On a General Approach to Free Vibrations Response of Integrally-Stiffened and/or Stepped-Thickness Rectangular Plates or Panels

2008 ◽  
Author(s):  
Umur Yuceoglu ◽  
Jaber Javanshir ◽  
O¨zen Guvendik
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Width Ratio ◽  
Al Alloy ◽  
Rectangular Plates ◽  
Stiffened Plate ◽  
Governing System ◽  
Stress Resultant ◽  
Method Of Solution

This study is mainly concerned with a “General Approach” to the “Theoretical Analysis and the Solution of the Free Vibrations Response of Integrally-Stiffened and/or Stepped-Thickness Plates or Panels with Two or more Integral Plate Stiffeners”. In general, the “Stiffened System” (regardless of the number of “Plate Stiffeners”) is considered to be composed of dissimilar “Orthotropic Mindlin Plates” with unequal thicknesses. The dynamic governing equations of the individual plate elements of the “System” and the stress resultant-displacement expressions are combined and algebraically manipulated. These operations lead to a new “Governing System of the First Order Ordinary Differential Equations” in “state vector” forms. The new “Governing System of Equations” facilitates the direct application of the present method of solution, namely, the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. As shown in the present study, the “MTMM” is sufficiently general to handle the “Free Vibrations Response” of the “Stiffened System” (with, at least, one or up to three or four “Integral Plate Stiffeners”). The present analysis and the method of solution are applied to the typical “Stiffened Plate or Panel System with Two Integral Plate Stiffeners”. The mode shapes with their natural frequencies are presented for the “Isotropic Al-Alloy” and “Orthotropic Composite” cases and for several sets of support conditions. As an additional example, the case of the “Stiffened Plate or Panel System with Three Integral Plate Stiffeners” is also considered and is shown in terms of the mode shapes and their natural frequencies for one set of the boundary conditions. Also, some parametric studies of the natural frequencies versus the “Stiffener Thickness Ratio” and the “Stiffener Length (or Width) Ratio” are investigated and are graphically presented.

Free Vibrations of Composite Shallow Circular Cylindrical Shell Panels With a Bonded Central Stiffening Shell Strip

2001 ◽  
Author(s):  
U. Yuceoglu ◽  
V. Özerciyes
Keyword(s):  
Shallow Shell ◽  
Natural Frequencies ◽  
Circular Cylindrical Shell ◽  
Adhesive Layer ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Order System ◽  
Theoretical Formulation ◽  
First Order ◽  
Stress Resultant

Abstract This study is concerned with the “Free Vibrations of Composite Shallow Circular Cylindrical Shells or Shell Panels with a Central Stiffening Shell Strip”. The upper and lower shell elements of the stiffened composite system are considered as dissimilar, orthotropic shallow shells. The upper relatively narrow stiffening shell strip is centrally located and adhesively bonded to the lower main shell element In the theoretical formulation, a “First Order Shear Deformation Shell Theory (FSDST)” is employed. The complete set of the shallow shell dynamic equations (including the stress resultant-displacement and the constitutive equations) and the equations of the thin flexible, adhesive layer are first reduced to a set of first order system of ordinary differential equations. This final set forms the governing equations of the problem. Then, they are integrated by means of the “Modified Transfer Matrix Method”. In the adhesive layer, the “hard” and the “soft” adhesive effects are considered. It was found that the material characteristics of the adhesive layer influence the mode shapes and the corresponding natural frequencies of the composite shallow shell panel system. Additionally, some parametric studies on the natural frequencies are presented.

Modal Characteristics of Swept Cantilevered Plates

Volume 2 ◽  
2004 ◽  
Author(s):  
Naveed A. Din ◽  
S. Olutunde Oyadiji
Keyword(s):  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Modal Data ◽  
Nodal Lines ◽  
Modes Of Vibration ◽  
Cantilevered Plates ◽  
Definition Of ◽  
Data Frequency

The aim of this paper is to produce modal data which can be used to synthesise assumed shapes for use in Rayleigh-Ritz approximations of the free vibrations of cantilevered swept plates. The modal data is generated via the use of the FEA technique to predict the natural frequencies and mode shapes of aluminium alloy plates of aspect ratio 2.0 and of swept angles varying from 0° to 20° in steps of 2°. The first fifty natural frequencies and mode shapes of swept cantilevered plates were calculated using ABAQUS FE programme which includes the ABAQUS/CAE pre-and post-processor. To classify each mode shape, the number of nodal lines i along the x-axis and the number of nodal lines j along the y-axis were defined. This definition worked fine for uniform rectangular plates of zero swept angles and also for the first few modes of the swept plates. But as the number of modes of the swept plates increased, this definition became difficult to apply. Similar mode shapes of the various swept angles were put into families and groups headed by the i and j definition of the uniform rectangular plate design. From this modal data, frequency charts, which showed the variation of the dimensionless natural frequencies of the swept plates with swept angle, were constructed. These charts can be used to deduce the types of modes of vibration, whether bending or torsion, of a vibrating swept plate and to synthesis accurately the assumed shapes for use in the prediction of the vibration characteristics of swept plates using the Rayleigh-Ritz approach.

