2020 China Postgraduate Mathematical Contest in Modeling F questions

2020 China Postgraduate Mathematical Contest in Modeling F questions

Research on Optimization of Air Vehicle Centroid Balance Fuel Supply Strategy

A certain type of aircraft carries multiple fuel tanks. During the flight, several fuel tanks are combined to supply fuel to meet flight mission requirements and engine work requirements. During mission execution, the change of the aircraft's center of mass has an important impact on the control of the aircraft. The distribution of fuel in each fuel tank and the fuel supply strategy will cause the change of the aircraft's center of mass, which in turn affects the control of the aircraft. Therefore, formulating the fuel supply strategy of each fuel tank is an important task for this type of aircraft control. Here, the fuel supply strategy of the fuel tank can be described by the speed curve of the fuel supply to the engine or other fuel tanks.
Assuming that this type of aircraft has a total of fuel tanks, the fuel supply diagram of each fuel tank is shown in Figure 1:

Figure 1: Schematic diagram of aircraft fuel tank fuel supply
The structure of the aircraft (such as the location, shape, size, fuel supply relationship, fuel supply speed limit, etc.) of the fuel tank affects the fuel supply strategy of the fuel tank and the center of mass change of the aircraft. In order to simplify the problem, the following assumptions and requirements are made for the structure of the aircraft and related fuel supply restrictions:

1. The fuel tanks are all cuboid and fixed inside the aircraft (as shown in Figure 1). Set the internal length, width, and height of the first fuel tank as, and, respectively. The three directions of length, width, and height are parallel to the x, y, and z axes of the aircraft coordinate system.
2. In the aircraft coordinate system (see the appendix for the description of the coordinate system), the center of mass of the aircraft (without fuel) is (0, 0, 0), and the center of the first empty fuel tank is. The total weight of the aircraft (without fuel) is.
3. The upper limit of the fuel supply speed of the i-th fuel tank is (>0). The duration of one fuel supply for each fuel tank is not less than 60 seconds.
4. The main oil tanks 2, 3, 4, and 5 can directly supply oil to the engine, and oil tank 1 and oil tank 6 serve as backup oil tanks to supply oil to oil tank 2 and oil tank 5 respectively, and cannot directly supply oil to the engine.
5. Due to the limitation of the aircraft structure, at most 2 fuel tanks can supply fuel to the engine at the same time, and at most 3 fuel tanks can supply fuel at the same time.
6. During the mission of the aircraft, the total amount of combined fuel supply from each fuel tank should at least meet the engine's fuel consumption requirements (if the fuel supply at a certain time is greater than the planned fuel consumption, the excess fuel can be discharged from the aircraft through other devices), The fuel consumption speed of the engine at each moment can be represented by a fuel consumption speed curve. Figure 2 shows the schematic diagram of the planned fuel consumption speed when the engine performs a certain task:

Figure 2: The planned engine fuel consumption speed curve in a certain mission.
7. The aircraft's attitude may change during flight, mainly due to the up and down pitch or left and right yaw of the flight heading. To simplify the problem, it is assumed that the change of the aircraft attitude in this topic only considers the situation of straight and straight flight and pitch. The pitch of the aircraft will cause the attitude of each fuel tank to tilt relative to the ground. Under the action of gravity, the fuel distribution of the fuel tank will also change, which will cause the center of mass of the aircraft to shift. The schematic diagram of the fuel tank attitude change is shown in Figure 3. The left picture shows the state of the fuel tank when the aircraft is on the ground, and the dotted line on the right represents the fuel level after the fuel tank attitude changes. Please refer to the appendix for the related coordinate system conventions of aircraft attitude changes.

