The Effect of the Longitudinal to Transverse Moduli Ratio on the Natural Frequencies of Symmetric Cross-Ply Laminated Cylindrical Helical Springs

1999 ◽  
Vol 121 (4) ◽  
pp. 634-639 ◽  
Author(s):  
Vebil Yildirim ◽  
Erol Sancaktar ◽  
Erhan Kiral
Keyword(s):  
Free Vibration ◽  
Young’S Modulus ◽  
Transfer Matrix Method ◽  
Parametric Study ◽  
Natural Frequencies ◽  
Young's Modulus ◽  
Vibration Frequencies ◽  
Axial Deformation ◽  
Helical Springs ◽  
Helix Pitch

The first six free vibration frequencies of symmetric cross-ply laminated cylindrical helical springs with fixed-fixed ends are theoretically computed based on the transfer matrix method. The rotary inertia, shear and axial deformation effects are taken into account in the solution. Considering different values for the helix pitch angles and the number of active turns, a parametric study is performed to analyze the effects of the ratio of the longitudinal Young’s modulus to the transverse Young’s modulus on the natural frequencies of such springs with square section. The results are given in dimensionless graphical forms.

Effect of the Material Types on the Fundamental Frequencies of Uniaxial Composite Conical Springs

1999 ◽  
Author(s):  
Vebil Yildirim ◽  
Erol Sancaktar ◽  
Erhan Kiral
Keyword(s):  
Transfer Matrix Method ◽  
Pitch Angle ◽  
Natural Frequencies ◽  
Axial Deformation ◽  
Helical Springs ◽  
Fundamental Frequencies ◽  
Helix Pitch ◽  
Constant Pitch ◽  
Α Helix

Abstract This paper deals with the effect of the material types (Graphite-Epoxies and Kevlar-Epoxy) on the fundamental frequencies of uniaxial constant-pitch composite conical helical springs with solid circle section and fixed-fixed ends. The transfer matrix method is used for the determination of the fundamental natural frequencies. The rotary inertia, the shear and axial deformation effects are taken into account in the solution. The free vibrational charts for each material presented in this study cover the following vibrational parameters: n (number of active turns) = 5–10, α = (helix pitch angle) = 5° and 25°, R2/R1, (minimum to maximum radii of the cylinder) = 0.1 and 0.9, and Dmax/d (maximum cylinder to wire diameters) = 5 and 15. These charts can be used for the design of uniaxial composite conical springs.

scholarly journals Improved Riccati Transfer Matrix Method for Free Vibration of Non-Cylindrical Helical Springs Including Warping

2012 ◽  
Vol 19 (6) ◽  
pp. 1167-1180 ◽  
Author(s):  
A.M. Yu ◽  
Y. Hao
Keyword(s):  
Free Vibration ◽  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Natural Frequencies ◽  
Free Vibration Analysis ◽  
Mode Shapes ◽  
Axial Deformation ◽  
Cross Sectional ◽  
Helical Springs

Free vibration equations for non-cylindrical (conical, barrel, and hyperboloidal types) helical springs with noncircular cross-sections, which consist of 14 first-order ordinary differential equations with variable coefficients, are theoretically derived using spatially curved beam theory. In the formulation, the warping effect upon natural frequencies and vibrating mode shapes is first studied in addition to including the rotary inertia, the shear and axial deformation influences. The natural frequencies of the springs are determined by the use of improved Riccati transfer matrix method. The element transfer matrix used in the solution is calculated using the Scaling and Squaring method and Pad'e approximations. Three examples are presented for three types of springs with different cross-sectional shapes under clamped-clamped boundary condition. The accuracy of the proposed method has been compared with the FEM results using three-dimensional solid elements (Solid 45) in ANSYS code. Numerical results reveal that the warping effect is more pronounced in the case of non-cylindrical helical springs than that of cylindrical helical springs, which should be taken into consideration in the free vibration analysis of such springs.

In-Plane Free Vibration of Uniform Circular Arches Made of Axially Functionally Graded Materials

2019 ◽  
Vol 19 (08) ◽  
pp. 1950084 ◽  
Author(s):  
Joon Kyu Lee ◽  
Byoung Koo Lee
Keyword(s):  
Free Vibration ◽  
Young’S Modulus ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Young's Modulus ◽  
Mass Density ◽  
Functionally Graded ◽  
Mode Shapes ◽  
Governing Equations ◽  
Direct Integral

This study focused on the in-plane free vibration of uniform circular arches made of axially functionally graded (AFG) materials. Based on the dynamic equilibrium of an arch element, the governing equations for the free vibration of an AFG arch are derived in this study, where arbitrary functions for the Young’s modulus and mass density are acceptable. For the purpose of numerical analysis, quadratic polynomials for the Young’s modulus and mass density are considered. To calculate the natural frequencies and corresponding mode shapes, the governing equations are solved using the direct integral method enhanced by the trial eigenvalue method. For verification purposes, the predicted frequencies are compared to those obtained by the general purpose software ADINA. A parametric study of the end constraint, rotatory inertia, modular ratio, radius parameter, and subtended angle for the natural frequencies is conducted and the corresponding mode shapes are reported.

