FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

VIBRATION ATTENUATION VIA CONSTRAINED LAYER DAMPING AND DASHPOT ON A SPRING-MASS-PLATE SYSTEM

The vibrations and damping characteristics of an annular plate with constrained layer damping (CLD) treatment subject to a traveling spring-mass-damper (SMD) are investigated. The equations of the CLD-treated plate are first derived from the energy principle. These equations are simplified via the Donnell-Mushtari-Vlasov assumptions. The response equations are eventually uncoupled for each mode and are in terms of a single-degree-of-freedom (s.d.o.f.) linear oscillator with hysteretic damping. The receptance method follows to joint the plate and the SMD, and the resulting change of natural frequencies and damping ratios are investigated. Individual effects due to the inertia and the stiffness are illustrated as well. The results shows that the damping ratios resulted from the viscoelastic core are more significant than that from the viscous damper. In addition, there exists a best design on the thickness of the viscoelastic material core to have the maximum damping ratios. The results also show that the attachment of SMD bifurcated the plate's natural frequencies for every mode but n = 0. The bifurcation becomes more obvious with the rotational speed. These results provide useful information for vibration suppression in engineering design.     

MODELLING OF TRANSVERSE VIBRATION OF SHORT BEAMS FOR CRACK DETECTION AND MEASUREMENT OF CRACK EXTENSION

The method of detection of location of crack in beams based on frequency measurements is extended here to short beams, taking into account the effects of shear deformation and rotational inertia through the Timoshenko beam theory and representing the crack by a rotational spring. Methods for solving both forward (determination of frequencies of beams knowing the crack parameters) and inverse (determination of crack location knowing the natural frequencies) problems are included. A method to estimate crack extension from a change in the first natural frequency is presented. Both numerical and experimental studies are given to demonstrate the accuracy of the methods. The accuracy of the results is quite encouraging.     

VIBRATION AND BUCKLING OF COMPOSITE THIN-WALLED BEAMS WITH SHEAR DEFORMABILITY

In this paper, a theoretical model is developed for the dynamic analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form, the shear flexibility (bending and warping shear) as well as a state of initial stresses. This allows to study the free vibration and buckling problems in a unified fashion. An analytical solution of the developed equations is obtained for the case of simply supported thin-walled beams. Numerical examples are given to demonstrate the importance of the shear flexibility on the vibration and buckling behavior of the considered structures.     

THE FORCED VIBRATION AND BOUNDARY CONTROL OF PRETWISTED TIMOSHENKO BEAMS WITH GENERAL TIME DEPENDENT ELASTIC BOUNDARY CONDITIONS

The governing differential equations and the general time-dependent elastic boundary conditions for the coupled bending-bending forced vibration of a pretwisted non-uniform Timoshenko beam are derived by Hamilton's principle. By introducing a general change of dependent variable with shifting functions, the original system is transformed into a system composed of four non-homogeneous governing differential equations and eight homogeneous boundary conditions. The transformed system is proved to be self-adjoint. Consequently, the method of separation of variables can be used to solve the transformed problem. The physical meanings of these shifting functions are explored. The orthogonality condition for the eigenfunctions of a pretwisted non-uniform beam with elastic boundary conditions is also derived. The relation between the shifting functions and the stiffness matrix is derived. The boundary control of a pretwist Timoshenko beam is studied. The effects of the total pretwist angle, the position of loading and the boundary spring constants on the energy required to control the performance of a pretwisted beam are investigated.     

VIBRATION AND BUCKLING OF INITIALLY STRESSED CURVED BEAMS

Equations of motion for curved beams in a general state of non-uniform initial stresses are derived using the principle of virtual work. The equations are adjusted to a generic expression by using appropriate transformations. The free vibration behaviours of the curved beams subjected to a combination of uniform initial tensile of compressive stresses and uniform initial bending stress are examined. The Galerkin method is employed in obtaining accurate values of free frequencies and initial buckling stresses. The curved beam is assumed to be vibrating in its plane. Natural frequencies and initial buckling stresses for hinged supported curved beams are presented for validation. Effects of arc segment angles, elastic foundation, and initial stresses on the natural frequencies are investigated. Effects of arc segment angles, elastic foundation, and initial bending stresses on the initial buckling stresses are explored in this paper.     

FREE VIBRATION ANALYSIS OF NON-UNIFORM BEAMS WITH AN ARBITRARY NUMBER OF CRACKS AND CONCENTRATED MASSES

An exact approach for free vibration analysis of a non-uniform beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Using the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a non-uniform beam with an arbitrary number of cracks and concentrated masses can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a non-uniform beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, the decrease in the determinant order as compared with previously developed procedures leads to significant savings in the computational effort and cost associated with dynamic analysis of non-uniform beams with cracks. Numerical examples are given to illustrate the proposed method and to study the effect of cracks on the natural frequencies and mode shapes of cracked beams.     

