On the Use of Beam Functions for Problems of Plates Involving Free Edges

1975 ◽  
Vol 42 (4) ◽  
pp. 858-864 ◽  
Author(s):  
S. F. Bassily ◽  
S. M. Dickinson
Keyword(s):  
Free Vibration ◽  
Vibration Mode ◽  
Ritz Method ◽  
Approximate Solutions ◽  
Mode Shapes ◽  
Static Deflection ◽  
Numerical Examples ◽  
Accurate Treatment ◽  
Beam Mode ◽  
Free Edges

The inadequacy of beam vibration mode shapes when used in the Ritz method to obtain approximate solutions for flexural problems concerning plates involving free edges is demonstrated. A new set of functions, related to beam mode shapes, is postulated which allows considerably more accurate treatment of such plates. Several numerical examples concerning static deflection and free vibration of plates involving free edges are examined and serve to illustrate the applicability and accuracy of the new functions and to further demonstrate the inadequacy of the ordinary beam functions.

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.

A Novel and Accurate Ritz Formulation for Free Vibration of Rectangular and Skew Plates

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
S. A. Eftekhari ◽  
A. A. Jafari
Keyword(s):  
Differential Equations ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Rectangular Plates ◽  
Natural Boundary ◽  
Boundary Points ◽  
Natural Boundary Conditions ◽  
Free Edges

One of the major limitations of the conventional Ritz method is its difficulty in implementation to the differential equations with natural boundary conditions at the boundary points/lines. Plates involving free edges/corners and irregularly shaped plates are two historical and classical examples which show that their solutions cannot be accurately approximated by the conventional Ritz method. To solve this difficulty, a simple, novel, and accurate Ritz formulation is introduced in this paper. It is revealed that the proposed methodology can produce much better accuracy than the conventional Ritz method for rectangular plates involving free edges/corners and skew plates.

Free Vibration of a Rotating Disk–Blade Coupled System With Shrouds

1996 ◽  
Author(s):  
Yukinori Kobayashi ◽  
Gen Yamada ◽  
Takahiro Tomioka
Keyword(s):  
Free Vibration ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Coupled System ◽  
Rotating Disk ◽  
Mode Shapes ◽  
Admissible Functions

The free vibration of rotating disk–blade coupled system is investigated by the Ritz method. Centrifugal effects due to rotation are taken into account for both of the disk and blades. The boundary and continuity conditions between the disk and blades are satisfied by means of artificial springs introduced at their joints, and the orthogonal polynomials generated by using the Gram–Schmidt process are employed as admissible functions for both of the disk and blades. Frequency parameters and mode shapes of vibration are obtained to investigate the vibration of the disk–blade coupled system.

Free Vibration Characteristics of SPR Joints

2006 ◽  
Author(s):  
Xiaocong He ◽  
Ian Pearson ◽  
Ken Young
Keyword(s):  
Aluminum Alloy ◽  
Free Vibration ◽  
Torsional Vibration ◽  
Natural Frequencies ◽  
Vibration Characteristics ◽  
Advanced Materials ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Numerical Examples ◽  
Different Characteristics

Self-piercing riveting (SPR) has drawn more attention in recent years because it can join some advanced materials that are hard to weld, such as aluminum alloy sheets. In this paper, the free torsional vibration characteristics of single lap-jointed encastre SPR beam are investigated in detail. The focus of the analysis is to reveal the influence on the torsional natural frequencies and mode shapes of the single lap-jointed encastre SPR beam of different characteristics of sheets to be jointed. Numerical examples show that the torsional natural frequencies increase significantly as the Young’s modulus of the sheets increase, but almost no change corresponding to the change in Poisson’s ratio of the sheets to be joint. The mode shapes show that there are different deformations in the jointed section of SPR beam compared with the reference encastre beam without joint. These different deformations may cause different natural frequency values and different stress distributions.

Buckling and Free Vibration of Nonuniformly Heated Functionally Graded Carbon Nanotube Reinforced Polymer Composite Plate

2017 ◽  
Vol 17 (06) ◽  
pp. 1750064 ◽  
Author(s):  
Nivish George ◽  
P. Jeyaraj ◽  
S. M. Murigendrappa
Keyword(s):  
Temperature Field ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Composite Plate ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Functionally Graded ◽  
Temperature Fields ◽  
Mode Shapes ◽  
Reinforced Polymer

Buckling and free vibration behavior of functionally graded carbon nanotube reinforced polymer composite plate subjected to nonuniform temperature fields have been investigated using finite element approach. The effective material constants of the plate are obtained using the extended rule of mixture along with efficiency parameters of the carbon nanotube (to include geometry-dependent material properties). Influence of boundary conditions, aspect ratio, functional grading of the carbon nanotube, nonuniform thermal loading on thermal buckling and free vibration behavior of the heated plate are analyzed. It is observed that temperature fields and functional grading are influenced on the critical buckling temperature of the plates. Further, nature of functional grading showed significant change in buckling mode shapes irrespective of the boundary conditions. The first few natural frequencies of the plate under thermal load decreases as the temperature increases and they are influenced significantly by the nature of temperature field. Variations in free vibration mode shapes of the square plates found with not significant change as temperature increases. However, free vibration modes of the rectangular plates are sensitive to the nature of temperature field whenever there is a free edge associated with the boundary condition. Influence of functional grading on the free vibration mode shapes is not significant in contrast with the free vibration natural frequencies. The magnitude of free vibration natural frequencies of functional grade-X type carbon nanotube reinforcement showed higher in comparison with other two types of reinforcements considered here.

