On the Free Vibration Analysis of a Sandwich Beam With Tip Mass

2018 ◽  
Author(s):  
E. F. Joubaneh ◽  
O. R. Barry
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Sandwich Beam ◽  
Free Vibration Analysis ◽  
Design Parameters ◽  
Governing Equations ◽  
Tip Mass

This paper presents the free vibration analysis of a sandwich beam with a tip mass using higher order sandwich panel theory (HSAPT). The governing equations of motion and boundary conditions are obtained using Hamilton’s principle. General Differential Quadrature (GDQ) is employed to solve the system governing equations of motion. The natural frequencies and mode shapes of the system are presented and Ansys simulation is performed to validate the results. Various boundary conditions are also employed to examine the natural frequencies of the sandwich beam without tip mass and the results are compared with those found in the literature. Parametric studies are conducted to examine the effect of key design parameters on the natural frequencies of the sandwich beam with and without tip mass.

scholarly journals Analysis of Laminated Shells by Murakami’s Zig-Zag Theory and Radial Basis Functions Collocation

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
D. A. Maturi ◽  
A. J. M. Ferreira ◽  
A. M. Zenkour ◽  
D. S. Mashat
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Vibration Analysis ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Basis Functions ◽  
Transverse Displacement ◽  
Laminated Shells ◽  
Radial Basis

The static and free vibration analysis of laminated shells is performed by radial basis functions collocation, according to Murakami’s zig-zag (ZZ) function (MZZF) theory . The MZZF theory accounts for through-the-thickness deformation, by considering a ZZ evolution of the transverse displacement with the thickness coordinate. The equations of motion and the boundary conditions are obtained by Carrera’s Unified Formulation and further interpolated by collocation with radial basis functions.

Free vibration analysis of a rectangular plate carrying multiple various concentrated elements

2003 ◽  
Vol 217 (2) ◽  
pp. 171-183 ◽  
Author(s):  
J-S Wu ◽  
H-M Chou ◽  
D-W Chen
Keyword(s):  
Finite Element ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Rectangular Plate ◽  
Normal Mode ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Equation Of Motion ◽  
Mode Shapes

The dynamic characteristic of a uniform rectangular plate with four boundary conditions and carrying three kinds of multiple concentrated element (rigidly attached point masses, linear springs and elastically mounted point masses) was investigated. Firstly, the closed-form solutions for the natural frequencies and the corresponding normal mode shapes of a rectangular ‘bare’ (or ‘unconstrained’) plate (without any attachments) with the specified boundary conditions were determined analytically. Next, by using these natural frequencies and normal mode shapes incorporated with the expansion theory, the equation of motion of the ‘constrained’ plate (carrying the three kinds of multiple concentrated element) were derived. Finally, numerical methods were used to solve this equation of motion to give the natural frequencies and mode shapes of the ‘constrained’ plate. To confirm the reliability of previous free vibration analysis results, a finite element analysis was also conducted. It was found that the results obtained from the above-mentioned two approaches were in good agreement. Compared with the conventional finite element method (FEM), the approach employed in this paper has the advantages of saving computing time and achieving better accuracy, as can be seen from the existing literature.

scholarly journals A Unified Approach of Free Vibration Analysis for Stiffened Cylindrical Shell with General Boundary Conditions

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Xue-Qin Li ◽  
Wei Zhang ◽  
Xiao-Dong Yang ◽  
Lu-Kai Song
Keyword(s):  
Cylindrical Shell ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Unified Approach ◽  
Stiffened Cylindrical Shell

A unified approach of free vibration analysis for stiffened cylindrical shell with general boundary conditions is presented in this paper. The vibration of stiffened cylindrical shell is modeled mathematically involving the first-order shear deformation shell theory. The improved Fourier series is selected as the admissible displacement function while the arbitrary boundary conditions are simulated by adjusting the equivalent spring stiffness. The natural frequencies and modal shapes of the stiffened shell are obtained by solving the dynamic model with the Rayleigh-Ritz procedure. Various numerical results of free vibration analysis for stiffened cylindrical shell are obtained, including natural frequencies and modes under simply supported, free, and clamped boundary conditions. Moreover, the effects of stiffener on natural frequencies are discussed. Compared with several state-of-the-art methods, the feasibility and validity of the proposed method are verified.

