A cantilever beam damper suppressing rectangular plate vibrations

The use of cantilever beams in suppressing excessive resonance amplitudes of rectangular cantilever plates is considered, and optimum values of their tuning and damping parameters are specified in graphical form. Because the cantilever plate problem, which is of strong industrial interest, does not lend itself to a Lévy-type solution, the Ritz method is used. Structural damping is incorporated into the main and auxiliary systems by treating them as having a complex elastic modulus. With appropriate selection of the parameters, the fundamental resonance of the plate is split into two new ones with considerably suppressed responses. In order to verify the analysis, an experimental investigation was carried out and the results obtained are compared with the theory developed.     

悬臂方薄板振动的辐射声场及竖裂纹对其频谱的影响

在将悬臂板挠度表示为正交多项式之和的基础上，利用瑞利-里兹方法求解了悬臂板的前几阶振动模态频率及挠度．进而利用瑞利积分求解了其自由振动辐射声场的分布规律。用分区技术的瑞利-里兹方法求解了竖裂纹对悬臂板低阶模态频率的影响，并将计算结果与有限元结果作了对照。实验上用小球撞击法测定了含竖裂纹悬臂板辐射声场的频谱，谱成分与有限元结果基本符合。     

Stability of cantilever plates subjected to biaxial subtangential loading

The stability of cantilever plates which, mathematically, comprises a non-self-adjoint problem is investigated. It is assumed that the plate is acted upon by a subtangential biaxial edge load embodying the dead loading and the follower type loading as its limiting states. The scheme of modal expansions, containing the constrained rigid modes, together with Galerkin's method is employed and the stability of the plate in terms of subtangency and load parameters is analysed. As an example the kinetic stability analysis of a square cantilever plate is carried out in detail.     

Interactions between a partially or totally immersed vibrating cantilever plate and the surrounding fluid

The dry and wet dynamic characteristics of a vertical and a horizontal cantilever square plate [1] immersed in fluid are discussed from the viewpoint of a linear hydroelasticity theory [2–5]. The surface piercing vertical plate is partially immersed in the fluid and the influence of submerged plate length on the resonance frequencies investigated. For the horizontal plate the influence of submerged depth below the free surface on the resonance frequencies is examined. Incorporated into the theoretical model is a free surface boundary condition allowing wave disturbances to be present. The interaction existing between the vibrating cantilever plate and the free surface is clearly exhibited in the calculated curves describing the generalized hydrodynamic coefficients. A limited comparison between predictions and experimental data [1] is also included.     

Analytical method for determining eigenvalues in the problem of vibrations of a plate moving in gas

An analytical method for determining eigenvalues for differential equations using special functions is presented. As an example, vibrations of an infinite-width cantilever plate moving with the supersonic velocity in a gas are described. For a two-dimensional problem, the analytical dependence between the plate eigenvibration characteristics and velocity was obtained, which allows us to make conclusions about motion stability. The developed technique can be applied to study many problems whose solution is associated with finding of eigenvalues. In contrast to similar techniques, the described method yields sufficiently accurate analytical expressions for eigenvalues, which allows one to use them in specific calculations.     

Study of broad bandwidth vibrational energy harvesting system with optimum thickness of PET substrate

