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**

1. Gao, J. and Xu, B. Applications of nanomaterials

inside cells", Nano Today, 4(1), pp. 37{51 (2009).

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 lm 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

owing

uid", 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 Oshore Research and

Engineering National University of Singapore (2004).

8. Pellicano, F. and Amabili, M. Dynamic instability

and chaos of empty and

uid-lled 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

uid 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. Eect of aspect

ratio on the dynamic response of a

uid-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

uid-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

uid", 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

uid-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

uid-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 eects on dynamic stability of double-walled

boron nitride nanotubes conveying viscose

uid based

on nonlocal shell theory", Sci. Iran., 20(6), pp. 2356{

2374 (2013).

18. Ghorbanpour Arani, A., Khoddami Maraghi, Z., and

Haghparast, E. The

uid structure interaction eect

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

ow 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

uid", J. Sound. Vib., 33(3), pp.

267{294 (1974).

21. Panda, L.N. and Kar, R.C. Nonlinear dynamics of

a pipe conveying pulsating

uid with combination,

principal parametric and internal resonances", Journal

of Sound and Vibration, 309, pp. 375{406 (2008).

V. Atabakhshian and A. Shooshtari/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 730{744 743

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

uid", 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

uid with nonlocal eect", 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

uid",

Ocean. Eng., 77, pp. 12{22 (2014).

25. Wang, L. A further study on the non-linear dynamics

of simply supported pipes conveying pulsating

uid",

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 Padoussis, M.P. Fluidstructure

interaction for nonlinear response of shells

conveying pulsatile

ow", J. Sound. Vib., 371, pp.

252{276 (2016).

28. Tubaldi, E., Amabili, M., and Paidoussis, M.P. Nonlinear

dynamics of shells conveying pulsatile

ow with

pulse-wave propagation: Theory and numerical results

for a single harmonic pulsation", J. Sound Vib., 396,

pp. 217{245 (2017).

29. Rai-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 nite 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 dierential 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 dierent 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

uid", J. Appl. Mech-T., ASME., 52(2),

pp. 389{396 (1985).

48. Karniadakis, G., Beskok, A., and Aluru, N., Micro

Flows Nano Flows: Fundamentals and Simulation,

Springer-Verlag (2005).

49. Rashidi, V., Mirdamadi, H.R., and Shirani, E. A

novel model for vibrations of nanotubes conveying

nano

ow", Comput. Mater. Sci., 51, pp. 347{352

(2012).

50. Shokouhmand, H., Isfahani, A.H.M., and Shirani, E.

Friction and heat transfer coecient in micro and

nano channels lled 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).

744 V. Atabakhshian and A. Shooshtari/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 730{744

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

ow", 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

uid

using molecular dynamics simulation", Comp. Mater.

Sci., 148, pp. 104{115 (2018).

Transactions on Mechanical Engineering (B)

March and April 2020Pages 730-744