Identifying and Computing Nonlinear Normal Modes

On the Proper Orthogonal Modes and Normal Modes of Continuous Vibration Systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, i.e., by denning proper normal modes of motion. These are defined in terms of invariant manifolds in the system’s phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach where the theory developed for discrete systems can be applied — are simultaneously applied to the same study case — an Euler-Bernoulli beam constrained by a non-linear spring —, and compared as regards accuracy and reliability, resulting in the abandonment of the continuous approach for lack of reliability. Numerical simulations of purely non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the nonlinear normal modes are demonstrated, and it is also found that, for a purely non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differentia] equation.

A study of the Nonlinear Normal Modes (NNMs) of a two degrees of freedom mechanical system with a bilateral elastic stop for one of them is considered. The issue related to the non-smoothness of the impact force is handled through a regularization technique. The Harmonic Balance Method (HBM) with a large number of harmonics, combined with the Asymptotic Numerical Method (ANM), is used to solve the regularized problem. The results are validated from periodic orbits obtained analytically in the time domain by direct integration of the non-regular problem. The first NNM shows an elaborate dynamics with the occurrence of multiple impacts per period, internal resonance and instabilities. On the other hand, the second NNM presents a more simple, almost linear, dynamics. The two NNMs converge asymptotically (for an infinite energy) toward two other Linear Normal Modes, corresponding to the system with a gap equal to zero.

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

Nonlinear normal mode continuation through a Proper Generalized Decomposition approach with modal enrichment

A complex-variable invariant-manifold approach is used to construct the normal modes of weakly nonlinear discrete systems with cubic geometric nonlinearities and either a one-to-one or a three-to-one internal resonance. The nonlinear mode shapes are assumed to be slightly curved four-dimensional manifolds tangent to the linear eigenspaces of the two modes involved in the internal resonance at the equilibrium position. The dynamics on these manifolds is governed by three first-order autonomous equations. In contrast with the case of no internal resonance, the number of nonlinear normal modes may be more than the number of linear normal modes. Bifurcations of the calculated nonlinear normal modes are investigated.

This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative spring–mass–pendulum system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities. Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then applied to the spring–mass–pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order–chaos–order sequence. This phenomenon is also presented and discussed.

A comprehensive understanding of molecular interactions and functions is imperative for unraveling the intricacies of viral protein behavior and conformational dynamics during cellular entry. Focusing on the SARS-CoV-2 spike protein (SARS-CoV-2 sp), a Principal Component Analysis (PCA) on a subset comprising 131 A-chain structures in presence of various inhibitors was conducted. Our analyses unveiled a compelling correlation between PCA modes and Anisotropic Network Model (ANM) modes, underscoring the reliability and functional significance of low-frequency modes in adapting to diverse inhibitor binding scenarios. The role of HR1 in viral processing, both linear Normal Mode Analysis (NMA) and Nonlinear NMA were implemented. Linear NMA exhibited substantial inter-structure variability, as evident from a higher Root Mean Square Deviation (RMSD) range (7.30 Å), nonlinear NMA show stability throughout the simulations (RMSD 4.85 Å). Frequency analysis further emphasized that the energy requirements for conformational changes in nonlinear modes are notably lower compared to their linear counterparts. Using simulations of molecular dynamics at constant pH (cpH-MD), we successfully predicted the pKa order of the interconnected residues within the HR1 mutations at lower pH values, suggesting a transition to a post-fusion structure. The pKa determination study illustrates the profound effects of pH variations on protein structure. Key results include pKa values of 9.5179 for lys-921 in the D936H mutant, 9.50 for the D950N mutant, and a slightly higher value of 10.49 for the D936Y variant. To further understand the behavior and physicochemical characteristics of the protein in a biologically relevant setting, we also examine hydrophobic regions in the prefused states of the HR1 protein mutants D950N, D936Y, and D936H in our study. This analysis was conducted to ascertain the hydrophobic moment of the protein within a lipid environment, shedding light on its behavior and physicochemical properties in a biologically relevant context.

Nonlinear Normal Mode backbone estimation with near-resonant steady state inputs

For the study of highly nonlinear, conservative dynamic systems, finding special periodic solutions which can be seen as generalization of the well-known normal modes of linear systems is very attractive. However, the study of low-dimensional invariant manifolds in the form of nonlinear normal modes is rather a niche topic, treated mainly in the context of structural mechanics for systems with Euclidean metrics, i.e., for point masses connected by nonlinear springs. In our previous research [1], [16], [17] we recognized, however, that a very rich structure of periodic and low-dimensional solutions exist also within nonlinear systems such as elastic multi-body systems encountered in the biomechanics of humans and animals or of humanoid and quadruped robots, which are characterized by a non-constant metric tensor. This letter briefly discusses different generalizations of linear oscillation modes to nonlinear systems and proposes a definition of strict nonlinear normal modes, which matches most of the relevant properties of the linear modes. The main contributions are a theorem providing necessary and sufficient conditions for the existence of strict oscillation modes on systems endowed with a Riemannian metric and a potential field as well as a constructive example of designing such modes in the case of an elastic double pendulum.

