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.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

Non-local effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation

Carbon nanotubes (CNTs) based NEMS with electrostatic sensing/actuation may be employed as sensors, in situations where it is fundamental to understand their dynamic behaviour. Due to displacements that are large in comparison with the thickness and to the non-linearity of the electrostatic force, these CNT based NEMS operate in the non-linear regime. The knowledge of the modes of vibration of a CNT provides a picture of what one may expect from its dynamic behaviour not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of CNTs actuated by electrostatic forces are investigated. For that purpose, a p-version finite element type formulation is implemented, leading to ordinary differential equations of motion in the time domain. The formulation takes into account non-local effects, which influence the inertia and the stiffness of CNTs, as well as the electrostatic actuation. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM) and then solved by an arc-length continuation method. Several harmonics are considered in the HBM. The importance of non-local effects, combined with the geometrical non-linearity and with the action of the electrostatic force, is analysed. It is found that different combinations of these effects can result in alterations of the natural frequencies, variations in the degrees of softening or hardening, changes in the frequency content of the free vibrations, and alterations in the mode shapes of vibration. It is furthermore found that the small scale, here represented by the non-local theory, has an effect on interactions between the first and higher order modes which are induced by the geometrical and material non-linearities of the system.     

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

Low-Dimensional Galerkin Models for Nonlinear Vibration and Instability Analysis of Cylindrical Shells

This paper discusses the derivation of discrete low-dimensional models for the non-linear vibration analysis of thin shells. In order to understand the peculiarities inherent to this class of structural problems, the non-linear vibrations and dynamic stability of a circular cylindrical shell subjected to dynamic axial loads are analyzed. This choice is based on the fact that cylindrical shells exhibit a highly non-linear behavior under both static and dynamic axial loads. Geometric non-linearities due to finite-amplitude shell motions are considered by using Donnell’s nonlinear shallow shell theory. A perturbation procedure, validated in previous studies, is used to derive a general expression for the non-linear vibration modes and the discretized equations of motion are obtained by the Galerkin method. The responses of several low-dimensional models are compared. These are used to study the influence of the modelling on the convergence of critical loads, bifurcation diagrams, attractors and large amplitude responses of the shell. It is shown that rather low-dimensional and properly selected models can describe with good accuracy the response of the shell up to very large vibration amplitudes.     

Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

The dynamics of a non-linear electro-magneto-mechanical coupled system is addressed. The non-linear behavior arises from the involved coupling quadratic non-linearities and it is explored by relying on both analytical and numerical tools. When the linear frequency of the circuit is larger than that of the mechanical oscillator, the dynamics exhibits slow and fast time scales. Therefore the mechanical oscillator forced (actuated) via harmonic voltage excitation of the electric circuit is analyzed; when the forcing frequency is close to that of the mechanical oscillator, the long term damped dynamics evolves in a purely slow timescale with no interaction with the fast time scale. We show this by assuming the existence of a slow invariant manifold (SIM), computing it analytically, and verifying its existence via numerical experiments on both full- and reduced-order systems. In specific regions of the space of forcing parameters, the SIM is a complicated geometric object as it undergoes folding giving rise to hysteresis mechanisms which create a pronounced non-linear resonance phenomenon. Eventually, the roles played by the electro-magnetic and mechanical components in the resulting complex response, encompassing bifurcations as well as possible transitions from regular to chaotic motion, are highlighted by means of Poincaré sections.     

Effect of non-linearities in elastomeric material dampers on torsional vibration control

The effect of non-linearities in clastomeric-material dampers used to quiet torsional oscillations of internal-combustion engines shafts is investigated. The method of multiple scales is used to solve the equations for the case of primary resonance. Steady-state solutions are obtained and their stability examined. A relationship that relates the elastomer properdes (for stable solutions) and other system parameters is found. The steady-state response shows a softening behavior of the system although the actual material non-linearity is of the hardening type.     

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 vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method

Geometrically non-linear forced vibrations of a shallow circular cylindrical panel with a complex shape, clamped at the edges and subjected to a radial harmonic excitation in the spectral neighborhood of the fundamental mode, are investigated. Both Donnell and the Sanders–Koiter non-linear shell theories retaining in-plane inertia are used to calculate the elastic strain energy. The discrete model of the non-linear vibrations is build using the meshfree technique based on classic approximate functions and the R-function theory, which allows for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries; Chebyshev orthogonal polynomials are used to expand shell displacements. A two-step approach is implemented in order to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for expanding the non-linear displacements. Lagrange approach is applied to obtain a system of ordinary differential equations on both steps. Different multimodal expansions, having from 15 up to 35 generalized coordinates associated with natural modes, are used to study the convergence of the solution. The pseudo-arclength continuation method and bifurcation analysis are applied to study non-linear equations of motion. Numerical responses are obtained in the spectral neighborhood of the lowest natural frequency; results are compared to those available in the literature. Internal resonances are also detected and discussed.     