Effects of Variable Non-Central Locations of Bonded Double Doubler Joint System on Free Flexural Vibrations of Orthotropic Composite Mindlin Plate or Panel Adherents

2012 ◽  
Author(s):  
U. Yuceoglu ◽  
Ö. Güvendik
Keyword(s):  
Natural Frequencies ◽  
Mode Shapes ◽  
Mindlin Plate ◽  
Central Position ◽  
Joint Region ◽  
Joint System ◽  
Adhesive Layers ◽  
Governing System ◽  
Plate Elements ◽  
Interpolation Polynomials

This study investigates the “Effects of Variable Non-Central Locations of Bonded Double Doubler Joint System on Free Flexural Vibrations of Orthotropic Composite Mindlin Plate or Panel Adherents”. The problem is theoretically analyzed and is numerically solved in terms of the natural frequencies and the corresponding mode shapes of the entire “System”. The “Bonded Double Doubler Joint System” and the “Plate of Panel Adherents” are considered as dissimilar “Orthotropic Mindlin Plates”. In all plate elements, the transverse shear deformations and the transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers in the “Bounded Joint Region” are assumed to be linearly elastic continua with transverse normal and shear deformations. The “damping effects” in the adhesive layers and in all plate elements of the “System” are neglected. The sets of the “Dynamic Mindlin Equations” of both upper and lower “Doubler Plates” and the “Plate or Panel Adherents” and the adhesive layer equations are combined together with the orthotropic stress resultant-displacement expressions resulting in a set of “Governing System of PDE’s” in a “special form”. By making use of the “Classical Levy’s Solutions”, in aforementioned “Governing PDE’s” and following some algebraic manipulations and combinations, the “Governing System of the First Order Ordinary Differential Equations” are obtained in compact “state vector” forms. Thus, the “Initial and Boundary Value Problem” at the beginning is finally converted into a “Multi-Point Boundary Value Problem” of Mechanics (and Physics). These analytical results developed facilitate the present method of solution that is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials)”. The final set of the “Governing System of ODE’s” is numerically integrated by means of the “MTMM with Interpolation Polynomials”. In this way, the natural frequencies and the mode shapes of the “Bonded System”, depending on the variable non-central location of the “Bonded Double Doubler Joint System” are computed for several sets of the far left and the far right “Boundary Conditions” of the “Orthotropic Plate or Panel Adherents”. It was observed that, based on the numerical results, the mode shapes and their natural frequencies are very much affected by the variable position (or location) of the “Bonded Double Doubler Joint” in the “System”. It was also found that as the “Bonded Double Doubler Joint” moves from the central position in the “System” towards the increasingly non-central position, the natural frequencies (in comparison with those of the central position) changes, respectively. The highly-stiff “Bonded Double Doubler Joint Region” becomes “almost stationary” in all modes in “Hard” Adhesive cases.

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.

scholarly journals An investigation into the free vibrations of carbon nanotubes using analytical and finite element methods

2021 ◽  
Author(s):  
Ishan Ali Khan
Keyword(s):  
Carbon Nanotubes ◽  
Finite Element ◽  
Eigenvalue Problem ◽  
Natural Frequencies ◽  
Nonlinear Eigenvalue Problem ◽  
Dynamic Stiffness ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Wide Range ◽  
Walled Cnts

Since their discovery, immense attention has been given to carbon nanotubes (CNTs), due to their exceptional thermal, electronic and mechanical properties and, therefore, the wide range of applications in which they are, or can be potentially, employed. Hence, it is important that all the properties of carbon nanotubes are studied extensively. This thesis studies the vibrational frequencies of double-walled and triple-walled CNTs, with and without an elastic medium surrounding them, by using Finite Element Method (FEM) and Dynamic Stiffness Matrix (DSM) formulations, considering them as Euler-Bernoulli beams coupled with van der Waals interaction forces. For FEM modelling, the linear eigenvalue problem is obtained using Galerkin weighted residual approach. The natural frequencies and mode shapes are derived from eigenvalues and eigenvectors, respectively. For DSM formulation of double-walled CNTs, a nonlinear eigenvalue problem is obtained by enforcing displacement and load end conditions to the exact solution of single equation achieved by combining the coupled governing equations. The natural frequencies are obtained using Wittrick-Williams algorithm. FEM formulation is also applied to both double and triple-walled CNTs modelled as nonlocal Euler-Bernoulli beam. The natural frequencies obtained for all the cases, are in agreement with the values provided in literature.

