Dynamic Analysis and Security Characteristics of Carrier-Based Aircraft Arresting in Yaw Condition

applied sciences

Article Dynamic Analysis and Security Characteristics of Carrier-Based Aircraft Arresting in Yaw Condition

Yiming Peng 1, Pengpeng Xie 2, Xiaohui Wei 1,* and Hong Nie 1,*

1 State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; [email protected] 2 Key Laboratory of Fundamental Science for National Defense-Advanced Design Technology of Flight Vehicle, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China; [email protected] * Correspondence: [email protected] (X.W.); [email protected] (H.N.)

Received: 22 January 2020; Accepted: 5 February 2020; Published: 13 February 2020

Abstract: In order to research the safety characteristics of carrier-based aircraft in yaw arrest, a complete dynamic model of the arresting system of a certain type of aircraft was developed to understand more about its dynamic properties. Based on the discrete kink-wave model, a simulation of centering arrest was conducted. The simulation results were compared with experimental data from the United States (US) military standards, demonstrating that the basic changing laws are almost the same. On the basis of centering arrest, a simulation of yaw arrest was carried out. The results show that in yaw state, the difference in the lengths of the arresting cables on either side of the hook is smaller in the early stage after the hook hangs on the rope, which leads to little influence on load fluctuation produced by the kink-wave. With the increase in arresting distance, the difference in the lengths of the arresting cables on either side becomes larger, resulting in a situation in which the cable tension on the departure side will gradually become greater than that on the opposite side. In this situation, yaw landing has a negative impact on the characteristics of arresting safety, and the excessive yaw angle causes the aircraft to rush out of the safe landing area.

Keywords: arresting system; kink-wave; yaw landing; dynamics; safety characteristics

1. Introduction In order to ensure the safety and accuracy of carrier-based aircraft, the United States (US) Navy took the lead after the 1970s in using the “all-weather electronic landing aid system” to assist the pilot in landing safely [1]. At present, the very short-term motion prediction technology of the aircraft carrier has become more and more mature [2]. However, because the carrier aircraft landing process is always affected by many complex factors, such as deck motion, deck airflow field, aerodynamic performance of carrier aircraft, and so on, the carrier aircraft is still not able to be completely yaw free and hook the arresting cable in the middle [3,4]. The United States has carried out a large number of flight tests and published relevant test result curves on the topic of arresting. Most of the test reports only refer to the situation of centering arrest under the condition of no off-center or yaw. With respect to yaw arrest, only a few fitting curves of measured data have been published; however, the mechanism and theoretical analyses of yaw arrest were not disclosed. In recent years, theoretical research of arresting devices is more comprehensive, but theoretical research and simulation analysis of arresting dynamic characteristics are not perfect. The relevant research on arresting dynamic characteristics mostly uses simplified spring linear force or the corresponding stress value curve [5–7] from the US military standard as input, and simplifies the arresting part (i.e., arresting rope and arresting machine) on the ship. In 1954, Thomlinson [8]

Appl. Sci. 2020, 10, 1253; doi:10.3390/app10041253 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 1253 2 of 18 established the kinematic equation between the fuselage and the arresting hook by analyzing and constructing the geometric connection relationship, and further analyzed the influence of the deck angle on the angular velocity of the rebound of the arresting hook. Lawrence [4] analyzed the main factors affecting the peak load of the arresting hook during the arresting process, including landing weight, landing speed, and initial eccentricity of landing, and based on the data statistics of the arresting system, the matching formula and curve of the arresting force were obtained. In 1959, Taylor [9] conducted a wind tunnel test with a speed reducer under the fuselage of the aircraft. Anderson et al. [10] studied the thrust reverser of a jet aircraft. In 1973, Hsin [11] analyzed the movement of the aircraft after grounding, and found that under the joint action of the arresting device and the arresting cable, the taxiing distance of the aircraft was significantly shortened and the aircraft could effectively be prevented from rushing out of the restricted area. The simulation results showed that the arresting device is expected to improve the controllability of the aircraft. In 2004, a US company published a safety report [12] in which the collision between the arresting hook and the arresting cable support was studied. Zhang et al. [13] optimized the aircraft dynamics in the arresting system and Zhang et al. [14] analyzed the result trend of eccentrically biased arresting. In the above studies, the function of the arresting system was introduced and proven to be effective in the landing process of carrier-based aircraft, but the influence of the kink-wave on the arresting force was ignored. In fact, when the carrier-based aircraft finishes landing and the arresting hook just touches the arresting cable, it is in the initial stage of engagement, and the arresting hydraulic system has not yet played a damping role. The arresting force in this stage is mainly produced by the initial tension of the arresting cable, which produces a kink-wave, which plays a major role in this early stage. When the transverse wave propagates and reflects between the engagement point of the hook cable and the deck pulley, it leads to the bending of the arresting cable, and then affects the arresting cable tension. Generally, the experimental or theoretical research on the arresting of carrier-based aircraft is mainly focused on the process of centering arrest, and the research on the kink-wave is also mainly focused on the process of centering arrest. Few scholars have studied the dynamics of yaw arrest while considering the kink-wave in the meantime. Peng et al. [15] explored the dynamic explanations for the lateral swing motion of the hook in yaw state but did not proceed to study the influence of yaw state on the whole arresting process. For this paper, based on a discrete kink-wave model, the arresting dynamic model was established and its accuracy was verified. On this basis, the simulation calculation of the yaw arrest process was carried out, and the influence of different yaw angles on the kink-wave, arresting cable tension, arresting dynamic characteristics, and arresting safety characteristics was analyzed.

