Thermo-mechanical nonlinear vibration analysis of fluid-conveying structures subjected to different boundary conditions using Galerkin-Newton-Harmonic balancing method

Document Type : Research Paper



2 University of Lagos, Nigeria.

3 Centre for Space Transport and Propulsion, National Space Research and Development Agency, Federal Ministry of Science and Technology, FCT, Abuja, Nigeria.


The development of mathematical models for describing the dynamic behaviours of fluid conveying pipes, micro-pipes and nanotubes under the influence of some thermo-mechanical parameters results into nonlinear equations that are very difficult to solve analytically. In cases where the exact analytical solutions are presented either in implicit or explicit forms, high skills and rigorous mathematical analyses were employed. It is noted that such solutions do not provide general exact solutions. Inevitably, comparatively simple, flexible yet accurate and practicable solutions are required for the analyses of these structures. Therefore, in this study, approximate analytical solutions are provided to the nonlinear equations arising in flow-induced vibration of pipes, micro-pipes and nanotubes using Galerkin-Newton-Harmonic Method (GNHM). The developed approximate analytical solutions are shown to be valid for both small and large amplitude oscillations. The accuracies and explicitness of these solutions were examined in limiting cases to establish the suitability of the method.


Main Subjects

[1] Iijima, S. Nature, London, Vol. 354, pp. 56(1991), 56–58.
[2] Benjamin. T. B. Dynamics of a system of articulated pipes conveying fluid. I. Theory. Proc R Soc A Vol. 261:pp. 487–99, 1961.
[3] Holmes, P. J. Pipe Supported at Both Ends cannot Flutter. Journal of Applied Mechanics. Vol. 45, pp. 669-672, 1978
[4] Housner, G. W., Dodds, H. L. and Runyan. H. Effect of High Velocity Fluid Flow in the Bending Vibration and Static Divergence of Simply Supported Pipes. National Aeronautics and Space Administration Report NASA TN D- 2870, June 1965.
[5] Naguleswaran, S. and Williams, C. J. H., Lateral Vibration of a Pipe Conveying Fluid. Journal of Mechanical Engineering Science.Vol. 10, pp. 228- 238, 1968.
[6] Paidoussis, M. P., Dynamics of Flexible Slender Cylinders in Axial Flow. Journal of Fluid Mechanics. Vol. 26, pp. 717-736, 1966
[7] Paidoussis, M. P. and Deksnis, E. B. Articulated Models of Cantilevers Conveying Fluid: the Study of Paradox. The Journal of Mechanical Engineering Science. Vol. 12, pp. 288-300, 1970.
[8] Paidoussis, M. P. and Issid, T. D. Dynamics Stability of Pipes Conveying Fluid. Journal of Sound and Vibration. Vol. 33, pp. 267-294, 1974.
[9] Paidoussis, M. P. and Laither, T. D. Dynamics of Timoshenko Beam Conveying Fluid. Journal of Mechanical Engineering Science, Vol. 18, pp. 210-220, 1976.
[10] Paidoussis, M. P., Luu, T. R. and Laither, B. E. Dynamics of Finite Length Tubular Beams Conveying Fluid. Journal of Sound and Vibration. Vol. 106 (2), pp. 311-331, 1986.
[11] Paidoussis, M. P. Fluid-structure interactions: slender structures and axial flow, vol. 1. London: Academic Press; 1998.
[12] Semler., C., Li, G. X. and Paidoussis, M. P. The Non-Linear Equations of Motion of Pipes Conveying Fluid. Journal of Sound and Vibration, Vol. 169, 577-599, 1999.
[13] Rinaldi, S., Prabhakar, S., Vengallatore, S. Paıdoussis, M. P. Dynamics of microscale pipes containing internal fluid flow: damping, frequency shift, and stability. J Sound Vib. Vol. 329, pp.1081–1088., 2010
[14] Akgoz, B. and Civalek, O. Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos. Struct., Vol. 98, pp. 314–22, 2013
[15] Wang L. Size-dependent vibration characteristics of fluid-conveying microtubes. J Fluid Struct. Vol. 26 pp. 675–84, 2010.
[16] Xia, W. and Wang, L.  Microfluid-induced vibration and stability of structures modeled as microscale pipes conveying fluid based on non-classical Timoshenko beam theory. Microfluid Nanofluid Vol. 9, pp. 955–62, 2010.
[17] Ahangar, S., Rezazadeh, G., Shabani, R., Ahmadi, G., Toloei, A. On the stability of a microbeam conveying fluid considering modified couple stress theory. Int. Journal Mech. Vol. 7, 327–42, 2011.
[18] Yin, L., Qian, Q. Wang. L. Strain gradient beam model for dynamics of microscale pipes conveying fluid. Appl Math Model. Vol. 35, pp. 2864–73, 2011.