In this paper, we present the development of a flexible PET-based (polyethylene terephthalate; PET) vibrational energy harvesting system with broad bandwidth. This broad bandwidth harvesting system comprises of four units of individual ZnO (zinc oxide) piezoelectric harvester in the form of a cantilever structure connected in parallel, and rectifying circuit with storage module. This system has ability to convert mechanical energy into electrical energy from the varying ambient vibration. The design and simulation of a piezoelectric cantilever plate was described by using commercial software ANSYS FEA (Finite Element Analysis) to determine the optimum thickness of PET substrate, internal stress distribution, operation frequency and electric potential. With the optimum thickness predicted by developed accurate analytical formula analysis, the one-way mechanical strain that is efficient to enhance the induced electric potential can be controlled within the piezoelectric ZnO layer. In addition, the relationship among the model solution of piezoelectric cantilever plate equation, vibration-induced electric potential and electric power was realized. An individual piezoelectric harvester consists of flexible PET substrate, piezoelectric ZnO thin film with (002) c-axis preferred orientation, and selectively deposited UV-curable resin lump structure which is used to change the resonant frequency of the harvester. In combination with multi-harvesters and rectifying with storage module together, an energy harvesting system with broad bandwidth can be fabricated. One individual harvester achieves a maximum OCV (open-circuit voltage) up to 4 V with power density of 1.247 μW/cm³. So far, we succeeded in accomplishing a broad bandwidth system with operating frequency range within 100 Hz-450 Hz to enhance powering efficiency. When the DC voltage (direct current voltage) across a storage module is charged up to 1.55 V after rectification, a flash LED (light emitting diode) is driven.     

Broadband energy harvesting via magnetic coupling between two movable magnets

Harvesting energy from ambient mechanical vibrations by the piezoelectric effect has been proposed for powering microelectromechanical systems and replacing batteries that have a finite life span. A conventional piezoelectric energy harvester （PEH） is usually designed as a linear resonator, and suffers from a narrow operating bandwidth. To achieve broadband energy harvesting, in this paper we introduce a concept and describe the realization of a novel nonlinear PEH. The proposed PEH consists of a primary piezoelectric cantilever beam coupled to an auxiliary piezoelectric cantilever beam through two movable magnets. For predicting the nonlinear response from the proposed PEH, lumped parameter models are established for the two beams. Both simulation and experiment reveal that for the primary beam, the introduction of magnetic coupling can expand the operating bandwidth as well as improve the output voltage. For the auxiliary beam, the magnitude of the output voltage is slightly reduced, but additional output is observed at off-resonance frequencies. Therefore, broadband energy harvesting can be obtained from both the primary beam and the auxiliary beam.     

Statistical damage identification of structures with frequency changes

Model updating methods based on structural vibration data have being rapidly developed and applied to detect structural damage in civil engineering. But uncertainties existing in the structural model and measured vibration data might lead to unreliable damage detection. In this paper a statistical damage identification algorithm based on frequency changes is developed to account for the effects of random noise in both the vibration data and finite element model. The structural stiffness parameters in the intact state and damaged state are, respectively, derived with a two-stage model updating process. The statistics of the parameters are estimated by the perturbation method and verified by Monte Carlo technique. The probability of damage existence is then estimated based on the probability density functions of the parameters in the two states. A higher probability statistically implies a more likelihood of damage occurrence. The presented technique is applied to detect damages in a numerical cantilever beam and a laboratory tested steel cantilever plate. The effects of using different number of modal frequencies, noise level and damage level on damage identification results are also discussed.     

各向异性矩形悬臂板稳态热传导解析

应用各向异性稳态热传导解析解分析了三边对流换热、另一边给定不均匀温度的各向异性矩形悬臂板温度场。讨论了各向异性角对温度场分布的影响。各向异性角的增大加剧了温度梯度。数值结果表明各向异性角为0°的悬臂板内温度梯度最小。     

MODAL CHARACTERISTIC OF A ROTATING RECTANGULAR CANTILEVER PLATE

Modal characteristics of a rotating cantilever plate are investigated in the present work. A dynamic modelling method for rectangular plates undergoing prescribed overall motion is employed to derive the equations of motion. The general equations are particularized for the modal analysis of a rotating cantilever plate and dimensionless parameters are identified through dimensional analysis. The effects of the dimensionless parameters on the modal characteristics of the rotating plate are investigated. Incidentally, eigenvalue loci veering and crossing phenomena along with the corresponding modeshape variations are exhibited and discussed.     

