Solving a System Dynamics Problem: Fishing Boat Displacement over Time

In summary: Set x = A cos wt + B sin wtThen x' = -Aw sin wt + Bw cos wtand x'' = -Aw^2 cos wt - Bw^2 sin wtInsert these into the diffeqIn summary, the problem involves a fishing boat being towed by a larger ship with a linearly elastic tow cable. The wave and viscous drag on the fishing boat are assumed to be linearly proportional to its velocity. The displacement of the fishing boat can be expressed as a function of time using a differential equation, with the constants M, B, and K representing the boat's weight, drag, and cable elasticity respectively. Solving the differential equation may involve non-real numbers, but
  • #1
leoflc
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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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  • #2
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
 
  • #3


Hello,

Thank you for sharing your attempt at solving the system dynamics problem. It seems like you have a good understanding of the problem and have correctly identified the mass, damping, and spring constants. However, I believe there may be a mistake in your differential equation.

The wave and viscous drag on the fishing boat should be a function of its velocity, not time. So the term Bx' should be replaced with Bx' = B(Vo - x') where x' is the velocity of the fishing boat. This will give you a more accurate equation to solve.

Additionally, when solving the differential equation, it is important to pay attention to the units of each term and make sure they are consistent. In this case, the units of the left side of the equation should be in Newtons (N) since it represents the forces acting on the fishing boat. On the right side, the units should be in meters per second squared (m/s^2) since it represents the acceleration of the fishing boat.

I hope this helps and good luck with your solution!
 

1. What is a System Dynamics problem?

A System Dynamics problem is a type of complex problem that involves understanding and analyzing the behavior of a system over time. It is a method of studying and modeling the behavior of dynamic systems, such as social, economic, or environmental systems, by using computer simulations and mathematical models.

2. How is System Dynamics different from other problem-solving approaches?

System Dynamics differs from other problem-solving approaches in that it takes a holistic, systems-thinking approach to understanding and solving problems. It considers the relationships and interactions between different components of a system, rather than just focusing on individual parts. It also takes into account the dynamic nature of systems, where changes in one part can have ripple effects throughout the entire system.

3. What are some common applications of System Dynamics?

System Dynamics has a wide range of applications, including in business, economics, public policy, and environmental sustainability. Some examples include analyzing supply chain dynamics, understanding the impact of economic policies on a country's economy, and predicting the effects of climate change on a particular region.

4. What are the key components of a System Dynamics model?

A System Dynamics model typically consists of stocks, flows, feedback loops, and time delays. Stocks represent the accumulation of a quantity within a system, such as population or capital. Flows represent the movement of a quantity between stocks. Feedback loops are the mechanisms that cause changes in one part of the system to affect other parts. Time delays represent the time it takes for changes to occur in the system.

5. How can System Dynamics be used to solve real-world problems?

System Dynamics can be used to solve real-world problems by helping to identify the underlying causes of complex issues and predicting the potential outcomes of different policies or interventions. By understanding the behavior of a system, decision-makers can make more informed and effective decisions to address the problem at hand.

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