1、ow Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For sale by the Office of Technical Services, Department of Commerce, Washington, D. C. 20230 - Price $1.50 PROBABILITY FUNCTIONS FOR RANDOM RESPONSES :
2、PREDICTION OF PEAKS, FATIGUE DAMAGE, AND CATASTROPHIC FAILURES By Julius S. Bendat Prepared under Contract No. NAS-5-4590 by MEASUREMENT ANALYSIS CORPORATION Los Angele s, California This report is reproduced photographically from copy supplied by the contractor. Provided by IHSNot for ResaleNo repr
3、oduction or networking permitted without license from IHS-,-,-PROBABILITY FUNCTIONS FOR RANDOM RESPONSES: PREDICTION OF PEAKS. FATIGUE DAMAGE. AND CATASTROPHIC FAILURES CONTENTS 1 . 2 . 3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . Introduction . Zero Crossings and Threshold Crossings Peak Probability Functions
4、for Narrow Band Noise . Expected Number and Spacing of Positive Peaks . Measurement of Peak Probability Functions . Expected Fatigue Damage and its Variance 6.1 6.2 E,q ected Fatinne namage S t ructur a1 Fatigue Problem s . Variance in Damage Estimates . 6-. Variance in Damage Estimate 7 . 1 7.2 Pea
5、k Probability Functions for Wideband Gaussian Noise . Expected Value in Damage Estimates Envelope Probability Density Functions . Probability of Catastrophic Failures References i Page 1 4 10 13 15 19 20 21 25 29 33 38 47 50 57 Provided by IHSNot for ResaleNo reproduction or networking permitted wit
6、hout license from IHS-,-,-PROBABILITY FUNCTIONS FOR RANDOM RESPONSES: PREDICTION OF PEAKS, FATIGUE DAMAGE, AND CATASTROPHIC FAILURES 13 PPC 1. INTRODUCTION This report reviews a number of theoretical matters in random process theory which can be applied to physical problems such as pre- dicting peak
7、s, structural fatigue damage, and catastrophic structural failures. The presentation emphasizes the basic assumptions which are involved, and discusses how to properly interpret the theoretical results. Various engineering examples are given as illustrations. fi UT/S -7- - - -_ - LUG uiatC;J*a? is d
8、i-v+dd int sectins 2s fdls: Section 2; Zero Crossings and Threshold Crossings, summarizes certain known important results which enable one to estimate the expected number of threshold crossings at any level per unit time. formulas are shown which apply only to Gaussian random processes. Section 3, P
9、eak Probability Functions for Narrow Band Noise, derives the familiar result that for narrow band Gaussian noise, the peak Simple quantitative I I probability density function follows a Rayleigh distribution. A more general result is derived for arbitrary non-Gaussian narrow band noise if the random
10、 process and its derivative random process are statistically independent. Section 4, Expected Number and Spacing of Positive Peaks, discusses pertinent formulas for estimating the expected number of positive peaks per unit time which lie above any level, and the average time between peaks at any lev
11、el. required to exceed a given peak level. A simple result is shown which applies only to Gaussian random processes. The next Section 5, Measurement of Peak Probability Functions, contains a new result not The latter quantity is equal to the average time 1 , Provided by IHSNot for ResaleNo reproduct
12、ion or networking permitted without license from IHS-,-,-appearing elsewhere which enables one to estimate the normalized standard error (defined here as the ratio of standard deviation of the measurement to the expected value of the measurement) in measuring a peak probability distribution function
13、 associated with a Gaussian narrow band random process. BT product for a sample record, where T is the record length and B is its equivalent noise bandwidth. sections where results are stated concisely. The result is expressed in terms of the Sections 2 through 5 are all short The next Section 6, Ex
14、pected Fatigue Damage and its Variance, discusses in some detail statistical criteria for estimating the expected value and the variance for the damage associated with typical narrow band stress records. Structural Fatigue Problems, to single degree-of -freedom engineering systems. to the response o
15、f the system. results, and as a reasonable approximation to many physical problems, it is assumed that the damage autocorrelation function is of a damped exponential form. formulas for estimating the standard error in structural fatigue measurements. These results are then applied in Section 7, It i
16、s assumed here that stress records are directly proportional For convenience in obtaining quantitative These assumptions lead to new useful practical The remaining three sections of the report take up special topics which are related to the previous material but which have important distinctions. Se
