Vibration Of Rectangular Plates Research Articles

Analytical solution of forced vibration of rectangular plates with part through surface crack based on wave propagation method

An analytical wave propagation framework is developed in this study for steady state forced vibration of plates with part through surface cracks for the first time. Rectangular thin plates with partial-width and full-width cracks are considered. The analytical waves are obtained by solving the dual equations of the plate established considering the line–spring model (LSM) of crack. The dual equations formed here are characterized by simultaneously solving kinematic variables and their dual dynamic variables. Both conventional and refined LSMs are considered. Two ways are proposed to generate the system equation in wave space based on the relationships of wave propagation, wave reflection and wave scattering. Compared with the traditional methods, most of them can only give exact analytical solutions under the condition of opposite-side simply supported boundaries, the developed analysis framework can treat arbitrary combinations of simple boundary condition of the cracked plate. Comparisons of results of the proposed method with those from the literature and finite element method (FEM) validate the effectiveness of the proposed method. A series of parametric studies for investigating the effects of length, position and depth of crack on the forced response of cracked plate are also presented. The proposed framework are effective for predicting the forced responses and natural frequencies of thin plates with part-through surface cracks.

A deviatoric couple stress Mindlin plate model and its degeneration

Based on the classical couple stress theory, the deviatoric couple stress theory (DCST), where both couple stress and curvature tensors are traceless, is developed. The deviatoric couple stress Mindlin plate is established based on a new hypothesis of the vanishing curvature of the transverse normal, and the governing equations and associated boundary conditions of the simplified size-dependent Mindlin plate are respectively derived by the principle of virtual work. In the absence of couple stresses, a simplified Mindlin plate model between the classical Kirchhoff plate model and classical Mindlin plate model is derived. To illustrate the simplified DCST-based Mindlin plate model, the static bending and free vibration of rectangular plates are studied. The dependence of the length scale parameter and curvature ratio on the deflection, rotation and natural frequency is analyzed. The numerical results show that the consideration of couple stresses and curvatures hardens the stiffness of the microstructure, resulting in a decrease in its deflection and rotation compared with that of the macroscopic structure under the same conditions, while the natural frequency increases, which is consistent with the size effect observed in experiments.

The Effect of Cracks on the Free Vibration of a Plate with Parabolic Thickness

This paper uses first-order shear deformation theory and the finite element method to analyze the vibrations of rectangular plates with one or more cracks. The study investigated the influence of cracks (length, angle of inclination), the number of cracks, and the ratio of plate thickness to the natural vibration frequency of the plate, using phase field simulation. Plate thickness varies nonlinearly with the parabolic function. The results of the proposed method were compared with reputable studies to verify its reliability. In addition, some pictures of the characteristic vibration patterns of the plate with varying thickness are presented when cracks appear.

Free vibrations of rectangular plates

The paper investigates free vibrations of rectangular metal plates. The finite element method and analytical calculation are particularly used to determine the vibration frequency. The analytical calculation is based on the equation of motion of a thin rectangular plate. The asymptotic method is applied to determine the dynamic edge effect. As a result, the free vibration frequency is determined for the rectangular metal plate. The finite element analysis is performed in Lira and SolidWorks software packages. For this, a solid plate model with sensors is created to measure the free vibration frequency; the effective mass participation factor was determined.The plate vibration tests were conducted to confirm the results of analytical calculations. The method of smooth sinusoidal vibrations is used. Spectrum graphs of the plate vibrations are suggested based on the vibration acceleration of sensors. Errors are identified in the free vibration frequencies depending on the applied method. The paper does not consider frequencies with the low effective mass participation factor.

Vibration of rectangular plate on fractional order viscoelastic foundation subjected to standing and moving loads

Vibration of rectangular plate on fractional order viscoelastic foundation subjected to standing and moving loads

Do we need to satisfy natural boundary conditions in energy approach to nonlinear vibrations of rectangular plates?

