Drop dynamic analysis of half-axle flexible aircraft landing gear

Xiao-Hui Wei1 , Cheng-Long Liu2 , Xiao-Chen Song3 , Hong Nie4 , Yi-Zhou Shao5

1, 4State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics, China

1, 2, 3, 4, 5Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight Vehicle, Nanjing University of Aeronautics & Astronautics, China

1Corresponding author

Journal of Vibroengineering, Vol. 16, Issue 1, 2014, p. 266-274.
Received 26 July 2013; received in revised form 4 November 2013; accepted 11 November 2013; published 15 February 2014

Copyright © 2014 JVE International Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Abstract.

Landing gear shock strut binding problem occurred during an unmanned aircraft’s flying test. The half-axle main landing gear of the unmanned aircraft was chosen to analyze the influences of shock strut flexibility on drop dynamics. The friction force was modeled based on the half-axle configuration and taking shock strut flexibility into account. Drop dynamic performances were analyzed and compared with those came from rigid strut model and drop test. Good correlation has been established between drop test data and the simulation predicated results. The results also showed that though the total axis force added merely 1 % when taking shock strut flexibility into account, the friction force added almost 45 %. A comprehensive deformation compatibility factor was presented to describe the actual deformation of shock strut bearings. Influence of deformation compatibility factor, flexibility of inner and outer cylinder were studied further.

Keywords: landing gear, drop, dynamics, half-axle, flexibility.

1. Introduction

There are four fundamental force elements, (air spring force, hydraulic force, friction force and structural limit force), in landing gear drop dynamic modeling usually [1]. For researches focus on whole aircraft or control performance, friction force is neglected or simplified frequently. In David H. Chester’s research on aircraft landing dynamics with emphasis on nose gear landing conditions [2], Phil Evans’s research on tricycle landing gear landing dynamics at normal and abnormal conditions [3], and David C. Batterbee’s research on magneto rheological oleo pneumatic landing gear drop dynamics [4], friction force of the shock strut were neglected. Anthony G. Gerardi’s landing gear model included friction force at first. However, the friction force was neglected when taking the symmetrical configuration of landing gear wheels into account [5]. Gu Hongbin simplified shock strut hydraulic damping and friction damping as a linear damping [6] and Yuan dong simplified them as a nonlinear damping [7]. Kapseong Ro [8],Jia Yuhong [9], Mu Rangke [10] mentioned that friction force were concluded in their dynamic models, but the friction force model were not presented in the articles.

For landing gear drop dynamics, friction force is usually taken into account. In Benjamin Milwitzky’s landing gear drop dynamic model [11], shock strut outer cylinder and inner cylinder were assumed as rigid. Shock strut friction force was modeled as the function of reaction forces at upper bearing and lower bearing. Francis E. Cook adopted the same friction force modeling method as Benjamin Milwitzky’s work [12]. Mahinder K. Wahi [13] and Wei Xiaohui [14, 15] inherited this model in their landing gear and aircraft landing dynamics.

Prashant Dilip Khapane neglected structural friction force and modeled the shock strut friction force as the function of inner air pressure, namely seal friction [16]. In fact, the more practical method is to add this seal friction force to structural friction force. James N. Daniels adopted this friction force model [17], and Archie B. Clark [18], Nie Hong [1], and Sui Fucheng [19] inherited this model in their researches respectively.

With the use of high-strength steel, such as 300M steel [20] and A100 steel [21], aircraft landing gear became more and more flexible. It seemed imperative that if the conventional rigid shock strut assumption still suitable for landing gear drop dynamics should be studied.

Frictional damping of the landing gear shock strut should be as small as possible, and it is usually no more than 5 percent of the shock strut axial force [22]. Shi Haiwen mentioned that tire longitudinal force has a great influence on the shock strut friction force [23]. But for the half-axle landing gear, tire vertical force also has an adverse effect on the shock strut friction force.

