Stability Analysis of Nonlinear Models of Nose Landing Gear Shimmy

Shimmy can reduce the service life of the nose landing gear, affect ride comfort, and even cause fuselage damage leading to aircraft crashes. Taking a light aircraft as the research object, the torsional freedom of landing gear around strut axis and lateral deformation of tire are considered. Since the landing gear shimmy is a nonlinear system, a nonlinear mechanical model of the front landing gear shimmy is established. Sobol index method is proposed to analyze the influence of structural parameters on the stability region of the nose landing gear, and Routh-Huritz criterion is used to verify the reliability of the analysis results of Sobol index method. We analyse the effect of torsional stiffness of strut, caster length, rated initial tire inflation pressure, rake angle, and vertical force on the stability region of theront landing gear. And the research shows that the optimization of the torsional stiffness of the strut and the caster length of the nose landing gear should be emphasized, and the influence of vertical force on the stability region of the nose landing gear should be paid attention to.

• Nose Landing Gear
• Shimmy Oscillations
• Stability
• Sobol Index Method

1. Introduction

The landing gear is suddenly excited by external excitation, which will produce certain deformation and torsion angle. When the external excitation disappears, the self-excited nonlinear instability phenomenon of the wheel torsion around the landing gear strut axis is called landing gear shimmy.

In the middle of the last century, Moreland [1] and Smiley [2] proposed the Point Contact Theory and Stretched String Theory with different assumptions based on tyre deformation. After a lot of theoretical analysis and experimental research, the two theories have been widely recognized and used. Boeckh [3] had measured the deformation of four different tires through experiments, and obtained the rotational inertia of the tire around the vertical axis and verified the Stretched String model. Besselink et al. [4] thought that with the change of motion state, the motion characteristics of tires would also change, and it changed greatly. Thota et al. [5] studied the relationship between the nose landing gear shimmy oscillation characteristics of passenger aircraft and tire inflation pressure, also conducting a bifurcation analysis of landing gear shimmy oscillation, then obtained two-parameter bifurcation diagrams for five different inflation pressures, which showed that the landing gear is not susceptible to shimmy oscillation at higher than nominal inflation pressure. Ran et al. [6] introduced the Tyre models with the Magic Formula and a non-constant relaxation length, and compared the two tire models using the energy flow method. The results show that the shimmy model with non-constant relaxation length tire model can obtain more accurate results when the amplitude is large.

Analysis of landing gear shimmy frequency domain is an important theme of landing gear shimmy. Sateesh et al. [7] pointed out that runway excitation leads to a significant reduction in the critical shimmy speed of the landing gear. The interaction of runway roughness and landing gear torsional free play adversely affects the lateral stability of the landing gear. Sura et al. [8] established a three-degree-of-freedom nose wheel landing gear shimmy model, and obtained the analytical expressions of shimmy velocity and shimmy frequency (shimmy instability), the shimmy frequency approximates the lowest natural frequency of the nose landing gear on the ground. Chuban [9] used the finite element method to obtain the natural frequencies and modes of landing gear shimmy.

In academic circles landing gear shimmy is generally considered to be a shimmy problem consisting of multiple degrees of freedom coupled together by tire deformation, strut torsional deformation and strut lateral deformation. Somieski [10] established a two-degree-of-freedom shimmy dynamics model for the nose landing gear and analyzed the stability problem of shimmy. Thota et al. [11] has established a three-degree-of-freedom shimmy dynamics model of the nose landing gear. The three degrees of freedom are the lateral bending of the strut, the torsion of the strut and the lateral deformation of the tire. The bifurcation analysis of the landing gear shimmy is carried out using the numerical continuation method, and the shimmy stability region was analyzed. Thota et al. [12] added the longitudinal bending of the strut on the basis of his existing research, and concluded that the longitudinal bending of the strut has very little impact on the stability area of the landing gear shimmy. Rahmani et al. [13] divided the torsion of the strut into three degrees of freedom (the lower strut rotates around the strut axis, the upper strut rotates around the strut axis, and the shimmy damper rotates around the strut axis) for analysis. Through analysis, it was found that smaller stiffness and larger damping of the shimmy damper can improve the performance of the shimmy damper.

With the in-depth study of landing gear oscillation, it is found that many details cannot be ignored. Collins et al. [14] divided the landing gear shimmy into “Tire-Yaw” shimmy and “Structural-Torsion” shimmy. Behdinani et al. [15] adopted the treatment method similar to that of Howcroft et al. [16] for the torque link apex freeplay, and established 1 the freeplay nose landing gear shimmy model including Coulomb friction factor. The results show that the stability of the nose landing gear system decreases with the increase of the freeplay. It is found that the increase of Coulomb friction factor will reduce the shimmy stability area dominated by torsion. Kewley et al. [17] added the lateral and vertical displacement of the fuselage on the basis of the model proposed by Thota et al. [18] , representing the yaw of the fuselage on the runway and the pitch of the fuselage respectively. Jiang et al. [19] established a nonlinear dynamic model of nose landing gear shimmy coupled with time-varying load and Coulomb friction, and showed that the shimmy reduction effect of Coulomb friction is weakened under time-varying load.

The paper is structured as follows. Firstly, a mathematical model of nose landing gear shimmy is given and a sensitivity analysis of the nose landing gear structural parameters using Sobol’s index is carried out in Section 2. The results of the Sobol index analysis are validated using Ross-Holwitz in Section 3. The conclusions are given in Section 4.

2. The Mathematical Model

A model diagram of a single-wheel nose landing gear for a light airplane is shown in Figure 1. The nose landing gear is a highly coupled nonlinear system, consisting mainly of strut sections, wheels, et al. The strut section consists of the upper strut and the lower strut, etc. (Table 1). The torque is transmitted through the upper and lower torque arms. This type of nose landing gear is from a light aircraft with retractable nose landing gear. It has a Caster length L, a Rake angle δ, and is taxiing on the runway at the same speed as the fuselage through wheels of radius R. The positive X-axis refers to the backward direction of the aircraft, the Y-axis has its positive direction determined by the right-hand coordinate system, and vertically up is the positive direction of the Z-axis.

In this paper, the nose landing gear shimmy is described by two degrees of freedom: the torsion angle (θ), which represents the rotation of the strut about its axis S, and the lateral deformation of the tire (Δ). It is assumed that the tires do not slip relative to the ground while the aircraft is taxiing on the runway without considering the effect of the fuselage on the front landing gear.

Because of the Rake angle δ , the swivel Angle ζ of the wheel is slightly lower than the torsion Angle ( θ ) of the nose landing gear strut, and its magnitude is equal to ζ = θ cos ( δ ) . The stretched string theory proposed by Von-Schlippe was adopted to simplify the tire model of the nose landing gear, as shown in Figure 2. The tyre will slip, when the lateral force exceeds a certain limit. The slip angle of each tyre is given by η = arctan ( Δ / σ ) ≈ Δ / σ .

Tire compression λ can be obtained from the following formula:

Table 1

Structural parameters of the nose landing gear

Symbols	Parameter	Value	Unit
Geometric parameters
δ	Rake angle	0.1571	rad
L	Caster length	0.07	m
Strut parameters
K θ	Torsional stiffness of strut	10,000	N·m/rad
C θ	Torsional damping of strut	10	N·m·s/rad
I Z	Torsional inertia of strut	1	Kg·m2
Wheel parameters
D	Wheel diameter	0.3	m
W	Wheel width	0.125	m
P 0	Rated initial tire inflation pressure	600000	pa
P r	Rated tire inflation pressure	600000	pa
C M η	Tire rotational stiffness coefficient	2	rad−1
C F η	Tire lateral stiffness coefficient	20	m/rad
K	Tire rotational damping coefficient	270	N·m2/rad
K n	Tire lateral stiffness coefficient	2.8	rad−1
η m	self-aligning moment Limit side angle	0.1744	rad
η F	Tire Lateral Force Limit side angle	0.0872	rad
external parameter
F Z	Vertical force	1800	N
V	velocity	0 - 100	m/s

λ = F Z 2.4 ( P 0 + 0.08 P r ) W D + 0.03 W (1)

Half of the Contact patch length a can be expressed by the amount of tire compression λ and the diameter of the wheel D:

a = 0.85 D λ D − ( λ D ) 2 (2)

P is the internal pressure of the tire when the corresponding compression amount is λ :

P = P 0 + 1.5 W D P 0 ( λ W ) 2 (3)

Relaxation length σ is obtained from the following formula:

