Bifurcations in the motion of articulated tubes conveying fluid, with an end mass

A system of articulated tubes conveying fluid has been studied for its nonlinear behaviour. The dynamics of the two tube system can be studied by varying two parameters: β corresponding to the ratio of masses of the tube and the fluid, and ρ, the dimensionless flow rate. For a particular β, as the flow rate is increased, the zero equilibrium becomes unstable via a Hopf bifurcation, leading to a limit cycle motion centered around the zero equilibrium. When a point mass is attached to the free end of the two tube system, a parameter α corresponding to the ratio of the mass of the point mass to the mass of the tubes and fluid, may be introduced. The obtained numerical solutions to the four first order differential equations suggest that as the end mass is increased, a symmetry-breaking bifurcation results in two stable limit cycle solutions neither centered around the zero equilibrium point. As α is further increased, this is followed by a period-doubling route to chaos. A Poincare section of the four dimensional motion is used to quantify the bifurcation, and is shown to be similar to that of the logistic map with α being the control parameter.