Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques
Introduction
Nonlinear normal modes (NNMs) offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena, yet they have a clear and simple conceptual relation to the classical linear normal modes (LNMs). The relevance of the NNMs for the structural dynamicist is addressed in Part I of this paper.
However, most structural engineers still view NNMs as a concept that is foreign to them, and they do not yet consider these nonlinear modes as a practical nonlinear analog of the LNMs. One reason supporting this statement is that most existing constructive techniques for computing NNMs are based on asymptotic approaches and rely on fairly involved mathematical developments. In this context, a significant contribution is that of Pesheck who proposed a meaningful numerical extension of the invariant manifold approach [1].
Algorithms for the numerical continuation of periodic solutions are really quite sophisticated and advanced (see, e.g., [2], [3], [4], [5], and the AUTO and MATCONT softwares). These algorithms have been extensively used for computing the forced response and limit cycles of nonlinear dynamical systems [6], [7], [8], [9], [10], [11]. Doedel and co-workers used them for the computation of periodic orbits during the free response of conservative systems [12], [13].
Interestingly, there have been very few attempts to compute the periodic solutions of conservative mechanical structures (i.e., NNM motions) using numerical continuation techniques. One of the first approaches was proposed by Slater [14] who combined a shooting method with sequential continuation to solve the nonlinear boundary value problem that defines a family of NNM motions. Similar approaches were considered in Lee et al. [15] and Bajaj et al. [16]. A more sophisticated continuation method is the so-called asymptotic-numerical method. It is a semi-analytical technique that is based on a power series expansion of the unknowns parameterized by a control parameter. It is utilized in [17] to follow the NNM branches in conjunction with finite difference methods, following a framework similar to that of [12].
In this study, a shooting procedure is combined with the so-called pseudo-arclength continuation method for the computation of NNM motions. We show that the NNM computation is possible with limited implementation effort, which holds promise for a practical and accurate method for determining the NNMs of nonlinear vibrating structures.
This paper is organized as follows. In the next section, the two main definitions of an NNM and their fundamental properties are briefly reviewed. In Section 3, the proposed algorithm for NNM computation is presented. Its theoretical background is first recalled, and the numerical implementation is then described. Improvements are also presented for the reduction of the computational burden. The proposed algorithm is then demonstrated using four different nonlinear vibrating systems in Section 4. In Part I, the relevance of the NNMs for the dynamicist is discussed.
Section snippets
Nonlinear normal modes (NNMs)
A detailed description of NNMs is given in the companion paper, Part I. For completeness, the two main definitions of an NNM and their fundamental properties are briefly reviewed in this section.
Numerical computation of NNMs
The numerical method proposed here for the NNM computation relies on two main techniques, namely a shooting technique and the pseudo-arclength continuation method. It is summarized in Fig. 6.
Numerical experimentations
In what follows, the proposed NNM computation method is demonstrated using four different nonlinear vibrating systems, namely a weakly nonlinear 2DOF system, a 2DOF system with an essential nonlinearity, a discrete model of a nonlinear bladed disk and a nonlinear cantilever beam discretized by the finite element method.
Conclusion and future work
In this paper, a numerical method for the computation of nonlinear normal modes (NNMs) of nonlinear mechanical structures is introduced. The approach targets the computation of the undamped modes of structures discretized by finite elements and relies on the continuation of periodic solutions. The proposed procedure was demonstrated using different nonlinear structures, and the NNMs were computed accurately in a fairly automatic manner. Complicated NNM motions were also observed, including a
References (37)
- et al.
Analysis of periodically excited nonlinear-systems by a parametric continuation technique
Journal of Sound and Vibration
(1995) - et al.
An algorithm for response and stability of large order non-linear systems—application to rotor systems
Journal of Sound and Vibration
(1998) Non-linear forced vibrations of thin/thick beams and plates by the finite element and shooting methods
Computers and Structures
(2004)- et al.
Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modelling of harmonically forced structures
Journal of Sound and Vibration
(2006) - et al.
Continuation of periodic orbits in conservative and hamiltonian systems
Physica D
(2003) - et al.
Complicated dynamics of a linear oscillator with a light, essentially nonlinear attachment
Physica D
(2005) - et al.
Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes
Computers and Structures
(2006) On nonlinear vibrations of systems with many degrees of freedom
Advances in Applied Mechanics
(1966)- et al.
Nonlinear normal modes and invariant manifolds
Journal of Sound and Vibration
(1991) - et al.
Normal modes for nonlinear vibratory systems
Journal of Sound and Vibration
(1993)
Normal modes of vibration for nonlinear continuous systems
Journal of Sound and Vibration
Normal modes and global dynamics of a 2-degree-of-freedom nonlinear-system; part I: low energies
International Journal of Non-Linear Mechanics
Normal modes and global dynamics of a 2-degree-of-freedom nonlinear-system; part II: high energies
International Journal of Non-Linear Mechanics
Presentation of the ECL benchmark
Mechanical Systems and Signal Processing
Numerical Methods in Bifurcation Problems
Practical bifurcation and stability analysis. From Equilibrium to Chaos
Cited by (482)
Insight into the influence of frictional heat on the modal characteristics and interface temperature of frictionally damped turbine blades
2024, Journal of Sound and VibrationBifurcation analysis of thin-walled structures trimming process with state-dependent time delay
2024, International Journal of Mechanical SciencesCombining the Asymptotic Numerical Method with the Harmonic Balance Method to capture the nonlinear dynamics of spur gears
2024, Mechanical Systems and Signal ProcessingNonlinear normal modes of multi-walled nanoshells with consideration of surface effect and nonlocal elasticity
2024, International Journal of Non-Linear MechanicsOn the detection of nonlinear normal mode-related isolated branches of periodic solutions for high-dimensional nonlinear mechanical systems with frictionless contact interfaces
2024, Computer Methods in Applied Mechanics and EngineeringShift manipulation of intrinsic localized mode in ac driven Klein Gordon lattice
2024, Physics Letters, Section A: General, Atomic and Solid State Physics