Abstract
This paper examines the free vibration and stability of an Euler-Bernoulli beam ejected from or, equivalently, drawn into an orifice at an arbitrary angle relative to gravity. A stability boundary for this system is presented in terms of two dimensionless, time varying parameters, one describing the beam bending stiffness and the other indicating the axial tension induced by gravity. This stability boundary is the limit of positive definiteness of a Lyapunov functional for the system. The Lyapunov functional is the Jacobi integral of the system, which qualifies as a Lyapunov functional for many gyroscopic systems. The ejected beam system is gyroscopic when the time varying coefficients in the system equation are held constant. It is also shown that initially, the free vibration problem for the ejected beam has the same vibration modes shapes as an ordinary cantilever beam but that the magnitude and period of vibration grow as the square root of time.
