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Effects of residual stress and viscous and hysteretic dampings on the stability of a spinning micro-shaft

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Abstract

This study examines the effects of the residual stress and viscous and hysteretic dampings on the vibrational behavior and stability of a spinning Timoshenko micro-shaft. A modified couple stress theory (MCST) is used to elucidate the size-dependency of the micro-shaft spinning stability, and the equations of motion are derived by employing Hamilton’s principle and a spatial beam for spinning micro-shafts. Moreover, a differential quadrature method (DQM) is presented, along with the exact solution for the forward and backward (FW-BW) complex frequencies and normal modes. The effects of the material length scale parameter (MLSP), the spinning speed, the viscous damping coefficient, the hysteretic damping, and the residual stress on the stability of the spinning micro-shafts are investigated. The results indicate that the MLSP, the internal dampings (viscous and hysteretic), and the residual stress have significant effects on the complex frequency and stability of the spinning micro-shafts. Therefore, it is crucial to take these factors into account while these systems are designed and analyzed. The results show that an increase in the MLSP leads to stiffening of the spinning micro-shaft, increases the FW-BW dimensionless complex frequencies of the system, and enhances the stability of the system. Additionally, a rise in the tensile residual stresses causes an increase in the FW-BW dimensionless complex frequencies and stability of the micro-shafts, while the opposite is true for the compressive residual stresses. The results of this research can be employed for designing spinning structures and controlling their vibrations, thus forestalling resonance.

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References

  1. POPPLEWELL, N. and CHANG, D. Free vibrations of a stepped, spinning Timoshenko beam. Journal of Sound and Vibration, 203, 717–722 (1997)

    Google Scholar 

  2. BANERJEE, J. and SU, H. Development of a dynamic stiffness matrix for free vibration analysis of spinning beams. Computers and Structures, 82, 2189–2197 (2004)

    Google Scholar 

  3. LIAO, C. L. and HUANG, B. W. Parametric instability of a spinning pretwisted beam under periodic axial force. International Journal of Mechanical Sciences, 37, 423–439 (1995)

    MATH  Google Scholar 

  4. HO, S. H. and CHEN, C. K. Free transverse vibration of an axially loaded non-uniform spinning twisted Timoshenko beam using differential transform. International Journal of Mechanical Sciences, 48, 1323–1331 (2006)

    MATH  Google Scholar 

  5. BANERJEE, J. and SU, H. Dynamic stiffness formulation and free vibration analysis of a spinning composite beam. Computers and Structures, 84, 1208–1214 (2006)

    Google Scholar 

  6. CIHAN, M., EKEN, S., and KAYA, M. O. Dynamic instability of spinning launch vehicles modeled as thin-walled composite beams. Acta Mechanica, 228, 4353–4367 (2017)

    MathSciNet  MATH  Google Scholar 

  7. OH, S. Y., LIBRESCU, L., and SONG, O. Vibration and instability of functionally graded circular cylindrical spinning thin-walled beams. Journal of Sound and Vibration, 285, 1071–1091 (2005)

    Google Scholar 

  8. GAYEN, D. and ROY, T. Finite element based vibration analysis of functionally graded spinning shaft system. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, 3306–3321 (2014)

    Google Scholar 

  9. SHIH, Y. S. and YEH, Z. F. Dynamic stability of a viscoelastic beam with frequency-dependent modulus. International Journal of Solids and Structures, 42, 2145–2159 (2005)

    MATH  Google Scholar 

  10. KARLIČIĆ, D., KOZIĆ, P., PAVLOVIĆ, R., and NEŠIĆ, N. Dynamic stability of single-walled carbon nanotube embedded in a viscoelastic medium under the influence of the axially harmonic load. Composite Structures, 162, 227–243 (2017)

    Google Scholar 

  11. MOHAMMADIMEHR, M. and MEHRABI, M. Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow. Applied Mathematical Modelling, 60, 255–272 (2018)

    MathSciNet  MATH  Google Scholar 

  12. APUZZO, A., BARRETTA, R., FAGHIDIAN, S., LUCIANO, R., and DE SCIARRA, F. M. Free vibrations of elastic beams by modified nonlocal strain gradient theory. International Journal of Engineering Science, 133, 99–108 (2018)

    MathSciNet  MATH  Google Scholar 

  13. FARAJI-OSKOUIE, M., ANSARI, R., and SADEGHI, F. Nonlinear vibration analysis of fractional viscoelastic Euler-Bernoulli nanobeams based on the surface stress theory. Acta Mechanica Solida Sinica, 30, 416–424 (2017)

    Google Scholar 

  14. JALAEI, M., ARANI, A. G., and TOURANG, H. On the dynamic stability of viscoelastic graphene sheets. International Journal of Engineering Science, 132, 16–29 (2018)

