Stress distribution and fatigue life of nonlinear vibration of an axially moving beam

  • Hu DingEmail author
  • Ling-Lu Huang
  • Earl Dowell
  • Li-Qun Chen


In this paper, the effects of the nonlinear vibration on stress distribution and fatigue life of the axially moving beam are studied. The parametric excitation of the flexible material is created by the pulsating moving speed. Three-to-one internal resonance condition is satisfied. The three-parameter model is adopted in the viscoelastic constitutive relation. The nonlinear vibration of the axially moving beam with parametric and internal resonance are studied by using the direct multiple scales method (MSM) with numerical simulation confirmation. Based on the approximate analytical solution, the distribution of tensile stress and bending stress on the axially moving beam is presented by adopting a V-belt as the prototype. Based on the maximum stable cyclic stress, the limit cycle response of the V-belt is utilized to evaluate the effect of the resonance on the fatigue life. Also, the influences of the internal resonance on the steady-state responses and the fatigue life of the V-belt are revealed. Numerical examples illustrate that large unwanted resonances occur and the second-order mode receives vibration energy from to the first-order mode. The numerical results demonstrate that the nonlinear vibration significantly reduces the fatigue life of the V-belt. The fatigue life analysis method in this paper can be applied to the excited vibration of other axially moving systems and even static continuum.

axially moving beam nonlinear vibration parametric excitation internal resonance stress distribution V-belt 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wickert J A, Mote Jr. C D. Travelling load response of an axially moving string. J Sound Vib, 1991, 149: 267–284CrossRefGoogle Scholar
  2. 2.
    Chen L Q. Analysis and control of transverse vibrations of axially moving strings. Appl Mech Rev, 2005, 58: 91–116CrossRefGoogle Scholar
  3. 3.
    Marynowski K, Kapitaniak T. Dynamics of axially moving continua. Int J Mech Sci, 2014, 81: 26–41CrossRefGoogle Scholar
  4. 4.
    Zhang L, Zu J W. Non-linear vibrations of viscoelastic moving belts, part I: Free vibration analysis. J Sound Vib, 1998, 216: 75–91CrossRefGoogle Scholar
  5. 5.
    Hu Y D, Hu P, Zhang J Z. Strongly nonlinear subharmonic resonance and chaotic motion of axially moving thin plate in magnetic field. J Comput Nonlinear Dynam, 2015, 10: 021010CrossRefGoogle Scholar
  6. 6.
    Chen L Q, Ding H. Steady-state responses of axially accelerating viscoelastic beams: Approximate analysis and numerical confirmation. Sci China Ser G-Phys Mech Astron, 2008, 51: 1707–1721CrossRefGoogle Scholar
  7. 7.
    Kesimli A, Özkaya E, Bağdatli S M. Nonlinear vibrations of spring-supported axially moving string. Nonlinear Dyn, 2015, 81: 1523–1534MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Ren Y, Chen X, Cai Y, et al. Attitude-rate measurement and control integration using magnetically suspended control and sensitive gyroscopes. IEEE Trans Ind Electron, 2018, 65: 4921–4932CrossRefGoogle Scholar
  9. 9.
    Lim C W, Li C, Yu J L. Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach. Acta Mech Sin, 2010, 26: 755–765MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Marynowski K. Dynamic analysis of an axially moving sandwich beam with viscoelastic core. Composite Struct, 2012, 94: 2931–2936CrossRefGoogle Scholar
  11. 11.
    Duan Y C, Wang J P, Wang J Q, et al. Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam. J Sound Vib, 2014, 333: 2885–2897CrossRefGoogle Scholar
  12. 12.
    Ding H, Zu J W. Steady-state responses of pulley-belt systems with a one-way clutch and belt bending stiffness. J Vib Acoust, 2014, 136: 041006CrossRefGoogle Scholar
  13. 13.
