Invariant Manifolds in Beam Dynamics: Free Vibrations and Nonlinear Normal Modes

2020 ◽  
pp. 1345-1352
Author(s):  
Rafael Palacios
Keyword(s):  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Free Vibrations ◽  
Beam Dynamics ◽  
Nonlinear Normal

Amplitude Nonlinear Normal Modes of Piecewise Linear Systems

2001 ◽  
Author(s):  
Dongying Jiang ◽  
Vincent Soumier ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Invariant Manifolds ◽  
Piecewise Linear ◽  
Autonomous Systems ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Flip Bifurcation ◽  
Strong Nonlinearity ◽  
Nonlinear Modes ◽  
Nonlinear Normal

Abstract A numerical method for constructing nonlinear normal modes for piecewise linear autonomous systems is presented. Based on the concept of invariant manifolds, a Galerkin based approach is applied here to obtain nonlinear normal modes numerically. The accuracy of the constructed nonlinear modes is checked by the comparison of the motion on the invariant manifold to the exact solution, in both time and frequency domains. It is found that the Galerkin based construction approach can represent the invariant manifold accurately over strong nonlinearity regions. Several interesting dynamic characteristics of the nonlinear modal motion are found and compared to those of linear modes. The stability of the nonlinear normal modes of a two-degree of freedom system is investigated using characteristic multipliers and Poincaré maps, and a flip bifurcation is found for both nonlinear modes.

scholarly journals Numerical Computation of Nonlinear Normal Modes of Nonconservative Systems

2013 ◽  
Author(s):  
L. Renson ◽  
G. Kerschen
Keyword(s):  
Periodic Solutions ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Techniques ◽  
Nonlinear Beam ◽  
Conservative Systems ◽  
Linear Damping ◽  
Nonlinear Normal ◽  
Element Technique

Since linear modal analysis fails in the presence of non-linear dynamical phenomena, the concept of nonlinear normal modes (NNMs) was introduced with the aim of providing a rigorous generalization of linear normal modes to nonlinear systems. Initially defined as periodic solutions, numerical techniques such as the continuation of periodic solutions were used to compute NNMs. Because these methods are limited to conservative systems, the present study targets the computation of NNMs for non-conservative systems. Their definition as invariant manifolds in phase space is considered. Specifically, the partial differential equations governing the manifold geometry are considered as transport equations and an adequate finite element technique is proposed to solve them. The method is first demonstrated on a conservative nonlinear beam and the results are compared to standard continuation techniques. Then, linear damping is introduced in the system and the applicability of the method is demonstrated.

Nonlinear Normal Modes for Vibratory Systems Under Periodic Excitation

2003 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Normal Mode ◽  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Beam Model ◽  
Periodic Excitation ◽  
Nonlinear Normal Mode ◽  
Nonlinear Normal

This paper considers the use of numerically constructed invariant manifolds to determine the response of nonlinear vibratory systems that are subjected to periodic excitation. The approach is an extension of the nonlinear normal mode formulation previously developed by the authors for free oscillations, wherein an auxiliary system that models the excitation is used to augment the equations of motion. In this manner, the excitation is simply treated as an additional system state, yielding a system with an extra degree of freedom, whose response is known. A reduced order model for the forced system is then determined by the usual nonlinear normal mode procedure, and an efficient Galerkin-based solution method is used to numerically construct the attendant invariant manifolds. The technique is illustrated by determining the frequency response for a simple two-degree-off-reedom mass-spring system with cubic nonlinearities, and for a discretized beam model with 12 degrees of freedom. The results show that this method provides very accurate responses over a range of frequencies near resonances.

Modal Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes*

2002 ◽  
Vol 124 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Eric Pesheck ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Degrees Of Freedom ◽  
Invariant Manifolds ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Nonlinear Coupling ◽  
Rotating Beam ◽  
Internal Resonances ◽  
Modal Expansion ◽  
Nonlinear Normal

A method for determining reduced-order models for rotating beams is presented. The approach is based on the construction of nonlinear normal modes that are defined in terms of invariant manifolds that exist for the system equations of motion. The beam considered is an idealized model for a rotor blade whose motions are dominated by transverse vibrations in the direction perpendicular to the plane of rotation (known as flapping). The mathematical model for the rotating beam is relatively simple, but contains the nonlinear coupling that exists between transverse and axial deflections. When one employs standard modal expansion or finite element techniques to this system, this nonlinearity causes slow convergence, leading to models that require many degrees of freedom in order to achieve accurate dynamical representations. In contrast, the invariant manifold approach systematically accounts for the nonlinear coupling between linear modes, thereby providing models with very few degrees of freedom that accurately capture the essential dynamics of the system. Models with one and two nonlinear modes are considered, the latter being able to handle systems with internal resonances. Simulation results are used to demonstrate the validity of the approach and to exhibit features of the nonlinear modal responses.

