Multibody systems with 3D revolute joints with clearances: an industrial case study with an experimental validation

Abstract

This article is devoted to the analysis of the influence of the joint clearances in a mechanism of a circuit breaker, which is a 42 degree-of-freedom mechanism made of seven links, seven revolute joints, and four unilateral contacts with friction. Spatial (3D) revolute joints are modeled with both radial and axial clearances taking into account contact with flanges. Unilateral contact, Coulomb’s friction and Newton impact laws are modeled within the framework of nonsmooth mechanics without resorting to some regularizations or compliance/damping at contact. The nonsmooth contact dynamics method based on an event-capturing time-stepping scheme with a second order cone complementarity solver is used to perform the numerical integration. Furthermore, the stabilization of the constraints at the position level is made thanks to the stabilized combined projected Moreau–Jean scheme. The nonsmooth modeling approach together with an event–capturing time-stepping scheme allows us to simulate, in an efficient and robust way, the contact and impacts phenomena that occur in joints with clearances. In particular, comparing with the event-detecting time-stepping schemes, the event-capturing scheme enables us to perform the time-integration with a large number of events (impacts, sliding/sticking transitions, changes in the direction of sliding) and possibly with finite-time accumulations with a reasonable time-step length. Comparing with compliant contact models, we avoid stiff problems related with high stiffnesses at contact which generate some issues in contact stabilization and spurious oscillations during persistent contact periods. In the studied mechanisms of the circuit breakers, the numerical method deals with more than 70 contact points without any problems. Furthermore, the number of contact parameters is small—one coefficient of restitution and one coefficient of friction. Though they are sometimes difficult to measure accurately, the sensitivity of the simulation result with respect to contact parameters is low in the mechanism of the circuit breaker. It is demonstrated that this method, thanks to its robustness and efficiency, allows us to perform a sensitivity analysis using a Monte Carlo method. The numerical results are also validated by careful comparisons with experimental data, showing a very good correlation.

Appendix A: Process capability and performance indices

If worst-case tolerances are specified for the inputs, a worst-case tolerance can be calculated for the output. Let us denote the output by $Y$ and the inputs by $X_{i}, i\in (1,\dots,n)$. We consider $Y = f(X_{i})$ and that the inputs have worst-case tolerances $N_{Xi}\pm T_{Xi}$. Then the resulting worst-case tolerance for $Y$ is denoted as $N_{Y}\pm T_{Y}$. The lower specification limit (LSL), the upper specification limit (USL), nominal ($N$), and the tolerance are calculated as:

$$\begin{aligned} \text{LSL}_{Y}&= \min_{\substack{{N_{Xi}-T_{Xi}~ \leqslant ~ X_{i}~ \leqslant ~N_{Xi} + T_{Xi}}\\[5pt]i~=~1,\dots,n}} f(X_{i}), \end{aligned}$$

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$$\begin{aligned} \text{USL}_{Y}&= \max_{\substack{{N_{Xi}-T_{Xi}~ \leqslant ~ X_{i}~ \leqslant ~N_{Xi} + T_{Xi}}\\[5pt]i~=~1,\dots,n}} f(X_{i}), \end{aligned}$$

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$$\begin{aligned} N_{Y}&= \bigg(\displaystyle\frac{\text{LSL}_{Y}+\text{USL}_{Y}}{2}\bigg), \end{aligned}$$

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$$\begin{aligned} T_{Y}&= \bigg(\displaystyle\frac{\text{USL}_{Y}-\text{LSL}_{Y}}{2}\bigg). \end{aligned}$$

(27)

The most common way of denoting the statistical tolerance is $N_{X}\pm T_{X}^{\langle ST\rangle}$. The statistical tolerance is always associated with a mean value ($\bar {m}$) and standard deviation ($\sigma$). The tolerance range is given as $T_{X}=3\sigma$. The statistical tolerances are more restrictive. They further require that the process capability ($C_{p}$) and process capability index ($C_{pk}$) are centered for a good process [38, 39, 60].

Process capability ($C_{p}$)

It evaluates the tolerance range (TR) compared to a standard six-sigma ($6\sigma$) production dispersion (see Fig. 22(a)). $C_{p}$ measures how close a process is running to its specification limits relative to the natural variability of the process (normally it is given by $6\sigma$):

$$ C_{p} = \displaystyle\frac{\text{Tolerance Range}~(\text{TR})}{6\sigma} = \displaystyle\frac{\text{USL}-\text{LSL}}{6\sigma} .$$

(28)

Fig. 22

Process capability $C_{p}$ and process capability index $C_{pk}$

Process capability index ($C_{pk}$)

It evaluates if the specification can be met given the process spread and tool shift from nominal. In other words, $C_{pk}$ is the capability of the process to see whether or not the mean is centered between the specification limits (see Figs. 22(b) and 23(a)–(b)):

$$ C_{pk} = \min \Bigg(\displaystyle\frac{\text{USL}-\bar{m}}{3\sigma},\ \displaystyle\frac{\bar{m} - \text{LSL}}{3\sigma}\Bigg) $$

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where $\bar{m}$ is the mean value. An index of absolute centering $C_{c}$ quantifies the shift of the mean from the nominal value, see Fig. 23(a)–(b). It is written as a ratio

$$ C_{c} = \displaystyle\frac{\text{Shift}}{0.5\,\text{TR}} .$$

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Fig. 23

Mean shift between desired (design) and actual (process) $C_{c}$