Reduced-order modeling and vibration transfer analysis of a fluid-delivering branch pipeline that consider fluid–solid interactions

doi:10.1007/s11465-024-0781-7

2024, Vol. 19

Issue (2) : 10 https://doi.org/10.1007/s11465-024-0781-7

Reduced-order modeling and vibration transfer analysis of a fluid-delivering branch pipeline that consider fluid–solid interactions

Wenhao JI^1,², Hongwei MA^1,², Wei SUN^1,²(

), Yinhang CAO³

¹. School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
². Key Laboratory of Vibration and Control of Aero-Propulsion Systems (Ministry of Education), Northeastern University, Shenyang 110819, China
³. College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China

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Abstract

The efficient dynamic modeling and vibration transfer analysis of a fluid-delivering branch pipeline (FDBP) are essential for analyzing vibration coupling effects and implementing vibration reduction optimization. Therefore, this study proposes a reduced-order dynamic modeling method suitable for FDBPs and then analyzes the vibration transfer characteristics. For the modeling method, the finite element method and absorbing transfer matrix method (ATMM) are integrated, considering the fluid–structure coupling effect and fluid disturbances. The dual-domain dynamic substructure method is developed to perform the reduced-order modeling of FDBP, and ATMM is adopted to reduce the matrix order when solving fluid disturbances. Furthermore, the modeling method is validated by experiments on an H-shaped branch pipeline. Finally, transient and steady-state vibration transfer analyses of FDBP are performed, and the effects of branch locations on natural characteristics and vibration transfer behavior are analyzed. Results show that transient vibration transfer represents the transfer and conversion of the kinematic, strain, and damping energies, while steady-state vibration transfer characteristics are related to the vibration mode. In addition, multiple-order mode exchanges are triggered when branch locations vary in frequency-shift regions, and the mode-exchange regions are also the transformation ones for vibration transfer patterns.

Keywords fluid-delivering branch pipeline vibration transfer analysis reduced-order modeling fluid–solid interactions finite element method absorbing transfer matrix method

Corresponding Author(s): Wei SUN

Just Accepted Date: 05 January 2024 Issue Date: 05 June 2024

Cite this article:

Wenhao JI,Hongwei MA,Wei SUN, et al. Reduced-order modeling and vibration transfer analysis of a fluid-delivering branch pipeline that consider fluid–solid interactions[J]. Front. Mech. Eng., 2024, 19(2): 10.

URL:

https://academic.hep.com.cn/fme/EN/10.1007/s11465-024-0781-7
https://academic.hep.com.cn/fme/EN/Y2024/V19/I2/10

Fig.1 Branch pipeline systems used in aero-engine and aerospace engineering: (a) aero-engine pipelines and (b) aerospace engine pipelines.

Fig.2 Absorbing transfer matrix method (ATMM)–finite element method (FEM) and FEM for fluid-delivering branch pipeline (FDBP): (a) dynamic modeling process of FDBP by using ATMM–FEM and (b) FEM of FDBP. FVAP: Fluid velocity and pressure.

Fig.3 Reduced-order modeling for a multibranch pipeline by absorbing transfer matrix method.

Fig.4 Procedure for the resonant frequency and vibration mode solution by using absorbing transfer matrix method (ATMM)–finite element method (FEM). FVAP: Fluid velocity and pressure.

Fig.5 Response solution by using the virtual force iteration method.

Domain	Density/(kg·m⁻³)	Elastic modulus/Pa	Poisson’s ratio	Velocity/(m·s⁻¹)	Pressure/Pa
Solid	$7800$	$2.04 × 1011$	0.285	?	?
Fluid (air)	$1.293$	?	?	0	0
Fluid (water)	$1000$	?	?	0	0

Tab.1 Parameters of the solid and fluid domains

Fig.6 H-shaped branch pipeline and experimental setup for modal test: (a) branch pipeline system and (b) experimental setup for modal test.

Fig.7 Frequency response functions of the (a) air-delivering and (b) water-delivering pipelines.

Tab.2 Natural frequencies of the branch pipeline

Fig.8 Velocity responses obtained by absorbing transfer matrix method: (a) air-delivering and (b) water-delivering pipelines.

Fluid medium	Stress response at point S1			Stress response at point S2
Fluid medium	$σ t$ /MPa	$σ s$ /MPa	$e σ = σ t − σ s σ t × 100 %$	$σ t$ /MPa	$σ s$ /MPa	$e σ = σ t − σ s σ t × 100 %$
Air	69.24	63.26	8.64	70.11	67.08	4.32
Water	64.32	63.55	1.20	76.12	68.52	9.98

Tab.3 Resonant stress responses at the measuring points

Fig.9 Experimental test rig for the response test.

Fig.10 Stress responses of the air-delivering pipeline at measuring points (a) S1 and (b) S2. Stress responses of the water-delivering pipeline at measuring points (c) S1 and (d) S2.

