Most authors have ignored the effect of rotatory inertia and shear deformation. This practice is justified for slender flexible arms. According to the Timoshenko beam theory, the deflection due to shear force and rotatory inertia should be taken into account in modelling for high speed and high precision requirement when the ratio of the cross-sectional dimensions to length increases. Based on Hamilton's principle and Timoshenko's flexible beam theory, the dynamic model of a single non-slender flexible link is derived, and it is shown that the elastic motion is governed by a pair of coupled partial differential equations with coupled boundary conditions. Then the abstract form of the dynamic equations is studied, and the properties of the spectrum of the elastic operator appearing in the evolution equation are given. Furthermore, the eigenvalue problem of the elastic operator is solved in explicit form. The formulation and well-posedness of the state-space equation, as well as the transfer function of the dynamic control system of the non-slender flexible link, are studied by spectral analysis. Spectral analysis is used to study the well-posedness of the dynamic control system. The tracking control problem is studied and a feedback control scheme that controls the rigid-body motion and elastic behaviors simultaneously is derived based on a n-modal model. Closed-loop configuration of a control system, equivalent circuit of a dc-motor and the overall system block diagram are proposed. The stabilization of the closed-loop system is studied analytically. Finally, the tracking control problem is studied, a stabilizing feedback control law based on a n-modal model to suppress vibrations of the flexible link is derived, and the necessary and sufficient conditions that can guarantee the stability of the closed-loop system, are given. Simulation results are given as well.

The disturbance estimated with the help of advanced mechatronic devices is used for realization of robust mechanical system, such as agricultural machines. In the actual application, the estimated disturbance is effective not only for the disturbance compensation but also for the parameter identification in the mechanical system. The identified external force is applicable to sensorless force feed-back control in mechanical system and is utilized for a realization of mechanical vibration control. The progress of robust control technologies makes it possible to realize high performance motion control. In the industrial drive system such as a modern agricultural machine with advanced mechatronic device system for motion control, however, the developed technology is not enough to obtain the stable and high speed motion response since the mechanical vibration arises under the high accuracy positioning control. The mechanical vibration control is also taken in the field of the motion control. In particular, a vibration control based on the external force feedback brings the sophisticated advantages to the mechatronic system. The paper introduces a vibration control strategy based on the external force feedback called "resonance ratio control" in multiple resonance system. In this case, the external force may be obtained by using the identification process.