Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 867012, 12 pages http://dx.doi.org/10.1155/2013/867012

Research Article Modeling and Simulation of Arresting Gear System with Multibody Dynamic Approach

Wenhou Shen,1,2 Zhihua Zhao,1 Gexue Ren,1 and Jiapeng Liu1

1 Tsinghua University, Beijing 100084, China 2 Naval Aviation Institute, Huludao 125001, China

Correspondence should be addressed to Wenhou Shen; [email protected]

Received 17 July 2013; Revised 19 September 2013; Accepted 19 September 2013

Academic Editor: Song Cen

Copyright © 2013 Wenhou Shen et al. 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.

The arresting dynamics of the aircraft on the aircraft carrier involves both a transient wave propagation process in rope and a smooth decelerating of aircraft. This brings great challenge on simulating the whole process since the former one needs small time-step to guarantee the stability, while the later needs large time-step to reduce calculation time. To solve this problem, this paper proposes a full-scale multibody dynamics model of arresting gear system making use of variable time-step integration scheme. Especially, a kind of new cable element that is capable of describing the arbitrary large displacement and rotation in three-dimensional space is adopted to mesh the wire cables, and damping force is used to model the effect of hydraulic system. Then, the stress of the wire ropes during the landing process is studied. Results show that propagation, reflection, and superposition of the stress wave between the deck sheaves contribute mainly to the peak value of stress. And the maximum stress in the case of landing deviate from the centerline is a little bit smaller than the case of landing along centerline. The multibody approach and arresting gear system model proposed here also provide an efficient way to design and optimize the whole mechanism.

1. Introduction by a smooth large overall motion of the cable and quick decel- eration of the aircraft. The whole process is so complicated The arresting gear system is an essential unit equipped by that some mechanical problems are hard to answer through aircraft carrier to assist landing of high-speed aircraft within theory analysis and even numerical simulation, such as when limited distance. Its first prototype was invented in 1911 by the maximum stress in rope will happen and where it is, what Hugh Robinson [1], the rope of which across the deck is is the key factor when the aircraft is landing along or deviate grasped and pulled out by the aircraft tailhook, raising sand- from the centerline, how to control the hydraulic forces to bagsatbothendsoftheropeviathepulleysandslowingdown make sure that the aircraft is arrested within limited distance the aircraft. Afterwards, the hydraulic arresting gears, which accompanying a steady deceleration process. Although many replaced the retarding force from the gravity of sandbags to scholars have been devoted on studying different aspects of hydraulic damping, appeared [2]. It soon became popular in arrestinggearsystem,therearestillmanyproblemsconcern- modern aircraft carriers since the hydraulic force is adjustable ing the deck pendant which result in several flight accidents to land aircrafts of different weights and velocities with the nowadays, the most recent of which was the fracture of the same mechanism. The most often equipped one is the MK7 ropes at USS George Washington in 2003 [3]. type hydraulic arresting gears. In literatures, Billec [4] tested experimentally the acceler- A typical arresting process contains two sequential stages ation of aircraft with different deck pendant length in 1967.He with completely different dynamical characters: the first stage pointed out that the maximum deceleration reduced about is dominated by the stress wave propagation and reflection 14percentwhenthependantincreasedfrom100feetto within rope, triggered by the impact between pendant and 130feet.In2010,Zhuokunetal.[5, 6]analyticallycalculated tailhook; the second stage is governed by the quasi-steady the maximum steady tension of the wire rope with quasi- hydraulicforceswhenthestresswaveisdecayedandfollowed steady landing assumption; they conclude that the maximum 2 Mathematical Problems in Engineering tension in the offline landing case will be about 3.3% smaller (2) with two purchase cables (3) on its both ends. Each pur- than the centerline landing case. In 2009, Mikhaluk et al. [7] chase cable is reeved through the fixed (5) and movable (8) proposedtobuildthewholearrestinggearsystemwithaidsof carriageofonegroup,andtheendpointisconnectedbythe commercial finite element software LS-DYNA. They meshed anchor damper (9). The movable carriage is forced to run the cables with beam element, defined the contact between towardthefixedcarriagewhentheropeispullingoutby cables and pulleys as Hertz contact model, and developed new aircraft. And the hydraulic oil in the main cylinder (6) flows hydraulic damping to model the hydraulic forces. The center- out since the plunger (7) of the hydraulic cylinder is fixed with line landing cases with an aircraft of a weight of 14 ton and themovablecarriage.Theflowisseparatedintotwobranches. a velocity ranging from 180 km/h to 240 km/h are simulated, The main one goes through the constant runout control and the achieved hydraulic pressure is similar to that of the valve (10) whose open area is passively adjusted during the experiment data. Later, Xinyu [8]andLihuaetal.[9, 10] landing processes to produce a steady retraction force for alsoadoptedtheLS-DYNAtoconstructthewholearresting slowing down the aircraft. The minor one goes to the anchor system, and the cables are modeled with concentrated spring dampers (9) to provide damping forces for absorbing shocks andmassunits.Theypointedoutthatthemaximumstress and vibrations of purchase cable. Besides, when the tailhook appeared at the moment that the stress wave reached the of the aircraft grasps the deck pendant at initial stage, there pulley and reflected from the pulley to the tailhook. appears a severe vibration of the rope. Two damper sheave Although the LS-DYNA model is competent to simulate installations(4)areintroducedtoreducetheamplitudeofthe the whole arresting process and capture the stress wave vibration in deck pendent. dynamics at initial impact stage, the explicit dynamics strat- Obviously, the arresting gear system is a complicated egy in LS-DYNA requires both fine mesh for the flexible mechanical and hydraulic coupled system which contains cablesandsmalltime-steptoguaranteethestabilityofinte- rigid bodies, flex ropes, and hydraulic units. All of the them gration algorithm. These two facts reduce the efficiency will be modeled here through multibody system dynamic greatlybothatthefirstimpactstageandthesecondsteady approach as illustrated in Figure 2.Andthegoverningequa- stage. Besides, the current available studies are focused on the tion of each part will be discussed in the left context of this centerline landing case, and no simulation has appeared in section. literature about the offline landing cases. Based on the above problems, in this paper a full-scale 2.1. Governing Equations of Rigid Bodies. The landing aircraft, MK7 model of the whole arresting gear system under the deck sheaves, fixed and movable carriage, plunger of the frame of the multibody system dynamics is presented. The hydraulic cylinder, and muffles are modeled as rigid bod- governing equations of the whole system are solved through ies. Generally speaking, a rigid body undergoing arbitrary an implicit backward differentiation formula to reduce the large three-dimensional motion owns six, three translational simulation time, since it can use larger time-step compared and three rotational, degrees of freedom. In the multibody r =[𝑥,𝑦,𝑧]𝑇 with explicit integration algorithm and the time-step is dynamicapproach,themasscenter rigid in adjustable according to the dynamical properties of problem. Cartesian space is usually adopted to describe the transla- The model includes an aircraft and its tailhook, hydraulic tional motion, where the superscript 𝑇 represents transpose system and dampers, deck pendant, and purchase wire. The transformation of vector or matrix hereafter. However, there cable element is introduced to simulate the arbitrary large are various ways to parameterize three-dimensional finite displacement and rotation of the wire ropes, and damping rotation, such as Euler angles or rotation vector with three forceisusedtomodeltheeffectofhydraulicsystem.The variables and Euler parameters with four variables. Here, 𝜆 =[𝜆0,𝜆1,𝜆2,𝜆3]𝑇 functionality of hydraulic parts is studied in detail. Further- we use Euler parameters rigid since it is more, the stress of the wire ropes in the along and deviate singularity free in contrast to each three-parameters parame- from centerline landing cases are simulated and analyzed. It is terization of finite rotation that involves singularity problem foundrepeatedlythatthereflectionandsuperpositionofthe as proved by [11]andreviewedby[12]. The definition of Euler stress wave at the deck sheave contribute mainly to the peak parameters is based on Euler’s rotation theorem, saying that valueofstress.Anditdrawsaconclusionthatthemaximum any rotation movement of a rigid body, or composition of stressintheofflinelandingcaseisalittlebitsmallerthanthe them, can be achieved by rotation by an angle 𝜃 with respect along centerline landing case. to an axis n. And the Euler parameters are defined as The following context is organized in this way. In 0 𝜃 1 𝜃 Section 2, we build the arresting gear system model through 𝜆 = ( ), 𝜆 =𝑛 ( ), cos 2 𝑥 sin 2 multibody system dynamics approach, verify some elements (1) at Section 3, and simulate the full scale MK7 type arresting 𝜃 𝜃 𝜆2 =𝑛 ( ), 𝜆2 =𝑛 ( ), gear system at Section 4. Conclusions are given in Section 5. 𝑦 sin 2 𝑧 sin 2 𝑛 𝑛 𝑛 2. Modeling MK7 Type Arresting Gear System where 𝑥, 𝑦,and 𝑧 arethethreecomponentsoftherotation axis n and 𝜃 istherotationangle.ThefourEulerparameters The basic structure and working principle of MK7 type arrest- are not independent, satisfying the following normalization ing gear system are shown in Figure 1.Duringthedeckland- constraint equation ing, the aircraft tailhook grasps the deck pendent (1) which is 𝜆𝑇 𝜆 −1=0. a cross the deck. The pendent is connected through muffles rigid rigid (2) Mathematical Problems in Engineering 3

