Calculating Frictional Losses in Belleville Springs by Geometrical Interpolation

Abstract

The use of Belleville springs has been proven to be beneficial in tackling the problems related to elastic interaction, creep, differential thermal expansion or in the isolation of seismic vibrations in bolted joints. Because of its high and easily variable spring rates, the use of these springs can also be observed in passive vibration assisted rotary drilling (VARD) tools. Because of relative movement of spring with respect to supporting surfaces and the mating spring surfaces, frictional losses in the spring take place during each compression and relaxation cycle leading to a slightly different load-deflection curve from what has been defined in the literature. The geometry of the spring combined with different stacking configurations complicate the study of frictional losses in these kinds of springs. This work presents a new method to calculate the displacement of different points of Belleville springs during its loading and unloading using geometrical interpolation method. The results of spring displacement are then used to develop a model to calculate frictional load as a function of spring deflection, which is used to analyse load-deflection curves of springs with different dimensional and frictional parameters. The developed methodology is used to plot and understand characteristics curves of four different kind of Belleville springs; High Load, Standard, Force Limiting and Force Adjusting Belleville springs by plotting the graphs for different free-height to thickness and diameter ratios. In the later section of the paper, the proposed methodology has been used to visualize the load-deflection characteristics of Belleville springs used in the p-VARD tool of the Large Drilling Simulator, one of the state-of-the-art drilling simulators at Memorial University of Newfoundland. Results show that understanding of the characteristic curves of different spring configuration helps to plan to drill with desired WOB using a p-VARD tool.