Stabilization of Nonhomogeneous Euler-Bernoulli Beam with a Point Controlled by Combined Feedback Force and Indefinite Damping

The transportation of masses or objects by cranes, necessary in the construction sector to increase productivity, often causes transverse vibrations of the system, which degrades production or process performance, and can even cause breakdowns. Actively controlling or mitigating these vibrations is becoming essential in many fields, such as the construction of satellite panels, etc. Therefore, many efforts have been made in recent years to find effective ways to eliminate these unwanted vibrations. Thus, the vibration behavior of a crane on a construction site, transporting a mass represented by beams, is translated into an equation using the theory of Euler-Bernoulli beam equations. These vibration effects are thus model led in a very reduced manner by a nonhomogeneous Euler-Bernoulli beam fixed at one end and subjected to a point mass at the free end. In this research article, we have generalized Wang’s results. We began by defining some fundamental properties of the closed-loop system and then analyzed its spectrum. Using the theory of perturbed problems, we obtained the basic Riesz property. The spectrum-determined growth condition and exponential stability were also derived. Moreover, when the damping was undefined, we provided a condition on the negative value of the damping without destroying the exponential stability of the system.