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

1999 ◽  
Vol 67 (3) ◽  
pp. 568-573 ◽  
Author(s):  
Y. Narita
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Numerical Study ◽  
Ritz Method ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Vibration Behavior ◽  
Edge Conditions ◽  
Important Field ◽  
Anisotropic Rectangular Plates

The free-vibration behavior of rectangular plates constitutes an important field in applied mechanics, and the natural frequencies are known to be primarily affected by the boundary conditions as well as aspect and thickness ratios. Any one of the three classical edge conditions, i.e., free, simply 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. 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 classical edge conditions and is used to numerically verify the numbers. In this numerical study the number of combinations in the free-vibration behavior is determined for some plate models by using the derived formulas. Results are corroborated by counting the numbers of different sets of the natural frequencies that are obtained from the modified Ritz method. [S0021-8936(00)02203-0]

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.

Hydroelastic Vibration of Rectangular Plates

1996 ◽  
Vol 63 (1) ◽  
pp. 110-115 ◽  
Author(s):  
Moon K. Kwak
Keyword(s):  
Boundary Conditions ◽  
Approximate Formula ◽  
Natural Frequencies ◽  
Mass Matrix ◽  
Ritz Method ◽  
Plate Boundary ◽  
Rigid Wall ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Virtual Mass

This paper is concerned with the virtual mass effect on the natural frequencies and mode shapes of rectangular plates due to the presence of the water on one side of the plate. The approximate formula, which mainly depends on the so-called nondimensionalized added virtual mass incremental factor, can be used to estimate natural frequencies in water from natural frequencies in vacuo. However, the approximate formula is valid only when the wet mode shapes are almost the same as the one in vacuo. Moreover, the nondimensionalized added virtual mass incremental factor is in general a function of geometry, material properties of the plate and mostly boundary conditions of the plate and water domain. In this paper, the added virtual mass incremental factors for rectangular plates are obtained using the Rayleigh-Ritz method combined with the Green function method. Two cases of interfacing boundary conditions, which are free-surface and rigid-wall conditions, and two cases of plate boundary conditions, simply supported and clamped cases, are considered in this paper. It is found that the theoretical results match the experimental results. To investigate the validity of the approximate formula, the exact natural frequencies and mode shapes in water are calculated by means of the virtual added mass matrix. It is found that the approximate formula predicts lower natural frequencies in water with a very good accuracy.

An exact solution for free vibration of thin functionally graded rectangular plates

2010 ◽  
Vol 225 (3) ◽  
pp. 526-536 ◽  
Author(s):  
A Hasani Baferani ◽  
A R Saidi ◽  
E Jomehzadeh
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Different Boundary Conditions

The aim of this article is to find an exact analytical solution for free vibration characteristics of thin functionally graded rectangular plates with different boundary conditions. The governing equations of motion are obtained based on the classical plate theory. Using an analytical method, three partial differential equations of motion are reformulated into two new decoupled equations. Based on the Navier solution, a closed-form solution is presented for natural frequencies of functionally graded simply supported rectangular plates. Then, considering Levy-type solution, natural frequencies of functionally graded plates are presented for various boundary conditions. Three mode shapes of a functionally graded rectangular plate are also presented for different boundary conditions. In addition, the effects of aspect ratio, thickness—length ratio, power law index, and boundary conditions on the vibration characteristics of functionally graded rectangular plates are discussed in details. Finally, it has been shown that the effects of in-plane displacements on natural frequencies of functionally graded plates under different boundary conditions have been studied.

Free Vibration Behavior of Angle-Ply Laminated Composite Stiffened Plates

2021 ◽  
pp. 2150187
Author(s):  
Leena Sinha ◽  
Amaresh Tripathy ◽  
Amar N. Nayak ◽  
Shishir K. Sahu
Keyword(s):  
Free Vibration ◽  
Numerical Investigation ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Mode Shapes ◽  
Stiffened Plates ◽  
Vibration Behavior ◽  
Fundamental Frequencies ◽  
Aspect Ratios ◽  
Support Conditions

This paper reports for the first time the experimental investigation on the free vibration characteristics of angle-ply laminated composite stiffened plates along with the numerical investigation. The natural frequencies of these plates are computed experimentally using an FFT analyzer and numerically by employing the finite element method with combination of isoparametric nine nodded plate and three nodded beam elements. Angle-ply laminated stiffened plates are fabricated using woven glass fiber fabrics and epoxy by hand layup technique. The effects of various support conditions; number, orientation and types of stiffeners; aspect ratios of plates and different fiber orientations on the fundamental frequencies of the angle-ply laminated stiffened plates are investigated. These parameters significantly influence the natural frequencies. It is found that there is a very good agreement observed between the fundamental frequencies obtained from both experimental and numerical investigation. Mode shapes are also presented for angle-ply laminated square stiffened plates with three different support conditions to justify the trend of increasing/decreasing of the modal frequency with the addition of stiffeners. It is observed that the enhancement of the frequencies of the angle-ply plates due to addition of stiffeners is influenced significantly by the position of stiffeners and consequently mode shapes. As the research on the free vibration behavior of angle-ply stiffened plates with experimental analysis is very rare, this study can be considered as the benchmark for future research.

