Vibrations Fundamentals and Practice Appe Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.
de Silva, Clarence W “Appendix E” Vibration: Fundamentals and Practice Clarence W de Silva Boca Raton: CRC Press LLC, 2000 Appendix E Reliability Considerations for Multicomponent Units In the practice of vibration (e.g., vibration monitoring, isolation, control, and testing), one depends on the proper operation of complex and multicomponent equipment Equipment that has several components that are crucial to its operation, can have more than one mode of failure Each failure mode of the overall system will depend on some combination of failure of the components Component failure is governed by the laws of probability Consider first some of the fundamentals of probability theory that are useful in the reliability or failure analysis of multicomponent units E.1 E.1.1 FAILURE ANALYSIS RELIABILITY The probability that a component will perform satisfactorily over a specified time period t (component age) under given operating conditions is called reliability It is denoted by R Hence, R(t ) = ℘(Survival) (E.1) where ℘ denotes “the probability of.” E.1.2 UNRELIABILITY The probability that the component will malfunction or fail during the time period t is called its unreliability, or its probability of failure It is denoted by F Hence, F(t ) = ℘(Failure) (E.2) Because it is known as a certainty that the component will either survive or fail during the specified time period t, one can write R(t ) + F(t ) = (E.3) The probability of survival of a component usually decreases with age Consequently, the typical R(t) is a monotonically decreasing function of t, as shown in Figure E.1 If it is known as a certainty that the component is good in the beginning, then R(0) = Because of manufacturing defects, damage during shipping, etc., however, one usually has R(0) ≤ For a satisfactory component, R(t) should not drop appreciably during its design life Td The drop is faster initially, however, because of infant mortality (again due to manufacturing defects and the like), and later on, as the component exceeds its design life because of old age (wear, fatigue, etc.) It is clear from equation (E.3) that the unreliability curve is completely defined by the reliability curve As shown in Figure E.1, transforming one to the other is a simple matter of reversing the axis ©2000 CRC Press FIGURE E.1 A typical reliability (unreliability) curve FIGURE E.2 Venn diagram illustrating the inclusion–exclusion formula E.1.3 INCLUSION–EXCLUSION FORMULA Consider two events, A and B, that are schematically represented by areas (as in Figure E.2) Each event consists of a set of outcomes The total area covered by the two sets denoted by A and B is given by adding the area of A to the area of B and subtracting the common area This procedure can be expressed as ℘( A or B) = ℘( A) +℘( B) −℘( A and B) (E.4) Example Consider the rolling of a fair die The set of total outcomes consists of six elements forming the space S = {1, 2, 3, 4, 5, 6} ©2000 CRC Press Each outcome has a probability of 1/6 Now consider the two events: A = {Outcome is odd} B = {Outcome is divisible by 3} Then, A = {1, 3, 5} B = {3, 6} Consequently, A or B = {1, 3, 5, 6} A and B = {3} It follows that ℘( A) = ; ℘( B) = ; ℘( A or B) = ; ℘( A and B) = These values satisfy equation (E.4) If the events A and B not have common outcomes, they are said to be mutually exclusive Then, the common area of intersection of sets A and B in Figure E.2 will be Hence, ℘( A and B) = (E.5) for mutually exclusive events E.2 BAYES’ THEOREM A simplified version of Bayes’ theorem can be expressed as ℘( A and B) = ℘( A B)℘( B) = ℘( B A)℘( A) (E.6) in which ℘(A/B) denotes the conditional probability that event A occurs, given the condition that event B has occurred In the previous example of rolling a fair die, if it is known that event B has occurred, the outcome must be either or Then, the probability that event A would occur is simply the probability of picking from the set {3, 6} Hence, ℘(A/B) = 1/2 Similarly, ℘(B/A) = 1/3 It should be noted that equation (E.6) holds for this example E.2.1 PRODUCT RULE FOR INDEPENDENT EVENTS If two events A and B are independent of each other, then the occurrence of event B has no effect whatsoever on determining whether event A occurs Consequently, ©2000 CRC Press ℘( A B) = ℘( A) (E.7) for independent events Then, it follows from equation (E.6) that ℘( A and B) = ℘( A)℘( B) (E.8) for independent events Equation (E.8) is the product rule, which is applicable to independent events It should be emphasized that although independence implies that the product rule holds, the converse is not necessarily true In the example on rolling a fair die, ℘(A/B) = ℘(A) = 1/2 Suppose, however, that it is not a fair die and that the probabilities of the outcomes {1, 2, 3, 4, 5, 6} are {1/3, 1/6, 1/6, 0, 1/6, 1/6} Then, ℘( A) = + + = whereas, ℘( A B) = 16 =1 +1 This shows that A and B are not independent events in this sample Furthermore, ℘(B) = 1/6 and ℘(A and B) = 1/6 It is seen that Bayes’ theorem is satisfied by this example E.2.2 FAILURE RATE The function F(t) defined by equation (E.2) is the probability-distribution function of the random variable T denoting the time to failure The rate functions can be defined as: r (t ) = dR(t ) dt (E.9) f (t ) = dF(t ) dt (E.10) where R(t) = ℘(T > t) F(t) = ℘(T ≤ t) In equation (E.10), f(t) is the probability-density function corresponding to the time to failure It follows that: ℘(Component survived up to t, failed within next duration dt ) = ℘( Failed within t, t + dt ) = dF(t ) = f (t )dt (E.11) Also, ℘(Component survived up to t ) = R(t ) ©2000 CRC Press (E.12) Define the function β(t) such that: ℘( Failed within next duration dt Survived up to t ) = β(t )dt (E.13) By substituting equations (E.11) through (E.13) into equation (E.6), one obtains f (t )dt = β(t )dtR(t ) or β(t ) = f (t ) f (t ) = R(t ) − F(t ) (E.14) Now suppose that there are N components If they all have survived up to t, then, on the average, Nβ(t)dt