RANDOM VIBRATION OF A ROTATING BLADE WITH EXTERNAL AND INTERNAL DAMPING BY THE FINITE ELEMENT METHOD

The finite element model is employed to investigate the mean-square response of a rotating blade with external and internal damping under stationary or non-stationary random excitation. The blade is considered to be subjected to white-noise and earthquake excitations. The effects of rotational speed, external and internal damping on the mean-square response are studied. It is found that the mean-square response decreases quickly when the external and internal damping increases within some scope. Moreover, the increment of rotational speed will reduce the mean-square response of a rotating blade. It is also found that the mean-square response decreases when the low natural frequency of base decreases. Inversely, the mean-square response increases when the high natural frequency of base (natural frequencies of base are over the first natural frequency of blade) decreases. The reliability of a rotating blade subjected to stationary or non-stationary excitations is also obtained.     

CONTROL DESIGN OF ACTIVE VIBRATION ISOLATION USING μ -SYNTHESIS

Numerical and experimental investigations on active vibration isolation system are presented in this paper. Two configurations are implemented for a statically balanced three-mount system. To reduce the influence of payload dynamics and coupling among the control actuators, intermediate masses are added to the system. Linear quadratic Gaussian control andμ -synthesis are employed in the controller synthesis. The controllers are implemented on the platform of a floating-point digital signal processor. The results obtained from simulations and experiments indicated that the optimal controllers achieved the desired performance under the constraint of robust stability.     

FREE VIBRATION OF A PARTIALLY LIQUID-FILLED AND SUBMERGED, HORIZONTAL CYLINDRICAL SHELL

The dynamic characteristics (i.e., natural frequencies and mode shapes) of a partially filled and/or submerged, horizontal cylindrical shell are examined. In this investigation, it is assumed that the fluid is ideal, and fluid forces are associated with inertial effects only: namely, the fluid pressure on the wetted surface of the structure is in phase with the structural acceleration. The in vacuo dynamic characteristics of the cylindrical shell are obtained using standard finite element software. In the “wet” part of the analysis, it is assumed that the shell structure preserves its in vacuo mode shapes when in contact with the contained and/or surrounding fluid and that each mode shape gives rise to a corresponding surface pressure distribution of the shell. The fluid-structure interaction effects are calculated in terms of generalized added masses, using a boundary integral equation method together with the method of images in order to impose an appropriate boundary condition on the free surface. To assess the influence of the contained and/or surrounding fluid on the dynamic behaviour of the shell structure, the wet natural frequencies and associated mode shapes were calculated and compared with available experimental measurements.     

TRANSVERSE VIBRATION OF ELASTIC-VISCOELASTIC-ELASTIC SANDWICH BEAMS: COMPRESSION-EXPERIMENTAL AND ANALYTICAL STUDY

Experimental and analytical results are presented from an investigation into the compressional vibration of an elastic-viscoelastic-elastic three-layer sandwich beam. Most analytical models make the fundamental assumption that shear deformation in the viscoelastic core yields the largest damping and compressional deformation is negligible. Experimental results from a cantilever beam with a constrained layer viscoelastic damping treatment driven with a sinusoidal input are given which show compressional deformation over a relatively wide driving frequency range. A new analytical model for compressional damping is presented and compared with experimental results, with the Mead and Markus shear damping model, and with the Douglas and Yang compressional damping model. These results indicate that the proposed compressional model is a better predictor of resonance frequencies for the cantilever beams tested and that all models show deficiencies in predicting damping     

AN EXACT METHOD FOR THE FREE VIBRATION ANALYSIS OF TIMOSHENKO-KELVIN BEAMS WITH OSCILLATORS

In this paper, a modern exact method is proposed for solving the problem of free vibrations of a Timoshenko-type viscoelastic beam with discrete rigid bodies, connected to the beam by means of viscoelastic constraints. The phenomenon of free vibrations of this discrete-continuous system is described by a set of three partial and two subsystem ordinary differential equations with generalized boundary conditions and initial conditions. Vector notation of the equations allows one to identify the self-adjoint linear operators of inertia, stiffness and damping. In this case, these operators are not homothetic hence a separation of variables in this set of equations is possible only in a complex Hilbert space. Such separation of variables leads to ordinary differential equations of motion with respect to time as well as to a set of three ordinary differential equations with respect to a spatial variable and two subsystem algebraical equations. Solution of the boundary-value problem was carried out in the classical way, but its results are complex conjugated. Using these results and the fundamental principle, describing the orthogonality property of complex eigenvectors, the problem of free vibrations of the system with arbitrary initial conditions has been finally solved exactly.     

