Thin Cantilever Plate Research Articles

Free Vibration of Rectangular Composite Cantilever Plate and Its Application in Material Degradation Assessment

Many composite cantilever plate-like structures have found engineering applications in different industries. For attaining a meaningful assessment of the plate vibration characteristics, it is important to have efficient and effective methods for determining the natural frequencies/mode shapes of composite cantilever plates. In this paper, a method formulated on the basis of the Ritz method and a simple first-order shear deformation theory (SFSDT) is presented to analyze the free vibration of thin as well as thick rectangular composite cantilever plates for determining their natural frequencies. In the SFSDT, the total deflection is the sum of two deflection components, namely, bending and through-thickness shear-deformation-induced deflections. The successful application of the Ritz method together with the SFSDT for the free vibration analysis of thick composite plates relies on the selection of two independent sets of characteristic functions for the bending and through-thickness shear-deformation-induced deflections, respectively, to satisfy the requirements for the displacement and force conditions at the fixed edge of the plate. The novelty of the proposed method is that two independent sets of characteristic functions, namely, polynomials and trigonometric functions, which satisfy the displacement and force conditions at the fixed edge have been identified and used in the variational method to construct the eigenvalue problem for extracting the modal characteristics (natural frequencies and mode shapes) of the plate. It has been shown that the uses of the selected characteristic functions can produce excellent natural frequencies for both thin and thick composite cantilever plates. Some existing theoretical and experimental natural frequencies of thin as well as thick composite plates have been used to demonstrate the accuracy of the proposed method in predicting natural frequencies. The significant effects of through-thickness shear deformation on the natural frequencies of composite cantilever plates are studied to show the merit of the present method. Finally, for illustrating the application of the proposed method in free vibration analysis, a novel procedure established on the basis of the sensitivity analysis of natural frequencies is presented to assess the material degradation of composite cantilever plates. The numerical examples have shown that fewer than 10 iterations are required in the identification process to produce a good estimation of the current value for each material constant.

Comparative Study of Beam and Plate Theories for Moderate Aspect Ratio Wings

The main objective of the present study is to identify the best structural model for both aeroelastic and structural analysis of unswept rectangular wings with aspect ratios between 2 to 5. The paper presents a detailed study of the similarities and differences between the classical beam and plate theories, which provides new insight into this classical problem. To perform the aeroelastic analysis, the three-dimensional Peters aerodynamic model is coupled with classical plate theory. This combination leads to a new simpler three-dimensional aeroelastic model that can reduce the computational time. The results have been evaluated using both the unpublished and highly cited experimental data. Interestingly, by performing both structural and aeroelastic analyses, a comment can be stated that, even for relatively low aspect ratios () of thin rectangular cantilever plates, the model based on the beam theory is closer to the experimental model compared to the plate theory. This finding may be unexpected by many researchers. Also, using a novel approach, it has been shown that the most significant difference between the beam and plate theories results arises from a subtle difference in the displacement field, which has not been given attention so far.

Static and Dynamic Nonlinear In-Plane Effect in the Response of a Cantilevered Plate With Tip Mass Load: Theory and Experiment

Abstract In this article, two nonlinear effects are investigated. One is the effect of the static stiffness nonlinearity in changing the linear dynamic natural frequency, and the other is the combination of nonlinear stiffness and nonlinear inertia effects in changing the nonlinear dynamic transient response due to a change in the initial release state of the system. A theoretical model has been developed for a cantilevered thin plate with a range of length-to-width ratio using beam theory and considering both stiffness and inertial nonlinearities in the model. Lagrange's equation was used to deduce nonlinear inertia and stiffness matrices for a modal representation. Some insights into how these nonlinear components influence the beam response are presented. Measurements with both a hammer test and also a release test of cantilevered thin plates were done using different configurations and tip mass values. Results from static and dynamic analyses using the linear and the nonlinear theoretical models show good agreement between theory and experiment for natural frequencies and the amplitude displacements versus time.

A dynamic stiffness improvement method for thin plate structures with laminated/embedded shape memory alloy actuators

Cantilever thin plates are commonly used as external structures in different aircraft, e.g., missile fins and aircraft rudders, which have important functions such as adjusting attitude and providing lift. However, when the aircraft is flying at a high speed, the elastic modulus of the material will be significantly reduced due to the thermal effect. This will decrease the dynamic stiffness of the aircraft and thereby lead to flutter and other serious problems. This study aims to investigate the dynamic stiffness improvement method for thin-walled structures at high temperatures with shape memory alloy (SMA) actuators. A theoretical model of an SMA laminated aluminum thin plate is established first. Then an improved stiffness enhancement method with distributed SMA actuators is proposed. The configuration of the actuator is designed and verified by the finite element method. A frequency prediction model of the thin plate structure is established with Back Propagation (BP) neural network and optimal placement of the actuator is achieved based on a genetic algorithm. The validity of the proposed method is verified through both numerical simulations and high-temperature frequency sweep experiments. The results show that the stiffness of the thin plate structure can be significantly improved by the SMA actuators. Specifically, the first-order bending frequency and first-order torsional frequency of the structure can be improved by nearly 10% at 200 °C.

