In this paper, the vibration model of an elastic beam, governed by the damped Euler-Bernoulli equation $\rho(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to the clamped boundary conditions $u(0,t)=u_x(0,t)=0$ at $x=0$, and the boundary conditions $\left(-r(x)u_{xx}\right)_{x=\ell}=k_r u_x(\ell,t)+k_a u_{xt}(\ell,t)$, $\left(-\left(r(x)u_{xx}\right)_{x}\right )_{x=\ell}$$=- k_d u(\ell,t)-k_v u_{t}(\ell,t)$ at $x=\ell$, is analyzed. The boundary conditions at $x=\ell$ 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 $\mu(x) \ge 0$, and the boundary springs $k_r,k_d \ge 0$ and dampers $k_a,k_v \ge 0$. Some numerical examples are given to illustrate the role of the damping coefficient and the boundary dampers.

翻译：本文分析了由阻尼Euler-Bernoulli方程 $\rho(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$ 描述的弹性梁振动模型，其边界条件为：在 $x=0$ 处固支 $u(0,t)=u_x(0,t)=0$；在 $x=\ell$ 处满足 $\left(-r(x)u_{xx}\right)_{x=\ell}=k_r u_x(\ell,t)+k_a u_{xt}(\ell,t)$ 和 $\left(-\left(r(x)u_{xx}\right)_{x}\right)_{x=\ell}$$=- k_d u(\ell,t)-k_v u_{t}(\ell,t)$。$x=\ell$ 处的边界条件分别对应由转动与角速度引起的阻尼力矩组合，以及由位移与速度引起的力的组合。基于经典的Lyapunov方法开展系统稳定性分析。在确保正则弱解存在的自然假设下，推导了系统能量的均匀指数衰减估计。该估计中的衰减率常数仅取决于梁的物理与几何参数（包括粘性外阻尼系数 $\mu(x) \ge 0$）以及边界弹簧系数 $k_r,k_d \ge 0$ 和阻尼器系数 $k_a,k_v \ge 0$。通过数值算例展示了阻尼系数与边界阻尼器的作用。