The idea of ​​May 1st Mathematical Modeling Question A has been updated, get it at the end of the article!

The complete idea of ​​question A:

Question A is a dynamics problem, which requires us to apply the concept of physics to real life. We can look at the question first

Question 1: Assuming that the UAV flies parallel to the horizontal plane, it drops materials in the air (the materials are spherical, with a radius of 20cm and a weight of 50kg) to the designated location on the ground.

(1) Establish a mathematical model to give the relationship between the drone's drop distance (the straight-line distance between the drone and the designated landing point of the ground material when delivering materials) and the drone's flight height, flight speed, air resistance, etc. .

We can use the principles of free fall and parabolic motion to build mathematical models . In this model, we need to consider the relationship between the drone's flight height (h), flight speed (v0), air resistance (k), and drop distance (d). Suppose the mass of the material is m, the acceleration of gravity is g, the horizontal velocity of the material is vx0, and the vertical velocity is vy0 at the moment of delivery.

First, let's consider motion in the vertical direction. Materials are affected by gravity and air resistance, and the equation of motion can be expressed as:

 We need to solve this differential equation to get the relationship between the velocity of the material in the vertical direction and the time change vy (t)

Then we can get the landing time of the material by solving the vertical motion equation of the material:

When the material lands, h(t) = 0, we can find out the landing time t.

Next, consider the movement of the material in the horizontal direction. Materials are affected by air resistance, and the equation of motion can be expressed as:

 We need to solve this differential equation to obtain the relationship vx(t) of the velocity of the material in the horizontal direction with time. Then we can get the displacement of the material in the horizontal direction by solving the horizontal motion equation of the material:

 Finally, we substitute t into d(t) to get the delivery distance d.

(2) Assume that the flying height of the UAV is 300m, the flying speed is 300km/h, the wind speed is 5m/s, and the wind direction is parallel to the horizontal plane. Establish a mathematical model to give the drone's launch distance when the flying direction of the drone is the same as the wind direction (the included angle is 0 degrees), opposite (the included angle is 180 degrees), and vertical (the included angle is 90 degrees).

According to the conditions given in the topic, we can substitute specific values ​​into the model to calculate the delivery distance under different wind direction conditions. Since wind speed and direction affect air resistance and the horizontal velocity of supplies, we need to adjust the parameters in the model according to the wind direction. Here are the calculations for the three cases:

When the included angle is 0 degrees (the wind direction is the same as the flying direction of the drone ): In this case, the wind speed will increase the horizontal speed of the material. Therefore, the initial velocity in the horizontal direction is:

 vx0=v0+ wind speed

The included angle is 180 degrees (the wind direction is opposite to the direction the drone is flying): In this case, the wind speed will reduce the horizontal speed of the material. Therefore, the initial velocity in the horizontal direction is:

vx0=v0-wind speed

The included angle is 90 degrees (the wind direction is perpendicular to the flying direction of the drone): In this case, the wind speed will not change the horizontal velocity of the material, but will cause a lateral displacement of the material. We can include the effect of lateral velocity in the model. Therefore, the initial velocity in the horizontal direction is:

vx0=v0

Question 2 : UAVs can not only drop materials at fixed points, but also launch explosives through the launch tube installed at the front of the UAV to clear the river. The general process is: the drone first flies horizontally close to the area where the obstacle is located, then dives to find the right time to launch the explosives, and then pulls up and flies away after the launch. A river is currently blocked by ice accumulation, and it is necessary to use a drone to launch explosives (the explosives are spherical, with a radius of 8cm and a weight of 5kg) to blast the target. Assume that the horizontal distance from the initial point of the UAV to the target is 10000m. Affected by the environment, the drone must dive and launch, and the launch direction is consistent with the flight direction of the drone.

Establish a mathematical model to give the relationship between the UAV launch distance (the straight-line distance between the launch point and the target) and the UAV's flight altitude, flight speed, dive angle and launch speed.

Answer: We can use a method similar to Question 1 to establish a mathematical model using physical principles. In this model, we need to consider the relationship between the UAV’s flight height (h), flight speed (v0), dive angle (θ), launch speed (v1), and launch distance (d). Assume that the horizontal velocity of the explosive is vx0 and the vertical velocity is vy0 at the moment of launch.

First, we need to convert the drone's flight speed and launch speed into horizontal and vertical velocity components. Let the UAV dive angle be θ, then:

Next, we can consider the horizontal and vertical motion of the explosive separately. Similar to problem 1, we need to solve the equation of motion of the explosive in the vertical direction to obtain the landing time t. Then, we can obtain the displacement of the explosive in the horizontal direction by solving the equation of motion of the explosive in the horizontal direction, that is, the launch distance d.

Finally, we substitute t into d to obtain the relationship between the launch distance d and the flight altitude, flight speed, dive angle and launch speed of the UAV.

(2) Assuming that the wind speed is 6m/s, the flying altitude of the drone is 800m, the flying speed is 300km/h when it approaches the target, and the launch speed of the explosive is 600km/h (relative to the speed of the drone). It is required that the distance between the UAV and the target is between 1000 m and 3000 m when launching explosives, and the height of the UAV is not lower than 300m. Please give the launch strategy for the UAV to hit the target.

According to the conditions given in the title, we substitute specific values ​​into the model. First select an appropriate dive angle θ so that the UAV meets the requirement of a height of not less than 300m when launching explosives. In this process, we can find the optimal θ by iterative method or other optimization methods.

After determining the appropriate dive angle θ, substitute it into the model to calculate the launch distance d. Next, we need to make sure that the distance between the UAV and the target is between 1000m-3000m. According to the given flight altitude, flight speed, launch speed and wind speed, we can calculate the distance range between the UAV and the target that meets the conditions through the model. After finding the distance range that satisfies the conditions, we can choose an optimal launch strategy according to the actual situation. For example, choose a launch point that is closer to increase the probability of hitting, or choose a launch point that is farther away to ensure the safety of the drone.

Question 3: The accuracy of the UAV launching explosives and hitting the target has a lot to do with the stability of the UAV flight. Under the same conditions, the more stable the UAV is when launching explosives, the higher the accuracy of hitting the target. After starting to dive, the UAV operator needs to constantly adjust the flight attitude of the UAV to correct the influence of wind direction and wind speed on the UAV.

1. In the case of a certain flight speed and launch speed, various factors are considered comprehensively, a mathematical model is established, the stability of the flight of the UAV is quantified, the relationship between the stability and the hit accuracy is given, and numerical simulation and other methods are used to analyze the unmanned aerial vehicles. The stability of the man-machine is analyzed and verified.

To quantify the stability of UAV flight, we introduce a stability parameter S. S can be described by the following factors: the flying speed of the drone (v0), the dive angle (θ), the wind speed (w), and the angle between the wind direction and the flying direction of the drone (α). The calculation method of the stability parameter S can use a kinetic model based on physical principles, or use data-driven methods such as machine learning to fit historical data.

The relationship between the stability parameter S and the hit accuracy can be described by establishing a probability model. For example, we can assume the following relationship between the hit precision P and the stability parameter S:

P = f(S)

Among them, f is the relationship function. We can determine the specific form of f by analyzing experimental data or using numerical simulation methods.

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