Orthogonality conditions and analytical response solutions of damped gyroscopic double-beam system: an example of pipe-in-pipe system

doi:10.1007/s10483-025-3247-6

摘要/Abstract

Abstract:

The double-beam system is a crucial foundational structure in industry, with extensive application contexts and significant research value. The double-beam system with damping and gyroscopic effects is termed as the damped gyroscopic double-beam system. In such systems, the orthogonality conditions of the undamped double-beam system are no longer applicable, rendering it impossible to decouple them in modal space using the modal superposition method (MSM) to obtain analytical solutions. Based on the complex modal method and state space method, this paper takes the damped pipe-in-pipe (PIP) system as an example to solve this problem. The concepts of the original system and adjoint system are introduced, and the orthogonality conditions of the damped PIP system are given in the state-space. Based on the derived orthogonality conditions, the transient and steady-state response solutions are obtained. In the numerical discussion section, the convergence and accuracy of the solutions are verified. In addition, the dynamic responses of the system under different excitations and initial conditions are studied, and the forward and reverse synchronous vibrations in the PIP system are discussed. Overall, the method presented in this paper provides a convenient way to analyze the dynamics of the damped gyroscopic double-beam system.

Key words: fluid-conveying pipe, transverse vibration, pipe-in-pipe (PIP) system, gyroscopic double-beam system, complex modal superposition method (MSM), analytical solution

中图分类号:

O321

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(5): 927-946.

Jinming FAN, Zhongbiao PU, Jie YANG, Xueping CHANG, Yinghui LI. Orthogonality conditions and analytical response solutions of damped gyroscopic double-beam system: an example of pipe-in-pipe system[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(5): 927-946.

图/表 13

Flow velocity	Mode	$ω n$	$ω − n$	$ω ˜ n$	$ω ˜ − n$
$u L = 1$	$n = 1$	11.909 9+0.026 6i	$−$ 11.909 9+0.026 6i	$−$ 11.909 9+0.026 6i	11.909 9+0.026 6i
	$n = 2$	23.231 6+0.117 0i	$−$ 23.231 6+0.117 0i	$−$ 23.231 6+0.117 0i	23.231 6+0.117 0i
	$n = 3$	41.244 2+0.065 3i	$−$ 41.244 2+0.065 3i	$−$ 41.244 2+0.065 3i	41.244 2+0.065 3i
	$n = 4$	59.864 0+0.078 8i	$−$ 59.864 0+0.078 8i	$−$ 59.864 0+0.078 8i	59.864 0+0.078 8i
$u L = 3$	$n = 1$	9.290 6+0.029 6i	$−$ 9.290 6+0.029 6i	9.290 6+0.029 6i	$−$ 9.290 6+0.029 6i
	$n = 2$	22.533 7+0.110 5i	$−$ 22.533 7+0.110 5i	22.533 7+0.110 5i	$−$ 22.533 7+0.110 5i
	$n = 3$	37.495 5+0.068 7i	$−$ 37.495 5+0.068 7i	37.495 5+0.068 7i	$−$ 37.495 5+0.068 7i
	$n = 4$	59.834 8+0.077 0i	$−$ 59.834 8+0.077 0i	59.834 8+0.077 0i	$−$ 59.834 8+0.077 0i

