CN109783953B - Novel aircraft landing dynamic load calculation method - Google Patents
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Abstract
The invention discloses a novel method for calculating the landing dynamic load of an airplane, which is used for researching the load response rule of a pavement slab under the actions of high tire pressure, large mass and short-distance take-off and landing, analyzing the coupling action of an undercarriage-an airplane wheel-a pavement according to the stress characteristic and the motion rule of the novel airplane, and establishing an airplane ground dynamic model with 5 degrees of freedom by considering the aerodynamic force change of the airplane. A differential equation set is solved through a fourth-order Runge-Kutta method, and the problem of change of the pavement load along with time and space is researched. The research result shows that: the coefficient of dynamic load of the pavement generated by the novel aircraft main landing gear can reach 1.4, and the sinking speed of the aircraft is a main factor influencing the landing load.
Description
Technical Field
The invention relates to the field of aviation technology and airport engineering, in particular to a novel method for calculating landing dynamic load of an airplane.
Background
With the rapid development of aviation and defense industry in China, a large number of novel airplanes such as J-20, Y-20, C919, an-225, J-31 and the like enter the field of vision of people. These new airplanes tend to have the characteristics of heavy weight, high tire pressure, and short take-off and landing distances. The ground load characteristic of the airplane is changed greatly compared with the prior art due to the change of airplane parameters and take-off and landing characteristics. In the current airport rigid pavement design standard in China, the constant dynamic load coefficient K is only used d The characteristics of the plane to the road surface load are reflected as follows: taking K when the tire pressure q is more than or equal to 1.08MPa at the end part of the road surface d ＝1.25；q<At 1.08MPa, take K d =1.20, the tire pressure of the novel airplane in China is far more than 1.25MPa. The method for equating dynamic load to static load and analyzing the structural stress of the pavement by relying on the theory of an elastic system and the statics knowledge is basically reasonable under the conditions of small weight of the airplane, small tire pressure and low sinking speed, but the static behavior representation of the novel airplane is obviously lack of scientificity and applicability. At present, scholars at home and abroad have certain research results on airplane ground dynamics and road (road) surface structure dynamics in respective fields, but still have a plurality of questions: if the ground load actual measurement data of the novel airplane is very lack, the coupling action mechanism of the airplane and the pavement system which accord with the actual condition is still unclear. Therefore, the research on the novel airplane load action rule and the mechanical response of the pavement is a fundamental problem to be solved urgently in response to the rapid development situation of the national defense aviation industry and the design requirement of the airport pavement.
Disclosure of Invention
Aiming at the problem of the load characteristic of the novel airplane landing on the airport runway, on the basis of analyzing the novel airplane landing characteristic and each subsystem of the airplane, a mathematical model of airplane ground dynamics is established, and the mechanism of the coupling action of the airplane and a pavement system under different landing conditions is considered.
The technical scheme is as follows:
a novel aircraft landing dynamic load calculation method comprises the following steps:
during the landing process of the airplane, the gliding speed of the airplane can be divided into the vertical direction and the horizontal direction, and the vertical sinking speed generates impact effect on the road surface to form dynamic load; the horizontal speed enables the airplane wheel and the road surface to generate relative motion to form friction force, and the friction force depends on the friction coefficient and the vertical load; when the airplane is in landing and taxiing, the airplane can generate vertical force F on the main landing gear under the conditions of self gravity W, air resistance D, lift force L and ground surface z1 Frictional force F x1 Vertical force F of the ground to the nose landing gear z2 Frictional force F x2 The speed is reduced to the normal sliding speed under the combined action of the two mechanisms, and then the runway system slides out; when the airplane runs on the runway, the acting force of the ground on the airplane is transmitted to the airplane body through the undercarriage, so that the undercarriage has an important influence on the ground dynamic characteristics of the airplane; the landing gear mainly comprises a strut, a buffer, a wheel system, a support or retraction system and the like, and has the main functions of supporting and buffering so as to improve the stress condition of the airplane in the vertical direction; during calculation, the landing gear system is simplified into a landing gear buffer and wheel system;
