Trapping vibratory energy of main linear structures by coupling light systems with geometrical and material non-linearities

Cubic potential and hysteresis behavior (Bouc–Wen type) of a non-linear energy sink are used to localize the vibratory energy of a linear structure. A general methodology is presented to deal with time evolutionary energy exchanges between two oscillators. Invariant manifold of the system and its stability borders are detected at fast time scale while traced equilibrium and singular points at slow time scale let us predict possible behaviors of the system during its pseudo-stationary regime(s). The paper is followed by an example that considers the Dahl model for representing the hysteresis behavior of the non-linear energy sink. All analytical developments and results are compared with those obtained by direct integration of system equations. Obtained analytical developments can be endowed for designing non-linear energy sink devices with hysteresis behavior to localize vibratory energy of main structures for the aim of passive control, energy harvesting and/or both of them.     

Non-linear longitudinal vibrations of non-slender piezoceramic rods

Characteristic non-linear effects can be observed, when piezoceramics are excited using weak electric fields. In experiments with longitudinal vibrations of piezoceramic rods, the behavior of a softening Duffing-oscillator including jump phenomena and multiple stable amplitude responses at the same excitation frequency and voltage is observed. Another phenomenon is the decrease of normalized amplitude responses with increasing excitation voltages. For such small stresses and weak electric fields as applied in the experiments, piezoceramics are usually described by linear constitutive equations around an operating point in the butterfly hysteresis curve. The non-linear effects under consideration were, e.g. observed and described by Beige and Schmidt [1,2], who investigated longitudinal plate vibrations using the piezoelectric 31-effect. They modeled these non-linearities using higher order quadratic and cubic elastic and electric terms. Typical non-linear effects, e.g. dependence of the resonance frequency on the amplitude, superharmonics in spectra and a non-linear relation between excitation voltage and vibration amplitude were also observed e.g. by von Wagner et al. [3] in piezo-beam systems. In the present paper, the work is extended to longitudinal vibrations of non-slender piezoceramic rods using the piezoelectric 33-effect. The non-linearities are modeled using an extended electric enthalpy density including non-linear quadratic and cubic elastic terms, coupling terms and electric terms. The equations of motion for the system under consideration are derived via the Ritz method using Hamilton's principle. An extended kinetic energy taking into consideration the transverse velocity is used to model the non-slender rods. The equations of motion are solved using perturbation techniques. In a second step, additional dissipative linear and non-linear terms are used in the model. The non-linear effects described in this paper may have strong influence on the relation between excitation voltage and response amplitude whenever piezoceramic actuators and structures are excited at resonance.     

Usefulness of passive non-linear energy sinks in controlling galloping vibrations

The suppression of vibration amplitudes of an elastically-mounted square prism subjected to galloping oscillations by using a non-linear energy sink is investigated. The non-linear energy sink consists of a secondary system with linear damping and non-linear stiffness. A representative model that couples the transverse displacement of the square prism and the non-linear energy sink is constructed. A linear analysis is performed to determine the impacts of the non-linear energy sink parameters (mass, damping, and stiffness) on the coupled frequency and onset speed of galloping. It is demonstrated that increasing the damping of the non-linear energy sink can result in a significant increase in the onset speed of galloping. Then, the normal form of the Hopf bifurcation is derived to identify the type of instability and to determine the effects of the non-linear energy sink stiffness on the performance of the aeroelastic system near the bifurcation. The results show that the non-linear energy sink can be efficiently implemented to significantly reduce the galloping amplitude of the square prism. It is also shown that the multiple stable responses of the coupled aeroelastic system are obtained as well as the periodic responses, which are dependent on the considered non-linear energy sink parameters.     

Vibration of a Non-Linear Self-Excited System with Two Degrees of Freedom under External and Parametric Excitation

Vibration analysis of a non-linear parametrically self-excited system with two degrees of freedom under harmonic external excitation was carried out in the present paper. External excitation in the main parametric resonance area was assumed in the form of standard force excitation or inertial excitation. Close to the first and second free vibrations frequency, the amplitudes of the system vibrations and the width of synchronization areas were determined. Stability of obtained periodic solutions was investigated. The analytical results were verified and supplemented with the effects of digital and analog simulations.     

