In English the term ‘mining’ usually means1: a. The act
b. A kind of process c. A branch of industry
all of which are connected with the extraction of useful mineral2. Here, we neglect its associa-tion with the military.
Mining engineering is an engineering discipline that involves the practice, the theory, the science, the technology and the application of extracting, hauling (sometimes dumping) and processing minerals from a naturally occurring environment. Mining engineering also com-prises the processing of minerals for additional value. Thus, in mining engineering the points of interest are the identified processes and their properties and also the properties of objects, understood here in a broad sense.
These processes are of different natures. They are connected with methods of winning rocks and the identification of mineral deposits, their extraction, haulage and ore dressing, dumping of overburden etc. Different processes are connected with the operation of the equipment involved in mining development and the point of interest here is the exploitation process of pieces of equipment, their parts and assemblies and also entire machinery systems.
In mining, the interesting processes are those that accompany mining development, i.e. the displacement of rocks due to rock extraction and all of the repercussions connected with this process.
The objects of interest are mainly of two kinds:
• Pieces of equipment that are a part of mine development and
• Surrounding rocks near a mine.
Changes in the properties of these two kinds of objects during a mining operation require the greatest attention of mining engineers.
Many problems considered in this book concern technical objects, and whether this object means a single item or a system does not matter. For this reason, it seems worth considering some aspects of the properties and life course.
Each technical object basically has three characteristic phases of its life.
It is presumed that the source of the birth of any technical object is ‘a need’3. There is a certain technological process and there is a need to design an object that will be able to real-ize this process or that will be of service in order to realreal-ize some phases of the process. In some cases ‘a need’ may be different. There is a piece of equipment which properly fulfils its duties but its parameters are not so advantageous compared to modern standards. This may concern the output that is usually attained by this item, its low effectiveness, the average level of its reliability and so on. All of these indicate that there is ‘a need’ to create a new item
1 In English books, one may find slightly different definitions, e.g. SME Mining Engineering Handbook (1973, 2011).
2 In some other languages the term ‘mining’ only means (b) and (c) (e.g. in Polish).
3 A need here is a concept of primary (primitive notion) as in economics. One can find a definition of a need in psychology, e.g. it is such a state of an individual that is a deviation from an optimal state.
Book.indb 49
Book.indb 49 12/9/2013 12:23:19 PM12/9/2013 12:23:19 PM
with better parameters. Sometimes, a competitive company has just presented a new piece of equipment with better characteristics than our product. Finally, pressure sometimes appears to get better achievements. Our factory has modern machines, well-trained personnel with the knowledge and experience to produce a given piece of equipment cheaper. There is ‘a need’ to create a new item.
In the first stage the object does not exist physically. It comes into being. It commences its existence in concepts, models, drafts, notes and in virtual notations. Later on it becomes more concretized; it appears as a list of parts and assemblies, a description of their mechanical and electrical connections and comes into being in a calculation procedure. This stage is ended when the design and construction documentation is completed. Actually, this is a virtual ver-sion. The non-existent object has got its properties—forecasted properties.
The second stage of an object’s life is its production. The item is formed physically. Its final properties are created during the production process, i.e. the features of the object which will characterize it in the third phase of its life. Properties—these real ones—are usually rather different than those given in the first stage of its life by designers and constructors; the pro-duction process is not an ideal realisation of their intentions. This stage is finished when the object exists physically and is ready to be transferred to the user.
When the object is purchased by its user, the third phase of its life usually begins4 and its usage commences. This is a process that continues over time and for the majority of objects is accompanied by the process of its maintenance5. These two processes interlace each other.
The object realizes the purpose of its existence. Also, the process of changes in its properties commences. Elements of the object begin to show wear and tear. Some of these occur in a significant way and failures occur. The object becomes more and more degraded. However, for objects that can be renewed periods of maintenance occur (repairs, prophylactic actions, adjustments) and the process of degradation is reduced. Periods of maintenance happen in either a random way or in a deterministic way if planned. The process of changes in an object’s properties used to be termed the exploitation process6.
There are two essential terms of exploitation theory associated with the term ‘exploitation process’. These are: the state of the object and the exploitation events.
During the object’s exploitation, i.e. during the process of the object’s utilisation and maintenance, the properties of the object change. For some features these changes will be of a continuous type, sometimes slow, sometimes transitional and sometimes drastic. There-fore, an object at a given moment in time is not identical to the object at a different moment in terms of its properties. In order to describe the process of these changes the term state is applied.
