Skip to main content
SpringerLink
Log in

A geometrically nonlinear spring element for structural analysis of helical springs

  • Original
  • Published:
Archive of Applied Mechanics Aims and scope Submit manuscript

Abstract

Helical springs belong to structures with spiral shapes and large curvatures, especially, traditional views that elemental shape functions are applied to interpolate shapes of structures, will result in plenty of elements and enormous computational cost. It is proved by this paper that selecting some characteristic parameters describing their shape regulations as variables can model the helical springs with almost no errors of model. Based on this thought, a geometrically nonlinear spring element for structural analysis of helical springs is proposed. First, strains irrelevant to rigid motions of cross sections and virtual deformation power of the curved beam with geometrical nonlinearity are derived. Next, parameters that generalized strains of spring elements depend on, namely helical radius, azimuth angles, height coordinates and torsion angles at one node of each coil are chosen as variables. A special shape function is built, in contrast of traditional shape functions, they can approximate the structure of helical springs accurately using less parameters. Then, nodal forces and generalized external forces as well as equilibrium equations are given, and in order to improve the computational efficiency, the Jacobian matrices are derived. Finally, two examples are considered to evaluate the accuracy of modeling and simulation against to ANSYS. Stiffness properties of cylindrical and conical springs are analyzed by the spring element. The proposed elements can give high-precision numerical results using less parameters from the comparison and be used an effective auxiliary tool for design of springs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27
Fig. 28
Fig. 29
Fig. 30
Fig. 31
Fig. 32
Fig. 33
Fig. 34
Fig. 35
Fig. 36

Similar content being viewed by others

Abbreviations

\({\varvec{g}}_{1} ,{\varvec{g}}_{2} ,{\varvec{g}}_{3}\) :

Base vectors of the global coordinate system

\({\varvec{e}}_{s} ,{\varvec{e}}_{t} ,{\varvec{e}}_{b}\) :

Base vectors of the cross sections’ coordinate system

\({{\varvec{\upkappa}}}\) :

The curvature vector

\({{\varvec{\upomega}}}\) :

The vector of angular velocity

\(\alpha ,\beta ,\gamma\) :

Cardan angles describing the rotations of cross sections

\(\kappa_{s} ,\kappa_{t} ,\kappa_{b}\) :

Components of curvature vectors with respect to the cross sections’ coordinate system

\(\omega_{s} ,\omega_{t} ,\omega_{b}\) :

Components of angular velocity with respect to the cross sections’ coordinate system

\(\varepsilon_{s}\) :

Stretch ratio of are-lengths

\(\sigma_{s}\) :

Normal stresses of cross sections

\(\tau\) :

Shear stresses of cross sections

\(\overline{s}\) :

Initial arc-length coordinates at nodes

\(s\) :

Current arc-length coordinates at nodes

\(\overline{\theta }\) :

Initial azimuth angles at nodes

\(\overline{\rho }\) :

Initial helical radius of coils at nodes

\(\overline{z}\) :

Initial height coordinates at nodes

\(\theta\) :

Current azimuth angles at nodes

\(\rho\) :

Current helical radius of coils at nodes

\(z\) :

Current height coordinates at nodes

\(\gamma\) :

Current torsion angles of cross sections at nodes

\(\user2{f}^{e}\) :

Nodal forces of spring elements

\(\user2{f}^{a}\) :

Generalized external forces of spring elements

\(\user2{G}\) :

Jacobian matrix of equilibrium equations of spring elements

\(\user2{q}\) :

Nodal parameters’ vector of spring elements

\(f_{a}\) :

The value of external forces applied to the top center of spring structures

\(d\) :

Spring wire’s diameters of spring structures

\(n\) :

The number of spring wire coils of spring structures

\(D\) :

Helical diameters of spring structures

\(EA,GJ,EI_{t} ,EI_{b}\) :

Constitutive modulus of spring structures

References

  1. Robert, H., Jan, C., Heinz, U.: An efficient spring model based on a curved beam with non-smooth contact mechanics for valve train simulations. SAE Int. J. Engines 3(1), 28–34 (2010)

    Article  Google Scholar 

  2. Pawar, H.B., Desale, D.D.: Optimization of three wheeler front suspension coil spring. Procedia Manuf. 20, 428–433 (2018)

    Article  Google Scholar 

  3. Jiang, Q., Qiao, Y., Zhao, F., et al.: Composite helical spring with skin-core structure: structural design and compression property evaluation. Polym. Compos. 42, 1292–1304 (2021)

    Article  Google Scholar 

  4. Thomson, W., Tait, P.G.: Treaties on natural philosophy, 2nd edn. Oxford, New York (1883)

    Google Scholar 

  5. Love, A.E.H.: A treatise on the mathematical theory of elasticity, 2nd edn. Cambridge at the University Press, Cambridge (1906)

    MATH  Google Scholar 

  6. Ancker, C.J., Goodier, J.N.: Pitch and curvature corrections for helical springs. J. Appl. Mech. 25, 466–470 (1958)

    Article  Google Scholar 

  7. Wahl, A.M.: Mechanical Springs. McGraw-Hill, New York (1963)

    Google Scholar 

  8. Wittrick, W.H.: On elastic wave propagation in helical springs. Int. J. Mech. Sci. 8, 25–47 (1966)

    Article  Google Scholar 

  9. Pearson, D., Wittrick, W.H.: An exact solution for the vibration of helical springs using a Bernoulli-Euler model. Int. J. Mech. Sci. 28, 83–96 (1986)

