The stability of timoshenko beams conveying a compressible fluid

The purpose of this study is to formulate and solve the stability problem associated with the equations of motion of a Timoshenko beam conveying a compressible fluid. The beam is assumed to be either cantilevered or simply supported. Shear deformation and rotary inertia are considered in the Timoshenko beam theory. The beam analysis that is carried out is quite general and allows for viscous damping external to the tube as well as both internal strain rate and hysteretic damping. Interactions between these three damping mechanisms, shear deformation, and rotary inertia also are accounted for. The compressible, isentropic flow through the tube is governed by Euler's equations of motion. Gravitational effects are neglected in the analysis. This is equivalent to assuming that the tube is horizontal. The motion of the fluid is coupled to that of the beam by a compatibility condition at the tube wall which requires that they vibrate together.

The equations of motion of the fluid and the compatibility condition are nonlinear. They are linearized using a perturbation about a reference solution. A velocity potential is introduced and separation of variables is applied to reduce the equation system to two simultaneous partial differential equations. A characteristic equation is derived by assuming harmonic solutions while a frequency equation is developed using the boundary conditions in conjunction with this harmonic solution. Numerical solutions to the resultant pair of coupled transcendental equations are computed using Muller's method.

The compressibility of the fluid is found to affect the stability of the fluid-tube system through the fluid sonic velocity and the tube aspect ratio. In the cantilevered case increased aspect ratio raises the critical velocity at which flutter occurs while reduced sonic velocity decreases it. The higher vibration modes become excited as the fluid sonic velocity is continually lowered. The stability of the system is significantly affected since instability occurs in the higher modes before it takes place in the lower ones. Any return-to-stability associated with an increase in flow velocity can be eliminated by inclusion of a finite sonic velocity in the analysis of cantilevered, fluid-conveying beams.

Euler's method of equilibrium is applied to the partial differential equation pair described above in the simply supported case. The resultant stability problem also is amenable to solution using Muller's method. Results indicate that the critical flow velocity at which the tube buckles increases when the aspect ratio is raised if the flow is subsonic. When the flow is supersonic an increase in aspect ratio decreases the critical velocity. Decreased sonic velocity always reduces the critical velocity in the simply supported case.