scholarly journals Vibration analysis of multi-stepped and multi-damaged parabolic arches using GDQ

2014 ◽  
Vol 2 (1) ◽  
Author(s):  
Erasmo Viola ◽  
Marco Miniaci ◽  
Nicholas Fantuzzi ◽  
Alessandro Marzani
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Internal Boundary ◽  
Radius Of Curvature ◽  
Inertia Effects ◽  
Circular Arch

AbstractThis paper investigates the in-plane free vibrations of multi-stepped and multi-damaged parabolic arches, for various boundary conditions. The axial extension, transverse shear deformation and rotatory inertia effects are taken into account. The constitutive equations relating the stress resultants to the corresponding deformation components refer to an isotropic and linear elastic material. Starting from the kinematic hypothesis for the in-plane displacement of the shear-deformable arch, the equations of motion are deduced by using Hamilton’s principle. Natural frequencies and mode shapes are computed using the Generalized Differential Quadrature (GDQ) method. The variable radius of curvature along the axis of the parabolic arch requires, compared to the circular arch, a more complex formulation and numerical implementation of the motion equations as well as the external and internal boundary conditions. Each damage is modelled as a combination of one rotational and two translational elastic springs. A parametric study is performed to illustrate the influence of the damage parameters on the natural frequencies of parabolic arches for different boundary conditions and cross-sections with localizeddamage.Results for the circular arch, derived from the proposed parabolic model with the derivatives of some parameters set to zero, agree well with those published over the past years.

Forced Vibrations of Viscoelastic Timoshenko Beams

1976 ◽  
Vol 98 (3) ◽  
pp. 820-826 ◽  
Author(s):  
C. C. Huang ◽  
T. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Attenuation Factor ◽  
Equations Of Motion ◽  
Bending Moment ◽  
Expansion Method ◽  
Free Vibrations ◽  
Forced Vibrations ◽  
Mode Shapes ◽  
Timoshenko Beams

In a previous paper, the correspondence principle has been applied to derive the differential equations of motion of viscoelastic Timoshenko beams with or without external viscous damping. To study free vibrations these equations are solved by Laplace transform and boundary conditions are applied to obtain the attenuation factor and the frequency of the damped free vibrations and mode shapes. The present paper continues to analyze this subject and deals with the responses in deflection, bending slope, bending moment and shear for forced vibrations. Laplace transform and appropriate boundary conditions have been applied. Examples are given and results are plotted. The solution of forced vibrations of elastic Timoshenko beams obtained as a result of reduction from viscoelastic case and by eigenfunction expansion method concludes the paper.

Theoretical Analysis of Free Vibrations Based on a New High Order Shell Theory for Cylindrical Shells Conveying Fluid

2017 ◽  
Author(s):  
Ming Ji ◽  
Kazuaki Inaba
Keyword(s):  
Boundary Conditions ◽  
Shell Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Cylindrical Shells ◽  
Fluid Pressure ◽  
Free Vibrations ◽  
High Order ◽  
Out Of Plane ◽  
Conveying Fluid

The natural frequencies of free vibrations for thick cylindrical shells with clamped-clamped ends conveying fluid are investigated. Equations of motion and boundary conditions are derived by Hamilton’s principle based on the new high order shell theory. The hydrodynamic force is derived from the linearized potential flow theory. Besides, fluid pressure acting on the shell wall is gotten by the assumption of non-penetration condition. The out-of-plane and in-plane vibrations are coupled together due to the existence of fluid-solid-interaction (FSI). Under the assumption of harmonic motion, the dispersion relationships are presented. Using the method of frequency sweeping, the natural frequencies of symmetric modes and asymmetric modes corresponding to each flow velocity are found by satisfying the dispersion relationship equations and boundary conditions. Several numerical examples with different flow velocities and thickness are presented compared with previous thin shell theory and FEM results and show reasonable agreement. The effects of thickness are discussed.