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]

Free and Forced Vibration Analysis of an Idealized Compliant Tower With Superstructure, due to Current Loads, Wave Excitation, and Earthquake Impulses

2014 ◽  
Author(s):  
Ankit ◽  
N. Datta
Keyword(s):  
Forced Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Mode Shapes ◽  
Wave Excitation ◽  
Uniform Beam ◽  
Compliant Tower ◽  
Forced Vibration Analysis ◽  
Beam Mode

A compliant tower (CT) is modeled as a partially dry, partially tapered, damped Timoshenko beam with the superstructure modeled as an eccentric tip mass, and a non-classical damped boundary at the base. The foundation is modeled as a combination of a linear spring and a torsional spring, along with linear and torsional dampers. The mean empty space factor due to the truss type structure of the tower is included. The effect of shear deformation and rotary inertia are included in the vibration analysis; with the non-uniform beam mode-shapes being a weighted sum of the uniform beam mode-shapes. The weights are evaluated by the Rayleigh-Ritz method, using the first ten modes and verified using Finite Element Method (FEM). The superstructure adds to the kinetic energy without affecting the stiffness of the beam, thereby reducing the natural frequencies. The weight of the superstructure acts as an axial compressive load on the beam, reducing its frequencies further. Kelvin-Voigt model of structural damping is included. A part of the structure being underwater, the virtual added inertia is included to calculate the wet natural frequencies. The CT is first subjected to steady current loads of a given velocity profile. The static deflection and overturning moment is estimated for current loads. The CT is then studied for wave excitation at various seas states. Morrison’s equation and Pierson-Moskowitz Spectrum are used to derive the forces for different sea states. The forced vibration analysis of the structure is done via Rayleigh-Ritz method and verified using FEM. The maximum horizontal deflection and shear stress of the base of the superstructure, and the normal/shear stresses at the foundation are analyzed. Finally, the CT is subjected to earthquake excitation, modeled as an arbitrary horizontal impact excitation at the base. The above forced vibration analysis is repeated.

scholarly journals Free Vibration Analysis of L-Shaped Folded Thin Plates

2019 ◽  
Vol 2 (1) ◽  
pp. 67-73
Author(s):  
Koji Sekine
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Energy Functional ◽  
Thin Plates ◽  
Mode Shapes ◽  
Width Ratio ◽  
Various Boundary Conditions

Free vibration analysis of L-shaped folded thin plates having various boundary conditions is presented. Vibration properties of the folded plates are analyzed by means of the Ritz method. Displacement functions satisfying the geometric boundary conditions are assumed in the form of double power series. The interconnection of plate elements of the folded plates is defined by translational and rotational coupling springs. The generalized eigenvalue problem, which is derived by means of minimizing the energy functional, is solved to determine the natural frequencies and mode shapes. The accuracy and validity of the present solutions are demonstrated through convergence studies and comparisons with the results from the literature and FEM (finite element method) analysis solutions. Numerical results are presented for different conditions, such as width ratio, length ratio and the four types of boundary condition.

Free Vibration of a Thin Shell Structure of Rectangular Cross-Section

2011 ◽  
Vol 486 ◽  
pp. 107-110 ◽  
Author(s):  
Yue Hua Chen ◽  
Guo Yong Jin ◽  
Zhi Gang Liu
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Shell Structure ◽  
Rectangular Cross Section ◽  
Ritz Method ◽  
Coupled Model ◽  
Double Series ◽  
General Boundary ◽  
Mode Shapes ◽  
The Finite Element Method

This paper presents an analysis on the free vibration of a shell structure of rectangular cross-section. The shell of rectangular cross-section is modeled by four rectangular panels elastically connected at right angles under general boundary conditions. With the general boundary and coupling conditions accounted for several groups of linear springs, the double series solutions for both flexural and in-plane vibrations are obtained by employing the Rayleigh-Ritz method and the validation of the calculations is proved by comparing the eigenpairs with the Finite Element Method results. It is shown that the mode shapes of the rigidly coupled model perform a symmetrical feature.

Generalized Hydrodynamic Mass for Beam Mode Vibration of Cylinders Coupled by Fluid Gap

1977 ◽  
Vol 44 (1) ◽  
pp. 172-173 ◽  
Author(s):  
M. K. Au-Yang
Keyword(s):  
Experimental Data ◽  
Free Vibration ◽  
Vibration Analysis ◽  
Nuclear Reactor ◽  
Cylindrical Shells ◽  
Numerical Examples ◽  
End Conditions ◽  
Mode Vibration ◽  
Beam Mode

The generalized hydrodynamic mass for the free vibration of two coaxial cylindrical shells coupled by a narrow fluid gap is derived by following a procedure similar to that mentioned in [1]; the end conditions of the shells are arbitrary. The equations are applied to evaluate the generalized hydrodynamic mass for the coupled beam mode vibration of two cylinders. Numerical examples that can be compared with experimental data are given for two specific cases. In both cases, agreement between computed and measured values is excellent. It is intended that the results of this study will be applied to the vibration analysis of nuclear reactor internal components.

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