2. Arresting Dynamics Model of Carrier Aircraft

2.1. Research Object The arresting system is shown in Figure1. After the carrier aircraft enters the site, according to the established method, the arresting hook acts as a bridge connecting the aircraft and the road surface, transmitting the arresting force to the fuselage, thus forcing the aircraft to decelerate rapidly, generally stopping in a safe area within 3 s. Therefore, the decelerating motion characteristic of the carrier aircraft is one of the most important factors in determining the deck length design, and is also an important factor in determining the arresting safety characteristics. Appl. Sci. Sci. 20202020,, 1010,, x 1253 FOR PEER REVIEW 3 3of of 18 18

FigureFigure 1. SchematicSchematic diagram diagram of of carrier carrier airc aircraftraft landing landing and hanging cable. The stress state of the aircraft is complex after the cable is hung with the arresting hook. In the 2.2. Drag Force Model of Carrier Aircraft process of arresting, the arresting hook is meshed with the braking arresting cable on the runway. The arrestingDue to thecable complexity is connected of the toforces the arrestingon the carrier machine aircraft on in both the sidesprocess of of the landing runway and through arresting, the andarresting in order belt. to Thefacilitate arresting the analysis, cable rebounds the force rapidly on the under aircraft the in tension the process provided of facing by the the arresting ship is simplifiedmachine, soas thatfollows the [16]: arresting resistance is transmitted to the aircraft. Therefore, calculating and 1.analyzing The runway the force is a provided plane, and by the the aircraft’s arresting head machine is parallel a short to timethe runway after the plane; carrier aircraft lands is 2.very In important. the process of arresting, the vibration after the engagement of the hook and cable is not considered, and the sum of forces in the cable (TL and TR) are shown in the exact opposite 2.2. Drag Force Model of Carrier Aircraft direction of aircraft movement; 3. TheDue influence to the complexity of the vertical of the forces height on of the the carrier carrier aircraft aircraft in the on process the arresting of landing resistance and arresting, is not andconsidered. in order to facilitate the analysis, the force on the aircraft in the process of facing the ship is simpliﬁed as follows [16]: When centering and arresting (without considering the eccentric yaw), the arresting resistance of1. theThe carrier runway aircraft is a is plane, as shown and thein Figure aircraft’s 2. head is parallel to the runway plane; 2. In the process of arresting, the vibration after the engagement of the hook and cable is not considered, and the sum of forces in the cable (TL and TR) are shown in the exact opposite direction of aircraft movement; 3. The inﬂuence of the vertical height of the carrier aircraft on the arresting resistance is not considered.

When centering and arresting (without considering the eccentric yaw), the arresting resistance of the carrier aircraft is as shown in Figure2.

Figure 2. Schematic diagram of arresting force. Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 18

Figure 1. Schematic diagram of carrier aircraft landing and hanging cable.

2.2. Drag Force Model of Carrier Aircraft Due to the complexity of the forces on the carrier aircraft in the process of landing and arresting, and in order to facilitate the analysis, the force on the aircraft in the process of facing the ship is simplified as follows [16]: 1. The runway is a plane, and the aircraft’s head is parallel to the runway plane; 2. In the process of arresting, the vibration after the engagement of the hook and cable is not considered, and the sum of forces in the cable (TL and TR) are shown in the exact opposite direction of aircraft movement; 3. The influence of the vertical height of the carrier aircraft on the arresting resistance is not considered.

Appl. Sci. 2020When, 10, 1253 centering and arresting (without considering the eccentric yaw), the arresting resistance 4 of 18 of the carrier aircraft is as shown in Figure 2.

FigureFigure 2. 2.Schematic Schematic diagram of of arresting arresting force. force.

AccordingAppl. Sci. 2020 to, Figure10, x FOR2 PEER, the REVIEW calculation formula of the blocking resistance is as follows:4 of 18  According to Figure 2, the calculation Fx = TformulaL cos β of+ theTR blockingcos β resistance is as follows:  S (1)  cos βFT==+cosββ T cos  xL√ 2+ 2 R  S D  β = S (1) cos S 22+ D where Fx refers to the arresting force acting on the carrier aircraft; TL refers to the pulling force of the left arrestingwhere 𝐹 refers cable; toT theR refers arresting to force the pulling acting on force the carrier of the aircraft; right T arrestingL refers to the cable; pullingβ refers force of to the the angle betweenleft the arresting line connecting cable; TR therefers hook to the cable pulling engagement force of the point right andarresting the pulley, cable; 𝛽 andrefers the to heading; the angleS refers to the displacementbetween the line of theconnecting carrier the aircraft; hook cable and Dengagement refers to halfpoint of and the the distance pulley, betweenand the heading; the two S pulleys on the deck.refers to the displacement of the carrier aircraft; and D refers to half of the distance between the two pulleys on the deck. In the case of yaw arrest, taking the right yaw as an example, the force of yaw arrest of the carrier In the case of yaw arrest, taking the right yaw as an example, the force of yaw arrest of the carrier aircraft isaircraft as shown is as shown in Figure in Figure3. 3.

FigureFigure 3. Schematic3. Schematic diagramdiagram of of yaw yaw arrest. arrest.

AccordingAccording to Figure to Figure3, the 3, calculation the calculation formula formula of of blocking blocking resistance resistance is as is follows: as follows: =+ϕϕ FTxLcos LR T cos R   ϕθϕϕθϕ=−, =+  Fx =LLTL cos ϕL RR+ TR cos ϕR   ϕ   θ = S cos  ϕL =cosθL ϕ, ϕR = θR + ϕ   ++ϕϕ22 (2)   − (DSS sinS cos )ϕ ( cos )  cosθL = q ϕ  θ = S cos2 2 (2)  cos R (D+S sin ϕ) +(S cos ϕ)   (D−+SS sinϕϕ )22 ( cos )   S cos ϕ  cos θR = q  (D S sin ϕ)2+(S cos ϕ)2 where θL is the angle between the left arresting −cable and the longitudinal direction, θR is the angle between the right arresting cable and the longitudinal direction, ϕL is the angle between the left arresting cable and the heading, and ϕR is the angle between the right arresting cable and the heading.

2.3. Calculation Model of Arresting Device To calculate the blocking resistance, it is necessary to obtain the change rule of the damping force output by the energy absorber of the blocking device, which is related to the movement displacement of the oblique flow control valve, the change rule of the sectional area of the oil groove on the valve, and the pressure in the accumulator [17]. The working principle of the energy absorber is shown in Figure 4. Appl. Sci. 2020, 10, 1253 5 of 18

where θL is the angle between the left arresting cable and the longitudinal direction, θR is the angle between the right arresting cable and the longitudinal direction, ϕL is the angle between the left arresting cable and the heading, and ϕR is the angle between the right arresting cable and the heading.

