Research on brake torque design of civil aircraft

Brake torque is one of the key parameters of aircraft brake system design. When braking, hydraulic oil enters the brake actuator assembly, the piston on the brake actuator is extended under the action of liquid pressure, the rotor and static brake discs are pressed together, and the friction between the brake discs forms the brake torque. The design of brake torque is too small, which will increase the deceleration distance of the aircraft and can not meet the requirements of the aircraft design field. Excessive brake torque will make the aircraft pause and transition during braking, make passengers feel uncomfortable, and make the aircraft carry unnecessary weight. Generally, the combined torque between the tire and the runway during the RTO or landing stage is called the dynamic brake torque. The torque required to satisfy 25.735 (d)parking brake item in airworthiness CCAR-25-R4 is called the static brake torque. Brake system designers need to calculate the dynamic brake torque and static brake torque requirements, and design brake actuator assembly to meet these requirements.


Introduction
The brake system is an important part of civil aircraft, which is mainly responsible for the ground deceleration function of aircraft in RTO (reject take off) and landing conditions.The braking system of civil aircraft generally converts the kinetic energy of the aircraft into heat energy by applying friction on the brake carbon disc to decelerate the aircraft.To make the aircraft be stopped on shorter airport runways, there are requirements for aircraft deceleration rate so that the aircraft can adapt to more airports and improve the competitiveness of the product.Brake system designers need to design appropriate brake torque to meet the requirements of aircraft deceleration while avoiding excessive brake torque design.Too large brake torque will make the aircraft pause and transition during braking, make passengers feel uncomfortable, and make the aircraft carry unnecessary weight.In addition, when designing the brake torque, it is also necessary to satisfy 25.735 (d)parking brake item in airworthiness CCAR-25-R4, that is, the aircraft cannot roll (can slide) on the runway when one engine is at maximum thrust and the other engines are at the most unfavorable combination of maximum idle thrust.Generally, the torque to meet the ground deceleration of the aircraft is called the dynamic brake torque, and the torque to meet the CCAR-25.735(d)parking brake item is called the static brake torque.[4] Brake system designers need to calculate the braking dynamic and static torque requirements, and design brake actuator assembly to meet the requirements.

Overview of dynamic brake torque
The braking system decelerates the aircraft on the ground during RTO or landing.During this period, a combined torque Mc is generated between the tire and the runway.When braking, the hydraulic oil enters the brake actuator, the piston on the brake actuator is extended under the action of liquid pressure, and the rotor and static brake discs are pressed together.The friction between the brake discs forms the dynamic brake torque Mb.The schematic diagram of the torque of the wheel in the braking process is shown in Figure 1.When Mc > Mb, the aircraft wheel is in a rolling state; When Mb > Mc, the aircraft wheel is in a sliding state; When Mc=Mb, the aircraft wheel is in a balanced state, which is an unstable state and easy to enter the sliding state of the wheel.Therefore, the brake system has an anti-skid function, through which it can adjust the brake pressure, and maintain the Mb slightly less than Mc to as far as possible keep the wheel in the rolling and not sliding state, so as to achieve higher braking efficiency and to obtain the minimum braking distance.Considering the dynamic response characteristics of the hydraulic system during the establishment of the brake command -hydraulic pressure -brake torque, the actual brake torque will be less than the brake command, so when designing the dynamic brake torque Mb, it is necessary to make it greater than the combined torque Mc.In the actual braking process, the brake pressure is adjusted through the anti-skid function, and the Mb is slightly less than the Mc so that the wheel is kept in the state of rolling and not sliding as much as possible, to obtain the minimum braking distance.[10] 2.1 Tire-runway combined torque analysis and calculation Tire-Runway combined torque is the torque generated by the contact between the aircraft tire and the runway during the taxiing process on the ground, and there are two calculation methods.The aircraft stress analysis diagram is shown in Figure 2. [8] Figure 2. Vertical force analysis of aircraft.
The first method is to calculate the Tire-Runway combined torque through the friction between the tire and runway in the braking process.
According to the aircraft stress analysis diagram, the vertical stress balance of the aircraft is calculated in Formula (1), and the loading force of the nose wheel is calculated in Formula (2).
where G is the aircraft gravity, L is the aircraft lifting force, F M is the main wheel loading force, and F N is the nose wheel loading force.[3] where W is the weight of the aircraft, a is the distance between the nose wheel and the center of gravity of the aircraft, b is the distance between the main wheel and the center of gravity of the aircraft, and g is the acceleration of gravity.
The loading force of the main wheel F M is obtained by bringing Formula (2) into Formula (1).F M is brought into Formula (3) to obtain the Tire-Runway combined torque M c of the single wheel.[ where N is the number of the main wheel, μ is the friction coefficient between the tire and the runway, and R is the main rolling radius of the tire (radial tire is the no-load radius minus 1/5 tire load deformation).[5] In this method, μ is a variable, which depends on factors such as wheel rolling speed, tire type, tire inflation pressure, tire wear, tread temperature of tire, runway grade, and so on.Civil aircraft generally take 0.3~0.5.[8] The second method is to use the aircraft deceleration rate to calculate the Tire-Runway combined torque.
Through the aircraft force analysis, the calculation Formula (4) of the aircraft horizontal force is as follows: From Formula (4), Formula (5) is obtained.F W a T D = ⋅ + − ( 5 ) where F A is the resultant force in the horizontal direction of the aircraft, F d is the ground friction force, D is the aerodynamic resistance, T is the engine thrust, and a ac is the aircraft deceleration rate.
Aerodynamic resistance D is calculated by Formula (6) as follows: where V is the aircraft speed, ρ is the air density, S is the wing area and C drag is the aerodynamic drag coefficient.
The aerodynamic resistance D obtained from Formula ( 6) is brought into Formula (5) to get the ground friction force F d and brought into Formula (7) to get the Tire-Runway combined torque of the single wheel where N is the number of main wheels, F N is the loading force of the nose wheel, μ N is the nose wheel rolling friction coefficient (generally 0.0025~0.0075),and R is the main starting tire rolling radius (radial tire rolling radius is no-load radius minus 1/5 tire load deformation).
Compared with Method 1 and Method 2, the friction coefficient between the tire and runway in Method 1 is a variable.Affected by many factors, it is difficult to accurately define the average friction coefficient between the tire and runway in the braking process.Through engineering practice, Method 2 is suggested to calculate the Tire-Runway combined torque.