Free Vibration of a Point-Supported Spherical Shell

1985 ◽  
Vol 52 (4) ◽  
pp. 890-896 ◽  
Author(s):  
T. Irie ◽  
G. Yamada ◽  
Y. Muramoto
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Lagrangian Multiplier ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Lagrangian Multiplier Method

An analysis is presented for the free vibration of an elastically or a rigidly point-supported spherical shell. For this purpose, the deflection displacements of the shell are written in a series of the products of the associated Legendre functions and the trigonometric functions. The dynamical energies of the shell are evaluated, and the frequency equation is derived by the Ritz method. For a rigidly point-supported shell, the Lagrangian multiplier method is conveniently employed. The method is applied to a closed spherical shell supported at equispaced four points located along a parallel of latitude; the natural frequencies and the mode shapes are calculated numerically, and the effects of the point supports on the vibration are studied.

A beam with arbitrarily placed lateral dampers: Evolution of complex modes with damping

2016 ◽  
Vol 24 (2) ◽  
pp. 379-392 ◽  
Author(s):  
Y Chen ◽  
DM McFarland ◽  
BF Spencer ◽  
LA Bergman
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Numerical Study ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Dynamic Features ◽  
Complex Modes ◽  
Orthogonality Conditions ◽  
Boundary Condition Method ◽  
Damped Systems

Free vibration of a beam with multiple arbitrarily placed lateral viscous dampers is investigated to gain insight into the intrinsic dynamic features of non-proportional damped systems. In terms of virtual boundary condition method, complex modes of a damper-beam system are achieved, and the solution is also suitable for the beams that have different boundary conditions. The features of the wave numbers satisfying the frequency equation were discussed in theory. The orthogonality analysis conducted in this paper provides two orthogonality conditions for complex modes. Pseudoundamped natural frequencies, damping ratios and complex modes are surveyed via numerical study. The analysis on the evolution of complex modes shows that the increasing damping would lead to over damped modes, and the mode shape that corresponds to the small one of a pair of real-valued natural frequencies is close to the static deformation shape of a beam subjected to static forces located at the positions of the dampers. For the rest modes that would never be over damped with increasing damping, the mode shapes and corresponding psuedoundamped natural frequencies will converge to that of a beam with rolling supports located at where dampers are placed. The exact solution of free vibration of a multiple-span beam is presented in addition.

scholarly journals Effects of Elliptical Hole on the Correlation of Natural Frequency with Buckling Load of Basalt Laminates Composite Plates

2020 ◽  
Vol 27 (1) ◽  
pp. 216-225
Author(s):  
Buntheng Chhorn ◽  
WooYoung Jung
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Basalt Fiber ◽  
Orientation Angle ◽  
Elliptical Hole ◽  
Composite Plates ◽  
Buckling Load ◽  
Mode Shapes ◽  
Geometric Ratio

AbstractRecently, basalt fiber reinforced polymer (BFRP) is acknowledged as an outstanding material for the strengthening of existing concrete structure, especially it was being used in marine vehicles, aerospace, automotive and nuclear engineering. Most of the structures were subjected to severe dynamic loading during their service life that may induce vibration of the structures. However, free vibration studied on the basalt laminates composite plates with elliptical cut-out and correlation of natural frequency with buckling load has been very limited. Therefore, effects of the elliptical hole on the natural frequency of basalt/epoxy composite plates was performed in this study. Effects of stacking sequence (θ), elliptical hole inclination (ϕ), hole geometric ratio (a/b) and position of the elliptical hole were considered. The numerical modeling of free vibration analysis was based on the mechanical properties of BFRP obtained from the experiment. The natural frequencies as well as mode shapes of basalt laminates composite plates were numerically determined using the commercial program software (ABAQUS). Then, the determination of correlation of natural frequencies with buckling load was carried out. Results showed that elliptical hole inclination and fiber orientation angle induced the inverse proportion between natural frequency and buckling load.

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