When defining a set of an object’s essential properties , = {c1, c2, …, cm}, we can say that the state of the object at time t is determined by a certain function:
(t) = f [ (t)] = f [c1(t), c2(t), …, cm(t)].
Kaźmierczak (2000, p. 119) gave a similar assessment of the term ‘state’: under the term state of object we are going to understand here a ‘photograph’ of values of object properties in a given moment of time.
In practical applications this function is not considered to be a continuous one. Discreti-sation occurs regularly and states are named. These names are usually associated with the
4 In some considerations more than three phases of an object’s life are taken into account, e.g. storage or montage in the final operation place.
5 There is a class of technical objects that cannot be renewed, e.g. hoist head ropes or balance ropes.
6 Some researchers are of the opinion that ‘the exploitation process of an object’ is everything that happens with the object from the moment of the end of its production until the moment of its final withdrawal from utilisation (Kaźmierczak 2000, p. 156).
Book.indb 50
Book.indb 50 12/9/2013 12:23:19 PM12/9/2013 12:23:19 PM
physical nature of the state, e.g. repair state, work state, stand-still state and so on. Notice that a simple conclusion can be made here: The exploitation process of an object can be understood as the sequence of the states of the object or—the usual formulation—as the process of the changes of states.
As a result of this discretisation, at each moment when a change of state occurs, an exploi-tation event takes place. Sometimes, such events are visible and to some extent perceptible, e.g.
a certain element of the object fails and the machine ceases its operation. Sometimes events are conventional ones—nothing physically happens apart from the fact that a certain object parameter exceeded its assumed limited value, e.g. a brake lining worn excessively. At this moment, it is assumed that the object is in a different state.
Based on the consideration above, different theoretical models of the exploitation proc-esses of technical objects can be constructed.
The simplest model is the one that describes the process of impulses (Figure 2.1). An object operates and as a parameter of the process a time is taken into account. At moments t1, t
2, t
3, ... interesting exploitation events occur, e.g. failures. In the process considered, only one state is distinguished and the process has one type of exploitation events. At first glance, this model looks very simple. However, when considering it more carefully, many significant problems arise.
Many essential questions connected with this process can be formulated; essential for the object’s user and for the object’s constructor and producer. Some of these questions are as follow:
– What kind of statistical properties does the observed sequence of times {ti+1 – ti}, i = 1, 2, 3, … have?
– If this sequence is a stationary one and it has no peculiar properties, then what kind of probability distribution can be used to associate it with the random variable: ‘time between neighbouring impulses’?
– If there is a possibility of two or more failures occurring at the same moment of time, then is this possibility stable or not?
– If the probability of the occurrence of failures is independent of time, then what is the probability distribution that describes the number of failures that can occur at a given moment in time?
– Until what moment the analysis of the course of this process makes sense?
To obtain answers to the above questions it is necessary to have a knowledge of the physi-cal nature of the object as well as to have the knowledge and skills to conduct a proper mathematical analysis. It is necessary to use the appropriate mathematical tools from the probabilistic area as well as from the field of mathematical statistics. All of these should be undertaken in order to identify the properties of the object exploitation process that is described by the theoretical model just presented.
Secondly, the most frequently used theoretical model is the model of the process of changes of states that is illustrated in Figure 2.2.
The object is used and in moments t
1, t
2, t
3, ... interesting exploitation events occur—
changes of states. This model is more complicated than the previous one. The exploitation
Figure 2.1. Exploitation process as a process of impulses.
0 t1 t2 t3 t4 t5 t
Book.indb 51
Book.indb 51 12/9/2013 12:23:19 PM12/9/2013 12:23:19 PM
repertoire is associated with this model. It is determined by the set of states of the object that can happen during the operation of the object:
= < 1, 2, ..., m> = < i ; i = 1, 2, ..., m >
The set of possible transitions between states is associated with this set:
= <λij ; i, j = 1, 2, ..., m >
where λij are the intensities of the transitions between states. For some stochastic processes the probabilities are considered instead of the intensities.
The determination of both sets can be achieved by logical analysis bearing in mind the operational reality. Notice that a special case of the above process is the process of changes in states: work-repair type, which is very well known in reliability theory (e.g. Gnyedenko et al.
