Inertia of pully system with wire that has a mass

Umbrako

1. The problem statement, all variables and given/known data
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

2. Relevant 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)?


3. 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

Hi Umbrako! Welcome to PF!
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!

yes …

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