Numerical modeling of the active damping of forced thermomechanical resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric inclusions

We consider the problem of active damping of forced resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric sensors and actuators. Here, the interaction of electromechanical and thermal fields is taken into account. For modeling of vibrations, we use the Kirchhoff–Love hypotheses as well as hypotheses adequate to them and describing the distribution of temperature and electric field quantities. The shell temperature increases as a result of dissipative heating. For the active damping of vibrations, piezoelectric sensors and actuators are used. It is supposed that the electromechanical characteristics of materials depend on the temperature. The solution of this complex nonlinear problem has been obtained by the iterative method and finite element method. We have investigated the influence of temperature of dissipative heating on the efficiency of active damping of vibrations of a viscoelastic cylindrical panel with rigid restraint of its edges.     

Influence of boundary conditions and temperature of dissipative heating on active damping of forced axisymmetric resonant bending vibrations of circular viscoelastic plates by piezoelectric sensors and actuators

The problem of forced monoharmonic, axisymmetric, bending vibrations and dissipative heating of circular viscoelastic plates with piezoelectric sensors and actuators is considered. We describe the viscoelastic behavior of a passive (without the piezoeffect) and a piezoactive material according to the concept of complex modules depending on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of boundary conditions and temperature of dissipative heating on the active damping of forced resonant vibrations of circular viscoelastic plates using piezoelectric sensors and actuators is investigated.     

Unilateral Constrained Bending Vibrations on a Torsional String

Vibrations in long torsional strings result in spatio‐temporal dynamics. In order to actively damp these vibrations the system has to be analysed analytically, numerically and experimentally. Stick‐slip‐effects result in torsional selfexcited vibrations of the string. These vibrations are coupled with bending vibrations which are constrained by the borehole. The straight string was modelled in an experimental setup. The control of the straight string and the unilateral constrained bending vibrations were treated seperately. The dynamics of straight strings were controlled using three different approaches: firstly, a simple PD‐controller with the parameters calculated based on a one‐degree‐of‐freedom oscillator, secondly, the parameters were optimized using a simplex‐method, thirdly, the Karhunen‐Loeve‐transformation was used in order to reduce the dimension of the system. A controller based on the reduced system was implemented and the parameters were optimized with the same simplex algorithm. The unilateral constrained bending motion were examined at a cantilever beam which was assumed to be constrained in one direction in the middle of the beam. First, the beam was modelled analytically as a continuous system. The two states (contact and no contact) were described separately. The transition between these states were modelled with energy assumptions. Second, the beam was modelled as a Finite‐Element‐System. The numerical results of both methods were compared with experimental data.     

Nonlinear vibrations of piezoelectric microcantilevers for biologically-induced surface stress sensing

In this paper, nonlinear vibrations of a piezoelectrically-driven microcantilever beam in presence of a biological monolayer are investigated and the corresponding equations of motion are derived and simulated. A part of the microcantilever beam surface is covered by a piezoelectric layer, which acts as an actuator. Inextensibility condition and the coupling between electrical and mechanical properties in piezoelectric materials are considered in the bending vibrations of the beam. The adsorbed biological layer is considered to be a monolayer and its adsorption induced surface stress is formulated from the molecular viewpoint. The nonlinear terms in the governing equations of motion of the beam appear in the quadratic form due to the presence of the piezoelectric layer, and the cubic form due to geometry of the beam and the adsorbed biological layer. Through extensive numerical simulations, it is demonstrated that the nonlinear effect of piezoelectric layer is significant in the microcantilever resonance sensing range. It is also shown that the effect of intermolecular attraction–repulsion on the surface stress is less dominant than other sources of surface stress (e.g., the electrostatic forces). Finally, it is observed that piezoelectrically-actuated microcantilever provides the ability of indirect measurement of vibrations and frequency response characteristics, instead of using bulky laser sensor.     

Static and dynamic analysis of smart cylindrical shell

Shell type components and structures are very common in many mechanical and structural systems. In smart structural applications, piezolaminated plates and shells are commonly used. In this paper a finite element formulation is presented to model the static and dynamic response of laminated composite shells containing integrated piezoelectric sensors and actuators subjected to electrical, mechanical and thermal loadings. The formulation is based on the first order shear deformation theory and Hamilton's principle. In this formulation, the mass and stiffness of the piezo-layers have been taken into account. A nine-noded degenerated shell element is implemented for the analysis. The model is validated by comparing with existing results documented in the literature. A simple negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to actively control the dynamic response of an integrated structure through a closed control loop. The influence of the stacking sequence and position of sensors/actuators on the response of the laminated cylindrical shell is evaluated. Numerical results show that piezoelectric sensors/actuators can be used to control the shape and vibration of laminated composite cylindrical shell.     

