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Continuation analysis of cam–follower mechanisms considering time-varying stiffness and loss of contact

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Abstract

The dynamic behaviour of cam–follower systems for automotive applications is investigated. This mechanism is modelled by means of a SDOF scheme, capable of capturing the fundamentals of the problem, considering external excitation, loss of contact and parametric excitation due to contact stiffness variation. The main novelty pertains to the qualitative characterization of the dynamic behaviour of cam–follower systems by means of continuation analysis considering both loss of contact and parametric excitation. Moreover, to the best knowledge of the authors, for the first time emphasis has been devoted to the evaluation and identification of the different internal and external forcing terms and their effect on the dynamic response and stability of the system. A parametric study is reported in the form of response diagrams to show how the variation of key parameters, such as stiffness and dam**, influences the system behaviour. Numerical results obtained by direct integration of the equation of motion are presented, demonstrating the existence of rich dynamics in the examined system. Finally, the actual dynamic loads are exhibited for different system modelling complexities, highlighting the necessity for the followed approach.

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Abbreviations

\(\beta \) :

Pressure angle between the cam and follower

\(\theta \) :

Cam rotation angle

\(\omega \) :

Angular velocity

\(c_1\) :

Equivalent dam** coefficient of the system

\(c_2\) :

Dam** coefficient of the return spring

\({f_\text {c}}\) :

Contact force between the cam and follower

\({f_\text {c}}^\text {o}\) :

Horizontal component of the contact force \(f_\text {c}\)

\({f_\text {c}}^\text {v}\) :

Vertical component of the contact force \(f_\text {c}\)

\(f_\text {p}\) :

Return spring preload force

\(k_1\) :

Equivalent stiffness of the system

\(k_2\) :

Stiffness of the return spring

m :

Follower mass

s :

Cam profile input displacement

\(s^\text {min}\) :

Minimum cam profile input displacement

t :

Time

x :

Follower displacement

\('\) :

Superscript indicating derivative with respect to angle

\(^.\) :

Symbol indicating derivative with respect to time

\(\hat{\cdot }\) :

Symbol indicating a non-dimensional quantity

References

  1. Norton, R.L.: Cam Design and Manufacturing Handbook. Industrial Press Inc., New York (2002)

    Google Scholar 

  2. Chen, F.Y.: A survey of the state of the art of cam system dynamics. Mechanism and machine theory 12(3), 201–224 (1977)

    Article  Google Scholar 

  3. Johnson, R.C.: The dynamic analysis and design of relatively flexible cam mechanisms having more than one degree of freedom. Journal of Engineering for Industry 81(4), 323–331 (1959)

    Article  Google Scholar 

  4. Freudenstein, F.: On the dynamics of high-speed cam profiles. International Journal of Mechanical Sciences 1(4), 342–349 (1960)

    Article  Google Scholar 

  5. Eiss, N., Jr.: Vibration of cams having two degrees of freedom. Journal of Manufacturing Science and Engineering, Transactions of the ASME 86(4), 343–349 (1964)

    Google Scholar 

  6. Hardy, S., M-Daragheh, M.: A parametric study using a single cam-follower system dynamic model. Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering 205(4), 273–277 (1991)

  7. Hoření, B.: Double-mass model of an elastic cam mechanism. Mechanism and machine theory 27(4), 443–449 (1992)

    Article  Google Scholar 

  8. Ünlüsoy, Y.S., Tümer, S.T.: Analytical dynamic response of elastic cam-follower systems with distributed parameter return spring. Journal of Mechanical Design, Transactions of the ASME 115(3), 612–620 (1993)

    Article  Google Scholar 

  9. Koster, M.: Effect of flexibility of driving shaft on the dynamic behavior of a cam mechanism. Journal of Manufacturing Science and Engineering, Transactions of the ASME 97(2), 595–602 (1975)

    Google Scholar 

  10. Felszeghy, S.F.: Steady-state residual vibrations in high-speed, dwell-type, rotating disk cam-follower systems. J. Vib. Acoust. 127(1), 12–17 (2005)

    Article  Google Scholar 

  11. Barkan, P.: Calculation of high-speed valve motion with a flexible overhead linkage. SAE Transactions, 687–700 (1953)

  12. Koster, M.P.: Vibrations of Cam Mechanisms. Macmillan International Higher Education, London (1974)

    Book  Google Scholar 

  13. Akiba, K., Shimizu, A., Sakai, H.: A comprehensive simulation of high speed driven valve trains. Technical report, SAE technical paper (1981)