Nonlinear normal modes (NNMs) are widely used as a tool for understanding the forced responses of nonlinear systems. However, the contemporary definition of an NNM also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped. As such, only a few NNMs are required to understand the forced dynamics. This paper firstly demonstrates the complexity that may arise from the NNMs of a simple nonlinear system—highlighting the need for a method for identifying the significance of NNMs. An analytical investigation is used, alongside energy arguments, to develop an understanding of the mechanisms that relate the NNMs to the forced responses. This provides insight into which NNMs are pertinent to understanding the forced dynamics, and which may be disregarded. The NNMs are compared with simulated forced responses to verify these findings.

Reduced-order models for large-amplitude vibrations of shells including in-plane inertia

The vibration characteristics of beams have been extensively studied due to their wide application across multiple fields (i.e. spacecraft antennae, aircraft wings, turbine blades, skyscrapers). Of particular interest, specific geometries of beams have been shown to induce coupling between the fundamental bending and torsion modes. This coupled motion can be observed in a beam’s linear normal modes can be avoided with the correct selection of geometric properties. This work investigates the coupled bending-torsion behaviour of a clamped-clamped beam that is coupled perpendicularly, mid-span to mid-span, to a second beam with tip masses within the nonlinear response regime. The first torsion mode of the beam system is tuned by modifying the mass distribution such that closely spaced bending and torsion linear normal modes can be realized. The nonlinear behaviour is presented using nonlinear normal mode backbone curves and forced responses in the vicinity of the modes of interest.

The sensitivity of the nonlinear dynamic response to damage is investigated in simply supported beams by computing the nonlinear normal modes (NNMs) and exploiting the rich information they unfold about the nonlinear system behavior. Damage is introduced as a reduction of the flexural stiffness within a small segment of the beam span. The problem is formulated in a piece-wise fashion and analytically tackled by using the method of multiple scales to compute the NNMs and the backbone curves. The latter describe the frequency versus free oscillation amplitude for each mode and as such they represent the skeleton of the nonlinear frequency response around each beam modal frequency. The bending of the backbones is regulated by the so-called effective nonlinearity coefficients associated with each mode. The comparison between the damaged and undamaged beams shows a sensitivity of the effective nonlinearity coefficients higher than the sensitivity of the linear natural frequencies. Moreover, the nonlinear frequency trends unfold an interesting dependence of the nonlinear free response on the stiffness reduction at the damage site. Such dependence yields useful information about the damage position. An effective identification strategy for the damage position is proposed by computing the damage-induced discontinuities in the second derivative of the NNMs (i.e., first order estimate of the bending curvature), without resorting to a baseline model of the undamaged beam. This goal is achieved by computing a high order derivative of the Nadaraya–Watson kernel estimator, which is evaluated from a finite set of sampled beam deflections associated with each individual NNM. The results show that the second derivatives of higher NNMs are more sensitive to damage-induced discontinuities than the linear modal curvatures. The robustness against noise of the damage identification process is also discussed.

Linear modal analysis has provided a robust and eminently useful framework for the analysis of structural dynamic systems. Considerable attention has been directed towards the development of a nonlinear variant of modal analysis that is effective in the presence of nonlinearities. Thus far, essentially two approaches to constructing nonlinear normal modes (NNM) have gained traction. The first was that of Rosenberg, whereby the modes are defined in terms of synchronous motions of the structure. The second is the geometrically more general approach of Shaw and Pierre, wherein modes are defined on invariant manifolds of the phase space of the system. A recent third approach from Worden and Green proposes a statistical definition of a nonlinear normal mode. Under this framework, the modal coordinates are defined by latent directions in the configuration space that result in statistically uncorrelated time series. This paper examines the properties of the NNMs generated by this framework, by applying techniques from nonlinear system identification (NLSI). Both linear and nonlinear time domain models are fitted to the physical and modal coordinates of a two degree-of-freedom simulated system with a cubic nonlinearity. It is demonstrated in this work, that the NNMs generated are able to decompose the system into independent functionals, and that the modal transformation generalises to lower excitation levels, while maintaining excellent reconstruction of the physical displacements.