Study on the Fast Extension Mechanism of Double-Cavity Shock Absorber with High-Pressure Piston

Double-cavity shock absorber with high-pressure piston is the core component of the nose landing gear of the carrier-based aircraft, and its fast-extension performance seriously affects the safety of the catapult-assisted takeoff. The design of a carrier-based aircraft in our country is carried out based on the traditional method of fast-extension dynamics, and it is found that the fast-extension capability is larger than designed. This paper analyzes the working principle of the high-pressure piston shock absorber and explains that the high-pressure air cavity pushes the piston rod to extend rapidly, which will cause the cavitation phenomenon in the main oil chamber. Thus, the cavitation in the main oil chamber makes the traditional modeling method of oil-liquid resistance force no longer applicable. Then, the axial force modeling method of shock absorber considering the cavitation effect is proposed. Based on the carrier-based aircraft, the dynamic response of the shock absorber in the process of fast extension is calculated and then it is compared with the calculation results of the traditional dynamic method. It is found that due to the cavitation effect caused by the forced fast extension section of the high-pressure air plug shock absorber, the fast extension work increases by 67.6%, thus, revealing the fast extension mechanism of the double-chamber shock absorber with high-pressure piston and successfully explaining the phenomenon of the fast extension ability exceeding the expectation of the shock absorber.


Introduction
The shock absorber is the key component of the aircraft landing gear. For catapult carrier-based aircraft [1], the shock absorber should not only meet the landing shock absorber requirements of carrier-based aircraft but also the catapult fast-extension performance. Therefore, the double cavity shock absorber with high-pressure piston which can meet the above contradictory functional requirements is one of the common configurations of the nose landing gear of carrier-based aircraft.
The dynamic characteristic analysis of the shock absorber is very important in the landing gear design. As early as 1951, Walls studied the parameter range of the variable index of air spring force [2] and contraction coefficient of oil hole [3] through the landing gear drop test. In 1976, Wahi studied the gas dissolution in the oil and the influence of oil compressibility on the dynamic characteristics of the shock absorber [4], and he also established a drop dynamic model including the dynamic characteristics of the shock absorber as well as the spin-up and spring back of the tire [5]. In 2009, Karam established the dynamic model of landing gear including the pressure of each cavity of shock absorber, the compressibility of oil, and the heat exchange between oil and gas [6], which has been verified by experiments. Chen et al. [7] found the abnormal fluctuation of vertical load in the landing gear drop test and judged that there was cavitation in the main oil chamber. Through theoretical calculation and laboratory test, it was proved that the existence of cavitation in the main oil chamber affects the landing gear shock absorber efficiency. Shin [8] used Adams to establish the dynamic model of single cavity shock absorber landing gear with cavitation effect, which proved that the simulation results were more consistent with the experimental results when the cavitation effect was taken into consideration.
The internal structure of the double-chamber shock absorber with high-pressure piston is complex, and the dynamic process of its fast extension is in the backward stroke, which is obviously different from the landing shock absorber.
Theoretical and experimental studies have been carried out on the dynamic characteristics of the nose landing gear of carrier-based aircraft. Shen and Huang [9] [10] established the dynamic model of fast extension of the dual cavity shock absorber of landing gear and optimized the oil needle section of the shock absorber. Liu and Li [11] obtained the landing gear double cavity shock absorber performance sample by ADAMS simulation analysis and established the surrogate model by using the nonlinear response surface method to optimize the landing gear dual cavity shock absorber performance. Zhang et al. [12] optimized the parameters of the landing gear dual cavity shock absorber by comprehensively considering the landing gear double cavity shock absorber's landing shock absorber and fast-extension performance. Dou et al. [13] and Liu and Cui [14] carried out an experimental study on the fast extension process of landing gear with double cavity shock absorber, studied the influence of internal structure and filling parameters of shock absorber on the fast extension performance, and pointed out that the improvement of shock absorber performance would be accompanied by the decrease of shock absorber efficiency.
In the above research on nose landing gear nose extension performance, the air spring force and oil damping force of shock absorber are modeled by traditional hydropneumatic shock absorber dynamics.
The design of a carrier-based aircraft is carried out according to the traditional method of fast extension dynamics, and it is found that the fast extension capability is larger than designed. Based on the study of the working condition of each cavity of the high-pressure piston shock absorber, it is found that the negative pressure appears in the main oil chamber of the shock absorber during the fast extension process, resulting in the cavitation phenomenon, which leads to the invalidity of some assumptions in the modeling method of force elements such as oil resistance force in the traditional method of fast extension dynamics. Then, the dynamic pressure modeling method of each cavity in the shock absorber based on the elastic oil model is proposed, and the axial force calculation model of high-pressure air plug shock absorber considering the cavitation effect is established. Finally, the dynamic analysis of the nose landing gear fast extension process is carried out.