the action of the buffer in the aircraft landing gear can be simplified into oil damping acting force and air spring force, the aircraft tire can also be divided into two parts, namely, the tire elastic force and damping force aircraft consists of an aircraft body system, a main landing gear system and a nose landing gear system, and m 0 ，m 1 ，m 2 Respectively the mass of the fuselage system, the mass of the main landing gear system, the mass of the nose landing gear system, J 0 Is the moment of inertia, s, of the aircraft fuselage system 1 ，s 2 The distances k from the center of gravity of the aircraft to the centers of the nose landing gear and the main landing gear axle 1 ，k 2 ，k 3 ，k 4 Respectively, the elasticity coefficient of the main landing gear suspension system, the elasticity coefficient of the main landing gear tire, the elasticity coefficient of the nose landing gear suspension system, the elasticity coefficient of the nose landing gear tire, c 1 ，c 2 ，c 3 ，c 4 Respectively, the damping coefficient of a main landing gear suspension system, the damping coefficient of a main landing gear tire, the damping coefficient of a nose landing gear suspension system and the damping coefficient of a nose landing gear tire, wherein theta is the pitch angle of an airplane body, and z is 0 ，z 1 ，z 2 Vertical displacements at the center of gravity of the aircraft body, the center of gravity of the main undercarriage suspension and the center of gravity of the nose undercarriage suspension are respectively obtained;
to aircraft fuselage system m 0 Equation (1) is obtained from the rigid body rotational differential equation:
in the formula (I), the compound is shown in the specification,
is the angular velocity of the aircraft fuselage system. />
Is the angular acceleration of the aircraft fuselage system; />
The vertical speed of the center of gravity of the aircraft body, the main landing gear suspension center of gravity and the nose landing gear suspension center of gravity; f h1 ，F a1 Damping acting force and air spring force of a main landing gear buffer are respectively adopted; f h2 ，F a2 Damping acting force and air spring force of a front landing gear buffer are respectively adopted;
and a differential equation (2) is formed in the vertical direction of the machine body:
in the formula, x 0 Is the horizontal displacement of the aircraft in the course,
the course speed of the center of gravity of the airplane body;
the vertical speed of the center of gravity of the aircraft body, the main landing gear suspension center of gravity and the nose landing gear suspension center of gravity and the course acceleration of the center of gravity of the aircraft body, g is the gravity acceleration, k is L For the overall coefficient of lift, take k L ＝0.5C L ρ S, wherein the lift factor C L ＝2π(θ-θ 0 )，θ 0 Is an airfoil zero lift angle of attack, rho is the atmospheric density, and S is the wing area;
for main landing gear suspension system m 1 Analytically, a differential equation of motion (3) is established in the vertical direction:
in the formula, F k1 ，F f1 Respectively the elastic force and the damping acting force of the main landing gear tire;
to nose landing gear suspension system m 2 Analytically, a differential equation of motion (4) is established in the vertical direction:
in the formula, F k2 ，F f2 Respectively the elastic force and the damping acting force of the front landing gear tire;
in the landing and running direction of the airplane, establishing a motion differential equation (5) for the whole airplane:
empirical formula (6), (7) combined with tyre elastic coefficient
k 2 ＝34.514+0.387p 0 +21.816p 0 z 1 (6)
k 4 ＝34.514+0.387p 0 +21.816p 0 z 2 (7)
In the formula, p 0 For the initial tire pressure of an aircraft, μ is the longitudinal sliding friction coefficient, and the relationship between it and the longitudinal slip ratio is relatively complex, and can be given by empirical formula (8):
in the formula, S g For slip ratio, defined as:
V x for aircraft taxi speed, V ω The linear velocity of the tire, r the radius of the tire, delta the deformation of the tire, and omega the rotational linear velocity of the tire;
and (3) simultaneous (1) to (9) differential equations obtain a second-order five-element differential equation set (10):
where sign is a sign function:
the dynamic load of the tire on the road surface is as follows:
q 1 ，q 2 in order to play a role in the excitation of the pavement,
the initial value problem of the high-order differential equation can be calculated by changing variable into the initial value problem of the first-order ordinary differential equation set,
let m =2,3,4 … …, there is an initial value problem of m-order ordinary differential equation:
let y 1 ＝z,y 2 ＝z’,……,y m ＝z (m-1) Then the above equation is normalized to a first order ordinary differential equation set:
for solving the problem of the first-order differential equation set, because the fourth-order Runge-Kutta method can change the step length according to the precision requirement of each stage, is easy to program and use, and has wide application in practical engineering, the method is adopted for solving the problem,
and (3) performing variable conversion on the dynamic differential equation set at the landing stage, and reducing the second-order five-element differential equation set in the step (2) into the following formula, so that:
θ＝y 1 ，
z 0 ＝y 3 ,/>
z 1 ＝y 5 ,/>
z 2 ＝y 7 ,/>
x 0 ＝y 9 ,/>
get q 1 ，q 2 And the second-order quinary differential equation set in the step 1 is converted into the following equation set:
and (4) calculating a differential equation set through numerical values to obtain a numerical solution.
