Vibration Of Thin Plates Research Articles

A semi-analytical method using auxiliary sine series for vibration and sound radiation of a rectangular plate with elastic edges

This paper proposes an efficient semi-analytical method using auxiliary sine series for transverse vibration and sound radiation of a thin rectangular plate with edges elastically restrained against translation and rotation. The formulation, constructed by two-dimensional sine and/or cosine series, can approximately express the bending displacement, and calculate vibration and sound radiation under excitation of point force, arbitrary-angle plane wave, or diffuse acoustic field with acceptable accuracy. It is also applied for baffled or unbaffled conditions. A post-process program is developed to predict vibrating frequencies and modes, mean square velocity spectrum, and sound transmission loss via reduced-order integrals of radiation impedances. The method is validated by experiment and simulation results, demonstrating accurate and efficient computation using a single program for transverse vibration and sound radiation of a plate under different elastic boundary conditions and different excitations. Formulas given in this paper provide a basis for the code development on transverse vibration and sound radiation analysis of thin plates.

Analytical Solutions Based on Hamiltonian System for the Free Vibration of Polyline Thin Plates

It is challenging to establish analytical solutions for conventional inverse and semi-inverse approaches because of the intricacy of standard mathematical calculations. In this study, the symplectic superposition approach is used to provide the analytical solution for the free vibration of a thin plate with a polyline shape. The polyline plate is separated into four sub-plates based on the superposition principle, and then the problem is solved by using the symplectic eigen-expansion method. Multiple sets of constants dependent on boundary conditions can be found by applying a specific mechanical quantity to the edge of each sub-plate. The four sub-plate solutions can be combined to provide the final analytical solution. By contrasting the numerical results obtained under various boundary restrictions, the correctness and effectiveness of the method are confirmed. The frequency values decrease with increasing a2/a1 due to the decrease of structure’s stiffness. The method proposed in this work can be extended to solve the vibration of anisotropic thin plate with other boundary conditions and irregular plates.

Vibration-Induced-Flow Mechanism and Its Application in Water Surface Robot.

Vibration is a common strategy for aquatic organisms to achieve their life activities, especially at the air-water interface. For the locomotion of small creatures, the organs with plate features are often used on water surfaces, which inspires relevant studies about using thin plates for robot propulsions. However, the influence of the general deformations of thin plates on the generated flow fields has not been considered. Here, a comprehensive investigation is conducted about the flow fields that arose by vibrations of thin plates and the potential as locomotion strategies are explored. It is discovered that as thin plates are subjected to vibration excitations on the water surface, the produced flow fields are mainly determined by the vibration shapes, and the influence rules of plate deformations on the flow fields are identified. The main factors producing asymmetric flow fields are analyzed to realize the morphology control of the flow fields. Then, to determine effective locomotion strategies on the water surface, the flow fields stimulated by integrated vibration exciters are explored, and 2 water surface robots are developed consequentially, which exhibit superior motion performance. This work reveals the basic rules of the vibration-induced-flow mechanism by thin plates and establishes new locomotion strategies for aquatic robots.

Study on the Self-Synchronization and Sommerfeld Effect of Rectangular Thin Plate Vibration System Excited by Double Induction Motors

Study on the Self-Synchronization and Sommerfeld Effect of Rectangular Thin Plate Vibration System Excited by Double Induction Motors

The Vibration Dynamic Model for Blister Detection in Medium-Density Fiberboard

Ultrasonic detection is currently used in the industry of medium-density fiberboard to detect blister defects. Due to the small detection area of a single sensor, multiple sensors need to be used, which results in high costs. Starting from elastic thin plate vibration theory, this paper builds a vibration dynamic model to detect blisters. The size and depth of the blister area can be established by determining the natural frequency of the thin plate vibration in the blister area. In this model, if the elastic modulus and density are known, the natural frequency of the thin plate vibration at the blister place is directly proportional to the blister depth and inversely proportional to the square of the blister radius. The size and depth of the blister can be determined by measuring the first third-order natural frequency of this area of research. A total of 25 specimens with blister sizes and depths were simulated, and the natural frequencies of the specimens were measured. Subsequently, the detection model was verified by comparing its experimental data with theoretical values. The theoretical value was highly consistent with the measured data. The measured values of the first, second, and third-order natural frequencies were slightly smaller than the theoretical calculated values, with average relative deviations of −1.6%, −1.34%, and −1.03%, respectively. As the order progressed, the deviation exhibited a downward trend, and the third-order natural frequency displayed the smallest deviation and highest accuracy. The proposed vibration dynamic model can detect larger blister areas by measuring the natural frequency, which can overcome the shortcomings of small ultrasonic detection areas in current actual industries. Thus, the practical online blister detection device is expected to be further developed.

