PSI - Issue 4

Ivo Černý / Procedia Structural Integrity 4 (2017) 35– 41 Author name / Structural Integrity Procedia 00 (2017) 000 – 000

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3 – 3l 2 x + x 3 ),

w(x) = F / 6EJ y (2l

(1)

where F is lateral force, l is beam length, x is distance of the force point, E is modulus of elasticity, J y is axle inertial moment. The lateral displacement is maximum at the point of force and zero at the hub edge, i.e. at the distance of x = l. For the above mentioned actual load and distance values, this displacement at the force point is 3.4 mm. The machine eccentric mass is not big. Therefore such quite small distance plays an important role, as centrifugal force is given by product m*a, where m is mass and a centrifugal acceleration, a =  2 * r, (2) r being rotation radius of the centre of mass (r = w(x) from Eq. 1 in this specific case), m mass of the specific axle segment being considered, contributing to a kind of axle self loading, and  is angle speed. As a simplified example, if axle segment of the length 0.5 m is considered (top quarter of the axle), its self loading centrifugal force, estimated at the first approach, would correspond approximately to lateral load 19 kN, which is 34 % of the lateral force applied during static calibration. An exact solution of the axle deflection could be calculated by analytical or numerical integration of contributions to centrifugal forces of infinitesimally small axle segments throughout the whole axle length, including the upper eccentric mass. The "self-loading" function is continuous and smooth, being highest near the axle top and zero at the hub edge. This axle "self-loading" effect results in the fact that the axle is eventually more linear in comparison with static calibration force, whereas bending is concentrated just to the area near the hub. The effect is not negligible and has to be considered during testing and subsequent use of the results. Note that in previous type of Sincotec machine with much robust head containing eccentric mass, test frequency only was ca. 15 Hz and the corresponding centrifugal force only 4.8 kN, i.e. just 8.6 % of the static calibration force. That is why the problem of centrifugal forces was not an important issue with the previous type of the machine. The deflection was evenly distributed throughout the whole axle unlike the tests at high frequencies, when the bending is much more concentrated to the near-press fitted area, as schematically shown in Fig. 10. The work was aimed at an evaluation of dynamic effects on redistribution of dynamic stresses along the full scale axle tested at rotation bending at high frequency, characteristic for new generation of sophisticated facilities with a high precision loading. Using a concrete example, the dynamic aspects are described and possible negative effects of the increased test frequency are discussed. The most important results of the study can be summarised as follows:  Static experimental stress analysis corresponded almost exactly to theoretical values at distances remote from the hub edge. There was a considerable stress redistribution near the hub edge, as expected.  Dynamic stresses corresponded to static stresses just at the distance of 250 mm from the hub edge, i.e. at the position of controlling strain gauges. However, the slope of the dynamic stress dependence on the distance from the hub significantly differed from the slope of the static loading, by approximately 20 %. The main source of the difference was a contribution of centrifugal forces of the axle mass itself to excitation forces introduced by the machine eccentric mass at quite high load frequency – 30 Hz.  Dynamic stresses measured along the whole axle circumference were almost ideally self consistent, within precision better than 2 %. 4. Conclusions

Acknowledgements

The work was performed supplementary to the German research project coordinated by TU Chemnitz. The collaboration of Dr. B. Brůžek, Dr. M. Latzer and Prof. Leidich is acknowledged.