Nonlinear Normal Vibration Modes Research Articles

Non-iterative Rauscher method for 1-DOF system: a new approach to studying non-autonomous system via equivalent autonomous one

Rauscher method becomes the matter of interest because in combination with the method of nonlinear normal vibration modes it allows to calculate steady forced vibrations in the system with multiple degrees of freedom (DOF) via reduction in the number of DOFs. However, modern realizations of that approach have drawbacks such as iterative nature and the need to have initial approximation for the solution. The primary principle of Rauscher method is in obtaining periodic solutions of a non-autonomous system via studying some equivalent autonomous one. In the paper, a new non-iterative variant of Rauscher method is considered. In its current statement, the method can be used in analysis of forced harmonic oscillations in a nonlinear system with one degree of freedom. The primary goals of the study were to find out what kind of equivalent autonomous systems could be built for a given non-autonomous one and how they can be used for the construction of periodic solutions and/or periodic phase plane orbits of the initial system. It is shown that three different types of equivalent autonomous dynamical systems can be built for a given 1-DOF non-autonomous one. The system of 1st type is a fourth-order dynamical system. Technically it can be considered as a 2-DOF system where additional “DOF” is explicitly “responsible” for forced oscillations. The system of 2nd type is a third-order dynamical system. Its periodic orbits are exactly the same as in the initial system. Using the invariant manifold of the system of 1st type, the system of 2nd type can be reduced to the form $$W(x,x')=0$$ (which is called here the equivalent system of the 3rd type). It is important that the function $$W(x,x')$$ can be built a priori. Once $$W(x,x')$$ is found: (i) one can obtain different periodical orbits corresponding to forced oscillations in the initial system; (ii) one can estimate amplitudes of vibrations for these regimes; (iii) one can track bifurcations of periodical regimes of the initial system with respect to change in amplitude of external excitation f. As shown in the paper, periodical orbits of the initial non-autonomous system can be obtained via two different approaches: (i) as set of points on phase plane satisfying the condition $$W(x,x')=0$$ ; (ii) via the application of harmonic balance method to the equivalent system of 1st type using system’s energy level as a continuation parameter. This approach has advantage over application of harmonic balance method to initial system because the latter requires good initial guess for expansion coefficients, while the new approach does not and always starts from zero initial guess.

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.

Evolution of ultrasonic impulses in chains of spheres using resonant excitation

It is demonstrated that broad-bandwidth ultrasonic signals containing frequency components in excess of 200 kHz can be created in spherical chains using harmonic excitation at 73 kHz. Multiple reflections created a periodic waveform containing both harmonics and sub-harmonics of the original forcing frequency, due to non-linear Hertzian contact. These discrete frequencies represented some of the many allowed non-linear normal modes of vibration of the whole chain. Excitation at a single fixed frequency could thus be used to produce wide-bandwidth impulses for different lengths of spherical chains. Experimental results were in good agreement with theoretical predictions.

A single pair-of-sensors technique for geometry consistent sensing of acceleration vector fields in beam structures: damage detection and dissipation estimation by POD modes

The acceleration vector field developed in a beam-like elastic structure is sampled over the space–time domain by a tri-axial accelerometer relocated over the beam surface while triggered by another mono-axial accelerometer fixed in space to attain synchronism. This technique leads to structure-geometry consistent spatio-temporal measurements of coupled structural dynamics. This result is established by extracting and interpreting the physics hidden in geometry consistent raw databases (mined by sweeping a single pair-of-sensors) with advanced proper orthogonal decomposition (POD) tools. For in-plane coupled vibrations, the dominant POD modes coincide with certain nonlinear normal modes of vibration belonging to the slow dynamics manifold structure. A parametric study reveals that they remain robust for increasing energy levels (nonlinearity) and changes in beam length-curvature (nonlinear geometry); and for out-of-plane bending and torsion vibration perturbations. Spatial local damage signatures due to a local mass and a plasticity defect are extracted from comparing logarithmically the POD modal shapes of healthy and damaged curved beams. In the presence of a local plasticity defect, certain dominant POD modes increases their internal dissipation index, the latter being modes independent for the healthy beam. This limited-sensor sensing technique is potentially useful for test and evaluation analysis of beam and cable like structural systems with complex geometries.

Non-linear normal forced vibration modes in systems with internal resonance

The new method of the forced resonance vibrations construction in mechanical systems with internal resonance is represented. According to this approach, the generalized theory of non-linear normal vibration modes by Shaw and Pierre, the modified Rauscher method and the harmonic balance method are combined with a new iterative computation procedure.The proposed approach is used in analysis of the single-disk rotor system with the isotropic-elastic shaft and the non-linear supports of Duffing type. Gyroscopic effects, asymmetrical disposition of the disk on the shaft and internal resonance are also taken into account. The NNM approach allows reducing the 8-DOF problem of the rotor dynamics to the 2-DOF non-linear system for each non-linear normal mode. Both the model of massless supports and the model of supports with inertial effects are considered. It is shown that in last case all resonance regimes are separated into two different kinds. First kind corresponds to cyclic symmetric trajectories in a system's configuration space; the second kind corresponds to centrally symmetric ones. Regimes of the first kind can be evaluated by the use of the simplified mathematical model proposed in this work. Simplified model consists only of four generalized coordinates instead of the eight initial ones.