Further, the results show that: the dynamic load coefficient of the pavement generated by the novel aircraft main landing gear can reach 1.4.
Furthermore, the acting force of the main landing gear of the airplane can reach more than 98%, and the landing load of the nose landing gear is small.
Further, the sinking speed of the airplane is a main factor influencing the landing load, and the landing quality is low, so that the dynamic load of the road surface is not greatly influenced by the tire pressure and the pitch angle.
The invention provides a simplified aircraft dynamic load calculation method for the acceptance of a baida, which is convenient for engineers to research and design and meets the engineering calculation requirements.
Drawings
FIG. 1 is a force analysis diagram of an aircraft;
FIG. 2 is an aircraft ground dynamics model;
FIG. 3 is a vertical displacement curve of the center of gravity of the fuselage;
FIG. 4 is an aircraft pitch angle variation curve;
FIG. 5 is a graph of glide distance variation;
FIG. 6 is a nose gear landing load change curve;
FIG. 7 is a main landing gear landing load change curve;
FIG. 8 is a main landing gear cushioning effect occupancy curve;
FIG. 9 is a change law of landing load of the main landing gear under the influence of landing quality;
FIG. 10 is a change rule of the dynamic load coefficient of the main landing gear under the influence of landing quality;
FIG. 11 is a change rule of a dynamic load coefficient of a main landing gear under the influence of tire pressure;
FIG. 12 is a variation law of the dynamic load coefficient of the main landing gear under the influence of the sinking speed;
FIG. 13 is a change law of the dynamic load coefficient of the main landing gear under the influence of the pitch angle;
FIG. 14 is a graph of the change law of the dynamic load factor of the nose landing gear under the influence of the pitch angle.
Detailed Description
The technical solutions of the present invention will be described in further detail with reference to the accompanying drawings and the detailed description.
In the landing process of the airplane, the gliding speed of the airplane can be divided into vertical and horizontal directions, and the sinking speed in the vertical direction generates impact effect on the road surface to form dynamic load; the horizontal speed makes the wheel and the road surface generate relative movement to form friction force, and the friction force depends on the friction coefficient and the vertical load.
When the airplane slides during landing, the airplane can slide under the conditions of self gravity W, air resistance D, lift force L and vertical force F of the ground to the main landing gear z1 Frictional force F x1 Vertical force F of the ground against the nose landing gear z2 Frictional force F x2 The aircraft decelerates to a normal taxi speed under the combined action of the two mechanisms, then slides out of a runway system, and the stress analysis of the aircraft is shown in figure 1.
When the airplane runs on the runway, the acting force of the ground to the airplane is transmitted to the airplane body through the landing gear, so that the landing gear has an important influence on the ground dynamics of the airplane. The landing gear mainly comprises a supporting column, a buffer, a wheel system, a supporting or retracting system and the like, and the landing gear mainly has the supporting and buffering functions so as to improve the stress condition of the airplane in the vertical direction. In the calculation, the landing gear system is simplified to a landing gear bumper and wheel system.
