Exponential stability of damped Euler-Bernoulli beam controlled by boundary springs and dampers

In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation ρ(x)utt+μ(x)ut+left(r(x)uxxright)xx=0, subject to the clamped boundary conditions u(0,t)=ux(0,t)=0 at x=0, and the boundary conditions left(-r(x)uxxright)x=ℓ=kr ux(ℓ,t)+ka uxt(ℓ,t), left(-left(r(x)uxxright)xright )x=ℓ=- kd u(ℓ,t)-kv ut(ℓ,t) at x=ℓ, is analyzed. The boundary conditions at x=ℓ correspond to linear combinations of damping moments caused by rotation and angular velocity and also, of forces caused by displacement and velocity, respectively. The system stability analysis based on well-known Lyapunov approach is developed. Under the natural assumptions guaranteeing the existence of a regular weak solution, uniform exponential decay estimate for the energy of the system is derived. The decay rate constant in this estimate depends only on the physical and geometric parameters of the beam, including the viscous external damping coefficient μ(x) ≥ 0, and the boundary springs kr,kd ≥ 0 and dampers ka,kv ≥ 0. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.