σ = ( 2.8 − 0.8 P P r ) ( 1 − 4.5 λ D ) W (4)

In summary, considering strut torsion and tire lateral deformation, the equation of nose landing gear nonlinear shimmy is given, as shown in Equation (5)-(6):

I Z θ ¨ + C θ θ ˙ + K θ θ + M F + M D = 0 (5)

Δ ˙ + V / σ Δ − V ζ − ( L e f f − a ) ζ ˙ = 0 (6)

Because of the Rake angle δ , the effective Caster length L e f f is no longer equal to L, which is given by:

L e f f = L cos ( δ ) + ( D / 2 + L sin ( δ ) ) tan ( δ ) (7)

The typical changing tendency of the nonlinear restoring force of the tire F_n and self-aligning moment M_n with lateral deformation of the tire is shown in Figure 3. The combined moment M_F is generated by the self-aligning moment M_n and the nonlinear restoring force of the tire F_n, which equals to

M F = ( M n + L e f f F n ) cos ( δ ) (8)

F n = K n tan − 1 ( 7.0 tan ( η ) ) cos ( 0.95 tan − 1 ( 7.0 tan ( η ) ) ) F Z (9)

M n = { C M η F Z ( η m 180 ∘ ) sin ( 180 ∘ η m η ) | η | ≤ η m 0 | η | ≥ η m (10)

The moment M_D is generated by the tire tread damping and is given as

M D = K V ζ ˙ (11)

3. Sensitivity Analysis of Structural Parameters

Sensitivity analysis is to determine the extent to which the parameters affect the model output, and provide a basis for the next model optimization, and also improve the efficiency of the optimization. Sensitivity analysis methods are divided into two categories: local sensitivity analysis and global sensitivity analysis. Local sensitivity is suitable for mathematical expressions that are relatively simple and easy to differentiate. Local sensitivity analyzes the effect of changes in only a single parameter on the system. Global sensitivity analysis can analyze the influence of a single parameter change on the model output, and can analyze the influence of multiple parameter interactions on the model output.

Due to the consideration of the nonlinear self-aligning moment and the restoring force of the tyre, the shimmy model is a system of nonlinear equations. At present, the critical velocity of shimmy cannot be given by the function analytical expression, but by numerical simulation (as shown in Figure 4). The number of structural parameters of the shimmy model is relatively large, and the global sensitivity analysis method is more suitable for the sensitivity analysis of this model than the local sensitivity analysis method. The Sobol index method in the global sensitivity analysis was chosen for comprehensive consideration; the sensitivity index cannot be calculated directly by the functional analysis method, but it can be calculated by the Monte Carlo method.

Structural parameter sensitivity analysis of the nose landing gear shimmy was performed, and the following structural parameters were selected: Torsional stiffness of strut, Caster length, Rated initial tire inflation pressure, Rake angle, and Vertical force. The range of values for each structural parameter is shown in Table 2.

For ease of representation, V_cr is simplified to V. The critical velocity of shimmy V is used as the output of the nose landing gear shimmy model:

Table 2

Range of values for relevant structural parameters

Symbols	range of values	Unit
K θ	[1000, 20,000]	N·m/rad
L	[0.001, 0.117]	m
P 0	[110,000, 1,200,000]	pa
δ	[0, 0.3]	rad
F Z	[1510, 3600]	N

V = f ( x ) (12)

Sampling is based on the range of the five independent variables described above (Figure 5), and the Sobol sequence sampling method is chosen for the sampling. This will generate a n × 2 k sample matrix:

M = ( x 11 … x 1 k … x 12 k x 21 … x 2 k … x 22 k ⋮ ⋮ ⋮ x n 1 ⋯ x n k ⋯ x n 2 k ) (13)

The first k columns of the extraction matrix M are set to the parameter matrix A, and the last k columns are set to the parameter matrix B:

A = ( x 11 x 12 … x 1 k x 21 x 22 … x 2 k ⋮ ⋮ ⋮ x n 1 x n 2 ⋯ x n k ) (14)

B = ( x 1 k + 1 x 1 k + 2 … x 12 k x 2 k + 1 x 2 k + 2 … x 22 k ⋮ ⋮ ⋮ x n k + 1 x n k + 2 ⋯ x n 2 k ) (15)

where: k is the number of covariates ( k = 5 ); n is the number of samples sampled ( n = 2000 ).

