Research Of Parametric Resonances Of Supported Pipes

In this paper, the pinned-pinned and clamped-clamped pipes conveying fluid are selected as the researching object. The stability and parametric resonances of pipes is chiefly considered. And the effects of parameters, such as the mean fluid velocity, damping, mass ratio, external applied tension, on the extent of parametric instability regions and the nonlinear dynamics of the system are analyzed.The partial differential governing equation of nonlinear vibrations of supported pipes is solved analytically by means of direct application of the method of multiple scales. The closed-form equations are obtained at order zero and order one. The natural frequencies are found analytically depending on mean fluid velocity and being effected by mass ratio and external applied tension for the first four modes. When pulsating frequency is nearly twice any a natural frequency or sum of any two natural frequencies, the sub-harmonic parametric resonance and combination resonance occur. From the solvability conditions, the stability boundaries are determined analytically for sub-harmonic parametric resonances and combination resonances in the first two modes. The stability of trivial solution and nontrivial solution are discussed. And nonlinear response curves of parametric resonances are obtained. The present result is compared with that obtained using the two-mode Galerkin's method, and it is found that the two results are in good agreement.With the Galerkin method the governing equation is truncated into a set of ordinary differential equations until order four. The numerical solution of these equations is obtained by using steplength-varied Runge-Kutta algorithm. The phase plane portrait and time history portrait are respectively investigated at different frequency to confirm the theoretical results.