The fundamental solution in the theory of shallow shells

A series representation for the fundamental solution of the shallow shell equations is obtained by means of a plane-wave decomposition of the Dirac δ-function. From this solution we can produce the singular solutions which correspond to concentrated forces, couples and thermal hot spots applied to a shallow shell with an arbitrary quadratic middle surface. The solutions converge for the entire range of the Gaussian curvature. Numerical results are presented for the case of a concentrated normal force acting on infinite shells having positive, zero or negative Gaussian curvature.     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

Approximate Methods for the Linear and Nonlinear Analysis of Plates and Shallow Shells

Abstract

Berger's equations for the large amplitude deformation of membranes are used to produce a simple approximate expression for the large amplitude deflection of plates. The deformation of shallow shells is also considered and two approximate methods are outlined. Several important problems are discussed, the obtained solution being in good agreement with both experimental data and other approximate results. The main advantage of this technique is its ease of application, as it requires comparatively little computational work. A simple approximate formula for computing the fundamental frequency of a vibrating shallow shell is also presented and is shown to yield very accurate values in the case of a shallow dome and a rectangular panel.     

Vibrations of rod-rigid element systems with a local non-linearity

The paper deals with vibrations of systems consisting of non-coaxial rods connected by rigid bodies and of a local non-linearity. The motion of the rods is described by classical wave equation and the solution of the d’Alembert type is applied in the study. This leads to solving ordinary differential equations with a retarded argument. The local non-linearity is described through irrational functions and in a special case it includes the polynomial of the third degree. Detailed considerations are given for a system consisting of three rods and two rigid bodies. In numerical analysis non-linear effects are discussed. The results concerning harmonic vibrations are presented for the local non-linearities having characteristics of a soft type as well as of a hard type.     

Robust optimization of the non-linear behaviour of a vibrating system

In this paper, one proposes to optimize the vibratory behaviour of an absorber of vibration related to a system subjected to a harmonic load, in the presence of uncertainties on the design parameters. The total system is modeled by two degrees of freedom (2 dof) with a shock absorber and a generalized non-linear stiffness. The resolution is carried out in the temporal field according to a traditional diagram.Two cases of non-linearity were considered. In the first case, one is interested in the study of the system comprising a combination of the two generalized non-linearities of quadratic and cubic type of stiffness and damping. The second case relates to a non-linearity of non-whole power (in this paper 1.5), combined with the cubic case. It is a question of seeking the optimal responses envelopes of the deterministic and stochastic case and this for the non-linear displacements, phases and forces.The multi-objective optimization step consists in seeking the first Pareto front of several linear and non-linear objective functions by using a genetic algorithm of type “NSGA” (Non-dominated Sorting Genetic Algorithm).The design parameters are: mass, linear and non-linear stiffness and damping of the absorber. To obtain solutions presenting a good compromise between optimality and the robustness, one introduces uncertainties on these design parameters. The robustness is then defined by the dispersion of the parameters (definite as the ratio: mean value/standard deviation) and it is introduced as additional objective function.The use of the clusters resulting from the Self-Organizing Maps of Kohonen (SOM) is also suggested for a rational management of the design space. A study of sensitivity a posteriori can be exploited in order to eliminate the non-significant design parameters.     

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.     

A stability in the large investigation for a non-linear second order two-degree-of-freedom system with periodic excitation

An extension to an algorithm due to Simpson has been developed for the analysis of a non-linear second order two-degree-of-freedom system with external periodic excitation. The form of equations considered arises from the study of mechanical systems with a single concentrated weak non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. The system considered is such that in the absence of external excitation, it possesses a stable equilibrium point and an unstable limit cycle arising from a sub-critical Hopf bifurcation. When forcing is applied, the stable equilibrium point may then be replaced by a stable periodic attractor, and the limit cycle by an unstable multi-periodic attractor. The method has been applied to the problem of locating these attractors, and if they exist, of finding the stable attractor's basin of attraction in terms of initial conditions. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution.The method was applied to three non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in those cases where the non-linearity was weak. However, the method would not be expected to give such accurate results if the non-linear effect was more significant. This was illustrated for a case involving the freeplay non-linearity.     

