Stability And Bifurcation Analysis Of Supportd Pipes Conveying Fluid By The Method Of Hamiltonian Mechanics

The nonlinear dynamics of pinned-pinned and clamped-clamped pipes conveying fluid are investigated in the presence of zero-to-one internal resonance with the exitation of pulsating flow and motion of supporting base.The nonlinear differential equation of the pinned-pinned and clamped-clamped pipes is established with the Hamiltonian mechanics. The behavior of the eigenvalues of the system is rather complex as the flow velocity u 0is varied. For some values of u 0, the eigenvalues of linear matrix have a double zero and a pair of pure imaginary eigenvalues, and hence, there is the possibility of presence of zero-to-one internal resonance in this system. Two different normal form methods are applied to obtain the normal form equation of motion which can be used to analyze the stability and bifurcation phenomenon of the system.The normal form equation is applied to analyze the subharmonic resonance of the system. The stability of the trivial and non-trivial solutions of the unperturbed system is calculated, and the possibility of the presence of homoclinic orbits is analyzed. Using the method described by Kovacic and Wiggins, the conditions under which a Silnikov-type homoclinic orbit may be present in a perturbed resonant system are determined. Numerical simalations are carried out to confirm the theoretical results obtained.