Runway

Aircraft

(1)

(2)

(4)

(11)

Air Fluid Air Fluid Air (10) Fluid Fluid Fluid (3) (9)

(5) (8)

(6) (7)

(1) Deck pendant (7) Plunger of hydraulic cylinder (2) Connecting muffle (8) Movable carriage (3) Purchase cable (9) Cable anchor damper (4) Deck sheave installation (10) Constant runout control valve (5) Fixed carriage (11) Accumulator (6) Hydraulic cylinder

Figure 1: Structure and working principle of MK7 type arresting gear system.

T c

Damper sheaver installation Sheave Crosshead Fs Purchase cable F1 Fixed sheave assembly

Deck pendant Aircraft Fs Tc Cable anchor damper

Figure 2: Multibody model of the arresting gear system. 4 Mathematical Problems in Engineering

r 𝜆 L The mass center rigid and Euler parameter rigid form the generalized coordinates of a rigid body as l

𝑇 𝑇 𝑇 N q =[r , 𝜆 ] . N rigid rigid rigid (3) 2 N1 f(s, t)

Assume that the whole arresting gear system contains 𝑛𝑏 𝑛 r1 N rigid bodies and they are interconnected through 𝑐 con- r 3 straints which can be expressed in the following unified form r2 Φ (q , q , ..., q ,𝑡)=0 𝑘=1,...,𝑛 . Y 𝑘 1 2 𝑛𝑏 𝑐 (4) r3 Then the governing equations of the whole system will be 0s0.5 1 𝑛 𝑐 O X M q̈ + Q + F + ∑Φ𝑇 𝜇 = 0 𝑖 = 1, 2, ..., 𝑛 Z 𝑖 𝑖 𝑖 𝑖 𝑘,q𝑖 𝑘 𝑏 𝑘=1 (5) Figure 3: Cable element. Φ𝑘 = 0 𝑘 = 1, 2, ...,𝑐 𝑛 , derived from the first kind Lagrange equations, where 𝜇𝑘 is Φ Φ The generalized elemental coordinate q𝑒 is the direct the Lagrange multiplier corresponding to constraint 𝑘, 𝑘,q𝑖 is the partial derivative of Φ𝑘 with respect to q𝑖,and union of three nodal positions, or 𝑇 𝑇 𝑇 𝑇 𝑚𝑖I3×3 0 𝑇 q𝑒 =[r1 , r2 , r3 ] . (7) M𝑖 =[ ], B𝑖 =4G𝑖 J𝑖G𝑖, 0 B𝑖 Lagrange functions are introduced as the shape functions to 1 0 3 2 r −𝜆𝑖 𝜆𝑖 𝜆𝑖 −𝜆𝑖 interpolate the elemental position filed. The position vector [ 2 3 0 1 ] at 𝑠 is G𝑖 = −𝜆𝑖 −𝜆𝑖 𝜆𝑖 𝜆𝑖 , (6) 3 2 1 0 [−𝜆 𝜆 −𝜆 𝜆 ] 𝑖 𝑖 𝑖 𝑖 r (𝑠,) 𝑡 = N (𝑠) q𝑒 (8) 1 𝜕(M q̇ ) N(𝑠) = [𝑁 I ,𝑁 I ,𝑁 I ] Q =(V − V𝑇) q̇ , V = 𝑖 𝑖 , with 1 3×3 2 3×3 3 3×3 ,where 𝑖 𝑖 2 𝑖 𝑖 𝑖 𝜕q 𝑖 1 𝑁1 (𝑠) =2(𝑠− ) (𝑠−1) , where 𝑚𝑖 and J𝑖 are the mass and inertia tensor of the 𝑖th rigid 2 I 3×3 𝑁2 (𝑠) =−4𝑠(𝑠−1) , (9) body and 3×3 is a identity matrix; the generalized force 1 F𝑖 contains all of the contribution from external forces such 𝑁 (𝑠) =2𝑠(𝑠− ). 3 2 as gravity and contact forces. Then the normal strain of the cable element is 2.2. Governing Equations of Flexible Cables. During the 1 𝑇 𝜀= (r󸀠 r󸀠 −1) arresting process, the deck pendant and purchase cable 2𝐿2 (10) are stretched and bended accompanying large displacement movement. To modele a cable with circular cross-section for according to Green’s strain definition in the theory of elastic- multibody dynamics, Shabana [13, 14] and several pioneers ity [15], where the prime represents the partial derivative with have proposed absolute nodal coordinate formulation, in respect to parameter s or which the global position and slope of cable are adopted 󸀠 𝜕r 󸀠 r = = N (𝑠) q . (11) as generalized coordinates to interpolate the displacement 𝜕𝑠 𝑒 field and calculate the normal strain and bending curvatures. Although this formulation is capable of modeling cables in A simple elastoplastic constitutive equation is adopted here to arresting system, we prefer to introduce nodal position only introduce damping force for elastic deformation only, which as generalized coordinate to reduce the calculation scale means that the stress 𝜎 is since the bending energy is negligible. As shown in Figure 3, 𝜎=𝐸(𝜀+𝛽𝜀)̇ , consider a cable element with a circular cross section area 𝐴 (12) 𝐿 𝑁 and length . It contains three uniformly located nodes 1, where 𝐸 is the Yong’s modulus, 𝛽 is the damping ratio of 𝑁 𝑁 r r r 2,and 3 with positions 1, 2,and 3 under the global material, and the overdot represents the partial derivative 𝑙 coordinate system OXYZ. Then the arc length equals to with respect to time 𝑡. 𝑁 𝐿 𝑁 𝐿 𝑁 0at 1,0.5 at 2,and at 3. A normalized elemental Then the virtual work done by inertial force, internal 𝑠=𝑙/𝐿 parameter is introduced for mathematical simplicity. elastic and damping force, and external forces should be zero, The cable is subjected to a distributed or concentrated which means that f(𝑠, 𝑡) 𝑠 external load which is a function of both space and 1 𝑡 𝑇 𝑇 time . The governing equation of this element will be derived 𝐿 ∫ (−𝜌𝐴r̈ 𝛿r −𝐴𝜎𝛿𝜀+f(𝑠,) 𝑡 𝛿r) d𝑠=0, (13) hereafter. 0 Mathematical Problems in Engineering 5

Collision detection points 2.4. Hydraulic Forces. To reduce the stress peak value of the deck pendant and vibration of the purchase cable and elimi- Node nate the phenomenon of slacking of the rope, damper sheave is installed between two deck sheaves and anchor damper is Cable set on the end of the rope, as is shown in Figure 5.

