Analytical prediction of the non-linear response of a self-excited structure

The time-dependent amplitude response of the self-excited harmonium reed vibrating at finite amplitudes is investigated both analytically and experimentally. The analysis contained herein is based on the assumption that all of the significant non-linear forces that act on the reed are of aerodynamic origin and that these forces influence the reed behavior through the system damping. The analysis is carried out, without the aid of empirical techniques, by deriving the induced aerodynamic pressure force as a non-linear function of the amplitude of the reed motion, via an unsteady potential flow field analysis, and then applying the resulting forcing function to the equation of motion of the reed. An approximate solution of the equivalent lumped parameter equation of motion yields functional relationships that predict various observable phenomena, including the limit cycle amplitude. A comparison of the results with experimental data serves to substantiate the analysis. Also presented is a brief discussion of some of the phenomenological similarities that exist between the results of the harmonium reed analysis and the observed behavior of a flat-plate wing undergoing torsional stall flutter.