scholarly journals Natural Frequencies of Rectangular Laminated Plates—Introduction to Optimal Design in Aeroelastic Problems

Aerospace ◽  
2018 ◽  
Vol 5 (3) ◽  
pp. 95 ◽  
Author(s):  
Aleksander Muc
Keyword(s):  
Free Vibration ◽  
Ritz Method ◽  
Laminated Plates ◽  
Mode Shapes ◽  
Maximization Problem ◽  
Unit Load ◽  
Element Analysis ◽  
Influence Coefficients ◽  
Design Variables ◽  
3D Solid

Free vibration (or eigenvalue analysis) is a prerequisite for aeroelastic analysis. For divergence analysis, slope influence coefficients (rotation at point i due to unit load at point j) are calculated using free vibration mode shapes and corresponding frequencies. The lowest eigenvalue is of interest and gives the divergence speed. The present paper considers the maximization problem of eigenfrequencies for composite panels. The influence of boundary conditions and constant or variable stiffnesses on optimization results are investigated herein. A new convenient set of design variables is employed in the analysis. The computations are carried out with the use of the Rayleigh–Ritz method and Finite Element analysis (2D quadrilateral and 3D solid elements).

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.

Vibration of Clamped Circular Symmetric Laminates

1994 ◽  
Vol 116 (2) ◽  
pp. 141-145 ◽  
Author(s):  
K. M. Liew
Keyword(s):  
Ritz Method ◽  
Numerical Procedure ◽  
Flexural Vibration ◽  
Boundary Function ◽  
Energy Functional ◽  
Laminated Plates ◽  
Mode Shapes ◽  
Circular Plates ◽  
Present Solution ◽  
Contour Plots

Treated in this paper is the free-flexural vibration analysis of symmetrically laminated thin circular plates. The total energy functional for the laminated plates is formulated where the pb-2 Ritz method is applied for the solution. The assumed displacement is defined as the product of (1) a two-dimensional complete polynomial function and (2) a basic boundary function. The simplicity and accuracy of the numerical procedure will be demonstrated by solving some plate examples. In the present study, the effects of material properties, number of layers and fiber stacking sequences upon the vibration frequency parameters are investigated. Selected mode shapes by means of contour plots for several 16-ply laminated plates with different fiber stacking sequences and composite materials are presented. This study may provide valuable information for researchers and engineers in design applications. In addition, the present solution plays an important role in increasing the existing data base for future references.

scholarly journals Investigation of Free Vibration Response of E-Glass/Epoxy Delaminated Composite Plates

2012 ◽  
Vol 21 (1) ◽  
pp. 096369351202100 ◽  
Author(s):  
Turan Ercopur ◽  
Binnur Goren Kiral
Keyword(s):  
Free Vibration ◽  
Natural Frequency ◽  
Epoxy Composite ◽  
Free Vibration Analysis ◽  
Composite Plates ◽  
Vibration Response ◽  
Mode Shapes ◽  
Element Analysis ◽  
The Finite Element Analysis ◽  
Free Vibration Response

This paper deals with the finite element analysis of free vibration response of the delaminated composite plates. Free vibration analysis is performed by using ANSYS commercial software developing parametric input files. Natural frequency values and associated mode shapes of E-glass/epoxy composite delaminated plates are determined. Effects of delamination shape, dimension and location on the natural frequency and associated mode shapes are investigated and for the purpose of the observing the effect of the boundary conditions, cantilever and clamped-pinned delaminated composite plates are taken into consideration. Comparisons with the results in literature verify the validity of the developed models in this study. It is observed that the natural frequency decreases in the existence of the delamination and level of the decrease depends on the dimension, shape and location of the delamination.

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.

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.

Nonlinear bending and free vibration response of SUS316-Al2O3 functionally graded plasma sprayed beams: theoretical and experimental study

2016 ◽  
Vol 24 (6) ◽  
pp. 1171-1184 ◽  
Author(s):  
Pravin Malik ◽  
Ravikiran Kadoli
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Maximum Error ◽  
Functionally Graded ◽  
Nonlinear Finite Element ◽  
Functionally Graded Beam ◽  
Element Analysis ◽  
Nonlinear Bending ◽  
Plasma Sprayed ◽  
3D Solid

Functionally graded SUS316-Al2O3 beams with ceramic content varying from 0 to 40% were prepared by a plasma spraying technique. Nonlinear finite element analysis was used to obtain the static deflection and free vibration of a clamped-free functionally graded beam. Von Kármán geometric nonlinearity and power law variation in material gradation through the beam thickness are considered in the analysis. The maximum error between the experimental and nonlinear finite element results for deflection is 6.68% and 14.31% on the fundamental frequency. Numerical results have also been attempted using ANSYS 3D solid element and they compare more closely with the experimental results.

scholarly journals Effect of Moment of Inertia of Attached Mass on Natural Frequencies of Cantilevered Symmetrically Laminated Plates

2018 ◽  
Vol 1 (2) ◽  
pp. 35-39
Author(s):  
Kenji Hosokawa
Keyword(s):  
Free Vibration ◽  
Natural Frequencies ◽  
Laminated Composite ◽  
Composite Plates ◽  
Vibration Characteristics ◽  
Laminated Plates ◽  
Moment Of Inertia ◽  
Mode Shapes ◽  
Attached Mass ◽  
Structural Applications

Since composite materials such as laminated composite plates have high specific strength and high structural efficiency, they have been usedin many structural applications. It is therefore very important to make clear the vibration characteristics of the laminated plates for the designand the structural analysis. Especially, the vibration characteristics of the laminated plates with attached mass are essential. However, wecannot find the theoretical or experimental approaches for the free vibration of laminated plates with attached mass. In the present study, theexperimental and numerical approaches are applied to the free vibration of cantilevered symmetrically laminated plates with attached mass.First, by applying the experimental modal analysis technique to the cantilevered symmetrically laminated plates with attached mass, thenatural frequencies and mode shapes of the plates are obtained. Next, the natural frequencies and mode shapes of the cantileveredsymmetrically laminated plates with attached mass are calculated by Finite Element Method (FEM). Finally, from the experimental andnumerical results, the effect of the moment of inertia of the attached mass to the natural frequencies and mode shapes of the cantileveredsymmetrically laminated plates are clarified.

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]

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.

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