Working Principle of High-Pressure Air Plug Shock Absorber and Causes of Cavitation
The internal structure of the double chamber shock absorber with high-pressure piston is shown in Figure 1(a). The shock absorber is mainly composed of high-pressure piston (1), piston retaining ring (2), side oil hole (3), main oil hole (4), and piston rod (5). Compared with the single chamber shock absorber shown in Figure 1(b), the shock absorber with high-pressure piston has added high-pressure piston and piston retaining ring. Each component of the shock absorber divides the internal space into four chambers, namely, high-pressure gas chamber (6), low-pressure gas chamber (7), side oil chamber (8), and main oil chamber (9). Com-pared with the single chamber shock absorber, highpressure gas chamber is added. Figure 2 shows the force analysis of shock absorber with high-pressure piston and conventional shock absorber in the piston extension. The compressed gas in the gas chamber of the conventional shock absorber (Figure 2(b)) compresses the oil in the gas chamber downward, and the oil flows to the main oil chamber through the main oil hole, squeezing the end face of the piston rod, so as to push the piston rod to extend outwards. In this process, the oil in the shock absorber remains squeezed, and there will be no negative pressure and cavitation. However, Figure 2(a) shows that there is a highpressure gas cavity in the shock absorber of medium and high-pressure piston. The piston rod is directly squeezed through the high-pressure piston downward (forced fast extension) to push the piston rod to extend outward. Due to the high air pressure in the high-pressure air chamber, the piston rod extends and the volume of the main oil chamber increases rapidly. The flow of oil into the main oil chamber under the action of the pressure in the low-pressure chamber is not enough to fill the main oil chamber. Consequently, a negative pressure appears when the pressure of the main chamber drops, which causes the dissolved gas in the oil to expand and form cavitation.

Traditional Modeling Method of Oil Damping Force and Its Problems
The traditional modeling method of the main oil hole resistance force F oz of conventional hydropneumatic shock absorber is as follows.
where ρ is oil density, A oz is oil pressure area of main oil chamber, C dz is shrinkage coefficient of main oil hole, A hz is area of main oil hole, and s is compression capacity of shock absorber.
When the shock absorber piston rod is extended, _ s and F oz both are negative, then, the above formula can be arranged as follows.
Meanwhile, according to the thin-walled hole flow algorithm [15], the flow formula of oil flowing through the shock absorber main oil hole during the piston rod extension can be written as follows: where q z represents the oil flow of the main oil hole (the flow from the main oil chamber to the low-pressure chamber 2 International Journal of Aerospace Engineering is positive), P d is the pressure of the low-pressure gas chamber, and P z is the pressure of the main oil chamber. From the analogy between formulas (2) and (3), it can be concluded that the oil damping force in formula (1) can represent the force generated by the pressure difference on the main pressure oil area. The difference is caused by the flow rate corresponding to the volume change rate of the main oil chamber flowing through the main oil hole. Then, the pre-mise of the modeling method in formula (1) is that the volume change rate of the main oil chamber is equal to the flow rate of the main oil hole.
However, when cavitation occurs in the main oil chamber, as shown in Figure 2(a), the cavitation occupies a part   (4), the flow rate of the main oil hole is larger than the actual flow rate, so it can be concluded the oil damping force calculated by formula (1) is larger than the actual situation.

Axial Force Model of Shock Absorber considering Cavitation Effect
For the purpose of studying the influence of cavitation effect on the calculation results of reverse stroke axial force of the shock absorber with high-pressure piston, the following work has been carried out. Based on the principle of the shock absorber, the force analysis of the shock absorber piston rod was carried out, and the calculation model of axial force of shock absorber was established by dynamically calculating the pressure of each chamber in shock absorber considering cavitation effect.

