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.

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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 centreline assumed to be extensible. Multiple scale perturbation method was used to resolve the governing equations, which resulted to an analytical approach for assessing the natural frequency, mode shape and the nonlinear coupled axial and transverse steady state response of the pipe. The analytical 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 behaviour, an increment in the void fractions of the two-phase flow resulted to a reduction in the pipe's transverse vibration frequencies and the coupled amplitude of the system. However, increasing 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.

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This work studied the nonlinear transverse vibrations of a cantilevered pipe conveying pulsatile two-phase flow. Internal flow induced parametric resonance is expected because of the time varying velocity of the conveyed fluid. This unsteady behaviour of the conveyed two-phase flow is considered in the governing equation as time dependent individual velocities with the harmonically varying components fluctuating about the constant mean velocities. Method of multiple scales analysis is adopted to study the nonlinear parametric resonance of dynamics of the cantilevered pipe. Contrary to the dynamics of pulsating single-phase flow, the assessment shows that if the frequencies of pulsation of the two phases are close, both can resonate with the pipe's transverse or axial frequencies together and both can also independently resonate with the pipe's transverse or axial frequencies distinctively. For the planar dynamics when only transverse frequencies are resonated, in the absence of internal resonance, numerical results show that the system exhibits softening nonlinear behavior. At post critical flow conditions, the system oscillates between subcritical and supercritical pitchfork bifurcation to simulate the nonlinear Mathieu's equation. However, in the presence of internal resonance, a nonlinear anti-resonance property is developed. Hence, the overall dynamics is quasi-periodic.