Thus, the MTBF = 1/0.015 = 66.7 weeks. Recall that the MTBF is the same as the scale factor η, in the constant hazard case. So η= 66.7 weeks. We are now going to use this value of η, vary the time t, and use different values of β, and see how the distribution changes as β changes.
Using expression 3.14, we compute the R(t) for the data in Appendix 3-1, namely, n= 66.7 weeks and for different values of β as t increases from 1 week to 100 weeks. From the R(t) value, we compute the cumulative failures F(t), which is = l-R(t). The F(t) values are given below.
At low values of b, the distribution of failures is skewed to the left, i.e., there are many more failures initially than toward the end of life. In our example, at the end of 10 weeks, let us see how the b value affects F(t) up to that point.
•When β =0.5, cumulative failures will be 32% of the total.
•When β =1.0, cumulative failures will be 14% of the total.
•When β =2.0, cumulative failures will be 2.2% of the total.
•When β =3.5, cumulative failures will be <0.2% of the total. we do not expect any significantfailures till about the 32nd week.
•When β =10, cumulative failures will be ~0% of the total, we do not expect any significant failures till about the 32nd week.
Also of interest is what happens after we exceed the characteristic life. In week 77, i.e., ~ 10 weeks after the characteristic life is passed,
•When β =0.5, cumulative failures will be 66%of the total.
•When β =1.0, cumulative failures will be 68% of the total.
•When β =2.0, cumulative failures will be 73% of the total.
•When β =3.5, cumulative failures willbe 80% of the total.
•When β =10, cumulative failures willbe 98% of the total.
From this sequence, you can see that the higher the β value, the more the clustering of failures towards the characteristic value, and hence the greater predictability of time of failure.
At t=66.7 weeks, for all values of β, the R(t) is the same. In other words, the shape factor does not affect the survival probability when t = scale factor.
Failure, Its Nature and Characteristics
In the last chapter we looked at aspects of reliability engineering that can be of use to the maintenance practitioner. We discussed some of the underlying principles that can help us identify reliability parameters from historical maintenance records. In order to apply this knowledge, it is useful to understand the nature of failure. In this chapter, we will examine the following.
•Failure in relation to the required performance standards; critical, degraded, and incipient failures;
•Significance of the operating context;
•Use of failures as a method of control of the process;
•Role of maintenance in restoration of desired performance;
•Incipiency and its use in condition-based maintenance;
•Age-related failures;
•System-level failures;
•Human errors and the effect of stress, sleep cycles, and shift patterns;
•Feelings and emotions; how these affect our reactions to situations.
4.1.1 Failure—a systems approach
Failure is the inability of an item of equipment, a sub-system, or system to meet a set of predetermined performance standards. This means that we have some expectations that we can express quantitatively. For example, we can expect the discharge pressure of a centrifugal pump to be 10 bar gauge at 1000 liters per minute. In some cases, we can define our expectations within a band of acceptable performance. For example, the discharge flow of this pump should be 950-1000 liters per minute at 10 bar gauge. The performance standard may be for the system, sub-system, equipment, or component in question. These standards relate to what we need to achieve and our evaluation of the item’s design capability and intrinsic reliability.
4.1.2 Critical and degraded failures
As a result of a failure, the system may be totally incapacitated such that there is a complete loss of function. For example, if a fire pump fails to start, it will result in the unavailability of water to fight fires. If there had been a real fire and only one fire pump installed, this failure could result in the destruction of the facility. In this case, the failure-to-start of the pump results in complete loss of function.
As a second example, let us say that we have a set of three smoke detectors in an enclosed equipment housing. The logic is such that an alarm will come on in the control room if any one of the three detectors senses smoke. If any two detectors sense smoke, the logic will activate the deluge system. It is possible that one, two, or all three detectors are defective, and are unable to detect smoke. When there is smoke, there is no effect if only one detector is defective, as the other two will activate the deluge. If two of them are in a failed state at the same time, the initiation of the deluge system will not take place when there is smoke in the housing.
Last, with the loss of all three, even the alarm will not initiate. The loss of all three units will result in total loss of function, so this is a critical failure. If two of the three fail, the third can still initiate the alarm on demand. The operator then has the ability to respond to the alarm and initiate the deluge system manually. The system can still be of use in raising the alarm, so it has partial or degraded functionality.
4.1.3 Evident failures
When the impeller of a pump wears out, the operator can see the change in flow or pressure and hence knows about the deterioration in its performance. We call it an evident failure as the operator knows its condition. Similarly, an increase in the differential pressure across a filter or exchanger indicates an increase in fouling. When we take bearing vibration readings and plot the changes, it is possible to predict when it needs replacement. In each case, the operators know the condition of the equipment, using their own senses or instruments. In this context, the operator is the person who is responsible for starting, running, and stopping the equipment. For example, the driver of an automobile is its operator.
4.1.4 Hidden failures
These failures, by contrast, are unknown to the operators during normal operation. Do you know if your automobile brake lights work? Similarly one does not know whether a smoke detector or a pressure relief valve is in a working condition at any point in time. A second event, such as a fire (causing smoke) will initiate the smoke detector, if it is in working condition. If the vessel pressure exceeds the relief valve’s set pressure, it should lift. The standby power generator must start when there is a power failure. Will the pressure relief-valve lift or the standby generator start?
Hidden failures are also observed with protective instruments. Once equipment complexity increases, the designer provides various protective devices to warn the operator, using alarms, or bring it to a safe condition, using