Nonlinear normal vibration modes in the dynamics of nonlinear elastic systems

Nonlinear normal modes (NNMs) are a generalization of the linear normal vibrations. By the Kauderer-Rosenberg concept in the regime of the NNM all position coordinates are single-values functions of some selected position coordinate. By the Shaw-Pierre concept, the NNM is such a regime when all generalized coordinates and velocities are univalent functions of a couple of dominant (active) phase variables. The NNMs approach is used in some applied problems. In particular, the Kauderer-Rosenberg NNMs are analyzed in the dynamics of some pendulum systems. The NNMs of forced vibrations are investigated in a rotor system with an isotropic-elastic shaft. A combination of the Shaw-Pierre NNMs and the Rauscher method is used to construct the forced NNMs and the frequency responses in the rotor dynamics.

Forced oscillations of beam with essentially nonlinear absorber

Forced oscillations of a beam are described by two-degrees-of-freedom discrete model, while an interaction of the beam and essentially nonlinear absorber is characterized by three-degrees-of-freedom model. We analyze motions of the beam which facilitate vibrations absorption presented in the form of nonlinear normal vibration modes in configuration space. The Rauscher method is used to investigate the motions of the beam. Stability of these motions is analyzed with the help of the Lyapunov method.

Determination of the Chaos Onset in Mechanical Systems with Several Equilibrium Positions

Determination of the chaos onset in some mechanical systems with several equilibrium positions are analyzed. Namely, the snap-through truss and the oscillator with a nonlinear dissipation force, under the external periodical excitation, are considered. Two approaches are used for the chaos onset determination. First, Pade and quasi-Pade approximants are used to construct closed homoclinic trajectories for a case of small dissipation. Convergence condition used earlier in the theory of nonlinear normal vibration modes as well conditions at infinity make possible to evaluate initial amplitude values for the trajectories with admissible precision. Mutual instability of phase trajectories is used as criterion of chaotic behavior in nonlinear systems for a case of not small dissipation. The numerical realization of the Lyapunov stability definition gives us a possibility to observe a process of appearance and fast enlargement of the chaotic behavior regions if some selected parameters of the dynamical systems under consideration are changing.

Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

The aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped nonlinear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms “periodic orbits” means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems.

Dynamical interaction of an elastic system and a vibro-impact absorber

The nonlinear two-degree-of-freedom system under consideration consists of the linear oscillator with a relatively big mass, which is an approximation of some continuous elastic system, and of the vibro-impact oscillator with a relatively small mass, which is an absorber of the linear system vibrations. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for the elastic vibration absorption, exists in a large region of the system parameters. In this regime, amplitudes of vibrations of the linear system are small, simultaneously vibrations of the absorber are significant.

Advanced Proper Orthogonal Decomposition Tools: Using Reduced Order Models to Identify Normal Modes of Vibration and Slow Invariant Manifolds in the Dynamics of Planar Nonlinear Rods

Reduced order models for the dynamics of geometrically exact planar rods are derived by projecting the nonlinear equations of motion onto a subspace spanned by a set of proper orthogonal modes. These optimal modes are identified by a proper orthogonal decomposition processing of high-resolution finite element dynamics. A three-degree-of-freedom reduced system is derived to study distinct categories of motions dominated by a single POD mode. The modal analysis of the reduced system characterizes in a unique fashion for these motions, since its linear natural frequencies are near to the natural frequencies of the full-order system. For free motions characterized by a single POD mode, the eigen-vector matrix of the derived reduced system coincides with the principal POD-directions. This property reflects the existence of a normal mode of vibration, which appears to be close to a slow invariant manifold. Its shape is captured by that of the dominant POD mode. The modal analysis of the POD-based reduced order system provides a potentially valuable tool to characterize the spatio-temporal complexity of the dynamics in order to elucidate connections between proper orthogonal modes and nonlinear normal modes of vibration.

Dynamical interaction of an elastic system and essentially nonlinear absorber

The nonlinear two-degrees-of-freedom system under consideration consists of a linear oscillator with a relatively big mass which is an approximation of some continuous elastic system, and an essentially nonlinear oscillator with a relatively small mass which is an absorber of the main linear system vibrations. Free and forced vibrations of the system are investigated. Analysis of nonlinear normal vibration modes shows that a stable localized vibration mode, which provides the vibration regime appropriate for an absorption, exists in a large region of the system parameters. In this regime amplitudes of vibrations of the main elastic system are small; simultaneously vibrations of the absorber are significant. Frequency response of the system under external periodic force is obtained. The dynamical interaction of elastic string under impact impulse and the essentially nonlinear absorber is considered too. Absorption of a longitudinal traveling wave in the system is analyzed.

