We designed a control strategy for cantilevered Timoshenko beams with free-end mass/inertia composed of boundary force and torque inputs. We initially develop an exact model knowledgecontrol law which exponentially stabilizes the beam flexiblestructuretion. We then show to redesign the control law as an adaptive controller to asymtotically stabilize the flexiblestructuretions while compensating for parametric uncertainty. Preliminary experimental results are included to illustrate the control performance.

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The demand for high speed, low cost, and low energy consuming systems has motivated the introduction of flexible parts in many mechanical systems (e.g., spacecraft structures, flexible link manipulators, flexible rotors, etc.). Mechanical systems composed of rigid and flexible parts are often mathematically described by a combination of ordinary differential equations (ODEs), partial differential equations (PDEs) (i.e., a distributed parameter model) and a set of boundary conditions. The design of high-performance control laws for such hybrid systems is complicated due to the dynamic coupling between the rigid and flexible subsystems; hence, the development of new control design/analysis tools for such systems has generated considerable interest.

In many hybrid mechanical systems, the flexible subsystem is modeled as a beam-type structure. The most commonly used beam model is based on the classical Euler-Bernoulli assumptions which neglect the rotary inertia of the beam. The Euler-Bernoulli model provides a good description of the beam's dynamic behaviour when the beam's cross sectional dimensions are small in comparison to its length. A more accurate beam model can be obtained by utilizing the Timoshenko theory which takes into account not only the rotary inertial energy but also the beam's deformation due to shear.

In this project, we considered the problem of controlling the vibrations of a cantilevered Timenshenko beam. We have developed a boundary control strategy for the hybrid dynamic model of a Timoshenko beam with mass/inertia at the free end. A Lyapunov-based design/analysis was utilized to first develop an exact model knowledge controlled, consisting of boundary force and torque inputs, which was shown to exponentially stabilize the beam vibration. The controller was then redesigned as an adaptive control law which asymptotically stabilized the vibrations while compensating for parametric uncertainty. Simulation results are presented here to illustrate the performance of the control laws.

## Experimental results. |

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