Modelling of the Piezoelectric Element Interaction for Actively Damped Laminated Plates

The paper deals with stationary vibrations of viscoelastic laminated rectangular plates controlled by distributed piezoelectric sensors and actuators. Assuming the perfect bonding, the model of interaction between piezoelements and the basic structure is developed by taking into account the inplane tension or compression of the plate caused by the two‐dimensional piezoelectric effect. Moreover, the governing equations are formulated for a non‐zero skew angle between the principal material axes of the piezoelectric patches and the laminate axes.     

Hybrid optimization procedure applied to optimal location finding for piezoelectric actuators and sensors for active vibration control

This paper presents an efficient hybrid optimization approach using a new coupling technique for solving the constrained optimization problems. This methodology is based on genetic algorithm, sequential quadratic programming and particle swarm optimization combined with a projected gradient techniques in order to correct the solutions out of domain and send them to the domain’s border. The established procedures have been successfully tested with some well known mathematical and engineering optimization problems, also the obtained results are compared with the existing approaches. It is clearly demonstrated that the solutions obtained by the proposed approach are superior to those of existing best solutions reported in the literature. The main application of this procedure is the location optimization of piezoelectric sensors and actuators for active control, the vibration of plates with some piezoelectric patches is considered. Optimization criteria ensuring good observability and controllability based on some main eigenmodes and residual ones are considered. Various rectangular piezoelectric actuators and sensors are used and two optimization variables are considered for each piezoelectric device: the location of its center and shape orientation. The applicability and effectiveness of the present methodological approach are demonstrated and the location optimization of multiple sensors and actuators are successfully obtained with some main modes and residual ones. The shape orientation optimization of sensors observing various modes as well as the local optimization of multiple sensors and actuators are numerically investigated. The effect of residual modes and the spillover reduction can be easily analyzed for a large number of modes and multiple actuators and sensors.     

Flexible-link robots with combined trajectory tracking and vibration control

This work deals with asymptotic trajectory tracking and active damping injection on a flexible-link robot by application of Multiple Positive Position Feedback. The flexible-link robot is modeled and validated by using finite element methods and experimental modal analysis, and then a reduced order model of the flexible-link robot dynamics, up to the first dominant vibration modes, is employed for experimental evaluation on a test rig. Then, a combined control scheme is synthesized in two parts: first, a Sliding-Mode Control based on a cascaded Proportional-Integral-Derivative for regulation and trajectory tracking tasks, via a direct current motor torque as the control input for the overall system dynamics, and, second, a Multiple Positive Position Feedback for active vibration control and attenuation of residual vibrations on the tip position, via the input voltage applied to a piezoelectric patch actuator attached directly on the flexible beam. The results are evaluated on an experimental platform, where the dynamic performance of the overall active vibration control scheme leads to fast and effective tracking results, with damping ratios increased up to 300%.     

Parameter estimation in a structural acoustic system with fully nonlinear coupling conditions

A methodology for estimating physical parameters in a class of structural acoustic systems is presented. The general model under consideration consists of an interior cavity which is separated from an exterior disturbance by an enclosing elastic structure. Piezoceramic patches are bonded to or embedded in the structure; these can be used both as actuators and sensors in applications ranging from the control of interior noise levels to the determination of structural flaws through nondestructive evaluation techniques. The presence and excitation of the patches, however, changes the geometry and material properties of the structure as well as involves unknown patch parameters, thus necessitating the development of parameter estimation techniques which are applicable in this coupled setting. In developing a framework for approximation, parameter estimation and implementation, strong consideration is given to the fact that the input operator is unbonded due to the discrete nature of the patches. Moreover, the model is weakly nonlinear as a result of the coupling mechanism between the structural vibrations and the interior acoustic dynamics. Within this context, an illustrating model is given, well-posedness and approximation results are discussed and an applicable parameter estimation methodology is presented. The scheme is then illustrated through several numerical examples with simulations modeling a variety of commonly used structural acoustic techniques for system excitation and data collection.     