Non-linear normal modes of a simply supported beam: continuous system and finite-element models

Non-linear normal modes of vibration for a hinged–hinged beam with fixed ends are evaluated considering both the continuous system and finite-element models. With regard to the latter ones, two alternative approaches are used, namely the invariant manifold technique and the method of multiple scales. Results from each one of the mathematical models are compared with data available in the literature.

Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions

A new approach for a construction of homo and heteroclinic trajectories of some principal non-linear dynamical systems is utilized here, namely the non-linear Schrodinger equation, non-autonomous Duffing equation and the equation of a parametrically excited damped pendulum are considered. Pade’ and quasi-Pade’ approximants and a convergence condition used earlier in the theory of non-linear normal vibration modes made possible to solve a boundary-value problems formulated for the orbits and to determine initial amplitude values of the trajectories with admissible precision. The approach proposed here is more exact than the generally accepted one because it is not necessary to use here separatrix trajectories of the corresponding autonomous equations.

Nonlinear Modal Analysis of Structural Systems Using Multi-Mode Invariant Manifolds

In this paper, an invariant manifold approach is introduced for the generationof reduced-order models for nonlinear vibrations of multi-degrees-of-freedomsystems. In particular, the invariant manifold approach for defining andconstructing nonlinear normal modes of vibration is extended to the case ofmulti-mode manifolds. The dynamic models obtained from this technique capture the essential coupling between modes of interest, while avoiding coupling fromother modes. Such an approach is useful for modeling complex systemresponses, and is essential when internal resonances exist between modes.The basic theory and a general, constructive methodology for the method arepresented. It is then applied to two example problems, one analytical andthe other finite-element based. Numerical simulation results are obtainedfor the full model and various types of reduced-order models, including theusual projection onto a set of linear modes, and the invariant manifoldapproach developed herein. The results show that the method is capable ofaccurately representing the nonlinear system dynamics with relatively fewdegrees of freedom over a range of vibration amplitudes.

Normal Vibrations in Near-Conservative Self-Excited and Viscoelastic Nonlinear Systems

A perturbation methodology and power series are utilizedto the analysis of nonlinear normal vibration modes in broadclasses of finite-dimensional self-excited nonlinear systems closeto conservative systems taking into account similar nonlinear normal modes.The analytical construction is presented for some concretesystems. Namely, two linearly connected Van der Pol oscillatorswith nonlinear elastic characteristics and a simplesttwo-degrees-of-freedom nonlinear model of plate vibrations in agas flow are considered.Periodical quasinormal solutions of integro-differentialequations corresponding to viscoelastic mechanical systems areconstructed using a convergent iteration process. One assumesthat conservative systems appropriate for the dominant elasticinteractions admit similar nonlinear normal modes.

ANALYTICAL CONSTRUCTION OF HOMOCLINIC ORBITS OF TWO- AND THREE-DIMENSIONAL DYNAMICAL SYSTEMS

An approach for the construction of homoclinic orbits of non-linear dynamical systems with phase spaces of dimensions equal to two or three is proposed here. The non-linear Schrodinger equation and Lorenz system are considered. Quasi-Pade' approximants are used for this construction. Potentiality and convergence conditions used earlier in the theory of non-linear normal vibration modes make it possible to solve the boundary-value problems formulated for the orbits and to evaluate initial amplitude values.

MODAL ANALYSIS OF A CRACKED BEAM

This paper addresses the problem of vibrations of a cracked beam. In general, the motion of such a beam can be very complex. This phenomenon can be attributed to the presence of the non-linearity due to the opening and closing of cracks. The focus of this paper is the modal analysis of a cantilever beam with a transverse edge crack. The non-linearity mentioned above has been modelled as a piecewise-linear system. In an attempt to define effective natural frequencies for this piecewise-linear system, the idea of a “bilinear frequency” is utilized. The bilinear frequency is obtained by computing the associated frequencies of each of the linear pieces of the piecewise-linear system. The finite element method is used to obtain the natural frequencies in each linear region.In order better to understand the essential non-linear dynamics of the cracked beam, a piecewise-linear two-degree-of-freedom model is studied. Perturbation methods are used to obtain the non-linear normal modes of vibration and the associated period of the motion. Results of this piecewise-linear model problem are shown to justify the definition of the bilinear frequency as the effective natural frequency. It is therefore expected that calculating piecewise mode shapes and bilinear frequencies is useful for understanding the dynamics of the infinite degree of freedom cracked beam.

An application of the inch algebraization to the stability of non-linear normal vibration modes

A normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by Ince (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained.

The normal modes of non-linear symmetric systems by group representation theory

Group representation theory is applied to analyze a symmetric spring mass system of finite degree of freedom for non-linear normal modes of vibration. A set of symmetry operations constituting a group characterises the symmetry of a system. The symmetry adapted basis vectors, corresponding to the irreducible representations present in the reducible representation, are determined. The equations of motion in the symmetry adapted coordinates have a lower order of coupling. An equipotential surface in each invariant subspace is determined from which the normal mode vectors are evaluated in terms of the symmetry adapted basis vectors.