Serpentine Belt Analysis Homework: Length & Motion Equations Explained

[chriskay301]

Homework Statement

Here is what the belt system looks like.

The first part of the problem is to prove that the length of the belt between pulleys is

assuming a negative radius for the backwrapped pulleys (idler and tensioner).

The second part is to derive the equations for motion, which are given so we can use them to check out answers.

Homework Equations

Listed above in pictures.

The Attempt at a Solution

So here are my attempts at both the length equation and the first motion equation (which will be the same concept for the first few)

For this length equation I've tried so many different things and I can't seem to get anything to work out. The picture I've posted is my last attempt that I've tried by just expanding out the formula and seeing if I can see where it is coming from. I've also tried subtracting/adding the radius when making the triangle, etc.

For the motion equation I know I have the correct equation. But I have that one spring written in compression to achieve it (the circles in the diagram). I'm not sure how I can validate that I know that spring is in compression, or if I just did the solution incorrectly. Any help would be appreciated..thanks!

 

[chriskay301]

Okay..so I figured out all of the motion equations. I'm still stuck on the length of the belt however. I've tried numerous approaches.. I am not sure if I'm defining the length wrong, or just not thinking about it properly. Or maybe over thinking it. But just drawing a triangle from center of circle A to the center of circle B would give you length of the belt wouldn't it?

 

[SteamKing]

Maybe you should make a sketch of the belt and show the sheaves which the belt drives. You know what xA, xB, rA, rB, etc. mean, but we do not.

 

[chriskay301]

I actually figured out all of this 2 days after I posted it.

THIS POST CAN BE CLOSED.

 

[rezeew]

I would first commend the student for their efforts in attempting to solve the problem and using various methods to try and understand the concepts. I would then offer some guidance and clarification on the equations and concepts involved.

For the length equation, it is important to remember that the belt is essentially a closed loop, so the length of the belt between the pulleys is equal to the circumference of the pulleys plus the distance between the centers of the pulleys. This can be expressed as L = 2πr + d, where r is the radius of the pulley and d is the distance between the centers. In this case, since we are assuming a negative radius for the backwrapped pulleys, the length equation would be L = 2π(-r) + d.

For the motion equations, it is important to understand the concept of tension and compression in a spring. When a spring is being pulled or stretched, it is in tension, and when it is being compressed, it is in compression. In this system, the spring that is being pulled by the belt is in tension, while the spring that is pushing against the belt is in compression. This is why the equations have different signs for the spring constants.

To validate that the spring is in compression, we can use Hooke's Law, which states that the force applied to a spring is directly proportional to the displacement of the spring from its equilibrium position. If we observe that the displacement of the spring is in the opposite direction of the applied force, we can confirm that the spring is in compression.

In conclusion, it is important to understand the concepts and assumptions involved in solving a problem like this. It may require multiple attempts and different approaches, but with perseverance and understanding, the correct solutions can be obtained.