Reliability of metal cutting tools:: Stochastic tool life modelling and optimization of tool replacement time

Abstract

This PhD thesis is based on six articles and proposes new approaches for modelling of the life of cutting tools and for determining the optimal tool replacement time. These issues are very closely related and play a critical role in machining economics. Replacing a tool too early means wasting of its potential and leads to high costs and reduced productivity. Late replacement poses a risk of wear-out and other types of tool failures, which can damage the component being produced and can cause expensive equipment downtimes. Therefore a lot of work has been done to develop models for predicting the life of a tool and to optimize its replacement time. Probably the best known of them is the Taylor’s tool life equation.

Developed in 1906 Taylor’s equation expresses the tool life in terms of the cutting speed. Despite being over a century old, this model is still widely used in practice. However, Taylor’s equation has a few drawbacks. For example, it ignores the effect of other, though less important, process parameters such as the depth of cut and the feed. To walk around this issue several extensions of Taylor’s equation have been proposed and are discussed in this thesis. Nevertheless all these models share another common flaw. They assume that tool life is deterministic, i.e., that given the process parameters the exact time to wear-out can be calculated. Unfortunately, in real machining processes there are a lot of sources of variation that affect the rate of tool wear and influence its life. As a result, deterministic models rarely give accurate estimates and are only valid as approximations.

To improve tool life predictions and assist process planners in choosing the optimal replacement time this PhD thesis proposes new methods. The underlying assumption is that tool life is a stochastic quantity and follows a certain probability distribution. With this in mind the reliability function is derived. Based on the physical analysis of machining processes it is assumed that a tool can fail due to the three main causes: (i) wear, (ii) internal defects and (iii) external stresses.

Tool wear depends on a number of factors, including the characteristics of the tool itself, such as its material, geometry and coating, properties of the workpiece material, cutting parameters, rigidity of the machine tool and the efficiency of the cooling process. This last factor is particularly important as most of the tool wear mechanisms depend on temperature. Therefore in this PhD thesis a lot of attention is given to high pressure cooling, which is an effective way to reduce the temperature in the cutting zone.

Internal defects are micro voids and cracks that develop inside the tool material during its manufacturing process or as a result of inappropriate handling. They act as stress concentrators and lead to shorter than normal tool life. External stresses are severe overloads that cause immediate tool failure regardless of its quality. They are random in nature and may originate from machine operator errors, failure of supporting equipment or some other external sources.

Considering all three failure modes total tool reliability function is found. It is assumed that in a given batch a certain percentage of tools are “bad”, i.e., they contain internal defects, while the rest are “good”. The life of the normal tools is modelled by a two-parameter Weibull distribution. Failures due to internal defects are also accounted for by the Weibull distribution, but with different parameters. Then the life of a tool chosen at random is predicted by the mixture model. In addition, tools of both types can fail due to external stresses, the occurrence of which is model by a homogeneous Poisson process.

The derived tool reliability function is used to determine the replacement time. Two models are proposed for this purpose. The first one is called the minimum acceptable reliability approach. The idea is to select such a replacement period that the reliability of the tool during it would not fall below a certain minimum level. We show that, when the reliability function is known, this can be done by using a simple graphical procedure.

The second model is based on the age replacement policy, which attempts to balance the costs of preventive and failure provoked tool changes. To solve this optimization problem the total time on test (TTT) transform of the reliability function is introduced, and a method for estimating it form the experimental data is proposed. Then, as in case of the first model, the replacement time is found by employing a simple graphical procedure.

For the above approach to be used in practice the expected costs of preventive and failure provoked replacements need to be known. It is shown that the former one can be determined by applying traditional formulas found in machining economics handbooks. The penalty cost, on the other hand, is not so well defined, and no good estimation models are available. Therefore, a new, probability tree-based approach is developed in this thesis.

The relevance and the applicability of the proposed models is tested in a few experimental and case studies described in the appended articles. In Article 1 reliability of machining systems as a whole is investigated, and the stochastic nature of the processes involved is clearly shown. In Article 2 it is demonstrated that a two-parameter Weibull distribution can be used to model the tool life, and a simple replacement model based on the reliability function is proposed. In Article 3 a more generic tool life model is developed, but a two-parameter Weibull distribution is still found to be a good approximation. The replacement time is than found by employing an optimization procedure based on the age replacement policy. In Article 4 an approach for estimating the penalty cost, which is a key input to the age replacement model, is developed. Finally in Articles 5 and 6 it is shown that high pressure cooling can help to extend the tool life and possibly to reduce its variation, which is the main reason why probabilistic models are needed.

Based on this experimental work and case studies the thesis concludes that stochastic approaches for tool life modelling and for determination of replacement time are relevant and applicable in practice. Therefore further work needs to be done to extend the use of these methods beyond the set-ups and conditions tested throughout the research described in this PhD thesis.