Document Type : Article

**Authors**

Department of Mechanical Engineering, Faculty of Engineering, Bu-Ali Sina University, Hamedan, Iran.

**Abstract**

In this study, the dynamical instabilities of an embedded smart micro-shell conveying pulsating fluid flow is investigated based on nonlocal piezoelasticity theory and nonlinear cylindrical shell model. The micro-shell is surrounded by an elastic foundation which is suitable for both Winkler spring and Pasternak shear modules. The internal fluid flow is considered to be purely harmonic, irrotational, isentropic, Newtonian and incompressible and it is mathematically modeled using linear potential flow theory, time mean Navier Stokes equations and Knudsen number. For more reality of the micro-scale problem the pulsating viscous effects as well as the slip boundary condition are also taken into accounts. Employing the modified Lagrange equations of motion for open systems, the nonlinear coupled governing equations are achieved and consequently the instability boundaries are obtained using the Bolotin’s method. In the numerical results section, a comprehensive discussion is made on the dynamical instabilities of the system (such as divergence; flutter and parametric resonance). It is found that applying positive electric potential field will improve the stability of the system as an actuator or as a vibration amplitude controller in the Micro Electro Mechanical Systems.

**Keywords**

**Main Subjects**

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28. Tubaldi, E., Amabili, M., and Paidoussis, M.P. "Nonlinear dynamics of shells conveying pulsatile flow with pulse-wave propagation: Theory and numerical results for a single harmonic pulsation", J. Sound Vib., 396, pp. 217-245 (2017).

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32. Civalek, O. and Demir, C . "A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method", Appl. Math. Comput., 289, pp. 335-352 (2016).

33. Ghorbanpour Arani, A., Atabakhshian, V., Loghman, A., Shajari, A.R., and Amir, S. "Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method", Physica B, 407, pp.2549-2555 (2012).

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35. Institute of Electrical and Electronics Engineers, Standard on Piezoelectricity, Std, IEEE, New York (1978). 36. Ding, H.J. and Chen, W.Q. "Three dimensional problems of piezoelasticity", Nova Science, New York(2001).

37. Eringen, A.C. "Nonlocal polar elastic continua", INT. J. ENG. SCI., 10(1), pp. 1-16 (1972).

38. Eringen, A.C., Nonlocal Continuum Field Theories, Springer-Verlag, New York (2002).

39. Eringen, A.C. "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, pp. 4703-4710 (1983).

40. Ke, L.L., Wang, Y.Sh., and Wang, Zh.D. "Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory", Compos. Struct., 94(6), pp. 2038- 2047 (2012).

41. Han, J.H. and Lee, I. "Analysis of composite plates with piezoelectric actuators for vibration control using layerwise displacement theory", Compos. Part B-Eng.,29(5), pp. 621-632 (1998).

42. Ke, L.L., Wang, Y.Sh., and Wang, Zh.D. "Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory", Compos. Struct., 94(6), pp. 2038- 2047 (2012).

43. Kurylov, Y. and Amabili, M. "Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with diff:erent boundary conditions", J. Sound. Vib., 329(9), pp. 1435-1449 (2010).

44. Alinia, M.M. and Ghannadpour, S. "Nonlinear analysis of pressure loaded FGM plates", Compos. Struct., 88(3), pp. 354-359 (2009).

45. Yang, J., An Introduction to the Theory of Piezoelectricity, 9th Ed., Springer, Lincoln (2005).

46. Fox, R.W., Pritchard, P.J., and McDonald, A.T., Introduction to Fluid Mechanics, 4th Ed., Wiley, New York, USA (2008).

47. Paidoussis, M.P., Misra, A.K., and Chan, S.P. "Dynamics and stability of coaxial cylindrical shells conveying viscous fluid", J. Appl. Mech-T., ASME., 52(2),pp. 389-396 (1985).

48. Karniadakis, G., Beskok, A., and Aluru, N., Micro Flows Nanoflows: Fundamentals and Simulation, Springer-Verlag (2005).