Despite a large literature, it seems still unclear if it is necessary to satisfy the natural boundary conditions in an energy approach to the study of nonlinear vibrations of rectangular plates. In the present study, which is based on Lagrange equations, the case of simply supported movable edges is considered. In the past, special nonlinear terms were added to the in-plane displacements to satisfy the natural boundary conditions. However, it was unclear if this was just helping the fast convergence of the series, reducing the number of degrees of freedom in the model, or it was actually necessary. In fact, it is well-known that for linear vibrations only geometric (essential) boundary conditions must be satisfied in variational energy approaches like the Rayleigh-Ritz method. Therefore, this doubt should be clarified. In the present study, accurate expansions of the plate displacements are introduced with additional in-plane trigonometric terms. They allow to satisfy the natural boundary conditions by energy minimization without the introduction of additional nonlinear terms. Nonlinear damping is also introduced in order to obtain results which agree with experiments.

Free Vibration Analysis of Rectangular Plate with Cutouts under Elastic Boundary Conditions in Independent Coordinate Coupling Method

Based on Kirchhoff plate theory and the Rayleigh-Ritz method, the model for free vibration of rectangular plate with rectangular cutouts under arbitrary elastic boundary conditions is established by using the improved Fourier series in combination with the independent coordinate coupling method (ICCM). The effect of the cutout is taken into account by subtracting the energies of the cutouts from the total energies of the whole plate. The vibration displacement function of the hole domain is based on the coordinate system of the hole domain in this method. From the continuity condition of the vibration displacement function at the cutout, the transition matrix between the two coordinate systems is constructed, and the mass and stiffness matrices are completely obtained. As a result, the calculation is simplified and the computational efficiency of the solution is improved. In this paper, numerical examples and modal experiments are presented to validate the effectiveness of the modeling methods, and parameters related to influencing factors of the rectangular plate are analyzed to study the vibration characteristics.

Influence of external load on forced vibration of rectangular plate

Rectangular plate is one typical structure with wild application in engineering, and the force vibration is the basis for analysis of sound radiation of plate. In this work, numerical simulation is conducted for force vibration of rectangular plate, and the influence of external load is also studied. Numerical results show that, the rectangular plate will resonate under external load at certain driven frequency, and it’s strictly related to structural modal frequencies. Structural vibration response will change obvious with the change of location of external load. When the location is fixed, with the increase of action region of external load, the acceleration of plate will decrease and some new resonance frequencies will be observed. Conclusions here will lay foundation for further work on sound radiation of plate.

Benchmark analytic in-plane vibration frequencies of orthotropic rectangular plates

The in-plane vibrations of rectangular plates is classic problem in mechanics. It involves the solution of two coupled differential equations with many combinations of possible edge restraints. Exact solution, for some cases, and high accuracy solution for few more cases, have been obtained along the years. In this work a new high accuracy solution, that covers all the possible combinations is presented. It is obtained by using carefully chosen series that solve the coupled partial differential equations of motion, for all possible combinations of edge conditions. Examples of the new solutions are given and compared with other known solutions and with approximate solutions for these cases. Benchmark results are given for square orthotropic plates with all possible combinations of boundary conditions.

Nonlinear Dynamic Response of a Thin Rectangular Plate Vibration System Excited by a Non-ideal Induction Motor

Nonlinear Dynamic Response of a Thin Rectangular Plate Vibration System Excited by a Non-ideal Induction Motor

Theoretical and Experimental Investigation of Free Vibrations of Rectangular Plates with Vertical Patterns

Theoretical and Experimental Investigation of Free Vibrations of Rectangular Plates with Vertical Patterns

Nonlinear Forced Vibrations of Plates with Oscillating Inclusions

These studies focus on the forced vibrations of rectangular plates incorporating oscillating inclusions uniformly distributed in the carrier elastic medium. Plate’s equations of motion consist of the Lamé equation for a carrier material and the equation for oscillators representing the inclusions. The oscillators are acted upon by the nonlinear friction force, described by Coulomb’s modified law, and a cubically nonlinear elastic force. The simply supported forced plate is considered. Using the Galerkin approximation, the problem of plate vibrations is reduced to the study of a system of ODEs. According to numerical and qualitative analyses, the application of harmonic forcing causes the appearance of periodic, quasiperiodic, and chaotic regimes in the system. In particular, the analytical expression is derived for the periodic regimes of the forcing frequency. The scenarios of vibration’s development is identified at the variation of forcing amplitude. The quasiperiodic and chaotic modes are studied by means of Poincaré section technique, and Fourier and Lyapunov spectra analyses. The statistical properties of the sequences of temporal intervals between maxima of system’s solutions are considered in more detail. The sequence extracted from the chaotic trajectory is shown to possess long-term correlations and approximately obeys the Weibull distribution.