    Google Scholar 

  14. Seidlitz, S.: Valve train dynamics-a computer study. Technical report, SAE technical paper (1989)

    Google Scholar 

  15. Pisano, A., Freudenstein, F.: An experimental and analytical investigation of the dynamic response of a high-speed cam-follower system. part 1: Experimental investigation. Journal of Mechanisms, Transmissions, and Automation in Design 105(4), 692–704 (1983)

  16. Pisano, A., 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. Journal of Mechanical Design, Transactions of the ASME 105(4), 699–704 (1983)

  17. Olmstead, E.H., Taylor, E.S.: Poppet valve dynamics. Journal of the Aeronautical Sciences 6(9), 370–375 (1939)

    Article  Google Scholar 

  18. Tümer, S.T., Ünlüsoy, Y.S.: Nondimensional analysis of jump phenomenon in force-closed cam mechanisms. Mechanism and machine theory 26(4), 421–432 (1991)

    Article  Google Scholar 

  19. Dresner, T., Barkan, P.: New methods for the dynamic analysis of flexible single-input and multi-input cam-follower systems. Journal of Mechanical Design, Transactions of the ASME 117(1), 150–155 (1995)

    Article  Google Scholar 

  20. Cardona, A., Géradin, M.: Kinematic and dynamic analysis of mechanisms with cams. Computer Methods in Applied Mechanics and Engineering 103(1–2), 115–134 (1993)

    Article  MATH  Google Scholar 

  21. Flores, P., Leine, R., Glocker, C.: Application of the nonsmooth dynamics approach to model and analysis of the contact-impact events in cam-follower systems. Nonlinear Dynamics 69(4), 2117–2133 (2012)

    Article  MathSciNet  Google Scholar 

  22. Midha, A., Turcic, D.A.: On the periodic response of cam mechanism with flexible follower and camshaft. ASME Journal of Dynamic Systems, Measurement, and Control 102, 255–264 (1980)

    Article  Google Scholar 

  23. Grewal, P., Newcombe, W.: Dynamic performance of high-speed semi-rigid follower cam systems-effects of cam profile errors. Mechanism and machine theory 23(2), 121–133 (1988)

    Article  Google Scholar 

  24. Sundar, S., Dreyer, J.T., Singh, R.: Rotational sliding contact dynamics in a non-linear cam-follower system as excited by a periodic motion. Journal of Sound and Vibration 332(18), 4280–4295 (2013)

    Article  Google Scholar 

  25. Suryawanshi, S., Sundar, S.: Nonlinear dynamics of system with combined rolling–sliding contact and clearance. Nonlinear Dynamics, 1–23 (2022)

  26. Theodossiades, S., Natsiavas, S.: Non-linear dynamics of gear-pair systems with periodic stiffness and backlash. Journal of Sound and vibration 229(2), 287–310 (2000)

    Article  Google Scholar 

  27. Natsiavas, S., Theodossiades, S., Goudas, I.: Dynamic analysis of piecewise linear oscillators with time periodic coefficients. International Journal of Non-Linear Mechanics 35(1), 53–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  28. Theodossiades, S., Natsiavas, S.: Periodic and chaotic dynamics of motor-driven gear-pair systems with backlash. Chaos, Solitons & Fractals 12(13), 2427–2440 (2001)

    Article  Google Scholar 

  29. Pizzolante, F., Battarra, M., D’Elia, G., Mucchi, E.: A rattle index formulation for single and multiple branch geartrains. Mechanism and Machine Theory 158, 104246 (2021)

    Article  Google Scholar 

  30. Alzate, R., Di Bernardo, M., Montanaro, U., Santini, S.: Experimental and numerical verification of bifurcations and chaos in cam-follower impacting systems. Nonlinear Dynamics 50(3), 409–429 (2007)

    Article  MATH  Google Scholar 

  31. Alzate, R., di Bernardo, M., Giordano, G., Rea, G., Santini, S.: Experimental and numerical investigation of coexistence, novel bifurcations, and chaos in a cam-follower system. SIAM Journal on Applied Dynamical Systems 8(2), 592–623 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Osorio, G., di Bernardo, M., Santini, S.: Corner-impact bifurcations: a novel class of discontinuity-induced bifurcations in cam-follower systems. SIAM Journal on Applied Dynamical Systems 7(1), 18–38 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zhang, W., Ye, M.: Local and global bifurcations of valve mechanism. Nonlinear Dynamics 6(3), 301–316 (1994)

