Vibrations Of Elastic Structures Research Articles

PARAMETRIC BENDING VIBRATIONS OF ELASTIC STRUCTURES WITH VARIABLE SECTIONS ON VIBRATION MOUNTINGS

The article is devoted to the study of the oscillatory motion of an elastic structure with variable sections on rolling bearings with straightened surfaces, which simulates a number of technical solutions, which has received its practical embodiment in the problem of ensuring the seismic resistance of building structures and vibration isolation of massive bodies. Equations of motion of an elastic structure with variable sections on rolling bearings bounded by high-order surfaces of revolution are obtained. The resonant modes of parametric vibrational motion of an elastic structure with variable sections are investigated using the Ritz variational method. In the resonant zone of the oscillatory movements of the base, the movement of the elastic structure will be small, and when the natural frequencies of the elastic structure coincide with the frequency of the disturbance, the phenomenon of resonance will appear. The amplitude of the parametric disturbance of the resonant zones of bending vibrations of elastic structures expands with increasing The amplitude-frequency characteristics of parametric oscillations depend on the parameters of the structure. An increase in the parameters of the base of wedge-shaped elastic structures leads to a shift of the resonance curves towards an increase in the disturbance frequencies

Vibration suppression of an elastic beam with boundary inerter-enhanced nonlinear energy sinks

Nonlinear vibration absorbers have been widely used for vibration suppression of elastic structures, but they were usually placed within the structures. However, designing such a vibration damping device within an engineering structure is possibly difficult. In this paper, an inertial nonlinear energy sinks (NES) is mounted on the boundaries of the elastic beam to suppress its vibration. Although this vibration suppression approach is more in line with engineering requirements, it introduces nonlinear oscillators at boundaries. This brings certain difficulties to the structural vibration analysis and the optimal absorber design. An approximate analytical approach for the steady-state response is developed in this work and verified by numerical solutions. The comparison with the uncontrolled system demonstrates the high-efficiency vibration suppression of the inertial NES installed on the boundary. Besides, the optimization of the NES parameters is performed. Resonance amplitude of the elastic structure can be reduced by 98% with the optimized NES. In summary, this paper proposes a novel approach to suppress the bending vibration of elastic structures through boundary NESs. The vibration reduction effect is very significant, and it is more feasible to implement. Therefore, this work is helpful to study the vibration of elastic structures with nonlinear boundaries and to promote the application of nonlinear vibration absorbers. A boundary damping strategy is proposed to solve the problem of the difficulty of installing the vibration absorber in engineering. Introduce an inertial nonlinear energy sinks (NES) to reduce the weight of the shock absorber. The nonlinear boundary is separated and merged into the elastic beam vibration control equation to realize the decoupling of the continuum model. Resonance peak of the elastic structure can be reduced by 98% with the optimized NES. This work is helpful to study the vibration of elastic structures with nonlinear boundaries and to promote the application of nonlinear vibration absorbers.

On possibility of local suppression of natural vibrations of elastic structures under shock effects

Manifestations of shock effects in the form of subsequent natural oscillations of structures for various purposes can be different in intensity, in directions of oscillatory processes (forms of natural oscillations) and in frequencies involved in oscillatory processes. At the same time, the tasks of suppressing or reducing the intensity of oscillations in certain specified directions have obvious relevance. Suppression of forced stationary oscillations in certain specified directions (obtaining zero amplitudes of oscillations in these directions) is possible by forming dynamic quenching effects implemented as the systems that satisfy a number of formalized requirements for the design parameters of such systems. In earlier papers of the authors it was shown, that the realization of such effects in the processes of natural oscillations is associated with the formation of conditions for the multiplicity of natural oscillations of elastic systems. In the proposed paper the conditions for the formation of the multiplicity of natural frequencies of multiply connected finite-dimensional systems of small dimension are considered, reflecting the specific manifestation of the multiplicity of natural oscillation frequencies of finite-dimensional dynamic systems of arbitrary dimension, the properties of such systems are applied to the suppression of natural oscillations in specified directions.