Free vibrations of rectangular plates simply supported at two opposite edges and elastically restrained at the two other edges. An analytical and a semi-analytical method

The present study provides a novelty in treating rectangular plates simply supported at two opposite edges and elastically restrained at the two other edges using two approaches. On one hand, the analytical solution of the plate vibration equation satisfying the rotational and translational spring boundary conditions is obtained for different plate aspect ratios leading to the corresponding modes and frequencies. On the other hand, the Rayleigh-Ritz method (RRM) is applied using plate functions defined as products of beam functions satisfying the beam appropriate end conditions in each direction. i.e. simply supported in the direction of the two opposite edges, and elastic end conditions in the other direction. The results obtained by the two methods coincide for all frequencies and modes examined, showing the effectiveness of the RRM applied to this problem in a general and systematic way. This makes the RRM ready for use in further studies concerning similar plates with added point masses or springs, or plate vibrating in the nonlinear regime. Also, useful numerical results corresponding to both the frequencies and mode shapes are presented for various plate modes, stiffness values and plate aspect ratios.

Nonlinear forced vibration of rectangular plates by modified multiple scale method

In the present study, the nonlinear forced vibration of a rectangular plate is investigated analytically using modified multiple scales method for the first time. The plate is subjected to transversal harmonic excitation, and the boundary condition is assumed to be simply supported. The von Karman nonlinear strain displacement relations are used. The extended Hamilton principle and classical plate theory are applied to derive the partial differential equations of motions. This research focuses on resonance case with 3:1 internal resonance. By applying Galerkin method, the nonlinear partial differential equations are transformed into time dependent nonlinear ordinary differential equations, which are then solved analytically by modified multiple scales method. This proposed approach is very simple and straightforward. The obtained results are then compared with both the traditional multiple scales method and previous studies, and excellent compatibility is noticed. The effect of some of the main parameters of the system is also examined.

A modified Fourier-Ritz method for free vibration of rectangular plates with elastic constrains

1. Abdulkerim S., Dafnis A., Riemerdes H., 2019, Experimental investigation of nonlinear vibration of a thin rectangular plate, International Journal of Applied Mechanics, 11, 267-289. CrossRef Google Scholar

Modified Fourier-Ritz Method to Analyze Transverse Vibration of Rectangular Plates with Additional Mass

In practical engineering, rectangular thin plate structures with concentrated mass points or equivalent to concentrated mass points have a wide range of applications in mechanical engineering, electronic engineering, and vehicle engineering, such as supporting workbenches, ship decks, PCB boards and so on. The modified Fourier Ritz method is used to establish the numerical analysis model of rectangular thin plate with lumped mass under general boundary conditions, which can avoid the problems of non-derivability or discontinuity at the boundary of thin plates in traditional methods. In addition, the Fourier expansion in the form of cosine function plus polynomial has better convergence than that in the form of sine function. The calculation method of mass matrix and stiffness matrix of rectangular thin plate with concentrated mass is given, the parameters of different boundary constraints are analyzed, and the influence of the size, position and quantity of concentrated mass on the modal of rectangular plate is discussed. The method and its results can be applied to the vibration analysis and vibration control of rectangular thin plates.

Vibration of rectangular plates stiffened by orthogonal beams

This paper presents a new analytical solution for the vibration response of plates stiffened by orthogonal beams using a double finite sine integral transform technique. The compatibility conditions at the coupling interfaces and the boundary conditions of the plate are automatically defined in the transform. It is shown that the analytical model can accommodate various combinations of clamped and simply supported boundary conditions. The analytical model is then used to study the reproduction of the vibration response of an orthotropic plate. It is shown that an orthogonally ribbed isotropic plate can produce a closely matching vibration response to that of an orthotropic plate by carefully choosing an appropriate number of ribs and the rib properties in each plate direction.

Vibration of a rectangular plate using finite continuous Radon transform

In this study, the finite continuous Radon transform the vibrations of rectangular thin plates is demonstrated. Finite continuous Radon transform using graphical representation GNU Octave, version 5.2.0 and solution for different functions are also studied. In order to check the validity of formulation, the numerical results are established using finite continuous Radon transform. In the concluding section the comparative study with approximations is demonstrated.

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions