Keywords

1 Introduction

Landing gear shimmy is generally considered as an unexpected self-excited vibration of wheel and landing gear motion coupling with each degree of freedom. Many researches omitted the nonlinear effects caused by freeplay. The structural freeplay, like mechanical transmission clearance of the landing gear, also has a great impact on the wheel shimmy. In this work, the torsional freeplay caused by torque links is taken into account, and Gaussian white noise will be also considered to approximately describe the stochastic lateral excitation.

2 The Mathematical Model

The motion of each tire under the torsional and lateral bending effects of struts is based on [2]. A typical dual wheel NLG of a midsized passenger aircraft is considered as sketched in Fig. 1. These two modes are coupled via the lateral deformation \(\lambda _L \) and \(\lambda _R \) of the left and right tyres. The torsional and lateral bending dynamics of the NLG can be described by Eq. (1a–b). The longitudinal mode is omitted here since it does not actively participate in the NLG dynamics [2]. The tire kinetmatic equation is obtained by the stretched string theory, which describes the wheel-ground interaction. The resultant lateral deformation can be expressed as an algebraic sum of the torsional and lateral bending modes where the natural frequencies of both modes are different. The motion of each tire under the torsional and lateral bending effects of struts is given by Eqs.(1a–b). Then the overall model of the dual wheel NLG system for torsional and lateral vibrational modes can be established as [2],

$$\begin{aligned} {{\dot{\lambda }}_L} + \frac{v}{\sigma }{\lambda _L} - v\sin (\theta ) - {l_g}\dot{\delta }\cos (\delta ) - ({e_{\textrm{eff}}} - h)\cos (\theta )\dot{\psi }\cos (\phi ) - \frac{D}{2}\dot{\psi }\sin (\theta )\cos (\phi ) = 0, \end{aligned}$$
(1a)
$$\begin{aligned} {{\dot{\lambda }}_R} + \frac{v}{\sigma }{\lambda _R} - v\sin (\theta ) - {l_g}\dot{\delta }\cos (\delta ) - ({e_{\textrm{eff}}} - h)\cos (\theta )\dot{\psi }\cos (\phi ) + \frac{D}{2}\dot{\psi }\sin (\theta )\cos (\phi ) = 0, \end{aligned}$$
(1b)

where D is the seperation distance between the centres of two wheels, and each tyre has the moment of inertia I about the spinning axis.

Fig. 1.
figure 1

Schematics of side and top views of the dual wheel NLG.

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The torsional freeplay originating from the torque links of the NLG is studied here. The gap between the torque links will cause a reduction on the torsional stiffness that is expressed as [3]

$$\begin{aligned} {M_{{K_\psi }}} = \left\{ {\begin{array}{*{20}{c}} {{k_\psi }\left( {\left| \psi \right| - {\theta _{FP}}} \right) {\mathop {\textrm{sgn}}} (\psi ),}& {\left| \psi \right| > {\theta _{FP}},}\\ {0,}& {\left| \psi \right| \le {\theta _{FP}}.} \end{array}} \right. \end{aligned}$$
(2)

Specifically, a Gaussian white noise \(\omega (t)\) is added to the lateral DOF as an additive noise excitation. The \(\omega (t)\) satisfies the following property

$$\begin{aligned} E\left[ {\omega (t)} \right] = 0,\quad E\left[ {\omega (t)\omega (t + \tau )} \right] = \bar{N}\eta (\tau ), \end{aligned}$$
(3)

where \(E[\cdot ]\) means expectation, \(\eta (\tau )\) is Dirac function, \(\bar{N}\) is the noise intensity.

3 Bifurcation Analysis

A bifurcation analysis is performed to explore the deterministic NLG system. The effects of the forward velocity and vertical loading force are to be analyzed regarded to the stability with freeplay.

Fig. 2.
figure 2

The single parameter bifurcation with freeplay (\(F_z\) = 500 kN).

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Figure 2 shows the bifurcation under high vertical load with the cases of freeplay (0.04/0.06 rad). Figure 2(a) indicates that the nonlinearity caused by freeplay still effects system at the high-speed stage. System does not return to the equilibrium after the second bifurcation, but oscillates in multiple periods as shown in Fig. 2(b). Increasing the freeplay will expand the shimmy area of the system. In addition, at the low speed stage, the freeplay will cause multi-period to single period and then to multi-period oscillation (Fig. 2(e)). System will oscillate in the negative axis in the torsional direction as shown in Figs. 2(a) and (d) and vibrate significantly in the lateral bending direction (Figs. 2(c) and (f)). The bifurcation diagrams with freeplay shown in Fig. 2 display multi-period and quasi-period motions substansially extend the results of existing works. The freeplay may lead to chaos under specific conditions, and the impact of freeplay on the dynamics is more complex than discussed above.

Fig. 3.
figure 3

The double parameter bifurcation (v-\(F_z\) plane).

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As shown in Fig. 3, the double parameter bifurcation diagram of the v-\(F_z\) plane of the double wheel NLG system with and without considering the structural freeplay are obtained respectively. It can be seen from the Fig. 3(a) that system shimmies when forward velocity reaches 5.439 m/s. The intersection of the two areas is the area where system shimmies in both torsional and lateral bending directions. The result without freeplay shown in Fig. 3(a) completely corresponds to the one in [2]. From Fig. 3 (b), comparing with the case without freeplay, the critical speed of system shimmy is advanced from 5.439 m/s to about 1.028 m/s. The shimmy point in the torsional direction is extended from 175 m/s to about 270 m/s. The range of vertical load impact shimmy area is also expanded from 550 KN to 900 KN, which means that the existence of the freeplay makes the shimmy region in the torsional direction expand significantly. In the lateral bending direction, due to the coupling effect, the existence of freeplay will make the shimmy curve move up, indicating the vertical load required for system to lateral shimmy increases. In other words, the torsional freeplay inhibits the lateral shimmy to a certain extent.

Fig. 4.
figure 4

The time histories with NES controller.

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Figure 4 shows the time histories in the determistic and stochastic cases respectively, the results indicate the shimmy can be well controlled by NES (Figs. 5 and 6).

Fig. 5.
figure 5

The UCRP of the system with an embedding of \(m=3\), \(\tau _1 =10 \) (\(F_z\) = 500 KN, v = 0.25 m/s). m is embedding dimension and \(\tau _1\) is delay time. The black line and red line on top of the UCRP represent the time history in the torsional direction and lateral bending direction respectively.

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Fig. 6.
figure 6

The quantification analysis diagrams of UCRP for the tortional direction (\(F_z\) = 500 KN, v = 0.25 m/s). The black lines show positive relation, the red lines show negative relation. The dash-dotted line in (a), (d), (g) marks the 5% confidence interval. The lag domain is [–100, 100].

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The unthreshold cross recurrence plot (UCRP) are obtained to compare the dynamics represented in torsional and lateral direction, showing the recursive structure characteristics from periodic to stochastic.

Moreover, the statistics on the cross correlation, recurrence rate and determinism are obtained It can be seen that the system has strong cross-correlation, high recurrence rate and intensive determinism in the case without freeplay, and reduced after considering the freeplay and noise, indicating the coupling effect and periodicity in the torsion and lateral bending directions decreased.

4 Conclusion

Shimmy dynamics of a dual wheel NLG with torsional freeplay under stochastic lateral disturbances are investigated. We found that the torsional freeplay leads to an enlargement of the shimmy area and an enhancement of the shimmy characteristics compared to the case without freeplay. The stochastic lateral excitation enhances the lateral bending direction shimmy and brings out the random switching phenomenon of amplitude in the torsional mode of the system with torsional freeplay.