Figure 3: Schematic diagram of fuel tank attitude changes.
Appendix 1 gives the relevant parameters of the aircraft. Appendix 2 – Appendix 5 gives the relevant data of the flight and control of this type of aircraft during the execution of a certain mission. Please your team establish mathematics according to the mission requirements Model, design algorithm, and analyze the effectiveness and complexity of the algorithm to complete the following questions:
Question 1. Attachment 2 gives the fuel supply speed of the six fuel tanks of the aircraft in a certain mission and the pitch angle change of the aircraft during flight Data, record a set of data per second (the same below). Please give the centroid change curve of the aircraft during the execution of this mission, and store the position data of its centroid in the aircraft coordinate system in order of time (one group per second) in the result table of Annex 6 "Results of the first question" "in.
Question 2. Attachment 3 gives the planned fuel consumption speed data of a certain mission and the ideal center of mass position data of the aircraft in the aircraft coordinate system. According to mission requirements, during the mission planning process where the aircraft always maintains level flight (pitch angle is 0), please develop a fuel supply strategy for the aircraft that meets the conditions (1) and (6) for this mission. maximum centroid and the centroid position over the Euclidean distance to a minimum, IE,
.
Please provide the fuel supply speed curve of each of the 6 fuel tanks and the total fuel supply speed curve of the 4 main fuel tanks (time interval is 1s) during the flight of the aircraft, the maximum distance between the aircraft's instantaneous center of mass and the ideal center of mass, and the 4 main fuel tanks The total fuel supply volume of the 6 fuel tanks, and the fuel supply speed data of the 6 fuel tanks are stored in the order of time (one group per second) in the result table "Second Question Results" in appendix 6.
Question 3. Assuming that the initial fuel quantity is not determined, other relevant parameters of the aircraft are shown in Appendix 1. Appendix 4 shows the planned fuel consumption speed data of a certain mission and the ideal center of mass position data of the aircraft in the aircraft coordinate system. During the mission planning process where the aircraft always maintains level flight (pitch angle is 0), please develop the initial fuel capacity and fuel supply strategy of the 6 fuel tanks that meet the conditions (1) and (6) for this mission. the total amount of fuel remaining at the end of at least six tank 1m3, and the maximum value of Euclidean distance from the ideal position of the centroid position of the centroid of the aircraft at each moment a minimum, IE,
.
Please give the initial fuel load of the 6 fuel tanks, the fuel supply speed curve of the 6 fuel tanks during the flight of the aircraft, the total fuel supply speed curve of the 4 main fuel tanks (time interval is 1s), and the distance between the center of mass of the aircraft and the ideal center of mass. Maximum and total fuel supply of 4 main fuel tanks. Please save the initial fuel quantity of the 6 fuel tanks into the prompt position in the “third question result” of the result table of appendix 6, and save the fuel supply speed data of the 6 fuel tanks in the order of time (one set per second) in appendix 6 In the result table "the third question result".
Question 4. In the actual mission planning process, the pitch angle of the aircraft changes with time. Attachment 5 shows the change data of the aircraft's pitch angle and fuel consumption speed data. Please formulate a fuel tank fuel supply strategy for this mission so that the maximum distance between the instantaneous center of mass of the aircraft and the center of mass of the aircraft (without fuel) is minimized, ie
.
Please draw the fuel supply speed curve of each of the 6 fuel tanks during the flight of the aircraft, and then plot the total fuel supply speed curve (time interval of 1s) and the planned fuel consumption speed curve of the 4 main fuel tanks in one graph, giving the aircraft The maximum distance deviation between the instantaneous center of mass and the center of mass of the aircraft (without fuel) and the total fuel supply of the 4 main fuel tanks, and store the fuel supply speed data of the 6 fuel tanks in order of time (one set per second) in the appendix 6 results In the table "Results of the fourth question".

Appendix
Two coordinate systems are agreed as follows:
Inertial coordinate system O-XYZ: When the aircraft is on the ground, the center of mass of the aircraft (without oil) is the origin O, and the longitudinal center axis of the aircraft is the X axis (the longitudinal center axis of the aircraft on the ground is Horizontal direction), taking the front of the aircraft as the positive direction and the opposite direction of the gravity direction as the positive direction of the Z-axis. The Y-axis is determined by the right-hand rule.
Aircraft coordinate system O(t)-X(t)Y(t)Z(t): At time t, the center of mass position of the aircraft (without fuel) is the origin O(t), and the longitudinal center axis of the aircraft is X(t) Axis, the front of the aircraft is the positive direction, the Y(t) axis is perpendicular to the longitudinal section of the aircraft where the X(t) axis is located, and O(t)-X(t)Y(t) forms a right-handed coordinate system, which is determined by the right-hand rule Z(t) axis.
The pitch angle of the aircraft at time t: the angle between the X(t) axis in the aircraft coordinate system O(t)-X(t)Y(t)Z(t) and the O-XY horizontal plane in the inertial coordinate system O-XYZ, X The positive direction of the (t) axis is positive when the gravity direction component is opposite to the gravity direction.
In this question, when on the ground (t=0), the aircraft coordinate system coincides with the inertial coordinate system. Since yaw and roll flight are not considered in this problem, the positive direction of the Y(t) axis in the aircraft coordinate system O(t)-X(t)Y(t)Z(t) and the inertial coordinate system O- The positive direction of the Y axis in XYZ is always consistent. Among the data related to the coordinate system in the attachment to this topic, except for the pitch angle, the others are given in the aircraft coordinate system.

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