Nonlinear Analysis of Flow-induced Vibration in Fluid-conveying Structures using Differential Transformation Method with Cosine-Aftertreatment Technique
AbstractIn this work, analytical solutions are provided to the nonlinear equations arising in thermal and flow-induced vibration in fluid-conveying structures using Galerkin-differential transformation method with cosine aftertreatment technique. From the analysis, it was established that increase of the length and aspect ratio of the fluid-conveying structures result in decrease the nonlinear vibration frequencies of the structure while increase in the fluid-flow velocity causes increase in nonlinear vibration frequencies of the structures. Also, increase in the slip parameter leads to decrease in the frequency of vibration of the structure and the critical velocity of the conveyed fluid while increase in the slip parameter leads to decrease in the dimensionless frequency ratio of vibration of the structure. As the Knudsen number increases, the bending stiffness of the nanotube decreases and in consequent, the critical continuum flow velocity decreases as the curves shift to the lowest frequency zone. Good agreement are established when the results of the differential transformation method are compared with the results of the numerical method and exact analytical method for the non-linear and linear models, respectively.
 Benjamin, T. B., “Dynamics of a System of Articulated Pipes Conveying Fluid”, I. Theory. Proc. R. Soc. A Vol. 261, pp. 487–499, (1961). 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). Holmes, P. J., “Pipe Supported at Both Ends Cannot Flutter”, Journal of Applied Mechanics, Vol. 45, pp. 669-672, (1978). 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, pp. 577-599, (1994). Paidoussis, M. P., “Dynamics of Flexible Slender Cylinders in Axial Flow”, Journal of Fluid Mechanics, Vol. 26, pp. 717-736, (1966). Paidoussis, M. P., and Deksnis, E. B., “Articulated Models of Cantilevers Conveying Fluid, the Study of Paradox”, Journal of Mechanical Engineering Science, Vol. 12, pp. 288-300, (1970). Rinaldi, S., Prabhakar, S., Vengallatore, S., and 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). 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. 314–322, (2013). 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. 955–962, (2010). Ahangar, S., Rezazadeh, G., Shabani, R., Ahmadi, G., and Toloei, A., “On the Stability of a Microbeam Conveying Fluid Considering Modified Couple Stress Theory”, Int. Journal Mech. Vol. 7, pp. 327–342, (2011). Yin, L., Qian, Q., and Wang, L., “Strain Gradient Beam Model for Dynamics of Microscalepipes Conveying Fluid”, Appl. Math. Model. Vol. 35, pp. 2864–2873, (2011). Sahmani, S., Bahrami, M., and 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. 219–230, (2014). Akgoz, B., and Civalek, O., “Buckling Analysis of Functionally Graded Microbeams Based on the Strain Gradient Theory”, Acta Mech. Vol. 224, pp. 2185–2201, (2013). Zhao, J., Zhou, S., Wang, B., and Wang, X., “Nonlinear Microbeam Model Based on Strain Gradient Theory”, Appl. Math. Model. Vol. 36, pp. 2674–2686, (2012). Kong, S. L., Zhou, S. J., Nie, Z. F., and Wang, K., “Static and Dynamic Analysis of Micro- beams Based on Strain Gradient Elasticity Theory”, Int. J. Eng. Sci. Vol. 47, pp. 487–498, (2009). Setoodeh, R., and Afrahhim, S., “Nonlinear Dynamic Analysis of FG Micro-pipes Conveying Fluid Based on Strain Gradient Theory”, Composite Structures, Vol. 116, pp. 128-135, (2014). 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, No. 9, pp. 1326–1336, (2005). Yan, Y., Wang, W.Q., and 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). 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). Yang, H.K., and Wang, X., “Bending Stability of Multi-wall Carbon Nanotubes Embedded in an Elastic Medium”, Modeling and Simulation in Materials Sciences and Engineering, Vol. 14, pp. 99–116, (2006). Yoon, J., Ru, C.Q., and Mioduchowski, A., “Vibration of an Embedded Multiwall Carbon Nanotube”, Composites Science and Technology, Vol. 63, No. 11, pp. 1533–1542, (2003). 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, No. 10, pp. 982–985, (2009). Zhang, Y., Liu, G., and Han, X., “Transverse Vibration of Double-walled Carbon Nanotubes under Compressive Axial Load”, Applied Physics Letter A, Vol. 340, No. 1-4, pp. 258–266, (2005).GhorbanpourArani, A., Zarei, M.S., Mohammadimehr, M., Arefmanesh, A., and 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). 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). Zhou, J.K., “Differential Transformation and its Applications for Electrical Circuits”, Huazhong University Press, Wuhan, China,(1986). Venkatarangan, S.N., and Rajakshmi, K., “A Modification of Adomian’s Solution for Nonlinear Oscillatory Systems”, Comput. Math. Appl. Vol. 29, pp. 67-73, (1995). Jiao, Y.C., Yamamoto, Y., Dang, C., and Hao, Y., “An Aftertreatment Technique for Improving the Accuracy of Adomian’s Decomposition Method”, Comput. Math. Appl. Vol. 43, pp. 783-798, (2002). Elhalim, A., and Emad, E., “A New Aftertratment Technique for Differential Transformation Method and its Application to Non-linear Oscillatory System”, International Journal of Non-linear Science, Vol. 8, No. 4, pp. 488-497, (2009). Eringen, A.C., “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves”, Journal of Applied Physics, Vol. 54, No. 9, pp. 4703–4710, (1983). Eringen, A.C., “Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves”, International Journal of Engineering Science, Vol. 10, No. 5, pp. 425–435, (1972). Eringen, A.C., and Edelen, D.G.B., “On Nonlocal Elasticity”, International Journal of Engineering Science, Vol. 10, No. 3, pp. 233–248, (1972). Eringen, A.C., “Nonlocal Continuum Field Theories”, Springer, New York, (2002). 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).