2.3. Calculation Model of Arresting Device To calculate the blocking resistance, it is necessary to obtain the change rule of the damping force output by the energy absorber of the blocking device, which is related to the movement displacement of the oblique ﬂow control valve, the change rule of the sectional area of the oil groove on the valve, and the pressure in the accumulator [17]. The working principle of the energy absorber is shown in FigureAppl. Sci.4 .2020, 10, x FOR PEER REVIEW 5 of 18

Figure 4. Principle diagram of the arresting gear energy absorber.

The mainmain didifferencefference between between yaw yaw arrest arrest and and centering centering arrest arrest lies inlies the in dithefference difference of impact of impact angle betweenangle between the left the and left right and arresting right arresting cables of cables the arresting of the arresting hook, which hook, leads which to aleads difference to a difference in arresting in cablearresting speed cable brought speed aboutbrough byt about the left by and the rightleft and pulleys, right pulleys, and finally and leadsfinally to leads a diff toerence a difference in piston in displacementpiston displacement and speed and onspeed either on side.either Forside. a certainFor a certain arresting arresting system, system, the output the output arresting arresting cable tensioncable tension is related is related to the to damper the damper stroke. stroke. When When the weight the weight of the of arrested the arrested aircraft aircraft is constant, is constant, the same the cablesame elongationcable elongation corresponds corresponds to the to same the arrestingsame arresting cable tensioncable tension [18]. [18]. When in centering arrest: When in centering arrest:  √ 2 2  = S +D D  Sr 22N+−−  = S S DD (3)  VrS=r V   N 2 f  N √S2+D (3)  = S VVrf When in right yaw:  22  q  NS+ D  (D+S sin ϕ)2+(S cos ϕ)2 D  SrL = − When in right yaw:  q N  2 2  (D S sin ϕ) +(S cos ϕ) D  S = − − (4) ⎧ rR (D + 𝑆 𝑠𝑖𝑛 𝜑) N+(𝑆𝑐𝑜𝑠𝜑) − D 𝑆 = V f cos ϕL ⎪ VrL = 𝑁 ⎪ N  (DV f −cos 𝑆ϕR 𝑠𝑖𝑛 𝜑) +(𝑆𝑐𝑜𝑠𝜑) − D ⎪ VrR = 𝑆 = N 𝑁 (4) where N is the number of movable⎨ pulleys𝑉 𝑐𝑜𝑠 in the 𝜑 movable pulley block, Sr is the displacement of the piston, v is the speed of the piston,⎪𝑉 V= is the velocity of the aircraft, S is the displacement of the left r ⎪ f 𝑁 rL piston, S is the displacement of⎪ the right𝑉 𝑐𝑜𝑠 piston, 𝜑 V is the movement speed of the left piston, and V rR 𝑉 = rL rR is the movement speed of the right⎩ piston.𝑁

where N is the number of movable pulleys in the movable pulley block, Sr is the displacement of the piston, vr is the speed of the piston, Vf is the velocity of the aircraft, SrL is the displacement of the left piston, SrR is the displacement of the right piston, VrL is the movement speed of the left piston, and VrR is the movement speed of the right piston.

3. Calculation Model of Kink-Wave Propagation

3.1. Model Assumptions When the arresting cable is impacted by the arresting hook, it produces a kink-wave in it. In the early stage of arresting, the kink-wave plays a major role. For this paper, a discrete kink-wave model was used. The model considers that the second kink-wave is generated at the engagement of hook and cable immediately after the first propagation of the kink-wave caused by the stress wave to the pulley, and then the third kink-wave is generated similarly. In the whole arresting process, the influence of kink-waves on the arresting load is gradually weakened. As the damping force of the arresting machine increases gradually, the influence of kink-waves is rapidly reduced. Therefore, the whole arresting system model only considers the first three kink-waves [19], and then the arresting process of the aircraft enters the stage of full hydraulic resistance force. Appl. Sci. 2020, 10, 1253 6 of 18

3. Calculation Model of Kink-Wave Propagation

3.1. Model Assumptions When the arresting cable is impacted by the arresting hook, it produces a kink-wave in it. In the early stage of arresting, the kink-wave plays a major role. For this paper, a discrete kink-wave model was used. The model considers that the second kink-wave is generated at the engagement of hook and cable immediately after the first propagation of the kink-wave caused by the stress wave to the pulley, and then the third kink-wave is generated similarly. In the whole arresting process, the influence of kink-waves on the arresting load is gradually weakened. As the damping force of the arresting machine increases gradually, the influence of kink-waves is rapidly reduced. Therefore, the whole arresting system model only considers the first three kink-waves [19], and then the arresting process of the aircraft enters the stage of full hydraulic resistance force. In the case of vertical impact, the stress on the rope can be regarded as only related to the impact velocity. In this case, the kink-wave velocity [20] is:

Appl. Sci. 2020, 10, x FOR PEER REVIEW q 6 of 18 2/3 4/3 2/3 4/3 w = c( 0.5 (v0/c) 0.5 (v0/c) ) (5) In the case of vertical impact, the stress on the rope− can be regarded as only related to the impact where wvelocity.is the kink-waveIn this case, the speed, kink-wavec is the velocity longitudinal [20] is: wave speed, and v0 is the contact point speed of the hook and cable. =−2/3()4/3 2/3 () 4/3 wc(0.5 vc00 / 0.5 vc / ) (5) When building the kink-wave and arresting system models, the following simpliﬁcations were adopted:where w is the kink-wave speed, c is the longitudinal wave speed, and v0 is the contact point speed of the hook and cable. 1. There isWhen no relativebuilding slidingthe kink-wave between and thearresting arresting system hook models, and the the following arresting simplifications cable; were adopted: 2. The component force of the arresting hook in the vertical direction is not considered; 3. At any1. time,There is the no tensionrelative sliding of the between arresting the arresting cable hook on one and the side arresting of the cable; arresting hook is the 2. The component force of the arresting hook in the vertical direction is not considered; same everywhere; 3. At any time, the tension of the arresting cable on one side of the arresting hook is the same 4. It is assumedeverywhere; that the arresting cable is the same cable with the same material; 5. The arresting4. It is assumed cable is that in the the lineararresting elastic cable is range the same during cable the with arresting the same process.material; 5. The arresting cable is in the linear elastic range during the arresting process. 3.2. Model Comparison of Centering and Yaw Arrest 3.2. Model Comparison of Centering and Yaw Arrest The formThe of form the of arresting the arresting cable cable relative relative to to the the centering centering arrest considering considering kink-wave kink-wave is shown is shown in Figure5.in Figure 5.

