4 Nonlinear Multiple-DOF Systems:
Local Analysis
4.1 Introduction
Certain nonlinear phenomena can occur only with systems having multiple degrees
of freedom. Thus, nonlinear interaction requires the presence of at least two
components that can interact. In this chapter we shall consider only 2-DOF systems,
since these will suffice for illustrating important properties of nonlinear multipleDOF systems in general. A 2-DOF model typically arises as an approximation to a
real multiple-DOF or continuous system, for which a single-DOF model fails to
describe the behavior to be studied.
To accustom ourselves to an additional degree of freedom the first example is
worked out in some detail, whereas for the remaining examples we focus on main
results. Nonlinearities are assumed to be weak, and perturbation methods (multiple
scales or averaging) are used to obtain first-order approximate solutions.
By contrast to a single degree of freedom system – which has only a single linear
natural frequency and a single mode of motion – a system having n degrees of
freedom system has n linear natural frequencies and n corresponding modes: the
linear normal modes. For linear systems these modes are uncoupled, that is, energy
imparted externally to any one mode remains in that mode; there is normally
1 no
mechanism by which energy can be shared with other modes. Nonlinearities,
however, may provide such a mechanism, so that energy imparted into one mode
may under certain conditions be exchanged with other modes. This is called nonlinear mode-coupling, or modal interaction.
1
With parametric/internal excitation this may be different: With time-varying coefficients even
linear systems may exhibit mode coupling, e.g., under conditions of parametric resonance or
anti-resonance (Dohnal 2008). The presence of asymmetrical phenomena like unidirectional flow
may also couple modes; this applies, e.g., to Coriolis flowmeters, where the measurement pipe is
typically excited to vibrate resonantly at the lowest mode of vibration, while fluid flowing through
the pipe excites motion in also the second mode of vibration (Thomsen and Fuglede 2020;
Thomsen and Dahl 2010; Enz et al. 2011b).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_4
209
Local Analysis
4.1 Introduction
Certain nonlinear phenomena can occur only with systems having multiple degrees
of freedom. Thus, nonlinear interaction requires the presence of at least two
components that can interact. In this chapter we shall consider only 2-DOF systems,
since these will suffice for illustrating important properties of nonlinear multipleDOF systems in general. A 2-DOF model typically arises as an approximation to a
real multiple-DOF or continuous system, for which a single-DOF model fails to
describe the behavior to be studied.
To accustom ourselves to an additional degree of freedom the first example is
worked out in some detail, whereas for the remaining examples we focus on main
results. Nonlinearities are assumed to be weak, and perturbation methods (multiple
scales or averaging) are used to obtain first-order approximate solutions.
By contrast to a single degree of freedom system – which has only a single linear
natural frequency and a single mode of motion – a system having n degrees of
freedom system has n linear natural frequencies and n corresponding modes: the
linear normal modes. For linear systems these modes are uncoupled, that is, energy
imparted externally to any one mode remains in that mode; there is normally
1 no
mechanism by which energy can be shared with other modes. Nonlinearities,
however, may provide such a mechanism, so that energy imparted into one mode
may under certain conditions be exchanged with other modes. This is called nonlinear mode-coupling, or modal interaction.
1
With parametric/internal excitation this may be different: With time-varying coefficients even
linear systems may exhibit mode coupling, e.g., under conditions of parametric resonance or
anti-resonance (Dohnal 2008). The presence of asymmetrical phenomena like unidirectional flow
may also couple modes; this applies, e.g., to Coriolis flowmeters, where the measurement pipe is
typically excited to vibrate resonantly at the lowest mode of vibration, while fluid flowing through
the pipe excites motion in also the second mode of vibration (Thomsen and Fuglede 2020;
Thomsen and Dahl 2010; Enz et al. 2011b).
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
J. J. Thomsen, Vibrations and Stability,
https://doi.org/10.1007/978-3-030-68045-9_4
209