Introduction
In this experiment we were ones again examining the spur gears with involute teeth. This time our goal was to determine maximal contact stresses in the contact zone and normal stresses at tooth root in the function of contact point position. We applied elastooptical method. The observations where provided in bright field with use of polar scope and camera (for magnification). The stand construction allows to apply different moment of force. The load can be determined by usage of tensometer-voltmeter set. This method is very accurate but quite time consuming, especially if taking into consideration time needed for manufacturing the gears, but we could easily observe the stress distribution in such gears.
The evaluation of stresses
Theoretical contact stresses
Based on the Hertz formulae we calculated the theoretical contact stresses σH
Where:
Pn - normal force between teeth,
L - length of line of contact equal to width of the teeth (4.6mm),
E - Young modulus of the resin (3200N/mm),
- Poisson ratio of the resin (0.36),
ρ - the radius of curvature at the contact point.
The respective values are put in the Table 9.2 for each set of gears and series of measurements. The results for both versions of gears are also put on the diagrams:
Version II
The linear regression of the measured data
Once we had read the order of fringes from the model, we applied linear regression model in order to determine the corrected value of the fringe order. With use of this approach, we removed the systematic error of the measurement.
The relation between N and σH is assumed linear and computed with use of the linear regression method.
The coefficients and are computed with the use of the following equations (the result from the assumption of the linearization of the data and minimizing the RMS error).
(Part I, section5,3)
Of course we do not use data from the end-points of the contact line.
We notice also the important relation:
The latter is based on the theory of elasticity and the fact that
The calculations yield:
a0= 2,933 a1= 0,095 6,316
The data used for the regression is included in the Table 9.3
i
1
2
3
4
5
6
7
8
9
Series No.
NiHmax
5,5
4,5
6,5
7,5
siH[N/mm2]
27,444
20,304
20,884
34,985
25,683
26,416
43,392
31,975
33,152
4,9
6,3
5,4
7,1
6,0
6,1
NiHmax -
0
-0,4
-0,8
0,1
-0,6
1,5
0,4
Table 9.3
The error analysis of the torque and the tooth contact force
In calculations of random errors of measurement, we will neglect the influence of geometric and material parameters, because these can only influence systematic error of measurement.
The torque was measured indirectly with use of the electric strain gauge transducer
Hence its variance takes the form:
The force between teeth of course depends of the torque and the radius of the circle
The variance is
The respective variances depend on the errors:
,...
MEiL