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

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.

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 Normal Modes and Localization in Elastic Vibro-Impact Systems With Multiple Constraints

2007 ◽  
Author(s):  
David Wagg
Keyword(s):  
Mathematical Formulation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Lumped Mass ◽  
Localized Modes ◽  
Impact Oscillator ◽  
Mass System ◽  
Nonlinear Normal ◽  
Two Degree Of Freedom

In this paper we consider the dynamics of compliant mechanical systems subject to combined vibration and impact forcing. Two specific systems are considered; a two degree of freedom impact oscillator and a clamped-clamped beam. Both systems are subject to multiple motion limiting constraints. A mathematical formulation for modelling such systems is developed using a modal approach including a modal form of the coefficient of restitution rule. The possible impact configurations for an N degree of freedom lumped mass system are considered. We then consider sticking motions which occur when a single mass in the system becomes stuck to an impact stop, which is a form of periodic localization. Then using the example of a two degree of freedom system with two constraints we describe exact modal solutions for the free flight and sticking motions which occur in this system. A numerical example of a sticking orbit for this system is shown and we discuss identifying a nonlinear normal modal basis for the system. This is achieved by extending the normal modal basis to include localized modes. Finally preliminary experimental results from a clamped-clamped vibroimpacting beam are considered and a simplified model discussed which uses an extended modal basis including localized modes.

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.

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.

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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