282 Z. Zhang et al. 33.2 Background To achieve an effective computational procedure, a stochastic variational method is proposed. Firstly, a variational formulation of a rotating beam with piezoelectric patches is derived from the generalised form of Hamilton’s principle for electromechanical systems, which involves the structural displacements and electrical voltage of piezoelectric patches. Then an inductance-resistance-shunt is included in the voltage variable by using the Kirchhoff’s second law. In order to accurately predict vibration characteristics of the rotating beam, the fully geometrically nonlinear beam theory is employed. The deterministic equations of motion are derived by Rayleigh-Ritz method based on the orthogonal polynomial bases. The generalized polynomial chaos expansion (gPCE) [5] is then employed to represent propagation of uncertainties (such as the resistance and the rotating speed) and to estimate the statistical characteristics of the responses. 33.3 Analysis Figure 33.1a shows the effect of Coriolis effects on the first in-plane natural frequency against the rotational speed. For small values of rotational speed, the Coriolis force has little influence on the dynamic characteristics. With the increase of the rotational speed, the influence becomes large and the Coriolis effects are counterbalance the stiffening effect of the centrifugal force due to the rotating motion. Figure 33.1b shows the twist displacement of the cantilever rotating beam with a tip force along the vertically bending direction. Due to the Coriolis effects and nonlinear effects, the in-plane and torsional motions are coupled with vertically bending and axial vibrations. The generalized-time integration method [6] is adopted to obtain the transient nonlinear dynamic responses for the rotating beam. It is found that the introducing the series single-mode shunt damping can significantly reduce the resonant magnitude of the selected resonant frequency. That is mainly due to that the strain energy in the beam and piezoelectric patch is transformed into the electrical energy of shunt, which will finally dissipate into heat. Uncertainty analysis is then performed to ascertain the influence of random parameters on responses. The results reveal that both the structure and piezoelectric uncertainties can affect the dynamic behaviors and hence the consideration of input uncertainties is necessary in analyses. 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 With CE Without CE Non-dimensional rotating speed Non-dimensional frequency (a) 0.0 0.2 0.4 0.6 -0.1 0.0 0.1 (b) with CE without CE Twist displacement J (rad) Time (s) Fig. 33.1 Effects of rotational speed on the dynamic characteristics of the rotating beam
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