49. Rashidi, V., Mirdamadi, H.R., and Shirani, E. "A novel model for vibrations of nanotubes conveying nanoflow", Comput. Mater. Sci., 51, pp. 347-352 (2012).

50. Shokouhmand, H., Isfahani, A.H.M., and Shirani, E. "Friction and heat transfer coefficient in micro and nano channels filled with potous media for wide range of Knudsen number", Int. Comm. Heat Mass, 37, pp. 890-894 (2010).

51. Irschik, H. and Holl, H. "The equations of Lagrange written for a non-material volume", Acta Mech, 153, pp. 231-248 (2002).

52. Yang, J., Ke, L.L., and Kitipornchai, S. "Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", Physica E, 42, pp. 1727-1735 (2010).

53. Bolotin, V.V., The Dynamic Stability of Elastic Systems, Holden-Day, Inc, San Francisco, USA (1964).

54. Amabili, M. and Graziera, R. "Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass. Part ii: Shells containing or immersed in axial flow", J. Fluid. Struct., 16(1), pp.31-51 (2002).

55. Mohammadi, K., Rajabpour, A., and Ghadiri, M. "Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation", Comp. Mater. Sci., 148, pp. 104-115 (2018).

2. Kong, J., Franklin, N.R., Zhou, C., Chapline, M.G., Peng, S., Cho, K., and Dai, H. "Nanotube molecular wires as chemical sensors", Science, 287(5453), pp. 622-625 (2000).

3. Dharap, P., Li, Z., Nagarajaiah, S., and Barrera, E.V. "Nanotube film based on single-wall carbon nanotubes for strain sensing", Nanotechnology, 15(3), p. 379 (2004).

4. Ashley, H. and Haviland, G. "Bending vibrations of a pipeline containing flowing fluid", J. Appl. Mech., 72(1), pp. 229-232 (1950).

5. Paidoussis, M.P., Fluid-Structure Interactions: Slender Structures and Axial Flow, 1, Academic Press, London, England (1998).

6. Amabili, M., Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, Parma, Italy (2008).

7. Reddy, J.N. and Wang, C.M., Dynamics of Fluid Conveying Beams: Governing Equations and Finite Element Models, Centre for Off:shore Research and Engineering National University of Singapore (2004).

8. Pellicano, F. and Amabili, M. "Dynamic instability and chaos of empty and fluid-filled circular cylindrical shells under periodic axial loads", J. Sound Vib., 293(1), pp. 227-252(2006).

9. Sadeghi, M.H. and Karimi-Dona, M.H. "Dynamic behavior of a fluid conveying pipe subjected to a moving sprung mass: an FEM-state space approach", Int. J. PressVessels Pip., 88, pp. 31-123 (2011).

10. Gu, J., Ma, T., and Menglan, D. "Eff:ect of aspect ratio on the dynamic response of a fluid-conveying pipe using the Timoshenko beam model", Ocean Eng, 114, pp. 185-191 (2016).

11. Kamm, R.D. and Pedley, T.J. "Flow in collapsible tubes: a brief review", J. Biomech. Eng., 111, pp. 177-179 (1989).

12. Paidoussis, M.P., Fluid-Structure Interactions: Slender Structures and Axial Flow, 2, Academic Press (2003).

13. Yan, Y., Wang, W.Q., and Zhang, L. X. "Dynamical behaviors of fluid-conveyed multi walled carbon nanotubes", Appl. Math. Modell., 33, pp. 1430-1440 (2009).

14. Kuang, Y.D., He, X.Q., Chen, C.Y., and Li, G.Q. "Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid", Int. J. Comput. Mater. Sci. Surf. Eng., 45, pp. 875-880 (2009).

15. Ghorbanpour Arani, A., Shajari, A.R., Amir, S., and Atabakhshian, V. "Nonlinear fluid-induced vibration and instability of an embedded piezoelectric polymeric microtube using nonlocal elasticity theory", J. Mech. Eng. Sci., 227(12), pp. 2870-2885 (2013).