Dynamic brake torque design of brake actuator assembly
The brake system of modern civil aircraft generally adopts the carbon disc brake.Its working principle is that the brake system controls the servo valve to produce hydraulic pressure, and extends the piston of the brake actuator to form an axial compression force, which then generates friction and brake torque between the brake rotor discs and static discs.[7] The torque is balanced with the Tire-Runway combined torque, which converts the kinetic energy of the aircraft into heat energy and decelerates the aircraft.[6] A typical brake actuator assembly configuration is shown in Figure 3.In order to achieve higher braking efficiency and minimum braking distance, the braking system adjusts the brake pressure through the anti-skid function, and maintains the dynamic brake torque M b slightly less than the Tire-Runway combined torque M c during the braking process, so that the wheel can be kept in the rolling and non-sliding state as much as possible.
In engineering applications, the design of dynamic brake torque needs to consider the margin, as shown in Formula (8).
where K is the coefficient, generally 1.08~1.12.The dynamic brake torque M b is calculated by combining Formula (7) and Formula (8).With M b as the design input, the braking piston area, the number of pistons, the layout of pistons, and the number of disks are designed.The brake actuator theoretically outputs dynamic brake torque M b , as shown in Formula (9).where P is the rated working pressure of the braking system, ΔP 1 is the system pressure drop (generally 10% to 15% of the rated working pressure), ΔP 2 is the anti-skid pressure control value when in RTO condition (generally 10% to 20% of the rated working pressure), S is the effective area of a single piston, m p is the number of pistons, and μ c1 is the friction coefficient of carbon disc during RTO condition, m f is the number of friction surfaces and L is the arm of force (distance from the piston midpoint to the wheel axle).
Among them, the effective piston area S, the number of pistons m p , the number of friction surfaces m f , and the arm of force L need to be preliminarily designed according to experience or similar model data (the number of pistons is also affected by the single cavity architecture or double cavity architecture of the brake system), and the final value is determined by comprehensive iteration according to performance calculation and simulation results.

Overview of static brake torque
The static brake torque, also known as the parking brake torque, refers to the torque required by the brake system to meet the CCAR25.735(d)parking brake item, which requires that the aircraft must have a parking brake device.When one engine is at maximum thrust and any or all of the other engines are at the most unfavorable combination of maximum idle thrust, it can be used by the pilot without further attention to prevent the aircraft from rolling on a dry and paved level runway.

Static brake torque analysis and calculation
The aircraft in the XY plane along the x-axis (Longitudinal direction) force balance: The torque balance of the aircraft around the center of gravity in the XY plane:  10), (11), and (12), the main wheel friction force f Mx is calculated, and the symmetrical thrust static brake torque M s1 is obtained, as shown in Formula (13) : In Formulas ( 10) -( 13), m is the weight of the aircraft, g is the acceleration of gravity, E F is the maximum thrust of the engine, α is the upper inclination angle of the engine installation, β is the internal inclination angle of the engine installation, N F is the loading force of the nose wheel, M F is the loading force of the main wheel, Nx f is the friction force of the nose wheel, Mx f is the friction force of the main wheel, a is the horizontal distance between the center of the nose wheel and the center of the aircraft, b is the horizontal distance between the ground point of the aircraft main landing gear and the center of the aircraft, c is the horizontal distance from the engine thrust operation point to the center of gravity of the aircraft, d is the vertical distance from the engine thrust operation point to the center of gravity of the aircraft, cg H is the height of the center of gravity of the aircraft, and r is the static pressure radius of the main landing gear tire.