Experimental and numerical analysis of vibration frequency in sandwich composites with different radii of curvature

2017 ◽  
Vol 21 (8) ◽  
pp. 2870-2886
Author(s):  
Melis Yurddaskal ◽  
Buket Okutan Baba
Keyword(s):  
Finite Element ◽  
Numerical Analysis ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Experimental Studies ◽  
Sandwich Panels ◽  
Mode Shapes ◽  
Sandwich Composite ◽  
Radius Of Curvature ◽  
Finite Element Software

In this study, free vibration responses of sandwich composite panels with different radius of curvature were presented numerically. The studies were carried out on square flat and curved sandwich panels made of E-glass/epoxy face sheets and polyvinyl chloride foam with three different radii of curvature. Experimental studies were used to verify the numerical results. Vibration tests were performed on flat and curved sandwich panels under free–free boundary conditions. The experimental data were then compared with finite element simulation, which was conducted by ANSYS finite element software and it was shown that the numerical analysis results agree well with the experimental ones. Effect of the curvature on natural frequencies under different boundary conditions (all edge free, simply supported, and fully clamped) was investigated numerically. Results indicated that the natural frequencies and corresponding mode shapes were affected by boundary conditions and curvature of the panel. For all boundary conditions, the variation of curvature had smaller effect on the natural frequency of the first mode than those of the other modes.

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 Vibrations of Symmetric Arch: Boundary Conditions of Stress Resultants Revisited

2020 ◽  
Vol 20 (09) ◽  
pp. 2071008
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Mode Shape ◽  
Natural Frequencies ◽  
Free Vibrations ◽  
Mode Shapes ◽  
All Solutions

This paper deals with analyzing free vibrations of the symmetric arch. The boundary conditions of the stress resultants are newly derived, which can be replaced by the conventional boundary conditions of the deflections. All solutions of the natural frequency with the mode shape, using the new boundary conditions, are the same as those of the conventional deflections. The boundary conditions mixed with new and conventional conditions act correctly to calculate natural frequencies. The mode shapes of the stress resultants using the new boundary conditions are reported in two types: symmetric and anti-symmetric modes.

scholarly journals Free Vibrations of Tapered Horseshoe Circular Arch with Constant Volume

2020 ◽  
Vol 10 (16) ◽  
pp. 5431
Author(s):  
Byoung Koo Lee ◽  
Gweon Sik Kim ◽  
Sang Jin Oh ◽  
Tae Eun Lee
Keyword(s):  
Natural Frequencies ◽  
Quadratic Function ◽  
Free Vibrations ◽  
Constant Volume ◽  
Mode Shapes ◽  
Cross Sectional ◽  
Circular Arch ◽  
Taper Function ◽  
Parametric Studies ◽  
Sectional Shape

This paper presents free vibrations of the tapered horseshoe circular arch with a constant volume. The volume of the arch is constant, and the cross-sectional shape of the arch is square and circular. The taper function of the arch is a quadratic function. Differential equations with the boundary conditions that govern the free vibration of such arches are derived and numerically solved to calculate natural frequencies and mode shapes. The natural frequencies of this study agree well with those of the finite element ADINA. Parametric studies of the geometrical and cross-sectional properties of the arch on frequencies and mode shapes are performed and discussed.

Analysis of Free Vibrations of Conical Shells Using Donnell’s Approximations

1995 ◽  
Vol 48 (11S) ◽  
pp. S84-S89
Author(s):  
V. C. M. de Souza ◽  
J. M. F. Saraiva
Keyword(s):  
Shell Theory ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Conical Shells ◽  
Classical Shell Theory ◽  
Clamped Edges

The free vibrations of conical shells, having two open rigidly clamped edges, are investigated by using a variational development of the equations of motion based upon the Classical Shell Theory, and results are compared with those obtained by using Donnell’s approximation in the development of these equations. Through suitable examples, the validity of Donnell’s approximation to compute natural frequencies and mode-shapes of conical shells is shown.