Nonlinear free vibration of reticulated shallow spherical shells taking into account transverse shear deformation

This paper deals with nonlinear free vibration of reticulated shallow spherical shells taking into account the effect of transverse shear deformation. The shell is formed by beam members placed in two orthogonal directions. The nondimensional fundamental governing equations in terms of the deflection, rotational angle, and force function are presented, and the solution for the nonlinear free frequency is derived by using the asymptotic iteration method. The asymptotic solution can be used readily to perform the parameter analysis of such space structures with numerous geometrical and material parameters. Numerical examples are given to illustrate the characteristic amplitude-frequency relation and softening and hardening nonlinear behaviors as well as the effect of transverse shear on the linear and nonlinear frequencies of reticulated shells and plates.     

Non-linear shear vibrations of piezoceramic actuators

A wide range of non-linear effects are observed in piezoceramic materials. For small stresses and weak electric fields, piezoceramics are usually described by linearized constitutive equations around an operating point. However, typical non-linear vibration behavior is observed at weak electric fields near resonance frequency excitations of the piezoceramics. This non-linear behavior is observed in terms of a softening behavior and the decrease of normalized amplitude response with increase in excitation voltage. In this paper the authors have attempted to model this behavior using higher order cubic conservative and non-conservative terms in the constitutive equations. Two-dimensional kinematic relations are assumed, which satisfy the considered reduced set of constitutive relations. Hamilton's principle for the piezoelectric material is applied to obtain the non-linear equation of motion of the piezoceramic rectangular parallelepiped specimen, and the Ritz method is used to discretize it. The resulting equation of motion is solved using a perturbation technique. Linear and non-linear parameters for the model are identified. The results from the theoretical model and the experiments are compared. The non-linear effects described in this paper may have strong influence on the design of the devices, e.g. ultrasonic motors, which utilize the piezoceramics near the resonance frequency excitation.     

Axial instability of rotating rods revisited

For strain sufficiently small such that Hooke's Law is valid, it is shown that only a linear model for axial deformation of rotating rods can be derived. As discussed in the literature, this linear model exhibits an instability when the angular speed reaches a certain critical value. However, unless this linear model is valid for large strain, it is impossible to determine whether this instability really exists; because, as the angular speed is increased, the strain becomes large well short of the critical speed. Next, axial deformation of rotating rods is analyzed using two strain energy functions to model non-linear elastic behavior. The first of these functions is the usual quadratic strain energy function augmented with a cubic term. With this model it is shown that no instability exists if the non-linearity is stiffening (i.e. if the coefficient of the cubic term is positive), although the strain can become large. If the non-linearity is of the softening variety, then the critical angular speed drops as the degree of softening increases. Still, the strains are large enough that, except for rubber-like materials, a non-linear elastic model is not likely to be appropriate. The second strain energy function is based on the square of the logarithmic strain and yields a softening model. It quite accurately models the behavior of certain rubber rods which exhibit the instability within the validated range of elongation.     

Stochastic non-linear response of a flexible rotor with local non-linearities

The effects of uncertainties on the non-linear dynamics response remain misunderstood and most of the classical stochastic methods used in the linear case fail to deal with a non-linear problem. So we propose to take into account of uncertainties into non-linear models, by coupling the Harmonic Balance Method (HBM) and the Polynomial Chaos Expansion (PCE). The proposed method called the Stochastic Harmonic Balance Method (Stochastic-HBM) is based on a new formulation of the non-linear dynamic problem in which not only the approximated non-linear responses but also the non-linear forces and the excitation pulsation are considered as stochastic parameters. Expansions on the PCE basis are performed by passing via an Alternate Frequency Time method with Probabilistic Collocation (AFTPC) for estimating the stochastic non-linear forces in the stochastic domain and the frequency domain. In the present paper, the Stochastic Harmonic Balance Method (Stochastic-HBM) that is applied to a flexible non-linear rotor system, with random parameters modeled as random fields, is presented. The Stochastic-HBM combined with an Alternate Frequency-Time method with Probabilistic Collocation (AFTPC) allows us to solve dynamical problems with non-regular non-linearities in presence of uncertainties. In this study, the procedure is developed for the estimation of stochastic non-linear responses of the rotor system with different regular and non-regular non-linearities. The finite element rotor system is composed of a shaft with two disks and two flexible bearing supports where the non-linearities are due to a radial clearance or a cubic stiffness. A numerical analysis is performed to analyze the effect of uncertainties on the non-linear behavior of this rotor system by using the Stochastic-HBM. Furthermore, the results are compared with those obtained by applying a classical Monte-Carlo simulation to demonstrate the efficiency of the proposed methodology.