Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities

1995 ◽  
Vol 117 (2) ◽  
pp. 199-205 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Quadratic Nonlinearities ◽  
Nonlinear Normal ◽  
Quadratic And Cubic Nonlinearities

We use two approaches to determine the nonlinear modes and natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions directly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized equations by using the method of multiple scales and the invariant manifold approach. Whereas one- and two-mode discretizations produce erroneous results for continuous systems with quadratic and cubic nonlinearities, all methods, in the present case, produce the same results because the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.

On Direct Methods for Constructing Nonlinear Normal Modes of Continuous Systems

1995 ◽  
Vol 1 (4) ◽  
pp. 389-430 ◽  
Author(s):  
Ali H. Nayfeh
Keyword(s):  
Normal Forms ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Direct Method ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Internal Resonances ◽  
Relief Valve ◽  
Nonlinear Normal

A direct method based on the method of normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the constructed nonlinear modes with all four methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and King and Vakakis are not applicable to problems with internal resonances.

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.

Some Observations of Mode Localizations in Flexibly Coupled Two Rotors

2005 ◽  
Author(s):  
Yohta Kunitoh ◽  
Hiroshi Yabuno ◽  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotor System ◽  
The Other ◽  
Whirling Motion ◽  
Averaged Equations ◽  
Weakly Coupled ◽  
Nonlinear Normal

Mode localizations in a weakly coupled two-span rotor system are theoretically and experimentally discussed. One rotor has a slight unbalance and the other one is well-assembled. First, the equations governing the whirling motions of the coupled rotors are expressed due to nonlinearity in each span and the weakness of the coupling. The averaged equations are obtained by the method of multiple scales and it is shown that the nonlinear normal modes are bifurcated from the linear normal modes. It results from this bifurcation that the number of nonlinear normal modes exceeds the equivalent degree of freedom of the two-span rotor system, i.e., 2-degree under the assumption that the trajectory of the whirling motion is circle. Also, it is theoretically clarified that whirling motion caused by the unbalance in the rotor is localized in the rotor with unbalance or in one without unbalance depending on the rotational speed. Furthermore, these mode localizations are experimentally confirmed.

Experimental Identification of Nonlinear Continuous Vibratory Systems Using Nonlinear Principal Component Analysis: Application to Structural Modification

2007 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yuichi Mizuno
Keyword(s):  
Principal Component Analysis ◽  
Structural Modification ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Principal Component ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Nonlinear Functions ◽  
Governing Equations ◽  
Nonlinear Normal

In identifying machines and structures, one sometimes encounters cases in which the system should be regarded as a nonlinear continuous system. The governing equations of motion of a nonlinear continuous system are described by a set of nonlinear partial differential equations and boundary conditions. Determining both of them simultaneously is a quite difficult task. Thus, one has to discretize the governing equations of motion, and reduce the order of the equations as much as possible. In analysis of nonlinear vibratory systems, it is known that one can reduce the order of the system by using the nonlinear normal modes while preserving the effect of the nonlinearity accurately. The nonlinear normal modes are description of motion by nonlinear functions of the coordinates for analysis. In identification it is expected that an accurate mathematical model with minimum degree of freedom can be determined if one can express the response as nonlinear functions of the coordinates for identification. Based on this idea, in a previous report the authors proposed an identification technique which uses nonlinear principal component analysis by a neural network. In this report, procedure to apply the identified result to structural modification is presented. It is shown via numerical example that when the structural modification is not so large, response prediction of the modified system with enough accuracy is possible.