The action of the bumpers in the aircraft landing gear can be simplified into oil damping action and air spring force. Aircraft tires may also be split into two-part applications, namely tire spring and damping forces, as shown in FIG. 2. The aircraft consists of a fuselage system, a main landing gear system and a nose landing gear system. m is 0 ，m 1 ，m 2 The mass of the aircraft fuselage system, the mass of the main landing gear system and the mass of the nose landing gear system are respectively. J is a unit of 0 Is the moment of inertia of the aircraft fuselage system. s 1 ，s 2 The distances from the center of gravity of the airplane to the centers of the nose landing gear and the main landing gear axle are respectively. k is a radical of 1 ，k 2 ，k 3 ，k 4 The elastic coefficients of a main landing gear suspension system, a main landing gear tire, a front landing gear suspension system and a front landing gear tire are respectively. c. C 1 ，c 2 ，c 3 ，c 4 The damping coefficient of the main landing gear suspension system, the damping coefficient of the main landing gear tire, the damping coefficient of the nose landing gear suspension system and the damping coefficient of the nose landing gear tire are respectively. And theta is the pitch angle of the airplane body. z is a radical of formula 0 ，z 1 ，z 2 The vertical displacements at the centre of gravity of the aircraft body, the centre of gravity of the suspension of the main landing gear and the centre of gravity of the suspension of the nose landing gear are respectively.
To aircraft fuselage system m 0 Equation (1) is obtained from the rigid body rotational differential equation:
in the formula (I), the compound is shown in the specification,
is the angular velocity of the aircraft fuselage system. />
Is the angular acceleration of the aircraft fuselage system. />
The vertical velocities at the center of gravity of the aircraft fuselage, the center of gravity of the main landing gear suspension, and the center of gravity of the nose landing gear suspension. F h1 ，F a1 Main landing gear bumper damping effort and air spring force, respectively. F h2 ，F a2 Respectively nose landing gear bumper damping effort and air spring force.
And a differential equation (2) is formed in the vertical direction of the machine body:
in the formula, x 0 Is the horizontal displacement of the aircraft in the course,
the course speed of the gravity center of the airplane body.
The vertical speed of the center of gravity of the aircraft body, the main landing gear suspension center of gravity, the front landing gear suspension center of gravity and the course acceleration of the center of gravity of the aircraft body are obtained, g is the gravity acceleration, k L Taking k as the coefficient of integrated lift L ＝0.5C L ρ S, wherein the lift factor C L ＝2π(θ-θ 0 )，θ 0 Is airfoil zero lift angle of attack, rho is atmospheric density, and S is wing area.
For main landing gear suspension system m 1 Analytically, a differential equation of motion (3) is established in the vertical direction:
in the formula, F k1 ，F f1 The elastic and damping forces of the main landing gear tires, respectively.
To nose landing gear suspension system m 2 Analytically, a differential equation of motion (4) is established in the vertical direction:
in the formula, F k2 ，F f2 Respectively the elastic force and the damping force of the nose landing gear tyre.
In the landing and running direction of the airplane, establishing a motion differential equation (5) for the whole airplane:
empirical formula (6), (7) combined with tyre elastic coefficient
k 2 ＝34.514+0.387p 0 +21.816p 0 z 1 (6)
k 4 ＝34.514+0.387p 0 +21.816p 0 z 2 (7)
In the formula, p 0 Is the initial tire pressure of the aircraft. μ is a longitudinal sliding friction coefficient, and the relationship between it and the longitudinal slip ratio is relatively complicated, and can be given by empirical formula (8):
in the formula, S g For slip ratio, defined as:
V x for aircraft taxi speed, V ω The tire linear velocity, r, δ, and ω are tire radius, tire deformation, and tire rotational linear velocity, respectively.
And (3) simultaneous (1) to (9) differential equations obtain a second-order five-element differential equation set (10):
where sign is a sign function:
the dynamic load of the tire on the road surface is as follows:
q 1 ，q 2 the function of the road surface stimulation is.
The initial value problem of the high-order differential equation can be calculated by changing variables into the initial value problem of the first-order ordinary differential equation set.
Let m =2,3,4 … …, have the initial value problem of m-order ordinary differential equation:
let y 1 ＝z,y 2 ＝z’,……,y m ＝z (m-1) Then the above equation is normalized to a first order ordinary differential equation set:
for the solving problem of the first-order differential equation set, the fourth-order Runge-Kutta (Runge-Kutta) method can change the step length according to the precision requirement of each stage, is easy to program and use, and has wide application in practical engineering, so the method is adopted for solving the problem.
