On Nonlinear Normal Modes of Systems With Internal Resonance

1996 ◽  
Vol 118 (3) ◽  
pp. 340-345 ◽  
Author(s):  
A. H. Nayfeh ◽  
C. Chin ◽  
S. A. Nayfeh
Keyword(s):  
Internal Resonance ◽  
Normal Modes ◽  
Discrete Systems ◽  
Nonlinear Normal Modes ◽  
Mode Shapes ◽  
Geometric Nonlinearities ◽  
Weakly Nonlinear ◽  
First Order ◽  
Nonlinear Normal ◽  
Nonlinear Discrete Systems

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.

Nonlinear Normal Modes of Buckled Beams: Three-to-One and One-to-One Internal Resonances

1997 ◽  
Author(s):  
Ali H. Nayfeh ◽  
Walter Lacarbonara ◽  
Char-Ming Chin
Keyword(s):  
Boundary Conditions ◽  
Internal Resonance ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Buckling Mode ◽  
Stable Mode ◽  
One To One ◽  
Nonlinear Normal

Abstract Nonlinear normal modes of a buckled beam about its first buckling mode shape are investigated. Fixed-fixed boundary conditions are considered. The cases of three-to-one and one-to-one internal resonances are analyzed. Approximate expressions for the nonlinear normal modes are obtained by applying the method of multiple scales to the governing integro-partial-differential equation and boundary conditions. Curves displaying variation of the amplitude with the internal resonance detuning parameter are generated. It is shown that, for a three-to-one internal resonance between the first and third modes, the beam may possess either one stable mode, or three stable normal modes, or two stable and one unstable normal modes. On the other hand, for a one-to-one internal resonance between the first and second modes, two nonlinear normal modes exist. The two nonlinear modes are either neutrally stable or unstable. In the case of one-to-one resonance between the third and fourth modes, two neutrally stable, nonlinear normal modes exist.

Nonlinear Modes in Systems With Strong Nonlinearities

1997 ◽  
Author(s):  
Melvin E. King
Keyword(s):  
Differential Equation ◽  
Normal Modes ◽  
Approximate Solutions ◽  
Discrete Systems ◽  
Nonlinear Normal Modes ◽  
Algebraic Equations ◽  
Conservation Of Energy ◽  
Singular Ordinary Differential Equation ◽  
Power Series Expansions ◽  
Nonlinear Normal

Abstract In this paper, a symbolic/numeric method is developed to compute nonlinear normal modes (NNMs) in conservative, two-degree-of-freedom (2-DoF) systems. Based upon the notion of NNMs, periodic motions are sought during which the two coordinates ‘vibrate-in-unison’. By parameterizing the response of one coordinate with respect to the response of the other (reference) coordinate and by imposing conservation of energy, we obtain a nonlinear, singular ordinary differential equation. Approximate solutions for these modal functions are obtained, for a given energy level, via truncated power-series expansions. The coefficients of the expansion, along with the maximum and minimum reference displacements, are then computed by (i) symbolically evaluating the singular differential equation at various (distinct) reference displacements, and then (ii) numerically solving the resulting set of nonlinear algebraic equations. Since the approximate solution inherently depends upon the order of the expansion, convergence studies must be performed in order to ensure sufficient accuracy. Note that even though the formulation presented herein is based on 2-DoF systems, the methodology is quite general and can readily be extended to higher-order discrete systems. Moreover, since it does not rely upon any ‘small-quantity’ assumptions, it can be used to investigate the dynamics of coupled, strongly nonlinear systems.

scholarly journals Nonlinear Normal Modes in a Three-Degree-of-Freedom Vibration Isolating System with Internal Resonance.

