Numerical analysis of nonlinear modes of piecewise linear systems torsional vibrations

The nonlinear modes of essentially nonlinear piecewise-linear finite degrees of freedom systems are calculated by the numerical methods, which are suggested in this paper. The basis of these methods is numerical solutions of the equations of the systems motions in configuration space. The numerical method for the nonlinear modes of essentially nonlinear piecewise-linear systems forced vibrations is suggested. The basis of this approach is the combination of the Rauscher method and the calculations of the autonomous system nonlinear modes. The nonlinear modes of the diesel engine transmission torsional vibrations are analyzed numerically. The vibrations are described by essentially nonlinear piecewise-linear system with fifteen degrees of freedom. The NNMs of this system forced vibrations are observed in the resonance regions. Both NNMs and the motions, which are essentially differ from NNMs, are observed in the distance from the resonances. NNMs of the forced vibrations of the systems with dissipation are close to NNMs of the system without dissipation.     

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

A system with piecewise linear restoring forces, typical of damaged beams with a breathing crack, exhibits bifurcations characterized by the onset of superabundant modes in internal resonance with a significantly different shape than that of modes on a fundamental branch. A 2-DOF frame with piecewise linear stiffness is analyzed by means of an experimental investigation; the frame is forced by an harmonic base excitation and the operative modal shapes as well as the response amplitude are directly measured; the results are compared with numerical outcomes for different damping values. This study shows that the shapes and the frequencies of certain nonlinear normal modes (NNMs) of the related autonomous system strongly affect the forced response, in both the numerical and the experimental environments. Therefore, it is possible to match the NNM with the forced response of the system, leading to the prospect of identifying the severity and position of the damage from experimental tests.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.     

Nonlinear Normal Modes and Their Bifurcations for an Inertially Coupled Nonlinear Conservative System

This work concerns the nonlinear normal modes (NNMs) of a 2 degree-of-freedom autonomous conservative spring–mass–pendulum system, a system that exhibits inertial coupling between the two generalized coordinates and quadratic (even) nonlinearities. Several general methods introduced in the literature to calculate the NNMs of conservative systems are reviewed, and then applied to the spring–mass–pendulum system. These include the invariant manifold method, the multiple scales method, the asymptotic perturbation method and the method of harmonic balance. Then, an efficient numerical methodology is developed to calculate the exact NNMs, and this method is further used to analyze and follow the bifurcations of the NNMs as a function of linear frequency ratio p and total energy h. The bifurcations in NNMs, when near 1:2 and 1:1 resonances arise in the two linear modes, is investigated by perturbation techniques and the results are compared with those predicted by the exact numerical solutions. By using the method of multiple time scales (MTS), not only the bifurcation diagrams but also the low energy global dynamics of the system is obtained. The numerical method gives reliable results for the high-energy case. These bifurcation analyses provide a significant glimpse into the complex dynamics of the system. It is shown that when the total energy is sufficiently high, varying p, the ratio of the spring and the pendulum linear frequencies, results in the system undergoing an order–chaos–order sequence. This phenomenon is also presented and discussed.     

Characterization of bifurcating non-linear normal modes in piecewise linear mechanical systems

The non-linear modal properties of a vibrating 2-DOF system with non-smooth (piecewise linear) characteristics are investigated; this oscillator can suitably model beams with a breathing crack or systems colliding with an elastic obstacle. The system having two discontinuity boundaries is non-linearizable and exhibits the peculiar feature of a number of non-linear normal modes (NNMs) that are greater than the degrees of freedom. Since the non-linearities are concentrated at the origin, its non-linear frequencies are independent of the energy level and uniquely depend on the damage parameter. An analysis of the NNMs has been performed for a wide range of damage parameter by employing numerical procedures and Poincaré maps. The influence of damage on the non-linear frequencies has been investigated and bifurcations characterized by the onset of superabundant modes in internal resonance, with a significantly different shape than that of modes on fundamental branch, have been revealed.     

Nonlinear dynamics of one disk asymmetrical rotor supported by two journal bearings

Asymmetrical one disk rotor interacting with fluid films of short journal bearings is considered. Gyroscopic moments acting on a disk are taken into account. The forces of the journal bearing fluid film are derived analytically. The system of four nonlinear ordinary differential equations is obtained to study the rotor vibrations. The origination of self-sustained vibrations of rotor is studied by means of Shaw–Pierre nonlinear modes. The harmonic balance method is applied to study the self-sustained vibrations with large amplitudes.     

Bifurcation of nonlinear internal resonant normal modes of a class of multi-degree-of-freedom systems

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases.     

