A nonlinear model for landing gear shimmy

546
G. Somieski tire damping moment from tread width [Nm] tire aligning moment [Nm] describing function describing function of tire moment M=/F, describing function of tire side force F,/Fz paraineter or parameter set yaw rate of landing gear [rad/s] Laplace variable time [s] input in dynamic system wheel forward velocity [m/s] state of dynamic system output of dynamic system lateral shift of leading tire contact point [m] slip angle or deflection angle of tire [rad] amplitude of limit cycle of a: [rad] limiting slip angle for aligning moment
dynamics of the landing gear and to enable a solution of the stability boundary and the limit cycle equations analytically. Extensive use of software is made, e.g. for computing eigenvalues and numerical simulation, obtaining analytical solutions for stability boundaries or limit cycles, and for graphical displaying. II - NONLINEAR MATHEMATICAL MODEL
Science and Technology, 1270.9638, 9710810 Elsevier, Paris
Jl (xl tt
k0 Ll(4
Lz (4 ~10;11;12 ~20;21,22 M Ml JJ2 M3
Fourier coefficients of steady oscillations moment of inertia about z-axis [kgm2] imaginary unit Bessel function of 1st order viscous friction moment coefficient [Nm/rad/s] dry friction moment coefficient [Nm] transfer function of linear dynamics of torsion transfer function of linear dynamics of elastic tire coefficients of transfer function L1 coefficients of transfer function Lz moment [Nm] spring moment [Nm] damping moment [Nm] total tire moment about z-axis [Nm]
Deutsche Forschungsanstalt fiir Luft- und Raumfahrt e. V Institut fiir Robotik und Systemdynamik, Postfach 1116, D-82230 Wepling, Germany Manuscript received February 3, 1991; revised version April 24, 1997.
NOTATIONS A a
a0,1,2,3
fl, I, i
f2
h,2,3 c
Cl...5 CFCY CMa &,2,3
TiF: F(u)
Aerospace
amplitude of limit cycles or system matrix half contact length of tire on ground [m] coefficients of parameter space equation coefficients of stability boundary equations torsional spring rate [Nm/rad] coefficients of characteristic equation tire side force derivative [l/r-ad] tire aligning moment derivative [m/r-ad] coefficients of side force Fy or moment Mz wheel caster length [m] tire side force [N] vertical force [N] nonlinearity in dynamic system
[deal
= A/S =amplitude of steady oscillation limiting slip angle for tire side force [deg] constant of tread width tire moment [Nm’/rad] eigenvalues of linear differential equations relaxation length of tire deflection [m] time constant of tire elasticity Es] yaw angle of landing gear [rad] frequency of a-oscillation [rad/s] frequency of limit cycle of Q [rad/s]
Aerospace
Science
and Technology,
1997,
no 8, 545-555
Shimmy Analysis of a Simple Aircraft Nose Landing Gear Model Using Different Mathematical Methods
Gerhard Somieski
Somieski Abstract
G., Aerospace
Science and Technology,
1997, no 8, 545-555.
Shimmy oscillations are still a problem in design and operation of aircraft landing gears, and accurate and appropriate analysis is required to master the task. Based on a nonlinear model of the mechanics of the landing gear and tire elasticity according to elastic string theory,பைடு நூலகம்some well known linear and nonlinear mathematical methods are applied to the shimmy analysis of a simple model of a nose gear: Computing eigenvalues, solving analytically the stability boundaries with a parameter space method, getting limit cycles by analytical formulae from describing functions, and last but not least numerical simulation of time histories. It seems that linear or quasi linear methods and analytical solutions are well suited to obtain extensive insight, respecting limitations of these methods. Numerical simulation on the other hand is a valuable tool for pointing out specific effects of a nonlinear system in large amplitude regions. Keywords: Analytical Shimmy analysis - Nose landing solutions. gear - Limit cycles - Instability - Describing functions -
The nonlinear mathematical shimmy model, compared to a similar model in [2, 5, lo], consists of the torsional dynamics of the landing gear, the forces and moments, and of approximations to describe the tire’s elastic lateral qualities, see Figure 1.
4aw angleW
tire side force Fy
Y
Fig. 1. - Top view of shimmy
lateral shift y1 of leading contact point