The Analysis of Nonlinear Vibrations of Top-Tensioned Cantilever Pipes Conveying Pressurized Steady Two-Phase Flow under Thermal Loading

Abstract

This paper studied the nonlinear vibrations of top-tensioned cantilevered pipes conveying pressurized steady two-phase flow under thermal loading. The coupled axial and transverse governing partial differential equations of motion of the system were derived based on Hamilton's mechanics, with the centerline assumed to be extensible. Using the multiple-scale perturbation technique, natural frequencies, mode shapes, and first order approximate solutions of the steady-state response of the pipes were obtained. The multiple-scale assessment reveals that at some frequencies the system is uncoupled, while at some frequencies a 1:2 coupling exists between the axial and the transverse frequencies of the pipe. Nonlinear frequencies versus the amplitude displacement of the cantilever pipe, conveying two-phase flow at super-critical mixture velocity for the uncoupled scenario, exhibit a nonlinear hardening behavior; an increment in the void fractions of the two-phase flow results in a reduction in the pipe's transverse vibration frequencies and the coupled amplitude of the system. However, increases in the temperature difference, pressure, and the presence of top tension were observed to increase the pipe's transverse vibration frequencies without a significant change in the coupled amplitude of the system.

Nonlinear vibrations of a cantilevered pipe conveying pulsating two phase flow

“…For the purpose of this analysis, two novel driving functions 1 and 2 are introduced to pulsate the two phases such that their velocities fluctuate harmonically at frequencies (Ω 1 and Ω 2 ) about constant mean values ( 0 1 and 0 2 ). This differs from the steady velocities considered in [24]. Precisely, the driving functions are expressed as;…”

“…Adopting the equations of motion of an extensible pipe conveying multi-phase flow as obtained by [24] (…”

“…( 1 ) = 0 (25) Adopting the assumed solutions in equation 28, the modulation equations are formed. With 'R' the real parts and 'I' the imaginary parts of M and N. The stationary solutions are obtained as: The solution of equation (24) will produce four roots of ( 1 ): With the stability condition fulfilled, the particular solution of equation (24) away from internal resonance condition is obtained as:…”

“…In addition, the critical velocity is observed to be increasing as the void fraction of the two-phase flow increases. Adegoke and Oyediran [24] studied the nonlinear vibrations of top-tensioned cantilevered pipes conveying pressurized steady two-phase flow under thermal loading. The multiple-scale assessment reveals that at some frequencies the system is uncoupled, while at other frequencies a 1:2 coupling exists between the axial and the transverse frequencies of the pipe.…”

Numerical Analysis on Flow Characteristics of Gas-liquid Two-phase Flow in a Vertical Pipe with Downward Stream

“…An investigation conducted by Adegoke and Oyediran [13] included the nonlinear dynamics behavior of cantilevered pipes that included the gas-liquid twophase flow. Through their studies, they applied different scales of methods to provide the axial and transverse vibrations.…”