The shock strut of half-axle landing gear suffered more bending moment than the landing gear with symmetric layout of wheels. If the shock strut is not rigid enough, the influences of shock strut bending are non-ignorable. Gao Zejiong presented that the shock strut’s bending deformation would add an additional moment on the bearings between inner and outer cylinder [22]. That would make the friction characteristic of the shock strut worse and possibly led to binding problem. But there are not any published researches of the effect of shock strut flexibility on the shock strut friction force and drop dynamics of landing gear. Then, the half-axle main landing gear of the aircraft was chosen to analyze the influences of shock strut flexibility on the shock strut friction force and drop dynamics.

2. Landing gear configuration and forces

The structural configuration of the main landing gear of the unmanned aircraft is a kind of half-axle shock strut. Figure 1 shows a schematic representation of the landing gear configuration. Figure 2 shows the forces on wheel and inner strut.

The ground coordinate system Oxyz is fixed to the ground. The origin O is a point located on the ground, axis x is in the forward longitudinal direction, axis y is in the upward direction, axis z is followed the right-hand rule.

Fig. 1. Landing gear configuration

 Landing gear configuration

Fig. 2. Forces on wheel and inner strut

 Forces on wheel and inner strut

In Figure 2: μ is the friction coefficient of bearings, a is the displacement between wheel axle and lower bearing, b is the displacement between lower bearing and upper bearing, Lt is the displacement between pivot point of torque arm and lower bearing, rK is the displacement between the center of wheel axle and shock strut axis, rT is the displacement between the line of torque arm action force and shock strut axis, Fxl and Fyl is the longitudinal and vertical ground reaction force respectively, N1x and N1z is the upper bearing reaction force in Oxy plane and Oyz plane respectively, N2x and N2z is the lower bearing reaction force in Oxy plane and Oyz plane respectively, Pxl and Pyl is the longitudinal and vertical wheel axle force respectively, T is the torque arm action force.

3. Drop dynamic model

The basic equations of motions are those used for a two-degree-of freedom system as shown in Figure 3. Figure 4 shows the schematic representation of the land gear’s oleo-pneumatic shock strut.

Fig. 3. System with two degrees of freedom

 System with two degrees of freedom

Fig. 4. Schematic representation of shock strut

 Schematic representation of shock strut

Equations of motion, tire forces and shock strut forces are modeled according to Nie’s research [1].

Equation of motions can be expressed by:

(1)
m 1 y ¨ 1 = m 1 g - F a - F h - F f ,
(2)
m 2 y ¨ 2 = - F V + m 2 g + F a + F h + F f ,

where m1 and m2 is the sprung and unsprung mass respectively, Fa is pneumatic force, Fh is hydraulic force, Ff is friction force.

Equations of motion, tire forces and shock strut forces are the same as Nie’s research, except that shock strut friction force is modeled as follow.

Friction in this gear comes mainly from two sources, friction due to tightness of the seal and friction due to the forces on tire(moment), and can be expressed by:

(3)
F f = F f 1 + F f 2 ,

where Ff is shock strut friction force, Ff1 is seal friction force and Ff2  is friction force due to the forces on tire.

The seal friction is assumed to be a function of internal air pressure and can be expressed by:

(4)
F f 1 = μ s e F a ,

where μse is the friction coefficient of seal cup, Fa is the shock strut air spring force.

The friction due to the forces on tire is the result of the moment produced by the nonaxially loaded piston within the cylinder. Considering flexibility of the shock strut, Figure 5 shows the forces in Oyz plane, then:

(5)
N 1 z - N 2 z + S + T = 0 ,
(6)
M 1 z + M 2 z + N 1 z b - V r K - S a E - T L T = 0 ,

where aE is the displicement between ground and lower bearing, d is the displacement between upper bearing and the end of outer cylinder, S is the side ground reaction force, M1z and M2z is the bending moment due to shock strut flexibility at upper and lower bearing in Oyz plane respectively.

Fig. 5. Forces in Oyz plane

 Forces in Oyz plane

Fig. 6. Forces in Oxy plane

 Forces in Oxy plane

Figure 6 shows the forces in the Oxy plane, then:

(7)
N 1 x - N 2 x + D = 0 ,
(8)
M 1 x + M 2 x + N 1 x b - D a = 0 ,

where M1x and M2x is the bending moment due to shock strut flexibility at upper and lower bearing in Oxy plane respectively.