旋转刚柔耦合系统的模型及控制

旋转刚柔耦合系统在航空航天、机器人、高速机构以及车辆等领域有着广泛的应用,主要描述负载在旋转刚体上的柔性梁的运动。对旋转刚柔耦合系统施加控制使得整个闭环系统达到:1)旋转刚体以预期的旋转角速度运动;2)负载在刚体上的柔性梁镇定。本文将从控制器设计的角度出发,介绍目前在旋转刚柔耦合系统控制方面取得的主要研究成果。     

Nonlinear finite element analysis of functionally graded plates integrated with patches of piezoelectric fiber reinforced composite

In this paper, a nonlinear static finite element analysis of simply supported smart functionally graded (FG) plates in the presence/absence of the thermal environment has been presented. The substrate FG plate is integrated with the patches of piezoelectric fiber reinforced composite (PFRC) material which act as the distributed actuators of the plate. The material properties of the FG substrate plate are assumed to be temperature dependent and graded along the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The derivation of this nonlinear thermo-electro-mechanical coupled finite element model is based on the first order shear deformation theory and the Von Karman type geometric nonlinearity. The numerical solutions of the nonlinear equations of the finite element model are obtained by employing the direct iteration method. The numerical illustrations suggest the potential use of the distributed actuator made of the PFRC material for active control of nonlinear deformations of smart FG structures. The effects of volume fraction index of the FG material of the substrate plates and the locations of the PFRC patches on the control authority of the patches are investigated. Emphasis has also been placed on investigating the effect of variation of piezoelectric fiber orientation angle in the PFRC patches on their actuation capability for counteracting the large deflections of FG plates.     

压电弯曲元的夹层梁解析模型

压电弯曲元是一类传感和作动器件,已得到广泛的应用．基于一阶剪切变形理论发展了压电弯曲元夹层梁解析模型,对梁截面采用统一转角并将耦合电势沿厚度的分布假设为二次函数,进一步修正了横向剪应变对电位移的影响．以弯曲元简支梁自由振动为例进行数值分析,解析模型解与二维精确解相比具有良好的精度,为分析弯曲元动力机电响应提供了良好的解析模型．     

Small free vibrations of an infinite cylindrical shell rotating on rollers

Small free vibrations of an infinitely long rotating cylindrical shell being in contact with rigid cylindrical rollers are considered. A system of linear differential equations for the vibrations of such a shell is derived. By using the Fourier transform of the solutions in the circumferential coordinate, a system of algebraic equations for approximately determining the vibration frequencies and mode shapes is obtained. It is shown that, for any number n of uniformly distributed rollers, the approximate values of the first n frequencies and mode shapes can be found explicitly. On the basis of the orthogonal sweep method, an algorithm for numerically solving the boundary value eigenvalue problem describing the vibrations of a rotating shell is developed. Analytical and numerical results are compared. The obtained approximate formulas for frequencies and the numerical algorithm can be used to design centrifugal concentrators for ore enrichment.     

Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell

In this paper, buckling and free vibration behavior of a piezoelectric rotating cylindrical carbon nanotube-reinforced (CNTRC) shell is investigated. Both cases of uniform distribution (UD) and FG distribution patterns of reinforcements are studied. The accuracy of the presented model is verified with previous studies and also with those obtained by Navier analytical method. The novelty of this study is investigating the effects of critical voltage and CNT reinforcement as well as satisfying various boundary conditions implemented on the piezoelectric rotating cylindrical CNTRC shell. The governing equations and boundary conditions have been developed using Hamilton's principle and are solved with the aid of Navier and generalized differential quadrature (GDQ) methods. In this research, the buckling phenomena in the piezoelectric rotating cylindrical CNTRC shell occur as the natural frequency is equal to zero. The results show that, various types of CNT reinforcement, length to radius ratio, external voltage, angular velocity, initial hoop tension and boundary conditions play important roles on critical voltage and natural frequency of piezoelectric rotating cylindrical CNTRC shell.     

Finite volume analysis of adaptive beams with piezoelectric sensors and actuators

This paper presents a finite volume (FV) formulation for the free vibration analysis and active vibration control of the smart beams with piezoelectric sensors and actuators. The governing equations based on Timoshenko beam theory are discretized using the finite volume method. For the purpose of forced vibration control of beam structures, the negative velocity feedback controller is designed for the single-input, single-output system. To achieve the best effect, the piezoelectric sensors and actuators are coupled with the host structure in different positions and then the performance of the designed control system is evaluated for each position. In the test examples, first the shear locking free feature of the present formulation is demonstrated. This has been performed by doing static and natural frequency analysis of some reference models. Then, the capability of the proposed method for the prediction of uncontrolled forced vibration response and active vibration control of a beam structure is studied.     

