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Durability of Springs pp 45–73Cite as

“Equivalent Columns” for Helical Springs

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Abstract

In this chapter the helical spring is substituted by a flexible rod that is located along the axis of helix. This rod possesses the same mechanical features, as the spring itself. Its bending, torsion and compression stiffness are equal to the corresponding stiffness of the helical spring. This rod is known as an “equivalent column” of the helical spring. The “equivalent column” equations are considerable easier to handle than the original equations of the helical spring. The integral spring properties, as an axial and transversal stiffness, buckling loads, fundamental frequencies could be directly determined using the “equivalent column” equations. In contrast, the local properties, like stresses in the wire or contact forces, could be evaluated only with the more complicated equations of the helical elastic rod.

In this chapter the stability and transversal vibrations of the spring are studied from the unified point of view, which is based on the “equivalent column” concept. Buckling refers to the loss of stability up to the sudden and violent failure of straight bars or beams under the action of pressure forces, whose line of action is the column axis. This concept is applied for the stability of helical springs.

An alternative approach method is based on the dynamic criterion for the spring stability. The equations for transverse (lateral) vibrations of the compressed coil springs were derived. This solution expresses the fundamental natural frequency of the transverse vibrations of the column as the function of the axial force, as well as the variable length of the spring.

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References

  • Andreeva, L.E.: Elastic Elements of Instruments (Russ.), 456 p. Mashgiz, Moscow (1962). [Transl.: Baruch, A., Alster, D.: Israel Program for Scientific Translation, Ltd., Jerusalem (1966)]

    Google Scholar 

  • Ayadi, S., Hadj-Taïeb, E.: Finite element solution of dynamic response of helical springs. Int. J. Simul. Model. 7(1), 17–28 (2008). doi:10.2507/IJSIMM07(1)2.094

    Article  Google Scholar 

  • Becker, L.E., Chassie, G.G., Cleghorn, W.L.: On the natural frequencies of helical compression springs. Int. J. Mech. Sci. 44, 825–841 (2002)

    Article  MATH  Google Scholar 

  • Biezeno, C.B., Koch, J.J.: Knickung von Schraubenfedern. Z. Angew. Math. Mech. 5, 279–280 (1925)

    Article  MATH  Google Scholar 

  • Bolotin, V.V.: The Dynamic Stability of Elastic Systems. Holden Day, San Francisco (1964)

    MATH  Google Scholar 

  • Chan, K.T., Wang, X.Q., So, R.M.C., Reid, S.R.: Superposed standing waves in a Timoshenko beam. Proc. R. Soc. A. 458, 83–108 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  • Chassie, G.G., Becker, L.E., Cleghorn, W.L.: On the buckling of helical springs under combined compression and torsion. Int. J. Mech. Sci. 39(6), 697–704. doi:10.1016/S0020-7403(96)00070-7 (1997)

  • Collins, J.A., Busby, H.R., Staab, G.H.: Mechanical Design of Machine Elements and Machines: A Failure Prevention Perspective. Wiley (2010)

    Google Scholar 

  • Costello, G.A.: Radial expansion of impacted helical springs. J. Appl. Mech. Trans. ASME. 42, 789–792 (1975)

    Article  Google Scholar 

  • Dick J.: On transverse vibrations of a helical spring with pinned ends and no axial load. Philos. Mag. Ser. 7. 33, 222, 513–519 (1942)

    Google Scholar 

  • DIN EN 13906-1:2013-11 Cylindrical Helical Springs Made from Round Wire and Bar—Calculation and Design—Part 1: Compression Springs. German version EN 13906-1:2013 (2013)

    Google Scholar 

  • Encyclopedia of Spring Design: Spring Manufacturers Institute, 2001 Midwest Road, Suite 106, Oak Brook, IL 60523-1335 USA (2013)

    Google Scholar 

  • Frikha, A., Treyssédee, F., Cartraud, P.: Effect of axial load on the propagation of elastic waves in helical beams. Wave Motion. 48(1), 83–92 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Godoy L.: Theory of Elastic Stability: Analysis and Sensitivity, 450 p. CRC Press (1999)

