The effect of nonlinear cross-coupling on reduced-order modelling

Tartaruga, Irene; Elliott, Alexander; Hill, Thomas L.; Neild, Simon A; Cammarano, Andrea

doi:10.1016/j.ijnonlinmec.2019.05.006

Cited by 11 publications

(7 citation statements)

References 19 publications

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“…Note that linear dependencies are imposed on the coefficients in equations (4.2), such that the energy in the system is conserved [31,32], similar to equations (3.1). The linear properties in equations (4.1) can be obtained directly through an eigenanalysis of the full system.…”

Section: Phase-varying Backbone Curves For a Cable Modelmentioning

confidence: 99%

Identifying phase-varying periodic behaviour in conservative nonlinear systems

et al. 2020

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Nonlinear normal modes (NNMs) are a widely used tool for studying nonlinear mechanical systems. The most commonly observed NNMs are synchronous (i.e. single-mode, in-phase and anti-phase NNMs). Additionally, asynchronous NNMs in the form of out-of-unison motion, where the underlying linear modes have a phase difference of 90°, have also been observed. This paper extends these concepts to consider general asynchronous NNMs , where the modes exhibit a phase difference that is not necessarily equal to 90°. A single-mass, 2 d.f. model is firstly used to demonstrate that the out-of-unison NNMs evolve to general asynchronous NNMs with the breaking of the geometrically orthogonal structure of the system. Analytical analysis further reveals that, along with the breaking of the orthogonality, the out-of-unison NNM branches evolve into branches which exhibit amplitude-dependent phase relationships. These NNM branches are introduced here and termed phase-varying backbone curves . To explore this further, a model of a cable, with a support near one end, is used to demonstrate the existence of phase-varying backbone curves (and corresponding general asynchronous NNMs) in a common engineering structure.

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Section: Phase-varying Backbone Curves For a Cable Modelmentioning

confidence: 99%

Identifying phase-varying periodic behaviour in conservative nonlinear systems

et al. 2020

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“…Note that the nonlinear coefficients are defined in this order for simplicity in later sections. Applying the Euler–Lagrange equation then leads to the following equations of motion: q¨1+ωn12q1+Ψ4q13+3Ψ1q12q2+Ψ3q1q22+Ψ2q23=0and q¨2+ωn22q2+Ψ1q13+Ψ3q12q2+3Ψ2q1q22+Ψ5q23=0.Note that the use of the Lagrangian in deriving these expressions restricts the number of nonlinear parameters to five, while ensuring that the equations of motion remain conservative [42,43].…”

Section: Breaking the Symmetry Of A Nonlinear 2-df Oscillatormentioning

confidence: 99%

“…Note that the use of the Lagrangian in deriving these expressions restricts the number of nonlinear parameters to five, while ensuring that the equations of motion remain conservative [42,43]. From equation (2.2b), when the coefficient of q 3 1 equals 0 (i.e.…”

Section: Breaking the Symmetry Of A Nonlinear 2-df Oscillatormentioning

confidence: 99%

Conditions for the existence of isolated backbone curves

2019

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Isolated backbone curves represent significant dynamic responses of nonlinear systems; however, as they are disconnected from the primary responses, they are challenging to predict and compute. To explore the conditions for the existence of isolated backbone curves, a generalized two-mode system, which is representative of two extensively studied examples, is used. A symmetric two-mass oscillator is initially studied and, as has been previously observed, this exhibits a perfect bifurcation between its backbone curves. As this symmetry is broken, the bifurcation splits to form an isolated backbone curve. Here, it is demonstrated that this perfect bifurcation, indicative of a symmetric structure, may be preserved when the symmetry is broken under certain conditions; these are derived analytically. With the symmetry broken, the second example—a single-mode nonlinear structure with a nonlinear tuned mass damper—is considered. The evolution of the system's backbone curves is investigated in nonlinear parameter space. It is found that this space can be divided into several regions, within which the backbone curves share similar topological features, defining the conditions for the existence of isolated backbone curves. This allows these features to be more easily accounted for, or eliminated, when designing nonlinear systems.

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“…In the displacement-based approach, which is often referred to as the enforced displacement procedure, the structure is constrained into a prescribed shape that is a linear combination of the basis functions, and the resulting reaction forces are extracted [32,34]. The main drawback of this approach lies in the fact that when only the low-frequency, dynamically important modes are included in the reduction basis, the ROM fails to capture the effect of membrane stretching, resulting in overly stiff behaviour [10,35]. To account for the induced in-plane motions, a number of membrane modes whose natural frequencies are well beyond the bandwidth of interest, must be explicitly included in the reduction basis [36,37].…”

Section: Introductionmentioning

confidence: 99%

Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy

2020

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Nonlinear dynamic analysis of complex engineering structures modelled using commercial finite element (FE) software is computationally expensive. Indirect reduced-order modelling strategies alleviate this cost by constructing low-dimensional models using a static solution dataset from the FE model. The applicability of such methods is typically limited to structures in which (a) the main source of nonlinearity is the quasi-static coupling between transverse and in-plane modes (i.e. membrane stretching); and (b) the amount of in-plane displacement is limited. We show that the second requirement arises from the fact that, in existing methods, in-plane kinetic energy is assumed to be negligible. For structures such as thin plates and slender beams with fixed/pinned boundary conditions, this is often reasonable, but in structures with free boundary conditions (e.g. cantilever beams), this assumption is violated. Here, we exploit the concept of nonlinear manifolds to show how the in-plane kinetic energy can be accounted for in the reduced dynamics, without requiring any additional information from the FE model. This new insight enables indirect reduction methods to be applied to a far wider range of structures while maintaining accuracy to higher deflection amplitudes. The accuracy of the proposed method is validated using an FE model of a cantilever beam.

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The effect of nonlinear cross-coupling on reduced-order modelling

Cited by 11 publications

References 19 publications

Identifying phase-varying periodic behaviour in conservative nonlinear systems

Identifying phase-varying periodic behaviour in conservative nonlinear systems

Conditions for the existence of isolated backbone curves

Indirect reduced-order modelling: using nonlinear manifolds to conserve kinetic energy

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