On the zero dynamics of helicopter rotor loads
type: Item
instance of: scholarly article
title: On the zero dynamics of helicopter rotor loads
zbMATH Open document ID: 0865.73037
DOI: 10.1016/S0947-3580(96)70029-8
published in: European Journal of Control
publication date: 1997-07-14T00:00:00Z
review text: 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 po…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 74F10
Mathematics Subject Classification ID: 93A30
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 938889
zbMATH Keywords: correction for bending component
zbMATH Keywords: stability
zbMATH Keywords: complex frequency plane
zbMATH Keywords: Euler-Bernoulli bending term
zbMATH Keywords: eigenvalue equation
zbMATH Keywords: Fourier series
zbMATH Keywords: eigenfunctions
zbMATH Keywords: fundamental mode eigenfunction
author: Sergio Bittanti
author: Marco Lovera
MaRDI profile type: Publication
cites work: An LQ Approach to Active Control of Vibrations in Helicopters
cites work: Zeros of modal models of flexible structures
cites work: Q4894836
cites work: Zeros of discrete-time linear periodic systems
description: scientific article; zbMATH DE number 938889