A Parametric Study on the Free Vibration of Noncylindrical Helical Springs

1998 ◽  
Vol 65 (1) ◽  
pp. 157-163 ◽  
Author(s):  
V. Yıldırım
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Parametric Study ◽  
Natural Frequencies ◽  
Rotary Inertia ◽  
Helical Springs ◽  
Wide Range ◽  
Stiffness Method

In the work based on the stiffness method reported in this paper, considering the rotary inertia, the axial and shear deformation terms, the natural frequencies of conical, barrel and hyperboloidal-type helical springs fixed at both ends are calculated. The results are presented in dimensionless graphical forms for the six lowest natural frequencies of all types of noncylindrical helices for a wide range of vibrational parameters which influence the natural frequencies. A discussion about the effects of vibrational parameters on the natural frequencies is also presented.

Effect of Centrifugal Force on Frequency Characteristic of Rotating Hollow Cylinders Using a Newly Designed Cylindrical Super Element

2006 ◽  
Author(s):  
M. Bonakdar ◽  
M. T. Ahmadian
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Centrifugal Force ◽  
Natural Frequencies ◽  
Rotating Cylinder ◽  
Finite Cylinder ◽  
Vibration Frequencies ◽  
Hollow Cylinders ◽  
Special Case

A sixteen node cylindrical super element is presented for evaluating the free vibration characteristics of a rotating laminated cylinder with conventional boundary conditions. It is shown that the natural frequencies are affected considerably when the centrifugal force is also taken into account. The vibration frequencies of rotating finite cylinder, obtained by conventional finite element are used to evaluate the accuracy of this approach. The special case of a stationary cylinder with zero spinning velocity is also considered as a check on this method. Results indicate only few number of cylindrical super elements are capable of predicting the natural frequency of the rotating cylinder within the same limit as many elements used in the conventional finite element method.

Research on Vibration and Instability of the Double-Span Beam Base on Transfer Matrix

2009 ◽  
Vol 16-19 ◽  
pp. 160-163 ◽  
Author(s):  
Ting Liu ◽  
Fei Feng ◽  
Ya Zhe Chen ◽  
Bang Chun Wen
Keyword(s):  
Free Vibration ◽  
Transfer Matrix ◽  
Axial Compression ◽  
Axial Force ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Natural Frequencies ◽  
Euler Beam ◽  
Intermediate Support ◽  
Critical Axial Force

The vibration and instability of a beam which is Double-span Euler Beam with axial force is studied by transfer matrix method. The transfer matrix of transverse free vibration and axial compression of the beam is derived. Then based on the assembled transferring matrix, the effect of the position of intermediate support on the natural frequencies and Euler critical axial force of the beam is discussed, which offered a useful method to start research of vibration of complicated framework.

Fundamental Frequencies of Uniaxial Composite Barrel and Hyperboloidal Springs

1999 ◽  
Author(s):  
Vebil Yildirim ◽  
Erol Sancaktar ◽  
Erhan Kiral
Keyword(s):  
Free Vibration ◽  
Transfer Matrix ◽  
Shear Deformation Theory ◽  
Variable Coefficients ◽  
Axial Deformation ◽  
Circular Section ◽  
First Order ◽  
Helical Springs ◽  
Fundamental Frequencies ◽  
Constant Pitch

Abstract The fundamental natural frequencies of uniaxial composite non-cylindrical helical springs (barrel and hyperboloidal types) are determined theoretically based on the transfer matrix method. The rotary inertia, shear and axial deformation effects are considered with the first order shear deformation theory. The overall transfer matrix is obtained by integrating the twelve scalar ordinary differential equations with variable coefficients governing the free vibration behavior of non-cylindrical helical springs made of an anisotropic material. Numerical results are verified with the reported values for isotropic non-cylindrical helices. A parametric study is performed to investigate the effects of the number of active coils (n = 5–0), the helix pitch angle (α = 5° and 25°), the ratio of the minimum to maximum cylinder radii (Rmin/Rmax), and the ratio of the maximum cylinder diameter to the wire diameter (Dmax/d) on the fundamental free vibration frequencies of constant-pitch composite barrel and hyperboloidal helical springs with circular section and fixed-fixed ends.

