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.

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.

A Numerical Study on the Free Vibration of Symmetric Cross-Ply Laminated Cylindrical Helical Springs

1999 ◽  
Vol 66 (4) ◽  
pp. 1040-1043 ◽  
Author(s):  
V. Yildirm
Keyword(s):  
Free Vibration ◽  
Deformation Theory ◽  
Numerical Study ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Continuous System ◽  
First Order ◽  
Helical Springs ◽  
Helix Pitch ◽  
The Mathematical Model

The free-vibration problem of (0°/90°/90°/0°) laminated composite cylindrical helical springs is modeled theoretically as a continuous system. The first-order shear deformation theory is employed in the mathematical model. The free-vibration equations are solved by the transfer matrix method. A nondimensional parametric study is performed to investigate the effects of the number of active coils, the ratio of the diameter of the cylinder to the thickness of section, the helix pitch angle, and material types on the first six natural frequencies of helical springs with square section and fixed-fixed ends.

scholarly journals A Semianalytical Approach for Free Vibration Characteristics of Functionally Graded Spherical Shell Based on First-Order Shear Deformation Theory

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Fuzhen Pang ◽  
Cong Gao ◽  
Jie Cui ◽  
Yi Ren ◽  
Haichao Li ◽  
...  
Keyword(s):  
Free Vibration ◽  
Spherical Shell ◽  
Shear Deformation ◽  
Deformation Theory ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Vibration Characteristics ◽  
Functionally Graded ◽  
Numerical Verification ◽  
First Order

This paper describes a unified solution to investigate free vibration solutions of functionally graded (FG) spherical shell with general boundary restraints. The analytical model is established based on the first-order shear deformation theory, and the material varies uniformly along the thickness of FG spherical shell which is divided into several sections along the meridian direction. The displacement functions along circumferential and axial direction are, respectively, composed by Fourier series and Jacobi polynomial regardless of boundary restraints. The boundary restraints of FG spherical shell can be easily simulated according to penalty method of spring stiffness technique, and the vibration solutions are obtained by Rayleigh–Ritz method. To verify the reliability and accuracy of the present solutions, the convergence and numerical verification have been conducted about different boundary parameters, Jacobi parameter, etc. The results obtained by the present method closely agree with those obtained from the published literatures, experiments, and finite element method (FEM). The impacts of geometric dimensions and boundary conditions on the vibration characteristics of FG spherical shell structure are also presented.

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