2.4.1. Damper Sheave Installation. The throttle of the damper sheave installation has an invariable cross section area aper- Rigid ture to let the oil flow goes through, producing the oil damping force. The relationship between damping force and the velocity of the piston motion is as follows [16]: Figure 4: Contact detecting between cable and rigid body. 2 𝐹𝑠 =𝑘𝑠V𝑠 +𝐴𝑠𝑃𝑠, (18)

where 𝑘𝑠 is the equivalent damping coefficient of the oil, V𝑠 is where symbol 𝛿 represents the variation operation. After sim- the velocity of the piston, 𝐴𝑠 isthecrosssectionareaofthe plifying, we got the governing equation of the element piston, and 𝑃𝑠 is the pressure of the gas in the air flask. Omitting the variation of the volume of the oil, the M𝑒q̈𝑒 + K𝑒 (q𝑒, q̇𝑒) q𝑒 = F𝑒 (14) displacement of the piston in the sheave damper is defined as 𝑢𝑠. As the gas experiences the isentropic process, 𝑃𝑠 satisfies with the relationship as follows 𝑉 𝛾 1 1 𝑃 =𝑃 ( 𝑠0 ) , M =𝜌𝐴𝐿∫ N𝑇N 𝑠, K =𝐸𝐴𝐿∫ (𝜀 + 𝛽𝜀)̇ N󸀠𝑇N󸀠 𝑠, 𝑠 𝑠0 (19) 𝑒 d 𝑒 d 𝑉𝑠0 +𝐴𝑠𝑢𝑠 0 0 1 where 𝑃𝑠0 is the initial pressure of the air flask, 𝑉𝑠0 is the initial 𝑇 𝛾 F𝑒 =𝐿∫ N f (𝑠,) 𝑡 d𝑠. volume of the air flask, and is the adiabatic coefficient of the 0 gas. (15) 2.4.2. Cable Anchor Damper. During the arrestment of the It is interesting to note that the elemental mass matrix M𝑒 is a aircraft, the cylinder of the cable anchor damper is connected constant matrix and the elemental stiffness matrix K𝑒 vanish with the main cylinder, so the pressure is equal for the two oncethestrainandthetimederivativeofstrainarezeros, parts. However, the oil pressure of cylinder is equal to that which means that cable has stiffness only when it is pre- of air of accumulator in the process of ram going back to its stressed. original position. The cross section area of piston of the cable anchor 2.3. Contact between Cable and Rigid Body. According to the damper is 𝐴𝑐,themassofthepistonis𝑚𝑐,itsdisplacement assumption of the cable elements, cable is simplified to an increment is 𝑥𝑐, and the force of the cable anchor damper is axial line in the process of collision between cable and rigid 𝑇𝑐; it can be obtained that body. When some detecting points are set on the surface of 𝑇 =𝑚𝑥̈ +𝑃𝐴 , the rigid body and the axial line of the cable, we transform 𝑐 𝑐 𝑐 𝑐 𝑐 (20) the detection between cable and rigid body to the detection where 𝑃𝑐 is the pressure of the oil in the cylinder of the cable between rigid body and points, as is illustrated in Figure 4. anchor damper. According to the Hertz contact theory, the contact force f𝑐 canbeexpressedas 2.4.3. Main Engine Cylinder. When the aircraft is landing, the kinetic energy of the aircraft is transferred to the arresting sys- f𝑐 =𝑓𝑛n𝑓 +𝑓𝑡t𝑓, (16) tem by the purchase cable, forcing the ram in the main engine holding the pressurized hydraulic fluid, and is changed to the where n𝑓 is the unit normal vector of the contact surface; t𝑓 heat of the hydraulic oil. Therefore, we can apply a damping is the corresponding unit tangential vector; 𝑓𝑛 is the normal force on the ram to simulate this procedure. collision force; 𝑓𝑡 is the tangential friction force. The main engine cylinder and the ram are illustrated in Collision force is exerted on the nodes of the cable Figure 6, the hydraulic pressure of the oil in the main engine element. Therefore, the generalized collision force can be cylinder is 𝑃1, the effective cross section area of the ram is 𝐴1, expressedbelowaccordingtothevirtualworkprinciple: the mass of the ram and crosshead is 𝑚1, the sliding velocity of the ram is V1, 𝐹1 is the force applied on the ram by the Q = N𝑇 (𝑠 ) f , 𝑝 𝑛 𝑛 (17) arresting rope, and 𝜇 isthedampingratioofthehydraulic oil.Itcanbegainedthat where N is the shape function of the cable element; 𝑠𝑛 is the 𝑑V element parameter of the detecting point; f𝑛 is the force 1 𝐹1 =𝑚1 +𝜇V1 +𝐴1𝑃1. (21) exerted on the detecting point. 𝑑𝑡 6 Mathematical Problems in Engineering

Accumulator A Cable anchor damper Cable c Tc Pc Piston Cylinder

Cooler

Damper sheave Fs installation As Ps Air s Fluid Piston rod

Figure 5: Damping mechanism in the arresting gear system.

Accumulator Cylinder Ram A 1 F1 m P1 1 1 Cam Control valve drive system Figure 6: Main engine cylinder. Plunger