An Analytical Method for Free Vibration Analysis of Composite Beams Subjected to Initial Thermal Stresses

2011 ◽  
Vol 482 ◽  
pp. 1-9
Author(s):  
A. Mahi ◽  
E.A. Adda-Bedia ◽  
A. Benkhedda
Keyword(s):  
Aspect Ratio ◽  
Free Vibration ◽  
Boundary Conditions ◽  
Thermal Stresses ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Beams ◽  
Ply Orientation

The purpose of this paper is to present exact solutions for the free vibration of symmetrically laminated composite beams. The present analysis includes the first shear deformation theory and the rotary inertia. The analytical solutions take into account the thermal effect on the free vibration characteristics of the composite beams. In particular, the aim of this work is to derive the exact closed-form characteristic equations for common boundary conditions. The different parameters that could affect the natural frequencies are included as factors (aspect ratio, thermal load-to-shear coefficient, ply orientation) to better perform dynamic analysis to have a good understanding of dynamic behavior of composite beams. In order to derive the governing set of equations of motion, the Hamilton’s principle is used. The system of ordinary differential equations of the laminated beams is then solved and the natural frequencies’ equations are obtained analytically for different boundary conditions. Numerical results are presented to show the influence of temperature rise, aspect ratio, boundary conditions and ply orientation on the natural frequencies of composite beams.

Small-amplitude free vibration analysis of active multistable shells under static multifield loading

2020 ◽  
pp. 107754632092393
Author(s):  
Dimitris Varelis
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Small Amplitude ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Free Vibration Analysis ◽  
Shell Element ◽  
Curvilinear Coordinates ◽  
Coupled Equations ◽  
Modal Response

This study considers the small-amplitude free vibrational response performed on top of the quasi-static snap through buckling, which is accompanied by large displacements and rotations of shallow doubly curved laminated piezoelectric shells under multifield loading. The mechanics incorporate coupling between mechanical, electric, and thermal fields and encompass geometric nonlinearity effects due to large quasi-static displacements and rotations. The governing equations are formulated explicitly in orthogonal curvilinear coordinates and combined with the kinematic assumptions of a mixed-field shear-layerwise shell laminate theory. Based on the above mechanics and adopting the finite element methodology, an eight-node nonlinear shell element is developed to yield the linearized discrete coupled small-amplitude dynamic equations of motion. Initially, the nonlinear coupled equations are linearized and solved quasi-statically using an extended cylindrical arc-length method in combination with the Newton–Raphson iterative technique, and subsequently the free vibration analysis is performed at each solution point. Validation and evaluation cases on laminated cylindrical shells demonstrate the accuracy of the present method and its robust capability to predict the modal response on top of the nonlinear quasi-static response of active multistable shells subject to combined thermo–piezo–electromechanical loads. Numerical cases show the feasibility to develop smart shell structures to detect, via the monitoring of natural frequencies, the onset of snap-through instability. The capability of smart shells to actively modify its natural frequencies such as to promote or mitigate snap-through instabilities is quantified. Additional results quantify the effect of thermomechanical loads on actuation capability. The influence of geometric parameters (curvature and thickness) on the modal response is finally investigated.

Free Vibration Analysis of Cracked Composite Beams Subjected to Coupled Bending-Torsion Loads Based on a First Order Beam Theory

2013 ◽  
Vol 325-326 ◽  
pp. 1318-1323 ◽  
Author(s):  
A.R. Daneshmehr ◽  
D.J. Inman ◽  
A.R. Nateghi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Beam Theory ◽  
Order Theory ◽  
Composite Beams ◽  
First Order ◽  
First Order Theory

In this paper free vibration analysis of cracked composite beams subjected to coupled bending-torsion loads are presented. The composite beam is assumed to have an open edge crack. A first order theory is applied to count for the effect of the shear deformations on natural frequencies as well as the effect of coupling in torsion and bending modes of vibration. Local flexibility matrix is used to obtain the additional boundary conditions of the beam in the crack area. After obtaining the governing equations and boundary conditions, GDQ method is applied to solve the obtained eigenvalue problem. Finally, some numerical results are given to show the efficacy of the method. In addition, to count for the effect of coupling on natural frequencies of the cracked beams, different fiber orientations are assumed and studied.

