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\({\mathcal {H}}_{2}\) optimization and numerical study of inerter-based vibration isolation system helical spring fatigue life

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Abstract

This paper presents an optimization and numerical analysis of vibration-induced fatigue in a two degree-of-freedom inerter-based vibration isolation system. The system is comprised of a primary, e.g. source body, and a secondary, e.g. receiving body, mutually connected through an isolator. The isolator includes a spring, a dashpot and an inerter. Inerter is a mechanical device which produces a force proportional to relative acceleration between its terminals. A broadband frequency force excitation of the primary body is imposed throughout the study. The goal of the proposed optimization is to prolong the fatigue life of the ground connecting helical spring of the secondary body. The optimization is based on minimizing separately the displacement and velocity amplitudes. Both optimization criteria are compared with regard to spring fatigue life improvement for fair benchmark comparison. The inerter-based optimized systems, in which the \({\mathcal {H}}_{2}\) index of the receiving body is minimized, are also compared with the optimized systems without inerter. Notable improvements are observed in inerter-based systems due to the inclusion of an optimally tuned inerter in the isolator. The proposed analytical vibration fatigue method optimization results are compared with the finite element method results, and a very good agreement is observed. Most accurate helical spring deflection and stress correction factors are discussed and determined. Furthermore, the inerter concept is successfully implemented into finite element-based dynamic solution.

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References

  1. Wahl, A.M.: Mechanical Springs, 1st edn. Penton Pub. Co., Cleveland (1944)

    Google Scholar 

  2. Čakmak, D., Wolf, H., Božić, Ž., Jokić, M.: Optimization of an inerter-based vibration isolation system and helical spring fatigue life assessment. Arch. Appl. Mech. 1–14 (2018). https://doi.org/10.1007/s00419-018-1447-x

  3. Rao, S.S.: Mechanical Vibrations, Sixth Edition in SI Units, Global Edition. Pearson, London (2017)

    Google Scholar 

  4. Smith, M.C.: Synthesis of mechanical networks: the inerter. IEEE Trans. Autom. Control 47(10), 1648–1662 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bishop, N.W.M., Sherratt, F.: Finite Element Based Fatigue Calculations. NAFEMS Ltd, Farnham (2000)

    Google Scholar 

  6. Todinov, M.T.: Maximum principal tensile stress and fatigue crack origin for compression springs. Int. J. Mech. Sci. 41(3), 357–370 (1999)

    Article  MATH  Google Scholar 

  7. Del Llano-Vizcaya, L., Rubio-González, C., Mesmacque, G., Cervantes-Hernandez, T.: Multiaxial fatigue and failure analysis of helical compression springs. Eng. Fail. Anal. 13(8), 1303–1313 (2006)

    Article  Google Scholar 

  8. Timoshenko, S.P.: Strength of Materials, Part I, Elementary Theory and Problems, 2nd edn. D. Van Nostrand. Company Inc, New York (1940)

    MATH  Google Scholar 

  9. Göhner, O.: Schubspannungsverteilung im Querschnitt einer Schraubenfeder. Ingenieur-Archiv 1(5), 619–644 (1930)

    Article  MATH  Google Scholar 

  10. Henrici, P.: On helical springs of finite thickness. Q. Appl. Math. 13(1), 106–110 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  11. Berry, W.R.: Practical problems in spring design. Proc. Inst. Mech. Eng. 139(1), 431–524 (1938)

    Article  Google Scholar 

  12. Ancker Jr., C.J., Goodier, J.N.: Pitch and curvature correction for helical springs. ASME J. Appl. Mech. 25(4), 466–470 (1958)

    MATH  Google Scholar 

  13. Research Committee on the Analysis of Helical Spring: Report of Research Committee on the Analysis of Helical Spring. Trans. Jpn. Soc. Spring Eng. 2004(49), 35–75 (2004)

  14. Calder, G.A., Jenkins, C.: Stress analysis of a helical coil automobile spring using rosettes. Exp. Tech. 12(2), 17–20 (1988)

    Article  Google Scholar 

  15. Lin, Y., Pisano, A.P.: General dynamic equations of helical springs with static solution and experimental verification. ASME J. Appl. Mech. 54(4), 910–917 (1987)

    Article  Google Scholar 

  16. Yazdani Sarvestani, H., Akbarzadeh, A.H.: Thick isotropic curved tubes: three-dimensional stress analysis. Arch. Appl. Mech. 87(6), 927–947 (2017)

    Article  Google Scholar 

  17. Bockwoldt, T.S., Munsick, G.A.: Correction to design equation for spring diametral growth upon compression. Trans. ASME J. Mech. Des. 135(12), 124503–124503-4 (2013)

    Article  Google Scholar 

  18. Burns, S.J.: The relation between helical spring compliances with free and fixed end rotations. ASME J. Appl. Mech. 78(6), 061005–061005-5 (2011)

    Article  Google Scholar 

  19. Krużelecki, J.: Parametrical optimization of compression helical springs against instability. Struct. Multidiscip. Optim. 13, 205–212 (1997)

    Article  Google Scholar 

  20. Mlikota, M., Schmauder, S., Božić, Ž.: Calculation of the Wöhler (S–N) curve using a two-scale model. Int. J. Fatigue 14, 289–297 (2018)

    Article  Google Scholar 

  21. Mlikota, M., Schmauder, S., Božić, Ž., Hummel, M.: Modelling of overload effects on fatigue crack initiation in case of carbon steel. Fatigue Fract. Eng. Mater. Struct. 40, 1182–1190 (2017)