Dynamics of a linear oscillator connected to a small strongly non-linear hysteretic absorber

The present investigation deals with the dynamics of a two-degrees-of-freedom system which consists of a main linear oscillator and a strongly non-linear absorber with small mass. The non-linear oscillator has a softening hysteretic characteristic represented by a Bouc-Wen model. The periodic solutions of this system are studied and their calculation is performed through an averaging procedure. The study of non-linear modes and their stability shows, under specific conditions, the existence of localization which is responsible for a passive irreversible energy transfer from the linear oscillator to the non-linear one. The dissipative effect of the non-linearity appears to play an important role in the energy transfer phenomenon and some design criteria can be drawn regarding this parameter among others to optimize this energy transfer. The free transient response is investigated and it is shown that the energy transfer appears when the energy input is sufficient in accordance with the predictions from the non-linear modes. Finally, the steady-state forced response of the system is investigated. When the input of energy is sufficient, the resonant response (close to non-linear modes) experiences localization of the vibrations in the non-linear absorber and jump phenomena.     

Damping for large-amplitude vibrations of plates and curved panels,Part 1: Modeling and experiments

Theoretical and experimental non-linear vibrations of thin rectangular plates and curved panels subjected to out-of-plane harmonic excitation are investigated. Experiments have been performed on isotropic and laminated sandwich plates and panels with supported and free boundary conditions. A sophisticated measuring technique has been developed to characterize the non-linear behavior experimentally by using a Laser Doppler Vibrometer and a stepped-sine testing procedure. The theoretical approach is based on Donnell's non-linear shell theory (since the tested plates are very thin) but retaining in-plane inertia, taking into account the effect of geometric imperfections. A unified energy approach has been utilized to obtain the discretized non-linear equations of motion by using the linear natural modes of vibration. Moreover, a pseudo arc-length continuation and collocation scheme has been used to obtain the periodic solutions and perform bifurcation analysis. Comparisons between numerical simulations and the experiments show good qualitative and quantitative agreement. It is found that, in order to simulate large-amplitude vibrations, a damping value much larger than the linear modal damping should be considered. This indicates a very large and non-linear increase of damping with the increase of the excitation and vibration amplitude for plates and curved panels with different shape, boundary conditions and materials.     

Response regimes in forced system with non-linear energy sink: quasi-periodic and random forcing

The system under investigation comprises a linear oscillator coupled to a non-linear energy sink (NES) under quasi-periodic forcing in the regime of 1:1:1 resonance. Interaction of the quasi-periodic excitation with the strongly modulated response (SMR) regime is studied in detail both analytically and numerically. Theoretical study developed in the paper allows establishing the threshold value for the amplitude of modulation beyond which SMR regime is excited. This phenomenon is of great practical use since applying the quasi-periodic excitation beyond the threshold results in elimination of possible undesired regimes causing high-amplitude oscillations of the main structure. Bifurcations of the SMR caused by quasi-periodic excitation were analyzed with the help of semi-analytical procedure based on two-dimensional maps. Numerical evidences for exciting the strongly modulated bursts in the response by a random, quasi-periodic narrow-band excitation are also provided. Fairly good correspondence was observed between analytical model and numerical simulations.     

Non-linear vibrations and frequency response analysis of piezoelectrically driven microcantilevers