Nonlinear vibration analysis of thin plate with periodically distributed tunable-stiffness nonlinear oscillators in subsonic flow

This paper provides a new passive control method for nonlinear vibration of subsonic thin plates. The control method uses periodically distributed tunable-stiffness nonlinear oscillators (NLO). By using von Karman’s nonlinear plate theory and subsonic aerodynamic model, the nonlinear dynamic model of thin plate is established based on Hamilton’s principle. Through analyzing the system’s nonlinear dynamics bifurcation, the impact of periodic distribution on the stability is discussed. The amplitude-frequency responses are discussed under different NLO installation modes and design parameters by using Matcont software. The results show that the periodically distributed tunable-stiffness nonlinear oscillators can improve the stability of the system and have a remarkable suppression effect. The optimal control scheme for vibration suppression by periodically distributed tunable-stiffness nonlinear oscillators is obtained. The results obtained from this research can offer a valuable understanding of nonlinear vibration control, thereby enabling the identification of various control approaches that can be implemented in practical applications during future design endeavors.

Modeling and mathematical approach for mechanical natural frequencies of structural plates with free edges

The current study aims to demonstrate an easy mathematical procedure and simple to be used by engineers to maintain the mechanical natural frequencies (Eigen values of free vibration) of thin rectangular structural plates with free edges included in boundary conditions. The reason for this easiness and simplicity is that new method of analysis depends mainly on using mechanical minimum energy concept represented in the Rayleigh–Ritz method with appropriate real polynomials as admissible functions. The new procedure guides these functions to satisfy the boundary conditions at the middle points on the free edges of the plate beside the nodal points of the vibration modes along each edge direction. Therefore, there are no complicated or imaginary terms in the developed functions while they lead to surprisingly acceptable and even accurate results when compared with other works and Ansys program models.

A meta-plate with radial rainbow reflection effect for broadband suppression of vibration and sound radiation

The suppression of vibration and sound radiation of thin plates has always been a concern in engineering fields such as aerospace, transportation, etc. The rainbow effect refers to that the waves with different frequencies can be stopped in different positions, where the wave energy is also enhanced, as the result of the spatial slowing-down of wave velocities. This effect exhibits great potential in suppressing structural vibration and sound radiation, but corresponding studies have not been reported yet. In this paper, a meta-plate composed of ring-type unit-cells with gradient heights is proposed to generate the radial rainbow reflection effect. Upon analyzing the dispersion relationship of unit-cells and revealing the rainbow reflection mechanism, the broadband vibration and sound radiation performances of a square meta-plate with free boundaries are studied. Numerical and experimental results demonstrate that the proposed meta-plate can suppress the vibration and sound radiation in broadband only with the extreme-low structural inherent loss. The wavenumber spectra of vibration velocity and supersonic intensity at typical frequencies including local and global resonant ones are analyzed to reveal the suppression mechanism of sound radiation, which is the vibration redistribution and systematic vibration reduction resulting from the rainbow reflection effect. Our design enhances the stiffness of plate structures and requires no additional damping, providing a novel approach for structural vibration and sound radiation suppression.

Dynamic characteristics of a relief valve captured by a combination of fluid–solid coupling and feedforward neural network (FNN)

Considering multidisciplinary characteristics of thin plate vibration, fluid–solid coupling, and other aspects of a relief valve controlled by annular thin plates, a dynamic finite element (FE) model in view of fluid–solid coupling is firstly established for capturing relationships between dynamic characteristics of crucial indexes and partial working conditions. Secondly, the partial dataset of FE model under different conditions is statistically analyzed, and it will be utilized to train the feedforward neural network (FNN) model. The training process of FNN could be completed if results drawn from the FNN model are highly consistent with those of the FE model. Thirdly, dynamic characteristics under more conditions will be predicted through such a trained model, and dynamic behaviors from the FE model for same conditions of the FNN model are also obtained. Finally, comparing with results from the FE model, the maximum absolute error of steady-state displacement from the FNN is 0.0052 mm in an instance, thus verifying the rationality of this combined method. Consequently, such a combination of the FE model and the FNN model presents high accuracy and avoids repeated calculations of FE model with long times.