And (3) performing variable conversion on the dynamic differential equation set in the landing stage, and reducing the second-order five-element differential equation set in the step (2) into the following formula. Order:
θ＝y 1 ，
z 0 ＝y 3 ,/>
z 1 ＝y 5 ,/>
z 2 ＝y 7 ,/>
x 0 ＝y 9 ,/>
get q 1 ，q 2 To 0, the second order quinary differential equation set in step 1 is converted into:
and (3) obtaining a numerical solution by numerically calculating a differential equation system, wherein the landing characteristic of the airplane is shown in figures 3-8, and the dynamic load characteristic change curve of the airplane under the influence of various factors is shown in figures 9-14.
The above description is only a preferred embodiment of the present invention, and the scope of the present invention is not limited thereto, and any simple modifications or equivalent substitutions of the technical solutions that can be obviously obtained by those skilled in the art within the technical scope of the present invention are within the scope of the present invention.
Claims (2)
1. A novel method for calculating the landing dynamic load of an airplane is characterized in that a fourth-order Runge-Kutta method is applied to solving the problem of airplane ground dynamics, and comprises the following steps:
step 1, analyzing the stress of an airplane in a landing and running stage;
in the landing process of the airplane, the gliding speed of the airplane can be divided into vertical and horizontal directions, and the sinking speed in the vertical direction generates impact effect on the road surface to form dynamic load; the horizontal speed enables the airplane wheel and the road surface to generate relative motion to form friction force, and the friction force depends on the friction coefficient and the vertical load; when the airplane slides during landing, the main landing gear is arranged on the ground under the conditions of self gravity W, air resistance D, lift force LVertical force F z1 Frictional force F x1 Vertical force F of the ground against the nose landing gear z2 Frictional force F x2 The speed is reduced to the normal sliding speed under the combined action of the two mechanisms, and then the runway system slides out; when the airplane runs on the runway, the acting force of the ground on the airplane is transmitted to the airplane body through the undercarriage, so that the undercarriage has an important influence on the ground dynamic characteristics of the airplane; the undercarriage mainly comprises a support column, a buffer, an airplane wheel system, a support or retraction system and the like, and has the main functions of supporting and buffering so as to improve the stress condition of the airplane in the vertical direction; during calculation, the landing gear system is simplified into a landing gear buffer and wheel system;
step 2, establishing a 5-degree-of-freedom airplane ground dynamics model;
the function of the buffer in the aircraft landing gear can be simplified into oil damping acting force and air spring force, the aircraft tire can also be divided into two parts, namely, the tire elastic force and damping force aircraft consists of an aircraft body system, a main landing gear system and a nose landing gear system, m 0 ，m 1 ，m 2 Respectively the mass of the fuselage system, the mass of the main landing gear system, the mass of the nose landing gear system, J 0 Is the moment of inertia of the aircraft fuselage system, s 1 ，s 2 The distances k from the center of gravity of the aircraft to the centers of the nose landing gear and the main landing gear axle 1 ，k 2 ，k 3 ，k 4 Respectively, the elastic coefficient of the main landing gear suspension system, the elastic coefficient of the main landing gear tire, the elastic coefficient of the nose landing gear suspension system, the elastic coefficient of the nose landing gear tire, c 1 ，c 2 ，c 3 ，c 4 Respectively, damping coefficient of a main landing gear suspension system, damping coefficient of a main landing gear tire, damping coefficient of a nose landing gear suspension system and damping coefficient of a nose landing gear tire, theta is a pitch angle of an airplane body, and z is 0 ，z 1 ，z 2 Vertical displacements at the center of gravity of the aircraft body, the center of gravity of the main undercarriage suspension and the center of gravity of the nose undercarriage suspension are respectively obtained;
to aircraft fuselage system m 0 Equation (1) is obtained from the rigid body rotational differential equation:
in the formula (I), the compound is shown in the specification,
is the angular velocity of the aircraft fuselage system; />
Is the angular acceleration of the aircraft fuselage system; />
The vertical speed of the center of gravity of the aircraft body, the main landing gear suspension center of gravity and the nose landing gear suspension center of gravity; f h1 ，F a1 Damping acting force and air spring force of a main landing gear buffer are respectively adopted; f h2 ，F a2 Damping acting force and air spring force of a front landing gear buffer are respectively adopted;