Microcantilevers have recently received widespread attentions due to their extreme applicability and versatility in both biological and non-biological applications. Along this line, this paper undertakes the non-linear vibrations of a piezoelectrically driven microcantilever beam as a common configuration in many scanning probe microscopy and nanomechanical cantilever biosensor systems. A part of the microcantilever beam surface is covered by a piezoelectric layer (typically ZnO), which acts both as an actuator and sensor. The bending vibrations of the microcantilever beam are studied considering the inextensibility condition and the coupling between electrical and mechanical properties in the piezoelectric materials. The non-linear terms appear in the form of quadratic expression due to presence of piezoelectric layer, and cubic form due to geometrical non-linearities. The Galerkin approximation is then utilized to discretize the equations of motion. In addition, the method of multiple scales is applied to arrive at the closed form solution for the fundamental natural frequency of the system. An experimental setup consisting of a commercial piezoelectric microcantilever attached on the stand of a state-of-the-art microsystem analyzer for non-contact vibration measurement is utilized to verify the theoretical developments. It is found that the experimental results and theoretical findings are in good agreement, which demonstrates that the non-linear modeling framework could provide a better dynamic representation of the microcantilever than the previous linear models. Due to microscale nature of the system, excitation amplitude plays an important role since even a small change in the amplitude of excitation can lead to significant vibrations and frequency shift.     

Non-linear dynamic responses of elastic two-story structures with partially filled liquid tanks

This paper deals with the non-linear vibrations of an elastic two-story structure with two liquid tanks installed under horizontal harmonic excitation. The influence of the configuration of the two rectangular tanks on the response of the structure is investigated. In the theoretical analysis, Galerkin's method is applied to derive the equations of motion for the structure and the modal equations for sloshing, while considering the non-linear liquid forces. Then, van der Pol's method is used to determine the frequency response curves. Three cases are investigated: In the first case two tanks are installed, one on the top and one on the second story of the structure, in the second case one tank is installed on top, and in the third case two tanks are installed on top. The theoretical results of the first case are compared with those of the second and third cases. In the numerical calculations, it is found that Hopf bifurcations occur near the tuning frequency and then amplitude modulated motion appears in both the first and third cases. It is thus concluded that multiple tanks yield less effectiveness in suppressing the vibrations of the structure. The experimental data confirm the validity of the theoretical results for the first and third cases.     

Energy Based Stochastic Estimation of Nonlinear Oscillators with Parametric Random Excitation

A new energy-based system identification method is developed, applicable in situations where the dynamic response of a structure is measurable but the excitation is unmeasurable and describable only in terms of a stochastic process. It is shown that, in the case of a non-linear single degree of freedom system subjected to purely parametric, non-white random excitation, the power spectrum of the excitation can be identified through an estimation of the diffusion coefficient relating to the energy envelope of the response process. Through an estimation of the drift coefficient an identification of the system damping is also possible. The method is validated through application to simulated data relating to a Duffing oscillator with non-linear damping.     

Dynamical behavior of a mechanical system including Saint-Venant component coupled to a non-linear energy sink

Multi-scale vibratory energy exchange between a main oscillator including Saint-Venant term and a cubic non-linear energy sink is studied. Analytically obtained invariant manifold of the system at a fast time scale and detected fixed points and/or fold singularities at a first slow time scale let us predict and explain different regimes that the system may face during the energy exchange process. The paper will be accompanied by some numerical results confirming our analytical predictions.     

Effects of a non-linear energy sink (NES) on vortex-induced vibrations of a circular cylinder

We investigate in detail the passive control of vortex-induced vibrations of a freely oscillating circular cylinder using a non-linear energy sink consisting of a secondary system having linear damping and an essential non-linear cubic stiffness. The loads on the cylinder are calculated using a direct numerical simulation of the incompressible flow over the cylinder using a parallel computational fluid dynamics code. A strongly coupled fluid structure control numerical model is used to determine the responses of the cylinder and the sink as well as the flow. We vary the sink parameters (mass and damping) and determine their effects on the response of the coupled system. We find multiple stable responses of the coupled system for different mass ratios and damping coefficient of the sink, depending on the initial conditions.     