Assuming that the deformation of outer cylinder and inner cylinder at upper and lower bearings is compatible, ideal situation is shown in Figure 7.

Fig. 7. Ideal deformation compatibility

 Ideal deformation compatibility

Then, the equation of deformation compatibility can be derived as:

(9)
θ A o = θ - θ A i ,
(10)
θ B o = θ B i + θ ,

where θA0 and θBo is the deformation angle of outer cylinder at upper and lower bearing respectively, θAi and θBi is the deformation angle of inner cylinder at upper and lower bearing respectively, θ is the angle between the line connecting A and B before and after the deformation.

In fact, there is a slight deflection angle due to bearing deflection or fit clearance as shown in Figure 8.

Fig. 8. Actual deformation compatibility

 Actual deformation compatibility

(a)

 Actual deformation compatibility

(b)

Then Eq. (9) and (10) can be revised as:

(11)
θ A o = θ - θ A i - θ A ,
(12)
θ B o = θ B i + θ - θ B ,

where θA and θB is the slight deflection angle due to bearing deflection or fit clearance at upper and lower bearing respectively.

θ A and θB are affected by many factors. This paper presents a comprehensive deformation compatibility factor Kθ=0~1 to describle actual deformation. The value of Kθ can be obtained from special test.

Then the total bending moment and reaction force at upper and lower bearing can be expressed as:

(13)
M 1 = ( M 1 z ) 2 + ( M 1 x ) 2 ,
(14)
M 2 = ( M 2 z ) 2 + ( M 2 x ) 2   ,
(15)
N 1 = ( N 1 z ) 2 + ( N 1 x ) 2   ,
(16)
N 2 = ( N 2 z ) 2 + ( N 2 x ) 2   ,

where M1 and M2 is the total bending moment at upper and lower bearing respectively, N1 and N2 is the total reaction force at upper and lower bearing respectively.

In order to calculate the equivalent reaction force, the forces at upper and lower bearing can be model as Figure 9.

Fig. 9. Equivalent forces

 Equivalent forces

According to Figure 9, then:

(17)
N A 1 - N A 2 = N 1 ,
(18)
( N A 1 + N A 2 ) d A 2 = M 1 ,
(19)
N B 2 - N B 1 = N 2 ,
(20)
( N B 1 + N B 2 ) d B 2 = M 2 ,

where dA and dB is the width of upper and lower bearing respectively, NA1 and NA2 is the equivalent reaction force at upper bearing, NB1 and NB2 is the equivalent reaction force at lower bearing.

Then, the total reaction force of bearings when taking shock strut flexibility into account can be expressed by:

(21)
N = N A 1 + N A 2 + N B 1 + N B 2 ,

where N is the total reaction force of bearings.

Then, the total friction force due to the forces on tire can be expressed by:

(22)
F f 2 = μ N .

Friction model can be expressed [24] by:

(23)
F f = F f s ˙ ,     s ˙   0 , F e , s ˙ = 0     and     F e < F f S , F f S sgn F e , other ,

where Fe is the sum of outer forces, FfS is shown as Figure 10.

Fig. 10. Schematic diagram of friction model

 Schematic diagram of friction model

4. Numerical calculation and analysis

Parameters of the landing gear used in the analysis are list in Table 1.

Table 1. Parameters of the landing gear

Parameters
Value
Parameters
Value
Parameters
Value
Parameters
Value
N m 1   / kg
692.5
a 0   / m
0.352
a E 0   / m
0.542
μ s e
0.06
N m 2   / kg
15
b 0   / m
0.134
r T 0   / m
0.126
μ x
0.3
V s i n k   / (m/s)
2.34
d 0   / m
0.266
μ
0.025
R 0 / m
0.19
V x   / (m/s)
36.7
r K   / m
0.165
L T 0   / m
0.127
k θ
0.2

In Table 1, Vsink is the sink speed of the landing gear, Vx is the longitudinal speed of the landing gear, μx is the friction coefficient between tire and ground, R is the tire radius, subscript 0 stands for the value at initial time.

