The May 1st Mathematical Modeling ABC question idea has been updated, get it at the end of the article!
Question ideas
Question A idea:
The complete idea of ABC questions has been updated, and you can get it for free
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.
- 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: UAV's flight speed ( v0 ), dive angle (θ), wind speed (w), and the angle between the wind direction and the UAV's flight direction (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.
Question B ideas:
Question 1: Attachment 1 is the courier transportation data recorded by the courier company between April 19, 2018 and April 17, 2019 (shipping city-receiving city). Consider multiple perspectives such as cargo volume, express delivery volume growth/decrease trend, and correlation, etc., establish a mathematical model, comprehensively rank the importance of each site city, and give the names of the top 5 site cities in terms of importance, and fill in the results in the table 1.
We first process the data , including extracting the shipping date, shipping city, receiving city, and calculating the shipping and receiving volume for each city. At the same time, the growth/decrease trend of the number of express delivery in each city is counted. Then for each city, we extract the following features:
- Shipment volume: the total shipment volume of the city;
- Receipt quantity: the total quantity of goods received by the city;
- Growth/decrease trend of courier volume: Linear regression analysis can be used to calculate the growth/decrease trend of courier volume for each city;
d. Correlation: The correlation between shipments and receipts between cities can be calculated, and the Pearson correlation coefficient can be used.
Finally, establish an evaluation model: for the extracted features, we can establish a weighted evaluation model. For example, Analytic Hierarchy Process (AHP) can be used to determine the weight of each feature, and then calculate the composite score of each city. According to the comprehensive score of each city, the ranking of the city's importance is obtained. Extract the top 5 cities and fill in Table 1.
Question 2 : Please use the data in Attachment 1 to establish a mathematical model to predict the number of express transportation between the cities of the "delivery-receipt" sites on April 18 , 2019 and April 19 , 2019 , as well as all "shipments " on that day - The total number of courier shipments between cities at the "receipt" site, and fill in the number of courier shipments between the designated site cities in Table 2, as well as the total number of courier shipments between all "delivery-receipt" site cities on the day quantity.
According to the data processed in question 1, for the express transportation quantity between cities of each "delivery-receipt" site, a time series analysis method (such as ARIMA) can be used to build a model and make predictions. The stationarity test is performed on the historical data of each city pair, and then an appropriate model is selected according to the stationarity. After the model training is completed, it is possible to predict the number of express transportation between the cities of each "delivery-receipt" site on April 18, 2019 and April 19, 2019. Add all the predicted express delivery volumes between the "shipping-receiving" site cities to get the total express delivery volume between all "shipping-receiving" site cities on that day. Fill in Table 2 with the predicted quantity of express delivery between cities at designated sites and the total delivery quantity of express delivery between cities at all "delivery-receipt" sites on that day .
Question 3: Attachment 2 is the number of express shipments recorded by the express company from April 28, 2020 to April 27, 2023. Due to the impact of emergencies, the express lines between some cities cannot be transported normally, resulting in the inability to deliver or receive goods normally between the site cities (no data indicates that delivery cannot be received normally, and 0 indicates that there is no delivery demand). Please use the data in Attachment 2 to establish a mathematical model to predict the city pairs (shipping city-receiving city) that can be normally "delivered-received" on April 28, 2023 and April 29, 2023, and judge the table Whether the site city pair specified in 3 can deliver normally, if it can deliver normally, give the corresponding express delivery quantity, and fill in the result in Table 3.
We first process the data in Attachment 2 to extract the delivery date, delivery city, receiving city, and the number of express delivery between each city. At the same time, find out the cities that cannot be delivered or received normally due to the impact of emergencies. For the quantity of express transportation between the cities of each "delivery-receipt" site, we use the time series analysis method (such as ARIMA) in question 2 to build a model and make predictions. The stationarity test is performed on the historical data of each city pair, and then an appropriate model is selected according to the stationarity. After the model training is completed, it is possible to predict the number of express transportation between the cities of each "delivery-receipt" site on April 28, 2023 and April 29, 2023. For the station-city pairs specified in Table 3, we need to check whether they are affected by emergencies. If it is found in the data in Attachment 2 that the goods cannot be shipped or received normally, fill in "No" in Form 3. If no failure to deliver or receive goods is found in the data in Attachment 2, then fill in "Yes" in Form 3. For the site city pairs that can deliver normally, fill in Table 3 with the predicted express delivery quantity. For site city pairs that cannot be shipped normally, leave the express delivery quantity column blank.
Question C ideas:
Question 1: Now there is a single-story flat-roofed building with a length of 4 meters, a width of 3 meters, and a height of 3 meters. Concrete pouring, the thickness is 30 cm (thermal conductivity 0.2W/㎡·K), the total area of doors and windows is 5 square meters (thermal conductivity 1.6W/㎡·K), and the ground is concrete (thermal conductivity 0.25W/㎡·K). See the table below for the monthly average temperature (unit: Celsius) of the geographical location of the building in a year (calculated on the basis of 365 days).