    Google Scholar 

  • Gramme, I.R.: Die Knickung von Schraubenfedern. Z. Angew. Math. Mech. 4, 384–389 (1924)

    Article  MATH  Google Scholar 

  • Guido, A.R., Della Pietra, L., della Valle, S.: Transverse vibrations of cylindrical helical springs. Meccanica. 13(2), 90–108 (1978)

    Article  MATH  Google Scholar 

  • Haktanir, V.: The complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes. Int. J. Numer. Methods Eng. 38(6), 1031–1056 (1995). doi:10.1002/nme.1620380611

    Article  MATH  Google Scholar 

  • Hamza, A., Ayadi, S., Hadj-Taieb, E.: Propagation of strain waves in cylindrical helical springs. J. Vib. Control. (2013a). doi:10.1177/1077546313503878

    Google Scholar 

  • Hamza, A., Ayadi, S., Hadj-Taieb, E.: Resonance phenomenon of strain waves in helical compression springs. Mech. Ind. 14, 253–265 (2013b). doi:10.1051/meca/2013069

    Article  Google Scholar 

  • Haringx, J.A.: On highly compressible helical springs and rubber rods, and their application for vibration-free mountings. Philips Res. Rep. 3, 401–449 (1948)

    Google Scholar 

  • Helical Springs: Engineering Design Guides. The United Kingdom Atomic Energy Authority and Oxford University Press (1974). ISBN 0-19-859142X

    Google Scholar 

  • Kessler, D.A., Rabin, Y.: Stretching instability of helical springs. Phys. Rev. Lett. 90, 024301 (2003)

    Article  Google Scholar 

  • Kobelev, V.: Effect of static axial compression on the natural frequencies of helical springs. Multidiscip. Model. Mater. Struct. 10(3), 379–398 (2014)

    Article  Google Scholar 

  • Kobelev, V.: Isoperimetric inequality in the periodic Greenhill problem of twisted elastic rod. Struct. Multidiscip. Optim. 54(1), 133–136 (2016)

    Article  MathSciNet  Google Scholar 

  • Kobelev, V.: Some exact analytical solutions in structural optimization. Mech. Des. Struct. Mach. 45(1). doi:10.1080/15397734.2016.1143374 (2017)

  • Kruzelecki, J., Zyczkowski, M.: On the concept of an equivalent column in the stability problem of compressed helical springs. Ing.-Archiv. 60, 367–377 (1990)

    Article  Google Scholar 

  • Leamy, M.J.: Intrinsic finite element modeling of nonlinear dynamic response in helical springs. In: ASME 2010 International Mechanical Engineering Congress and Exposition Volume 8: Dynamic Systems and Control, Parts A and B, Vancouver, BC, Canada, November 12–18, Paper No. IMECE2010-37434, pp. 857–867; 11. doi:10.1115/IMECE2010-37434 (2010)

  • Lee, J.: Free vibration analysis of cylindrical helical springs by the pseudospectral method. J. Sound Vib. 302, 185–196 (2007)

    Article  Google Scholar 

  • Lee, J., Thompson, D.J.: Dynamic stiffness formulation, free vibration and wave motion of helical springs. J. Sound Vib. 239, 297–320 (2001)

    Article  Google Scholar 

  • Lee, C.-Y., Zhuo, H.-C., Hsu, C.-W.: Lateral vibration of a composite stepped beam consisted of SMA helical spring based on equivalent Euler–Bernoulli beam theory. J. Sound Vib. 324, 179–193 (2009)

    Article  Google Scholar 

  • Leung, A.Y.T.: Vibration of thin pre-twisted helical beams. Int. J. Solids Struct. 47, 177–1195 (2010)

    Article  MATH  Google Scholar 

  • Majkut, L.: Free and forced vibrations of timoshenko beams described by single difference equation. J. Theor. Appl. Mech. 47(1), 193–210 (2009)

    Google Scholar 

  • Michalczyk, K.: Analysis of lateral vibrations of the axially loaded helical spring. J. Theor. Appl. Mech. 53(3), 745–775. doi:10.15632/Jtam-Pl.53.3.745 (2015)