2002 ◽  
Vol 68 (671) ◽  
pp. 1950-1958
Author(s):  
Tetsuro TOKOYODA ◽  
Noriaki YAMASHITA ◽  
Hiroyuki OISHI ◽  
Takeshi YAMAMOTO ◽  
Masatsugu YOSHIZAWA
Keyword(s):  
Internal Resonance ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Nonlinear Normal ◽  
Three Degree Of Freedom

Nonlinear Normal Modes of a Cantilever Beam

1995 ◽  
Vol 117 (4) ◽  
pp. 477-481 ◽  
Author(s):  
A. H. Nayfeh ◽  
C. Chin ◽  
S. A. Nayfeh
Keyword(s):  
Cantilever Beam ◽  
Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Mode Shapes ◽  
Nonlinear Modes ◽  
Linear Mode ◽  
Complete Set ◽  
Nonlinear Normal

Two approaches for determination of the nonlinear planar modes of a cantilever beam are compared. In the first approach, the governing partial-differential system is discretized using the linear mode shapes and then the nonlinear mode shapes are determined from the discretized system. In the second approach, the boundary-value problem is treated directly by using the method of multiple scales. The results show that both approaches yield the same nonlinear modes because the discretization is performed using a complete set of basis functions, namely, the linear mode shapes.

A Semi-Analytical Method for Calculation of Strongly Nonlinear Normal Modes of Mechanical Systems

2018 ◽  
Vol 13 (4) ◽  
Author(s):  
S. Mahmoudkhani
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Internal Resonances ◽  
Oscillatory Systems ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

A new scheme based on the homotopy analysis method (HAM) is developed for calculating the nonlinear normal modes (NNMs) of multi degrees-of-freedom (MDOF) oscillatory systems with quadratic and cubic nonlinearities. The NNMs in the presence of internal resonances can also be computed by the proposed method. The method starts by approximating the solution at the zeroth-order, using some few harmonics, and proceeds to higher orders to improve the approximation by automatically including higher harmonics. The capabilities and limitations of the method are thoroughly investigated by applying them to three nonlinear systems with different nonlinear behaviors. These include a two degrees-of-freedom (2DOF) system with cubic nonlinearities and one-to-three internal resonance that occurs on nonlinear frequencies at high amplitudes, a 2DOF system with quadratic and cubic nonlinearities having one-to-two internal resonance, and the discretized equations of motion of a cylindrical shell. The later one has internal resonance of one-to-one. Moreover, it has the symmetry property and its DOFs may oscillate with phase difference of 90 deg, leading to the traveling wave mode. In most cases, the estimated backbone curves are compared by the numerical solutions obtained by continuation of periodic orbits. The method is found to be accurate for reasonably high amplitude vibration especially when only cubic nonlinearities are present.

The Construction of Nonlinear Normal Modes for Systems With Internal Resonance: Application to Rotating Beams

2002 ◽  
Author(s):  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Internal Resonance ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Reduced Order Model ◽  
Degree Of Freedom ◽  
Beam Model ◽  
Order Model ◽  
Reduced Order ◽  
Nonlinear Normal

A numerical method for constructing nonlinear normal modes for systems with internal resonances is presented based on the invariant manifold approach. In order to parameterize the nonlinear normal modes, multiple pairs of system state variables involved in the internal resonance are kept as ‘seeds’ for the construction of the multi-mode invariant manifold. All the remaining degrees of freedom are constrained to these ‘seed’ variables, resulting in a system of nonlinear partial differential equations governing the constraint relationships, which must be solved numerically. The solution procedure uses a combination of finite difference schemes and Galerkin-based expansion approaches. It is illustrated using two examples, both of which focus on the construction of two-mode models. The first example is based on the analysis of a simple three-degree-of-freedom example system, and is used to demonstrate the approach. An invariant manifold that captures two nonlinear normal modes is constructed, resulting in a reduced-order model that accurately captures the system dynamics. The methodology is then applied to a more large system, namely an 18-degree-of-freedom rotating beam model that features a three-to-one internal resonance between the first two flapping modes. The accuracy of the nonlinear two-mode reduced-order model is verified by time-domain simulations.