Valve stem The velocity of the oil in the main engine cylinder is equal 𝜙 x to the sliding velocity of the ram; 𝑃1, which is the hydraulic pressure in the main engine cylinder, is related to the design d of the constant runout control valve. If the proposal of the small aperture is adopted [16], the pressure is as follows: Cylinder 𝜌𝐴2 𝑃 = 1 V2 +𝑃, Figure 7: Constant runout control valve. 1 2 2 1 2 (22) 2𝑐𝑑𝐴2 where 𝑐𝑑 is the coefficient of the flow which is related to the where 𝑘1 and 𝑘2 are the functions describing the shape of areaoftheapertureandthemassofliquid,andsoforth,𝐴2 𝑥 𝑃 the surface of the cam; 0 is initial area of the throttle whose is the cross section area of the hole of the throttle, and 2 is relationship with diameter of the valve is as follows: the pressure of the energy accumulator. 𝑑 𝑥0 = , (25) 2.4.4. Constant Runout Valve. Constant runout valve is 𝑘𝑚 locatedbetweenthemainenginecylinderandtheaccumula- tor, controlling the arresting force to arrest the aircraft within where 𝑘𝑚 is the weight adjusting coefficient. We can change aprescribeddistanceinthatitcanadjusttheincrementof the initial area of the throttle by changing 𝑘𝑚,sothearresting the oil in the main engine cylinder by changing the area of gear system can capture the aircrafts of different weights. the hole of the throttle. Constant runout valve is composed of The control valve drive system is installed between cam, valve stem, plunger, and the control valve drive system, crosshead and the fixed sheaves, so the relationship between as is illustrated in the Figure 7. the position of the crosshead 𝑢 and the rotation angle of the 𝛼 The cross section area of the hole of the throttle is 𝐴2, cam of the throttle is as follows: whose relationship with the position of the valve stem 𝑥 is as 𝑢 follows: 𝛼= , (26) 𝑘𝑢 𝑥 𝐴 =𝜋𝑑𝑥 𝜙(1− 2𝜙) , 2 sin 2𝑑 sin (23) where 𝑘𝑢 is the transfer coefficient between crosshead posi- tion and the rotation angle of cam. When the association where 𝜙 is half of the top angle of valve stem; 𝑑 is the diameter of the area of the throttle and the position of the crosshead of the valve which connects the main oil cylinder and the is established, the arresting distance of the aircraft can be throttle, and the relationship between diameter and the 𝛼 controlled. whichistherotationangleofthecamisasfollows: 𝑥=𝑥 (1 − 𝑘 𝛼−𝑘 𝛼2), 2.4.5. Integration Method of the Whole System. Basedonthe 0 1 2 (24) above description, the governing equations of the arresting Mathematical Problems in Engineering 7 gear system form a set of differentiation and algebraic equa- tions (DAEs). They can be generally and uniformly expressed T2 as E (𝑡, q, q̇, q̈) =0, (27) Cylinder where q is a ne-dimensional vector of variables, to be solved in the whole system, including rigid bodies’ position, Euler parameters, cable node position, and Lagrangian multipliers of constraints, E is the implicit equations with same dimen- sion ne. One popular numerical algorithm for solving DAEs with contact problems is the implicit first-order backward Cable differentiation formula (BDF). Although the details of first- orderBDFcouldbefoundinShampineandReichelt’spaper [17] or Hairer and Wanner’s monograph [18], it is preferred to T1 outline the basic idea here for the integrity of this paper. Assume that we are going to solve q𝑛+1 at time 𝑡𝑛+1,which satisfies Figure 8: Model of a cable wrapped around a fixed cylinder.

E (𝑡𝑛+1, q𝑛+1, q̇𝑛+1, q̈𝑛+1) = 0. (28)

Given the values of q at time 𝑡𝑛 and 𝑡𝑛−1 as q𝑛 and q𝑛−1, (3) If the Newton-Raphson iteration success with solu- q ∗ the first and second time derivative of 𝑛+1 could be derived tion q𝑛+1 the integration tolerance 𝜏𝑖 is one order through numerical discrete as smaller than the settled tolerance 𝜏 or 𝜏𝑖 <𝜏/10,then 1 1 the current step size will be increased. q̇𝑛+1 = q𝑛+1 + 𝛼, 𝛼 =− q𝑛, ℎ ℎ In this way, the time-step of BDF is automatically adjusted 1 to fit the dynamic properties of the system. The integration q̈ = q + 𝛽, 𝑛+1 ℎ2 𝑛+1 (29) method will chose small time-step for the initial impact stage at arresting process and large time-step for the stable 1 1 1 1 1 deceleration stage. This saves calculation time a lot compared 𝛽 = ( − ) q𝑛 + q𝑛−1, ℎ 𝑡𝑛 −𝑡𝑛−1 ℎ ℎ 𝑡𝑛 −𝑡𝑛−1 with fixed time-step integration method. where ℎ=1/(𝑡𝑛+1 −𝑡𝑛) is the current time-step. After sub- stituting (29)into(28), the DAEs are simplified to nonlinear 3. Verification Example q algebraic equations of variables 𝑛+1 only In order to verify the established cable element and the colli- 1 1 sion between the cable and the rigid body, a simple multibody E (𝑡 , q , q + 𝛼, q + 𝛽)=0 𝑛+1 𝑛+1 ℎ 𝑛+1 ℎ2 𝑛+1 (30) example is presented. As shown in Figure 8, considering a cable wrapped around a fixed cylinder with two full circles, 𝛼 𝛽 since both and are known. This nonlinear equation it is subjected to vertical forces 𝑇1 and 𝑇2 at two ends, respec- (30) can be solved in terms of the classical Newton-Raphson tively. The friction coefficient between cable and cylinder is q0 q iteration method, and one possible initial guess 𝑛+1 of 𝑛+1 ] = 0.1. Then the analytical solution shows that the cable will for iteration could be get by assuming that q is linear variable 4𝜋] be balanced when 𝑇2 =𝑇1/𝑒 .Thevelocityresponseofthe and extending q𝑛 and q𝑛−1 to time 𝑡𝑛+1 node at the upper end under the external forces 𝑇1 =40Nand 4𝜋] q − q 𝑇2 =𝑇1/𝑒 = 11.3844 NisshowninFigure 9. It is almost q0 = q + 𝑛 𝑛−1 ℎ. 𝑛+1 𝑛 (31) stable which agrees with the analytical prediction greatly. 𝑡𝑛 −𝑡𝑛−1 ℎ There are three cases in which the current time-step will be 4. Simulation of Arresting Gear System adjusted. (1) If the Newton-Raphson iteration fails to get a con- Making use of the modeling methodology developed in the verged solution, then the ℎ will be reduced and the Section 2, we build a full-scale MK7 type hydraulic arresting nonlinear equation (30)willberesolved. gear system with parameters shown in Table 1. To determine the mesh size of the cables in arresting gear system, we build (2) If the Newton-Raphson iteration successfully gives a a simple supported cable with same physical and geometrical q∗ converged solution 𝑛+1 and the integration tolerance parameters of the purchase cable as shown in Figure 10.It 𝜏 𝜏 𝜏 >𝜏 𝑖 is larger than the settled tolerance or 𝑖 ,then is subject to impact with initial velocity 20 m/s. The cable ℎ the current time-step will be reduced, where is uniformly meshed by 300 elements and 600 elements, 󵄨 ∗ 0 󵄨 respectively. The maximum displacement difference between 󵄨q𝑛+1 − q𝑛+1 󵄨 1 𝜏 = (󵄨 󵄨) . (32) these two meshes at center point within the first second is 𝑖 max 󵄨 q 󵄨 ℎ2 󵄨 𝑛 󵄨 1.3%, and the maximum difference of stress is 4.1%. Since 8 Mathematical Problems in Engineering