1969, Kopociński 1973). The estimation of measures associated with the individual nonzero elements of the sets requires an extensive analytical procedure and here again one may for-mulate a list of questions that are worth answering.
– What kind of statistical properties does each sequence of times of states have?
– If a given sequence is stationary and has no peculiar properties, then what kind of prob-ability distribution may be used to describe well, in a statistical sense, the random variable
‘time of given state’?
– Are times of states independent of each other?
– If some states are stochastically dependent, then what kind of random dependence is it?
– Until what moment the analysis of the course of this process makes sense?
The above list is similar to the previous one. Nevertheless, the necessary analysis connected with a trial to get answers to these questions is more complicated and more comprehensive.
In both lists of questions is the presumption stating ‘if this sequence is a stationary one and it has no peculiar properties then …’. Let us explain what this means.
When a statistical observation has been made and a sample has been taken, one obtains a sequence of the times of a given state {t1, t2, ..., tn}. And here a fundamental question should be formulated: What kind of stochastic properties characterize the sequence? These proper-ties contain rich information about the running physical processes in the object being inves-tigated and about the repercussions of these processes. An exploitation reality is determined by the particular realisation of a given sequence. It is worth remembering that three elements determine the course of the exploitation process of a technical object, namely:
• Properties of the object given by the designer, the constructor and set up by the producer
• Properties of the surroundings of the object (recall: surroundings of an object includes everything that is around it and that remains in a certain interaction with it)
• Executed policy of use and maintenance of the object.
Figure 2.2. Model of an exploitation process as the process of changes of states.
1
0 9(t)
2 3
t1 t2 t3 t4 t5 t6 t
Book.indb 52
Book.indb 52 12/9/2013 12:23:19 PM12/9/2013 12:23:19 PM
Each sequence observed contains encoded information. By performing an adequate analy-sis and using suitable statistical tools in an appropriate sequence, we are able to decode it and translate it into engineering language. Usually, we are looking for answers, among other things, to the following questions:
– Is the observed sequence homogeneous or does it perhaps have untypical elements that distinctly differ from the others?
– Is the observed sequence stationary or is there a trend in it?
– Is there a stable dispersion of values in the sequence or does it depend on time?
Obviously, the above list can be extended according to our needs.
A trial to answer the above listed questions is the subject of consideration in the next chapter, which is the first part of an analysis conducted in the book. The next step will be the statistical synthesis: The estimation of the parameters of random variables that are being investigated as well as finding the theoretical probability distributions that describe the empirical distributions well 7.
A further part of the considerations will concern a case in which two or more random vari-ables are observed. Two problems are important here—the investigation of whether random variables are independent of each other and—if not—an examination of the interdepend-ence (in the shape of a correlation) between the variables. The second problem comprises a much broader scope of analysis than the first one.
The next part, in turn, comprises the second stage of the statistical analysis. If there is information that random variables are stochastically dependent, the problem is: What kind of relationship exists between them? This part will present a description in the form of a func-tion illustrating this stochastic interdependence. Considerafunc-tion commences from a simple linear regression analysis and linear transformations up to multidimensional models. Next, more advanced models will be presented starting from autocorrelation and autoregression models, through classical linear regression for many variables and regressions when errors in the values of random variables are traced. This part of the consideration concludes by tak-ing into account that in some cases there is additional information on the random variables examined and this information should be included in the study in order to improve statistical inference conducted.
Chapter 7 contains a special topic—statistical prediction. There are many problems con-nected with any inference about the future. They concern terminology, definitions in use, the areas of study and so on. In this chapter some order is presented in this regard and a few examples are presented based on data taken from practice.
The penultimate chapter is a supplement where basic statistical terms are defined in order to better understand the considerations presented in the book.
The book concludes with Chapter 9 in which a set of tables to carry out statistical infer-ence is included.
7 In this book the phrase ‘…describe well …’ will be found in many places. This term ‘well’ does not have the commonly understood meaning in this context but it is the use of ‘well’ in a statistical sense. This means further that the statistical investigation was conducted and a positive result was obtained. Thus, we are authorized to state that, for example: ‘This model describes well the empirical data.’
Book.indb 53
Book.indb 53 12/9/2013 12:23:20 PM12/9/2013 12:23:20 PM
55