Axisymmetric Free Vibrations of a Transversely Isotropic Finite Cylindrical Rod

1988 ◽  
Vol 55 (4) ◽  
pp. 855-862 ◽  
Author(s):  
C. P. Lusher ◽  
W. N. Hardy
Keyword(s):  
Boundary Conditions ◽  
Elastic Constants ◽  
Transversely Isotropic ◽  
Acoustic Vibration ◽  
Elastic Cylinder ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Hexagonal Symmetry ◽  
Hexagonal System ◽  
Better Than

The frequencies of free vibration and mode shapes are calculated for axisymmetric modes of an elastic cylinder of finite length having hexagonal symmetry with the crystallographic c-axis coincident with the axis of the cylinder (a transversely isotropic finite cylindrical rod). A series solution is used which satisfies term-by-term the differential equations of linear elasticity and the boundary conditions on the shear stress; the boundary conditions on the normal stresses are satisfied by using an orthogonalization procedure. As an example, the method is applied to sapphire, with one of the six elastic constants (c14) taken to be zero. The other five elastic constants are those of the hexagonal system. The calculated acoustic vibration frequencies agree to better than 1 percent with measurements made on sapphire at room temperature, for a cylinder of half-height to radius ratio ∼ 1.

scholarly journals In-Plane free Vibration Analysis of an Annular Disk with Point Elastic Support

2011 ◽  
Vol 18 (4) ◽  
pp. 627-640 ◽  
Author(s):  
S. Bashmal ◽  
R. Bhat ◽  
S. Rakheja
Keyword(s):  
Boundary Conditions ◽  
Orthogonal Polynomials ◽  
Ritz Method ◽  
Free Vibration Analysis ◽  
Free Vibrations ◽  
Outer Edge ◽  
Mode Shapes ◽  
Annular Disk ◽  
Different Boundary Conditions ◽  
Boundary Characteristic

In-plane free vibrations of an elastic and isotropic annular disk with elastic constraints at the inner and outer boundaries, which are applied either along the entire periphery of the disk or at a point are investigated. The boundary characteristic orthogonal polynomials are employed in the Rayleigh-Ritz method to obtain the frequency parameters and the associated mode shapes. Boundary characteristic orthogonal polynomials are generated for the free boundary conditions of the disk while artificial springs are used to account for different boundary conditions. The frequency parameters for different boundary conditions of the outer edge are evaluated and compared with those available in the published studies and computed from a finite element model. The computed mode shapes are presented for a disk clamped at the inner edge and point supported at the outer edge to illustrate the free in-plane vibration behavior of the disk. Results show that addition of point clamped support causes some of the higher modes to split into two different frequencies with different mode shapes.

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.

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.

A new semi-analytical solution method for free vibration analysis of composite rectangular plates with general edge constraints coupled with single piezoelectric layer

2018 ◽  
Vol 29 (20) ◽  
pp. 3873-3889 ◽  
Author(s):  
Mehdi Baghaee ◽  
Amin Farrokhabadi ◽  
Ramazan-Ali Jafari-Talookolaei
Keyword(s):  
Boundary Conditions ◽  
Piezoelectric Layer ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Composite Plates ◽  
Energy Functional ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Rectangular Plates ◽  
Generalized Displacements

In this article, a new approach is presented to study the free vibrations of rectangular composite plates coupled with single piezoelectric layer. The laminated plate with general stacking sequences is subjected to the elastic edge restraints. Based on the first-order shear deformation theory and Hamilton’s principle, the equations of the motion along with boundary conditions of the problem are deduced. To solve the problem, generalized displacements as well as general electric potentials are expanded using the Legendre polynomial series as the base functions. Then, the kinetic and potential energies of the problem are obtained. Afterwards, by means of Lagrange multipliers all the boundary conditions have been added to the energies to form the functional. This energy functional is extremised to get the natural frequencies and mode shapes of the problem through generalized eigenvalue problem. Credibility of the proposed method is verified by comparing the obtained results with those achieved by other theories and finite element method.

