Calculating Machine Friction
Let’s consider the above possibilities with a particular machine with the following notations:
P = Actual effort required for producing an action at the output of the machine, with friction taken into consideration
P’ = Ideal effort, same as above, but having virtually no friction involved
W = Actual actions implemented by the machine over an output load, with friction being taken into account
W’ = Ideal implementations by the machine, just as W, but without any frictional resistance into its path.
It can be simply envisaged that in the above discussion P and W will be greater than P’ and W’ respectively due to the additional frictional resistances.
Therefore it becomes obvious that (P – P’) will be equal to the additional effort required to overcome the additional frictional resistance (W’ – W) and will be equivalent to the additional actions done by the machine over the load at the output.
Now according to the formula of machine efficiency, η = M.A./V.R., where M.A. is the Mechanical Advantage and V.R. is the Velocity Ratio and η is the efficiency of the machine.
We know that with ideal machines the efficiency will be always equal to 1.
Substituting 1 in the above equation gives:
η = M.A./V.R.,
1 = M.A./V.R.,
Or V.R. = M.A.
However since M.A. is actually the ratio of weight lifted upon to the effort applied, may also be expressed as W/P, the above equation becomes:
V.R. = W/P,
As discussed above, with an ideal machine the required effort is P’, substituting this in the above equation gives:
P’ = W/V.R.
We have already studied that (P – P’) is equal to the magnitude of the generated friction, if expressed in terms of effort we get:
P – P’ = P – W/V.R.
Therefore the required additional effort to overcome friction may be finally expressed as:
F (effort) = P – W/V.R.------------------------------ ( i )
We also know that an effort P will operate a load W’ in case the friction is neglected and the same will operate a load W in case the friction is taken into account.
Since W/P = V.R.
Considering the machine to be an ideal one (without friction), we substitute W’ in place of W in the above expression and get:
W’/P = V.R.
Or W’ = P × V.R.
But since (W’ – W) is actually the friction, if expressed in terms of load we gat:
W’ – W = (P × V.R.) – W,
Therefore the additional frictional resistance that needs to be overcome may be finally expressed as:
F(load) = (P × V.R.) – W---------------------------- ( ii )
Equations ( i ) and (ii ) provides us with expressions and simple solutions regarding how to measure friction in machines.