    Cao D X, Zhang W. Global bifurcations and chaotic dynamics for a string-beam coupled system. Chaos Solitons Fractals, 2008, 37: 858–875MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fung R F, Wu J W, Wu S L. Stabilization of an axially moving string by nonlinear boundary feedback. J Dyn Sys Meas Control, 1999, 121: 117–121CrossRefzbMATHGoogle Scholar
  15. 15.
    Lee U, Kim J, Oh H. Spectral analysis for the transverse vibration of an axially moving Timoshenko beam. J Sound Vib, 2004, 271: 685–703CrossRefzbMATHGoogle Scholar
  16. 16.
    Yang X D, Zhang W, Melnik R V N. Energetics and invariants of axially deploying beam with uniform velocity. AIAA J, 2016, 54: 2183–2189CrossRefGoogle Scholar
  17. 17.
    Yao G, Zhang Y. Reliability and sensitivity analysis of an axially moving beam. Meccanica, 2016, 51: 491–499MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ding H, Zhang G C, Chen L Q, et al. Forced vibrations of super-critically transporting viscoelastic beams. J Vib Acoust, 2012, 134: 051007CrossRefGoogle Scholar
  19. 19.
    Chen L Q, Lim C W, Hu Q Q, et al. Asymptotic analysis ofa vibrating cantilever with a nonlinear boundary. Sci China Ser G-Phys Mech Astron, 2009, 52: 1414–1422CrossRefGoogle Scholar
  20. 20.
    Yang X D, Zhang W. Nonlinear dynamics of axially moving beam with coupled longitudinal-transversal vibrations. Nonlinear Dyn, 2014, 78: 2547–2556CrossRefGoogle Scholar
  21. 21.
    An C, Su J. Dynamic response ofaxially moving Timoshenko beams: Integral transform solution. Appl Math Mech-Engl Ed, 2014, 35: 1421–1436CrossRefzbMATHGoogle Scholar
  22. 22.
    Tan C A, Yang B, Mote C D. Dynamic response of an axially moving beam coupled to hydrodynamic bearings. J Vib Acoust, 1993, 115: 9–15CrossRefGoogle Scholar
  23. 23.
    Ghayesh M H, Farokhi H. Nonlinear dynamical behavior of axially accelerating beams: Three-dimensional analysis. J Comput Nonlinear Dynam, 2016, 11: 011010CrossRefGoogle Scholar
  24. 24.
    Öz H R, Pakdemirli M, Özkaya E. Transition behaviour from string to beam for an axially accelerating material. J Sound Vib, 1998, 215: 571–576CrossRefGoogle Scholar
  25. 25.
    Ravindra B, Zhu W D. Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime. Archive Appl Mech (Ingenieur Archiv), 1998, 68: 195–205CrossRefzbMATHGoogle Scholar
  26. 26.
    Yan Q Y, Ding H, Chen L Q. Periodic responses and chaotic behaviors of an axially accelerating viscoelastic Timoshenko beam. Nonlinear Dyn, 2014, 78: 1577–1591MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang L H, Hu Z D, Zhong Z, et al. Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length. Acta Mech, 2010, 214: 225–244CrossRefzbMATHGoogle Scholar
  28. 28.
    Chakraborty G, Mallik A K. Parametrically excited non-linear moving beams with and without external forcing. Nonlinear Dyn, 1998, 17: 301–324CrossRefzbMATHGoogle Scholar
  29. 29.
    Ding H, Zu J W. Periodic and chaotic responses of an axially accelerating viscoelastic beam under two-frequency excitations. Int J Appl Mech, 2013, 05: 1350019CrossRefGoogle Scholar
  30. 30.
    Özhan B B. Vibration and stability analysis of axially moving beams with variable speed and axial force. Int J Str Stab Dyn, 2014, 14: 1450015MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Öz H R, Pakdemirli M, Boyaci H. Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. Int J Non-Linear Mech, 2001, 36: 107–115CrossRefzbMATHGoogle Scholar
  32. 32.
