A general method with shaping filters to study random vibration statistics of lifting rotors with feedback controls
This paper is concerned with the random vibration statistics of linear or perturbed linear dynamic systems with variable parameters, in particular of lifting rotors with feedback controls. By introducing shaping filters to random inputs, the response variance matrix is solved directly from another set of linear equations. Therefore, with the stipulation of stationary Gaussian random inputs, the computational scheme of response threshold crossing statistics under steady-state conditions is essentially no more involved than generating the state vector to step inputs. For certain types of feedback controls of lifting rotors having coupling effects between blades, the equations of motion studied by other means using blade coordinates are adopted. Treated for illustrative purposes are the rigid flapping oscillations of multibladed rotors at a high advance ratio, having a rigid hub with elastically restrained and centrally arranged flapping hinges. The stochastic input refers to free atmospheric vertical turbulence at the centre of the rotor disk. The numerical results of the flapping response statistics include the root-mean-square description and the expected values of threshold up-crossings per unit time. When compared to earlier related studies, the proposed method requires negligible core storage requirement, directly permits variable step sizes to guarantee preset accuracy criteria, and offers substantial saving in machine time. In the parametric studies of system and feedback constants, the available core space could be used in storing discrete values of aerodynamic damping, spring and input modulating functions which otherwise involve lengthy Fourier series approximations at high advance ratios or repeated computations. The method in general offers considerable promise in the routine analysis of lifting rotor models having blades with many degrees of freedom of motion and having other combinations of feedback control systems.