Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

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.

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.

scholarly journals Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

2004 ◽  
Vol 10 (4) ◽  
pp. 319-335 ◽  
Author(s):  
Mathias Legrand ◽  
Dongying Jiang ◽  
Christophe Pierre ◽  
Steven W. Shaw
Keyword(s):  
Invariant Manifold ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotating Shaft ◽  
Nonlinear Mode ◽  
Large Amplitude Motions ◽  
Nonlinear Normal ◽  
The Individual

The nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig–Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single-d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and nonconservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well the simulations from the reference Craig–Bampton model.

Computing Nonlinear Normal Modes Using Numerical Continuation and Force Appropriation

2012 ◽  
Author(s):  
Robert J. Kuether ◽  
Matthew S. Allen
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Numerical Continuation ◽  
Gradient Based ◽  
Nonlinear Normal ◽  
Definition Of ◽  
Nonlinear Equations Of Motion ◽  
Mass Spring

Many structures can behave nonlinearly, exhibiting behavior that is not captured by linear vibration theory such as localization and frequency-energy dependence. The nonlinear normal mode (NNM) concept, developed over the last few decades, can be quite helpful in characterizing a structure’s nonlinear response. In the definition of interest, an NNM is a periodic solution to the conservative nonlinear equations of motion. Several approaches have been suggested for computing NNMs and some have been quite successful even for systems with hundreds of degrees of freedom. However, existing methods are still too expensive to employ on realistic nonlinear finite element models, especially when the Jacobian of the equations of motion is not available analytically. This work presents a new approach for numerically calculating nonlinear normal modes by combining force appropriation, numerical integration and continuation techniques. This method does not require gradients, is found to compute the NNMs accurately up to moderate response amplitudes, and could be readily extended to experimentally characterize nonlinear structures. The method is demonstrated on a nonlinear mass-spring-damper system, computing its NNMs up to a 35% shift in frequency. The results are compared with those from a gradient based algorithm and the relative merits of each method are discussed.

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.

scholarly journals Comparison of Galerkin, POD and Nonlinear-Normal-Modes Models for Nonlinear Vibrations of Circular Cylindrical Shells

2006 ◽  
Author(s):  
M. Amabili ◽  
C. Touze´ ◽  
O. Thomas
Keyword(s):  
Nonlinear Vibrations ◽  
Circular Cylindrical Shell ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Harmonic Excitation ◽  
Nonlinear Responses ◽  
Nonlinear Normal ◽  
The Galerkin Method ◽  
Proper Orthogonal

The aim of the present paper is to compare two different methods available to reduce the complicated dynamics exhibited by large amplitude, geometrically nonlinear vibrations of a thin shell. The two methods are: the proper orthogonal decomposition (POD) and an asymptotic approximation of the Nonlinear Normal Modes (NNMs) of the system. The structure used to perform comparisons is a water-filled, simply supported circular cylindrical shell subjected to harmonic excitation in the spectral neighbourhood of the fundamental natural frequency. A reference solution is obtained by discretizing the Partial Differential Equations (PDEs) of motion with a Galerkin expansion containing 16 eigenmodes. The POD model is built by using responses computed with the Galerkin model; the NNM model is built by using the discretized equations of motion obtained with the Galerkin method, and taking into account also the transformation of damping terms. Both the POD and NNMs allow to reduce significantly the dimension of the original Galerkin model. The computed nonlinear responses are compared in order to verify the accuracy and the limits of these two methods. For vibration amplitudes equal to 1.5 times the shell thickness, the two methods give very close results to the original Galerkin model. By increasing the excitation and vibration amplitude, significant differences are observed and discussed.

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.

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