components will fail during the next dt Consequently, Nβ(t) corresponds to the rate of failure for the collection of components at time t For a single component (N = 1), the rate of failure is β(t) For obvious reasons, β(t) is sometimes termed conditional failure Other names for this function include intensity function and hazard function, but failure rate is the most common name In view of equation (E.10), one can write equation (E.14) as a first-order linear, ordinary differential equation with variable parameters: dF(t ) + β(t ) F(t ) = β(t ) dt (E.15) Assuming a good component initially, one has F(0) = (E.16) The solution of equation (E.15) subject to equation (E.16) is t F(t ) = − exp − β( τ)dτ ∫ (E.17) where τ is a dummy variable Then, from equation (E.3), t R(t ) = exp − β( τ)dτ ∫ (E.18) It is observed from equation (E.18) that the reliability curve can be determined from the failurerate curve, and the reverse A typical failure-rate curve for an engineering component is shown in Figure (E.3) It has a characteristic “bathtub” shape, which can be divided into three phases, as in the figure These phases might not be so distinct in a real situation The initial burn-in period is characterized by a sharp drop in the failure rate Because of such reasons as poor workmanship, material defects, and poor handling during transportation, a high degree of failure can occur during a short initial period of design life Following that, the failures typically will be due to random causes The failure rate is approximately ©2000 CRC Press FIGURE E.3 A typical failure-rate curve constant in this region Once the design life is exceeded (third phase), rapid failure can occur because of wearout, fatigue, and other types of cumulative damage, and eventual collapse will result It is frequently assumed that the failure rate is constant during the design life of a component In this case, equation (E.18) gives the exponential reliability function: R(t ) = exp( −βt ) (E.19) This situation is represented in Figure E.4 This curve is not comparable to the general reliability curve shown in Figure E.1 As a result, the constant failure rate should not be used for relatively large durations of time (i.e., for a large segment of the design life) unless it has been verified by tests For short durations, however, this approximation is normally used and it results in considerable analytical simplicity E.2.3 PRODUCT RULE FOR RELIABILITY For multicomponent equipment, if it is assumed that the failure of one component is independent of the failure of any other, the product rule given by equation (E.8) can be used to determine the overall reliability of the equipment The reliability of an N-component object with independently failing components is given by: R(t ) = R1 (t ) R2 (t ) K RN (t ) (E.20) where Ri(t) is the reliability of the ith component If there is no component redundancy, which is assumed in equation (E.20), none of the components should fail (i.e., Ri(t) ≠ for i = 1, 2, …, N) for the object to operate properly (i.e., R(t) ≠ 0) This follows from equation (E.20) In vibration testing, a primary objective is to maximize the risk of component failure when subjected to the test environment (so that the probability of failure is less in the actual in-service environment) One way of achieving this is by maximizing the test-strength-measure function given by ©2000 CRC Press FIGURE E.4 Reliability curve under constant failure rate r TS = ∑ F (T )Φ i i (E.21) i =1 in which Fi(T) is the probability of failure (unreliability) of the ith component for the test duration T, and Φi is a dynamic-response measure at the location of the ith component The parameters of optimization may be the input direction and the frequency of excitation for a given input intensity Regarding component redundancy, consider the simple situation of ri identical subcomponents connected in parallel (ri th-order redundancy) to form the ith component The component failure requires the failure of all ri subcomponents The failure of one subcomponent is assumed to be independent of the failure state of other subcomponents Then, the unreliability of the ith component can be expressed as Fi = ( F0 i ) ri (E.22) in which F0i is the unreliability of each subcomponent in the ith component This simple model for redundancy might not be valid in some situations There are two basic types of redundancy: active redundancy and standby redundancy In active redundancy, all redundant elements are permanently connected and active during the operation of the equipment In standby redundancy, only one of the components in a redundant group is active during equipment operation If that component fails, an identical second component will be automatically connected For standby redundancy, some form of switching mechanism is needed, which means that the reliability of the switching mechanism itself must be accounted for Component aging is relatively less, however, and the failure of components within the redundant group is mutually independent In active redundancy, there is no need for a switching mechanism; but the failure of one component in the redundant group can overload the rest, thereby increasing their probability of failure (unreliability) Consequently, component failure within the redundant group is not mutually independent in this case Also, component aging is relatively high because the components are continuously active ©2000 CRC Press ... redundancy: active redundancy and standby redundancy In active redundancy, all redundant elements are permanently connected and active during the operation of the equipment In standby redundancy, only... Consequently, A or B = {1, 3, 5, 6} A and B = {3} It follows that ℘( A) = ; ℘( B) = ; ℘( A or B) = ; ℘( A and B) = These values satisfy equation (E.4) If the events A and B not have common outcomes,... intersection of sets A and B in Figure E.2 will be Hence, ℘( A and B) = (E.5) for mutually exclusive events E.2 BAYES’ THEOREM A simplified version of Bayes’ theorem can be expressed as ℘( A and B) = ℘(