Fig.11 1st-order transient vibration transfer analysis of fluid-delivering branch pipeline. Vibration transfer laws at (a)

1 × 10 − 4

, (b)

2.6 × 10 − 3

, (c)

5.1 × 10 − 3

, and (d)

7.6 × 10 − 3 s

Fig.12 Energy transformation relationship in the 1st-order transient vibration transfer. Energy transformation relationship of (a) pipeline segments 1–3, (b) pipeline segment 2, (c) pipeline segments 4 and 5, and (d) pipeline system. CR: change rate.

Fig.13 Second-order transient vibration transfer analysis of fluid-delivering branch pipeline. Vibration transfer laws at (a)

1 × 10 − 4

and (b)

1.8 × 10 − 3 s

Fig.14 Energy transformation relationship in the second-order transient vibration transfer. Change rates of (a) strain, (b) kinematic, and (c) damping energies.

Fig.15 1st- and 2nd-order steady-state vibration transfer of fluid-delivering branch pipeline: (a) 1st-order steady-state vibration transfer law, (b) 2nd-order steady-state vibration transfer law, (c) 1st-order vibration mode, and (d) 2nd-order vibration mode.

Fig.16 Effects of branch location on natural frequencies: (a) the 1st- to 7th-order natural frequencies and (b) the 8th- to 13th-order natural frequencies.

Fig.17 Vibration mode exchanges in the frequency-shift regions: (a) 2nd- and 3rd-order vibration modes, (b) 6th- and 7th-order vibration modes, (c) 8th- and 9th-order vibration modes, and (d) 11th- and 12th-order vibration modes.

Fig.18 First-order steady-state vibration transfer behavior of fluid-delivering branch pipeline that corresponds to different branch locations. Steady-state vibration transfer characteristics when branch locations (

L b

) are (a)

0.08

, (b)

0.12

, (c)

0.27

, and (d)

0.30 m

Fig.19 Energy changes that correspond to different branch locations: (a) change rate (CR) of strain energy, (b) CR of kinematic energy, (c) CR of damping energy, (d)

W ˙ k ∗ − W ˙ s ∗

curves, (e)

W ˙ k − W ˙ d

curves, and (f)

W ˙ s − W ˙ d

curves.

Fig.20 1st-order steady-state vibration transfer characteristics of fluid-delivering branch pipeline under subcritical fluid velocity and pressure. Vibration transfer characteristics when (a) fluid velocity is 120 m/s and (b) fluid pressure is 30 MPa.

Fig.21 Energy changes of fluid-delivering branch pipeline under subcritical fluid velocity and pressure: (a) change rate (CR) of kinematic energy, (b) CR of damping energy, and (c)

W ˙ k − W ˙ d

curves.

Abbreviations
ATMM	Absorbing transfer matrix method
CR	Change rate
DOF	Degree of freedom
FDBP	Fluid-delivering branch pipeline
FEM	Finite element method
FVAP	Fluid velocity and pressure
ROM	Reduced-order modeling
SAM	Semi-analytical method
SIM	Structural intensity method
TMM	Transfer matrix method
Variables
A_f	Area of the fluid
$A ⌢$	Transfer matrix between the master and slave branches
b	Number of boundary DOFs
C_r, c	Damping matrices of the structure and fluid, respectively
f_x, f_y, f_z	External forces at a position in the x, y, and z directions, respectively
f	Natural frequency
G_c, G	Constraint matrix and transformation matrix, respectively
$i ~$	Number of internal DOFs
I_i(t)	Structural intensity response
$K p s$ , $k$ , $K c$ , K	Stiffness matrices of the pipeline, fluid, coupling element, and pipeline system, respectively
$K^$	Reduced stiffness matrix
M(n − 1)	General transfer matrices used in TMM
$M p s$ , $m$ , $M$	Mass matrices of the pipeline, fluid, and pipeline system, respectively
m_x, m_y, m_z	External moments at a position in the x, y, and z directions, respectively
$M^$	Reduced mass matrix
P_f	Fluid pressure
$Q^$	Reduced external force
$Q^(L, s)$	Transfer matrix used in TMM
$R^(L, s)$	External excitation vector used in TMM
$\| u ˙ ¯ \|$	Amplitudes of the velocity responses
$u ˙$ , $v ˙$ , $w ˙$	Velocity responses at a position in the x, y, and z directions, respectively
u_n	Node displacement response
$u ˙ ~$ , $σ ~$	Steady-state velocity and stress
V_f	Fluid velocity
$W ˙ k$ , $W ˙ s$ , $W ˙ d$	CRs of kinetic, strain, and damping energies, respectively
x	Reduced DOFs
$y (z^, t)$	Time-domain state vectors
$Θ ~ s$ , $Γ^s$ , $Ψ^s$ , $D^s$	Coefficient matrices used in TMM
$ϕ ˙ x$ , $ϕ ˙ y$ , $ϕ ˙ z$	Angular velocity responses at a position in the x, y, and z directions, respectively
$Φ^$	Frequency-domain state vectors
$\| σ ¯ \|$	Amplitudes of stress responses
σ	Stress response
ψ, γ	Phase angles of stress and velocity responses, respectively
ω	Angular frequency
Ξ	State matrix of the branch
ξ₁, ξ₂	First- and second-order damping ratios, respectively

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