6.3. Simplified fluid lubricant wear model

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6.3.1. Model description

Fluid lubricant wear modelling is described on the example of a ball bearing. The modelling is applicable to other cases of fluid lubricant wear, however, the number of revolution has to be substituted with another measure of sliding distance. For fluid lubricant reservoir wear, the limit state function is formulated as follows:

\[ g\left( K, \Theta \right) = M_0 - M_{\text{inact}} - \Theta\frac{N_b(1-b_d/p_d\cos{\vartheta})}{4K\exp{(-3.35 p_m)}}r \]

Where \(M_0\) denotes the initial mass of the lubricant in the bearing, \(M_{\text{inact}}\) is the mass not participating in the contact, \(K\) is the empirical constant of the lubricant life model, \(N_b\) is the number of balls in the bearing, \(p_d\) ist the pitch diameter, \(b_d\) is the ball diameter, \(\vartheta\) is the contact angle, \(p_m\) is the contact pressure, \(r\) is the number of revolutions, and \(\Theta\) is the model uncertainty. A summary of these variables and their meaning is given in Table 6.3.1.

Table 6.3.1 Input variables for reliability analysis

Name	Description	Unit	Type
\(M_0\)/M_0	Initial mass of the lubricant in the bearing	\(mg\)	deterministic
\(M_{\text{inact}}\)/M_inact	Mass of lubricant not participating in the contact (e.g. due to evaporation)	\(mg\)	deterministic
\(K\)/K	Empirical constant of the lubricant life model	\(1/(mg\cdot\exp(GPa))\)	uncertain
\(N_b\)/N_b	Number of balls in the bearing	\(-\)	deterministic
\(p_d\)/p_d	Pitch diameter	\(mm\)	deterministic
\(b_d\)/b_d	Ball diameter	\(mm\)	deterministic
\(\vartheta\)/theta	Contact angle	\(deg\)	deterministic
\(p_m\)/p_m	Contact pressure	\(GPa\)	deterministic
\(r\)/r	Number of revolutions	\(-\)	deterministic
\(r_h\)/r_h	Revolutions per hour	\(1/h\)	deterministic
\(\Theta\)/Theta	Model uncertainty	\(-\)	uncertain

6.3.2. Interactive reliability prediction

This page offers an interactive reliability prediction that lets the user specify the properties of all variables listed in Table 6.3.1. The value of deterministic variables can be selected with a slider. Uncertain variables are characterized by:

• Distribution denoted by “Dist” and can be choosen from a set of parametric probability distributions;

• Mean value denoted by “E” and can be selected with a slider;

• Coefficient of variation denoted by “C.o.V.” and can be selected with a slider.

The variable \(r_h\) denotes the average number of revolutions per hour and is only required for plotting the probability of failure as a function of time.

Note

To run the interactive reliability prediction on this page, click the –> Live Code button on the top of the page. Wait a few seconds until the Kernel has loaded and run the cell below with Run.

from nrpmint.booktools import fluid_lubricant_wear

# start the web user-interface
fluid_lubricant_wear.web_ui()