[19] Sahmani, S. Bahrami, M, Ansari. R. Nonlinear free vibration analysis of functionally graded third-order shear deformable microbeams based on the modified strain gradient elasticity theory. Compos Struct. Vol. 110, pp. 219–30, 2014.
[20] Akgoz, B. and Civalek, O. Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech. Vol. 224, pp. 2185–201, 2013.
[21] Zhao, J., Zhou, S. Wang, B. Wang, X. Nonlinear microbeam model based on strain gradient theory. Appl Math Model. Vol. 36, 2674–86, 2012.
[22] Akgoz, B. and Civalek, O. Shear deformation beam models for functionally graded microbeams with new shear correction factors. Compos Struct. Vol.112: pp. 214–25, 2014.
[23] Kong, S. L. Zhou, S. J. , Nie,Z. K., Wang, K. Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci. Vol. 47, pp. 487–98, 2009.
[24] Asghari, M., Ahmadian, M., Kahrobaiyan, M., Rahaeifard, M. On the size-dependent behavior of functionally graded micro-beams. Mater Des. Vol. 31, pp. 2324–2329, 2010.
[25] Yang, T. Z., Ji, S., Yang, X. D., Fang, B. Microfluid-induced nonlinear free vibration of microtubes. Int J Eng Sci. Vol. 76, pp. 47–55, 2014.
[26] Ke, L.L. Wang, Y.S. Yang, J., Kitipornchai, S. Nonlinear free vibration of size dependent functionally graded microbeams. Int J Eng Sci. Vol. 50, pp. 256–67, 2010.
[27] Salamat-Talab, M., Shahabi, F., Assadi, A. Size dependent analysis of functionally graded microbeams using strain gradient elasticity incorporated with surface energy. Appl Math Model Vol. 37, pp. 507–26, 2013.
[28] Setoodeh, R., Afrahhim, S. Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory. Composite Structures, Vol. 116, pp. 128-135, 2014.
[29] Yoon, G., Ru,C. Q. and Mioduchowski, A. Vibration and instability of carbon nanotubes conveying fluid, Journal of Applied Mechanics, Transactions of the ASME, Vol. 65(9), pp. 1326–1336, 2005.
[30], Y. Wang, W. Q. Zhang, L. X. Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field, Journal of Applied Math and Modelling, Vol. 34, pp. 3422–3429, 2010
[31] Murmu, T. and Pradhan, S.C. Thermo-mechanical vibration of Single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory, Computational Material Science, Vol. 46 , pp. 854–859, 2009.
[32] Yang, H. K. Wang, X. Bending stability of multi-wall carbon nanotubes embedded in an elastic medium, Modeling and Simulation in Materials Sciences and Engineering, Vol. 14, 99, 99–116, 2006.
[33] Yoon, J., Ru, C. Q., Mioduchowski, A. Vibration of an embedded multiwall carbon nanotube, Composites Science and Technology, Vol. 63(11) , pp. 1533–1542, 2003.
[34] Chang, W. J. and Lee, H. L. Free vibration of a single-walled carbon nanotube containing a fluid flow using the Timoshenko beam model, Physics Letter A, Vol. 373 (10), pp. 982–985, 2009.
[35] Lu, P., Lee, H. P., Lu, C., Zhang, P. Q. Application of nonlocal beam models for carbon nanotubes, International Journal of Solids and Structures, Vol. 44(16), pp.5289–5300, 2007.
[36] Zhang, Y., Liu, G., Han, X. Transverse vibration of double-walled carbon nanotubes under compressive axial load, Applied Physics Letter A, Vol. 340 (1-4), pp. 258–266, 2005.
[37] Arani, G., Zarei, M. S., Mohammadimehr, M., Arefmanesh, A., Mozdianfard, M. R. The thermal effect on buckling analysis of a DWCNT embedded on the Pasternak foundation, Physica E, Vol. 43, pp. 1642–1648, 2011.
[38] Sobamowo, M. G. Thermal analysis of longitudinal fin with temperature-dependent properties and internal heat generation using Galerkin’s method of weighted residual. Applied Thermal Engineering, Vol. 99 pp. 1316–1330, 2016
[39] Zhou, J. K. Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press: Wuhan, China (1986)
[40] Ho, S. H. and Chen, C. K. Analysis of general elastically end restrained non-uniform beams using differential transform. Appl. Math. Modell. Vol. 22, pp. 219-234, 1998.
[41] Chen, C. K. and Ho, S. H. Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform. Int. J. Mechanical Sciences. Vol. 41, pp. 1339-1356, 1999.
[42] Chen, C. K. and Ho., S. H. Application of Differential Transformation to Eigenvalue Problems. Journal of Applied Mathematics and Computation, Vol. 79, pp. 173-188, 1996.
