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.

Modal Characteristic of Cantilevered Beams and Plates: A Study of Transition From Beam-Like to Plate-Like Behaviour

1999 ◽  
Author(s):  
K. Tangchaichit ◽  
S. O. Oyadiji
Keyword(s):  
Transition Point ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Fe Analysis ◽  
Mode Shapes ◽  
Structural Components ◽  
Cantilevered Plates ◽  
The Difference ◽  
Cantilevered Beams ◽  
High Length

Abstract The paper presents the finite element (FE) analysis of the free vibrations of cantilevered aluminium alloy beams and plates of 5mm thickness and of length-to-breadth ratios ranging from a ratio of 20 for a beam down to a ratio of 0.25 for a plate. The analysis was carried out using the ABAQUS FE programme. For each ratio, a total of 20 natural frequencies and mode shapes were predicted. The objective of the paper was to demonstrate that a transition zone for beam-like to plate-like behaviour of structural components can be approximately defined for various length-to-breadth ratios. It is shown that the frequency parameters of cantilevered plates asymptotically approach the frequency parameters of cantilevered beams at high length-to-breadth ratios. In addition, it is shown that at the transition point for beam-like to plate-like behaviour, which occurs at small length-to-breadth ratios, the difference between the frequency parameters of cantilevered beams is less than the frequency parameters of cantilevered plates about 3 %.

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.

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.

Free Vibrations of Constrained Beams

1952 ◽  
Vol 19 (4) ◽  
pp. 471-477
Author(s):  
Winston F. Z. Lee ◽  
Edward Saibel
Keyword(s):  
Natural Frequencies ◽  
Degree Of Approximation ◽  
Free Vibrations ◽  
Frequency Equation ◽  
Continuous Beam ◽  
Concentrated Mass ◽  
Simple Manner ◽  
Natural Modes ◽  
Modes Of Vibration ◽  
Rigid Supports

Abstract A general expression is developed from which the frequency equation for the vibration of a constrained beam with any combination of intermediate elastic or rigid supports, concentrated masses, and sprung masses can be found readily. The method also is extended to the case where the constraint is a continuous elastic foundation or uniformly distributed load of any length. This method requires only the knowledge of the natural frequencies and natural modes of the beam supported at the ends in the same manner as the constrained beam but not subjected to any of the constraints between the ends. The frequency equation is obtained easily and can be solved to any desired degree of approximation for any number of modes of vibration in a quick and simple manner. Numerical examples are given for a beam with one concentrated mass, for a beam with one sprung mass, and a continuous beam with one sprung mass.

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.

Updating of Non-Conservative Systems Using Inverse Methods

1997 ◽  
Author(s):  
Ladislav Starek ◽  
Milos Musil ◽  
Daniel J. Inman
Keyword(s):  
Analytical Model ◽  
Degrees Of Freedom ◽  
Ad Hoc ◽  
Natural Frequencies ◽  
Eigenvalue Problems ◽  
Model Updating ◽  
Analytical Models ◽  
Mode Shapes ◽  
Element Analysis ◽  
Modal Data

Abstract Several incompatibilities exist between analytical models and experimentally obtained data for many systems. In particular finite element analysis (FEA) modeling often produces analytical modal data that does not agree with measured modal data from experimental modal analysis (EMA). These two methods account for the majority of activity in vibration modeling used in industry. The existence of these discrepancies has spanned the discipline of model updating as summarized in the review articles by Inman (1990), Imregun (1991), and Friswell (1995). In this situation the analytical model is characterized by a large number of degrees of freedom (and hence modes), ad hoc damping mechanisms and real eigenvectors (mode shapes). The FEM model produces a mass, damping and stiffness matrix which is numerically solved for modal data consisting of natural frequencies, mode shapes and damping ratios. Common practice is to compare this analytically generated modal data with natural frequencies, mode shapes and damping ratios obtained from EMA. The EMA data is characterized by a small number of modes, incomplete and complex mode shapes and non proportional damping. It is very common in practice for this experimentally obtained modal data to be in minor disagreement with the analytically derived modal data. The point of view taken is that the analytical model is in error and must be refined or corrected based on experimented data. The approach proposed here is to use the results of inverse eigenvalue problems to develop methods for model updating for damped systems. The inverse problem has been addressed by Lancaster and Maroulas (1987), Starek and Inman (1992,1993,1994,1997) and is summarized for undamped systems in the text by Gladwell (1986). There are many sophisticated model updating methods available. The purpose of this paper is to introduce using inverse eigenvalues calculated as a possible approach to solving the model updating problem. The approach is new and as such many of the practical and important issues of noise, incomplete data, etc. are not yet resolved. Hence, the method introduced here is only useful for low order lumped parameter models of the type used for machines rather than structures. In particular, it will be assumed that the entries and geometry of the lumped components is also known.

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.

Non-linear free transverse vibration of thin rotating discs

2007 ◽  
Vol 221 (3) ◽  
pp. 467-473 ◽  
Author(s):  
H R Hamidzadeh
Keyword(s):  
Steady State ◽  
Free Vibration ◽  
Natural Frequencies ◽  
Mode Shapes ◽  
Stress Distributions ◽  
Rotating Speed ◽  
Rotating Discs ◽  
Non Linear ◽  
Modes Of Vibration ◽  
Free Transverse Vibration

An analytical method is adopted to determine modal characteristics of non-linear spinning discs. The disc is assumed to be isotropic and rotating under steady-state conditions. The effects of amplitude and rotating speed on natural frequencies are determined. The developed procedure is also capable of analysing natural frequencies of linear free vibration, which is independent of amplitude. Attention is confined to determine natural frequencies, mode shapes, stress distributions, and critical speeds for different numbers of nodal diameters. The developed procedure does not consider modes of vibration corresponding to nodal circles. Validity of this procedure is verified by comparing some of the computed results with those established for certain cases.

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.

Sign in / Sign up

Export Citation Format

Share Document