My final project (undergrad thesis) involves modeling the pulling force of a winch.

As the problem involves a rope under tension and a block being pulled on a surface (friction considered), I can model the problem as a damped spring mass system (due to the internal friction of the rope). The force on the string is expected to peak and then decrease to its nominal value.

As a guide, I used the example of Machine Design - An Integrated Approach, 3rd edition - Robert L Norton.

**MODELING METHOD [/ B]**

1. Define the differential equation

Figure 1 shows the Dynamic System, the Lumped Model and the Free Body Diagram (From the book problem)

(My differential equation is analogous to the Norton example. The difference is that the load to be pulled is horizontal)

Figure 2 shows the model on my problem.

2. Define Boundary Conditions

The initial conditions reflect the fact that the system is initially stopped. Speeds, displacements and accelerations are all equal to zero.

Figure 3 shows the boundary conditions of the Norton problem, which are analogous to mine the resulting equation, with physical constants already substituted. Note that g does not fit into my equation. In my problem, g will be replaced by the ratio of \(\displaystyle (F_friction + F_resistance) / m [/ MATH]

3. Solving the equation and plotting the graphs.

Figure 4 shows the results of the book, which are congruent with the expected response to the actual physical situation.

1. Define the differential equation

Figure 1 shows the Dynamic System, the Lumped Model and the Free Body Diagram (From the book problem)

(My differential equation is analogous to the Norton example. The difference is that the load to be pulled is horizontal)

Figure 2 shows the model on my problem.

2. Define Boundary Conditions

The initial conditions reflect the fact that the system is initially stopped. Speeds, displacements and accelerations are all equal to zero.

Figure 3 shows the boundary conditions of the Norton problem, which are analogous to mine the resulting equation, with physical constants already substituted. Note that g does not fit into my equation. In my problem, g will be replaced by the ratio of \(\displaystyle (F_friction + F_resistance) / m [/ MATH]

3. Solving the equation and plotting the graphs.

Figure 4 shows the results of the book, which are congruent with the expected response to the actual physical situation.

**MY RESULTS [/ B]**

Figure 5 shows the equation with my values already plugged in. (Equation-A)

Solving Equation - A on https://www.symbolab.com/solver/second-order-differential-equation-calculator,

We have the solution - Figure 6

Plotting its second derivative in Wolphram, we obtain the graph of Figure 7. So far, all okay.

However, I do not know how to plot the force graph.

I tried to multiply the mass of the block by the acceleration equation (second derivative of the solution of equation A) and plot the result and I got what we see in Figure 8

It does not match the reality of the problem.

Can you tell me where I'm going wrong? The force should tend to the nominal force (F_winch or F_friction _ F_resistance), as in the Norton problem, not zero.

Please help guys!\)Figure 5 shows the equation with my values already plugged in. (Equation-A)

Solving Equation - A on https://www.symbolab.com/solver/second-order-differential-equation-calculator,

We have the solution - Figure 6

Plotting its second derivative in Wolphram, we obtain the graph of Figure 7. So far, all okay.

However, I do not know how to plot the force graph.

I tried to multiply the mass of the block by the acceleration equation (second derivative of the solution of equation A) and plot the result and I got what we see in Figure 8

It does not match the reality of the problem.

Can you tell me where I'm going wrong? The force should tend to the nominal force (F_winch or F_friction _ F_resistance), as in the Norton problem, not zero.

Please help guys!