270. VIBRATION CONTROL OF CANTILEVER BEAM. J. KOVÁŘOVÁ 1, A , M. SCHLEGEL 2,B AND J. DUPAL 3,C © VIBROMECHANIKA. JOURNAL OF VIBROENGINEERING. 2007 APRIL/JUNE, VOLUME 9, ISSUE 2, ISSN 1392-8716 45 270. Vibration control of cantilever beam J. Kovářová 1, a , M. Schlegel 2,b and J. Dupal 3,c 1 Department of Mechanics, West Bohemia University in Pilsen, Univerzitní 8, 306 14 Pilsen, Czech Republic 2 Department of Cybernetics, West Bohemia University in Pilsen, Univerzitní 8, 306 14 Pilsen, Czech Republic 3 Department of Mechanics, West Bohemia University in Pilsen, Univerzitní 8, 306 14 Pilsen, Czech Republic E-mail: a jkovarov@kme.zcu.cz, b schlegel@kky.zcu.cz, c dupal@kme.zcu.cz (Received 2 April 2007; accepted 15 June 2007) Abstract. The paper describes two approaches to problem of active damping of vibrations of cantilever beam. First one uses standard LTI (linear time invariant) mathematical model of the system and state feedback with the state observer designed by pole placement method. The incomplete pole assignment method is used instead of the standard full assignment. The second one is based on experimental identification of the first mode shape and design dynamic compensator. Experimental results of both methods are compared. The problem of robustness of the compensator by frequency domain method based on the unstructured uncertainty of the model is also addressed. Keywords: cantilever beam, incomplete pole assignment, robustness, self tuning controller, vibration control. Introduction Vibration control of flexible structures is an important issue in many engineering applications. Balancing the stringent performance objectives of modern structures such as superior strength and minimal weight introduces a dynamic component that needs to be considered. Depending on the applications, low structural damping can lead to problems such as measurement inaccuracy of attached equipment, transmission of acoustic noise or structural failure. Various methods to suppress vibrations have been developed and these commonly include active, passive, semi-active and hybrid vibration control systems. This paper addresses the vibration control of cantilever beam by the methods of linear feedback control. It is concerned with state feedback designed by pole placement method (in our case modification of this method – incomplete pole assignment) and by self tuning controller [1], [2]. An optimal position is chosen for sensor and actuator by the method shown in [3] which is based on mode shapes – amplitudes and nodal points. The nominal mathematical model is used for design of the controller based on incomplete pole assignment. Due to this, the problem of robustness needs to be solved to avoid the instability of the closed loop because of the design inaccuracies and truncation errors. Mathematical model Mathematical model results from the equation of motion ( ) ( ) () () , , , t x t x t x γ = + F q q K M   (1) where M is the inertial forces operator, K stiffness operator, ( ) x F vector of forces space distribution, ( ) t γ time function of excitation and ( ) t x, q is the vector of displacement. Solution in frequency domain with respecting orthonormalized eigenfunctions by M-norm ( ) ( ) () () , , , , 2 i ij j i ij j i x x x x Ω = = δ v v δ v v K M (2) where ( ) x i v is eigenfunction for i subscript, leads to the displacement vector expressed in frequency domain ( ) ( ) () ( ) () . , , 2 2 x x x x j j j j v Γ F v Q ∑ ω ω − Ω = ω (3) For single force Eq. 3 takes the form