FigureFigure 5. 5.Form Form of of cablecable in in centering centering arrest. arrest.

The arresting cable configuration on one side at the set time is shown in Figure 5. To establish a

coordinate system, O is the origin, (𝑥,𝑦) are the coordinates of the bending point Q, (𝑥, 𝑦) are the real-time coordinates of the aircraft, (𝑥̅,𝑦) are the coordinates of the aircraft at the end of the previous kink-wave, l1 and l2 are the lengths of the cable from the arresting hook to the bending point and the bending point to the pulley, and lw is the distance of the heavy kink-wave. When yaw arrest (right yaw) takes the arresting form of the kink-wave into account, its general form is shown in Figure 6. Appl. Sci. 2020, 10, 1253 7 of 18

The arresting cable conﬁguration on one side at the set time is shown in Figure5. To establish a coordinate system, O is the origin, (xw, yw) are the coordinates of the bending point Q, (x, y) are the real-time coordinates of the aircraft, (x, y) are the coordinates of the aircraft at the end of the previous kink-wave, l1 and l2 are the lengths of the cable from the arresting hook to the bending point and the bending point to the pulley, and lw is the distance of the heavy kink-wave. When yaw arrest (right yaw) takes the arresting form of the kink-wave into account, its general Appl.form Sci. is 2020 shown, 10, x in FOR Figure PEER6 .REVIEW 7 of 18

FigureFigure 6. FormForm of of cable cable in in yaw yaw arrest. arrest. 3.3. Calculation Method of Kink-Wave 3.3. Calculation Method of Kink-Wave In the calculation of the kink-wave, the trapezoid formula of the Euler method was used, and the In the calculation of the kink-wave, the trapezoid formula of the Euler method was used, and convergence can be guaranteed when the simulation step length is small enough. Assuming that the the convergence can be guaranteed when the simulation step length is small enough. Assuming that acceleration, speed, and displacement of the aircraft at the previous time are a , v , and S , and the the acceleration, speed, and displacement of the aircraft at the previous time are0 𝑎 0, 𝑣 , and0 𝑆 , and acceleration, speed, and displacement of the aircraft at the current time are a , v , andS , taking the the acceleration, speed, and displacement of the aircraft at the current time are1 𝑎 1, 𝑣 , and1 𝑆 , taking time step as ∆t = 0.0001 s, then: the time step as Δ=t 0.0001s , then: ( v = v + (a + a )∆t/2 1 0 0 1 (6) s = =+s + (v()+ +v )∆ Δt/2 1vv100 aat0 011 /2  (6) From Figure5 above: ss=+() vvt + Δ/2  q 10 01  2 2  l1 = (xw x) + (y yw) From Figure 5 above:  q − −   l = (D x )2 + y2  2 w w (7)  ⎧𝑙 = (𝑥−−𝑥) +(𝑦−𝑦)  xw = lw cos θ + x, yw = y lw sin θ   ⎪𝑙 =Rt(D−𝑥) +𝑦 −y  lw = w dt, θ = arctan( ) 𝑥 =𝑙t0 𝑐𝑜𝑠̅ 𝜃+𝑥 ,𝑦 =𝑦D̄+−𝑙x 𝑠𝑖𝑛 𝜃 (7) ⎨ where t and y are the starting time of the kink-wave and the displacement𝑦̄ of the aircraft at that time, 0 ⎪𝑙 =𝑤 d𝑡, 𝜃 = 𝑎𝑟𝑐 𝑡𝑎𝑛( ) D+𝑥̅ respectively, and θ is the angle between⎩ the line connecting the hook cable engagement point and the pulley, and the transverse. where t0 and 𝑦 are the starting time of the kink-wave and the displacement of the aircraft at that According to Hooke’s law, the tension of the arresting cable can be obtained: time, respectively, and θ is the angle between the line connecting the hook cable engagement point and the pulley, and the transverse.  Eq  T = δ According to Hooke’s law, the tension of lthe arresting cable can be obtained: (8)  δ = l1 + l2 D 2Nz 𝐸𝑞 − − 𝑇= 𝛿 𝑙 (8) 𝛿=𝑙 +𝑙 −𝐷−2𝑁𝑧 where E and q are the elastic modulus and cross-sectional area of the arresting cable, respectively, l is the length of the cable from the unilateral pulley to the engagement point of the hook cable, and z is the moving distance of the moving pulley block. Then, the drag of the arresting hook acting on the aircraft can be obtained, that is, the hook load: xx−− xx HT= T cos(arctan(wL )−+ϕϕ ) T cos(arctan( wR ) + ) LR−− (9) yywL yy wR Appl. Sci. 2020, 10, 1253 8 of 18 where E and q are the elastic modulus and cross-sectional area of the arresting cable, respectively, l is the length of the cable from the unilateral pulley to the engagement point of the hook cable, and z is the moving distance of the moving pulley block. Then, the drag of the arresting hook acting on the aircraft can be obtained, that is, the hook load:

xwL x x xwR HT = TL cos(arctan( − ) ϕ) + TR cos(arctan( − ) + ϕ) (9) y ywL − y ywR − − where (xwL, ywL) and (xwR, ywR) are the coordinates of the left and right bending points at the starting time ofAppl. the Sci. kink-wave, 2020, 10, x FOR respectively, PEER REVIEW and HT is the hook load. 8 of 18 In different periods of kink-waves, the initial moment and initial bending angle on either side are where (𝑥,𝑦) and (𝑥,𝑦) are the coordinates of the left and right bending points at the different,starting which time makes of the thekink-wave, calculation respectively, formula and of HT some is the lengths hook load. diff erent. Therefore, it is very difficult to establishIn a different continuous periods model of kink-waves, of kink-waves the initial to moment describe and the initial whole bendin arrestingg angle processon either [side19]. In this paper, theare discretedifferent, methodwhich makes was usedthe calculation to build theformula model, of some and thelengths accumulation different. Therefore, sum was it usedis very to replace the integral,difficult so to that establish the initial a continuous state is model known of kink-w and theaves whole to describe solution the whole model arresting is determined. process [19]. In this paper, the discrete method was used to build the model, and the accumulation sum was used 4. Modelto replace Checking the integral, so that the initial state is known and the whole solution model is determined.