Question 1: Assume that the temperature in the building needs to be kept at 18-26 degrees all the time. When the temperature is not suitable, the temperature must be adjusted by electricity. The consumption of one degree of electricity is equivalent to 0.28 kilograms of carbon emissions . Please calculate the annual carbon emissions of the building that adjusts the temperature through air conditioning (assuming that the air-conditioning heating performance coefficient COP is 3.5, and the cooling performance coefficient EER is 2.7). (Try to use the conditions given in this question to calculate carbon emissions, without considering other losses)
In order to calculate the annual carbon footprint of a building that is air-conditioned, we need to first calculate the monthly heating and cooling needs. The heat loss or heat gain of a building is related to the thermal conductivity of walls, roofs, doors, windows, and ground, as well as the temperature difference between indoors and outdoors. We can calculate heat loss or heat gain using the following formula:
Heat loss or heat gain = surface area x thermal conductivity x temperature difference
We need to calculate the surface area of the wall, roof, doors, windows and ground first , the surface area of the wall = (4m × 3m × 2 + 3m × 3m × 2), the surface area of the roof = 4m × 3m, the total area of the doors and windows has been given as 5 square meters , the ground surface area = 4m × 3m.
The indoor temperature needs to be kept at 18-26 degrees, and the indoor and outdoor temperature difference can be calculated according to the average temperature of each month. For example, the indoor and outdoor temperature difference in January is: 18 - (-1) = 19 degrees (heating), similar calculations for other months.
Use the formula to calculate the heat loss or gain for each month, then add the heat loss or gain for each section. And according to the COP and EER of the air conditioner, the monthly heat loss or heat gain is converted into electricity demand. For example, the power demand in January is: heat loss/COP, similar calculations for other months. Add the monthly electricity demand to get the annual total electricity demand, and then multiply it by the carbon emissions per kWh (0.28 kg/kWh) to get the annual carbon emissions.
Question 2 : In the entire life cycle of residential buildings ( construction, operation, demolition ) , there are many factors that affect carbon emissions, such as architectural design standards, climate, production and transportation of building materials, regional differences, energy consumption for construction and demolition, decoration style, usage Energy consumption, building type, etc. Please search and analyze data, establish a mathematical model, and find indicators that are highly correlated with the above factors and are easy to quantify, and based on these indicators, conduct a comprehensive evaluation of the carbon emissions of the entire life cycle of residential buildings.
In order to comprehensively evaluate the carbon emissions of the entire life cycle of residential buildings, we can establish a linear weighted model, which takes into account the correlation of each indicator and its ease of quantification. Firstly, the weight of each indicator needs to be determined, and then the value of each indicator is multiplied by the corresponding weight and summed to obtain a comprehensive evaluation value. Identify indicators that are highly relevant to carbon emissions and easily quantifiable: Based on the factors you provide, we can consider the following indicators:
Building design standards: e.g. building energy ratings
Climate: e.g. mean annual temperature, annual precipitation
Production and transport of construction materials: e.g. carbon emissions per square meter of materials required for construction
Building energy consumption: e.g. heating and cooling energy consumption per square meter per year
Building Type: For example, single-story, multi-storey, or high-rise buildings
The weights are assigned according to the degree of impact of each indicator on carbon emissions and the degree of ease of quantification. For example, building energy consumption may have a higher impact on carbon emissions and may therefore be assigned a larger weight. The weights should sum to 1. The value of each indicator is multiplied by the corresponding weight and summed to obtain the comprehensive evaluation value.
Comprehensive evaluation value = w1 * building design standard + w2 * climate + w3 * production and transportation of building materials + w4 * building energy consumption + w5 * building type
Among them, w1, w2, w3, w4, w5 are the weights of each index respectively.
This linear weighted model is simple and easy to understand, and can provide decision makers with intuitive evaluation results. However, there may be mutual influence among the indicators, which cannot be reflected in the linear weighted model.
Question 3: On the basis of Question 2, consider the carbon emissions of the three stages of the building life cycle, search for relevant information, establish a mathematical model, and conduct a comprehensive evaluation of the carbon emissions of residential buildings in 13 prefecture-level cities in Jiangsu Province in 2021. And the validity of the built evaluation model is verified.
On the basis of Question 2, we can divide the carbon emissions of residential buildings into three stages: construction, operation, and demolition. First, we need to identify relevant and easily quantifiable indicators for each stage. Then, assign weights to the indicators of each stage. Finally, the value of each stage index is multiplied by the corresponding weight and summed to obtain the comprehensive evaluation value.
Identify indicators that are highly relevant and easily quantifiable to carbon emissions:
a. Construction phase:
Architectural design standards , production and transportation of building materials ; carbon emissions per square meter of building materials ; building types , single-storey, multi-storey or high-rise buildings
b. Run phase:
What are the climates , the annual average temperature and annual precipitation ; the annual heating and cooling energy consumption per square meter ; the influence of interior design style on energy consumption
c. Demolition phase:
Carbon emissions per square meter of energy required for building demolition ; degree of recycling of building materials after demolition
Determine the weight of indicators at each stage: assign weights according to the degree of impact of each indicator on carbon emissions and the degree of ease of quantification. The weights should sum to 1.
Calculate the comprehensive evaluation value: Multiply the value of each stage index by the corresponding weight and sum to obtain the comprehensive evaluation value.
Comprehensive evaluation value = Σ(stage weight * Σ(index weight of each stage * index value))
Validation of the evaluation model: Collect the carbon emission data of residential buildings in 13 prefecture-level cities in Jiangsu Province, and use the established model to predict. Then, the forecast results are compared with the actual data to calculate the forecast error. If the prediction error is within the acceptable range, it indicates that the model has good validity.
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