  This paper addresses the analysis of a boundary feedback system involving a
non-homogeneous Euler-Bernoulli beam governed by the equation
$m(x)u_{tt}+\mu(x)u_{t}$$+\left(r(x)u_{xx}\right)_{xx}=0$, subject to the
initial $u(x,0)=u_0(x)$, $u_t(x,0)=v_0(x)$ and boundary conditions $u(0,t)=0$,
$\left (-r(x)u_{xx}(x,t)\right )_{x=0}=-k^{-}_r u_{x}(0,t)-k^{-}_a
u_{xt}(0,t)$, $u(\ell,t)=0$, $\left (-r(x)u_{xx}(x,t)\right )_{x=\ell}=-k^{+}_r
u_{x}(\ell,t)-k^{+}_a u_{xt}(\ell,t)$, with boundary control at both ends
resulting from the rotation and angular velocity. The approach proposed in this
study relies on the utilization of regular weak solutions, energy identity, and
a physically motivated Lyapunov function. By imposing natural assumptions
concerning physical parameters and other inputs, which ensure the existence of
a regular weak solution, we successfully derive a uniform exponential decay
estimate for the system's energy. The decay rate constant featured in this
estimate is solely dependent on the physical and geometric properties of the
beam. These properties encompass crucial parameters such as the viscous
external damping coefficient $\mu(x)$, as well as the boundary springs
$k^{-}_r,k^+_r $ and dampers $k^{-}_a,k^+_a$. To illustrate the practical
effectiveness of our theoretical findings, numerical examples are provided.
These examples serve to demonstrate the applicability and relevance of our
derived results in real-world scenarios.