The new parameter matrix A B i ( i = 1 , 2 , ⋯ , k ) is obtained by replacing the ith column in matrix A with the value of the ith column of matrix B:

A B i = ( x 11 x 1 k + i … x 1 k x 21 x 2 k + i … x 2 k ⋮ ⋮ ⋮ x n 1 x n k + i ⋯ x n k ) (16)

The parameter matrices A, B and A B i are introduced into the pendulum oscillation model to obtain the system output vectors V A 、 V B and V A B i for each parameter matrix. According to the Monte Carlo method, the first order sensitivity (main effect) index S x i and the full effect index S x i T for each variable are denoted as

S x i = V a r x i ( E x − i ( V | x i ) ) V a r ( V ) (17)

S x i T = E x − i ( V a r x i ( V | x i ) ) V a r ( V ) (18)

where V a r x i ( E x − i ( V | x i ) ) 、 E x − i ( V a r x i ( V | x i ) ) and V a r ( V ) are respectively:

V a r x i ( E x − i ( V | x i ) ) ≈ 1 n V B T ( V A B i − V A )

E x − i ( V a r x i ( V | x i ) ) ≈ 1 2 n ( V A T V A + V A B i T V A B i − 2 V A T V A B i )

V a r ( V ) = V a r ( [ V A V B ] )

x − i is the remaining variable with x i removed; E x − i ( V | x i ) is a series of fixed values for x i , with x − i taken multiple times over a range of variation at each fixed value, to find the mean of these outputs V; V a r x i ( E x − i ( V | x i ) ) seeks the variance of E x − i ( V | x i ) ; V a r x i ( V | x i ) is a series of fixed values for x i , with x − i taken multiple times over a range of variation at each fixed value, to find the variance; and E x − i ( V a r x i ( V | x i ) ) seeks the mean of V a r x i ( V | x i ) .

The greater the value of the first-order sensitivity indicator S x i , the greater the effect of the univariate x i on the output response of the system. The full effect indicator S x i T has the following role: contains the first-order sensitivity of the variable x i ; also contains the effect of the interaction between x i and other variables on the output response, and the greater the difference between its value and the first-order sensitivity indicator, the more pronounced the interaction.

The histograms of the first-order sensitivity and the sensitivity of the full effects metrics for each structural parameter are shown in Figure 6(a), and the difference between these two metrics is shown in Figure 6(b). Through the sensitivity analysis of these structural parameters, it is found that the first-order sensitivity of the torsional stiffness of strut, the caster length and the vertical force are greater than the other two structural parameters, and the difference between the same first-order sensitivity index and the full effect index is also greater. Therefore, the above three structural parameters have a greater influence on the critical velocity of shimmy, and the optimization of structural parameters needs to focus on the torsional stiffness of strut, the caster length and the vertical force.

4. Validation of Results

In order to verify the above results, the nonlinear shimmy model of the nose landing gear is simplified into a linear model for processing, and the influence law of the relevant structural parameters on the stability of the nose landing gear is kept constant.

Express the above nonlinear restoring force of the tire F_n expression as a segmented function:

F n = { C F η F Z η | η | ≤ η F C F η F Z η F s i g n ( η ) | η | ≥ η F (19)

When small torsional displacements of the landing gear structure are considered, the nonlinear restoring force of the tire F_n and self-aligning moment M_n can be reduced to a function of the Vertical force proportional to the corresponding coefficients.

{ F n = C F η F Z η M n = C M η F Z η (20)

Therefore, the set of equations for the nonlinear dynamics of the nose landing gear shimmy can be represented by a linear set of state equations:

x 1 = θ , x 2 = θ ˙ , x 3 = Δ

{ x ˙ 1 = x 2 , x ˙ 2 = ( − K θ x 1 − C θ x 2 − ( L e f f C F η + C M η ) F Z x 3 σ cos ( δ ) − K V cos ( δ ) x 2 ) / I Z , x ˙ 3 = − V / σ x 3 + V cos ( δ ) x 1 + ( L e f f − a ) cos ( δ ) x 2 , (21)

The system of linear equations of state is changed to matrix form A, and the frequency of the nose landing gear shimmy is found by finding the imaginary

part of the complex eigenvalues of the matrix. Figure 7 depicts the strut torsional vibration frequency corresponding to a change in speed. At low speeds, the effect of velocity on the vibration frequency is extremely significant, and the change in vibration frequency becomes relatively flat near the takeoff speed.