    Lv H, Li Y, Li L, et al. Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity. Appl Math Model, 2014, 38: 2558–2585MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Chen L H, Zhang W, Yang F H. Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations. J Sound Vib, 2010, 329: 5321–5345CrossRefGoogle Scholar
  34. 34.
    Ding H. Steady-state responses ofa belt-drive dynamical system under dual excitations. Acta Mech Sin, 2016, 32: 156–169MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Riedel C H, Tan C A. Coupled, forced response of an axially moving strip with internal resonance. Int J Non-Linear Mech, 2002, 37: 101–116CrossRefzbMATHGoogle Scholar
  36. 36.
    Lu L F, Wang Y F, Liu Y X. Internal resonance of coupled non-planar vibrations of an axially moving cable with small sag. Int J Nonlin Sci Num, 2009, 10: 341–352CrossRefGoogle Scholar
  37. 37.
    Yao M, Zhang W, Zu J W. Multi-pulse Chaotic dynamics in non-planar motion of parametrically excited viscoelastic moving belt. J Sound Vib, 2012, 331: 2624–2653CrossRefGoogle Scholar
  38. 38.
    Tang Y Q, Chen L Q. Primary resonance in forced vibrations of inplane translating viscoelastic plates with 3:1 internal resonance. Nonlinear Dyn, 2012, 69: 159–172CrossRefzbMATHGoogle Scholar
  39. 39.
    Wang Y Q, Liang L, Guo X H. Internal resonance of axially moving laminated circular cylindrical shells. J Sound Vib, 2013, 332: 6434–6450CrossRefGoogle Scholar
  40. 40.
    Li H Y, Li J, Liu Y J. Internal resonance of an axially moving unidirectional plate partially immersed in fluid under foundation displacement excitation. J Sound Vib, 2015, 358: 124–141CrossRefGoogle Scholar
  41. 41.
    Gültekin Sinir B. Infinite mode analysis of a general model with external harmonic excitation. Appl Math Model, 2015, 39: 1823–1836MathSciNetCrossRefGoogle Scholar
  42. 42.
    Chen L H, Zhang W, Liu Y Q. Modeling of nonlinear oscillations for viscoelastic moving belt using generalized Hamilton’s principle. J Vib Acoust, 2007, 129: 128–132CrossRefGoogle Scholar
  43. 43.
    Burak Özhan B, Pakdemirli M. A general solution procedure for the forced vibrations of a continuous system with cubic nonlinearities: Primary resonance case. J Sound Vib, 2009, 325: 894–906CrossRefzbMATHGoogle Scholar
  44. 44.
    Yu W, Chen F. Multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam. Arch Appl Mech, 2013, 83: 647–660CrossRefzbMATHGoogle Scholar
  45. 45.
    Chen S H, Huang J L, Sze K Y. Multidimensional Lindstedt-Poincaré method for nonlinear vibration of axially moving beams. J Sound Vib, 2007, 306: 1–11CrossRefGoogle Scholar
  46. 46.
    Sze K Y, Chen S H, Huang J L. The incremental harmonic balance method for nonlinear vibration of axially moving beams. J Sound Vib, 2005, 281: 611–626CrossRefGoogle Scholar
  47. 47.
    Huang J L, Su R K L, Li W H, et al. Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J Sound Vib, 2011, 330: 471–485CrossRefGoogle Scholar
  48. 48.
    Ghayesh M H, Amabili M. Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance. Nonlinear Dyn, 2013, 73: 39–52MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Ghayesh M H, Amabili M. Nonlinear stability and bifurcations of an axially moving beam in thermal environment. J Vib Control, 2015, 21: 2981–2994MathSciNetCrossRefGoogle Scholar
  50. 50.
    Ding H, Huang L, Mao X, et al. Primary resonance of traveling viscoelastic beam under internal resonance. Appl Math Mech-Engl Ed, 2017, 38: 1–14MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Sahoo B, Panda L N, Pohit G. Two-frequency parametric excitation and internal resonance of a moving viscoelastic beam. Nonlinear Dyn, 2015, 82: 1721–1742MathSciNetCrossRefGoogle Scholar
  52. 52.
    Sahoo B, Panda L N, Pohit G. Combination, principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation. Int J Non-Linear Mech, 2016, 78: 35–44CrossRefGoogle Scholar
  53. 53.
    Tang Y Q, Zhang D B, Gao J M. Parametric and internal resonance of axially accelerating viscoelastic beams with the recognition of longitudinally varying tensions. Nonlinear Dyn, 2016, 83: 401–418MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Chen L Q, Wu J, Zu J W. The chaotic response of the viscoelastic traveling string: An integral constitutive law. Chaos Solitons Fractals, 2004, 21: 349–357CrossRefzbMATHGoogle Scholar
  55. 55.
    Chen L Q, Chen H, Ding H, et al. Nonlinear combination parametric resonance ofaxially accelerating viscoelastic strings constituted by the standard linear solid model. Sci China Technol Sci, 2010, 53: 645–655CrossRefzbMATHGoogle Scholar
  56. 56.
    Wang B. Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model. Appl Math Mech-Engl Ed, 2012, 33: 817–828MathSciNetCrossRefGoogle Scholar
  57. 57.
    Saksa T, Jeronen J. Dynamic analysis for axially moving viscoelastic poynting-thomson beams. In: Mathematical Modeling and Optimization of Complex Structures. Computational Methods in Applied Sciences. Cham: Springer, 2016. 40: 131–151CrossRefGoogle Scholar
  58. 58.
    Moobola R, Hills D A, Nowell D. Stress intensity factors and fatigue life of beams in reversed bending. J Strain Anal Eng Des, 1997, 32: 401–409CrossRefGoogle Scholar
  59. 59.
    Mars W, Fatemi A. A literature survey on fatigue analysis approaches for rubber. Int J Fatigue, 2002, 24: 949–961CrossRefzbMATHGoogle Scholar
  60. 60.
    Quadrini E, Bergmann J P, et al. Influence of residual stresses on the fatigue behaviour of welded beams. Damage Fracture Mech, 1998, 19: 639–648Google Scholar
  61. 61.
    Kim B, Kim Y, Chun D M, et al. Durability improvement of automotive v-belt pulley. IntJ Automot Technol, 2009, 10: 73–77CrossRefGoogle Scholar
  62. 62.
    Luo R K, Mortel W J, Wu X P. Fatigue failure investigation on anti-vibration springs. Eng Failure Anal, 2009, 16: 1366–1378CrossRefGoogle Scholar
  63. 63.
    Jones K W, Dunn M L. Fatigue crack growth through a residual stress field introduced by plastic beam bending. Fatigue Fract Eng Mater Struct, 2008, 31: 863–875CrossRefGoogle Scholar
  64. 64.
    Ngolemasango F E, Nkeng G E, O’Connor C, et al. Effect of nature and type of flaw on the properties of a natural rubber compound. Polymer Testing, 2009, 28: 463–469CrossRefGoogle Scholar
  65. 65.
    Spotts M F. Design of Machine Elements. 6th ed. Englewood Cliffs: Prentice-Hall Inc, 1985Google Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Hu Ding
    • 1
    • 2
    Email author
  • Ling-Lu Huang
    • 1
  • Earl Dowell
    • 3
  • Li-Qun Chen
    • 1
    • 2
    • 4
  1. 1.Shanghai Institute of Applied Mathematics and MechanicsShanghai UniversityShanghaiChina
  2. 2.Shanghai Key Laboratory of Mechanics in Energy EngineeringShanghai UniversityShanghaiChina
  3. 3.Department of Mechanical Engineering and Materials ScienceDuke UniversityDurhamUSA
  4. 4.Department of MechanicsShanghai UniversityShanghaiChina

Personalised recommendations