4.1. Model4. Model Solving Checking Process After4.1. Model the calculation Solving Process of the first three kink-waves, the arrest entered the period of full hydraulic damping. DuringAfter the this calculation period, of Simulink the first three was kink-waves used to build, the arrest the calculation entered the period model of of full the hydraulic arresting force, and ode113damping. solver During was this used period, for the Simulink model. was This used solver to build is the a variable calculation step model size of solver the arresting with high force, accuracy and ode113 solver was used for the model. This solver is a variable step size solver with4 high accuracy when the error tolerance is strict. The relative error tolerance was set to 1 10− and the maximum when the error tolerance is strict. The relative error tolerance was set to 1 × 10×−4 and the maximum step size to 0.001 s. The state variables at this period were used as outputs (the displacement and speed step size to 0.001 s. The state variables at this period were used as outputs (the displacement and of the aircraft,speed of thethe aircraft, angle between the angle thebetween left andthe left right and side right of side the of arresting the arresting cable cable and and the the initial initial position, the displacementposition, the displacement and speed of and the speed two of sides the two of sides the piston,of the piston, etc.), etc.), then then the the dynamic dynamic calculation of the arrestingof the process arresting continued process continued until the until aircraft the aircra stoppedft stopped andthe and simulation the simulation ended. ended. The The specificspecific flow of the modelflow solution of the model is shown solution in is Figureshown 7in. Figure 7.

Figure 7. Diagram of calculation chart. Figure 7. Diagram of calculation chart.

4.2. Model Validation For comparison and verification, taking the MK7-3 arresting device as an example for calculation, the model calculation parameters [21] are shown in Table 1. Appl. Sci. 2020, 10, 1253 9 of 18

4.2. Model Validation For comparison and veriﬁcation, taking the MK7-3 arresting device as an example for calculation, the model calculation parameters [21] are shown in Table1.

Table 1. Parameters of the carrier aircraft and landing gear.

Parameter Value Parameter Value Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 18 Accumulator air bag Mass of aircraft (kg) 22,680 2 volume (m3) Table 1. Parameters of the carrier aircraft and landing gear. 1 Initial pressure of Landing speed (m s− ) 63 3 Parameter· Value accumulator Parameter (MPa) Value Maximum stopping Accumulator piston 104 3 10 distanceMass (m) of aircraft (kg) 22,680 Accumulatormass (kg) air bag volume (m ) 2 Piston areaLanding of energy speed (m·s−1) 63 InitialAccumulator pressure piston of accumulator area (MPa) 3 0.28 1 Maximumabsorber (m stopping2) distance (m) 104 Accumulator(m2) piston mass (kg) 10 Piston stroke of energy Distance between Piston area of energy absorber (m52) 0.28 Accumulator piston area (m2) 34 1 absorber (m) pulleys on deck (m) Piston stroke of energy absorber (m) 5 Distance between pulleys on deck (m) 34

WhenWhen inin centeringcentering arrest,arrest, the the curve curve of of the the arresting arresting resistance resistance with with the the travel travel is is shown shown in in Figure Figure8. It8. canIt can be seenbe seen from from Figure Figure8 that 8 when that consideringwhen considering the kink-wave, the kink-wave, the arresting the arresting resistance resistance has obvious has ﬂuctuationobvious fluctuation in the initial in the stage, initial which stage, shows which thatshows the that eﬀect the of effect the kink-waveof the kink-wave in the arrestingin the arresting cable cannotcable cannot be ignored. be ignored.

FigureFigure 8.8. Curve of arresting force–displacementforce–displacement inin centeringcentering arrest.arrest.

AsAs shownshown inin FigureFigure9 9,, and and from from the the comparison comparison results results of of the the two two groups groups of of data, data, it it can can be be seen seen thatthat inin thethe initialinitial stage,stage, thethe arrestingarresting forceforce fluctuatesfluctuates underunder thethe influenceinfluence ofof thethe kink-wave,kink-wave, andand bothboth simulationsimulation andand testtest datadata [[6]6] showshow thethe featurefeature ofof kink-waves.kink-waves. The loadload fluctuationfluctuation isis thethe resultresult ofof repeatedrepeated propagationpropagation ofof stressstress waveswaves betweenbetween thethe deckdeck pulleypulley andand hookhook cablecable engagementengagement point. ThereThere areare some some di differencesfferences between between the the simulation simulation results results and and the the test test data data in thein the middle middle stage, stage, but thebut formthe form of the of the change change of theof the arresting arresting force force is consistentis consistent in in the the process process of of arresting, arresting, so so the the simulationsimulation resultsresults havehave goodgood consistencyconsistency andand credibilitycredibility inin general.general. Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 18 Appl. Sci. 2020, 10, 1253 10 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 18

Figure 9. Curves comparison of arresting force–displacement in simulation and test. Figure 9. 5. CalculationFigure and 9. Analysis CurvesCurves comparison comparison of of arresting arresting force–displacement force–displacement in in simulation simulation and and test. test.

