Solving a System Dynamics Problem: Fishing Boat Displacement over Time

Click For Summary
SUMMARY

The discussion centers on solving a system dynamics problem involving a fishing boat being towed by a larger ship. The fishing boat has a weight of 147,150 N, and the tow cable exhibits a linear elasticity of 0.0278 m for every 1000 N of tension. The damping force acting on the boat is 55,000 N-s/m, and the initial velocity of the towing ship is 2 m/s. The differential equation derived for the fishing boat's displacement is Mx'' + Bx' + Kx = k(V_o)t, where M, B, and K are constants derived from the problem parameters.

PREREQUISITES
  • Understanding of linear elasticity and Hooke's Law
  • Familiarity with differential equations and their solutions
  • Knowledge of damping forces in mechanical systems
  • Basic concepts of complex numbers and their applications in physics
NEXT STEPS
  • Study the method of solving second-order linear differential equations
  • Learn about the application of boundary conditions in mechanical systems
  • Explore the use of complex numbers in engineering problems
  • Investigate numerical methods for solving differential equations
USEFUL FOR

Students and professionals in mechanical engineering, physics, and applied mathematics who are dealing with dynamics and system modeling, particularly in scenarios involving towing and displacement analysis.

leoflc
Messages
54
Reaction score
0

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!
 
Physics news on Phys.org
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