Stability of a Vibrating and Rotating Beam

Abstract

Deformations of a rotating beam (of a circular cross-section and the lenght L = 1) may be described (after an appropriate choice of a time scale) by the differential equation

$$ {{z}_{{tt}}} + {{z}_{{xxxx}}} + a(t,x){{z}_{{xxx}}} + b(t,x){{z}_{{xx}}} + c(t,x){{z}_{x}} + d(t,x)z + + e(t,x){{z}_{t}} = f(t,x,z,{{z}_{t}},{{z}_{x}},{{z}_{{xx}}},{{z}_{{xxx}}}) $$

((1))

where z is a complex function of variables t and x defined for t ≥ 0 and x ∈ <0, 1> (see [2], [3]). The coefficients a, b, c, d, e as well as the nonlinear function f represent forces and moments acting on the beam. The function z is supposed to satisfy boundary conditions

$$ z(t,0) = {{v}_{1}}{{z}_{{xx}}}(t,0) + {{v}_{2}}{{z}_{x}}(t,0) = 0, $$

((20))

$$ z(t,1) = {{v}_{3}}{{z}_{{xx}}}(t,1) + {{v}_{4}}{{z}_{x}}(t,1) = 0 $$

((21))

for t ≥ 0. We assume that | v₁| + |v₂| > 0, |v₃| + |v₄| > 0. The function f is assumed to be continuous and bounded together with its second derivatives with respect to x, z, z _t , z _x , z _xx , z _xxx on the set A = = <0,+ ∞) × <0,1> } <- R, R>⁵ (where R is a positive real number).