5.5. Calculation CalculationThe yaw andangle and Analysis Analysis was set in the model to simulate the yaw arrest dynamics. The yaw of all calculation examples is right deviation, and other parameters are consistent with the simulation of TheThe yawyaw angleangle was was set set in thein modelthe model to simulate to simula thete yaw the arrest yaw dynamics. arrest dynamics. The yaw ofThe all yaw calculation of all centering arrest. calculationexamples is examples right deviation, is right and deviation, other parameters and other are parameters consistent are with consistent the simulation with ofthe centering simulation arrest. of centering arrest. 5.1. The Influence Inﬂuence of Yaw Angle on Kink-Wave

5.1. TheIn the Influence case of of right Yaw yawAngle arrest, on Kink-Wave the triple kink-wav kink-wavee on the right side ends first, first, and then the third kink-wave on the left side ends. The influence influence of yaw angle on the resistance of both sides during the In the case of right yaw arrest, the triple kink-wave on the right side ends first, and then the third firstfirst triple kink-wave was analyzed, and thethe resultsresults areare shownshown inin FiguresFigures 1010 andand 1111.. kink-wave on the left side ends. The influence of yaw angle on the resistance of both sides during the first triple kink-wave was analyzed, and the results are shown in Figures 10 and 11.

Figure 10. Port tension during kink-wave period. Figure 10. Port tension during kink-wave period. Appl. Sci. 2020, 10, 1253 11 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 18

Figure 11. Starboard tension duri duringng kink-wave period.

The results of the time length of the triple kink-wavekink-wave on both sides at different diﬀerent yaw angles are listed in Table2 2 for for comparative comparative analysis. analysis.

Table 2. Time length comparison of kink-wave in yaw arrest. Table 2. Time length comparison of kink-wave in yaw arrest. First Kink-Wave Second Third Kink-Wave Yaw AngleYaw FirstAngle Kink-Wave (s) Second Kink-Wave (s) Third Kink-Wave (s) (s) Kink-Wave (s) (s) Port 0.0994 0.2055 0.3320 0° Port 0.0994 0.2055 0.3320 Starboard0 0.0994 0.2055 0.3320 ◦ Starboard 0.0994 0.2055 0.3320 Port 0.0993 0.2071 0.3360 3° Port 0.0993 0.2071 0.3360 Starboard3 0.0992 0.2035 0.3264 ◦ Starboard 0.0992 0.2035 0.3264 Port 0.0993 0.2083 0.3390 5° Port 0.0993 0.2083 0.3390 Starboard5 0.0992 0.2023 0.3227 ◦ Starboard 0.0992 0.2023 0.3227 Port 0.0993 0.2094 0.3419 7° Port 0.0993 0.2094 0.3419 Starboard7 0.0992 0.2010 0.3189 ◦ Starboard 0.0992 0.2010 0.3189 Port 0.0993 0.2106 0.3450 9° Port 0.0993 0.2106 0.3450 Starboard9 0.0992 0.1998 0.3152 ◦ Starboard 0.0992 0.1998 0.3152

It can be seen from the results that there is no difference in the time length of the first kink-wave betweenIt can the be left seen and from right the sides, results both that of there which is noare di aboutfference 0.0993 in the s. timeAfter length the propagation of the first kink-waveof the first betweenkink-wave, the the left time and length right sides, of the both left ofkink-wave which are increases about 0.0993 slightly s. Afterwith the increase propagation of eccentricity, of the first kink-wave,and the time the length time of length the right of the kink-wave left kink-wave decreas increaseses slightly slightly with the with increase the increase of yaw ofangle. eccentricity, and theThe time main length reason of is the that right when kink-wave the first decreases kink-wave slightly propagates, with the the increase initial length of yaw of angle. the arresting cablesThe on maineither reason side is is the that same, when and the the first kink-wave kink-wave velocity propagates, is slightly the initial different, length so ofthe the time arresting of the cablesfirst kink-wave on either propagation side is the same, is basically and the the kink-wave same. After velocity the end is slightlyof the first diff kink-wave,erent, so the the time length of the of firstthe arresting kink-wave cables propagation on either isside basically is different. the same. When After the right the end yaw of arrests, the first the kink-wave, right arresting the length cable ofis theshorter arresting than the cables left, on the either kink-wave side is propagation different. When time the is rightshorter, yaw and arrests, the time the rightdifference arresting of the cable same is shorterkink-wave than on the either left, side the kink-waveis obvious. propagation time is shorter, and the time difference of the same kink-waveWhen onthe eitheryaw angle side isis obvious.larger, the fluctuation of the first kink-wave on the right side is obviously weakened.When theWhen yaw the angle yaw is angle larger, reaches the fluctuation 7°, the fluc oftuation the first is kink-wave smaller. When on the the right yaw side angle is obviously reaches weakened.9°, the fluctuation When theof the yaw basic angle first reaches kink-wave 7◦, the is fluctuation not obvious. is smaller. When the yaw angle reaches 9◦, the fluctuation of the basic first kink-wave is not obvious. 5.2. The Influence of Yaw Angle on the Tension of Arresting Cable Appl. Sci. 2020, 10, 1253 12 of 18

Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 18 5.2. The Inﬂuence of Yaw Angle on the Tension of Arresting Cable The maximum yaw angle was taken as 9 degrees, and the variation trend of the tension of the The maximum yaw yaw angle angle was was taken taken as as 9 degrees, an andd the variation trend of the tension of the unilateral arresting cable when the yaw angle was 0, 3, 5, 7, and 9 degrees was analyzed. The results unilateral arresting cable when the yaw angle was 0, 3, 5, 7, and 9 degrees was analyzed. The results are shown in Figures 12 and 13. are shown in Figures 12 and 1313..

Figure 12. The port tape tension. Figure 12. The port tape tension.

Figure 13. The starboard tape tension. Figure 13. The starboard tape tension. The results of the maximum belt tension on the left side and the maximum belt tension on the The results of the maximum belt tension on the left side and the maximum belt tension on the right side under didifferentﬀerent yawyaw anglesangles areare listedlisted inin TableTable3 3for for comparative comparative analysis. analysis. right side under different yaw angles are listed in Table 3 for comparative analysis.