Practical industrial method of increasing structural damping in machinery,I: Squeeze-film damping with air

There is frequently a need to reduce sound radiation due to resonant flexural motion of stiff machinery panels. This can be achieved by applying squeeze-film damping to the vibrating panel by attaching an auxiliary plate parallel to the surface, thereby trapping a thin layer of air. Relative vibration of the plates pumps this air at high velocities, resulting in energy loss due to the air viscosity. In this study the damping below the critical frequency of the “thick plate” with an “attached plate” and air layer has been investigated by using an impedance approach. This model is incorporated into a two element Statistical Energy Analysis (SEA) model to predict the damping well above the critical frequency of the thick plate. The agreement between the predicted and measured results is remarkably good. Below the critical frequency the damping is pumping controlled, while above critical the plate couplings are the controlling factor.     

MODAL ANALYSIS OF ROTATING COMPOSITE CANTILEVER PLATES

A modelling method for the modal analysis of a rotating composite cantilever plate is presented in this paper. A set of linear ordinary differential equations of motion for the plate is derived by using the assumed mode method. Two in-plane stretch variables are employed and approximated to derive the equations of motion. The equations of motion include the coupling terms between the in-plane and the lateral motions as well as the motion-induced stiffness variation terms. Dimensionless parameters are identified and the explicit mass and the stiffness matrices for the modal analysis are obtained with the dimensionless parameters. The effects of the dimensionless angular velocity and the fiber orientation angles of rotating composite cantilever plates on their modal characteristics are investigated. Natural frequency loci veering and crossing along with associated mode shape variations are observed.     

A note on vibration of a cantilever plate immersed in water

The added mass of the fluid surrounding it plats an important role in the dynamic behaviour of a submerged structure. The first few mode shapes and the respective natural frequencies of a submerged cantilever plate are found by using a finite element procedure, eigenvalues being obtained by a simultaneous iteration technique. The influence of the water depth below the plate and also of the water's lateral extent is considered, in order to test the convergency of the results. Results on the effects of the depth of immersion on the natural frequencies and mode shapes of the cantilever plate for different aspect ratios are presented.     

Performance of a cantilever piezoelectric energy harvester impacting a bump stop

Piezoelectric cantilever beam energy harvesters are commonly used to convert ambient vibration into electrical energy. In practical applications, energy harvesters are subjected to large shocks which can shorten the service life by causing mechanical failure. In this work, a bump stop is introduced into the design of a piezoelectric cantilever beam energy harvester to limit the maximum displacement of the cantilever and prevent excessively high bending stresses developing as a result of shocks. In addition to limiting the maximum displacement of the beam, it is inevitable that the deflected shape of the beam and the electrical output are modified. A theoretical model for a piezoelectric cantilever beam harvester impacting against a stop is derived, which aims to develop an understanding of the vibration characteristics of the cantilever and quantify how the electrical output of the harvester is affected by the stop. An experiment is set up to measure the dynamics and the electrical output of a bimorph energy harvester and to validate the theoretical model. Numerical simulation results are presented for energy harvesters with different initial gaps and different stop locations, and it is found that the reduction in maximum bending stress is at the expense of the electrical power of the harvester.     

Experimental model validation for a nonlinear energy harvester incorporating a bump stop

In some practical applications, cantilever beam piezoelectric energy harvesters are subjected to large amplitude base excitations which induce nonlinear behaviour in the harvester that affects their performance. In this paper, a cantilever piezoelectric energy harvester model is developed which takes account of geometric nonlinearity arising through the inextensible beam condition and material nonlinearity arising in the piezoelectric layers of the harvester. The model is validated against experimental measurements for different base accelerations and load resistances, and an investigation into the nonlinear behaviour indicates that nonlinear softening is caused predominantly by material nonlinearity. To reduce the beam amplitude and the resulting bending stress in the cantilever harvester, a bump stop is incorporated into the harvester design and the influence of the bump stop is modelled. Comparisons of theoretical predictions with experimental measurements indicate that taking account of the nonlinear behaviour improves the prediction significantly in some cases. Parameter studies are also conducted to investigate how the stop location and initial gap size between the harvester and stop affect the performance of the nonlinear energy harvester.     