Experimental and Numerical Analysis of a Sandwich Beam With Tip Mass

2018 ◽  
Author(s):  
E. F. Joubaneh ◽  
O. R. Barry ◽  
D. C. D. Oguamanam
Keyword(s):  
Numerical Analysis ◽  
Sandwich Panel ◽  
Equations Of Motion ◽  
Mathematical Formulation ◽  
Sandwich Beam ◽  
Design Parameters ◽  
Governing Equations ◽  
Parametric Studies ◽  
Tip Mass ◽  
Good Agreement

This paper presents experimental and numerical analyses of a vibrating sandwich beam with a tip mass. The mathematical formulation is based on higher order sandwich panel theory (HSAPT) and the governing equations of motion and boundary conditions are obtained using Hamilton’s principle. General Differential Quadrature (GDQ) is employed to solve the system governing equations of motion. Experiments are carried out to validate the proposed formulation and the results show very good agreement. Parametric studies are conducted to investigate the influence of key design parameters on the natural frequency and vibration response of the system.

A Theoretical Approach for Free Vibration Analysis of the Nano-Plates Considering the Small Scale Effect

2010 ◽  
Author(s):  
Emad Jomehzadeh ◽  
Ali Reza Saidi
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Scale Effect ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Small Scale ◽  
Edge Zone ◽  
Wide Range ◽  
Small Scale Effect

The free vibration analysis of a nano-plate is investigated based on the first order shear deformation theory considering the small scale effect. The governing equations of motion are obtained using Hamilton’s principle by considering the nonlocal constitutive equations of Eringen. These coupled partial differential equations are reformulated into two new equations called the edge-zone and interior equations. Analytical solutions are obtained for a nano-plate with Levy boundary conditions. In order to find the natural frequencies of the nano-plate, the various boundary conditions at one direction of the plate should be imposed. Applying these conditions and setting the determinant of the six order coefficient matrix equal to zero, the natural frequencies of the nano-plate are evaluated. Non-dimensional frequency parameters are presented for over a wide range of nonlocal parameters and different boundary conditions. In addition, the effects of nonlocal parameter on the natural frequency of a nano-plate are discussed in details.

DISCRETE SINGULAR CONVOLUTION (DSC) FOR FREE VIBRATION ANALYSIS OF CONICAL SHELLS WITH VARIOUS BOUNDARY CONDITIONS

2007 ◽  
Vol 04 (01) ◽  
pp. 81-108 ◽  
Author(s):  
ÖMER CIVALEK
Keyword(s):  
Free Vibration ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Shell Theory ◽  
Free Vibration Analysis ◽  
Governing Equations ◽  
Conical Shells ◽  
Various Boundary Conditions ◽  
Discrete Singular Convolution ◽  
Thin Shell Theory

This paper gives a relatively novel computational approach, the discrete singular convolution (DSC) algorithm, for the free vibration analysis of isotropic and orthotropic conical shells with different boundary conditions. The governing differential equations of vibration of the shell are formulated using Love's first approximation classical thin shell theory. In the proposed approach, the derivatives in both the governing equations and the boundary conditions are discretized by the method of DSC. Typical numerical results are presented illustrating the effect of various geometric and material parameters. The influence of boundary conditions on the frequency characteristics is also discussed. The obtained results are in excellent agreement with those in the literature.

Free Vibration Analysis the Cracked FGM Beam under Bending-Torsion Loading, Using GDQ Method

2013 ◽  
Vol 330 ◽  
pp. 942-947 ◽  
Author(s):  
Alireza Daneshmehr ◽  
D.J. Inman ◽  
A. Mohammadi Fakhar
Keyword(s):  
Free Vibration ◽  
Vibration Analysis ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Algebraic Equations ◽  
Governing Equations ◽  
Functionally Graded Beam ◽  
Fg Beam ◽  
Fgm Beam

This paper presents a theoretical investigation of free vibration analysis of a functionally graded beam (FGM) under the bending-torsion loading using a classical elasticity theory. The FG beam is assumed to have an open edge crack. It is assumed that the material properties of the simply-supported cracked beam, vary along the beam thickness following a polynomial distribution in the thickness direction. This analysis is based on the linear fracture mechanics. First of all, governing equations and boundary conditions of the FG beam are derived using Hamilton's principle. The governing equations are solved using generalized differential quadrature (GDQ) method. By applying GDQ method, the governing differential equations convert to a linear system of algebraic equations. Then solving the eigenvalue problem, natural frequencies of the FG beam can be found. The results indicate that natural frequencies in the presence of a crack are affected by the crack ratio and location.

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