17、ction 8, Peak Probability Functions for Wideband Gaussian Noise, reviews some important not widely known formulas, which extend the familiar narrow band Rayleigh result. It is shown that the peak probability density function for determining the proba- bility that a positive peak will be found among
18、the population of all positive peaks, is in general neither Rayleigh nor Gaussian but a mixture of them both. A criteria for establishing the precise nature 2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-of the peak probability density function is
19、 the ratio of the expected number of zero crossings per unit time to the expected number of maxima per unit time. Section 9, Envelope Probability Density Functions, discusses briefly the topic of envelope probability density functions where the probability in question represents the probability per
20、unit time that the envelope will fall inside different envelope levels. It is shown that envelope probability density functions are equivalent to peak probability density functions for narrow band Gaussian processes. The final Section 10, Probability of Catastrophic Failures, explains how to formula
21、te these questions mathematically, and derives basic probability relations. of the expected number of threshold crossings per unit time, the topic discussed in Section 2. also and interpreted as the reliability of the structure to perform properly for a specified length of time. Its reciprocal yield
22、s the mean time failure for catastropic events. Results are shown to depend upon knowledge The probability of nonfailure is calculated 3 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2. ZERO CROSSINGS AND THRESHOLD CROSSINGS Let x(t) be a random re
23、cord from a stationary random process b(t) whose instantaneous amplitude probability density function is defined by p(x). No assumption is made that p(x) is necessarily Gaussian. However, for simplicity, it will be assumed that the mean value is zero. At an arbitrary threshold level x = a, the expec
24、ted number of crossings per unit time through the interval (a, a+da), where da is arbitrary small, will be denoted by N . The expected number of crossings per unit time through the interval (a, a + da) with positive slope will be denoted by N . Since, on the average, there should be an equal number
25、of crossings with positive and negative slope, N = (1/2)N . See Figures 1 and 2. a t CY t a CY Denote the time derivative of x(t) by v(t) = (dx/dt). Let p(a, (3) represent the joint probability density function of x(t) and v(t). By definition p(a, (3) da d(3 2 Probability a a o! P and It should be n
26、oted that quantity l - Pp(a) , which defines the probability that a peak amplitude is less than a, is often called the distribution function for peaks. Of course, since p (a) is a probability density over (0, a), P The quantity N gives an indication of the “apparent frequency“ 0 of the noise record.
27、 For example, if x(t) were a sine wave of frequency fo cps, then N 60 cps sine wave has 120 zeros per second). No = (1/2)N0 estimates the expected number of cycles per unit time. For example, if x(t) were a sine wave of frequency fo cps, then N = f cps. If each cycle leads to a single positive peak,
28、 as occurs for extremely narrow band noise processes, then N estimates the would be 2f zeros per second (e. g. , a 0 0 The quantity t t 00 t a 10 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-expected number of cycles per unit time with peaks above
29、 the level x(t) = a. fraction of cycles having peaks greater than x(t) = a Thus, for narrow band noise processes, an estimate of the is given by A generalization of this result for arbitrary Gaussian noise processes which are not necessarily narrow band is given in Section 8. I Comparing the results
30、 in Eqs. (13) and (23), the following simple result is obtained for the peak probability for narrow band noise, I p(cr) P(0) P (a) = Prob positive peak a= P By taking the derivative of Eq. (24) with respect to the amplitude level x(t) = a, the following result is obtained for the peak probability de
31、nsity function for narrow band noise, using Eq. I (21), i Prob 1y CY = e (27) P From Eqs. (25) and (26), the corresponding peak probability density function for narrow band Gaussian noise is 22 -CY 12u X X Thus, for the special case of narrow band noise where the probability density function for the
32、 instantaneous amplitudes, p(x), is the Gaussian function given in Eq. (26), the resulting probability density function for the peak amplitudes, Eq. (28). pP(.), will be the Rayleigh function shown in Example Consider a narrow u = 1 volt. Assuming of a peak occurring with X From Eq. (27) band random
33、 signal with an rms amplitude of the signal is Gaussian, what is the probability an amplitude greater than CY = 4 volts? -8 P (4) = Probpositive peak 4 = e = 0.00033 P Hence, there is about one chance in 3000 that any given peak will have an amplitude greater than CY = 4 volts. 12 Provided by IHSNot