Ultrasonic radiation in dynamic force microscopy

In dynamic force microscopy the cantilever of an atomic force microscope is vibrated at ultrasonic frequencies in the range of several 10 kHz up to several MHz while scanning a sample surface. The amplitude and phase of the cantilever vibration as well as the shift of the cantilever resonance frequencies provide information about local sample surface properties. In several operation modes of dynamic force microscopy, for example force modulation microscopy, tapping mode or atomic force acoustic microscopy, the sensor tip is in contact with the sample at least during a fraction of its vibration cycle. The periodic indentation of the tip with the sample surface generates ultrasonic waves. In this paper, the ultrasonic radiation of a vibrating cantilever into a sample and its contribution to the damping of the cantilever vibration are calculated. The theoretical results are compared to experiments.     

Thermal transitions of polymers measured by atomic force microscopy

A new method has been developed to measure thermal transitions by atomic force microscopy in the non-contact mode, using it as a dynamic mechanical analyser on a local scale. In this method the cantilever is oscillating above the polymer surface and the resonance frequency is measured as a function of the temperature. Thermal transitions of a polymer are clearly visible as a change in the characteristic-frequency behaviour of the cantilever. This paper introduces a simple model to explain the response of the cantilever caused by the transitions in the polymer and the related form of the frequency/temperature curves. This new technique adds a new dimension to the standard thermal analysis techniques, with which the thermal transitions of different polymer phases can be resolved individually for polymer blends or copolymers, for example in structured multiphase polymers. Received: 1 November 2001 / Accepted: 7 November 2001 / Published online: 23 January 2002     

A hybrid method of modal synthesis using vibration tests

The assembly of structures along continuous boundaries presents great difficulty in the context of modal synthesis. In order to solve such problems, a method is proposed in which a hybrid model is defined reflecting the dynamic behavior of a structure loaded along a boundary. It is based on Weinstein's method and corresponds to a generalization of the impedance matrix method. Generalized boundary co-ordinates are defined from branch modes obtained by introducing mass loading along the boundary. Thus, the hybrid model can be derived from testing as a result of two independent modal identifications. The method permits high precision prediction of the influence of strong structural modifications. Thus, in the case of rectangular plates, it has been possible to find the modes of a cantilever plate and of a plate with stiffeners from the free modes.     

横向三角脉冲磁场下悬臂导电薄板的动力稳定性分析

对于在仅有外加横向三角脉冲磁场下的悬臂导电薄板,给出了受到脉冲磁场影响时其动力稳定性的定量分析方法。揭示了在三角脉冲磁场下由于受到涡电流与总磁场相互作用所产生的面内磁体压力的作用,悬臂导电薄板会发生失稳。得到了在不同稳态磁场作用下受到三角脉冲磁场作用时导电薄板的磁弹性动力响应的数值模拟结果。在此基础上,给出了关于稳态磁场值、横向脉冲磁场峰值和脉冲持续时间等参数范围的磁弹性动力稳定性区域。     

FREE VIBRATION ANALYSIS OF CANTILEVER PLATES WITH STEP DISCONTINUITIES IN PROPERTIES BY THE METHOD OF SUPERPOSITION

Utilizing the superposition method, an analytical type solution is obtained for the free vibration eigenvalues and mode shapes of a cantilever plate with step discontinuities in plate properties. Property discontinuity lines run parallel to the clamped edge of the plate. Verification tests are performed for limiting cases by comparing computed eigenvalues with known eigenvalues for plates with uniform properties. Very good agreement is also obtained when computed results are compared with those obtained experimentally utilizing a test plate with discontinuities in thickness. Computed eigenvalues and mode shapes are presented for the benefit of other researchers. Besides the general interest, the problem has an application in the modelling of certain multi-story buildings during seismic studies.