According to the drop dynamic model constructed above, an analysis program based C++ was developed to calculate the dynamical response of the landing gear during drop process. The results are shown as Figure 11, Figure 12 and Table 2.

Fig. 11. Time history of shock strut axis force

 Time history of shock strut axis force

Fig. 12. Time history of friction force

 Time history of friction force

Drop dynamic performances came from rigid strut model were presented in Figure 11, Figure 12 and Table 2 in addition.

Table 2. Comparison of numerical calculated and test results

F S m a x   / N
V m a x   / N
D m a x   / N
F f m a x   / N
n
Test results
/
22400
6630
/
3.30
Rigid strut model
Calculated results
21643
22038
6611
3296
3.25
Relative error
/
–1.6 %
–0.2 %
/
–1.6 %
Flexibility strut model
Calculated results
21855
22245
6674
4764
3.28
Relative error
/
–0.6 %
0.5 %
/
–0.6 %

In Table 2, Fs is the total axis force of the shock strut, subscript max stands for the maximum value.

According to Figure 11 and Table 2, good correlation has been established between drop test data and the simulation predicated results. Compared with the rigid model, the total axis force added merely 1 % when taking shock strut flexibility into account. Though shock strut flexibility has a tiny influence on shock strut axis force, the friction force added enormously according to Figure 12 and Table 2. The friction force added almost 45 % when taking shock strut flexibility into account. Shock strut friction force is the main factor that leads to shock strut binding problem.

Comprehensive deformation compatibility factor have a great influence on shock strut friction force according to Eq. (11) and Eq. (12). In order to learn how will this comprehensive deformation compatibility factor affects shock strut friction force, comparisons were presented in Figure 13 and Table 3.

Fig. 13. Effect of deformation compatibility factor

 Effect of deformation compatibility factor

Table 3. Comparison of deformation compatibility factor

k θ
0.2
0.1
0.3
Relative variation
/
–50 %
50 %
Max friction force
4764
2777
7529
Relative variation
/
–41.7 %
58.0 %

According to Figure 13 and Table 3, shock strut fricton force will decrease 41.7 % when deformation compatibility factor equals 0.1, and will increase 58.0 % when deformation compatibility factor equals 0.3. Shock strut fricton force vary almost linearly with comprehensive deformation compatibility factor.

In order to learn how will shock strut flexibility affects shock strut friction force, comparisons were presented in Figure 14, Figure 15 and Table 4.

According to Figure 14, Figure 15 and Table 4, shock strut friction force will be increased with the increasing of the flexibility of outer cylinder or decreasing the flexibility of inner cylinder. 30 % decreasing of the flexibility of inner cylinder leads to 15.1 % increasing of shock strut friction force.

Fig. 14. Effect of the flexibility of outer cylinder

 Effect of the flexibility of outer cylinder

Fig. 15. Effect of the flexibility of inner cylinder

 Effect of the flexibility of inner cylinder

Table 4. Comparison of shock strut flexibility

Flexibility of outer cylinder
Flexibility of inner cylinder
Relative variation
/
–30 %
30 %
–30 %
30 %
Max friction force
4764
4339
5170
5284
4671
Relative variation
/
–8.9 %
8.5 %
15.1 %
–2.0 %

5. Conclusions

(1) Shock strut flexibility has a tiny influence on the total shock strut axis force during drop process. However, it has a enormous influence on shock strut friction force. Though the total axis force added merely 1 % when taking shock strut flexibility into account, the friction force added almost 45 %.

(2) Comprehensive deformation compatibility factor has a great influence on shock strut friction force. Shock strut fricton force vary almost linearly with Comprehensive deformation compatibility factor.

(3) Increasing the flexibility of outer cylinder or decreasing the flexibility of inner cylinder will increase the shock strut friction force. In order to eliminate the binding problem of this landing gear, the relative flexibility of outer cylinder and inner cylinder should be decreased suitably.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (Grant Nos: 51105197 and 51075203) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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