Advertisement

Acta Mechanica Solida Sinica

, Volume 23, Issue 1, pp 85–94 | Cite as

Formulation and evaluation of an analytical study for cylindrical helical springs

  • Aimin Yu
  • Changjin Yang
Article

Abstract

The free vibration analysis of cylindrical helical springs is carried out by means of an analytical study. In the governing equations of the motion of the springs, all displacement functions are defined at the centroid axis and also the effects of the rotational inertia, axial and shear deformations are included in the proposed model. Explicit analytical expressions which give the vibrating mode shapes are derived by rigorous application of the symbolic computing package MATHEMATICA and a process of searching is used to determine the exact natural frequencies. Numerical examples are provided for fixed-fixed boundary conditions. The free vibrational parameters are chosen as the number of coils (n = 4 ∼ 14), the helix pitch angle (α = 5 ∼ 30°) and as the ratio of the diameters of the cylinder and the wire (D/d = 4 ∼ 18) in a wide range. Validation of the proposed model has been achieved through comparison with a finite element model using two-node standard beam elements and the results available in published literature, which in these cases indicates a very good correlation.

Key words

cylindrical helical spring free vibration frequency mode shape 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Wittrick, W.H., On elastic wave propagation in helical springs. International Journal of Mechanical Sciences, 1966, 8(1): 25–47.CrossRefGoogle Scholar
  2. [2]
    Pearson, D., The transfer matrix method for the vibration of compressed helical springs. International Journal of Mechanical Engineering Sciences, 1982, 24(1): 163–171CrossRefGoogle Scholar
  3. [3]
    Nagaya, K., Takeda, S. and Nakata, Y., Free vibration of coil springs of arbitrary shape. International Journal for Numerical Methods in Engineering, 1986, 23(1): 1081–1099.CrossRefGoogle Scholar
  4. [4]
    Yildirim, V., Investigation of parameters affecting free vibration frequency of helical springs. International Journal for Numerical Methods in Engineering, 1996, 39(1): 99–114.CrossRefGoogle Scholar
  5. [5]
    Yildirim, V., Free vibration analysis of non-cylindrical coil springs by combined used of the transfer matrix and the complementary functions method. Communications in Numerical Method in Engineering, 1997, 13(1): 487–494.CrossRefGoogle Scholar
  6. [6]
    Yildirim, V., Expression for predicting fundamental natural frequencies of non-cylindrical helical springs. Journal of Sound and Vibration, 2002, 252(1): 479–491.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Becker, L.E., Chassie, G.G. and Cleghorn, W.L., On the natural frequencies of helical compression springs. International Journal of Mechanical Sciences, 2002, 44(1): 825–841.CrossRefGoogle Scholar
  8. [8]
    Lee, J. and Thompson, D.J., Dynamic stiffness formulation, free vibration and wave motion of helical springs. Journal of Sound and Vibration, 2001, 239(1): 297–320.CrossRefGoogle Scholar
  9. [9]
    Lee, J., Free vibration analysis of cylindrical helical springs by the pseudospectral method. Journal of Sound and Vibration, 2007, 302(1): 185–196.CrossRefGoogle Scholar
  10. [10]
    Mottershead, J.E., Finite elements for dynamical analysis of helical rods. International Journal of Mechanical Sciences, 1980, 22(1): 267–283.CrossRefGoogle Scholar
  11. [11]
    Stander, N. and Du Preez, R.J., Vibration analysis of coil springs by means of isoparametric curved beam finite elements. Communications in Applied Numerical Methods, 1992, 8(1): 373–383.CrossRefGoogle Scholar
  12. [12]
    Yu, A.M., Fang, M.X. and Ma, X., Theoretical research on naturally curved and twisted beams under complicated loads. Computers and Structures, 2002, 80(32): 2529–2536.CrossRefGoogle Scholar
  13. [13]
    Yu, A.M., Yang, X.G. and Nie, G.H., Generalized coordinate for warping of naturally curved and twisted beams with general cross-sectional shapes. International Journal of Solids and Structures, 2006, 43(10): 2853–2867.CrossRefGoogle Scholar