0.01 peaks,andthemaximumoneis3.98gwhichisalsothe maximum acceleration of the whole arresting process. The character of entering the second stage is that the vibration of impact diminished, and acceleration gots stable. It ends when thevelocityreducestozero,andtheelasticforcegeneratedby 0 deck pendent leads to the third backward stage. 0.0002 m/s It is interesting to note that the stress at the middle of Velocity (m/s) Velocity deck pendent, shown in Figure 12,performssimilartothe acceleration of the aircraft. It also contains three peaks at the initialstage.Itrunstoslowlyvaryingstage.Thissimilarity −0.01 012implies that the acceleration of aircraft comes from the stress Time (s) of cable at contact point. The stress gets its first peak point A once the impact between tailhook and cable happens. Then Figure 9: The velocity of the cable node. the stress wave generated by this shock propagates toward andreachesthedecksheaveat0.128s;itisreflectedbackward and results in the second stress peak B immediately after Table 1: Parameters of the arresting gear system. superposing with the original propagating one. When the backward wave reaches tailhook again at 0.197 s, its reflection Physical parameters Symbol Unit Value and superposition result in the third stress peak C with value 𝑚 Aircraft mass 𝑎 kg 25,000 of 609.3 MPa which is also the maximum stress during the Landing speed 𝑉𝑎 m/s 65 whole arresting process. Later, the stress wave decays, and Sheave mass 𝑚𝑝 kg 277.8 the hydraulic force mainly generated by constant runout valve 11 Yong’s modulus 𝐸𝑐 Pa 2×10 leads to a smooth variable period of stress. We will see later 3 3 that the decay of the stress wave is caused by the absorbing Cable density 𝜌𝑐 kg/m 7.8 × 10 −6 effect of damper sheave installation and cable anchor damper. Damping ration 𝛽𝑐 — 1×10 8 Stiffness coefficient K N/m 3×10 4 Damping coefficient C N⋅s/m 1×10 4.2. Stress in Cable at Different Locations. Five points are cho- Collision index e —1sen as the detection points on the deck pendant and purchase cable of the arresting gear system, as is shown in Figure 13.A Friction coefficient V — 0.02 isthemiddlepointofthedeckpendant;Bisanarbitrarypoint in the purchase cable; C and D are the arbitrary two points inthepurchasecablebetweencrossheadandfixedsheaves both the displacement and stress converged to acceptable whereCisclosetopointBandDisclosetopointE;Eis accuracies, we decided to use 300 elements every 30 meters an arbitrary point in the cable anchor installation. The results to mesh the cables in arresting gear system. show that the maximum along the whole cable happens at the In the following context, the aircraft landing along the middle point of deck pendent. Each stress at the second stage runway centerline and offset the centerline will be simulated from point A to E varies a little due to the friction between and analyzed, respectively. The acceleration, velocity, and cable and sheaves but shares the same smooth varying position of aircraft and the stress within rope are presented. process. At the first stage, the point E which is far away And the functionalities of cable anchor damper and damper from contact point is not notable, influenced by the impact sheave installation on the rope stress are analyzed in detail. between deck pendent and cable.

4.1. Landing along Centerline of Runway. At the most ideal 4.3. Functionality of Damper Sheave Installation and Cable situation, the landing aircraft velocity is parallel to the runway Anchor Damper. Tounderstandandevaluatethefunction- and the anchor hooks the cable right at the middle point. In ality of damper sheave installations (DSI) and cable anchor this case, the response of aircraft from the initial arrestment damper (CAD), we build four models with parameters listed to stop and the snapshots of the whole system are shown in in Table 1,andtheonlydifferenceis:(1)removeDSIandCAD, Figure 11.Ittakesabout105mand3.52secondstodecelerate (2)removeCADonly,(3)removeDSIonly,or(4)keepboth the velocity from full speed to zero. And then the aircraft is DSI and CAD. The stresses at the middle of deck pendent with pulled back a little because of the retraction force generated these four cases are shown in Figure 14. from the elastic energy stored in deck pendant. Figure 14 shows the following cases. The acceleration vibrates severely initially then stays (1) Without any damper, the stress of cable vibrates around 2 g for a while and reduces to zero finally. Based on violently not only at the first stage but also at the this, we divide the arresting process into three stages: (1) second stage as shown by the red dashed line, and the captureshockstage,(2)effectivearrestingstage,and(3)back- maximum stress reaches 808 MPa. ward stage. The first stage is caused by the impact between tailhook and deck pendent which happened at 0.016 s because (2) After installing DSI only, the maximum stress at the of the initial gap between them. It contains three acceleration first stage is reduced by 24.6 percent to 609 MPa, and Mathematical Problems in Engineering 9

1.5 Δ% = 1.3% 1

0.5

0

Displacement (m) Displacement −0.5

−1

m 0 0.2 0.4 0.6 0.8 1.0 Time (s)

l=30 1200 m/s 1000 Δ% = 4.1% Measuring point Measuring =20 0

800

600

Stress (MPa) 400

200

0 0 0.2 0.4 0.6 0.8 1.0 Time (s)

Node 300 nodes 600 nodes

Figure 10: The mesh convergence test.

