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

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 In such systems, the orthogonality conditions of the undamped double-beam system are no longer applicable, rendering it impossible to decouple them in modal space using the modal superposition method (MSM) to obtain analytical solutions.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

2010 MSC Number:

O321

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

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Figures/Tables 13

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Table 1

Eigenvalues of the original and adjoint systems of the damped PIP systems with different flow velocities"

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

Table 1

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