OUT-OF-PLANE DYNAMIC ANALYSIS OF PLANAR CIRCULAR CURVED BEAMS USING THE STATE SPACE METHOD

By selecting the appropriate generalized displacements and their conjugated forces as state variables, the state space formula is properly set up to investigate the out-of-plane dynamic behavior of planar circular curved beams. On the basis of the established state space formula, the out-of-plane natural frequencies and corresponding vibration modes of circular curved beams under four common boundary conditions in engineering practice are then solved. The state space formula and the conception of sympletic inner product are utilized to derive two orthogonality relations of vibration modes. One is with respect to the mass and rotary inertia properties, and another is with respect to the stiffness property. Based on the established orthogonality relation with respect to the mass and rotary inertia properties, the mode superposition method is used to obtain the analytical solution of the inhomogeneous state equation, i.e. the governing equation for out-of-plane forced vibration, and the dynamic responses of circular curved beams under a vertical moving concentrated load are given. Several numerical examples are presented and the results are compared with those available in the literature with good agreement, which demonstrates the accuracy and reliability of the proposed method.