16. Ghorbanpour Arani, A., Shajari, A.R., Atabakhshian, V., Amir, S., and Loghman, A. "Nonlinear dynamical response of embedded fluid-conveyed micro-tube reinforced by BNNTs", Compos. Part B-Eng., 44(1), pp. 424-432 (2013).

17. Ghorbanpour Arani, A. and Hashemian, M. "Surface stress effects on dynamic stability of double-walled boron nitride nanotubes conveying viscose fluid based on nonlocal shell theory", Sci. Iran., 20(6), pp. 2356-2374 (2013).

18. Ghorbanpour Arani, A., Khoddami Maraghi, Z., and Haghparast, E. "The fluid structure interaction effect on the vibration and instability of a conveyed doublewalled boron nitride nanotube", Sci. Iran., 22(2), pp.436-447 (2015).

19. Atabakhshian, V., Shoshtari, A.R., and Karimi, M. "Electro-thermal vibration of a smart coupled nanobeam system with an internal flow based on nonlocalel asticity theory", Physica B: Condensed Matter, 456, pp. 375-382 (2015).

20. Paidoussis, M.P. and Issid, N.T. "Dynamic stability of pipes conveying fluid", J. Sound. Vib., 33(3), pp. 267-294 (1974).

21. Panda, L.N. and Kar, R.C. "Nonlinear dynamics of a pipe conveying pulsating fluid with combination, principal parametric and internal resonances", Journal of Sound and Vibration, 309, pp. 375-406 (2008).

22. Azrar, A., Azrar, L., and Aljinaidi, A.A. "Numerical modeling of dynamic and parametric instabilities of single-walled carbon nanotubes conveying pulsating and viscous fluid", Compos. Struct, 125(8), pp. 127- 143 (2015).

23. Liang, F. and Su, Y. "Stability analysis of a singlewalled carbon nanotube conveying pulsating and viscous fluid with nonlocal effect", Appl. Math. Model., 37, pp. 6821-6828 (2013).

24. Da, H.L., Wang, L., Qian, Q., and Ni, Q. "Vortexinduced vibrations of pipes conveying pulsating fluid", Ocean. Eng., 77, pp. 12-22 (2014).

25. Wang, L. "A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid", Int. J. Non. Linear Mech., 44, pp. 115-121 (2009).

26. Yang, K.S., Cheng, Y.C., Liu, M.C., and Shyu, J.C. "Micro pulsating heat pipes with alternate microchannel widths", Appl. Therm. Eng., 83, pp. 131-138 (2015).

27. Tubaldi, E., Amabili, V., and Paidoussis, M.P. "Fluidstructure interaction for nonlinear response of shells conveying pulsatile flow", J. Sound. Vib., 371, pp. 252-276 (2016).

28. Tubaldi, E., Amabili, M., and Paidoussis, M.P. "Nonlinear dynamics of shells conveying pulsatile flow with pulse-wave propagation: Theory and numerical results for a single harmonic pulsation", J. Sound Vib., 396, pp. 217-245 (2017).

29. Rafii-Tabar, H., Ghavanloo, E., and Fazelzadeh, S.A. "Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures", Physics Reports, 638, pp. 1-97 (2016).

30. Mercan, K. and Civalek, O. "DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix", Compos. Struct., 143, pp. 300-309 (2016).

31. Akgoz, B. and Civalek, O. "Bending analysis of embedded carbon nanotubes resting on an elastic foundation using strain gradient theory", Acta Astronaut, 119, pp. 1-12 (2016).

32. Civalek, O. and Demir, C . "A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method", Appl. Math. Comput., 289, pp. 335-352 (2016).

33. Ghorbanpour Arani, A., Atabakhshian, V., Loghman, A., Shajari, A.R., and Amir, S. "Nonlinear vibration of embedded SWBNNTs based on nonlocal Timoshenko beam theory using DQ method", Physica B, 407, pp.2549-2555 (2012).