−4 Arresting landing on centre C 700 − 3 C 600 −2 B B Process 500 −1 A A(0.016;−0.72) B(0.128;−1.79) C(0.197;−3.98)

Acceleration (g) Acceleration A 0 400 Point A B C Time 0.016 s 0.128 s 0.197 s Stress 60 300 400.2 MPa 535.2 MPa 609.3 MPa

40 (MPa) Stress 200 Runway centre-line 20 (3.524; 0) 100 Velocity (m/s) Velocity 0 0 120 0 0.5 1 1.5 2 2.5 3 3.5 4 (3.524; 105.1) 80 Time (s)

40 Figure 12: Stress of cable in the process of landing along center.

Distance (m) Distance 0 0 0.51 1.5 2 2.5 3 3.5 4 the stress vibration at the second stage is reduced Time (s) greatly although it still exists. (3) After installing CAD only, the maximum stress at the first stage is the same as without any damper, but the stress vibration at the second stage almost vanished as shownbytheblackdashedline. (4) After installing both DSI and CAD, the first stage 0 s 1 s2 s3 s4 s maximum stress is reduced to 609 MPa, and the sec- ondstagevibrationisalsovanishedasshownbythe Figure 11: Movement of aircraft. pink solid line. 10 Mathematical Problems in Engineering

700 ×104 600 A2 A3 Point (time/s; stress/MPa) B3 500 A1(0.016; 400.2) 2.0 600 A1 B2 C3 B1(0.032; 274.2) 400 C1(0.052; 206.4) Sheave 300 B1 C2 E3 𝜎 500 C1 A2(0.128; 535.2) 1.6 Cable anchor 0 =300 200 B2(0.136; 431.5) damper D3 C2(0.152; 358) MPa 100 Crosshead l=25 400 A3(0.197; 609.3) m Damper sheave 0 B3(0.208; 504.2) 1.2 installation 0.05 0.1 0.15 0.2 0.25 C3(0.224; 396.8) 300 D3(0.236; 320.6) 3.75 Hz E3(0.276; 268) Fixed sheave assembly 0.8 Amplitude

Stress (MPa) Stress 200 B 100 A C E 0.4 7.5 Hz 10.75 Hz 0 D 0 −100 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 Frequency (Hz) Time (s) Without any damper With damper sheave installation A D With cable anchor damper With all damper B E C Figure 15: Cable stress of four cases at frequency domain. Figure 13: Stress of cable at different locations.

Arresting landing off centre 900 600 900 D 800 808 MPa 24.6% Process E Without any damper 609 MPa 500 C 700 F Point AB CDE F A Time 0.016 s 0.1 s 0.14 s 0.164 s 0.22 s 0.248 s 600 400

Stress (MPa) Stress Stress 403.2 MPa 403.2 MPa 473.9 MPa 538.7 MPa 492.6 MPa 452.1 MPa 500 0 B 0 Time (s) 0.3 300 400 v Stress (MPa) Stress 300 With damper sheave (MPa) Stress dx 200 Runway centre-line installation only 200 With all damper 100 100 With cable anchor damper only 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Time (s) Time (s)

Figure 14: Cable stress of four cases at time domain. Figure 16: Stress of cable in the process of landing off center.