DYNAMIC MODELLING AND VIBRATION ANALYSIS OF A FLEXIBLE CABLE-STAYED BEAM STRUCTURE

In this paper, the non-linear vibration of a cable-stayed beam with time-varying length and tension in the cable is investigated. A set of non-linear, time-varying differential equations describing this coupling system is derived by Hamilton's principle and the finite element method. According to the results of numerical simulation, the tension of the cable is related to the cable length, which in turn is a function of the longitudinal and transverse displacements of the cable. Furthermore, it is shown that the tension and length of the cable can be considerably different by using linear and non-linear models.     

部分潜入水中任意形状等截面柱体的振动分析

本文研究部分潜入水中的任意形状等截面柱体的弯曲自由振动问题,应用Fourier级数匹配法,提供了一个精确计算这类柱水耦联系统振动特性的解析方法,结果可由计算机解出。本文最后取水中椭圆柱体为例,给出了一些具体的数值计算结果。     

金属和压电陶瓷构成的两层复合薄圆板的强迫振动

本文分析了电压策动或均匀声压策动下的由半径不同的压电陶瓷和金属板构成的两层复合薄圆板的强迫振动。给出了简支边界条件和固定边界条件下的严格解和串联共振频率方程与并联共振频率方程。用Rayleigh-Ritz方法给出了简支边界条件下电压策动的强迫振动的近似解和串联共振频率方程。数字结果给出了能得到较大机电耦合系数的无量纲几何参量的最佳值。本文还讨论了弹性耦合和惯性耦合对共振频率的影响。     

ANALYTICAL AND NUMERICAL STUDY ON SPATIAL FREE VIBRATION OF NON-SYMMETRIC THIN-WALLED CURVED BEAMS

For spatial free vibration of non-symmetric thin-walled circular curved beams, an accurate displacement field is introduced by defining all displacement parameters at the centroidal axis and three total potential energy functionals are consistently derived by degenerating the potential energy for the elastic continuum to that for thin-walled curved beams. The closed-form solutions are newly obtained for in-plane and out-of-plane free vibration analysis of monosymmetric curved beams respectively. Also, two thin-walled curved beam elements are developed using the third and fifth order Hermitian polynomials. In order to illustrate the accuracy and the practical usefulness of the present method, analytical and numerical solutions by this study are presented and compared with previously published results or solutions by ABAQUS' the shell element. Particularly, effects of the thickness curvature as well as the inextensional condition are investigated on free vibration of curved beams with monosymmetric and non-symmetric cross-sections.     

VIBRATIONS OF PARTIALLY FILLED CYLINDRICAL TANKS WITH RING-STIFFENERS AND FLEXIBLE BOTTOM

The dynamics of a tank partially filled with a liquid having a free surface is investigated. In the analysis, the effect of free surface waves is taken into account, so that both bulging and sloshing modes are studied. The structure is completely flexible, and is composed of a vertically standing circular cylindrical shell, with ring-stiffeners, and a flexible bottom that consists of a circular plate generally resting on an elastic Winkler foundation. The plate edge is joined to the shell by coupling rotational springs distributed around the edge. The interaction between the plate and the shell via the fluid is included. Application to the particular cases of (i) a flexible shell with a rigid bottom and (ii) a flexible bottom plate with a rigid cylinder are also presented, as well as a comparison with approximate theories. Ring-stiffeners on the shell are also considered in order to investigate their influence on the modal behaviour of the tank. Solution of this problem is achieved via the method of artificial springs within the framework of the Rayleigh–Ritz theory, while considering strong fluid–structure interaction.     

MODELLING AND CONTROL WITH PIEZOACTUATORS FOR A SIMPLY SUPPORTED BEAM UNDER A MOVING MASS

In the paper, the dynamic modelling and control are presented for a simply supported beam under a moving mass. The equations of motion are obtained based on the Euler-Bernoulli beam theory by including the dynamic effect of a moving mass travelling along a vibrating path. The equations of motion are discretized by using the assumed modes method with the static deflection of the beam. In order to reduce the deflection of the beam under a moving mass, a controller with full state feedback is designed based on linearized equations of motion. Two piezoelectric actuators are bonded along the bottom of the beam at different locations determined by the minimization of an optimal cost functional. Numerical simulations are performed with respect to different constant velocities and different moving masses. The controller with two piezoelectric actuators shows excellent performance under unknown disturbances to the system.     

透射式GaAs光电阴极AlGaAs窗层和GaAs光电发射层界面应变状况的X射线衍射研究

本文介绍了应变和弛豫的概念以及倒易点在倒易空间的分布,阐明了GaAs光电阴极AlGaAs窗层和GaAs光电发射层界面应变状况的X射线衍射的分析方法,最后给出了实例.