[43] Cansu, Ü. and Özkan, O. Differential Transform Solution of Some Linear Wave Equations with Mixed Nonlinear Boundary Conditions and its Blow up. Applied Mathematical Sciences Journal, Vol. 4(10), 467-475, 2010.
[44] Chang, S. H. and Chang. I. L. A New Algorithm for Calculating Two-dimensional Differential Transform of Nonlinear Functions. Journal of Applied Mathematics and Computation, Vol. 215, pp. 2486-2494, 2009.
[45] Chen, C. L. and Liu, Y. C. Two-dimensional Differential Transform for Partial Differential Equations. Journal of Applied Mathematics and Computation. Vol. 121, pp. 261-270, 2001.
[46] Jafari, H., M., Alipour, H. M. and H. Tajadodi. Two-dimensional Differential Transform Method for Solving Nonlinear Partial Differential Equations. International Journal of Research and Reviews in Applied Sciences, Vol. 2(1), pp. 47-52, 2010.
[47] Kaya, M. O. Free Vibration Analysis of a Rotating Timoshenko Beam by Differential Transform Method. Aircraft Engineering and Aerospace Technology Journal, Vol. 78(3), pp. 194-203, 2006.
[48] Jang, M. J., Chen, C. L. Liy, Y. C. On solving the initial-value problems using the differential transformation method. Appl. Math. Comput. Vol. 115, pp. 145-160, 2000.
[49] K¨oksal, M., and Herdem, S. Analysis of nonlinear circuits by using differential Taylor transform. Computers and Electrical Engineering. Vol. 28, pp.513-525, 2008.
[50] Hassan, I. H. A. H. Different applications for the differential transformation in the differential equations. Appl. Math. Comput. Vol.129, pp. 183-201, 2002.
[51] Ravi, A. S. V. Kanth, K. Aruna: Solution of singular two-point boundary value problems using differential transformation method. Phys. Lett. A. Vol. 372, pp. 4671-4673, 2008.
[52] Chen, C. K. and Ho S.H. Solving partial differential equations by two-dimensional differential transform method. Appl. Math. Comput. Vol. 41, pp. 171-179, 1999.
[53] Ayaz, F. On two-dimensional differential transform method. Appl. Math. Comput.Vol. 143, pp. 361-374, 2003.
[54] Ayaz, F. Solutions of the system of differential equations by differential transform method. Appl. Math. Comput.147: pp. 547-567, 2004.
[55] Chang, S. H. and Chang, I.L., A new algorithm for calculating one-dimensional differential transform of nonlinear functions. Appl. Math. Comput. Vol. 195, pp. 799-808, 2008.
[56] Momani, S. and Ert¨urk, V. S. Solutions of non-linear oscillators by the modified differential transform method. Computers and Mathematics with Applications. Vol. 55(4), 833-842, 2008.
[57] Momani, S., 2004. Analytical approximate solutions of non-linear oscillators by the modified decomposition method. Int. J. Modern. Phys. C, Vol. 15(7), pp. 967-979, 2004.
[58] El-Shahed, M. Application of differential transform method to non-linear oscillatory systems. Communic.Nonlin.Scien.Numer.Simul. Vol. 13, pp. 1714-1720, 2008.
[59] Venkatarangan, S. N., Rajakshmi, K. A modification of adomian’s solution for nonlinear oscillatory systems. Comput.Math. Appl. Vol. 29, pp. 67-73, 1995.
[60] Jiao, Y. C. Yamamoto, Y. C. Dang, C. Hao, C. An aftertreatment technique for improving the accuracy of Adomian’s decomposition method. Comput.Math. Appl. Vol. 43, pp. 783-798, 2002.
[61] Elhalim, A. and Emad, E. A new aftertratment technique for differential transformation method and its application to non-linear oscillatory system. Internation Journal of Non-linear Science, Vol. 8(4), pp. 488-497, 2009.
[62] Lai, S. K., Lim, C. W., Wu, B. S. Newton-harminic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators.Applied Math.Modeling, Vol. 33, pp. 852-866, 2009.
[63] Eringen, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, Journal of Applied Physics, Vol. 54 (9), pp. 4703–4710, 1983.
[64] Eringen, A. C. Linear theory of nonlocal elasticity and dispersion of plane waves, Inter- national Journal of Engineering Science, Vol. 10 (5), pp. 425–435, 1972.
[65] Eringen, A. C and D.G., B. Edelen, D. G. B. On nonlocal elasticity, International Journal of Engineering Science, Vol. 10 (3), pp. 233–248, 1972.
[66] Eringen, A. C. Nonlocal continuum field theories, Springer, New York 2002.
[67] Shokouhmand, H., Isfahani A. H. M. and Shirani, E. Friction and heat transfer coefficient in micro and nano channels with porous media for wide range of Knudsen number, International Communication in Heat and Mass Transfer, Vol. 37, pp. 890-894, 2010.