Perturbation Analysis of Nonlinear Wheel Shimmy

Abstract

A perturbation analysis of nonlinear wheel shimmy in aircraft landing gear is presented for nonlinear models that include terms due to coulomb friction between the oleo struts and freeplay in the torque links. The method of multiple timescales is used to obtain general expressions for the limit-cycle amplitude and the frequency that are functions of ground speed. The analysis shows that stable or unstable limit cycles can exist for taxi speeds above or below a critical value with stability of the limit cycles being determined by the sign of a computed coef cient. When only coulomb friction is present, an unstable limit cycle exists. When only freeplay is present, a stable limit cycle exists. When both coulomb friction and freeplay are present, it is shown that stable and unstable limit cycles and a turning point can exist depending on the system parameter values. The solution method is applied to a simple shimmy model, and results from the perturbation analysis are shown to be in good agreement with those obtained by direct numerical integration of the nonlinear shimmy equations. Nomenclaturevectors of nonlinear terms f cf = vector of coulomb friction terms f fp = vector of nonlinear stiffness terms including freeplay effects f .n/ j ; f .n/ k = Fourier coef cient vectors i = imaginary unit, p ¡1 K; N K = stiffness matrices K .0/ = ² 0 -order term in expansion for K K .1/ = ² 1 -order term in expansion for K M; N M = mass matrices q = vector of state variables q m = coordinate vector in expansion for q V = system parameter (ground speed) V i = i th-order term in expansion for V V 0 = critical value of V ®= complex coef cient = complex coef cient°= complex coef cient ± = complex coef cient ² = small parameter ¿ = independent variable time ¿ m = multiple timescales à = phase ! = frequency, dependent on amplitude A ! 0 = eigenvalue for V 0 and det L 0

Landing Gear Design/Maintenance Analysis for Nonlinear Shimmy

“…Thus, friction by itself causes the system response to decay even above V cr , unless there is a sufficiently large disturbance, consistent with the time-marching results in figure 1 of [6]. When friction accompanies freeplay, it tends to alleviate the subcritical LCO response due to freeplay, even reducing the unbounded response at the nominal V cr to a finite amplitude (see figure 4 of [6]). The HB methodology indicates that this tendency will hold if V cr increases monotonically with C θ , which is plausible for most systems.…”

“…Among the common nonlinearities, freeplay is often present in the torque collar, wheel hub, and steer-damp mechanism, and so is found to be the most detrimental, with instances of limit-cycle oscillation (LCO) reported well within the operating velocity range for main as well as nose gear of civil and military aircraft [1,5]. Coulomb friction appears to be generally beneficial, and nonlinear tire dynamics is reported to be important mainly when there is high slippage [6].…”

Investigation on the effect of coulomb friction on nose landing gear shimmy

“…The MBD model is assembled in a modular manner and it can be modified without the necessity to manually update equations of motion. Other methods also employed in shimmy studies include quasi-linear analysis (Krüger and Morandini, 2011), describing functions (Burton, 1981), multiple timescales (Gordon, 2002), and harmonic balance (Zhou and Zhang, 2005; Padmanabhan and Dowell, 2015).…”

“…A third-order LG model by Somieski (1997) and a similar one in a study by Airbus (Coetzee, 2007) include the rotational DoF and lateral deflection of the tire patch. A study by Boeing (Gordon, 2002) employs a similar model but adds rotational freeplay and a fixed Coulomb torque. Atabay and Ozkol (2013) reused Somieski’s model and incorporated a magnetorheological damper for shimmy suppression.…”

Hopf bifurcation character of an interactive vehicle–road shimmy system under bisectional road conditions

“…Similar to the work of Karanam and Ghosal 19 , the landing gear of aircraft can be considered as a three-wheeled vehicle when travelling on the runway. Gordon 20 studied the front-wheel shimmy system of the landing gear of aircraft. The general expression for the amplitudes of the limit cycles and their corresponding frequencies was deduced using the perturbation method and the multiple-scales method.…”