Axial Static Load Dependence Free Vibration Analysis of Helical Springs Based on the Theory of Spatially Curved Bars

Abstract This work addresses an accurate and detailed axial static load dependence linearly elastic free vibration analysis of cylindrical helical springs based on the theory of spatially curved bars and the transfer matrix method. For a continuous system, governing equations comprise coupled vibration modes namely transverse vibrations in two orthogonal planes, torsional and axial vibrations. The axial and shear deformation effects together with the rotatory inertia effects are all considered based on the first order shear deformation theory and their effects on the frequencies are investigated. The effects of the initial stress resultants on the frequencies are also studied. After buckling, forward-shifting phenomenon of higher frequencies is noticeably demonstrated. It is also revealed that a free/forced vibration analysis with an axial static load should not be performed individually without checking buckling loads

Vibration behavior of composite beams with rectangular sections considering the different shear correction factors

7th International Conference on Vibration Problems (ICOVP 2005) -- SEP 05-09, 2005 -- Isik Univ, Sile Campus, Istanbul, TURKEYWOS: 000246655700075As is well known, there are the first and higher order shear deformation theories that involve the shear correction factor (k- factor), which appears as a coefficient in the expression for the transverse shear stress resultant, to consider the shear deformation effects with a good approximation as a result of non-uniform distribution of the shear stresses over the cross-section of the beam. Timoshenko's beam theory (TBT) accounts both the shear and rotatory inertia effects based upon the first order shear deformation theory which offers the simple and acceptable solutions. The numerical value of the k- factor which was originally proposed by Timoshenko depends upon generally both the Poisson's ratio of the material and the shape of the cross-section. Recently, especially the numerical value of the k-factor for rectangular sections is examined by both theoretical and experimental manners. Although there are no large numerical differences among the most of the theories, a few of them says that the k-factor varies obviously with the aspect ratio of rectangular sections while Timoshenko's k-factor is applicable for small aspect ratios. In this study, the effect of the different k-factors developed by Timoshenko, Cowper and Hutchinson on the in-plane free vibration of the orthotropic beams with different boundary conditions and different aspect ratios are studied numerically based on the transfer matrix method. For the first six frequencies, the relative differences of among the theories are presented by charts

A Closed-Form Buckling Formula for Open-Coiled and Properly Supported Circular-Bar Helical Springs

As a continuation of the author’s previous studies on the buckling analysis of helical springs, a closed-form formula having been obtained with the help of the artificial neural network (ANN) is proposed and discussed in detail for the first time for a cylindrical close/open-coiled helical spring with fixed ends and having a solid circular section. As far as the author knows there is no such a formula in the open-literature to consider the effects of all stress resultants (torsional and bending moments, axial and shearing forces), large helix pitch angles together with the axial and shear deformations on the buckled state. The present formula may be used in a wide range of the total number of active turns, the ratio of the free axial length to the mean helix diameter, and the spring index. It is yet again revealed that it is not appropriate to use the elementary theory to determine the critical buckling loads for open-coiled springs. The present formula may allow the deeper understanding of spring buckling mechanism and to be used directly and safely in the design processes of such closely/open-coiled springs

Free vibration behavior of unidirectional composite cylindrical helical springs with circular section

ASMEProceedings of the 1998 ASME International Mechanical Engineering Congress and Exposition --15 November 1998 through 20 November 1998 -- Anaheim, CA, USA --The free vibration problem of unidirectional composite cylindrical helical springs is modeled theoretically as a continuous system considering the rotary inertia, shear and axial deformation effects. The first order shear deformation theory is employed in the mathematical model. The twelve scalar ordinary differential equations governing the free vibration behavior of cylindrical helical springs made of an anisotropic material are solved simultaneously by the transfer matrix method. The overall transfer matrix of the helix is computed up to any desired accuracy by using the effective numerical algorithm available in the literature. The theoretical results are verified with the reported values, which were obtained theoretically and experimentally for straight beams. A parametric study is performed to investigate the effects of the number of active coils, the helix pitch angle and material types on the fundamental natural frequencies of helical springs with circular section and fixed-fixed ends

Free vibration/buckling analyses of noncylindrical initially compressed helical composite springs

WOS: 000386159000004This article presents the use of the stiffness matrix method based on the first-order shear deformation theory to predict the fundamental natural frequencies and buckling loads of noncylindrical unidirectional composite helical springs subjected to initial static axial force and moment. This theoretical study about such springs with circular/rectangular cross-sections and large pitch angles is performed for the first time in the literature. The validity of the present results is verified by the benchmark studies related with initially compressed isotropic cylindrical springs

Free vibration of symmetric cross-ply laminated cylindrical helical springs

ASMEProceedings of the 1998 ASME International Mechanical Engineering Congress and Exposition --15 November 1998 through 20 November 1998 -- Anaheim, CA, USA --The first three free vibration frequencies of symmetric cross-ply laminated cylindrical helical springs with square section and 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. A parametric study is performed, to analyze the effects of the ratio of the extensional modulus to the shearing modulus, helix pitch angle and the number of active turns on the natural frequencies of such springs. The results are given in graphical forms

The effect of the longitudinal to transverse moduli ratio on the natural frequencies of symmetric cross-ply laminated cylindrical helical springs

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

Application of Artificial Neural Networks in the Prediction of Critical Buckling Loads of Helical Compression Springs

WOS: 000281888500008This paper proposes the use of artificial neural networks (ANN) to perfectly predict the critical buckling loads of cylindrical isotropic helical spring with fixed ends and with circular sections, and with large pitch angles. The buckling equations of cylindrical isotropic helical springs loaded axially consist of a set of twelve linear differential equations. As finding a solution in an analytical manner is too difficult, numerical solution in an exact manner based on the transfer-matrix method to collect consistent dimensionless numerical data for the training process is used. In this way almost perfect weight values are obtained to predict the non-dimensional buckling loads. A good agreement is observed with the data available in the literature. (C)2010 Journal of Mechanical Engineering. All rights reserved.Technical Research Council of Turkey (TUBITAK)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [106M307]The material covered in this paper is based upon work which was supported by the research grant 106M307 from Technical Research Council of Turkey (TUBITAK)