Active suppression of nonlinear composite beam vibrations by selected control algorithms

This paper is focused on application of different control algorithms for a flexible, geometrically nonlinear beam-like structure with Macro Fiber Composite (MFC) actuator. Based on the mathematical model of a geometrically nonlinear beam, analytical solutions for Nonlinear Saturation Controller (NSC) are obtained using Multiple Scale Method. Effectiveness of different control strategies is evaluated by numerical simulations in Matlab–Simulink software. Then, the Digital Signal Processing (DSP) controller and selected control algorithms are implemented to the physical system to compare numerical and experimental results. Detailed analysis for the NSC system is carried out, especially for high level of amplitude and wide range of frequencies of excitation. Finally, the efficiency of the considered controllers is tested experimentally for a more complex autoparametric “L-shape” beam system.     

Multi-harmonic measurements and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions

This study presents a direct comparison of measured and predicted nonlinear vibrations of a clamped–clamped steel beam with non-ideal boundary conditions. A multi-harmonic comparison of simulations with measurements is performed in the vicinity of the primary resonance. First of all, a nonlinear analytical model of the beam is developed taking into account non-ideal boundary conditions. Three simulation methods are implemented to investigate the nonlinear behavior of the clamped–clamped beam. The method of multiple scales is used to compute an analytical expression of the frequency response which enables an easy updating of the model. Then, two numerical methods, the Harmonic Balance Method and a time-integration method with shooting algorithm, are employed and compared one with each other. The Harmonic Balance Method enables to simulate the vibrational stationary response of a nonlinear system projected on several harmonics. This study then proposes a method to compare numerical simulations with measurements of all these harmonics. A signal analysis tool is developed to extract the system harmonics’ frequency responses from the temporal signal of a swept sine experiment. An evolutionary updating algorithm (Covariance Matrix Adaptation Evolution Strategy), coupled with highly selective filters is used to identify both fundamental frequency and harmonic amplitudes in the temporal signal, at every moment. This tool enables to extract the harmonic amplitudes of the output signal as well as the input signal. The input of the Harmonic Balance Method can then be either an ideal mono-harmonic signal or a multi-harmonic experimental signal. Finally, the present work focuses on the comparison of experimental and simulated results. From experimental output harmonics and numerical simulations, it is shown that it is possible to distinguish the nonlinearities of the clamped–clamped beam and the effect of the non-ideal input signal.     

Structural health monitoring of hybrid composite structure with piezoelectric patches

Structural health monitoring is performed on a hybrid structure specially designed for simple assembly and disassembly or replacement of components. The net-like structure is composed of composite beams connected with aluminum pads and it is fixed to a frame in selected locations. Modal analysis is performed both experimentally and numerically. The experiment is carried out using impact hammer, accelerometer and spectrum analyzer and the frequency and damping characteristics are thus obtained. The numerical model using finite element method is validated by comparison of the calculated and measured eigenfrequencies and eigenshapes. Furthermore, piezoelectric patches are applied on selected beam. The modal analysis is carried out again using the piezoelectric patches and with some of the composite beams replaced by damaged beam. The variation in modal characteristics between original and damaged configurations is analyzed both experimentally and numerically. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterisation and Modelling of Piezoceramic Actuator Hystereses

Piezoelectric ceramics are often used as actuators in smart structures technology. In the vast majority of papers dealing with this topic only linear constitutive relations are used. However, the electric field-strain relations of such actuators show hysteretic behaviour, which means that the piezoelectric coupling coefficient is not constant. In this study the hysteresis of a mechanically unconstrained actuator is obtained using the Michelson interferometry. The hysteretic behaviour is modelled by a Preisach model. Using these experimental data, for the modelling of an active structure with embedded piezoelectric actuators the actual coupling coefficient can then be determined with the help of the Preisach model. With this procedure the actuation strain of an embedded actuator, including the physical nonlinearities, can be calculated using the material characteristics obtained for an unconstrained actuator. For an experimental validation of the method outlined above, a Lead Zirconate Titanate (PZT) actuator is characterised experimentally and then glued to a cantilever beam. Then, the tip displacement of the actuated beam is determined experimentally and simulated numerically using the above method. The experimental and numerical results agree reasonably well if the shear lag due to the bonding layer between the actuator and the structure is taken into consideration. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)