Free Vibration of Helicoidal Beams of Arbitrary Shape and Variable Cross Section

2002 ◽  
Vol 124 (3) ◽  
pp. 397-409 ◽  
Author(s):  
Wisam Busool ◽  
Moshe Eisenberger
Keyword(s):  
Free Vibration ◽  
Cross Section ◽  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Free Vibration Analysis ◽  
Rotational Inertia ◽  
Variable Cross ◽  
Dynamic Stiffness Matrix ◽  
Helical Springs

In this study, the dynamic stiffness method is employed for the free vibration analysis of helical springs. This work gives the exact solutions for the natural frequencies of helical beams having arbitrary shapes, such as conical, hyperboloidal, and barrel. Both the cross-section dimensions and the shape of the beam can vary along the axis of the curved member as polynomial expressions. The problem is described by six differential equations. These are second order equations with variable coefficients, with six unknown displacements, three translations, and three rotations at every point along the member. The proposed solution is based on a new finite-element method for deriving the exact dynamic stiffness matrix for the member, including the effects of the axial and the shear deformations and the rotational inertia effects for any desired precision. The natural frequencies are found as the frequencies that cause the determinant of the dynamic stiffness matrix to become zero. Then the mode shape for every natural frequency is found. Examples are given for beams and helical springs with different shape, which can vary along the axis of the member. It is shown that the present numerical results agree well with previously published numerical and experimental results.

scholarly journals A Stochastic Galerkin Cell-based Smoothed Finite Element Method (SGCS–FEM)

2019 ◽  
Vol 17 (08) ◽  
pp. 1950054
Author(s):  
Tittu Varghese Mathew ◽  
Lars Beex ◽  
Stéphane PA Bordas ◽  
Sundararajan Natarajan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Free Vibration ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Numerical Study ◽  
Mesh Size ◽  
Statistical Characteristics ◽  
Smoothed Finite Element Method ◽  
Element Method

In this paper, the cell-based smoothed finite element method is extended to solve stochastic partial differential equations with uncertain input parameters. The spatial field of Young’s Modulus and the corresponding stochastic results are represented by Karhunen-Loéve expansion and polynomial chaos expansion, respectively. Young’s Modulus of structure is considered to be random for stochastic static as well as free vibration problems. Mathematical expressions and the solution procedure are articulated in detail to evaluate the statistical characteristics of responses in terms of the static displacements and the free vibration frequencies. The feasibility and the effectiveness of the proposed SGCS–FEM method in terms of accuracy and lower demand on the mesh size in the solution domain over that of conventional FEM for stochastic problems are demonstrated by carefully chosen numerical examples. From the numerical study, it is inferred that the proposed framework yields accurate results.

Frequency Control of Light-Activated Shape Memory Polymer Laminated Beams: Characterization and Experiments

2016 ◽  
Author(s):  
Huiyu Li ◽  
Xufang Zhang ◽  
Hornsen Tzou
Keyword(s):  
Experimental Data ◽  
Constitutive Model ◽  
Shape Memory ◽  
Young’S Modulus ◽  
Natural Frequency ◽  
Natural Frequencies ◽  
Young's Modulus ◽  
Frequency Control ◽  
Reverse Reaction ◽  
Flexible Beam

Light-activated shape memory polymers (LaSMPs) exhibit stiffness variations when exposed to ultraviolet (UV) lights. Thus, LaSMP could manipulate structural natural frequencies with UV light exposures when laminated on structures. This study aims to experimentally demonstrate the effectiveness of LaSMP frequency control of a flexible beam. The natural frequency of a three-layered Euler-Bernoulli beam composed of LaSMP, adhesive tape and the flexible beam is analyzed and its frequency formulation exhibits the LaSMP stiffness influence. As the LaSMP adopted in this study is a new spiropyran based composition, a generic Young’s modulus model is proposed and then simplified to model the present LaSMP composition. To make sure the experiment is carried out in a homogenous light field, the light intensities of the UV surface light source at different positions are tested. The temperature change of the LaSMP sample under UV exposures is also measured. The time constant of the reverse reaction and the threshold intensity of the reverse reaction are measured. LaSMP Young’s modulus variation is tested with a uniaxial tension experiment. The constitutive model of LaSMP’s Young’s modulus is validated by experimental data. With these preparations, the LaSMP laminated flexible beam model is exposed to the UV lights and its natural frequencies are identified with a data acquisition and analysis system. The maximum natural frequency variation ratio achieves 5.67%. Comparing both theoretical and experimental data of natural frequency control, this study also validates the LaSMP Young’s modulus constitutive model.

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