with small strains, certain assumptions in the derivation have been introduced to make the problem linear. But accurate
representation of the displacement may involve geometric nonlinearity. For instance,
membrane stresses in the neutral surface which are neglected in the beam analysis
may reduce displacements significantly, though the value of the displacement may
remain quite small. These types of problems are also referred to as large deflection problems, and this will result in nonlinear equations. There is another class of
problem, where the stress–strain relationship of the material of the structure is not
linear. For a material like steel, the stress–strain relation is linear up to yield point,
but beyond it, the curve deviates from linearity. So, for a structure made of steel, if an
analysis is to be carried out beyond yield point, it will involve the nonlinear material
behaviour. For certain other materials, the stress varies in a curvilinear manner with
the strain.
© 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_14
515
Nonlinear Vibration
14.1 Introduction
In all the preceding chapters, we have discussed different linear systems. All the
differential equations that have been formed and dealt with so far, whether a single
equation in the case of SDF system or a set of coupled equations in the case of
MDF system, are linear. But there are certain vibration phenomena, which cannot be
predicted by the linear theory. The vibration of a string, the belt friction drive, the
shimming of automobile wheels and the inclusion of nonlinear damping are some of
the examples of nonlinear vibration [1–5].
The principle of superposition does not hold good for nonlinear systems. If the
excitation to a system is trebled, it does not mean that the response will also be trebled.
Thus, Duhamel’s integral cannot be applied to determine the response of these SDF
systems. Another characteristic of nonlinear system is that, it can have more than
one equilibrium position, whereas a linear system can have only one equilibrium
position.
Nonlinear problems are classified into two categories: geometric nonlinearity
and material nonlinearity. So far we have dealt with small strains, certain assumptions in the derivation have been introduced to make the problem linear. But accurate
representation of the displacement may involve geometric nonlinearity. For instance,
membrane stresses in the neutral surface which are neglected in the beam analysis
may reduce displacements significantly, though the value of the displacement may
remain quite small. These types of problems are also referred to as large deflection problems, and this will result in nonlinear equations. There is another class of
problem, where the stress–strain relationship of the material of the structure is not
linear. For a material like steel, the stress–strain relation is linear up to yield point,
but beyond it, the curve deviates from linearity. So, for a structure made of steel, if an
analysis is to be carried out beyond yield point, it will involve the nonlinear material
behaviour. For certain other materials, the stress varies in a curvilinear manner with
the strain.
© 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_14
515