    MathSciNet  MATH  Google Scholar 

  15. DENG, J., LIU, Y., ZHANG, Z., and LIU, W. Size-dependent vibration and stability of multispan viscoelastic functionally graded material nanopipes conveying fluid using a hybrid method. Composite Structures, 179, 590–600 (2017)

    Google Scholar 

  16. KULKARNI, P., BHATTACHARJEE, A., and NANDA, B. Study of damping in composite beams. Materials Today: Proceedings, 5, 7061–7067 (2018)

    Google Scholar 

  17. LIU, H., LIU, H., and YANG, J. L. Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation. Composites Part B: Engineering, 155, 244–256 (2018)

    Google Scholar 

  18. MOHAMMADIMEHR, M., MONAJEMI, A., and MORADI, M. Vibration analysis of viscoelastic tapered micro-rod based on strain gradient theory resting on visco-Pasternak foundation using DQM. Journal of Mechanical Science and Technology, 29, 2297–2305 (2015)

    Google Scholar 

  19. MOHAMMADIMEHR, M., FARAHI, M., and ALIMIRZAEI, S. Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory. Applied Mathematics and Mechanics (English Edition), 37, 1375–1392 (2016) https://doi.org/10.1007/s10483-016-2138-9

    MathSciNet  MATH  Google Scholar 

  20. TALIMIAN, A. and BEDA, P. Dynamic stability of a size-dependent micro-beam. European Journal of Mechanics-A/Solids, 72, 245–251 (2018)

    MathSciNet  MATH  Google Scholar 

  21. GHAYESH, M. H. Dynamics of functionally graded viscoelastic microbeams. International Journal of Engineering Science, 124, 115–131 (2018)

    MathSciNet  MATH  Google Scholar 

  22. REDDY, J. Microstructure-dependent couple stress theories of functionally graded beams. Journal of the Mechanics and Physics of Solids, 59, 2382–2399 (2011)

    MathSciNet  MATH  Google Scholar 

  23. GHAYESH, M. H. and FAROKHI, H. On the viscoelastic dynamics of fluid-conveying microtubes. International Journal of Engineering Science, 127, 186–200 (2018)

    MathSciNet  MATH  Google Scholar 

  24. GHAYESH, M. H., FAROKHI, H., and HUSSAIN, S. Viscoelastically coupled size-dependent dynamics of microbeams. International Journal of Engineering Science, 109, 243–255 (2016)

    MathSciNet  MATH  Google Scholar 

  25. BAHAADINI, R. and SAIDI, A. R. On the stability of spinning thin-walled porous beams. Thin-Walled Structures, 132, 604–615 (2018)

    MATH  Google Scholar 

  26. CHEN, C., LI, S., DAI, L., and QIAN, C. Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces. Communications in Nonlinear Science and Numerical Simulation, 19, 1626–1637 (2014)

    MathSciNet  MATH  Google Scholar 

  27. ARVIN, H. Free vibration analysis of micro rotating beams based on the strain gradient theory using the differential transform method: Timoshenko versus Euler-Bernoulli beam models. European Journal of Mechanics-A/Solids, 65, 336–348 (2017)

    MathSciNet  MATH  Google Scholar 

  28. MELANSON, J. and ZU, J. Free vibration and stability analysis of internally damped rotating shafts with general boundary conditions. Journal of Vibration and Acoustics, 120, 776–783 (1998)

    Google Scholar 

  29. MERRETT, C. G. Time to flutter theory for viscoelastic composite aircraft wings. Composite Structures, 154, 646–659 (2016)

    Google Scholar 

  30. ZHU, K. and CHUNG, J. Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions. Applied Mathematical Modelling, 66, 362–382 (2019)

    MathSciNet  MATH  Google Scholar 

  31. VATTA, F. and VIGLIANI, A. Internal damping in rotating shafts. Mechanism and Machine Theory, 43, 1376–1384 (2008)

    MATH  Google Scholar 

  32. ILKHANI, M. and HOSSEINI-HASHEMI, S. Size dependent vibro-buckling of rotating beam based on modified couple stress theory. Composite Structures, 143, 75–83 (2016)

    Google Scholar 

  33. WANG, J., LI, D., and JIANG, J. Coupled flexural-torsional vibration of spinning smart beams with asymmetric cross sections. Finite Elements in Analysis and Design, 105, 16–25 (2015)

    MathSciNet  Google Scholar 

  34. XU, T., RONG, J., XIANG, D., PAN, C., and YIN, X. Dynamic modeling and stability analysis of a flexible spinning missile under thrust. International Journal of Mechanical Sciences, 119, 144–154 (2016)

    Google Scholar 

  35. HOSSEINI-HASHEMI, S. and ILKHANI, M. R. Exact solution for free vibrations of spinning nanotube based on nonlocal first order shear deformation shell theory. Composite Structures, 157, 1–11 (2016)

