Abstract
Linear vibration theory uses the concept of defining a specific set of modes of vibration for the system under consideration. Physically, each mode relates to a particular geometric configuration in the system, such as two lumped masses oscillating either in- or out-of-phase with each other. For linear systems, the superposition principle means that the complete vibration response can be computed as a summation of the responses from each mode. In general terms, modal analysis has come to mean considering the response of a system by studying its vibration modes; modal decomposition is the process of transforming the system from a physical to a modal representation. This is particularly useful in linear systems, because each mode has an associated resonance, and understanding where resonances could occur in a structure is a key part of analysing vibration problems. In this chapter the use of modal analysis for nonlinear systems is also considered. First, the decomposition of discrete and continuous linear systems into modal form is reviewed and the effect of nonlinear terms on this analysis is discussed. Following this, methods for decomposing nonlinear systems are considered. Initially a brief discussion of nonlinear normal modes is given and a special case system, in which there is nonlinear but no linear coupling between two oscillators, is analysed using the harmonic balance approach. Following this, attention is turned to the main technique for carrying out nonlinear modal decomposition, which is the method of normal forms that was introduced in Sect. 4.5. This is a technique that transforms the system to the simplest form possible. The approach described here uses linear modal decomposition as the first step in the process. The main advantage of using normal forms is that information about nonlinear (also called internal) resonances in the system can be obtained. As a result, a normal form analysis can be used to obtain information about both linear and nonlinear resonances in a nonlinear vibration problem. The modal decomposition techniques are used to find backbone curves that represent the undamped vibration response of the system in the frequency domain. The reason for taking this approach is that modal analysis is most relevant for lightly damped systems, where multiple resonant peaks can occur in the response. Just like linear systems, nonlinear systems with light damping have a forced response that is determined by the underlying undamped characteristics (It’s possible to define systems that don’t have this property, but we will restrict our discussion to systems that do.). In the frequency domain this is captured by the backbone curves. As a result, defining the backbone curves for the system gives a nonlinear modal model. The nonlinear examples considered in this chapter are confined to two degrees-of-freedom, but can be extended to higher degrees-of-freedom, and a short discussion of relevant literature on this topic is given at the end of the chapter.
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Notes
- 1.
These assumptions are sometimes referred to as separation of space and time variables.
- 2.
For each frequency value there is a single amplitude value in the resonance peak—it is a single-valued function, which increases monotonically up to the resonant frequency and then monotonically decreases after the resonant frequency.
- 3.
Note that a bold subscript is used to indicate that the whole term is a vector, not that the subscript is a vector.
- 4.
See Caughey (1963) for necessary and sufficient conditions for simultaneous diagonalisation of the M, C and K matrices.
- 5.
There are situations where Jordan normal form is preferable to diagonal matrices, but these are not considered here.
- 6.
Note this should not be confused with complex modes which arise for systems with non-proportional damping. See, Ewins (2000) for a more detailed discussion.
- 7.
- 8.
This technique is often used for beams and cables, however it cannot be used for more complex structures such as plates. In these cases, approximate techniques are used, see Chap. 8.
- 9.
See Finlayson (1972) for a description of the Galerkin and related methods.
- 10.
This is analogous to the approach used to calculate a Fourier series approximation to a function.
- 11.
Note that if the modes are scaled such that when \(k=n\) the integral equals one, the modes are said to be orthonormal.
- 12.
Note that the related problem of nonlinear system identification is not considered here. See Kerschen et al. (2006) for a comprehensive review of these techniques.
- 13.
The damping ratios are the other key set of quantities that is required, but in this simplified example, no damping is assumed.
- 14.
Internal resonance is a form of nonlinear resonance.
- 15.
In fact the manifold in Fig. 5.5 is not exactly the same as that considered by Shaw and Pierre , but the concept is similar.
- 16.
Note that depending on the method adopted and assumptions that are made a range of such forms can be derived, Murdock (2002).
- 17.
- 18.
The term mode refers to a mode of the linearised system, whereas a nonlinear normal mode, or NNM, is for the full nonlinear system. A mixed-mode backbone curve defines a response made up of multiple linear modes.
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Wagg, D., Neild, S. (2015). Modal Analysis for Nonlinear Vibration. In: Nonlinear Vibration with Control. Solid Mechanics and Its Applications, vol 218. Springer, Cham. https://doi.org/10.1007/978-3-319-10644-1_5
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