step 3, establishing a mechanical balance equation in each degree of freedom direction;
and a differential equation (2) is formed in the vertical direction of the machine body:
in the formula, x 0 Is the horizontal displacement of the aircraft in the course,
the course speed of the center of gravity of the airplane body; />
The vertical speed of the center of gravity of the aircraft body, the main landing gear suspension center of gravity and the nose landing gear suspension center of gravity and the course acceleration of the center of gravity of the aircraft body, g is the gravity acceleration, k is L For the overall coefficient of lift, take k L ＝0.5C L ρ S, wherein the lift factor C L ＝2π(θ-θ 0 )，θ 0 Is an airfoil zero lift angle of attack, rho is the atmospheric density, and S is the wing area;
for main landing gear suspension system m 1 Analytically, a differential equation of motion (3) is established in the vertical direction:
in the formula, F k1 ，F f1 Respectively the elastic force and the damping acting force of the main landing gear tire;
to nose landing gear suspension system m 2 Analytically, a differential equation of motion (4) is established in the vertical direction:
in the formula, F k2 ，F f2 Respectively the elastic force and the damping acting force of the front landing gear tire;
in the landing and running direction of the airplane, establishing a motion differential equation (5) for the whole airplane:
step 4, solving a differential equation set by a fourth-order Runge-Kutta method;
empirical formula (6), (7) combined with tyre elastic coefficient
k 2 ＝34.514+0.387p 0 +21.816p 0 z 1 (6)
k 4 ＝34.514+0.387p 0 +21.816p 0 z 2 (7)
In the formula, p 0 For the initial tire pressure of an aircraft, μ is the longitudinal sliding friction coefficient, and the relationship between it and the longitudinal slip ratio is relatively complex, and can be given by empirical formula (8):
in the formula, S g Is slip ratio, defined as:
V x for aircraft taxi speed, V ω The linear velocity of the tire, r the radius of the tire, delta the deformation of the tire, and omega the rotational linear velocity of the tire;
and (3) simultaneous (1) to (9) differential equations obtain a second-order five-element differential equation set (10):
where sign is a sign function:
the dynamic load of the tire on the road surface is:
q 1 ，q 2 in order to realize the exciting action of the road surface,
the initial value problem of the high-order differential equation can be calculated by changing variable into the initial value problem of the first-order ordinary differential equation set,
let m =2,3,4 … …, there is an initial value problem of m-order ordinary differential equation:
let y 1 ＝z,y 2 ＝z’,……,y m ＝z (m-1) Then the above equation is normalized to a first order ordinary differential equation set:
for solving the problem of the first-order differential equation set, because the fourth-order Runge-Kutta (Runge-Kutta) method can change the step length according to the precision requirement of each stage, is easy to program and use, and has wide application in practical engineering, the method is adopted for solving the problem,
and (3) performing variable conversion on the dynamic differential equation set at the landing stage, and reducing the second-order five-element differential equation set in the step (2) into the following formula, so that:
θ＝y 1 ，
z 0 ＝y 3 ,/>
z 1 ＝y 5 ,/>
z 2 ＝y 7 ,/>
x 0 ＝y 9 ,/>
get q 1 ，q 2 And the second-order quinary differential equation set in the step 1 is converted into the following equation set:
and (4) calculating a differential equation set through numerical values to obtain a numerical solution.
2. The method for calculating the landing dynamic load of the novel airplane according to claim 1, wherein the following research conclusions are obtained:
(1) the coefficient of the dynamic load of the pavement generated by the main landing gear of the airplane can reach more than 1.4;
(2) the acting force of the main landing gear of the airplane accounts for more than 98%, and the landing load of the nose landing gear is smaller;
(3) the sinking speed of the airplane is a main factor influencing the landing load, and the second is the landing quality, and the dynamic load of the road surface is not greatly influenced by the tire pressure and the pitch angle.
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| CN110341980B (en) * | 2019-07-11 | 2022-11-01 | 中国人民解放军空军工程大学 | Airplane plateau take-off and landing load reduction use analysis method |
| CN112173158A (en) * | 2020-09-25 | 2021-01-05 | 中国直升机设计研究所 | Landing/ship-borne load calculation method for wheeled landing gear helicopter |
| CN112597641B (en) * | 2020-12-10 | 2022-07-22 | 上海宇航系统工程研究所 | Carrier landing stability optimization method |
| CN116167249B (en) * | 2023-04-23 | 2023-06-27 | 民航机场规划设计研究总院有限公司 | Dynamic load calculation method, device and storage medium for asymmetric landing of airplane |
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