Precise machining based on Ritz non-uniform allowances for titanium blade fabricated by selective laser melting

Selective laser melting (SLM) is an increasingly concerned trend in Ti-6Al-4V blade manufacturing, while the SLMed Ti-6Al-4V blade cannot be used directly because of poor surface integrity and high residual stresses. Precise machining after SLM is a feasible solution but also a challenge. The low rigidity of the blade will lead to deformation when machining. The deformation can lead to surface error and may make defect parts. Two-step machining processes to address the problems were proposed in this paper. First, a non-uniform allowances distribution was allocated and optimized in semi-finishing based on Ritz solution to elastic deformation. The blade was simplified as a cantilever thin plate with various thicknesses, and the thicknesses of finishing allowances were designed and optimized on the premise of ensuring the thin-wall stiffness of the blade, so as to realize the design of Ritz non-uniform allowances. Then, finishing machining was conducted to achieve precise parts. A blade deformation model was established to evaluate part distortions with cutting force moving and changing. Finite element analysis (FEA) and experimental validation in ball-end milling of a blade were conducted. FEA results and experimental results showed dimensional errors have been reduced up to 50%. Further surface tests demonstrated that the mean surface roughness reduced from 7.88 to 0.815 μm. And the residual surface stresses of the SLM samples changed after semi-finishing machining due to the residual stresses relaxation and redistribution. The results demonstrated that the proposed method enhanced the surface quality of the blade fabricated by SLM.

Effect of crack on free vibration of a pre-stressed curved plate

In this study, the effect of crack on free vibration of a large deflected cantilever plate, which forms the case of a pre-stressed curved plate, is investigated. A distributed load is applied at the free edge of a thin cantilever plate. Then, the loading edge of the deflected plate is fixed to obtain a pre-stressed curved plate. The large deflection equation provides the non - linear deflection curve of the large deflected flexible plate. The thin curved plate is modeled by using the finite element method with a four-node quadrilateral element. Three different aspect ratios are used to examine the effect of crack. The effect of crack and its location on the natural frequency parameter is given in tables and graphs. Also, the natural frequency parameters of the present model are compared with the finite element software results to verify the reliability and validity of the present model. This study shows that the different mode shapes are occurred due to the change of load parameter, and these different mode shapes cause a change in the effect of crack.

Vibration stability of a cantilevered thin plate with inner fluid circulation

The vibration of a cantilevered thin plate with inner fluid circulation is a typical situation in engineering. The plate’s vibration can be considered a combination of bending motion and torsional motion. With the force generated by the inner fluid circulation on the plate, these two types of motion affect each other and raise the problem of stability. In the present study, the vibration stability of different vibrating pattern thus generated is investigated. It is found that with the increase of the flow velocity, torsional mode frequencies decrease synchronously, whereas their modal shapes remain unchanged, which results in the buckling of the plate. Under the circumstance of coupled vibration, because both bending and torsional modal frequencies vary with the flow velocity, a phenomenon of coupled-mode flutter occurs between a certain pair of modes with frequency orders of (2k) and (2k − 1) and vibration becomes unstable.

Research on theory, simulation and measurement of stress behavior under regenerated roof condition

To determine ground stress behavior under the special condition of a regenerated roof, we established a model of elastic rectangular cantilever thin plates. Moreover, the critical conditions for bending and fracturing the regenerated roof during mining were analysed. Meanwhile, by applying continua FLAC-3D numerical simulation, this research simulated changes in the stress and strain on a regenerated roof during mining and proposed prevention and control methods for dynamic disasters. The results show that: (1) the thinner the regenerated roof, the larger the tensile stress on the roof based on analysis using the theoretical model. Furthermore, the longer the advance distance during mining, the greater the tensile stress on the regenerated roof. (2) By analysing simulation results, during the fracturing of the regenerated roof, roof displacement firstly suddenly increases and then gradually decreases to be stable. Floor-heave-induced displacement presents a divergent state, that is, increases outwards in an elliptical manner. (3) For control of the regenerated roof, monitoring on activities of the roof should be strengthened and stress should be relieved timeously. Moreover, effective support methods should be taken to prevent development of hazards on working faces and roadways caused by the widespread behavior of the roof.