Non-linear vibrations of doubly curved shallow shells

Large amplitude (geometrically non-linear) vibrations of doubly curved shallow shells with rectangular base, simply supported at the four edges and subjected to harmonic excitation normal to the surface in the spectral neighbourhood of the fundamental mode are investigated. Two different non-linear strain-displacement relationships, from the Donnell's and Novozhilov's shell theories, are used to calculate the elastic strain energy. In-plane inertia and geometric imperfections are taken into account. The solution is obtained by Lagrangian approach. The non-linear equations of motion are studied by using (i) a code based on arclength continuation method that allows bifurcation analysis and (ii) direct time integration. Numerical results are compared to those available in the literature and convergence of the solution is shown. Interaction of modes having integer ratio among their natural frequencies, giving rise to internal resonances, is discussed. Shell stability under static and dynamic load is also investigated by using continuation method, bifurcation diagram from direct time integration and calculation of the Lyapunov exponents and Lyapunov dimension. Interesting phenomena such as (i) snap-through instability, (ii) subharmonic response, (iii) period doubling bifurcations and (iv) chaotic behaviour have been observed.     

One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models

The mechanism of cable end angle-variation induced oscillations in the non-linear interactions between beams and cables in stayed-systems is first explained by a proposed analytical model. It is then verified by both experimental and finite element models. The non-linear interaction maximizes its effects for cable oscillations when inherent quadratic coupling between local and global modes produces energy transfer from low to high frequency vibrations by means of a one-to-two global-local autoparametric resonance. The response of the analytical model is fully described using a continuation method applied directly to the reduced two degree of freedom discrete model showing that, for a selected one-to-two global-local resonant system, primary harmonic excitation of the global mode produces large oscillations of the local mode at twice the excitation frequency. Detailed comparisons between the responses of the analytical model, experimental results and finite element simulations show excellent agreement both in the qualitative behaviour and in the calculated/measured response amplitudes.     

Broadband passive targeted energy pumping from a linear dispersive rod to a lightweight essentially non-linear end attachment

We examine non-linear resonant interactions between a damped and forced dispersive linear finite rod and a lightweight essentially non-linear end attachment. We show that these interactions may lead to passive, broadband and on-way targeted energy flow from the rod to the attachment, which acts, in essence, as non-linear energy sink (NES). The transient dynamics of this system subject to shock excitation is examined numerically using a finite element (FE) formulation. Parametric studies are performed to examine the regions in parameter space where optimal (maximal) efficiency of targeted energy pumping from the rod to the NES occurs. Signal processing of the transient time series is then performed, employing energy transfer and/or exchange measures, wavelet transforms, empirical mode decomposition and Hilbert transforms. By computing intrinsic mode functions (IMFs) of the transient responses of the NES and the edge of the rod, and examining resonance captures that occur between them, we are able to identify the non-linear resonance mechanisms that govern the (strong or weak) one-way energy transfers from the rod to the NES. The present study demonstrates the efficacy of using local lightweight non-linear attachments (NESs) as passive broadband energy absorbers of unwanted disturbances in continuous elastic structures, and investigates the dynamical mechanisms that govern the resonance interactions influencing this passive non-linear energy absorption.     

On the accuracy of the multiple scales method for non-linear vibrations of doubly curved shallow shells

Non-linear free and forced vibrations of doubly curved isotropic shallow shells are investigated via multi-modal Galerkin discretization and the method of multiple scales. Donnell’s non-linear shallow shell theory is used and it is assumed that the shell is simply supported with movable edges. By deriving two different forms of the stress function, the equations of motion are reduced to a system of infinite non-linear ordinary differential equations with quadratic and cubic non-linearities. A quadratic relation between the excitation and the fundamental frequency is considered and it is shown that, although in case of hardening non-linearities the results resemble those found via numerical integration or continuation softwares, in case of softening non-linearity the solution breaks down as the amplitude becomes larger than the thickness. Results reveal that, expressing the relation between the excitation and fundamental frequency in this form, which was considered by many researchers as a useful tool in analyzing strong non-linear oscillators, yields in spurious results when the non-linearity becomes of softening type.     

Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells

A geometrically non-linear theory is developed for shells of generic shape allowing for third-order thickness and shear deformation and rotary inertia by using eight parameters; geometric imperfections are also taken into account. The geometrically non-linear strain–displacement relationships are derived retaining full non-linear terms in all the 8 parameters, i.e. in-plane and transverse displacements, rotations of the normal and thickness deformation parameters; these relationships are presented in curvilinear coordinates, ready to be implemented in computer codes. Higher order terms in the transverse coordinate are retained in the derivation so that the theory is suitable also for thick laminated shells. Three-dimensional constitutive equations are used for linear elasticity. The theory is applied to circular cylindrical shells complete around the circumference and simply supported at both ends to study initially static finite deformation. Both radially distributed forces and displacement-dependent pressure are used as load and results for different shell theories are compared. Results show that a 6 parameter non-linear shell theory is quite accurate for isotropic shells. Finally, large-amplitude forced vibrations under harmonic excitation are investigated by using the new theory and results are compared to other available theories. The new theory with non-linearity in all the 8 parameters is the only one to predict correctly the thickness deformation; it works accurately for both static and dynamics loads.     

On the coupling of non-linear normal modes

The phenomenon of internal resonance is known as the exchange of energy between the modes and the existence of coupled-mode response under a single-mode excitation. This phenomenon is observed whenever a non-linear normal mode loses its stability, called the modal coupling. The details of modal coupling are formulated in the free vibrations of two-degree-of-freedom systems, and compared with internal resonance. The theory is based on the structural change in Poincaré map due to the stability change of normal modes. It is shown that every change in stability of normal modes gives rise to a pitchfork or a period-doubling bifurcation. The functional form is derived to compute the coupled modes by the method of harmonic balance. Examples are given to describe the procedure of stability analysis of non-linear normal modes, to compute the coupled modes, and then to demonstrate that results of internal resonances can be derived by model coupling. Other examples are given to demonstrate that the results of some modal couplings cannot be obtained by internal resonances.     

Non-linear dynamics of wind turbine wings

The paper deals with the formulation of non-linear vibrations of a wind turbine wing described in a wing fixed moving coordinate system. The considered structural model is a Bernoulli-Euler beam with due consideration to axial twist. The theory includes geometrical non-linearities induced by the rotation of the aerodynamic load and the curvature, as well as inertial induced non-linearities caused by the support point motion. The non-linear partial differential equations of motion in the moving frame of reference have been discretized, using the fixed base eigenmodes as a functional basis. Important non-linear couplings between the fundamental blade mode and edgewise modes have been identified based on a resonance excitation of the wing, caused by a harmonically varying support point motion with the circular frequency ω. Assuming that the fundamental blade and edgewise eigenfrequencies have the ratio of ω₂/ω₁?2, internal resonances between these modes have been studied. It is demonstrated that for ω/ω₁?0.66,1.33,1.66 and 2.33 coupled periodic motions exist brought forward by parametric excitation from the support point in addition to the resonances at ω/ω₁?1.0 and ω/ω₂?1.0 partly caused by the additive load term.     

Non-linear vibrations of cantilever beams with feedback delays

A comprehensive investigation of the effect of feedback delays on the non-linear vibrations of a piezoelectrically actuated cantilever beam is presented. In the first part of this work, we examine the linear and non-linear free responses of a beam subjected to a delayed-acceleration feedback. We show that the trivial solution loses stability via a Hopf bifurcation leading to limit-cycle oscillations. We analyze the stability of the dynamic response in the postbifurcation, close to the stability boundaries by examining the nature of the Hopf bifurcation and away from the stability boundaries by using the method of harmonic balance and Floquet theory. We find that, increasing the gain for certain feedback delays may culminate in quasiperiodic and chaotic oscillations of the beam.In the second part, we analyze the effect of feedback delays on a beam subjected to a harmonic base excitations. We find that the nature of the forced response is largely defined by the stability of the trivial solutions of the unforced response. For stable trivial solutions (i.e., inside the stability boundaries of the trivial solutions), the homogeneous response emanating from the feedback diminishes, leaving only the particular solution resulting from the external excitation. In this case, delayed feedback acts as a vibration absorber. On the other hand, for unstable trivial solutions, the response contains two co-existing frequencies. Depending on the excitation amplitude and the commensurability of the delayed-response frequency to the excitation frequency, the response is either periodic or quasiperiodic.