Component Mode Synthesis Using Nonlinear Normal Modes

This paper describes a methodology for developing reduced-order dynamic models of structural systems that are composed of an assembly of nonlinear component structures. The approach is a nonlinear extension of the fixed-interface component mode synthesis (CMS) technique developed for linear structures by Hurty and modified by Craig and Bampton. Specifically, the case of nonlinear substructures is handled by using fixed-interface nonlinear normal modes (NNMs). These normal modes are constructed for the various substructures using an invariant manifold approach, and are then coupled through the traditional linear constraint modes (i.e., the static deformation shapes produced by unit interface displacements). A class of systems is used to demonstrate the concept and show the effectiveness of the proposed procedure. Simulation results show that the reduced-order model (ROM) obtained from the proposed procedure outperforms the ROM obtained from the classical fixed-interface linear CMS approach as applied to a nonlinear structure. The proposed method is readily applicable to large-scale nonlinear structural systems that are based on finite-element models.     

Performances of dynamic vibration absorbers for beams subjected to moving loads

The goal of this work is a general assessment regarding the performances of linear and nonlinear dynamic vibration absorbers (DVAs) applied to the specific problem of moving loads or vehicles. The problem consists of a simply supported linear Euler–Bernoulli beam excited with a moving load/vehicle; a DVA is connected to the beam in order to reduce the vibrations. The moving vehicle is modeled by a single degree of freedom mass spring system. The partial differential equations governing the beam dynamics is reduced to a set of ordinary differential equations by means of the Bubnov–Galerkin method. A parametric analysis is carried out to find the optimal parameters of the DVA that minimize the maximum vibration amplitude of the beam. For the case of a moving vehicle, the energy absorbed by the DVA is evaluated. Comparisons among the performances of different types of linear and DVAs are carried out. The goal is to clarify if the use of nonlinearities in the DVAs can effectively improve their performances. The study shows that the most effective type of DVA for the test cases considered is the piecewise linear elastic restoring force.     

Numerical calculation of nonlinear normal modes in structural systems

This paper presents two methods for numerical calculation of nonlinear normal modes (NNMs) in multi-degree-of-freedom, conservative, nonlinear structural dynamics models. The approaches used are briefly described as follows. Method 1: Starting with small amplitude initial conditions determined by a selected mode of the associated linear system, a small amount of negative damping is added in order to “artificially destabilize” the system; numerical integration of the system equations of motion then produces a simulated response in which orbits spiral outward essentially in the nonlinear modal manifold of interest, approximately generating this manifold for moderate to strong nonlinearity. Method 2: Starting with moderate to large amplitude initial conditions proportional to a selected linear mode shape, perform numerical integration with the coefficient ε of the nonlinearity contrived to vary slowly from an initial value of zero; this simulation methodology gradually transforms the initially flat eigenspace for ε = 0 into the manifold existing quasi-statically for instantaneous values of ε. The two methods are efficient and reasonably accurate and are intended for use in finding NNMs, as well as interesting behavior associated with them, for moderately and strongly nonlinear systems with relatively many degrees of freedom (DOFs).     

Analysis of forced vibrations by nonlinear modes

The combination of Rausher method and nonlinear modes is suggested to analyze the forced vibrations of nonlinear discrete systems. The basis of the Rausher method is iterative procedure. In this case, the analysis of a nonautonomous dynamical system reduces to the multiple solutions of the autonomous ones. As an example, the forced vibrations of shallow arch close to equilibrium position are considered in this paper. The results of the analysis are shown on the frequency response.     

A Volterra Series Approach to the Approximation of Stochastic Nonlinear Dynamics

A response approximation method for stochastically excited, nonlinear, dynamic systems is presented. Herein, the output of the nonlinear system isapproximated by a finite-order Volterra series. The original nonlinear system is replaced by a bilinear system in order to determine the kernels of this series. The parameters of the bilinear system are determined by minimizing, in a statistical sense,the difference between the original system and the bilinear system. Application to a piecewise linear modelof a beam with a nonlinear one-sided supportillustrates the effectiveness of this approach in approximatingtruly nonlinear, stochastic response phenomena in both the statistical momentsand the power spectral density of the response of this system in case ofa white noise excitation.     

Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods

The aim of the present paper is to compare two different methods available for reducing 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. The response is investigated also for a fixed excitation frequency by using the excitation amplitude as bifurcation parameter for a wide range of variation. Bifurcation diagrams of Poincaré maps obtained from direct time integration and calculation of the maximum Lyapunov exponent have been used to characterize the system.     

On stability of relative equilibria of a double pendulum with vibrating suspension point

We consider the motions of a double pendulum consisting of two hinged identical rods. The pendulum suspension point is assumed to perform harmonic vibrations of arbitrary frequency and arbitrary amplitude in the vertical direction. We carry out a complete nonlinear analysis of the stability of the four pendulum relative equilibria on the vertical. The problem on the stability of the relative equilibria of the mathematical pendulum in the case where the suspension point performs vertical harmonic vibrations of arbitrary frequency and arbitrary amplitude was considered in a linear setting [1–3] and a nonlinear setting [4, 5]. In the case of small-amplitude rapid vertical vibrations of the suspension point, linear and (mathematically not fully rigorous) nonlinear stability analysis of the relative equilibria was carried out for an ordinary pendulum [6–9] and a double pendulum [10, 11]. In [12], for the same case of rapid vibrations, stability conditions in the linear approximation were obtained for the four relative equilibria of a system consisting of two physical pendulums. In the special case of a system consisting of two identical rods, the problem was solved in the nonlinear setting.     