Nonlinear vibrations of a slightly curved beam with nonlinear boundary conditions

The existing studies of nonlinear vibration of elastic structures are usually focused on straight structures with homogeneous linear boundaries. Differently, this paper investigates the nonlinear transverse vibrations of a slightly curved beam with nonlinear boundary conditions. By using the generalized Hamilton's principle, the governing equation with geometric nonlinearity is obtained for the dynamics of the curved beam. A method of dealing with nonlinear boundaries is proposed, which is considered as a nonlinear concentrated force at the boundary. The normal modes and natural frequencies of the curved beam are determined using two different hypothetical modes based on the derived system. The harmonic balance method in combination with the pseudo arc-length method is employed to obtain the primary resonance response and 1/2 super-harmonic resonance response of the slightly curved beam. It is found that the initial curvature plays a significant role in the characteristics of the nonlinear vibrations of the curved beam. With an increase of the initial curvature, the nonlinear characteristics of softening and hardening types can coexist in the steady-state amplitude-frequency response. Moreover, the results show that the initial curvature can induce 1/2 super-harmonic resonance. Furthermore, it is also found that the nonlinear boundary has a significant influence on the nonlinear vibration of the curved structure. Therefore, the obtained results provide useful information for further studying the nonlinear vibrations of the curved beam with nonlinear and time-dependent boundary conditions.

A polynomial chaos‐based approach to risk‐averse piezoelectric control of random vibrations of beams

SummaryThis paper proposes a risk‐averse formulation for the problem of piezoelectric control of random vibrations of elastic structures. The proposed formulation, inspired by the notion of risk aversion in economy, is applied to the piezoelectric control of a Bernoulli‐Euler beam subjected to uncertainties in its input data. To address the high computational burden associated to the presence of random fields in the model and the discontinuities involved in the cost functional and its gradient, a combination of a nonintrusive anisotropic polynomial chaos approach for uncertainty propagation with a Monte Carlo sampling method is proposed. In the first part, the well‐posedness of the control problem is established by proving the existence of optimal controls. In the second part, an adaptive gradient‐based method is proposed for the numerical resolution of the problem. Several experiments illustrate the performance of the proposed approach and the significant differences that may occur between the classical deterministic formulation of the problem and its stochastic risk‐averse counterpart.

Instance of hidden instability traps in intermittent transition of moving masses along a flexible beam

The problem of an elastic beam under the periodic loading of successive moving masses is investigated as a pragmatic case for studying dynamic stability of linear time-varying systems. This model serves to highlight the odds of multi-solutions coexistence, a form of hidden instability which reveals dangerous as it may be precipitated by the slightest disturbance or variation in the model. Since no engineering model perfectly represents a physical system, such situations for which Floquet theory naively predicts stability are potentially inevitable. The harmonic balancing method is used in order to thoroughly explore the stability diagrams for detecting these instability gaps. Although this phenomenon has also been described in other physical systems, it has not been addressed for beam–moving mass systems. This result may find particular importance in applications involving self-induced vibrations of elastic structures and hence also appears of practical relevance.

DGFEM for dynamical systems describing interaction of compressible fluid and structures

The paper is concerned with the numerical solution of flow-induced vibrations of elastic structures. The dependence on time of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian–Eulerian) formulation of the compressible Navier–Stokes equations. The deformation of the elastic body, caused by aeroelastic forces, is described by the linear dynamical elasticity equations. These two systems are coupled by transmission conditions. The flow problem is discretized by the discontinuous Galerkin finite element method (DGFEM) in space and by the backward difference formula (BDF) in time. The structural problem is discretized by conforming finite elements and the Newmark method. The fluid–structure interaction is realized via weak or strong coupling algorithms. The developed technique is tested by numerical experiments and applied to the simulation of vibrations of vocal folds during phonation onset.