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.

scholarly journals A Study on the Effect of Carbon Nanotubes’ Distribution and Agglomeration in the Free Vibration of Nanocomposite Plates

2020 ◽  
Vol 6 (4) ◽  
pp. 79
Author(s):  
D. S. Craveiro ◽  
M. A. R. Loja
Keyword(s):  
Carbon Nanotubes ◽  
Elastic Properties ◽  
Natural Frequencies ◽  
Mechanical Performance ◽  
Negative Impact ◽  
Free Vibrations ◽  
Mode Shapes ◽  
The Finite Element Method ◽  
Agglomeration Effect ◽  
Performance Predictions

The present work aimed to characterize the free vibrations’ behaviour of nanocomposite plates obtained by incorporating graded distributions of carbon nanotubes (CNTs) in a polymeric matrix, considering the carbon nanotubes’ agglomeration effect. This effect is known to degrade material properties, therefore being important to predict the consequences it may bring to structures’ mechanical performance. To this purpose, the elastic properties’ estimation is performed according to the two-parameter agglomeration model based on the Eshelby–Mori–Tanaka approach for randomly dispersed nano-inclusions. This approach is implemented in association with the finite element method to determine the natural frequencies and corresponding mode shapes. Three main agglomeration cases were considered, namely, agglomeration absence, complete agglomeration, and partial agglomeration. The results show that the agglomeration effect has a negative impact on the natural frequencies of the plates, regardless the CNTs’ distribution considered. For the corresponding vibrations’ mode shapes, the agglomeration effect was shown in most cases not to have a significant impact, except for two of the cases studied: for a square plate and a rectangular plate with symmetrical and unsymmetrical CNTs’ distribution, respectively. Globally, the results confirm that not accounting for the nanotubes’ agglomeration effect may lead to less accurate elastic properties and less structures’ performance predictions.

scholarly journals Numerical and Experimental Approach for Identifying Elastic Parameters in Sandwich Plates

2002 ◽  
Vol 9 (4-5) ◽  
pp. 193-201 ◽  
Author(s):  
Sergio Ferreira Bastos ◽  
Lavinia Borges ◽  
Fernando A. Rochinha
Keyword(s):  
Experimental Approach ◽  
Natural Frequencies ◽  
Free Edge ◽  
Free Vibrations ◽  
Elastic Parameter ◽  
Sandwich Plates ◽  
Rectangular Plates ◽  
Elastic Parameters ◽  
Non Destructive ◽  
Parameter Problem

This article deals with the identification of elastic parameters (engineering constants) in sandwich honeycomb orthotropic rectangular plates. A non-destructive method is introduced to identify the elastic parameters through the experimental measurements of natural frequencies of a plate undergoing free vibrations. Four elastic constant are identified. The estimation of the elastic parameter problem is solved by minimizing the differences between the measured and the calculated natural frequencies. The numerical method to calculate the natural frequencies involves the formulation of Rayleigh-Ritz using a series of characteristic orthogonal polynomials to properly model the free edge boundary conditions. The analysis of the results indicates the efficiency of the method.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

Effects of Position (or Location) of Non-Centrally Bonded Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) on Bending Vibrations of Composite Mindlin Plates or Panels

2010 ◽  
Author(s):  
U. Yuceoglu ◽  
O¨. Gu¨vendik
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Shear Stresses ◽  
Lap Joint ◽  
Joint System ◽  
Adhesive Layers ◽  
Present Solution ◽  
Stress Resultant ◽  
Bonded Joint

In the present study, the “Effects of Position (or Location) of Non-Centrally Bonded Symmetric Double Doubler Joint in Composite Mindlin Plates or Panels” are theoretically analyzed and are numerically solved in some detail. The “Plate Adherends” and the upper and lower “Doubler Plates” of the “Bonded Joint System” are considered as dissimilar, orthotropic “Mindlin Plates” joined through the dissimilar upper and lower very thin adhesive layers. The transverse and rotary moments of inertia are included in the analysis. The relatively very thin adhesive layers are assumed to be linearly elastic continua with transverse normal and shear stresses. The “damping effects” in the entire “Bonded Joint System” are neglected. The sets of the dynamic “Mindlin Plate” equations of the “Plate Adherends”, the “Double Doubler Plates” and the thin adhesive layers are combined together with the orthotropic stress resultant-displacement expressions in a “special form”. This system of equations, after some further manipulations, is eventually reduced to a set of the “Governing System of the First Order Ordinary Differential Equations” in terms of the “state vectors” of the problem. Hence, the final set of the aforementioned “Systems of Equations” together with the “Continuity Conditions” and the “Boundary Conditions” facilitate the present solution procedure. This is the “Modified Transfer Matrix Method (MTMM) (with Interpolation Polynomials). The present theoretical analysis and the present method of solution are applied to a typical “Non-Centrally Positioned (or Located) Symmetric Double Lap Joint (or Symmetric Double Doubler Joint) System”. The effects of the location (or position) of the “Bonded Joint System” and also of the relatively “Stiff (or “Hard”) and the relatively “Flexible” (or “Soft”) adhesive properties, on the natural frequencies and mode shapes are considered in some detail. The very interesting mode shapes with their dimensionless natural frequencies are presented for various sets of “Boundary Conditions”. From the numerical results obtained, some important conclusions are drawn for the “Bonded Joint System” studied here.

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