Free Vibration of Annular Plates Constrained by Translational and/or Rotational Springs on Outer and Inner Edges

This paper presents comprehensive lists of natural frequencies for vibration of thin isotropic annular plates with outer and inner edges elastically constrained by translational and rotational springs. A method is extended from a previously presented Ritz approach to include effects of the springs. In numerical examples, sixteen models are considered to cover all general cases of locating translational and/or rotational springs on outer and/or inner edges in addition to classical boundary conditions of free and simple supported edges. After convergence study and comparison test in specific cases are made to validate solution accuracy, frequency parameters of the sixteen models are summarized and frequency variations with increasing spring stiffness are illustrated.

Unravelling the excitation mechanism of very long-period (VLP) tremors in the Gulf of Guinea: evidence for vibrations of thin surface crustal plates

SUMMARY The Gulf of Guinea exhibits a continuous emission of narrow-band and long-period signals (16, 26 and 27 s) on teleseismic records, yet the underlying excitation mechanism remains unclear. This study establishes a connection between these tremors and the vibration of thin, decoupled crustal plates at unexplored volcanoes in the gulf. We first formulate the damped plate oscillation equation, by incorporating the vibration of the thin surface crustal plate and magma flow in the subsurface sill. The findings reveal that a fundamental-mode vibration with a period of several dozen seconds can be induced by a crustal plate that is less than 1.0 km thick but extends over tens of kilometres in both length and width, given a subsurface sill depth exceeding 10.0 cm. The thin plate hypothesis also allows for excitation of a few overtone modes, but such waves in higher frequencies diminish over long distances, leaving only the monotonous fundamental-mode vibration at teleseismic stations. The long duration of Guinea tremors at each recurrence is attributed to the presence of low viscosity basaltic magma, which influences the damping factor. Direct wave loads at the shallow gulf serve as the primary vibration source, accounting for seasonal variations and recurring patterns. Sporadic energy bursts may also occur due to large storms. Radiation patterns of Guinea tremors are linked to the geometric structure of the thin plate. Our theoretical estimates of tremor spectra closely align with observed data, confirming the model’s accuracy in capturing reported Guinea tremor characteristics. This study provides valuable insights into the origins of very long-period tremors at continental volcanoes.

Demonstration of nonlinear tuning curve vibration response using a homemade analog sweep spectrum analyzer

A super-heterodyne sweep spectrum analyzer has been designed (using analog circuitry), built and tested to cover the 5 to 1000 Hz range. The analyzer employs a linear ramp voltage versus time, driving a voltage-controlled oscillator VCO that generates a sweep from 32.768 kHz to 33.768 kHz in 70 s. An analog AD734 multiplies the VCO signal by a sinusoidal signal (generated by a watch crystal 32.768 kHz oscillator). The “mixer” signal is low-pass filtered and amplified to generate a 5-1000 Hz swept tone using a 2-in. speaker. The accelerometer (mounted on a thin circular clamped acrylic plate) vibration response involves multiplying this signal by the VCO, then filtering this “mixer” signal using a 4-stage watch crystal ladder filter with a 1 Hz bandwidth. This signal is squared and low-pass filtered to generate the time (converted to frequency) vs. mean-squared voltage response on an oscilloscope. In the demonstration, 300 g of 6 mm diameter glass beads are supported by the 11.4 cm diam, 3.2 mm thick plate and upper rigid cylindrical wall column. The nonlinear tuning curve vibration response is recorded for various incremental drive amplitudes to demonstrate that the tuning curve resonant frequency significantly decreases with increasing drive amplitude.

Frequency dependent exact trial functions in Galerkin boundary method for free vibration analysis of thin plate

Method of frequency dependent exact trial functions for thin plate vibration based on fundamental solutions of Voigt is suggested. Contrary to other methods, where one of the functions is chosen as the scale and specific boundary condition dependent, it employs only the frequency dependent functions for both space coordinates. Thus they are scale independent, so the same functions can be used for different boundary conditions and plate dimensions. General rule for the choice of exact trial functions is formulated. The boundary conditions are satisfied according to the Galerkin boundary method, which actually minimizes the energy residuals with respect to each weight function (fundamental solution).The verification of method is performed on examples of rectangular plate with various boundary conditions. Several arbitrarily chosen sets of trial functions were employed in calculation of the same tasks, and it is shown that accuracy of the method is almost independent from the choice of them. Even the small number of trial functions provides the very accurate results for natural frequencies.

