Shimmy dynamics in a dual-wheel nose landing gear with freeplay under stochastic wind disturbances

Abstract

Shimmy dynamics of a dual-wheel nose landing gear system with torsional freeplay under stochastic lateral wind disturbances is studied. Dynamic characteristics of the deterministic case are numerically analysed, especially the shimmy of the landing gear through bifurcation analysis. Meanwhile, the influences of the freeplay nonlinearity on shimmy behaviours are examined in detail. We found that the freeplay leads to an enlargement of the shimmy area and an enhancement of the shimmy characteristics compared to the case without freeplay. Furthermore, impacts of stochastic lateral wind disturbances on the shimmy of the landing gear system are estimated via time history and recurrence plots. We find that the stochastic excitation enhances shimmy of the lateral bending direction. More interestingly, the stochastic excitation strengthens the effect of the freeplay nonlinearity, which causes random intermittent large-amplitude oscillations in the torsional direction. Our results show that the interaction between the freeplay nonlinearity and the random load induces a significant reduction in the critical shimmy velocity, which has an adverse impact on the stability of the nose landing gear of an aircraft. This work will provide an insightful guidance for the design of landing gear parameters in engineering practice.

Appendix

(1):

Recurrence plot

RP is especially suitable for short time series data, and can test the stationarity and internal similarity of time series. Following Takens’ embedding theorem [40], the dynamics can be appropriately presented by a reconstruction of the phase space trajectory ${{\textbf{x}}_i}$ from a time series $u_k$ (with a sampling time t) by using an embedding dimension m and a time delay $\tau _1$. The RP measures recurrences of a trajectory ${\textbf{x}_i} \in {\mathbb {R}} ^d$ in phase space, and CRP compares dynamics of two trajectories ${\textbf{x}_i}$ and ${\textbf{y}_j}$, which can be formally expressed as [39],

$$\begin{aligned}{} & {} {R_{i,j}}\left( \varepsilon \right) = \Theta \left( {\varepsilon - \left\| {{{\textbf{x}}_i} - {{\textbf{x}}_j}} \right\| } \right) , i,j = 1,...,{\mathcal {N}}, \\{} & {} {CR_{i,j}}\left( \varepsilon \right) = \Theta \left( {\varepsilon - \left\| {{{\textbf{x}}_i} - {{\textbf{y}}_j}} \right\| } \right) ,\\{} & {} i= 1,...,{\mathcal {N}}, j = 1,...,{\mathcal {M}} \end{aligned}$$

where ${\mathcal {N}}$ and ${\mathcal {M}}$ are the number of measured points and $\varepsilon$ is a predefined threshold distance. Consider a series of intervals with a given colour corresponding to each interval to this type of recursive graphs, called the unthresholded recurrence plot. $\Theta \left( \cdot \right)$ is the Heaviside function (i.e. $\Theta \left( x \right) = 0$, if $x < 0$, and $\Theta \left( x \right) = 1$ otherwise), and $\left\| \cdot \right\|$ is a norm. For $\varepsilon$-recurrent states, i.e. for states which are in an $\varepsilon$-neighbourhood, we have ${\textbf{x}_i} \approx {\textbf{x}_j} \Leftrightarrow {R_{i,j}} \equiv 1$, and here, we use the most frequently norm $L_2$-norm (Euclidean norm).

(2):

Recurrence rate

The recurrence rate for CRP is defined as [41]

$$\begin{aligned} RR\left( t \right) = \frac{1}{{N - t}}\sum \limits _{l = 1}^{N - t} {l{P_t}\left( l \right) }, \end{aligned}$$

where ${P_t}\left( l \right)$ are the distributions of the diagonal line lengths. The index $t \in \left[ { - T,T} \right]$ marks the number of the diagonal line. $t = 0$ marks the main diagonal, $t > 0$ the diagonals above and $t < 0$ the diagonals below the main diagonal, which represent positive and negative time delays, respectively.

(3):

Determinism

The determinism is the proportion of recurrence points forming long diagonal structures of all recurrence points [41], that is,

$$\begin{aligned} DET\left( t \right) = \frac{{\sum \nolimits _{l = {l_{\min }}}^{N - t} {l{P_t}\left( l \right) } }}{{\sum \nolimits _{l = 1}^{N - t} {l{P_t}\left( l \right) } }}. \end{aligned}$$

Stochastic as well as heavily fluctuating data cause none or only short diagonals, whereas deterministic systems cause longer diagonals. If both deterministic systems have the same or similar phase space behaviour, i.e. parts of the phase space trajectories meet the same phase space regions during certain times, the amount of longer diagonals increases and the amount of smaller diagonals decreases. High values of DET represent a long time span of the occurrence of a similar dynamics in both systems. DET are sensitive to fast and highly fluctuating data.

(4):

Sample entropy

Sample entropy is a nonlinear dynamic parameter used to quantify the regularity and unpredictability of time series fluctuations. It uses a non-negative number to represent the complexity of a time series, reflecting the possibility of new information occurring in the time series.

$$\begin{aligned} SampEn\left( {{m_1},{r_1},S} \right) = \mathop {\lim }\limits _{S \rightarrow \infty } \left\{ { - \ln \left[ {\frac{{{A^{{m_1}}}\left( {{r_1}} \right) }}{{{B^{{m_1}}}\left( {{r_1}} \right) }}} \right] } \right\} , \end{aligned}$$

where ${m_1}$ is dimension, ${r_1}$ is the set distance, S is the number of data, ${A^{{m_1}}}\left( {{r_1}} \right) = \frac{1}{{S - {m_1} - 1}}A$, and ${B^{{m_1}}}\left( {{r_1}} \right) = \frac{1}{{S - {m_1} - 1}}B$. A and B are the number of two sets of sequences whose distance is less than or equal to $r_1$. The more complex the time series, the greater the approximate entropy corresponding to it [42].