Inertia of pully system with wire that has a mass

[Umbrako]

Homework Statement

A pully system with a hanging mass is connected to a winch. The goal is to find acceleration of the mass for a given motor torque. The pully sheaves is considered with no friction, but has a given inertia. I have found the inertia of the winch, but I have trouble calculating the inertia for the wire on the pully.
The given information is:
n: number of lines in the pully
m1: Mass of wire per length
dh: difference in height between the lower and upper pully
r: radius of the sheaves
Is: Inertia of a singel pully wheel

Homework Equations

How do I calculate the inertia of the wire in the pully system?

If I have n lines, I have n+1 number of pully wheels. (As each end of the wire is either connected to the winch or to a fixed point on the ground). Can I just add the constant inertia of each wheel to a total inertia, even if they will move at different speeds (and accelerations)?

The Attempt at a Solution

I have solved the problem with the hanging mass related to the motor toque on the winch. I have found the Inertia of the winch drum, which was just cylinder.

But the total inertia should be the sum of the inertias of the winch, the wire and the pullys. And I don't know where to start.

 

[tiny-tim]

Welcome to PF!

Hi Umbrako! Welcome to PF!

A pully system with a hanging mass is connected to a winch. The goal is to find acceleration of the mass for a given motor torque.
…
The given information is:
…
Is: Inertia of a singel pully wheel
…
How do I calculate the inertia of the wire in the pully system?

If I have n lines, I have n+1 number of pully wheels. (As each end of the wire is either connected to the winch or to a fixed point on the ground). Can I just add the constant inertia of each wheel to a total inertia, even if they will move at different speeds (and accelerations)?

The Attempt at a Solution

I have solved the problem with the hanging mass related to the motor toque on the winch. I have found the Inertia of the winch drum, which was just cylinder.

But the total inertia should be the sum of the inertias of the winch, the wire and the pullys. And I don't know where to start.

first, when you say that you are given "Inertia of a single pulley wheel",

do you mean the moment of inertia?

if so, you must say so … "inertia" (on its own) means something completely different

you cannot add moments of inertia of different bodies unless they share the same rotation

if (as in this case), they don't, then you will need a separate F=ma or τ=Iα equation for each body ​

 

[Umbrako]

Sorry, moment of inertia I mean! (translated problem from my nativ language...)

After working some more on this I see that I'm on the wrong track.

To help me along on how to attack the problem, I can tell you what else I know:

Jd : Moment of inertia of the Winch drum
Tm : Motor torque
ng : Gear ratio from motor to drum
rd : Drum radius (to where the wire is attached)
dH : Difference in height between uper and lower pulley

So a few equations I have set ut
Td = Tm*ng Torque on drum
Fw = Td/rd Wire tension (at standstill, motor toque holds the weight)
Fb = Fw*n Force on hanging mass from pully (wiretension times number of strings)
Ftot= Fb-M*g Total forces on hanging mass. (Equals zero for standstill)

This equations are based on that the system is at standstill.

I have the moment of inertia of the drum, but I'm unsure how to set up the problem when the wire also has a mass, and the pully sheaves have a moment of inertia.

Do you have any sugestions on how to attack the problem?

 

[tiny-tim]

Hi Umbrako!

I have the moment of inertia of the drum, but I'm unsure how to set up the problem when the wire also has a mass, and the pully sheaves have a moment of inertia.

Do you have any sugestions on how to attack the problem?

yes …

call the tension "T" and write a separate F=ma or τ=Iα equation for each body

 

[asteriatic]

As a scientist, it is important to carefully consider all aspects of a problem and to approach it with a systematic and logical approach. In this case, the problem involves calculating the inertia of a pully system with a hanging mass and a winch. It is important to first define and understand the terms and variables involved in the problem.

In this case, the given information includes the number of lines in the pully system (n), the mass of the wire per length (m1), the difference in height between the lower and upper pully (dh), the radius of the sheaves (r), and the inertia of a single pully wheel (Is). It is also mentioned that there are n+1 pully wheels in the system, as each end of the wire is either connected to the winch or to a fixed point on the ground.

To calculate the total inertia of the system, it is important to consider the individual inertias of each component - the winch, the wire, and the pully wheels. The inertia of the winch can be calculated using the known properties of a cylinder. The inertia of the wire can be calculated using the formula for the inertia of a rod. However, the calculation of the inertia for the pully wheels may be more complex, as it involves multiple pully wheels moving at different speeds and accelerations.

One approach to solving this problem could be to consider the pully system as a system of particles, with each pully wheel representing a particle. The total inertia of the system would then be the sum of the inertias of each individual particle. This approach would take into account the different speeds and accelerations of the pully wheels.

Another approach could be to use the parallel axis theorem, which states that the inertia of a body rotating about an axis parallel to its own is equal to its inertia about its center of mass plus the product of its mass and the square of the distance between the two axes. This could potentially simplify the calculation of the inertia for the pully wheels.

In summary, calculating the inertia of a pully system with a hanging mass and a winch involves considering the individual inertias of each component and potentially using mathematical concepts such as the parallel axis theorem. It is important to carefully define and understand the problem before attempting to solve it, and to approach it with a logical and systematic approach.