A symplectic analytical wave propagation model for forced vibration of thin plate with acoustic black hole indentation

In recent years, acoustic black holes (ABHs) have been revealed as an effective bending wave control technology and widely used to regulate the vibration and noise reduction performance of plate structure. However, the isolation effect of most traditional ABH plate can only be aroused at frequencies above the cut-off frequency. In order to widen the frequency range of the ABH effect, this paper embedded single and multiple ABH indentations at the end of plate. For the forced vibration response of the ABH indentation plate, an analytical wave propagation model was proposed based on the symplectic method. The governing equations of bending vibration of thin plate was first introduced into the symplectic dual system, then the symplectic analytical solution of wave propagation parameters and waveforms were obtained. A Gaussian-discretization scheme was used to discretize the ABH indentation, whereas the proposed method was used to evaluate the wave amplitude. The vibration response was finally obtained using the superposition principle and wave propagation, reflection, and transmission relationships. The effectiveness of the proposed method was verified using the finite element results. The results showed that the ABH indentation plate, especially the plate with multiple ABH indentations, had significant vibration reduction performance. Considering that embedding ABH indentation can weaken the structural rigidity of the plate, the ABH indentation plate with reinforcing stiffeners was then proposed to alleviate this limitation, and the influence of the width and quantity of reinforcing stiffeners on the vibration response was studied. The results indicated that the increase of width will to some extent weaken the ABH effect, and the quantity of stiffeners had a significant impact on the ABH effect.

Nonlinear vibrations of auxetic honeycomb thin plates based on the modified Gibson functions

Auxetic honeycombs possess many excellent properties, making them ideal cores for sandwich structures that can absorb energy effectively. This paper investigates the nonlinear vibrations of auxetic honeycomb composite plates, focusing on the primary resonance, super-/sub-harmonic resonances of the composite plates. Two pure panels sandwich an auxetic honeycomb core to form the honeycomb composite plate. The proposed modified Gibson function enables the derivation of effective material properties. By considering geometric nonlinearity, the nonlinear motion equations are derived through the implementation of the Hamilton principle, and then nonlinear ordinary differential equations are given by introducing stress functions and the Galerkin method. The nonlinear response curves are then determined by the multiple scale method. The impact of important parameters on the nonlinear vibration of auxetic honeycomb composite plates is evaluated in this study.

Two-dimensional orthotropic plate problems in a thermal environment: Refined crack modelling

Three two-dimensional (2D) refined models are devised for the analysis of orthotropic thin plates with part-through surface and internal cracks under in-plane forces due to temperature differences. The refined models are used to formulate the governing equations for cracked orthotropic thin plates in a thermal environment. Subsequently, the refined models and their ensuing modified forms can be directly applied to all-over and finite-length cracks, respectively. Using an interface-fitted numerical scheme (i.e., matched interface and boundary technique), the resulting equations of motion for the buckling and free vibration of cracked orthotropic thin plates are solved. The effects of various crack types and dimensions on the critical buckling temperatures and vibration frequencies of orthotropic thin plates are investigated. The present models are also validated by comparing their numerical results with the line-spring model and three-dimensional finite-element solutions. Illustrative examples, including boron fiber/epoxy and glass-fibre-reinforced composite materials, are considered to show the superiority of the proposed models over the conventional one, especially in terms of understanding the effect of in-plane forces arising from temperature differences at the cracked location.

Dynamic modelling and chaos control for a thin plate oscillator using Bubnov–Galerkin integral method

The utilization of thin plate systems based on acoustic vibration holds significant importance in micro-nano manipulation and the exploration of nonlinear science. This paper focuses on the analysis of an actual thin plate system driven by acoustic wave signals. By combining the mechanical analysis of thin plate microelements with the Bubnov–Galerkin integral method, the governing equation for the forced vibration of a square thin plate is derived. Notably, the reaction force of the thin plate vibration system is defined as f = α|w|, resembling Hooke’s law. The energy function and energy level curve of the system are also analyzed. Subsequently, the amplitude–frequency response function of the thin plate oscillator is solved using the harmonic balance method. Through numerical simulations, the amplitude–frequency curves are analyzed for different vibration modes under the influence of various parameters. Furthermore, the paper demonstrates the occurrence of conservative chaotic motions in the thin plate oscillator using theoretical and numerical methods. Dynamics maps illustrating the system’s states are presented to reveal the evolution laws of the system. By exploring the effects of force fields and system energy, the underlying mechanism of chaos is interpreted. Additionally, the phenomenon of chaos in the oscillator can be controlled through the method of velocity and displacement states feedback, which holds significance for engineering applications.