Dynamics of inverted flags: Experiments and comparison with theory

The study of the dynamics of cantilevered thin flexible plates in reverse axial flow – also known as inverted flags – has become of significant interest, partly due to their energy harvesting potential. This paper presents fresh experimental results, aiming to enhance our understanding of the dynamics of inverted flags, particularly concerning stability and global dynamics sensitivity to various system parameters, such as the aspect ratio (i.e. height-to-length ratio) and flag material. This is achieved through testing flags of various dimensions, made from different materials. The dynamics of the system is presented in the form of bifurcation diagrams in which the flag tip angle is shown as a function of the dimensionless flow velocity. The frequency content of oscillatory motions is presented in the form of spectrograms. Also, time-histories, phase-plane portraits, as well as power spectral density and probability density function plots are presented for different flow regimes. Interesting dynamical features, such as small- and large-amplitude flapping, static divergence, large-amplitude buckling, and chaotic motions, have been observed experimentally, some for the first time. It is shown that for flags of very low aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of large aspect ratio, on the other hand, the undeflected static equilibrium becomes unstable via a supercritical pitchfork bifurcation leading to static divergence (buckling) at a sufficiently high flow velocity; at higher flow velocities, past the pitchfork bifurcation, a supercritical Hopf bifurcation materializes leading to flapping motion around the deflected static equilibrium; at even higher flow velocities, flapping motion becomes symmetric around the undeflected static equilibrium. Interestingly, it is found that heavy flags may exhibit large-amplitude flapping right after the initial static equilibrium, provided that they are subjected to a sufficiently large disturbance. In addition to experiments, nonlinear analytical models for the two asymptotic cases of zero and infinite aspect ratios are outlined, the results of which are used for comparison with the experimental results. Theory and experiments are found to be in reasonably good agreement.

Resonance Analysis of Cantilever Plates Subjected to Moving Forces by a Semi-Analytical Method

Cantilever plate structures are widely used in civil and aerospace engineering. Here, a semi-analytical method is proposed to calculate the dynamic responses of cantilever plates subjected to moving forces. The Rayleigh–Ritz method is used to obtain the semi-analytical modal frequencies and shapes of a thin, isotropic, and rectangular cantilever plate using the assumed mode shapes that fulfill the boundary conditions of the plate. The modal superposition method is used to decouple the motion equations of the cantilever plate to obtain a series of modal equations. Then, the generalized forces are transformed into a Fourier series in terms of discrete harmonic forces. The dynamic responses of the cantilever plate are obtained by superimposing the analytical responses of a number of single-degree-of-freedom modal systems under discrete harmonic forces. The proposed semi-analytical method is verified through comparison with the numerical method. Then, the vibration of the cantilever plate under the action of moving forces is investigated based on the semi-analytical results. It is found that the contribution of the high-order modes to the dynamic responses of the plate cannot be ignored. In addition, the wavelengths of the mode shapes not only affect the magnitude of the modal forces but also the dominant frequency of the modal forces. Resonant responses of the plate are produced by the moving forces when the load interval equals the wavelength of the mode shape of a high-order mode and the exciting frequency of the moving forces equals the natural frequency of this mode.

Coating position optimization for a hard-coating thin-plate structure based on resonance response

The ultimate goal of vibration reduction using hard coating is to suppress the resonance peaks of the structure. Thus, the optimal position of the coating as a function of the resonance response will better satisfy the design requirements. Based on a full consideration of the continuous coating area, a method for optimizing the coating position with the objective of minimizing resonance response was developed for the hard-coating thin plate. A semi-analytical analysis model of the partially coated cantilever thin plate was created, and the formula for solving the vibration response was identified by the mode superposition method. An optimization model was established. In the model, the position coordinates of coating patches are the design variables, and the objective function is the maximizing of the reciprocals of the resonance peaks, which is equivalent to minimizing the resonance response. A multi-population genetic algorithm (MPGA) has been proposed to solve the coating position optimization problem. Finally, a cantilever titanium plate coated with NiCrAlCoY + YSZ hard coating was chosen to demonstrate the presented method. The results show that the obtained optimized results can guarantee that the resonance peaks of the hard-coating thin plate are always less than those of general cases, whether it is for single-order or multi-order optimization objectives.

