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In this paper, the vibration model of an elastic beam, governed by the dampedEuler-Bernoulli equation$\rho(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to theclamped boundary conditions $u(0,t)=u_x(0,t)=0$ at $x=0$, and the boundaryconditions $\left(-r(x)u_{xx}\right)_{x=\ell}=k_r u_x(\ell,t)+k_au_{xt}(\ell,t)$, $\left(-\left(r(x)u_{xx}\right)_{x}\right )_{x=\ell}$$=- k_du(\ell,t)-k_v u_{t}(\ell,t)$ at $x=\ell$, is analyzed. The boundary conditionsat $x=\ell$ correspond to linear combinations of damping moments caused byrotation and angular velocity and also, of forces caused by displacement andvelocity, respectively. The system stability analysis based on well-knownLyapunov approach is developed. Under the natural assumptions guaranteeing theexistence of a regular weak solution, uniform exponential decay estimate forthe energy of the system is derived. The decay rate constant in this estimatedepends only on the physical and geometric parameters of the beam, includingthe viscous external damping coefficient $\mu(x) \ge 0$, and the boundarysprings $k_r,k_d \ge 0$ and dampers $k_a,k_v \ge 0$. Some numerical examplesare given to illustrate the role of the damping coefficient and the boundarydampers.

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