scholarly journals A generalized solution procedure for in-plane free vibration of rectangular plates and annular sectorial plates

2017 ◽  
Vol 4 (8) ◽  
pp. 170484 ◽  
Author(s):  
Siyuan Bao ◽  
Shuodao Wang
Keyword(s):  
Free Vibration ◽  
Generalized Solution ◽  
Solution Procedure ◽  
Tangential Displacement ◽  
Rectangular Plates ◽  
Unified Framework ◽  
Stiffness Matrices ◽  
Different Shapes ◽  
Radial Variable ◽  
Admissible Functions

A generalized solution procedure is developed for in-plane free vibration of rectangular and annular sectorial plates with general boundary conditions. For the annular sectorial plate, the introduction of a logarithmic radial variable simplifies the basic theory and the expression of the total energy. The coordinates, geometric parameters and potential energy for the two different shapes are organized in a unified framework such that a generalized solving procedure becomes feasible. By using the improved Fourier–Ritz approach, the admissible functions are formulated in trigonometric form, which allows the explicit assembly of global mass and stiffness matrices for both rectangular and annular sectorial plates, thereby making the method computationally effective, especially when analysing annular sectorial plates. Moreover, the improved Fourier expansion eliminates the potential discontinuity of the original normal and tangential displacement functions and their derivatives in the entire domain, and accelerates the convergence. The generalized Fourier–Ritz approach for both shapes has the characteristics of generality, accuracy and efficiency. These features are demonstrated via a few numerical examples.

scholarly journals Free Vibration of Rectangular Plates with Porosity Distributions under Complex Boundary Constraints

2019 ◽  
Vol 2019 ◽  
pp. 1-16 ◽  
Author(s):  
Yuan Du ◽  
Siyu Wang ◽  
Liping Sun ◽  
Yanhe Shan
Keyword(s):  
Fourier Series ◽  
Free Vibration ◽  
Metal Foam ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Current Method ◽  
Rectangular Plates ◽  
Modal Shape ◽  
Boundary Constraints ◽  
Admissible Functions

Free vibration of rectangular plates with three kinds of porosity distributions and different boundary constraints has been performed by means of a semianalytical method. The distribution of porous varies along the thickness of the plate, in which the mechanical properties are defined by open-cell metal foam. Regardless of boundary conditions, displacement admissible functions are represented by combination of standard cosine Fourier series and auxiliary sine series. The kinetic energy and potential energy of plates are also expressed on the basis of first-order shear deformation theory (FSDT) and displacement admissible functions. Finally, the coefficients in the Fourier series which determine natural frequencies and modal shape are derived by means of the Rayleigh–Ritz method. Convergence and dependability of the current method are verified by comparing with the results of FEM and related literatures. In addition, some new results considering geometry parameters under classical and elastic boundary constraints are listed. The effects of geometry parameters, material parameters, and boundary constraints have been discussed in detail.

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

2020 ◽  
Vol 71 (7) ◽  
pp. 853-867
Author(s):  
Phuc Pham Minh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Phase Field ◽  
Deformation Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
Free Vibration Frequency

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

scholarly journals A Parametric Study on Free Vibration of Multi-perforated Rectangular Plates

2016 ◽  
Vol 144 ◽  
pp. 60-67 ◽  
Author(s):  
Keya Ghonasgi ◽  
Kalpit Bakal ◽  
Kiran D. Mali
Keyword(s):  
Free Vibration ◽  
Parametric Study ◽  
Rectangular Plates

Combinations for the Free Vibration Behaviors of Anisotropic Rectangular Plates Under General Edge Conditions

1999 ◽  
Author(s):  
Yoshihiro Narita
Keyword(s):  
Free Vibration ◽  
Structural Design ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Technical Information ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Plate Models ◽  
Edge Conditions ◽  
Anisotropic Rectangular Plates

Abstract The free vibration behavior of rectangular plates provides important technical information in structural design, and the natural frequencies are primarily affected by the boundary conditions as well as aspect and thickness ratios. One of the three classical edge conditions, i.e., free, simple supported and clamped edges, may be used to model the constraint along an edge of the rectangle. Along the entire boundary with four edges, there exist a wide variety of combinations in the edge conditions, each yielding different natural frequencies and mode shapes. For counting the total number of possible combinations, the present paper introduces the Polya counting theory in combinatorial mathematics, and formulas are derived for counting the exact numbers. A modified Ritz method is then developed to calculate natural frequencies of anisotropic rectangular plates under any combination of the three edge conditions and is used to numerically verify the numbers. In numerical experiments, the number of combinations in the free vibration behaviors is determined for some plate models by using the derived formulas, and are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the Ritz method.

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