    Article  Google Scholar 

  34. Hamidzadeh, H., Dehghani, M.: Dynamic stability of flexible cam follower systems. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 213(1), 45–52 (1999)

    Google Scholar 

  35. Mahyuddin, A., Midha, A., Bajaj, A.: Evaluation of parametric vibration and stability of flexible cam-follower systems. Journal of Mechanical Design, Transactions of the ASME 116(1), 291–297 (1994)

    Article  Google Scholar 

  36. Mahyuddin, A., Midha, A.: Influence of varying cam profile and follower motion event types on parametric vibration and stability of flexible cam-follower systems. Journal of Mechanical Design, Transactions of the ASME 116(1), 298–305 (1994)

    Article  Google Scholar 

  37. Cveticanin, L.: Stability of motion of the cam-follower system. Mechanism and machine theory 42(9), 1238–1250 (2007)

    Article  MATH  Google Scholar 

  38. Hu, B., Zhou, C., Chen, S.: Elastic dynamics modelling and analysis for a valve train including oil film stiffness and dry contact stiffness. Mechanism and Machine Theory 131, 33–47 (2019)

    Article  Google Scholar 

  39. Doedel, E.J., Champneys, A.R., Dercole, F., Fairgrieve, T.F., Kuznetsov, Y.A., Oldeman, B., Paffenroth, R., Sandstede, B., Wang, X., Zhang, C.: Auto-07p: Continuation and bifurcation software for ordinary differential equations (2007)

  40. Keller, H.B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. P. H. Rabinowitz, ed., “Applications of Bifurcation Theory”, 359–384 (1977)

  41. Doedel, E., Keller, H.B., Kernevez, J.P.: Numerical analysis and control of bifurcation problems (ii): Bifurcation in infinite dimensions. International Journal of Bifurcation and Chaos 1(04), 745–772 (1991)

  42. Ermentrout, B., Mahajan, A.: Simulating, analyzing, and animating dynamical systems: a guide to xppaut for researchers and students. Appl. Mech. Rev. 56(4), 53–53 (2003)

    Article  Google Scholar 

  43. Vetterling, W.T., Vetterling, W.T., Press, W.H., Press, W.H., Teukolsky, S.A., Flannery, B.P., Flannery, B.P.: Numerical Recipes: Example Book C. Cambridge University Press, New York (1992)

    MATH  Google Scholar 

  44. Shampine, L.F., Reichelt, M.W.: The matlab ode suite. SIAM journal on scientific computing 18(1), 1–22 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. Norton, R.L., Gillis, C.A., Maynard, C.N.: Dynamic modeling of the typical industrial cam-follower system: Part 1-single-degree-of-freedom models. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 36533, pp. 297–304 (2002). American Society of Mechanical Engineers

  46. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. John Wiley & Sons, New York (2008)

    MATH  Google Scholar 

  47. Hundal, M.: Aid of digital computer in the analysis of rigid spring-loaded valve mechanisms. Technical report, SAE Technical Paper (1962)

    Google Scholar 

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Acknowledgements

The authors wish to acknowledge the support of the TUCEP Erasmus+ for Traineeship Scheme.

Funding

This work was supported by the TUCEP Erasmus+ for Traineeship Scheme.

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Authors and Affiliations

  1. Engineering Department, University of Ferrara, Via G. Saragat, 1, 44122, Ferrara, Italy

    Caterina Natali & Mattia Battarra

  2. Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, UK

    Stephanos Theodossiades

Authors
  1. Caterina Natali
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  2. Mattia Battarra
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  3. Stephanos Theodossiades
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Contributions

CN was involved in investigation, methodology, software, formal analysis, writing—original draft and visualization. MB was involved in conceptualization, methodology and writing—review and editing. ST was involved in conceptualization, methodology, supervision and writing—review and editing.

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Correspondence to Caterina Natali.

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Natali, C., Battarra, M. & Theodossiades, S. Continuation analysis of cam–follower mechanisms considering time-varying stiffness and loss of contact. Nonlinear Dyn 111, 16921–16938 (2023). https://doi.org/10.1007/s11071-023-08684-4

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