参考文献 30

[1]	LIU, S. B. and YANG, B. G.A closed-form analytical solution method for vibration analysis of elastically connected double-beam systems. Composite Structures, 212, 598–608 (2019)
[2]	HAN, F., DAN, D. H., and DENG, Z. C.A dynamic stiffness-based modal analysis method for a double-beam system with elastic supports. Mechanical Systems and Signal Processing, 146, 106978 (2021)
[3]	MATIN-NIKOO, H., BI, K., and HAO, H.Passive vibration control of cylindrical offshore components using pipe-in-pipe (PIP) concept: an analytical study. Ocean Engineering, 142, 39–50 (2017)
[4]	KUANG, Y. D., HE, X. Q., CHEN, C. Y., and LI, G. Q.Analysis of nonlinear vibrations of double-walled carbon nanotubes conveying fluid. Computational Materials Science, 45, 875–880 (2009)
[5]	VU, H. V., ORDÓÑEZ, A. M., and KARNOPP, B. H.Vibration of a double-beam system. Journal of Sound and Vibration, 229, 807–822 (2000)
[6]	LI, Y. X., HU, Z. J., and SUN, L. Z.Dynamical behavior of a double-beam system interconnected by a viscoelastic layer. International Journal of Mechanical Sciences, 105, 291–303 (2016)
[7]	LI, Y. X. and SUN, L. Z.Transverse vibration of an undamped elastically connected double-beam system with arbitrary boundary conditions. Journal of Engineering Mechanics, 142, 04015070 (2016)
[8]	LI, Y. X., XIONG, F., XIE, L. Z., and SUN, L. Z.State-space approach for transverse vibration of double-beam systems. International Journal of Mechanical Sciences, 189, 105974 (2021)
[9]	PALMERI, A. and ADHIKARI, S.A Galerkin-type state-space approach for transverse vibrations of slender double-beam systems with viscoelastic inner layer. Journal of Sound and Vibration, 330, 6372–6386 (2011)
[10]	GUO, Y., ZHU, B., ZHAO, X., CHEN, B., and LI, Y. H.Dynamic characteristics and stability of pipe-in-pipe system conveying two-phase flow in thermal environment. Applied Ocean Research, 103, 102333 (2020)
[11]	DENG, H., CHEN, K., CHENG, W., and ZHAO, S.Vibration and buckling analysis of double-functionally graded Timoshenko beam system on Winkler-Pasternak elastic foundation. Composite Structures, 160, 152–168 (2017)
[12]	ZHAO, X., CHEN, B., LI, Y. H., ZHU, W. D., NKIEGAING, F. J., and SHAO, Y. B.Forced vibration analysis of Timoshenko double-beam system under compressive axial load by means of Green's functions. Journal of Sound and Vibration, 464, 115001 (2020)
[13]	CHEN, B., LIN, B. C., ZHAO, X., ZHU, W. D., YANG, Y. K., and LI, Y. H.Closed-form solutions for forced vibrations of a cracked double-beam system interconnected by a viscoelastic layer resting on Winkler-Pasternak elastic foundation. Thin-Walled Structures, 163, 107688 (2021)
[14]	CHEN, B., LIN, B. C., LI, Y. H., and TANG, H. P.Exact solutions for steady-state dynamic responses of a laminated composite double-beam system interconnected by a viscoelastic layer in hygrothermal environments. Composite Structures, 268, 113939 (2021)
[15]	ONISZCZUK, Z.Forced transverse vibrations of an elastically complex simply supported double-beam system. Journal of Sound and Vibration, 264, 273–286 (2003)
[16]	WU, Y. X. and GAO, Y. F.Dynamic response of a simply supported viscously damped double-beam system under a moving oscillator. Journal of Sound and Vibration, 384, 194–209 (2016)
[17]	ZHAO, X., MENG, S. Y., ZHU, W. D., ZHU, Y. L., and LI, Y. H.A closed-form solution of forced vibration of a double-curved-beam system by means of the Green's function method. Journal of Sound and Vibration, 561, 117812 (2023)
[18]	LI, J. and HUA, H. X.Spectral finite element analysis of elastically connected double-beam systems. Finite Elements in Analysis and Design, 43, 1155–1168 (2007)
[19]	ZHAO, X. Z. and CHANG, P.Free and forced vibration of double beam with arbitrary end conditions connected withviscoelastic layer and discrete points. International Journal of Mechanical Sciences, 209, 106707 (2021)
[20]	LANGTHJEM, M. A. and SUGIYAMA, Y.Vibration and stability analysis of cantilevered two-pipe systems conveying different fluids. Journal of Fluids and Structures, 13, 251–268 (1999)
[21]	ZHANG, Y. F., YAO, M. H., ZHANG, W., and WEN, B. C.Dynamical modeling and multi-pulse chaotic dynamics of a cantilevered pipe conveying pulsating fluid in parametric resonance. Aerospace Science and Technology, 68, 441–453 (2017)
[22]	LIANG, F., YANG, X. D., QIAN, Y. J., and ZHANG, W.Transverse free vibration and stability analysis of spinning pipes conveying fluid. International Journal of Mechanical Sciences, 137, 195–204 (2018)
[23]	GUO, Y., LI, J. A., ZHU, B., and LI, Y. H.Flow-induced instability and bifurcation in cantilevered composite double-pipe systems. Ocean Engineering, 258, 111825 (2022)
[24]	LÜ, L., HU, Y. J., WANG, X. L., LING, L., and LI, C. G.Dynamical bifurcation and synchronization of two nonlinearly coupled fluid-conveying pipes. Nonlinear Dynamics, 79, 2715–2734 (2014)
[25]	NI, Q., ZHANG, Z. L., WANG, L., QIAN, Q., and TANG, M.Nonlinear dynamics and synchronization of two coupled pipes conveying pulsating fluid. Acta Mechanica Solida Sinica, 27, 162–171 (2014)
[26]	LIU, Z. Y., JIANG, T. L., WANG, L., and DAI, H. L.Nonplanar flow-induced vibrations of a cantilevered PIP structure system concurrently subjected to internal and cross flows. Acta Mechanica Sinica, 35, 1241–1256 (2019)
[27]	CHANG, X. P., FAN, J. M., HAN, D. Z., CHEN, B., and LI, Y. H.Stability and modal conversion phenomenon of pipe-in-pipe structures with arbitrary boundary conditions by means of Green's functions. International Journal of Structural Stability and Dynamics, 22, 2250034 (2022)
[28]	CHANG, X. P., FAN, J. M., QU, C. J., and LI, Y. H.Coupling vibration of a composite pipe-in-pipe structure subjected to gas-liquid mixed transport by means of Green's functions. Mechanics of Advanced Materials and Structures, 30, 1604–1623 (2022)
[29]	CHUNG, C. H. and KAO, I. I.Modeling of axially moving wire with damping: eigenfunctions, orthogonality, and applications in slurry wiresaws. Journal of Sound and Vibration, 330, 2947–2963 (2011)
[30]	ZHANG, H. J., MA, J., DING, H., and CHEN, L. Q.Vibration of axially moving beam supported by viscoelastic foundation. Applied Mathematics and Mechanics (English Edition), 38(2), 161–172 (2017) https://doi.org/10.1007/s10483-017-2170-9