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1 Introduction
An airplane (informally plane) is a powered, fixed-wing aircraft that is propelled for-ward by thrust from a jet engine or propeller. Its main feature is fast and safe. Typi-cally, air travel is approximately 10 times safer than travel by car, rail or bus. Howev-er, when using the deaths per journey statistic, air travel is significantly more danger-ous than car, rail, or bus travel. In an aircraft crash, almost no one could survive [1]. Furthermore, the wreckage of the lost plane is difficult to find due to the crash site may be in the open ocean or other rough terrain.
Thus, it will be exhilarating if we can design a model that can find the lost plane quickly. In this paper, we establish several models to find the lost plane in seawater and develop an op-timal scheme to assign search planes to model to locate the wreckage of the lost plane.
1.1 Restatement of the Problem
We are required to build a mathematical model to find the lost plane crashed in open water. We decompose the problem into three sub-problems:
? Work out the position and distributions of the plane?s wreckage ? Arrange a mathematical scheme to schedule searching planes
In the first step, we seek to build a model with the inputs of altitude and other factors to locate the splash point on the sea-level. Most importantly, the model should reflect the process of the given plane. Then we can change the inputs to do some simulations. Also we can change the mechanism to apply other plane crash to our model. Finally, we can obtain the outputs of our model.
In the second step, we seek to extend our model to simulate distribution of the plane wreckage and position the final point of the lost plane in the sea. We will consider more realistic factors such as ocean currents, characteristics of plane.
We will design some rules to dispatch search planes to confirm the wreckage and de-cide which rule is the best.
Then we attempt to adjust our model and apply it to lost planes like MH370. We also consider some further discussion of our model.
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1.2 Literature Review
A model for searching the lost plane is inevitable to study the crashed point of the plane and develop a best scheme to assign search planes.
According to Newton's second law, the simple types of projectile motion model can work out the splash point on the seafloor. We will analyze the motion state of
the plane when it arrives at the seafloor considering the effect of the earth's rotation,
After the types of projectile motion model was established, several scientists were devoted to finding a method to simulate the movement of wreckage. The main diffi-culty was to combine natural factors with the movement. Juan Santos-Echeandía introduced a differential equation model to simplify the difficulty [2]. Moreover, A. Boultif and D. Lou?r introduced a dichotomy iteration algorithm to circular compu-ting which can be borrowed to combine the motion of wreckage with underwater ter-rain [3]. Several conditions have to be fulfilled before simulating the movement: (1) Seawater density keeps unchanged despite the seawater depth. (2) The velocity of the wreck stay the same compared with velocity of the plane before it crashes into pieces. (3) Marine life will not affect our simulation. (4) Acting force
of seawater is a function of the speed of ocean currents.
However the conclusion above cannot describe the wreckage zone accurately. This inaccuracy results from simplified conditions and ignoring the probability distribution of wreckage. In 1989, Stone et.al introduced a Bayesian search approach for searching problems and found the efficient search plans that maximize the probability of finding the target given a fixed time limit by maintaining an accurate target location probabil-ity density function, and by explicitly modeling the target?s process model [4].
To come up with a concrete dispatch plan. Xing Shenwei first simulated the model with different kinds of algorithm. [5] In his model, different searching planes are as-sessed by several key factors. Then based on the model established before, he use the global optimization model and an area partition algorithm to propose the number of aircrafts. He also arranged quantitative searching recourses according to the maxi-mum speed and other factors. The result shows that search operations can be ensured and effective.
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Further studies are carried out based on the comparison between model and reality. Some article illustrate the random error caused by assumptions.
2 Assumptions and Justifications
To simplify the problem, we make the following basic assumptions, each of which is properly justified.
? Utilized data is accuracy. A common modeling assumption.
? We ignore the change of the gravitational acceleration. The altitude of an
aircraft is less than 30 km [6]. The average radius of the earth is 6731.004km, which is much more than the altitude of an aircraft. The gravitational accele-ration changes weakly.
? We assume that aeroengine do not work when a plane is out of contact.
Most air crash resulted from engine failure caused by aircraft fault, bad weather, etc.
? In our model, the angle of attack do not change in an air crash and the
fuselage don’t wag from side to side. We neglect the impact of natural and human factors ? We treat plane as a material point the moment it hit the sea-level. The
crashing plane moves fast with a short time-frame to get into the water. The shape and volume will be negligible.
? We assume that coefficient of air friction is a constant. This impact is neg-ligible compared with that of the gravity.
? Planes will crash into wreckage instantly when falling to sea surface.
Typically planes travel at highly speed and may happen explosion accident with water. So we ignore the short time.
3 Notations
All the variables and constants used in this paper are listed in Table 1 and Table 2.
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Table 1 Symbol Table–Constants Symbol
Definition
ω Rotational angular velocity of the earth g Gravitational acceleration r The average radius of the earth
CD Coefficient of resistance decided by the angle of attack ρ Atmospheric density
φ Latitude of the lost contact point μ Coefficient of viscosity
S0 Area of the initial wrecking zone S Area of the wrecking zone ST Area of the searching zone K Correction factor
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