Table 3. Comparison of the port and starboard maximum belt tensions of different eccentricity. Table 3. Comparison of the port and starboard maximum belt tensions of different eccentricity. Parameters Centering Yaw 3° Yaw 5° Yaw 7° Yaw 9° Parameters Centering Yaw 3° Yaw 5° Yaw 7° Yaw 9° Port maximum belt tension (kN) 373.76 381.63 397.25 420.06 445.31 Port maximum belt tension (kN) 373.76 381.63 397.25 420.06 445.31 Starboard maximum belt tension (kN) 373.76 365.74 360.35 354.93 349.50 Starboard maximum belt tension (kN) 373.76 365.74 360.35 354.93 349.50

It can be seen from the results that with the increase of the yaw angle, the cable tension of the It can be seen from the results that with the increase of the yaw angle, the cable tension of the left-side arresting system increases gradually compared with that of the centering arresting system, left-side arresting system increases gradually compared with that of the centering arresting system, and the maximum belt tension of the left side with the yaw angle of 9 degrees is 445.31 kN. The belt and the maximum belt tension of the left side with the yaw angle of 9 degrees is 445.31 kN. The belt tension of the right-side arresting system decreases gradually compared with that of the centering tension of the right-side arresting system decreases gradually compared with that of the centering Appl. Sci. 2020, 10, 1253 13 of 18

Table 3. Comparison of the port and starboard maximum belt tensions of diﬀerent eccentricity.

Parameters Centering Yaw 3◦ Yaw 5◦ Yaw 7◦ Yaw 9◦ Port maximum belt tension 373.76 381.63 397.25 420.06 445.31 (kN) Starboard maximum belt 373.76 365.74 360.35 354.93 349.50 tension (kN)

It can be seen from the results that with the increase of the yaw angle, the cable tension of the left-side arresting system increases gradually compared with that of the centering arresting system, and the maximum belt tension of the left side with the yaw angle of 9 degrees is 445.31 kN. The belt tensionAppl. of theSci. 2020 right-side, 10, x FOR PEER arresting REVIEW system decreases gradually compared with that of13 the of centering18 arresting system, and the maximum belt tension of the right side with the yaw angle of 9 degrees is 349.50 kN.arresting system, and the maximum belt tension of the right side with the yaw angle of 9 degrees is The349.50 change kN. trend of the tension of the two sides of the arresting belt was then analyzed, when the The change trend of the tension of the two sides of the arresting belt was then analyzed, when yaw anglethe yaw was angle 0, 3, 5,was 7, 0, and 3, 5, 9 7, degrees. and 9 degrees. The resultsThe results are are shown shown in in Figure Figure 1414..

Figure 14. The comparison of port and starboard tape tensions: (a) when yaw angle is 0°; (b) when Figure 14. The comparison of port and starboard tape tensions: (a) when yaw angle is 0◦;(b) when yaw angle is 3°; (c) when yaw angle is 5°; (d) when yaw angle is 7°; (e) when yaw angle is 9°. yaw angle is 3◦;(c) when yaw angle is 5◦;(d) when yaw angle is 7◦;(e) when yaw angle is 9◦. It can be seen from Figure 14 that with the increase of yaw angle, the tension difference between the two side belts is larger, and the tension of the left-side belt is larger than that of the right-side belt. Compared with the centering arresting, due to the different elongation of the arresting cables on the left and right sides, the speed of the pistons on either side is different, and the tension of the arresting belts on either side is different. The upper limit of cable tension is 784 kN, and yaw arrest does not cause drag to exceed this limit.

5.3. The Influence of Yaw Angle on the Performance of the Arresting System The response of aircraft speed, displacement, acceleration, and arresting force when the yaw angle changes was then analyzed, as shown in Figures 15–18. Appl. Sci. 2020, 10, 1253 14 of 18

It can be seen from Figure 14 that with the increase of yaw angle, the tension difference between the two side belts is larger, and the tension of the left-side belt is larger than that of the right-side belt. Compared with the centering arresting, due to the different elongation of the arresting cables on the left and right sides, the speed of the pistons on either side is different, and the tension of the arresting belts on either side is different. The upper limit of cable tension is 784 kN, and yaw arrest does not cause drag to exceed this limit.

5.3. The Inﬂuence of Yaw Angle on the Performance of the Arresting System The response of aircraft speed, displacement, acceleration, and arresting force when the yaw angle changesAppl. Sci. 20202020 was,, 1010 then,, xx FORFOR analyzed, PEERPEER REVIEWREVIEW as shown in Figures 15–18. 1414 ofof 1818

Figure 15. Displacement–time in simulation. Figure 15. Displacement–time in simulation.

Figure 16. Velocity–time. Figure 16. Velocity–time.

Figure 17. Longitudinal acceleration–time. Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 18

Figure 15. Displacement–time in simulation.

Appl. Sci. 2020, 10, 1253 Figure 16. Velocity–time. 15 of 18

Appl. Sci. 2020, 10, x FOR PEER REVIEWFigure 17. Longitudinal acceleration–time. 15 of 18 Figure 17. Longitudinal acceleration–time.

FigureFigure 18. 18. ArrestingArresting force–displacement. force–displacement.

TheThe calculation calculation results results of of the the arresting arresting system system an andd aircraft aircraft response response under under different diﬀerent yaw yaw angles angles areare listed listed in in Table Table4 4 for for comparativecomparative analysis.analysis.

TableTable 4. 4. ComparisonComparison of of performance performance of of the the arresting arresting system system of of different diﬀerent eccentricity.

ParametersParameters Centering Centering Yaw 3◦ YawYaw 3° 5◦Yaw 5° YawYaw 7 ◦7° Yaw 9°Yaw 9◦ MaximumMaximum arresting arresting force (kN) 698.78 698.76 698.72 698.68 698.61 698.78 698.76 698.72 698.68 698.61 forceMaximum (kN) acceleration (m·s−2) 30.81 30.81 30.81 30.80 30.80 Maximum Arresting distance30.81 (m) 30.81 86.76 86.62 30.81 86.39 30.8086.06 85.38 30.80 acceleration (m s 2) · − Arresting distance 86.76 86.62 86.39 86.06 85.38 It can(m) be seen from the comparison results that the peak value of the arresting force increases slightly with the increase of the yaw angle, and has little influence on the arresting distance and the peak value of the aircraft acceleration. It can be seen from the comparison results that the peak value of the arresting force increases slightly with the increase of the yaw angle, and has little inﬂuence on the arresting distance and the 5.4. The Influence of Yaw Angle on Arresting Safety peak value of the aircraft acceleration. The landing area of the flight deck as shown in Figure 19 is actually the area surrounded by safety warning lines on both sides of the inclined deck, which is often referred to as the inclined landing runway. The landing runway should not only meet the safe landing requirements of the carrier aircraft, but also ensure that there is enough distance to complete the missed approach after confirming the failure of the hooking rope [22]. In addition, there should be a long enough turning area to meet the turning operation of the carrier aircraft.