Nonlinear Parametric Vibrations of Elastic Structures Containing Two Cylindrical Liquid-Filled Tanks

Nonlinear vibrations of elastic structures containing two cylindrical liquid-filled tanks subjected to vertical harmonic excitation are investigated. When a 2:1:1 ratio of internal resonance is satisfied between the natural frequencies of the structure and the sloshing in the two liquid tanks, nonlinear parametric vibrations can be observed. The equation of motion for the structure is derived considering the nonlinearity of the hydrodynamic force, and the modal equations for sloshing are derived by using Galerkin's method. Then, frequency response curves are determined by using van der Pol's method. The respective influences of the liquid levels and the deviation of the internal resonance ratio 2:1:1 on the frequency response curves are discussed. The results of the response curves differed from those obtained in the case where a structure had only one liquid-filled tank installed. It is found that two types of vibrations may occur depending on the excitation frequency and the liquid levels: a single-mode solution (sloshing occurs in either tank) and a double-mode solution (sloshing occurs in both tanks). Amplitude and phase modulated motion appears in the single-mode solution when some deviation of the internal resonance ratio exists. The validity of the theoretical results was verified by the experiments.

Nonlinear Vibrations of Elastic Structures Containing a Cylindrical Liquid Tank under Vertical Excitation

This study investigates the nonlinear vibrations of an elastic structure, with a liquid-filled cylindrical tank, which is subjected to a vertical sinusoidal excitation. This structure-tank system behaves as an autoparametric system. Modal equations governing the coupled motions of the structure and liquid sloshing are derived when the natural frequency of the structure is equal to twice the natural frequency of the first axisymmetric sloshing mode. Van der Pol's method is applied to the modal equations to determine the theoretical resonance curves. The theoretical results can be concluded as follows: (1) As the liquid level decreases, the resonance curve for the liquid sloshing changes from a soft spring type to a hard spring type. (2) The structure's resonance curve flattens out at small amplitude when the liquid level is appropriate. (3) Amplitude-modulated motions appear for the negative and positive values of the internal resonance ratio's deviation (the detuning parameter) in the high and low liquid levels, respectively. (4) Furthermore, the results of the bifurcation analysis, Poincare maps and Lyapunov exponents reveal that amplitude-modulated motions and chaotic oscillations can occur in the system. In experiments, the theoretical resonance curves were quantitatively in agreement with the experimental data.

Autoparametric resonances in elastic structures carrying two rectangular tanks partially filled with liquid

Nonlinear vibrations of elastic structures carrying two liquid-filled tanks under vertical harmonic excitation are investigated. A 2:1:1 ratio of internal resonance is satisfied between the natural frequency of the structure and the lowest natural frequencies of sloshing in two rectangular liquid tanks. The theoretical analysis here is restricted to two-dimensional flow and three-dimensional Faraday waves are not considered. The finite fluid depths are also assumed. Modal equations of motion are derived by using Galerkin's method, taking into account the nonlinearity of the hydrodynamic force. Then, frequency response curves are determined by using van der Pol's method. The respective influences of the liquid levels and the deviation of the internal resonance ratio (i.e., the mistuning) on the frequency response curves are demonstrated. As a result, we have obtained results different from those obtained in the case of a structure with a single liquid-filled tank. It is found that two types of vibrations can occur depending on the liquid levels and the excitation frequency: a single-mode solution (sloshing in either tank) and a double-mode solution (sloshing in both tanks). Amplitude- and phase-modulated motion including chaotic vibrations appears on the branches of the frequency response curves, which correspond to the single mode solution, when some deviation of the internal resonance ratio exists. The validity of the theoretical results was verified by the experiments.