    Article  Google Scholar 

  10. Lin, Y., Pisano, A.P.: General dynamic equations of helical springs with static solution arid experimental verification. ASME J. Appl. Mech. 54, 910–917 (1987)

    Article  Google Scholar 

  11. Lin, Y., Pisano, A.P.: The differential geometry of the general helix as applied to mechanical springs. ASME J. Appl. Mech. 55, 831–836 (1988)

    Article  MathSciNet  Google Scholar 

  12. Okarmus, M., Keribar, R., Dascalescu, D., Zdrodowski, R.: An efficient, one-dimensional, finite element helical spring model for use in planar multi-body dynamics simulation. SAE Int. J. Engines 6(2), 979–989 (2013)

    Article  Google Scholar 

  13. Frendo, F., Vitale, E., Carmignani, L., et al.: Development of a lumped-parameter model for the dynamic analysis of valve train systems. Small Engine Technol. Conf. Expos. 1, 32–51 (2004)

    Google Scholar 

  14. Pisano, A.P., Freudenstein, F.: An experimental and analytical investigation of the dynamic response of a high-speed cam-follower system. Part 2: A combined, lumped/distributed parameter dynamic model. J. Mech. Des. 105, 699–704 (1983)

  15. Guo, J., Zhang, W., Zou, D.: Investigation of dynamic characteristics of a valve train system. Mech. Mach. Theory 46, 1950–1969 (2011)

    Article  Google Scholar 

  16. Rob, J., Arnold, M.: Analysis of dynamic interactions in valve train systems of IC-engines by using a simulation model. Int. Congr. Expo. 1, 930616 (1993)

  17. Dalpiaz, G., Rivola, A.: A non-linear elastodynamic model of a desmodromic valve train. Mech. Mach. Theory 35, 1551–1562 (2000)

    Article  Google Scholar 

  18. Rivolaa, A., Troncossia, M., Dalpiazb, G., Carlinic, A.: Elastodynamic analysis of the desmodromic valve train of a racing motorbike engine by means of a combined lumped/finite element model. Mech. Syst. Signal Process. 21, 735–760 (2007)

    Article  Google Scholar 

  19. Gatti, G., Mundo, D.: On the direct control of follower vibrations in cam-follower mechanisms. Mech. Mach. Theory 45, 23–35 (2010)

    Article  Google Scholar 

  20. Mottershead, J.E.: Finite elements for dynamical analysis of helical rods. Int. J. Mech. Sci. 22(5), 267–283 (1980)

    Article  Google Scholar 

  21. Iritani, T., Shozaki, A., Sheng, B., et al.: Prediction of the dynamic characteristics in valve train design of a diesel engine. C R Acad. Sci. Hebd. Seances Acad. Sci. D. 1(1), 1–7 (2002)

    Google Scholar 

  22. Ke, Q., Pan, J.Y., et al.: Geometric nonlinear dynamic analysis of curved beams using curved beam element. Acta. Mech. Sin. 27(6), 1023–1033 (2011)

    Article  MathSciNet  Google Scholar 

  23. Zeng, S., Chen, S., Wang, H., et al.: Finite element analysis of spatial curved beam in large deformation. Southeast Univ. (English Ed.) 26(4), 591–596 (2010)

    MathSciNet  Google Scholar 

  24. Stander, N., Preez, R.: Vibration analysis of coil springs by means of isoparametric curved beam finite elements. Commun. Appl. Numer. Methods. 8, 373–383 (1992)

    Article  Google Scholar 

  25. Taktak, M., Dammak, F., Abid, S., et al.: A mixed-hybrid finite element for three-dimensional isotropic helical beam analysis. Int. J. Mech. Sci. 47(2), 209–229 (2005)

    Article  Google Scholar 

  26. Fakhreddine, D., Mohamed, T., Said, A., et al.: Finite element method for the stress analysis of isotropic cylindrical helical spring. Eur. J. Mech. 24(6), 1068–1078 (2005)

    Article  Google Scholar 

  27. 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 

  28. Waki, Y., Mace, B.R., Brennan, M.J.: Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides. J. Sound Vib. 327(1–2), 92–108 (2009)

    Article  Google Scholar 

  29. Yang, C.J., Zhang, W.H., Ren, G.X., et al.: Modeling and dynamics analysis of helical spring under compression using a curved beam element with consideration on contact between its coils. Meccanica 49(4), 907–917 (2014)

    Article  MathSciNet  Google Scholar 

  30. Gu, Z., Hou, X., Ye, J.: Advanced static and dynamic analysis method for helical springs of non-linear geometries. J. Sound Vib. 513, 1114 (2021)

    Article  Google Scholar 

  31. Zhang, Y.H., Liu, H.H., Wang, D.C.: Spring manual. China Machine Press, Beijing (2008)

    Google Scholar 

Download references

Acknowledgements

This work has been supported by the National Natural Science Foundation of China(Grant No. 11802048, No. 11872137 and No. 91748203).

Author information

Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian, 116024, China

    Yingpeng Zhuo, Zhaohui Qi & Jian Zhang

  2. School of Ocean Science and Technology, Dalian University of Technology, Panjin, 124221, China

    Gang Wang

Authors
  1. Yingpeng Zhuo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhaohui Qi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhaohui Qi.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhuo, Y., Qi, Z., Zhang, J. et al. A geometrically nonlinear spring element for structural analysis of helical springs. Arch Appl Mech 92, 1789–1821 (2022). https://doi.org/10.1007/s00419-022-02147-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00419-022-02147-9

Keywords

Search

Navigation