Passive vibration control of plate using nodal patterns of mode shape

Passive constrained layer damping (PCLD) is nowadays a general method for reducing vibrations. Constrained layer damping (CLD) is a very good technology in terms of reducing vibration energy. This paper deals with optimal damping of first three modes of a thin square cantilever plate by constrained viscoelastic layers which are positioned according to the nodal patterns of the mode. In essence, the CLD patches are cut according to the nodal patterns and then applied on to the bare plate. Thus, this approach uses prior information of the deformation patterns of the plate. The ABAQUS finite element analysis software was used to create viscoelastically damped plate models subjected to forced vibration. The direct steady-state dynamic analysis was used to obtain the frequency response functions from which the resonance frequencies of the models were derived. Vibration control using nodal pattern CLD patches is a better way to apply CLD treatment to a plate. This is because the mode shapes give good predictions of where the maximum strains are in the plate as well as the design and size of the damping patches required. It is believed that the methodology proposed in this research work is capable of providing good results.

Wake vortex regimes of a pitching cantilever plate in quiescent air and their correlation with mean flow generation

Jet flows induced by oscillating cantilever plates enable power-efficient cooling enhancement and fluid acceleration. We report a combined experimental and numerical study of the vortex regimes present in the wake of a harmonically oscillating thin cantilever plate in a quiescent fluid and analyze their effect on the flow generation downstream. We use Particle Image Velocimetry (PIV) in conjunction with two-dimensional numerical simulations to investigate the vortex evolution around the trailing edge of the plates with different geometries and vibrational properties. Our observations suggest the existence of three distinct regimes in the wake: non-propagating, intermediate and propagating. Comparing the temporal decay of the vortex circulation in different regimes shows that different mechanisms are involved in the formation of these vortical patterns. A regime map is proposed next, denoting the incident of each vortex regime as a function of relevant dimensionless parameters. Our analysis of the mean jet on the normal mid-plane, as quantified by the momentum-averaged Reynolds number Rejet, shows that the induced jet downstream the trailing edge is tightly correlated with the identified vortex regimes. The present study improves our understanding of vortex generation and propagation in oscillating cantilevers and facilitates optimized design and operation of piezoelectric fans and similar devices.

Modelling time-domain vibratory deflection response of thin-walled cantilever workpieces during flank milling

While machining thin-walled workpieces, the chatter and corresponding damage to the machined surface can be avoided if that is predicted well in advance. Currently, investigations of machining vibrations mainly rely on experiments, which are costly and difficult to perform and FEM simulation methods that are many times computationally cumbersome. This work presents a mathematical model to obtain time domain forced vibration deflection response of a thin rectangular cantilever plate for any time and space varying external loads. The model is adapted for a milling process to obtain machining vibration response of a thin impeller blade by simplifying its geometry as a thin rectangular cantilever plate. The developed model was verified by checking the natural frequencies and mode shapes obtained by the model with the Finite Element simulation results for a thin cantilever plate. Peripheral milling experiments were performed to machine a thin rectangular cantilever plate of Ti-6Al-4V alloy on a vertical 3-axis CNC milling machine. Milling forces and workpiece acceleration were acquired during machining. Workpiece material properties and moving machining forces obtained by Linear Edge Force Model are input to the model. The model was validated by comparing vibratory deflection of thin cantilever plate obtained by the model and by the acquired experimental data. In general, the developed model can be used to obtain time-domain forced vibration response of thin rectangular cantilever plate workpieces.

On new analytic free vibration solutions of rectangular thin cantilever plates in the symplectic space

In this paper, we obtain accurate analytic free vibration solutions of rectangular thin cantilever plates by using an up-to-date rational superposition method in the symplectic space. To the authors’ knowledge, these solutions were not available in the literature due to the difficulty in handling the complex mathematical model. The Hamiltonian system-based governing equation is first constructed. The eigenvalue problems of two fundamental vibration problems are formed for a cantilever plate. By symplectic expansion, the fundamental solutions are obtained. Superposition of these solutions are equal to that of the cantilever plate, which yields the analytic frequency equation. The mode shapes are then readily obtained. The developed method yields the benchmark analytic solutions with fast convergence and satisfactory accuracy by rigorous derivation, without assuming any trial solutions; thus, it is regarded as rational, and its applicability to more boundary value problems of partial differential equations represented by plates’ vibration, bending and buckling may be expected.

