This work focuses on the control problem related to the active damping of thin mechanical structures that are under exogenous disturbances. Such problems arise for example within the mechanical structures composing the embedded systems of various vehicles. It is the case of the electronic boards which are constantly under vibrations that can be caused, in the case of aircrafts, by wind, crossing of atmospheric turbulences, working engines, and so on. These vibrations are responsible of several failures affecting the embedded electronic boards, such that fretting wear of the connectors, copper track breaking and so on. The vibration damping appears as a good way to limit such deteriorations. Active solution with the use of sensors, actuators and output feedback controllers is an appealing approach to vibration damping. It allows to transform a mechanical structure, once equipped with sensors and actuators, into an input-output dynamical system which has the particularity to contain an infinite number of vibrating modes, each of which having a resonant frequency. Several works have been done on this topic and most of them were based on the design of linear controllers which have proven their efficiency for applications of several flexible structures. In the case of embedded systems, the take into account of realistic running conditions matters particularly in the design process of controllers. As a consequence, one can cite the importance of low complexity controllers since these last should be implemented in embedded real-time computers, and for which the computation tasks should be reduced as much as possible. In the proposed work, the problem of vibration damping for a thin beam, with an Euler-Bernoulli configuration, is addressed thanks to an original output feedback controller based on delayed proportional actions. The considered beam is equipped with two piezoelectric patches: one of them works as a sensor and the other as an actuator. Each one of them is bonded on one side of the beam and both are collocated. The model of this system has been obtained numerically with a finite element modeling, leading to a linear state-space model. Then, the design of a controller based on proportional and delayed-proportional actions on both the input and output signals is tackled. The purpose of this work is to investigate the properties of the proposed output feedback controller in terms of damping performances and of robustness to uncertainties due to an unperfect modeling. This controller is based on a rightmost-characteristic root assignment approach. Simulation results show the effectiveness of this controller design method and its ability to damp several peaks of resonance while ensuring robustness against neglected dynamics at high frequency.