IN-PLANE VIBRATION OF CIRCULAR ARCHES WITH VARYING CROSS-SECTIONS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350003 ◽  
Author(s):  
EKREM TUFEKCI ◽  
OZNUR OZDEMIRCI YIGIT
Keyword(s):  
Exact Solution ◽  
Cross Section ◽  
Cross Sections ◽  
Free Vibrations ◽  
Transverse Shear Deformation ◽  
Mode Transition ◽  
Rotatory Inertia ◽  
Initial Value ◽  
Inertia Effects ◽  
Circular Arch

The in-plane free vibration of circular arches with continuously varying cross-sections is studied by means of the exact solution. The exact solution can be obtained only for a circular arch with constant cross-section. As an approximation, the circular arch with varying cross-sections is divided into a number of arch elements with constant cross-sections. The cross-section of each arch element is determined by averaging the upper and lower cross-sections. Then, the exact solution of free vibrations for each arch element can be obtained by using the initial value method. The axial extension, transverse shear deformation and rotatory inertia effects are included in the analysis. As the number of the arch elements increases, the fast convergence of the frequencies to those of the original arch is observed. Clamped–clamped (CC), hinged–hinged (HH), hinged–clamped (HC), clamped–free (CF) and free–free (FF) boundary conditions are studied for different opening angles, taper types and taper ratios. A detailed parametric study is performed, by which the mode transition phenomenon is observed. The results obtained are compared with those available in the literature.

A Spectral-Tchebychev Solution for Three-Dimensional Vibrations of Parallelepipeds Under Mixed Boundary Conditions

2012 ◽  
Vol 79 (5) ◽  
Author(s):  
Sinan Filiz ◽  
Bekir Bediz ◽  
L. A. Romero ◽  
O. Burak Ozdoganlar
Keyword(s):  
Boundary Conditions ◽  
Natural Frequencies ◽  
Mixed Boundary ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mixed Boundary Conditions ◽  
Integral Form ◽  
Mode Shapes ◽  
Different Boundary Conditions ◽  
Practical Applications

Vibration behavior of structures with parallelepiped shape—including beams, plates, and solids—are critical for a broad range of practical applications. In this paper we describe a new approach, referred to here as the three-dimensional spectral-Tchebychev (3D-ST) technique, for solution of three-dimensional vibrations of parallelepipeds with different boundary conditions. An integral form of the boundary-value problem is derived using the extended Hamilton’s principle. The unknown displacements are then expressed using a triple expansion of scaled Tchebychev polynomials, and analytical integration and differentiation operators are replaced by matrix operators. The boundary conditions are incorporated into the solution through basis recombination, allowing the use of the same set of Tchebychev functions as the basis functions for problems with different boundary conditions. As a result, the discretized equations of motion are obtained in terms of mass and stiffness matrices. To analyze the numerical convergence and precision of the 3D-ST solution, a number of case studies on beams, plates, and solids with different boundary conditions have been conducted. Overall, the calculated natural frequencies were shown to converge exponentially with the number of polynomials used in the Tchebychev expansion. Furthermore, the natural frequencies and mode shapes were in excellent agreement with those from a finite-element solution. It is concluded that the 3D-ST technique can be used for accurate and numerically efficient solution of three-dimensional parallelepiped vibrations under mixed boundary conditions.

Experimental Modal Analysis of Rectangular and Circular Beams

2003 ◽  
Author(s):  
B. H. Emory ◽  
W. D. Zhu
Keyword(s):  
Boundary Conditions ◽  
Cross Sections ◽  
Natural Frequencies ◽  
Beam Theory ◽  
Mode Shapes ◽  
Euler Beam ◽  
Modal Assurance Criterion ◽  
Rectangular Beam ◽  
Circular Beam ◽  
A New Technique

Analytical and experimental methods are used to determine the natural frequencies and mode shapes of Aluminum 6061-T651 beams with rectangular and circular cross sections. A unique test stand is developed to provide the rectangular beam with different boundary conditions including clamped-free, clamped-clamped, clamped-pinned, and pinned-pinned. The first 10 bending frequencies and mode shapes for each set of boundary conditions are measured. The effects of the bolt torque on the measured frequencies of the rectangular beam are examined. The material properties of the circular beam, including the elastic modulus, shear modulus, and Poisson’s ratio, are determined by measuring its first 20 natural frequencies. A new technique is used to mount an accelerometer to measure the torsional modes of the circular beam. A roving hammer test is conducted to measure the first 10 mode shapes. The measured mode shapes of the circular and rectangular beams are compared with their thoretical predictions using the modal assurance criterion. The Timoshenko beam theory is shown to provide better predictions of the natural frequencies for the higher modes of the circular beam than the Bernoulli-Euler beam theory. The use of the rectangular and circular beam test stands as a teaching tool for undergraduate and graduate students is discussed.

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