Boundary stabilization of nonlinear vibrations of a flexible structure in a bounded domain in [formula omitted

We study a problem of boundary stabilization of the vibrations of elastic structure governed by the nonlinear integro-differential equation u ″ = ( a 2 + b ∫ Ω | ∇ u | 2 d x ) Δ u + f , in a bounded domain Ω in R n with a smooth boundary Γ, under mixed boundary conditions. To stabilize this system, we apply a velocity feedback control only on a part of the boundary. We prove that the solution of such system is stable subject to some restriction on the uncertain disturbing force f. We also estimate the total energy of the system over any time interval [ 0 , T ] , with a tolerance level of the disturbances. Finally, we establish the uniform decay of solution by a direct method, with an explicit form of exponential energy decay estimate, when this disturbing force f is insignificant.

Free vibration of cellular structures using continuum modes

This paper is concerned with free vibration of elastic structures made up of cellular material. A network of cells in a plane is considered. Cell walls are modelled as beams. The problem of calculating the first few natural frequencies and the corresponding mode shapes is addressed. An approximate method based on the assumption that cellular solids behave as a continuum for low-frequency dynamics is presented. A direct application of the assumed modes method incorporating continuum modes as basis leads to difficulties. The source of this difficulty is identified. A method that uses inverse power iterations is proposed to pre-condition the assumed modes appropriately. The proposed method of using continuum modes as basis for model order reduction leads to substantial computational saving while maintaining good accuracy. Two examples are given to illustrate the proposed procedure of using the continuum modes in conjunction with pre-conditioning.

Transient transport equations for high-frequency power flow in heterogeneous cylindrical shells

We use some recent mathematical results obtained for the high-frequency asymptotics of hyperbolic partial differential equations to derive exact transient power flow equations for vibrations of randomly heterogeneous cylindrical shells. The theory shows that the angularly resolved energy densities of an heterogeneous, elastic medium satisfy transient transport equations at higher frequencies. The behaviour of solutions of such equations short of their diffusion limit—if any—is fundamentally different from that of the solution of a diffusion equation, although the latter one is often invoked in the analyses of high-frequency vibrations of elastic structures. A condition by which diffusion equations can be obtained from transport equations is the presence of reflectors or heterogeneities such that scattering mean free paths are short with respect to the characteristic dimensions of the structure. The diffusion limit is reached in this study taking account of scattering by random heterogeneities of the background medium at the scale of the wavelength. This approach fills the gap between transport theory and the diffusion approach in structural dynamics, and clarifies the range of validity of the latter. Our results can be extended to fully coupled dynamic equations for compression, shear and bending of Timoshenko beams or Mindlin plates.

A new approach for simulating aerodynamic vibrations of structures in a wind tunnel—development of an experimental system by means of hybrid vibration technique

The present paper describes a newly proposed technique for simulating aerodynamic vibration of structures, which is referred to as the hybrid vibration technique or HVT. This technique is a combination of step-by-step computer calculation and measurement of the aerodynamic force acting on a model structure in a wind tunnel test. Even though the HVT can be applied effectively, problems still exist with respect to obtaining accurate simulations. These problems include response delay in controlling the model behavior and the inertia force component present in the measured aerodynamic force. Techniques for compensating for response delay and for eliminating inertia force from measured aerodynamic force are applied to a developed system based on HVT simulations of aerodynamic vibration of elastic structures and structures exhibiting elasto-plastic behavior are performed by means of the newly developed system. The effectiveness of the techniques for compensating for the response delay and eliminating the inertia force can be confirmed from the results of the simulations. In addition, the possibility and applicability of HVT is indicated.