Differential equations of oscillation of thin plates with point bonding

Abstract The relevance of the research results presented in this article lies in the general concept of elasticity theory, which integrates the bases of theoretical-empirical physics, practical mathematics and the natural implementation of modelling results in the fields of industrial (to a large extent in the design and manufacture of aircraft and naval aircraft shells and fuselages), construction (more so in the design and formation of multi-layer building structures), electronics and other areas of the science and industry complex. The aim of the study is to form a mathematical model of thin plate vibration based on a system of differential equations for the computational case of point bonding. The method of scientific search (Multilocal Literature Review) is used to achieve the set goal, which made it possible to establish the actual scientific-theoretical basis of the investigated problem, the method of mathematical modelling allowing to systematize the systems of differential equations developed earlier and formed in the framework of the present study, both for the general concept of the theory of elasticity of thin plates and for a selected calculation situation with partial constraints in the form of point bond imposing. As a result of the investigations conducted in the framework of this study, a mathematical model of the oscillations of thin plates bounded by special point-coupling conditions has been obtained, consisting of a system of differential equations obtained by successive iterations of mathematical transformations for the generated local boundary conditions. The mathematical model obtained is of practical scientific interest. The developed model environment forms a complete mathematical theory of elasticity for the formulated problem of the oscillatory process of thin plates with bounding point couplings. This problem has not received a satisfactory mathematical apparatus because of the complexity and cumbersomeness of analytical methods to describe the investigated elastic object.

Research on the vibration characteristics of an embedded elastic metamaterial plate

For a class of embedded elastic metamaterial plate, a general vibration analysis process is given based on the idea of equivalent analysis, which combines the equivalent medium theory and the classical thin plate vibration theory. The embedded elastic metamaterial plate is treated as an equivalent homogeneous plate with abnormal effective medium parameters by using the equivalent medium theory, and then the natural vibration characteristics and steady-state vibration characteristics are investigated. The results show that the semi-analytic form of the vibration analysis process is effective, which can obtain the vibration characteristics of such heterogeneous and complex structures quickly and accurately within the medium and low frequency regions. The results also reveal the plate’s anomalous behaviors that are quite different from the traditional homogeneous plates. For the natural vibration, there exists no natural frequency in the band gap and they can be formed only in the pass band. These modes tend to gather gradually when approaching the initial frequency of the band gap, and the closer to the lower band edge of the band gap, the smaller the distance between natural frequencies and the greater the modal density. Additionally, the modes of the corresponding orders before and after the band gap are identical on the macro level but completely different on the unit-cell level. For the steady-state vibration, there is no resonance peak in the band gap, the resonance peaks are distributed denser and denser when approaching the initial frequency of the band gap, and the distribution of the displacement field at corresponding order of resonance peaks before and after the band gap is also similar on the macro level. This work not only facilitates the design and analysis of elastic metamaterial plate structures, but also deepens the understanding of their extraordinary vibration characteristics and expands the design space of mode characteristics.

Analytical solutions of free vibration for rectangular thin plate and right-angle triangle plate on the Winkler elastic foundation based on the symplectic superposition method

Abstract Based on the symplectic superposition method, the free vibration models of rectangular and right-angle triangle plates on the Winkler elastic foundation are established in the present paper, and the modes and frequencies are studied. In addition, the theoretical calculation model and finite element analysis model of rectangular thin plate and right-angle triangle plate on elastic foundation are established by using Mathematica software and ABAQUS software. It proves that the symplectic superposition method converges very fast and has a good consistency with the finite element simulation results. Analytical results show that foundation stiffness, aspect ratio, and boundary condition have great influences on vibration frequency and mode shape for structures. This paper solved the free vibration problem of rectangular plate and right-angle triangle plate on elastic foundation by using symplectic superposition method. Compared with the inverse or semi-inverse method, this method avoids the process of assuming the form about the solution, hence the result of this method is completely rational.