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 Resonant Ultrasound Spectroscopy: Sensitivity Analysis for Anisotropic Materials with Hexagonal Symmetry

2021 ◽  
pp. 1-20
Author(s):  
Christopher Sevigney ◽  
Onome Scott-Emuakpor ◽  
Farhad Farzbod
Keyword(s):  
Elastic Constants ◽  
Material Properties ◽  
Free Vibrations ◽  
Hexagonal Symmetry ◽  
Resonant Ultrasound Spectroscopy ◽  
Cubic Symmetry ◽  
Complete Set ◽  
Resonant Ultrasound ◽  
Non Destructive

Abstract Resonance ultrasound spectroscopy (RUS) is a non-destructive technique for evaluating elastic and an-elastic material properties. The frequencies of free vibrations for a carefully crafted sample are measured, and material properties can be extracted from this. In one popular application, the determination of monocrystal elasticity, the results are not always reliable. In some cases, the resonant frequencies are insensitive to changes in certain elastic constants or their linear combinations. Previous work has been done to characterize these sensitivity issues in materials with isotropic and cubic symmetry. This work examines the sensitivity of elastic constant measurements by the RUS method for materials with hexagonal symmetry, such as titanium-diboride. We investigate the reliability of RUS data and explore supplemental measurements to obtain an accurate and complete set of elastic constants.

The compression of anisotropic fibre monofilaments. II

1966 ◽  
Vol 291 (1425) ◽  
pp. 267-278 ◽  
Keyword(s):  
Elastic Constants ◽  
Poisson’S Ratio ◽  
Synthetic Fibre ◽  
Transversely Isotropic ◽  
Elastic Cylinder ◽  
Poisson's Ratio ◽  
Elastic Cylinders ◽  
Complete Specification ◽  
Microscope Stage

A synthetic fibre monofilament was compressed between transparent flats mounted on a microscope stage. The change in the monofilament diameter parallel to the plane of the flats was determined as a function of load for polyethylene terephthalate and nylon monofilaments. These monofilaments can be regarded as transversely isotropic elastic cylinders. A theoretical solution was therefore derived for the change in diameter of a transversely isotropic elastic cylinder compressed between two rigid parallel planes. The solution for the change in diameter in terms of the elastic constants and the applied load was verified experimentally. Measurements of the change in diameter as a function of load were combined with measurements of the width of the contact zone between the cylinder and the upper transparent flat. The results together with subsidiary measurements of extensional compliance and extensional Poisson’s ratio were then used to derive the transverse compliance and the transverse Poisson’s ratio of the monofilaments. This compression problem provides the first reported determination of the transverse Poisson’s ratio for synthetic fibre monofilaments. It is therefore now possible to obtain all five independent elastic constants, four of which are related to the measurements described in this paper. For the sake of completeness, results for the torsional compliance of these monofilaments have been undertaken thus giving complete specification of the elastic properties.

Axisymmetric Compression Wave Propagation in a Magneto-Electro-Elastic Cylinder

2011 ◽  
Vol 66-68 ◽  
pp. 776-781
Author(s):  
Jin Jing
Keyword(s):  
Wave Propagation ◽  
Boundary Conditions ◽  
Compression Wave ◽  
Transversely Isotropic ◽  
Elastic Cylinder ◽  
Numerical Results ◽  
Propagation Properties

In this paper the propagation properties of axisymmetric compression wave in an infinite and transversely isotropic magneto-electro-elastic cylinder are investigated. Numerical results show that mechanical boundary conditions have obvious and different influences on the propagation properties of axisymmetric compression wave in a magneto-electro-elastic cylinder.

Free Vibrations of Viscoelastic Timoshenko Beams

1971 ◽  
Vol 38 (2) ◽  
pp. 515-521 ◽  
Author(s):  
T. C. Huang ◽  
C. C. Huang
Keyword(s):  
Boundary Conditions ◽  
Laplace Transform ◽  
Correspondence Principle ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Viscous Damping ◽  
Complex Frequency ◽  
Timoshenko Beams ◽  
Numerical Illustration ◽  
End Conditions

The correspondence principle has been applied to derive the differential equations of viscoelastic Timoshenko beams with external viscous damping. These equations are solved by Laplace transform and boundary conditions are applied to obtain complex frequency equations and mode shapes for beams of any combination of end conditions. For beams without external damping, the correspondence principle can be applied directly to the available solutions of elastic Timoshenko beams. Numerical illustration is given.

scholarly journals Vibration of rotating circular cylindrical shells with distributed springs

2021 ◽  
Vol 37 ◽  
pp. 346-358
Author(s):  
Fuchun Yang ◽  
Xiaofeng Jiang ◽  
Fuxin Du
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Shell Theory ◽  
Rotation Speed ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Governing Equations ◽  
Natural Response ◽  
Circular Cylindrical Shells ◽  
The Galerkin Method

Abstract Free vibrations of rotating cylindrical shells with distributed springs were studied. Based on the Flügge shell theory, the governing equations of rotating cylindrical shells with distributed springs were derived under typical boundary conditions. Multicomponent modal functions were used to satisfy the distributed springs around the circumference. The natural responses were analyzed using the Galerkin method. The effects of parameters, rotation speed, stiffness, and ratios of thickness/radius and length/radius, on natural response were also examined.

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