Non-linear vibrations and stability analysis of tensioned pipes conveying fluid with variable velocity

In this study, the non-linear transverse vibrations of highly tensioned pipes with vanishing flexural stiffness and conveying fluid with variable velocity are investigated. The pipe is on fixed supports and the immovable end conditions result in the extension of the pipe during vibration and hence introduce further non-linear terms to the equation of motion. The velocity is assumed to be a harmonic function about a mean velocity. These systems experience a Coriolis acceleration component which renders such systems gyroscopic. The equation of motion is solved analytically by direct application of the method of multiple scales (a perturbation technique). Principal parametric resonance cases are investigated in detail. Non-linear frequencies are derived depending on amplitude. For frequencies close to two times the natural frequency, stability and bifurcations of steady-state solutions are analyzed. For frequencies close to zero, it is shown that the amplitudes are bounded in time.