Force Analysis of Shock Absorber Piston Rod.
When the high-pressure piston shock absorber works in reverse stroke, it first goes through the high-pressure stroke condition, as shown in Figure 3(a), and then goes through the lowpressure stroke condition, as shown in Figure 3(b). In Figure 3, the mechanical behaviors of piston rod in highpressure stroke condition and low-pressure stroke condition are different. As a result, the axial force of shock absorber is calculated in low-pressure condition and high-pressure condition separately.
where s d is the stroke of shock absorber low-pressure structure.
As shown in Figure 3(a), during the high-pressure stroke condition, the piston rod is affected by the pressure of highpressure gas chamber, low-pressure gas chamber, main oil chamber, and side oil chamber. The expression of resultant force along the axial direction of shock absorber is as follows.
where P g is the pressure of high-pressure gas chamber, P d is the pressure of low-pressure gas chamber, P z is the oil pressure of main oil chamber, P c is the oil pressure of side oil chamber, A g is the annular cross-sectional area of high-pressure piston, A d is the cross-sectional area of low-pressure gas chamber, A z is the cross-sectional area of main oil chamber, and A h is the external cross-sectional area of piston rod.
The axial force on the piston rod during the low-pressure stroke condition of the shock absorber is shown in Figure 3. The piston rod is separated from the high-pressure piston and only bears the pressure of the low-pressure gas chamber, main oil chamber, and side oil chamber. The expression of the shock absorber force on the piston rod is as follows.   International Journal of Aerospace Engineering the oil hole flow, the internal pressure of each cavity should be calculated according to the dynamic state of oil elastic modulus.
In equation (7), the calculation expression of the pressure in the low-pressure chamber is as follows.
where the expression of high-pressure piston stroke is defined below.
In equation (6), the calculation expression of highpressure gas chamber pressure is as follows.
In equation (7), without considering the elastic deformation of the piston rod, the expression of the time derivative of the oil pressure P z in the main oil chamber is defined below.
where V z0 is the initial volume of the main oil chamber under the fully extended shock absorber state, E z is the bulk modulus of the oil in the main oil chamber, and q z is the flow rate of the main oil chamber flowing to the low-pressure gas cavity through the main oil hole.
The expression of oil bulk modulus E in oil chamber [15] is as follows.
where E l is the bulk modulus of pure oil, α is the volume content of gas in the oil, γ o is the polytropic index of gas in the oil, and P is the oil pressure in the oil chamber.
In equation (7), the expression of the time derivative of the oil pressure in the side oil chamber is as follows.
where V c0 is the initial volume of the side oil chamber, E c is the volume modulus of the oil in the side oil chamber, and q c is the flow rate of the side oil cavity flowing to the lowpressure gas cavity through the side oil hole.

Flow Calculation Model of Damping Orifice.
In equation (8), the expression of the flow rate of the main oil hole is as follows.
In equation (11), the expression of side oil hole flow is as follows.
where C dc is the side oil hole shrinkage coefficient, and A hc is the side oil hole area.

Analysis of Dynamic Response Results of Shock Absorber during Fast Extension
Based on the test method of nose landing gear in reference [13], this paper studies the fast extension performance of nose landing gear shock absorber with high-pressure air plug.

Dynamic Model of Nose Landing Gear
Extension. The dynamic model of nose landing gear nose extension test method is modeled by two mass models [14]. Take the basket, counterweight, landing gear strut and strut as elastic mass, and shock absorber piston rod, wheel, and tire as inelastic mass. The dynamic equation is as follows.
where m tx is the elastic support mass, m ft is the inelastic support mass, y tx is the elastic mass vertical displacement, k v is the tire radial stiffness, y ft is the inelastic mass vertical displacement, and g is the gravity acceleration. The dynamic response model is used to calculate the dynamic response of the nose landing gear of a carrierbased aircraft in the fast extension. The total structural stroke of the nose landing gear shock absorber of the carrier-based aircraft is 550 mm, and the structural stroke of the low pressure is 350 mm. At the initial moment of nose landing gear extension, the shock absorber compression is 478 mm. The specific model parameters are shown in Table 1.

Calculation Results of Axial Force Component of Shock
Absorber (Traditional Modeling Method). Figures 4 and 5 show the oil resistance force F oz of main oil hole, air spring force F ad of low-pressure air chamber, and air spring force F ag of high-pressure air chamber in the process of fast extension using the traditional modeling method. The nose landing gear starts to extend at the zero point of the time axis. During the process of extension, the maximum reverse force F oz of shock absorber is 52.5 kN at the time 0.091 s, and the corresponding F ad value at this time is only 19.2 kN.

International Journal of Aerospace Engineering
According to the principle of oil resistance force of main oil hole, in the process of shock absorber elongation, the reverse damping force of main oil hole is generated when oil flows from low-pressure chamber through main oil hole to main oil chamber. As a result, the value of reverse damping force of main oil hole shall not be greater than that of air spring force of low-pressure chamber. From this, we can see that the results of main oil hole reverse damping force and low-pressure chamber air spring force are contradictory. The reason is that the damping force of the main oil hole is not produced by air spring force of high-pressure chamber at 45.1 kN (Figure 5), while the damping force is considered by traditional modeling method.