    Google Scholar 

  36. SHABANLOU, G., HOSSEINI, S. A. A., and ZAMANIAN, M. Free vibration analysis of spinning beams using higher-order shear deformation beam theory. Iranian Journal of Science and Technology, Transactions of Mechanical Engineering, 42, 363–382 (2018)

    MATH  Google Scholar 

  37. LI, X., LI, Y., and QIN, Y. Free vibration characteristics of a spinning composite thin-walled beam under hygrothermal environment. International Journal of Mechanical Sciences, 119, 253–265 (2016)

    Google Scholar 

  38. PAI, P. F., QIAN, X., and DU, X. Modeling and dynamic characteristics of spinning Rayleigh beams. International Journal of Mechanical Sciences, 68, 291–303 (2013)

    Google Scholar 

  39. CHEN, W. R. On the vibration and stability of spinning axially loaded pre-twisted Timoshenko beams. Finite Elements in Analysis and Design, 46, 1037–1047 (2010)

    Google Scholar 

  40. TORABI, K. and AFSHARI, H. Optimization of flutter boundaries of cantilevered trapezoidal functionally graded sandwich plates. Journal of Sandwich Structures and Materials, 21, 503–531 (2019)

    Google Scholar 

  41. MUSTAPHA, K. and ZHONG, Z. Spectral element analysis of a non-classical model of a spinning micro beam embedded in an elastic medium. Mechanism and Machine Theory, 53, 66–85 (2012)

    Google Scholar 

  42. HOSSEINI, S. A. A., ZAMANIAN, M., SHAMS, S., and SHOOSHTARI, A. Vibration analysis of geometrically nonlinear spinning beams. Mechanism and Machine Theory, 78, 15–35 (2014)

    Google Scholar 

  43. TORABI, K., AFSHARI, H., and NAJAFI, H. Whirling analysis of axial-loaded multi-step Timoshenko rotor carrying concentrated masses. Journal of Solid Mechanics, 9, 138–156 (2017)

    Google Scholar 

  44. FANG, J., GU, J., and WANG, H. Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory. International Journal of Mechanical Sciences, 136, 188–199 (2018)

    Google Scholar 

  45. EFTEKHARI, M., DASHTI-RAHMATABADI, A., and MAZIDI, A. Magnetic field effects on the nonlinear vibration of a rotor. Applied Mathematics and Mechanics (English Edition), 41, 289–312 (2020) https://doi.org/10.1007/s10483-020-2567-6

    Google Scholar 

  46. CHOI, S. T., WU, J. D., and CHOU, Y. T. Dynamic analysis of a spinning Timoshenko beam by the differential quadrature method. AIAA Journal, 38, 851–856 (2000)

    Google Scholar 

  47. ANITESCU, C., ATROSHCHENKO, E., ALAJLAN, N., and RABCZUK, T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Material and Continua, 59, 345–359 (2019)

    Google Scholar 

  48. GUO, H., ZHUANG, X., and RABCZUK, T. A deep collocation method for the bending analysis of Kirchhoff plate. Computers, Material and Continua, 59, 433–456 (2019)

    Google Scholar 

  49. SAMANIEGO, E., ANITESCU, C., GOSWAMI, S., NGUYEN-THANH, V. M., GUO, H., HAMDIA, K., ZHUANG, X., and RABCZUK, T. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 362, 112790 (2020)

    MathSciNet  MATH  Google Scholar 

  50. VU-BAC, N., LAHMER, T., ZHUANG, X., NGUYEN-THOI, T., and RABCZUK, T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 100, 19–31 (2016)

    Google Scholar 

  51. YANG, F., CHONG, A., LAM, D. C. C., and TONG, P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, 2731–2743 (2002)

    MATH  Google Scholar 

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Acknowledgements

The authors would like to thank the referees for their valuable comments. Additionally, they are thankful to the Iranian Nanotechnology Development Committee for the financial support and the University of Kashan for supporting this work (No. 682561/18).

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Authors and Affiliations

  1. Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-53153, Iran

    A. A. Monajemi & M. Mohammadimehr

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  1. A. A. Monajemi
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  2. M. Mohammadimehr
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Correspondence to M. Mohammadimehr.

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Project supported by the Iranian Nanotechnology Development Committee and the University of Kashan (No. 682561/18)

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Monajemi, A.A., Mohammadimehr, M. Effects of residual stress and viscous and hysteretic dampings on the stability of a spinning micro-shaft. Appl. Math. Mech.-Engl. Ed. 41, 1251–1268 (2020). https://doi.org/10.1007/s10483-020-2640-8

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  • DOI: https://doi.org/10.1007/s10483-020-2640-8

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