Localized and Non-Localized Nonlinear Normal Modes in a Multi-Span Beam With Geometric Nonlinearities

1996 ◽  
Vol 118 (4) ◽  
pp. 533-542 ◽  
Author(s):  
J. Aubrecht ◽  
A. F. Vakakis
Keyword(s):  
Coupling Parameter ◽  
Galerkin Approximation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Finite Amplitude ◽  
Torsional Stiffness ◽  
Mode Localization ◽  
Weakly Nonlinear ◽  
Weakly Coupled ◽  
Nonlinear Normal

The nonlinear normal modes of a geometrically nonlinear multi-span beam consisting of n segments, coupled by means of torsional stiffeners are examined. Assuming that the stiffeners possess large torsional stiffness, the beam displacements are decomposed into static and flexible components. It is shown that the static components are much smaller in magnitude than the flexible ones. A Galerkin approximation is subsequently employed to discretize the problem, whereby the computation of the nonlinear normal modes of the multi-span beam is reduced to the study of the periodic solutions of a set of weakly coupled, weakly nonlinear ordinary differential equations. Numerous stable and unstable, localized and non-localized nonlinear normal modes of the multi-span beam are detected. Assemblies consisting of n = 2, 3, and 4 beam segments are examined, and are found to possess stable, strongly localized nonlinear normal modes. These are free synchronous oscillations during which only one segment of the assembly vibrates with finite amplitude. As the number of periodic segments increases, the structure of the nonlinear normal modes becomes increasingly more complicated. In the multi-span beams examined, nonlinear mode localization is generated through two distinct mechanisms: through Pitchfork or Saddle-node mode bifurcations, or as the limit of a continuous mode branch when a coupling parameter tends to zero.

Constant-Gain Linear Feedback Control of Piecewise Linear Structural Systems via Nonlinear Normal Modes

2004 ◽  
Vol 10 (10) ◽  
pp. 1535-1558 ◽  
Author(s):  
E. A. Butcher ◽  
R. Lu
Keyword(s):  
Feedback Control ◽  
Degrees Of Freedom ◽  
Piecewise Linear ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Pole Placement ◽  
Mode Shapes ◽  
Linear Feedback ◽  
Structural Systems ◽  
Nonlinear Normal

We present a technique for using constant-gain linear position feedback control to implement eigen-structure assignment of n-degrees-of-freedom conservative structural systems with piecewise linear nonlinearities. We employ three distinct control strategies which utilize methods for approximating the nonlinear normal mode (NNM) frequencies and mode shapes. First, the piecewise modal method (PMM) for approximating NNM frequencies is used to determine n constant actuator gains for eigenvalue (pole) placement. Secondly, eigenvalue placement is accomplished by finding an approximate single-degree-of-freedom reduced model with one actuator gain for the mode to be controlled. The third strategy allows the frequencies and mode shapes (eigenstructure) to be placed by using a full n × n matrix of actuator gains and employing the local equivalent linear stiffness method (LELSM) for approximating NNM frequencies and mode shapes. The techniques are applied to a two-degrees-of-freedom system with two distinct types of nonlinearities: a bilinear clearance nonlinearity and a symmetric deadzone nonlinearity.

An Energy-Based Approach to Computing Resonant Nonlinear Normal Modes

1996 ◽  
Vol 63 (3) ◽  
pp. 810-819 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Internal Resonance ◽  
Natural Frequencies ◽  
Linear Expansion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Modal Interactions ◽  
Internal Resonances ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

A formulation for computing resonant nonlinear normal modes (NNMs) is developed for discrete and continuous systems. In a canonical framework, internal resonance conditions are immediately recognized by identifying commensurable linearized natural frequencies of these systems. Additionally, a canonical formulation allows for a single (linearized modal) coordinate to parameterize all other coordinates during a resonant NNM response. Energy-based NNM methodologies are applied to a canonical set of equations and asymptotic solutions are sought. In order to account for the resonant modal interactions, it will be shown that high-order terms in the O(1) solutions must be considered (in the absence of internal resonances, a linear expansion at O(1) is sufficient). Two applications (‘3:1’ resonances in a two-degree-of-freedom system and ‘3:1’ resonance in a hinged-clamped beam) are then considered by which to demonstrate the resonant NNM methodology. It is shown that for some responses, nonlinear modal relations do not exist in the context of physical coordinates and thus a transformation to a canonical framework is necessary in order to appropriately define NNM relations.

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