D = [ 0 1 0 − K θ − ( C θ + K V cos ( δ ) ) / I Z − ( L e f f C F η + C M η ) F Z cos ( δ ) σ I Z V cos ( δ ) ( L e f f − a ) cos ( δ ) − V / σ ] (22)

The Routh-Huritz criterion is used to solve for the critical velocity of the nose landing gear shimmy. Firstly, the characteristic Equation (23) is derived from the system of linear Equation (22), and the characteristic equation is written in the form of a polynomial (24), and secondly, the Routh array of Equation (24) is written as shown in Table 3.

| I Z s 2 + K θ + C θ s + K V s cos ( δ ) ( L e f f C F η + C M η ) F Z σ cos ( δ ) − V cos ( δ ) − ( L e f f − a ) cos ( δ ) s s + V σ | = 0 (23)

a 3 s 3 + a 2 s 2 + a 1 s + a 0 = 0 (24)

where the coefficients are given by

a 3 = I Z

a 2 = I Z V 2 + C θ σ V + σ K cos ( δ ) V σ

a 1 = C θ V + K θ σ + K cos ( δ ) + ( − F Z C M η a + F Z L e f f 2 C F η + F Z L e f f C M η − F Z L e f f C F η a ) cos ( δ ) 2 σ

a 0 = K θ V + ( F Z V C M η + F Z L e f f C F η V ) cos ( δ ) 2 σ

According to the Ross criterion, the nose landing gear shimmy system is stable if the first column of the Ross array is all positive. Figure 8 depicts the effect

Table 3

Ross array of characteristic polynomials

	First column	Second column
s 3	a 3	a 1
s 2	a 2	a 0
s 1	a 1 a 2 − a 3 a 0 a 2	0
s 0	a 0	0

of Torsional stiffness of strut, Caster length, Rated initial tire inflation pressure, Rake angle, and Vertical force on the critical velocity of the nose landing gear Critical speed of shimmy, based on the reference case (Table 1).

There are three nose landing gear structural parameters that have a significant effect on shimmy: strut torsional stiffness, Torsional stiffness of strut, Caster length and Vertical force. As the torsional stiffness of strut increases the critical velocities of shimmy increase rapidly, from 9.7 m/s at the beginning to 41.1 m/s. The effect of the Caster length on the critical velocity of shimmy is a small decrease followed by a sharp increase, but the magnitude of the uncontrolled increase negatively affects the forces on the nose landing gear and the turning of the wheels. The increase of vertical force will greatly reduce the velocity of the shimmy, which has a great impact on the take-off and landing of the aircraft. When the rated initial inflation pressure of the tire is very small, the critical velocity of the shimmy will be reduced, and the effect of increasing the inflation pressure near the rated inflation pressure of the tire is very small. With the increase of the rake angle, the critical velocity of the shimmy also increases gradually. Compared with the other four structural parameters, the overall rate of change is relatively stable and gentle.

Through the above analysis, the reliability of Sobol index method to judge the influence of structural parameters on the critical velocity of nose landing gear shimmy is verified.

5. Conclusions

In this paper, a nonlinear shimmy model of landing gear is established. The influence of structural parameters related to the nose landing gear of light aircraft on the shimmy stability is analyzed. The main conclusions are as follows:

The velocity of the aircraft can affect the vibration frequency of the shimmy. At low velocity, the influence of velocity on the vibration frequency is very large.

The Sobol index method is in good agreement with the Routh-Huritz criterion to analyze the influence of structural parameters on the stability region of the nose landing gear. The torsional stiffness of strut is positively correlated with the critical velocity of the shimmy, while the vertical force is negatively correlated. The larger caster length can greatly increase the critical shimmy velocity, but it also has negative effects (such as wheel turning). Rated initial tire inflation pressure and rake angle relative to the first three parameters, the impact is small.

In future work, the Sobol index method can be used to analyze the parameters that have a large impact on the shimmy. It can save the analysis time of each parameter and improve the optimization efficiency.

Acknowledgements

This work is supported by the National Nature and Science Foundation of China under Grant No. 12072140, Natural Science Foundation of Jiangxi Province of China under Grant No. 20202ACBL201003.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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