On the zero dynamics of helicopter rotor loads
This paper contains some basic analysis of helicopter blade dynamics. The authors comment that many textbooks such as recent book by \textit{A. R. S. Bramwell} [Helicopter dynamics, Edward Arnold Publishers, London (1996)] are primarily concerned with stability. Application of the ``black box'' identification techniques leads consistently to non-minimum phase characteristics. Consistently, in the past flexibility of the rotor blade has been ignored, leading to the so called ``flapping angle'' description of a rigid beam representing mathematically the rotor blade. Also, since the main objective was a study of stability, only the poles of the system's response in the frequency plane were of interest. The authors observe that a zero is also located in the right hand side of the complex frequency plane. For a rigid blade analysis this zero is always positioned on the real axis at the point \(\sqrt {8}\Omega\), where \(\Omega\) is the angular velocity of the helicopter blade, \(\Omega\) is an assumed constant in this study. The authors use the following mathematical system to model more accurately the behavior of the blade: \(d^2/dr^2[E(r)I(r)d^2z(r,t)/dr^2]-d/dr[\int m(\rho)\Omega^2\rho d\rho]+d^2mz(r,t)/dz^2=L_s(r,t)\). The first term can be easily recognized as the Euler-Bernoulli bending term, the second is a correction for the bending component of the horizontal action of the centrifical force. The authors separate variables obtaining routinely the eigenvalue equation, then expand solution of the homogeneous equation in Fourier series in the corresponding eigenfunctions. Since \(\nu_1=\Omega\) turns out to be the lowest natural frequency, the authors insert this value, obtaining so-called fundamental mode eigenfunction \(\eta_1\). The other modal equations are easily derived. Now a more detailed study follows a sound engineering practice of rewriting all equations in dimensionless variables. Forward motion and hovering are considered separately. Zeros as well as poles of the system are found. Then the authors revert to state space formulation. Finally, back in the frequency domain, they study the positions of poles and zeros for continuous time periodic systems. They come close to considering the root locus technique, but they manage to avoid it, and finish with an analogy of the time invariant case. An important contribution of this paper is the authors' study of effects of the real zero or zeros in the right hand plane of the frequency domain.
scientific article; zbMATH DE number 938889