Figure 19. The compositions of angled deck. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 18

Figure 18. Arresting force–displacement.

The calculation results of the arresting system and aircraft response under different yaw angles are listed in Table 4 for comparative analysis.

Table 4. Comparison of performance of the arresting system of different eccentricity.

Parameters Centering Yaw 3° Yaw 5° Yaw 7° Yaw 9° Maximum arresting force (kN) 698.78 698.76 698.72 698.68 698.61 Maximum acceleration (m·s−2) 30.81 30.81 30.81 30.80 30.80 Arresting distance (m) 86.76 86.62 86.39 86.06 85.38

It can be seen from the comparison results that the peak value of the arresting force increases

Appl.slightly Sci. 2020with, 10 the, 1253 increase of the yaw angle, and has little influence on the arresting distance and16 of the 18 peak value of the aircraft acceleration.

5.4. The Influence Influence of Yaw AngleAngle onon ArrestingArresting SafetySafety The landinglanding areaarea of of the the flight flight deck deck as shownas shown in Figurein Figure 19 is 19 actually is actually the area the surroundedarea surrounded by safety by warningsafety warning lines on lines both on sides both of sides the inclinedof the inclined deck, which deck, iswhich often is referred often referred to as the to inclined as the inclined landing runway.landing Therunway. landing The runway landing should runway not should only meet not the only safe meet landing the requirementssafe landing ofrequirements the carrier aircraft, of the butcarrier also aircraft, ensure but that also there ensure is enough that there distance is enough to complete distance the to missed complete approach the missed after confirmingapproach after the failureconfirming of the the hooking failure ropeof the [22 hooking]. In addition, rope [22]. there In should addition, be athere long should enough be turning a long areaenough to meet turning the turningarea to meet operation the turning of the carrieroperation aircraft. of the carrier aircraft.

Figure 19.19. The compositions of angled deck.

In the picture: ( B = B + 2B a T S (10) Sa = d + ∆x1 + ∆x2 + ∆x3 + Ss + St + SS where Ba is the width of the landing runway of the aircraft deck, BT is the total width of the aircraft with the largest wing span, BS is the width of the safety margin between the safety warning sign and the aircraft’s wing tip, Sa is the total length of the landing area, d is the length of the landing area, ∆x1 + ∆x2 + ∆x3 is the length of the blocking area, Sd is the length of the deceleration braking area, St is the length of the turning area, and SS is the length of the safety reserve area. It is assumed that the deck landing runway width of an aircraft carrier is 40 m and the wingspan of an aircraft carrier is 24.56 m; then, BS is 7.72 m. Under diﬀerent yaw conditions, the deviation of carrier aircraft stopping is shown in Table5.

Table 5. Comparison of oﬀsets.

Parameters Centering Yaw 3◦ Yaw 5◦ Yaw 7◦ Yaw 9◦ Arresting oﬀset 0 4.53 7.53 10.49 13.36 distance (m)

According to the analysis of arresting oﬀset, when the yaw angle is 5 degrees, the carrier aircraft stops just at the edge of the safe landing area. If the yaw angle is bigger than 5 degrees, the aircraft will rush out of the safe landing area, which may lead to safety accidents. Therefore, the yaw angle should be reduced as much as possible to reduce the lateral oﬀset of the carrier aircraft, so as to prevent the aircraft from colliding with the aircraft or the ship island building outside the safe landing area in the process of arresting and improve the arresting safety.

6. Conclusions In order to solve the problem of asymmetric arresting, a complete dynamic model of an arresting system was established. The accuracy of the model was verified by the experimental results of Appl. Sci. 2020, 10, 1253 17 of 18 the US military. The dynamic simulation of yaw arrest under different conditions was carried out. The comprehensive analysis of the simulation results showed the following: (1) In the case of yaw arrest, there is almost no difference in the length of the first kink-wave on either side. After the propagation of the first kink-wave, the time length of the kink-wave of the departure side increases slightly with the increase of the yaw angle, and the time length of the kink-wave of the deflection side decreases slightly. The influence of the yaw on the time length of the kink-wave is small. (2) For yaw arrest, the tension of the deflection-side arresting belt is smaller than that of the middle, and the tension of the departure-side arresting belt is larger than that of the middle. With the increase of yaw angle, the difference between the peak values of the tension of the arresting belt on either side becomes increasingly larger. When the yaw angle is too large, the peak value of the tension of the departure-side arresting belt may exceed the allowable pulling force of the arresting cable, resulting in arresting failure. (3) Yaw arrest has little influence on arresting distance, aircraft deceleration process, peak value of arresting resistance, and peak value of acceleration. With the increase of yaw angle, the arresting distance slightly decreases, and the peak values of the acceleration and arresting force also decrease slightly. (4) For yaw arrest, when the yaw angle is less than 5 degrees, the carrier aircraft is stopped within the safe range of the landing area, and the excessive yaw angle has an adverse effect on the arresting safety characteristics.

Author Contributions: Conceptualization, X.W. and H.N.; Data curation, Y.P.; Formal analysis, Y.P.; Investigation, Y.P. and P.X.; Methodology, Y.P.; Project administration, X.W. and H.N.; Resources, X.W. and H.N.; Software, P.X.; Supervision, X.W. and H.N.; Validation, Y.P.; Visualization, P.X.; Writing—original draft, Y.P.; Writing—review and editing, Y.P. All authors have read and agreed to the published version of the manuscript. Funding: This work was financially supported by the China Postdoctoral Science Foundation (2019M651827), Jiangsu Planned Projects for Postdoctoral Research Funds (2018K042B), the National Defense Excellence Youth Science Fun of China (2018-JCJQ-ZQ-053), the Fundamental Research Funds for the Central Universities of China (NP2017401), and the Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) (MCMS-0217G01). A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. Conflicts of Interest: The authors declare no conflict of interest.

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