34、 for ResaleNo reproduction or networking permitted without license from IHS-,-,-4. EXPECTED NUMBER AND SPACING OF POSITIVE PEAKS Let M denote the total expected number of positive peaks of x(t) per unit time, and M peaks per unit time which lie above x(t) = a. denote the expected number of positive
35、a Then M = MP (D) a P where P (a) is the probability that a positive peak exceeds x(t) = a, as defined in Eq. (20). Hence, if T is the total time during which x(t) is observed, the expected number of positive peaks which exceed the level a in time T is given by P M T= MP (a)T a P Clearly, the averag
36、e time between positive peaks above the level a will be equal to the reciprocal of the expected number of peaks above that level per unit time. That is, where T is the average time between positive peaks above the level a. (Y Consider now the special case where x(t) is a narrow band random signal. F
37、or this case, each peak above the level x(t) = a will be associated with a crossing of the level a. Then, the average time between crossings (with positive slope) of the level a is T as given in Eq. (31), where P (a) is as given by Eq. (23). a P Again for narrow band noise, the expected number of po
38、sitive peaks of x(t) per unit time, denoted by M, is equal to one-half of the expected number of zeros of v(t) = (t) per unit time; that is, the nmber of crossings by v(t) of the level v(t) = 0. The factor one-half 13 Provided by IHSNot for ResaleNo reproduction or networking permitted without licen
39、se from IHS-,-,-stems from the observation that half of the zeros of v(t), on the average, represent negative peaks. By analogy with Eq. (ll), if a(t) = $(t) = g(t) , and if x(t) , v(t) and v(t) , a(t) are pairwise independent, have zero means , and follow normal distributions, then I 2 a where m is
40、 the variance associated with a(t). A general expression to determine M which is valid for arbi- trary probability density functions is given by where CY, 0, y) with x(t) = CY, v(t) = 0, and a(t) = y. This result is discussed in Ref. is the third-order probability density function associated Example
41、 Consider a narrow band random signal with an rms amplitude of r = 1 volt and a center frequency of f = 100 cps. Assuming the signal is Gaussian, what is the expected number of positive peaks per second with an amplitude greater than CY = 4 volts, and what is the average time between such peaks? x b
42、 From the example in Section 2, the expected number of positive t Ob peaks per second is M = N = f = 100 cps. From the example in Section 3, the peak probability P (CY) for CY = 4 is P (4) = 0.00033. Then, the expected number of positive peaks per second above CY= 4 is P P M = MP (4) = 0.033 (4) P H
43、ence, the average time between positive peaks above CY= 4 is T = 1/M(4) = 30 seconds (4) 14 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-5. MEASUREMENT OF PEAK PROBABILITY FUNCTIONS Referring to Eq. (24), the peak probability distribution function
44、 for narrow band noise is given by P (a) = Probpositive peak a- - P(ff) P P(0) (34) Hence, the probability of peaks above any given amplitude level x(t) = a may be determined from measurements of the amplitude probability density function p(x) at the levels x(t) = a and x(t) = 0. The amplitude proba
45、bility density p(x) at any amplitude level x(t) = a is easurec! iising the following relationship. Here, t (a) is the total time spent by the signal x(t) within a narrow amplitude interval between (Y and atax, and T is the total observa- tion time. The hat (A ) over $(a) means that this is only an e
46、stimate of p(a). Ax-0 and T+m. Ax An exact measurement would be obtained in the limit as The expected deviation of $(a) from pia) may be defined in terms of a normalized variance, E (a), for the measurement as follows. 2 2 u2 $4 E (a) = The quantity r2 * represents the variance of the term in the br
47、ackets. The positive square root of the normalized variance is the normalized standard deviation E (a), which is often called the normalized standard error of the measurement. 15 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-For the case where p(x)
48、 is approximately Gaussian, it has been shown by previous theoretical and experimental work Here, B is the noise bandwidth for the random signal being measured, T is the total observation time, and Ax is the amplitude interval for the measurement. What is the variance associated with a peak probability measurement P (cy) P based upon measurements of $(cy) and $(O), as shown in Eq. (34)? The question that now arises is as follows. A Let the normalized variance associated with a measurement h From Eq. (34), the variance in a measurement P (cy)
