BIFURCATION CONTROL OF PARAMETRIC RESONANCE IN AXIALLY EXCITED CANTILEVER BEAM H. Yabuno and M. Hasegawa Graduate School of Systems and Information Engineering University of Tsukuba Tsukuba, 305-8573 JAPAN ABSTRACT Bifurcation control of parametric resonance in an axially excited cantilever beam is theoretically investigated. In the cases when the excitation frequency is in the neighborhood of twice the natural frequencies, the parametric resonance is produced through the trans-critical bifurcation and the cantilever overcomes also saddle-node bifurcation at the finite amplitude which is a discontinuous bifurcation. In state of the passage through resonance, jumping phenomena occurs due to the discontinuity of the bifurcation. Our control objectives are to avoid the occurrence of the jump phenomena in the state of the passage through resonance by proposing a bifurcation control to change the discontinuous bifurcation into continuous one. Introduction Bifurcation control of parametric resonance in an axially excited cantilever beam is theoretically investigated. In the cases when the excitation frequency is in the neighborhood of twice the natural frequencies, the parametric resonance is produced through the trans-critical bifurcation and the cantilever overcomes also saddle-node bifurcation at the finite amplitude which is a discontinuous bifurcation [1] [3]. In state of the passage through resonance, jumping phenomena occurs due to the discontinuity of the bifurcation. Our control objectives are to avoid the occurrence of the jump phenomena in the state of the passage through resonance. We have proposed a bifurcation control method for parametric resonance in a single degree of freedom system. In this research, the method is expanded to the parametric resonance produced in a continuous system, i.e., cantilever beam. Then, a control method for the avoidance of the parametric resonance is proposed. First, we derive the equation of motion of the cantilever beam by taking into account the effect of the curvature nonlinearlity [4]. The governing equation is theoretically analyzed by using the method of multiple scales [6]. The obtained averaged equation is autonomous for which the bifurcation analysis is easily performed. The above mentioned discontinuous bifurcations (saddle- node bifurcation and subcritical pitchfork bifurcation) are investigated with respect to the excitation frequency and their bifurcation points are detected. Next, we consider a control method for the avoidance by using a piezo actuator. We derive the equation of motion under the effect of the bending moment by the actuator. It is assumed that the bending moment is proportional to the input voltage. The averaged equation under the control input is derived again by the method of multiple scales. We set the control input to be proportional to the velocity of the beam and the cubic nonlinear term with respect to deflection. The feedback gains are designed based on the averaging equation. Analytical Model and Basic Equation The system under investigation consists of a beam with a piezo actuator for control as shown in Fig. 1. The clamped end of the beam is parametrically excited in the axially direction, and it is assumed that the beam behaves like an Euler-Bernoulli beam. The beam has a length l， stiffness EI , and mass per unit length ρA. The excitation of the beam, ξ(t), is expressed as follows: ξ = a b cos Nt (1) We introduce a static Cartesian coordinate system x − y − z, whose origin is put at the clamped end of the beam in the initial state, and also a coordinate along the elastic axis of the beam, s. The inextensibility condition of the beam can be written as  ∂v ∂s  2 +  1+ ∂u ∂s  2 =1, (2)