Internally Resonant Nonlinear Free Vibrations of Horizontal/Inclined Sagged Cables

2005 ◽  
Author(s):  
G. Rega ◽  
N. Srinil ◽  
S. Chucheepsakul
Keyword(s):  
Multiple Scales ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Original System ◽  
Nonlinear Normal Modes ◽  
Free Vibrations ◽  
Order Effects ◽  
Internal Resonances ◽  
Closed Form Solutions ◽  
Nonlinear Normal

Internally resonant dynamics in the nonlinear free vibrations of suspended cables are analytically investigated by means of a multi-mode Galerkin-based discretization and second-order multiple scales. Emphasis is placed on planar 2:1 internal resonances. The equations of motion of a general inclined cable model, which account for the dynamic extensibility effects and the system asymmetry due to inclined equilibrium, are considered. By considering higher-order effects due to quadratic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations associated with the resonant nonlinear normal modes reveal the dependence of cable nonlinear response on different resonant and non-resonant modes. Based on the modal convergence properties performed on the resonantly activated cables, the illustrative results provide hints for proper reduced-order model selections from the asymptotic solution. The underlying effects of cable inclination and cable sag are presented. The theoretical predictions are validated by finite difference numerical time laws of the original system equations of motion.

scholarly journals Nonlinear normal modes and their application in structural dynamics

2006 ◽  
Vol 2006 ◽  
pp. 1-15 ◽  
Author(s):  
Christophe Pierre ◽  
Dongying Jiang ◽  
Steven Shaw
Keyword(s):  
Large Scale ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Realistic Model ◽  
Galerkin Projection ◽  
Engineering Structures ◽  
Nonlinear Modes ◽  
Nonlinear Modal Analysis ◽  
Nonlinear Normal

Recent progress in the area of nonlinear modal analysis for structural systems is reported. Systematic methods are developed for generating minimally sized reduced-order models that accurately describe the vibrations of large-scale nonlinear engineering structures. The general approach makes use of nonlinear normal modes that are defined in terms of invariant manifolds in the phase space of the system model. An efficient Galerkin projection method is developed, which allows for the construction of nonlinear modes that are accurate out to large amplitudes of vibration. This approach is successfully extended to the generation of nonlinear modes for systems that are internally resonant and for systems subject to external excitation. The effectiveness of the Galerkin-based construction of the nonlinear normal modes is also demonstrated for a realistic model of a rotating beam.

Nonlinear Normal Modes in a Class of Nonlinear Continuous Systems

1993 ◽  
Author(s):  
Melvin E. King ◽  
Alexander F. Vakakis
Keyword(s):  
Reference Point ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Partial Differential ◽  
Analytical Methodology ◽  
Conservation Of Energy ◽  
Linearized Stability ◽  
Nonlinear Normal

Abstract A general methodology is developed for computing the nonlinear normal modes of a class of undamped vibratory systems governed by nonlinear partial differential equations of motion. A nonlinear normal mode is defined as free motion during which all points of the system vibrate equiperiodically, reaching their extremum positions at the same instants of time. The analytical methodology is based on a previous work by Shaw and Pierre (1992b), where the displacements and velocities at any point of a structure were expressed as functions of the displacement and velocity of a single reference point. The dynamics of the continuous system were then restricted to invariant manifolds of the phase space. Motivated by the methodology presented by Shaw and Pierre, we express the displacement of an arbitrary point of the structure as a function of the displacement of a single reference point. Assuming undamped oscillations (and thus conservation of energy), a singular partial differential equation for the function relating the displacements is derived, and is subsequently solved using an asymptotic, power series methodology. Applications of the general theory are then given by computing the nonlinear normal modes of a simply supported beam resting on a nonlinear elastic foundation, and of a cantilever beam having geometric nonlinearities. The stability of the detected modes is then investigated by a linearized stability analysis.

scholarly journals Nonlinear Normal Modes in a Two-Stage Isolator Using a Modified Finite-Element Galerkin Method

2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Cheng Li ◽  
Hongguang Li
Keyword(s):  
Finite Element ◽  
Galerkin Method ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Fourier Series Expansion ◽  
Periodic Motions ◽  
Two Stage ◽  
System Components ◽  
Nonlinear Normal

A modified Galerkin method is proposed to approximate the nonlinear normal modes in a new type of a two-stage isolator. Besides the displacement of payload and the force transmissibility of this typical nonlinear dynamic system, the nonlinear normal modes defined as invariant manifolds can provide more information about the nonlinear coupling between the system components when periodic motions corresponding to the normal modes of the system occur. The presented approach applies a combination of finite-element discretization and Fourier series expansion for the approximate invariant manifolds. A Galerkin projection of the governing equations for the approximate invariant manifolds yields a set of nonlinear algebraic equations in expansion coefficients, which can be solved numerically with a general choice of zero as initial guess for the cases in this work. The resultant approximate solutions for the invariant manifolds can accurately describe the nonlinear interactions between system components in periodic motions of the specific nonlinear normal modes. In addition, one can solve the invariant manifolds for an annular domain of interest directly by this method, without considering other domain that includes the origin of phase space.

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