Rewriting an engineering report
pedro000
report1..pdf
Submarine
Abstract
The objectives of our experiment were to gain understanding of dynamical systems and
modeling experience through MATLAB and SIMULINK simulation and understanding second
order system concepts such as damping ratio and natural frequency in dynamic systems. The
problem was to simulate a submarine that is towed by ship using a 300 ft cable. The cable
manufacturer indicated that the cable has elastic properties so we assumed that the cable is acting
like a spring. Therefore, we modeled the problem using SIMULINK in MATLAB after
determining the equation of motion of the submarine so that we could get what was asked in the
problem.
The maximum force in the tow cable which was 8194.1 𝑙𝑏/𝑓𝑡! and it occurs at 1.76 seconds after the towing starts. The maximum force occurs when the submarine goes through its maximum acceleration because force is equal to mass times acceleration, and since the mass is constant, force is at maximum when the acceleration is maximum. Also, the elongation in the tow cable at steady state is 3.59 ft. All of the results we got agrees with the theory which means that we used the one.
Theory
This problem is a simple dynamics problem. We have a submarine that is towed by a
ship, so we can consider the submarine mass is m1, the ship is m2 and the cable stiffness is k. The
stiffness k can be obtained using thing realtion:
k = ! !!
[1]
Also the wave and viscous drag on the submarine can be assumed to be linearly proportional to
its forward velocity under tow so we can consider it the damping coefficient b.
After drawing the free body diagram (figure 1 below) we can see that the problem is a
multi-degree of freedom problem and can be modeled in MATLAB’s SIMULINK to get the
reponse.
Figure 1
Both the submarine and the ship are moving to the right (positive x1 and x2 direction) with the ship pulling the submarine. While they are moving to the right, there is an opposing force acting on them due to the waves and viscous drag. Since we are focusing on the submarine, we will only focus on the submarine equation of motion to get our answers.
We can obtain the equation of motion using this equation
𝐹 = 𝑚𝑥 [2]
There are two forces acting on the submarine. The force of the spring which is 𝑘 𝑥! − 𝑥! in the positive direction and the forced of the waves and viscous drag 𝑏𝑥! in the negative direction. Knowing that we can rewrite eqn[2] to
𝑚!𝑥! = 𝑘 𝑥! − 𝑥! − 𝑏𝑥! [3]
Since 𝑥! = v, we can integrate 𝑥! to get x2 = vt. So [3] becomes
𝑚! 𝑚!
𝑥! 𝑥!
𝑏�̇�! 𝑏�̇�!
k
𝑚!𝑥! = 𝑘 𝑣𝑡 − 𝑥! − 𝑏𝑥! [4]
Dividing m1 on both sides gives us
𝑥! = ! !"!!! !!!!
!! [5]
𝑥! = !"#!!!!!!!!
!! [6]
For finding the length of the cable these equations are used
𝑘 = ∝ ! [7]
ζ = ! ! !"
[8]
Discussion and conclusion
In this experiment, we learned how to simulate a dynamics problem, which was a second order differential equation, into MATLAB’s SIMULINK. All of the results obtained in this report
matched the original theory. We got the similar results in the sample calculations and the results
in the graph. For example we 3.6 ft elongation in the sample calculation and in figure 4 in the
results section of the report. Same goes when calculating the force, velocity, and displacement.
These matching results show that we can simulate dynamics problems in MATLAB’s
SIMULINK and get the same results as theoretical values. However, using SIMULINK is much
faster in compare with doing it by hand.
Also, changing the length of the cable had a great effect on the submarine. The 300 ft
cable had a maximum force of 8194.1 lb, a velocity of 5.96 ft/s and a displacement of 121.4 ft.
However, changing the length to 694.34 ft would minimize the force to 6397.8 lb, the velocity to
5.22 ft/s, and the displacement to 116.67 ft. the only thing that would increase is the elongation
from 3.6 ft to 8.3 ft. All of that is the effect of changing the cable length to one that has a
damping ratio of 0.707. You can see from figure 1 that the submarine settles really quickly in
compared to the 300 ft cable.
