Application of the differential transformation method to vibration analysis of pipes conveying fluid. (English) Zbl 1346.74038
Summary: In this paper, a relatively new semi-analytical method, called differential transformation method (DTM), is generalized to analyze the free vibration problem of pipes conveying fluid with several typical boundary conditions. The natural frequencies and critical flow velocities are obtained using the DTM. The results are compared with those predicted by the differential quadrature method and with other results reported in the literature. It is demonstrated that the DTM has high precision and computational efficiency in the vibration analysis of pipes conveying fluid.
MSC:
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
References:
| [1] | Païdoussis, M. P.; Li, G. X., Pipes conveying fluid: a model dynamical problem, J. Fluids Struct., 7, 137-204 (1993) |
| [2] | Rinaldi, S.; Prabhakar, S.; Vengallator, S.; Païdoussis, M. P., Dynamics of microscale pipes containing internal fluid flow: damping, frequency shift, and stability, J. Sound Vib., 329, 1081-1088 (2010) |
| [3] | Whitby, M.; Quirke, N., Fluid flow in carbon nanotubes and nanopipes, Nat. Nanotechnol., 2, 87-94 (2007) |
| [4] | Wang, L., Vibration and instability analysis of tubular nano- and micro-beams conveying fluid using nonlocal elastic theory, Physica E, 41, 1835-1840 (2009) |
| [5] | Wang, L., Size-dependent vibration characteristics of microtubes conveying fluid, J. Fluids Struct., 26, 675-684 (2010) |
| [6] | Païdoussis, M. P., Fluid-Structure Interactions: Slender Structures and Axial Flow, vol. 1 (1998), Academic Press: Academic Press London |
| [7] | Païdoussis, M. P.; Issid, N. T., Dynamic stability of pipes conveying fluid, J. Sound Vib., 33, 267-294 (1974) |
| [8] | Païdoussis, M. P., Flow-induced instabilities of cylindrical structures, Appl. Mech Rev., 40, 163-175 (1987) |
| [9] | Païdoussis, M. P.; Semler, C., Nonlinear and chaotic oscillations of a constrained cantilevered pipe conveying fluid: a full nonlinear analysis, Nonlinear Dyn., 4, 655-670 (1993) |
| [10] | Sarkar, A.; Païdoussis, M. P., A cantilever conveying fluid: coherent modes versus beam modes, Int. J. Non-Linear Mech., 39, 467-481 (2004) · Zbl 1348.76111 |
| [11] | Païdoussis, M. P.; Sarkar, A.; Semler, C., A horizontal fluid-conveying cantilever: spatial coherent structures, beam modes and jumps in stability diagram, J. Sound Vib., 280, 141-157 (2005) |
| [12] | Jin, J. D.; Song, Z. Y., Parametric resonances of supported pipes conveying pulsating fluid, J. Fluids Struct., 20, 763-783 (2005) |
| [13] | Wang, L., A further study on the non-linear dynamics of simply supported pipes conveying pulsating fluid, Int. J. Non-Linear Mech., 44, 115-121 (2009) |
| [14] | Qian, Q.; Wang, L.; Ni, Q., Instability of simply supported pipes conveying fluid under thermal loads, Mech. Res. Commun., 36, 3, 413-417 (2009) · Zbl 1258.74117 |
| [15] | Wang, L.; Ni, Q., In-plane vibration analyses of curved pipes conveying fluid using the generalized differential quadrature rule, Comput. Struct., 86, 1-2, 133-139 (2008) |
| [16] | Ni, Q.; Wang, L.; Qian, Q., Bifurcations and chaotic motions of a curved pipe conveying fluid with nonlinear constraints, Comput. Struct., 84, 708-717 (2006) |
| [17] | Pramila, A., On the gyroscopic terms appearing when the vibration of fluid conveying pipe is analyzed using the FEM, J. Sound Vib., 105, 515-516 (1986) |
| [18] | Zhang, Y. L.; Gorman, D. G.; Reese, J. M., A finite element method for modelling the vibration of initially tensioned thin-walled orthotropic cylindrical tubes conveying fluid, J. Sound Vib., 245, 93-112 (2001) · Zbl 1237.74140 |
| [19] | Pramila, A.; Laukkanen, J.; Liukkonen, S., Dynamics and stability of short fluid-conveying Timoshenko element pipes, J. Sound Vib., 144, 421-425 (1991) |
| [20] | Olson, L. G.; Jamison, D., Application of a general purpose finite element method to elastic pipes conveying fluid, J. Fluids Struct., 11, 207-222 (1997) |
| [21] | Kuiper, G. L.; Metrikine, A. V., Dynamic stability of a submerged, free-hanging riser conveying fluid, J. Sound Vib., 280, 1051-1065 (2005) |
| [22] | Zhou, J. K., Differential Transformation and Its Applications for Electrical Circuits (1986), Huazhong University Press: Huazhong University Press Wuhan, China |
| [23] | Chen, C. K.; Ho, S. H., Application of differential transformation to eigenvalue problems, Appl. Math. Comput., 79, 173-188 (1996) · Zbl 0879.34077 |
| [24] | Chen, C. K.; Ho, S. H., Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform, Int. J. Mech. Sci., 41, 1339-1356 (1999) · Zbl 0945.74029 |
| [25] | Mei, C., Application of differential transformation technique to free vibration analysis of a centrifugally stiffened beam, Comput. Struct., 86, 1280-1284 (2008) |
| [26] | Balkaya, M.; Kaya, M. O.; Saglamer, A., Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method, Arch. Appl. Mech., 79, 135-146 (2009) · Zbl 1177.74179 |
| [27] | Chen, C. K.; Chen, S. S., Application of the differential transformation method to a non-linear conservative system, Appl. Math. Comput., 154, 431-441 (2004) · Zbl 1134.65353 |
| [28] | Odibat, Z. M.; Bertelle, C.; Aziz-Alaoui, M. A.; Duchamp, G. H.E., A multi-step differential transform method and application to non-chaotic or chaotic systems, Comput. Math. Appl., 59, 1462-1472 (2010) · Zbl 1189.65170 |
| [29] | Zou, L.; Wang, Z.; Zong, Z., Generalized differential transform method to differential-difference equation, Phys. Lett. A, 373, 4142-4151 (2009) · Zbl 1234.35235 |
| [30] | Chen, Y. F.; Chai, Y. H.; Li, X.; Zhou, J., An extraction of the natural frequencies and mode shapes of marine risers by the method of differential transformation, Comput. Struct., 87, 1384-1393 (2009) |
| [31] | Chen, S. S.; Chen, C. K., Application of the differential transformation method to the free vibrations of strongly non-linear oscillators, Nonlinear Anal. RWA, 10, 881-888 (2009) · Zbl 1167.70328 |
| [32] | Kurnaz, A.; Oturanç, G.; Kiris, M. E., \(n\)-Dimensional differential transformation method for solving PDEs, Int. J. Comput. Math., 82, 369-380 (2005) · Zbl 1065.35011 |
| [33] | Thomson, W. T., Theory of Vibration with Applications (1988), Unwin Hyman Ltd.: Unwin Hyman Ltd. London |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