Experimental Identification of Nonlinear Continuous Vibratory Systems Using Nonlinear Principal Component Analysis

2006 ◽  
Author(s):  
Keisuke Kamiya ◽  
Yuichi Mizuno ◽  
Kimihiko Yasuda
Keyword(s):  
Principal Component Analysis ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Principal Component ◽  
Continuous System ◽  
Nonlinear Normal Modes ◽  
Component Analysis ◽  
Nonlinear Functions ◽  
Governing Equations ◽  
Nonlinear Normal

In identifying machines and structures, one sometimes encounters cases in which the system should be regarded as a nonlinear continuous system. The governing equations of motion of a nonlinear continuous system are described by a set of nonlinear partial differential equations and boundary conditions. Determining both of them simultaneously is a quite difficult task. Thus, one has to discretize the governing equations of motion, and reduce the order of the equations as much as possible. In analysis of nonlinear vibratory systems, it is known that one can reduce the order of the system by using the nonlinear normal modes preserving the effect of the nonlinearity accurately. The nonlinear normal modes are description of motion by nonlinear functions of the coordinates for analysis. In identification it is expected that an accurate mathematical model with minimum degree of freedom can be determined if one can express the response as nonlinear functions of the coordinates for identification. Based on this idea, this paper proposes an identification technique which uses nonlinear principal component analysis by a neural network. Applicability of the proposed technique is confirmed by numerical simulation.

On Nonlinear Modes of Continuous Systems

1994 ◽  
Vol 116 (1) ◽  
pp. 129-136 ◽  
Author(s):  
A. H. Nayfeh ◽  
S. A. Nayfeh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Continuous Systems ◽  
Partial Differential ◽  
Nonlinear Beam ◽  
Nonlinear Modes ◽  
One Dimensional

We use several methods to study the nonlinear modes of one-dimensional continuous systems with cubic inertia and geometric nonlinearities. Invariant manifold and perturbation methods applied to the discretized system and the method of multiple scales applied to the partial-differential equation and boundary conditions are discussed and their equivalence is demonstrated. The method of multiple scales is then applied directly to the partial-differential equation and boundary conditions governing several nonlinear beam problems.

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.

An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 332-340 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Conservation Of Energy ◽  
Nonlinear Modes ◽  
One Dimensional ◽  
Linear Differential ◽  
Nonlinear Normal ◽  
The Stability ◽  
Nonlinear Elastic

The nonlinear normal modes of a class of one-dimensional, conservative, continuous systems are examined. These are free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. During a nonlinear normal mode, the motion of an arbitrary particle of the system is expressed in terms of the motion of a certain reference point by means of a modal function. Conservation of energy is imposed to construct a partial differential equation satisfied by the modal function, which is asymptotically solved using a perturbation methodology. The stability of the detected nonlinear modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and analyzing the resulting set of linear differential equations with periodic coefficients by Floquet analysis. Applications of the general theory are given by computing the nonlinear normal modes of a simply-supported beam lying on a nonlinear elastic foundation, and of a cantilever beam possessing geometric nonlinearities.

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.

Numerical Computation of Nonlinear Normal Modes in Multi-Mode Models of an Inertially Coupled Elastic Structure

2009 ◽  
Author(s):  
Fengxia Wang ◽  
Anil K. Bajaj
Keyword(s):  
Small Amplitude ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Elastic Structure ◽  
Internal Resonances ◽  
Reduced Order ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom ◽  
Quadratic And Cubic Nonlinearities

There are many techniques available for the construction of nonlinear normal modes. Most studies for systems with more than one degree of freedom utilize asymptotic techniques or the method of multiple time scales, which are valid only for small amplitude motions. Previous works of the authors have investigated nonlinear normal modes in elastic structures with essential inertial nonlinearities, and considered two degree-of-freedom reduced-order models that exhibit 1:2 resonance. For small amplitude oscillations with low energy, this reduced analysis is acceptable, while for higher energy vibrations and vibrations that are away from internal resonances, this may not provide an accurate representation of NNMs. For high energy vibration and vibrations away from internal resonances, two natural issues to be addressed are the dimension of the reduced-order model used for constructing NNMs, and the order of nonlinearities retained in the truncated models. To address these issues, a comparison of NNMs computed for three different reduced degree of freedom models for the elastic structure is reported here. The reduced models considered are: (i) A two degree-of-freedom reduced model with only quadratic nonlinearities; (ii) A two degree-of-freedom reduced model with both quadratic and cubic nonlinearities; (iii) A five degrees-of-freedom model with both quadratic and cubic nonlinearities. A numerical method based on shooting technique is used for constructing the NNMs and results for system near 1:2 internal resonances between the two lowest modes and away from any internal resonance are compared.

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