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Team # 35565 Page 13 of 47
5.1.3 Modeling
Considering the effect caused by earth rotation and air draught to plane when crashing to sea level, we analyze the force on the X axis by using Newton?s second law, the differential equation on x y and axis, we can conclude:
m x ?? ? 2 ? m ? w ? y ? ? ? ? F rx sin
?w m y ?? ? ? 2 ? m ?x ? ? sin ? ? z ?cos ? ? ? F ry
m z ?? ? 2 ? m ? w ? y ? ? cos ? rz ? ? Fmg
In conclusion, we establish the earth rotation and types of projectile second order dif-ferential model:
According to Coriolis theorem, we analyze the force of the plane on different direc-tions. By using the Newton?s laws of motion, we can work out the resultant accelera-tion on all directions:
???22???cD?0.04?x??F????S??y????7.3?10?5rad?s?????w???sin??j???cos??k?1?frx?F??cD???x??(x?)2?(y?)2?(z?)22??1f?F??cD???y??(x?)2?(y?)2?(z?)2?ry2?1?222????f?F??c???z?(x)?(y)?(z)rzD?2?
CD is the angle of attack of a plane flew in the best state, w is the angular speed of a moving object, vector j and k are the unit vector on y and z direction respectively,μis
?mx???2m?w?y??sin??f1?obj.?my????2m?w??x??sin??z??cos???f2?mz???2m?w?y??cos??f?mg3?Team # 35565 Page 14 of 47
the coefficient of viscosity of the object.
5.1.4 Solution of the Model
When air flows through an object, only the air close to layer on the surface of the ob-ject in the laminar airflow is larger, whose air viscosity performance is more noticea-ble while the outer region has negligible viscous force [8]. Typically, to simplify cal-culation, we ignore the viscous force produced by plane surface caused by air resis-tance.
Step 1: the examination of dimension in model
To verify the validity of the model based on Newton?s second theorem, first, we standardize them respectively, turn them into the standardization of dimensionless data to diminish the influence of dimensional data. The standard equation is:
x?xyi?
s
Step 2: the confirmation of initial conditions
In a space coordinate origin based on plane, we assume the earth's rotation direc-tion for the x axis, the plane's flight heading as y axis, the vertical downward di-rection for z axis. Space coordinate system are as follows:
Figure 1 Space coordinate system
Step 3: the simplification and solution
After twice integrations of the model, ignoring some of the dimensionless in the
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integral process, we can simplify the model and get the following:
??x???2?w?y??sin??cD???s????y??2?w?z?cos???(v0?y?)2?2m?cD???s?2????z?2?w?y?cos???(z)?g?2m?
We can calculate the corresponding xyz by putting in specific data to get the in-formation about the point of losing contact.
Step 4: the solution of the coordinate
The distance of every latitude on the same longitude is 111km and the distance of
every longitude on the same latitude is 111*cos (the latitude of this point) (km). Moreover, the latitude distance of two points on the same longitude is r×cos(a×pi/180) and the longitude distance of two points on the same latitude is: r×sin(a×pi/180)[9].
We assume a as the clockwise angle starting with the due north direction and r as the distance between two points; X、Y are the latitude and longitude coordinates of the known point P respectively; Lon, Lat are the latitude and longitude coordi-nates of the unknown point B respectively.
Therefore, the longitude and latitude coordinates of the unknown point Q is:
r?sin(a?pi/180)?Lon?X??111?cos(Y?pi/180)???Lat?Y?r?cos(a?pi/180)?111?
Thus, we can get coordinates of the point of splash by putting in specific data.
5.2 Modeling For Locating Wreckage
In order to understand how the wreckage distributes in the sea, we have to understand the whole process beginning from the plane crashing into water to reaching the seaf-loor. One intuition for modeling the problem is to think of the ocean currents as a
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stochastic process decided by water velocity. Therefore, we use a differential equation method to simulate the impact on wreckage from ocean currents.
A Bayesian Searching model is a continuous model that computing a probability dis-tribution on the location of the wreckage (search object) in the presence of uncertain-ties and conflicting information that require the use of subjective probabilities. The model requires an initial searching zone and a set of the posterior distribution given failure of the search to plan the next increment of search. As the search proceeds, the subjective estimates of the detection will be more reliable.
5.2.1 Assumptions of the Model
The following general assumptions are made based on common sense and we use them throughout our model.
? Seawater density keeps unchanged despite the seawater depth.
Seawater density is determined by water temperature, pressure, salinity etc. These factors are decided by or affected by the seawater density. Considering the falling height, the density changes slightly. To simplify the calculation, we consider it as a constant. ? The velocity of the wreck stay the same compared with velocity of the
plane before it crashes into pieces. The whole process will end quickly with a little loss of energy. Thus, we simplify the calculation.
? Marine life will not affect our simulation. Most open coast habitats are
found in the deep ocean beyond the edge of the continental shelf, while the falling height of the plane cannot hit.
? Acting force of seawater is a function of the speed and direction of ocean
currents. Ocean currents is a complicated element affected by temperature, wide direction, weather pattern etc. we focus on a short term of open sea. Acting force of seawater will not take this factors into consideration.
5.2.2 Preparation of the Model
? The resistance of objects of different shapes is different. Due to the continuity
of the movement of the water, when faced with the surface of different shapes,
the water will be diverted, resulting in the loss of partial energy. Thus the pressure of the surface of objects is changed. Based on this, we first consider the general object, and then revise the corresponding coefficients.