34. Alibeigi, B., Beni, Y.T., and Mehralian, F. "On the thermal buckling of magneto-electro-elastic piezoelectric nanobeams", Eur. Phys. J. Plus., 133, pp. 133- 138 (2018).

35. Institute of Electrical and Electronics Engineers, Standard on Piezoelectricity, Std, IEEE, New York (1978). 36. Ding, H.J. and Chen, W.Q. "Three dimensional problems of piezoelasticity", Nova Science, New York(2001).

37. Eringen, A.C. "Nonlocal polar elastic continua", INT. J. ENG. SCI., 10(1), pp. 1-16 (1972).

38. Eringen, A.C., Nonlocal Continuum Field Theories, Springer-Verlag, New York (2002).

39. Eringen, A.C. "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, pp. 4703-4710 (1983).

40. Ke, L.L., Wang, Y.Sh., and Wang, Zh.D. "Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory", Compos. Struct., 94(6), pp. 2038- 2047 (2012).

41. Han, J.H. and Lee, I. "Analysis of composite plates with piezoelectric actuators for vibration control using layerwise displacement theory", Compos. Part B-Eng.,29(5), pp. 621-632 (1998).

42. Ke, L.L., Wang, Y.Sh., and Wang, Zh.D. "Nonlinear vibration of the piezoelectric nanobeams based on the nonlocal theory", Compos. Struct., 94(6), pp. 2038- 2047 (2012).

43. Kurylov, Y. and Amabili, M. "Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with diff:erent boundary conditions", J. Sound. Vib., 329(9), pp. 1435-1449 (2010).

44. Alinia, M.M. and Ghannadpour, S. "Nonlinear analysis of pressure loaded FGM plates", Compos. Struct., 88(3), pp. 354-359 (2009).

45. Yang, J., An Introduction to the Theory of Piezoelectricity, 9th Ed., Springer, Lincoln (2005).

46. Fox, R.W., Pritchard, P.J., and McDonald, A.T., Introduction to Fluid Mechanics, 4th Ed., Wiley, New York, USA (2008).

47. Paidoussis, M.P., Misra, A.K., and Chan, S.P. "Dynamics and stability of coaxial cylindrical shells conveying viscous fluid", J. Appl. Mech-T., ASME., 52(2),pp. 389-396 (1985).

48. Karniadakis, G., Beskok, A., and Aluru, N., Micro Flows Nanoflows: Fundamentals and Simulation, Springer-Verlag (2005).

49. Rashidi, V., Mirdamadi, H.R., and Shirani, E. "A novel model for vibrations of nanotubes conveying nanoflow", Comput. Mater. Sci., 51, pp. 347-352 (2012).

50. Shokouhmand, H., Isfahani, A.H.M., and Shirani, E. "Friction and heat transfer coefficient in micro and nano channels filled with potous media for wide range of Knudsen number", Int. Comm. Heat Mass, 37, pp. 890-894 (2010).

51. Irschik, H. and Holl, H. "The equations of Lagrange written for a non-material volume", Acta Mech, 153, pp. 231-248 (2002).

52. Yang, J., Ke, L.L., and Kitipornchai, S. "Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", Physica E, 42, pp. 1727-1735 (2010).

53. Bolotin, V.V., The Dynamic Stability of Elastic Systems, Holden-Day, Inc, San Francisco, USA (1964).

54. Amabili, M. and Graziera, R. "Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass. Part ii: Shells containing or immersed in axial flow", J. Fluid. Struct., 16(1), pp.31-51 (2002).

55. Mohammadi, K., Rajabpour, A., and Ghadiri, M. "Calibration of nonlocal strain gradient shell model for vibration analysis of a CNT conveying viscous fluid using molecular dynamics simulation", Comp. Mater. Sci., 148, pp. 104-115 (2018).

Transactions on Mechanical Engineering (B)

March and April 2020Pages 730-744