  • Ponomarev, S.D.: Stability of helical springs under compression and torsion (in Russian). In: Chudakov, E. A. (ed.) Mashinostr, Vol. 2. Moscow, pp 683–685 (1948)

    Google Scholar 

  • Ponomarev, S.D., Andreeva, L.E.: Calculation of Elastic Elements of Machines and Instruments. Moscow (1980)

    Google Scholar 

  • Renno, J.M., Mace, B.R.: Vibration modelling of helical springs with non-uniform ends. J. Sound Vib. 331(12), 2809–2823 (2012)

    Article  Google Scholar 

  • Satoh, T., Kunoh, T., Mizuno, M.: Buckling of coiled springs by combined torsion and axial compression. JSME Int. J. Ser. 1(31), 56–62 (1988)

    Google Scholar 

  • Skoczeń, B., Skrzypek, J.: Application of the equivalent column concept to the stability of axially compressed bellows. Int. J. Mech. Sci. 34(11), 901–916. doi:10.1016/0020-7403(92)90020-H (1992)

  • Sorokin, S.V.: Linear dynamics of elastic helical springs: asymptotic analysis of wave propagation. Proc. R. Soc. A. 465, 1513–1537 (2009). doi:10.1098/rspa.2008.0468

    Article  MathSciNet  MATH  Google Scholar 

  • Stephen, N.G., Puchegger, S.: On the valid frequency range of Timoshenko beam theory. J. Sound Vib. 297, 1082–1087 (2006)

    Article  Google Scholar 

  • Tabarrok, B., Xiong, Y.: A spatially curved and twisted rod element for buckling analysis. Int. J. Solids Struct. 29(23), 3011–3023. doi:10.1016/0020-7683(92)90155-M. ISSN:0020-7683 (1992)

  • Taktak, M., Dammak, F., Abid, S., Haddar, M.: A finite element for dynamic analysis of a cylindrical isotropic helical spring. J. Mech. Mater. Struct. 3(4), (2008)

    Google Scholar 

  • Wittrick, W.H.: On elastic wave propagation in helical springs. Int. J. Mech. Sci. 8, 25–47. doi:10.1016/0020-7403(66)90061-0 (1966)

  • Yildirim, V.: Free vibration analysis of non-cylindrical coil springs by combined used of the transfer matrix and the complementary functions method. Commun. Numer. Methods Eng. 13, 487–494 (1997)

    Article  MATH  Google Scholar 

  • Yildirim, V.: Expression for predicting fundamental natural frequencies of non-cylindrical helical springs. J. Sound Vib. 252, 479–491 (2002)

    Article  Google Scholar 

  • Yildirim, V.: On the linearized disturbance dynamic equations for buckling and free vibration of cylindrical helical coil springs under combined compression and torsion. Meccanica. 47(4), 1015–1033 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Yu, A.M., Yang, C.J., Nie, G.H.: Analytical formulation and evaluation for free vibration of naturally curved and twisted beams. J. Sound Vib. 329, 1376–1389 (2010)

    Article  Google Scholar 

  • Yun, A.M., Hao, Y.: Free vibration analysis of cylindrical helical springs with noncircular cross-sections. J. Sound Vib. 330, 2628–2639 (2011)

    Article  Google Scholar 

  • Ziegler, H.: Arguments for and against Engesser’s formulas. Ing. Arch. 52, 105–113 (1982)

    Article  MATH  Google Scholar 

  • Ziegler, H., Huber, A.: Zur Knickung der gedrückten und tordierten Schraubenfeder. Z. Angew. Math. Phys. 1, 183–195 (1950)

    MATH  Google Scholar 

Download references

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  1. Faculty of Engineering, University of Siegen, Siegen, Germany

    Vladimir Kobelev

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  1. Vladimir Kobelev
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Kobelev, V. (2018). “Equivalent Columns” for Helical Springs. In: Durability of Springs. Springer, Cham. https://doi.org/10.1007/978-3-319-58478-2_3

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  • DOI: https://doi.org/10.1007/978-3-319-58478-2_3

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