    Article  Google Scholar 

  22. Božić, Ž., Schmauder, S., Mlikota, M., Hummel, M.: Multiscale fatigue crack growth modelling for welded stiffened panels. Fatigue Fract. Eng. Mater. Struct. 37, 1025–1033 (2014)

    Article  Google Scholar 

  23. Rahman, M.M., Ariffin, A.K., Abdullah, S.: Finite element based vibration fatigue analysis of a new twostroke linear generator engine component. Int. J. Mech. Mater. Eng. 2(1), 63–74 (2007)

    Google Scholar 

  24. Halfpenny, A.: A frequency domain approach for fatigue life estimation from finite element analysis. Key Eng. Mater. 167–168, 401–410 (1999)

    Article  Google Scholar 

  25. Mršnik, M., Slavič, J., Boltežar, M.: Multiaxial vibration fatigue: a theoretical and experimental comparison. Mech. Syst. Signal Process. 76–77, 409–423 (2016)

    Article  Google Scholar 

  26. Mršnik, M., Slavič, J., Boltežar, M.: Frequency-domain methods for a vibration-fatigue-life estimation: application to real data. Int. J. Fatigue 47, 8–17 (2013)

    Article  Google Scholar 

  27. Česnik, M., Slavič, J.: Vibrational fatigue and structural dynamics for harmonic and random loads. Strojniški vestnik J. Mech. Eng. 60(5), 339–348 (2014)

    Article  Google Scholar 

  28. Braccesi, C., Cianetti, F., Lori, G., Pioli, D.: A frequency method for fatigue life estimation of mechanical components under bimodal random stress process. Int. J. Struct. Integrity Durab. 1(4), 277–290 (2005)

    MATH  Google Scholar 

  29. Bonte, M.H.A., de Boer, A., Liebregts, R.: Prediction of mechanical fatigue caused by multiple random excitations. Proc. ISMA 2004, 697–708 (2004)

    Google Scholar 

  30. Zhou, Y., Fei, Q., Wu, S.: Utilization of modal stress approach in random-vibration fatigue evaluation. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng. 231(14), 2603–2615 (2016)

    Article  Google Scholar 

  31. Warburton, G.B.: Optimum absorber parameters for various combinations of response and excitation parameters. Earthq. Eng. Struct. Dyn. 10(3), 381–401 (1982)

    Article  Google Scholar 

  32. Alujević, N., Zhao, G., Depraetere, B., Sas, P., Pluymers, B., Desmet, W.: \({\cal{H}}_{2}\) optimal vibration control using inertial actuators and a comparison with tuned mass dampers. J. Sound Vib. 333(18), 4073–4083 (2014)

    Article  Google Scholar 

  33. Zhao, G., Alujević, N., Depraetere, B., Sas, P.: Dynamic analysis and \({\cal{H}}_{2}\) optimisation of a piezo-based tuned vibration absorber. J. Intell. Mater. Syst. Struct. 26(15), 1995–2010 (2014)

    Article  Google Scholar 

  34. Cheung, Y.L., Wong, W.O., Cheng, L.: Optimization of a hybrid vibration absorber for vibration control of structures under random force excitation. J. Sound Vib. 332(3), 494–509 (2013)

    Article  Google Scholar 

  35. Alujević, N., Čakmak, D., Wolf, H., Jokić, M.: Passive and active vibration isolation systems using inerter. J. Sound Vib. 418, 163–183 (2018)

    Article  Google Scholar 

  36. Alujević, N., Wolf, H., Gardonio, P., Tomac, I.: Stability and performance limits for active vibration isolation using blended velocity feedback. J. Sound Vib. 330, 4981–4997 (2011)

    Article  Google Scholar 

  37. Alujević, N., Gardonio, P., Frampton, K.D.: Smart double panel for the sound radiation control: blended velocity feedback. AIAA J. 49(6), 1123–1134 (2011)

    Article  Google Scholar 

  38. Caiazzo, A., Alujević, N., Pluymers, B., Desmet, W.: Active control of turbulent boundary layer-induced sound transmission through the cavity-backed double panels. J. Sound Vib. 422, 161–188 (2018)

    Article  Google Scholar 

  39. Senjanović, I., Alujević, N., Ćatipović, I., Čakmak, D., Vladimir, N.: Vibration analysis of rotating toroidal shell by the Rayleigh–Ritz method and Fourier series. Eng. Struct. 173, 870–891 (2018)

    Article  Google Scholar 

  40. Dassault Systèmes: Abaqus 6.9 User’s Guide and Theoretical Manual. Hibbitt, Karlsson & Sorensen Inc., Providence (2009)

    Google Scholar 

  41. SafeTechnology, Fe-Safe 6 User Manual (2011)

  42. Dassault Systèmes: Catia V5R19 Documentation: Finite Element Reference Guide (2007)

  43. Kong, Y.S., Omar, M.Z., Chua, L.B., Abdullah, S.: Fatigue life prediction of parabolic leaf spring under various road conditions. Eng. Fail. Anal. 46, 92–103 (2014)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, I. Lučića 5, 10 000, Zagreb, Croatia

    D. Čakmak, Z. Tomičević, H. Wolf & Ž. Božić

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  1. D. Čakmak
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  2. Z. Tomičević
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  3. H. Wolf
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  4. Ž. Božić
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Correspondence to Z. Tomičević.

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Čakmak, D., Tomičević, Z., Wolf, H. et al. \({\mathcal {H}}_{2}\) optimization and numerical study of inerter-based vibration isolation system helical spring fatigue life. Arch Appl Mech 89, 1221–1242 (2019). https://doi.org/10.1007/s00419-018-1495-2

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