The variational theory of complex rays: a predictive tool for medium-frequency vibrations

Today, the main numerical modeling techniques for the analysis of medium-frequency vibrations are all based on finite element or boundary element approaches. In order to represent small-wavelength phenomena in complex structures such as car chassis, satellites or ships, these techniques require a huge number of degrees of freedom (at least seven elements per wavelength are required to represent oscillating solutions). In addition, the solution obtained is highly sensitive to material properties and boundary conditions. The use of high-frequency approaches such as the statistical energy analysis (SEA) or any of its improvements does not appear to be suitable for medium-frequency vibrations: the vibrational behavior becomes too smooth and, in general, the coupling loss factor cannot be calculated in a predictive way. In this paper, a new and multiscale approach to the calculation of the vibrations of elastic structures in the medium-frequency range, called the “variational theory of complex rays”, is presented. The feasibility and effectiveness of the method are demonstrated for structures made of plate assemblies. Several 3D examples are analyzed and compared to the results from a classical finite element approach and from a SEA code. This comparison shows that our method is able to predict the effective quantities at a very low numerical cost.

La Théorie Variationnelle des Rayons Complexes pour le calcul des vibrations moyennes fréquences

ABSTRACT A new approach named the “Variational Theory of Complex Rays” is introduced for computing the vibrations of elastic structures weakly damped in the medium frequency range. Emphasis has been placed here on the most fundamental aspects. The effective quantities (elastic energy, vibration intensity…) are evaluated after computing a small system of equations which does not derive from a finite element dicretization of the structure. Numerical examples related to plates show the interest and the possibilities of the VTRC.

Mechanical vibrations induced on elastic structures by piezopolymer transducers

Theoretical and experimental results are presented to analyze the mechanical vibrations of elastic structures activated by piezopolymer transducers. Elastic thin plates and cylindrical shell structures are activated by polyvinylidene fluoride piezoelectric films. Vibrational behavior of activated structures are observed in gaseous and liquid surrounding media. These structures have been considered in view of applications for protection of oceanographic sensors against marine fouling by mechanical surface vibrations.

Homology Design for Eigenvector of Bending Vibration Using Finite Element Method

A method of homology design is presented for linear and undamped vibration of elastic structures in eigenvalue problems based on the finite element method. It is aimed at to realize the homologous mode shape, which maintain a prescribed geometrical property in a structure both in standstill and vibrant state. The original eigenvalue problem is separated into two modified eigenvalue problems with square and rectangular coefficient matrices by introducing the homologous mode shape constraint. The governing equation of design change for homologous mode shape is derived by means of the finite element sensitivity analysis of the modified eigenvalue problems. The Moore-Penrose generalized inverse is used for the solution since the coefficient matrix of the governing equation is rectangular. The validity and versatility of the proposed method are demonstrated by the numerical examples of a planar lattice frame.

Noncollocated Feedback for the Control of Elastic Structures With Friction in the Actuators

Investigated are PID-type controllers that use noncollocated sensory feedback for damping vibrations of elastic structures with high friction in the actuators. Based on a two-degree-of-freedom spring-mass system, it is demonstrated how the control-lability of such systems can be considerably improved by feedback of noncollocated measurements. Friction in the actuators leads to undamped, residual vibrations that cannot be controlled because the occurring forces do not suffice to overcome static friction. The boundaries of this uncontrollable domain are derived in the state space. A positive feedback of noncollocated sensory signals renders a stronger coupling of the motions of the actuators to structural deformations, resulting in a considerable reduction of the amplitudes of uncontrollable vibrations. Additionally, steady state errors can be significantly reduced because higher controller gains can be realized. It is shown that the time behavior of the deformations of an elastic structure can be predicted in good approximation by a linear model that neglects friction if the proposed controller is used.

Vibrational response of a simple oscillator attached to an elastic structure

The effect of many internal degrees of freedom on the vibration of elastic structures has been of great recent interest; several theories have predicted damping effects arising from associated internal resonances. As an initial effort to explore this phenomena experimentally, the behavior of a single such oscillator attached to ribbed cylindrical shell section has been measured. The single oscillator has been found to give rise to a fairly broadband resonance structure, with a bandwidth that paradoxically increases with decreasing frequency. These results, along with the physical interpretation and modeling, will be discussed.