Isogeometric analysis of nonlinear Euler–Bernoulli beam vibrations

In this paper we analyze the vibrations of nonlinear structures by means of the novel approach of isogeometric finite elements. The fundamental idea of isogeometric finite elements is to apply the same functions, namely B-Splines and NURBS (Non-Uniform Rational B-Splines), for describing the geometry and for representing the numerical solution. In case of linear vibrational analysis, this approach has already been shown to possess substantial advantages over classical finite elements, and we extend it here to a nonlinear framework based on the harmonic balance principle. As application, the straight nonlinear Euler–Bernoulli beam is used, and overall, it is demonstrated that isogeometric finite elements with B-Splines in combination with the harmonic balance method are a powerful means for the analysis of nonlinear structural vibrations. In particular, the smoother k-method provides higher accuracy than the p-method for isogeometric nonlinear vibration analysis.     

Order reduction of structural dynamic systems with static piecewise linear nonlinearities

A technique for order reduction of dynamic systems in structural form with static piecewise linear nonlinearities is presented. By utilizing two methods which approximate the nonlinear normal mode (NNM) frequencies and mode shapes, reduced-order models are constructed which more accurately represent the dynamics of the full model than do reduced models obtained via standard linear transformations. One method builds a reduced-order model which is dependent on the amplitude (initial conditions) while the other method results in an amplitude-independent reduced model. The two techniques are first applied to reduce two-degree-of-freedom undamped systems with clearance, deadzone, bang-bang, and saturation stiffness nonlinearities to single-mode reduced models which are compared by direct numerical simulation with the full models. It is then shown via a damped four-degree-of-freedom system with two deadzone nonlinearities that one of the proposed techniques allows for reduction to multi-mode reduced models and can accommodate multiple nonsmooth static nonlinearities with several surfaces of discontinuity. The advantages of the proposed methods include obtaining a reduced-order model which is signal-independent (doesn’t require direct integration of the full model), uses a subset of the original physical coordinates, retains the form of the nonsmooth nonlinearities, and closely tracks the actual NNMs of the full model.     

Energy Pumping with Various Nonlinear Structures: Numerical Evidences

Numerical investigations are carried out on a linear structure, weakly coupled to a small nonlinear attachment. The essential nonlinearity of the attachment enables it to resonate with any of the linearized modes of the structure leading to energy pumping, i.e. passive, one-way, irreversible transfer of energy from the structure to the attachment. Different nonlinear structures (piecewise linear system, chaotic system) and efficiency of energy pumping are studied in each case in order to be able to apply it to civil engineering. As a specific application, attenuation of vibrations of a building is studied with two building models. In particular, the case of stochastic excitations is analyzed to examine if it is possible to process energy pumping when a seism occurs and an indicator of efficiency has been introduced.     

Normal Modes and Boundary Layers for a Slender Tensioned Beam on a Nonlinear Foundation

In this paper, the nonlinear normal modes (NNMs) of a thin beamresting on a nonlinear spring bed subjected to an axial tension isstudied. An energy-based method is used to obtain NNMs. In conjunction with amatched asymptotic expansion, we analyze, through simple formulas, thelocal effects that a small bending stiffness has on the dynamics, alongwith the secular effects caused by a symmetric nonlinearity. Nonlinearmode shapes are computed and compared with those of the unperturbedlinear system. A double asymptotic expansion is employed to compute theboundary layers in the nonlinear mode shape due to the small bendingstiffness. Satisfactory agreement between the theoretical and numericalbackbone curves of the system in the frequency domain is observed.     

A Degenerate Bifurcation Structure in the Dynamics of Coupled Oscillators with Essential Stiffness Nonlinearities

We study the degenerate bifurcations of the nonlinear normal modes(NNMs) of an unforced system consisting of a linear oscillator weaklycoupled to a nonlinear one that possesses essential stiffnessnonlinearity. By defining the small coupling parameter , we study thedynamics of this system at the limit 0. The degeneracy in the dynamics ismanifested by a 'bifurcation from infinity' where a bifurcation point isgenerated at high energies, as perturbation of a state of infiniteenergy. Another (nondegenerate) bifurcation point is generated close tothe point of exact 1:1 internal resonance between the linear andnonlinear oscillators. The degenerate bifurcation structure can bedirectly attributed to the high degeneracy of the uncoupled system inthe limit 0, whose linearized structure possesses a double zero, and aconjugate pair of purely imaginary eigenvalues. First we construct localanalytical approximations to the NNMs in the neighborhoods of thebifurcation points and at other energy ranges of the system. Then, we`connect' the local approximations by global approximants, and identifyglobal branches of NNMs where unstable and stable mode and inverse modelocalization between the linear and nonlinear oscillators take place fordecreasing energy.