Comparison of Dynamic Response Calculation Results of
Two Axial Force Models. Figure 6 shows the pressure change of each shock absorber chamber during the fast extension process calculated by the shock absorber model considering cavitation effect. From the pressure curve of the main oil chamber  International Journal of Aerospace Engineering in the figure, the pressure in the main oil chamber has been negative after 0.053 seconds, which indicates that cavitation has been existing in the main oil chamber after 0.053 s. The comparison results of the axial stroke of the shock absorber calculated by two modeling methods of the nose landing gear during extension are given in Figure 7. It can be seen that there is obvious difference between the traditional modeling method and the shock absorber model with cavitation effect. The shock absorber model with cavitation effect reaches the fast extension at 0.1 s. However, it takes 0.130 s for the traditional modeling method to complete the high-pressure stroke. Moreover, due to the cavitation effect, the damping force of the main oil hole in the high-pressure stroke stage is invalid. The axial force calculated by the shock absorber model considering the cavitation effect is larger than that calculated by the traditional modeling method, and the work diagram of the shock absorber is also fuller (Figure 8). In Figure 8, the output work of the shock absorber model with cavitation effect is 9.05 kJ, while that of the shock absorber model with traditional modeling method is 5.40 kJ, with a difference of 67.6%.
In Figure 8, when the shock absorber is in the lowpressure stroke stage (s < 350 mm), there is no obvious difference in the axial direction calculated by the two modeling methods. The reason is that the working principle of the high-pressure piston shock absorber in the low-pressure stroke stage is not significantly different from that of the conventional single cavity shock absorber. It reveals that the traditional modeling method of oil damping force is accurate for the calculation of single cavity shock absorber. Figure 9 shows the vertical velocity of elastic support mass during nose landing gear fast extension. The peak velocity value calculated by shock absorber model with cavitation effect is 2.20 m/s, while that calculated by traditional modeling method is 2.07 m/s, with a difference of 6.3%. At the same time, there are also obvious differences in the vertical displacement calculated by the two shock absorber modeling methods (Figure 10). At the time of 0.3 s, the elastic support mass displacement value calculated by the shock absorber model considering the cavitation effect is 416.5 mm, while that calculated by the traditional modeling method is 338.0 mm, with a difference of 23.2%.

Conclusions
(1) It is concluded that for the shock absorber with highpressure piston, in the forced fast extension condition, the high-pressure air cavity pushes the piston rod to extend rapidly, which will cause the main oil chamber to appear negative pressure and lead to cavitation (2) It is clarified that cavitation in the main oil chamber of the shock absorber high-pressure piston will lead to the hypothesis that the volume change rate of the main oil chamber is equal to the flow rate of the main oil hole, which is the premise of the traditional oil damping force modeling method for sudden extension dynamics (3) Based on the elastic oil model, the dynamic pressure modeling method of each chamber in the shock absorber is proposed, and the axial force model considering cavitation effect is established to calculate the dynamic pressure change in each chamber (4) Based on the study of the dynamics of the shock absorber axial force modeling method considering the cavitation effect, it is found that compared with the traditional modeling method, the output work of the fast extension is increased by 67.6%

Nomenclature
A oz : Oil pressure area of main oil chamber C dz : Shrinkage coefficient of main oil hole F oz : Main oil hole resistance force q z : Oil flow of the main oil hole P z : Pressure of the main oil chamber. P g : Pressure of high-pressure gas chamber A g : Annular cross-sectional area of high-pressure piston A z : Cross-sectional area of main oil chamber s g : High-pressure piston stroke E: Oil bulk modulus α: Volume content of gas in the oil P: Oil pressure in the oil chamber E c : Volume modulus of the oil in the side oil chamber A hc : Side oil hole area m ft : Inelastic support mass y ft : Inelastic mass vertical displacement A hz : Area of main oil hole s: Compression capacity of shock absorber ρ: Oil density P d : Pressure of the low-pressure gas chamber s d : Stroke of shock absorber P c : Oil pressure of side oil chamber A d : Cross-sectional area of low-pressure gas chamber A h : External cross-sectional area of piston rod V z0 : Initial volume of the main oil chamber E l : Bulk modulus of pure oil γ o : Polytropic index of gas in the oil V c0 : Initial volume of the side oil chamber C dc : Side oil hole shrinkage coefficient m tx : Elastic support mass y tx : Elastic mass vertical displacement g: Gravity acceleration.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declared that they have no conflicts of interest to this work.