Solving a System Dynamics Problem: Fishing Boat Displacement over Time

AI Thread Summary
The discussion centers on solving a system dynamics problem involving a fishing boat being towed by a larger ship, with specific parameters for weight, drag, and cable elasticity. The user has derived a differential equation but encounters non-real numbers while attempting to solve it. Responses clarify that the general solution can involve complex numbers, but the specific conditions of the problem require focusing on the real parts. The importance of separating real and imaginary components in the solution is emphasized, particularly using Euler's formula. The conversation highlights the need to ensure that the particular solution remains real, aligning with the physical context of the problem.
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Homework Statement


The problem:
A fishing boat weighing 147,150 N is towed by a much larger ship. The tow cable is linearly elastic and elongates 0.0278 m for each 1000 N of tension in it. The wave and viscous drag on the fishing boat can be assumed to be linearly proportional to its velocity, and equal to 55,000 N-s/m. At time t=0, the larger tow ship starts moving with constant velocity, V_o = 2 m/s. There is no initial slack in the cable.

Homework Equations



Fing an expression for the fishing boat displacement, x, as a function of time. Plot the displacement of both boats on the same graph.

The Attempt at a Solution


So I have:
M=147150
B=55000
K=(1000/0.0278)=35971 N/m

The diff-eq I found:
Mx''+Bx'+Kx=k(V_o)t

but when I try to solve the diff-eq, I have some non-real number, which doesn't seem right. What should I do?

Thank you very much!
 
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The general solution of the equation is valid for complex numbers.

The boundary conditions x(0) and x'(0) are real and all the constants M B K k are real, so the particular solution for this problem has the imaginary part equal to zero.

To separate the real and imaginary parts, remember that
e^{iwt} = cos wt + i sin wt
 

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