Static brake torque of asymmetrical thrust.
The static brake torque of the asymmetrical thrust (maximum thrust on one side, idle thrust on one side) of the twin engines is calculated.The XY plane force diagram of the aircraft is shown in Figure 4, and the XZ plane force diagram is shown in Figure 5. Assuming that, along the course, the left engine is the maximum thrust, and the right engine is the idle thrust, then: The aircraft is force-balanced along the z-axis in the YZ plane (without considering the left and right main landing gear side friction), as shown in Formula (14): The torque balance of the aircraft around the ground point of the right main landing gear in the XZ plane is calculated, as shown in Formula (15): By combining Formula (14) and Formula (15), the left main wheel friction force f LMx is obtained, and then the asymmetric thrust static brake torque M s2 is calculated, as shown in Formula (16).[9]

Static torque design of brake actuator assembly
According to the braking area, number of pistons, layout of pistons, and number of brake disks designed in Section 1.2, the Formula (17) is taken to analyze whether the brake actuator assembly designed according to the dynamic brake torque requirement meets the static torque requirement.
[ ] where P is the rated working pressure of the brake system, 1 ΔP is the system pressure drop (generally 10% to 15% of the rated working pressure), S is the effective area of a single piston, m p is the number of pistons, μ c2 is the minimum friction coefficient of the carbon disc between the lowest ambient temperature and the dispatch temperature, m f is the number of friction surfaces, and L is the arm of force (distance from the piston midpoint to the wheel axle).

Engineering case
After calculation and analysis, the brake torque design requirements of a certain type of civil wide-body aircraft are shown in Table 1.In order to meet the requirement for brake torque, the design parameters of the brake actuator assembly are shown in Table 2. Compared with civil aircraft of the same level globally, such as Airbus A350-800 and Boeing B787-9, through some public information, it is verified that the brake actuator assembly design parameters, such as the number of pistons, piston diameter, piston arm of force, and the number of friction surfaces are within a reasonable design range.About other data, including the working pressure of the brake system and the friction coefficient of the carbon disk, it is necessary to obtain accurate data through the brake system integration test and carry out further design confirmation or optimization iteration work.
Aircraft-level performance requirements need to be verified in subsequent flight tests and approved by the Airworthiness Authority.These aircraft-level performance requirements include aircraft deceleration performance and deceleration distance, and they are integrated performance requirements related to other aircraft systems in addition to the brake system, such as the spoiler design of the flight control system and the reverse thrust design of the engine.

Conclusion
Based on the analysis and calculation of dynamic brake torque and static brake torque, and combined with engineering application experience, two methods of Tire-Runway combined torque are compared, some key links and suggested values of key parameters of brake torque analysis process and brake actuator assembly design process are given, and the brake torque requirements and brake actuator assembly design parameters of a wide-body aircraft are calculated.
The actual working pressure of the braking system during landing or RTO brake is much less than that during parking brake.Therefore, the design drive of the effective piston area, the number of pistons, the layout of pistons, and the number of carbon disks in the brake actuator assembly are the requirements of the braking dynamic torque.In addition, in order to meet the compliance of CCAR25.735(d)parking brake airworthiness, it is necessary to carry out the verification work to confirm whether the brake actuator assembly design meets the static brake torque during the parking brake.
The design of brake torque is a part of the integrated design of the brake system, which requires multiple iterations.Different problems may be encountered in each development stage, such as brake vibration, high brake temperature, brake dynamic response problems, etc.To solve these problems, the design parameters of the brake actuator assembly may be changed.The brake torque analysis and calculation method proposed in this paper can shorten the iteration cycle.

Figure 1 .
Figure 1.Schematic diagram of wheel torque during braking.

3. 1 . 1
Static brake torque of symmetrical thrust.The static brake torque of the symmetrical thrust (maximum thrust on both sides) of the twin engines is calculated, and the XY plane force diagram of the aircraft is shown in Figure4.

Figure 4 .
Figure 4. XY plane force diagram of the aircraft.Formulas (10), (11), and (12) are listed according to force balance and torque balance.The aircraft in the XY plane along the y-axis (Vertical direction) force balance:

Figure 5 .
Figure 5. XZ plane force diagram of the aircraft.

F 1 d 2 d 3 d 4 d
is the maximum thrust of the engine, ER F is the idle thrust of the engine, α is the upper inclination angle of the engine installation, β is the internal inclination angle of the engine installation, Nz f is the lateral friction force of the nose landing gear, LMx fis the friction force of the left main landing gear, is the distance (Longitudinal direction) between the engine thrust application point and the grounding center point of the main landing gear, is the distance between the right engine thrust application point and the right main landing gear grounding center point (Lateral direction), is the distance between the left and right landing gear grounding center point, and is the distance between the left engine and the right landing gear grounding center point (Lateral direction).

Table 1 .
Brake Torque Requirements for a Type of Civil Wide-body Aircraft.

Table 2 .
Design Parameters of Brake Actuator Assembly for a Type of Wide-body Aircraft.