  14. [14]
    Sanchez-Hubert, J. and Sanchez-Palencia, E., Statics of curved rods on account of torsion and flexion. European Journal of Mechanics, A/Solids, 1999, 18(3): 365–390.MathSciNetCrossRefGoogle Scholar
  15. [15]
    Kapania, R.K. and Li, J., On a geometrically exact curved/twisted beam theory under rigid cross-section assumption. Computational Mechanics, 2003, 30(5-6): 428–443.CrossRefGoogle Scholar
  16. [16]
    Ignat, A., Sprekels, J. and Tiba, D., A model of a general elastic curved rod. Mathematical Methods in the Applied Sciences, 2002, 25(10): 835–854.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Tadjbakhsh, I.G. and Lagoudas, D., Variational theory of motion of curved, twisted and extensible elastic rods. International Journal of Engineering Science, 1994, 32(4): 569–577.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Tabarrok, B., Sinclair, A.N., Farshad, M. and Yi, H., On the dynamics of spatially curved and twisted rods — A finite element formulation. Journal of Sound and Vibration, 1988, 123(2): 315–326.CrossRefGoogle Scholar
  19. [19]
    Karami, G., Farshad, M. and Yazdchi, M., Free vibrations of spatial rods — A finite-element analysis. Communications in Applied Numerical Methods, 1990, 6(6): 417–428.CrossRefGoogle Scholar
  20. [20]
    Benedetti, A., Deseri, L. and Tralli, A., Simple and effective equilibrium models for vibration analysis of curved rods. Journal of Engineering Mechanics — ASCE, 1996, 122(4): 291–299.CrossRefGoogle Scholar
  21. [21]
    Farshad, M., Wave propagation in prestressed curved rods. ASCE. Journal of Engineering Mechanics Division, 1980, 106(2): 395–408.Google Scholar
  22. [22]
    Shih, Y.D., Chen, J.K., Chang, Y.C. and Sheng, J.P., Natural vibration of arbitrary spatially curved rectangular rods with pre-twisted angles. Journal of Sound and Vibration, 2005, 285(4-5): 925–939.CrossRefGoogle Scholar
  23. [23]
    Tsay, H.S. and Kingsbury, H.B., Vibrations of rods with general space curvature. Journal of Sound and Vibration, 1988, 124(3): 539–554.CrossRefGoogle Scholar
  24. [24]
    Yildirim, V. and Ince, N., Natural frequencies of helical springs of arbitrary shape. Journal of Sound and Vibration, 1997, 204(2): 311–329.CrossRefGoogle Scholar
  25. [25]
    Yu, A.M. and Qu, Z.H., The study of variational principle in naturally curved and twisted slender beams. Journal of Shanghai Technical College of Metallurgy, 1994, 15(3): 1–8 (in Chinese).Google Scholar
  26. [26]
    Yu, A.M. and Yang, C.J., Generalized variational principle of dynamic analysis on naturally curved and twisted box beams for anisotropic materials. Meccanica, 2008, 43(6): 611–622.MathSciNetCrossRefGoogle Scholar
  27. [27]
    Loïc Brancheriau, Influence of cross section dimensions on Timoshenko’s shear factor — Application to wooden beams in free-free flexural vibration. Annals of Forest Science, 2006, 63(3): 319–321.CrossRefGoogle Scholar
  28. [28]
    Timoshenko, S.P., On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Philos. Mag. J. Sci., 1921, 41: 744–746.CrossRefGoogle Scholar
  29. [29]
    Yildirim, V., An efficient numerical method for predicting the natural frequencies of cylindrical helical springs. International Journal of Mechanical Sciences, 1999, 41(8): 919–939.CrossRefGoogle Scholar
  30. [30]
    Della Pietra, L. and Della Valle, S., On the dynamic behavior of axially excited helical springs. Meccanica, 1982, 17(1): 31–43.CrossRefGoogle Scholar
  31. [31]
    Wahl, A.M., Mechanical Springs. New York: McGraw-Hill Book Company, 1963.Google Scholar
  32. [32]
    Haktanir, V. and Kiral, E., Determination of free vibration frequencies of helical springs by the Myklestad method. In: Proc. of 4th National Symp. On Machinery Theory, Istanbul, September, 1990, 479-488 (in Turkish).Google Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics and Technology 2010

Authors and Affiliations

  1. 1.School of Aerospace Engineering and Applied MechanicsTongji UniversityShanghaiChina
  2. 2.School of Urban Railway TransportationSoochow UniversitySuzhouChina

Personalised recommendations