The variance in SH can be found using eqn 4.82.
2 2
u m^ooopm)=u (CH ) a m +I2 (VHooo*) V^2ooom
O CH ) aVHIOOOpm
eqn 4.83
The evaluated partial derivatives in eqn 4.83 are expressed in eqns 4.84 and 4.85.
-HI00=m Mooo1m = CH eqns 4.84,4.85
aCH avHIOOOpm
This procedure is repeated for the vertical displacement delta.
VIooopm = CVvnooopm eqn 4.86
The variance in Sv can be found using eqn 4.86.
-( voo,, 2 2 ( a8f o. 2 (
Vc ) VlO00pm
The evaluated partial derivatives in eqn 4.87 are expressed in eqns 4.88 and 4.89.
-v," = Vvlooon', -vC--- = Cv eqns 4.88,4.89
acV VO VIO00Pm
Each one of the uncertainties in the measured values is then propagated through their
relations according to Figure 4-8. This standard procedure is used to find the friction
coefficient, I, and the uncertainty in the friction coefficient u((g) for any reciprocating
test run on the microtribometer.
This analysis allows for the uncertainty in the friction coefficient to be evaluated
over a range of combinations of normal loads and friction coefficient values that may be
realized during testing. This uncertainty space can be displayed graphically using plots
of lines of constant uncertainty in the friction coefficient as a function of friction
coefficient and applied normal load, as shown in Figure 4-11.