Partial Surface Damper to Suppress Vibration for Thin Walled Plate Milling

The material removal rate and required workpiece surface quality of thin-walled structure milling are greatly limited due to its severe vibration, which is directly associated with the dynamic characteristics of the system. Therefore, the suppression of vibration is an unavoidable problem during milling. A novel partial surface damping method is proposed to modify the mode of the thin walled cantilever plate and to suppress vibration during milling. Based on classical plate theory, the design criterion is analyzed and configuration of the partial surface damper is introduced, in which viscoelastic plate and constraining plate are attached to the surface of the plate to increase the system’s natural frequency and loss factor. In order to obtain the energy expression of the cutting system, the Ritz method is used to describe the unknown displacements. Then, with Lagrange’s equation, the natural frequency and loss factor are calculated. In addition, the plate is divided into a finite number of square elements, and the regulation of treated position is studied based on theoretic and experimental analysis. The milling tests are conducted to verify its damping performance and the experimental results show that with treatment of partial surface damper, the deformation of the bare plate is reduced from 0.27 mm to 0.1 mm, while the vibration amplitude of the bare plate is reduced from 0.08 mm to 0.01 mm. The proposed research provides the instruction to design partial surface damper.

Damping Optimization of Hard-Coating Thin Plate by the Modified Modal Strain Energy Method

Due to the medium and small damping characteristics of the hard coating compared with viscoelastic materials, the classical modal strain energy (CMSE) method cannot be applied to the prediction of damping characteristics of hard-coating composite structure directly. In this study, the CMSE method was modified in order to be suitable for this calculation, and then the damping optimization of the hard-coating thin plate was carried out. First, the solution formula of modified modal strain energy (MMSE) method was derived and the relevant calculation procedure was proposed. Then, based on the principle that depositing the hard coating on the locations where modal strain energy is higher, the damping optimization method and procedure were presented. Next, a cantilever thin plate coated with Mg-Al hard coating was taken as an example to demonstrate the solution of the modal damping parameters for the composite plate. Finally, the optimization of coating location was studied according to the proposed method for the cantilever thin plate, and the effect of the coating area on the damping characteristics of hard-coating plate was also discussed.

基于频响函数矩阵法的薄板结构阻尼系数识别

机械系统的粘性或结构阻尼系数可以从空间上描述系统的阻尼分布，这对于阻尼结构减振设计具有重要的意义。本文研究基于频响函数矩阵法来辨识悬臂薄板结构系统阻尼系数。详细叙述了频响函数矩阵法的原理和测试流程。在此基础上组建了辨识结构系统阻尼系数的测试系统，对薄板结构进行了测试分析。从识别的阻尼矩阵可以看出，无论是粘性阻尼还是结构阻尼，薄板根部的数值均大于自由端的数值。比较不同频带的阻尼系数，可以看出在低频段粘性阻尼系数较大，而在高频段结构阻尼系数较大。 The distribution of system damp could be described by the internal friction and structure damped coefficient. It was important for design of reducing vibration of the damping structure. The damping coefficient of cantilever thin plate was identified by the frequency response function matrix method in this paper. The principle and test process were introduced. The test system was built to identify the damping coefficient. The thin plate was analyzed by the system. The data of the plate root were all bigger than that of freedom end no matter the damping matrix was inherent friction or the structural damping coefficient. The inner friction coefficient of low frequency band was bigger, while the structural damping coefficient of high frequency band was bigger.

Actively Tuned Vibration Absorber using Smart Adaptive Composites

Shape memory materials (SMMs) are known for their excellent vibration absorption properties. The current work is focused on study of vibration absorption characteristics of Smart Adaptive Composites (Shape Memory Composites) (SACs), as tuned vibration absorbers. Thin cantilever plate of SAC works as the absorber system in the configuration.

Sparse regularization for force identification using dictionaries

The classical function expansion method based on minimizing l2-norm of the response residual employs various basis functions to represent the unknown force. Its difficulty lies in determining the optimum number of basis functions. Considering the sparsity of force in the time domain or in other basis space, we develop a general sparse regularization method based on minimizing l1-norm of the coefficient vector of basis functions. The number of basis functions is adaptively determined by minimizing the number of nonzero components in the coefficient vector during the sparse regularization process. First, according to the profile of the unknown force, the dictionary composed of basis functions is determined. Second, a sparsity convex optimization model for force identification is constructed. Third, given the transfer function and the operational response, Sparse reconstruction by separable approximation (SpaRSA) is developed to solve the sparse regularization problem of force identification. Finally, experiments including identification of impact and harmonic forces are conducted on a cantilever thin plate structure to illustrate the effectiveness and applicability of SpaRSA. Besides the Dirac dictionary, other three sparse dictionaries including Db6 wavelets, Sym4 wavelets and cubic B-spline functions can also accurately identify both the single and double impact forces from highly noisy responses in a sparse representation frame. The discrete cosine functions can also successfully reconstruct the harmonic forces including the sinusoidal, square and triangular forces. Conversely, the traditional Tikhonov regularization method with the L-curve criterion fails to identify both the impact and harmonic forces in these cases.