Throughtheaboveanalysis,weconcludethattheCADis far away from the tailhook and it can not generate any effect which agrees with the frequency of the first spark 3.75 Hz onthefirstshockstage,whiletheDSIiseffectiveatboththe considering the frequency resolution of FFT. In addition, first and the second stages. Besides, it is also inspirational these sparks vanished after installing CAD or DSI since the to check the cable stress at frequency domain despite that damping force absorbed the vibration in cables severely. the dynamic behavior of the arresting system keeps changing during arresting process. The stress at time range [0.5, 2.5] 4.4. Landing off Centerline of Runway. Now we discuss the is analyzed via fast Fourier transformation (FFT); then the situationofthattheaircraftvelocityisparallelwiththerun- frequency resolution is 0.5 Hz since the sampling time lasts way centerline, while the landing point is 0.5 m left off the two seconds. As shown in Figure 15, there are three sharp centerline. In this case, the stress at the middle point of the sparksat3.75Hz,7.5Hz,and10.75Hzwhennodamperis deck pendant during the whole process is shown in Figure 16. installed at all. This corresponds to the significant stress vibra- It is shown from the simulation result that the deck tion in time domain. Further, it is interesting to note that the pendant has 5 stress peaks at the first stage. The first is caused frequencies of second and third sparks are roughly two and by the instant collision between the arresting hook and the three times that of the first spark. This behavior is very similar deck pendant at 0.016 s, and the second peak is caused by with the eigenfrequency character of a tensioned string. In the reflection and superposition from the left deck sheave fact, the cable length from deck sheave to fixed shave is always of the stress wave after it reached there. However, the third 25 m during the whole arresting process, and the stable stress peak is caused by the reflection and superposition from the within cable is around 300 Mpa from Figure 13. Therefore, the right deck sheave of the stress wave after it reached there. The fundamental frequency of this tensioned string is fourthoneiscausedbythesuperpositionofthetransverse wave at the position of the arresting hook of the aircraft after √𝜎/𝜌 √300 × 106/7860 it is reflected from the left deck sheaves. The fifth one iscaused 𝑓0 = = = 3.9 Hz (33) 2𝑙 2×25 by the superposition of the transverse wave at the position of Mathematical Problems in Engineering 11 the arresting hook of the aircraft after it is reflected from the 𝜎:Thestressofcable right deck sheaves. N: The shape function of the cable element Compared with along centerline landing case, we got five r:Thepositionvectorofcablenode peak values here instead of three. And the maximum stress M𝑒: The elemental mass matrix of cable here is 539MPa which is smaller than 609MPa in case of element along centerline. This means that the offline landing case K𝑒: The elemental stiffness matrix of cable could a little bit reduce the stress peak. element F𝑒: The generalized force of cable element f 5. Conclusion 𝑐:Thecontactforce Q𝑝: The generalized collision force The arresting process is a complicated coupling dynamics 𝑠𝑛: The element parameter of the detecting between rigid bodies, flex bodies, and hydraulic units. It is point strong nonlinear and involves both a transient wave propa- f𝑛: The force exerted on the detecting point gation process in rope and a smooth decelerating of aircraft. 𝐹𝑠: The oil damping force in damper sheave This fact discounts the calculate efficiency of explicit method installation since the former process needs small time-step to capture the 𝑘𝑠: The equivalent damping coefficient of the propagation events, while the later one needs large time-step oil in damper sheave installation to speed up. To solve this problem, in this paper, we proposed V𝑠:Thevelocityofthepistonindamper to build a multibody dynamic model of a full-scale MK7 type sheave installation arresting gear system. And the resulting governing algebraic 𝑢𝑠: The displacement of the piston in the and differentiation equations are solved through a time- sheave damper installation variable implicit backward differentiation formula. In the 𝐴𝑠: The cross section area of the piston in model, a kind of new cable element that is capable of describ- damper sheave installation ing the arbitrary large displacement and rotation in three- 𝑃𝑠: The pressure of the gas in damper sheave dimensional space is developed to model the wire cables and installation damping force is used to simulate the effect of hydraulic 𝛾: The adiabatic coefficient of the gas system. 𝑇𝑐: The force of the cable anchor damper Dynamic simulation shows that the cable stress is dom- 𝐴𝑐: The cross section area of piston of the inated by the propagation, refection, and superposition of cable anchor damper stress waves during the early stage of the arresting process, 𝑚𝑐: The mass of the piston of the cable anchor and later the shock is quickly dissipated by the damper sheave damper installations and cable anchor dampers. Simulation results 𝑃𝑐: Thepressureoftheoilinthecylinderof repeat that maximum stress value happens when the stress the cable anchor damper wave is reflected and superposed between the deck sheaves. 𝑃1: The hydraulic pressure of the oil in the And the maximum stress in the off centerline landing case is main engine cylinder a little bit smaller than the along centerline landing case. In 𝐴1: Theeffectivecrosssectionareaoftheram addition,themultibodyapproachandarrestinggearsystem 𝑚1: Themassoftheramandcrosshead model proposed here also provide an efficient way to design V1: The sliding velocity of the ram and optimize the whole mechanism. 𝐹1: Theforceappliedontherambythe arresting rope 𝜇 Nomenclature :Thedampingratioofthehydraulicoil 𝑐𝑑: The coefficient of the hydraulic oil q 𝐴 rigid: The generalized coordinate of rigid body 2: The cross section area of the hole of the r rigid: Cartesian coordinate of rigid body’s throttle position 𝑃2: The pressure of the energy accumulator 𝜆 : Euler parameter of rigid body’s attitude 𝑥: The position of the valve stem rigid 𝜙 𝜇𝑘: Lagrange multiplier : Halfofthetopangleofvalvestem 𝑑 Φ𝑘: Constraint equations :Thediameterofthevalve 𝛼 𝑚𝑖:Masstensorofthe𝑖th rigid body :Therotationangleofthecam 𝑘 𝑘 J𝑖: Inertia tensor of the 𝑖th rigid body 1, 2: The functions describing the shape of the F𝑖: The generalized force surface of the cam 𝑥 q𝑒: The generalized elemental coordinates of 0: Initial area of the throttle cable 𝑘𝑚: The weight adjusting coefficient f(𝑠, 𝑡): Concentrated external load on cable 𝑢: The position of the crosshead element 𝑘𝑢: The transfer coefficient between crosshead 𝜀: Thenormalstrainofthecableelement position and the rotation angle of cam 𝐸: The Yong’s modulus of material q: Generalized coordinates of the whole 𝛽: Thedampingratioofmaterial arresting system. 12 Mathematical Problems in Engineering

Acknowledgments [16]L.Yongtang,L.Bufang,andG.Yuzhuo,Modeling and Simula- tion of the Hydraulic System, Chapter 2, Metallurgical Industry This work is partially supported by a China Postdoctoral Press, Beijing, China, 2003. Science Foundation (2012M510417). And the authors would [17] L. F. Shampine and M. W. Reichelt, “The MATLAB ODE suite,” also like to thank Dr. Jiawei He for his kind help on visualizing SIAM Journal on Scientific Computing,vol.18,no.1,pp.1–22, simulation results. The anonymous reviewers also provided 1997. some very useful suggestions on improving this paper; it is [18] E. Hairer and G. Wanner, Solving Ordinary Differential Equa- highly appreciated. tions. II: Stiff and Differential-Algebraic Problems,vol.14of Springer Series in Computational Mathematics,Chapter5, Springer, Berlin, Germany, 2nd edition, 1996. References

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