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Moments of the first-passage time for a narrow-band process
Journal of Sound attd Vibration.(1974) 32(4), 449--458
M O M E N T S O F T H E FIRST-PASSAGE TIME F O R A . NARROW-BAND
PROCESS
B. SAHAY
Department of Meckanical Engineering, Indian lnstitnte of Technology, Kanpur, htdia AND
W. LENNOX
Department of Civil Enghzeerhtg, University of Waterloo, Waterloo, Ontario, Canada (Received 10 April 1973, and in revisedform 25 September 1973) The moments of the first-passage time of a narrow-band stochastic process produced by a linear oscillator excited by wide-band noise are presented for the case of a "safe" region defined by a circular barrier in the phase plane. As long as damping is small the moments produced will be approximately the same as those for the two-sided barrier problem. An iterative set of partial differential equations governing the moments, the first being the Pontriagin-Vitt equation for the mean first-passage time, are solved by using a Galerkin approximation. No restrictions are required on the barrier size.
1. INTRODUCTION Many problems in the reliability of systems subJected to random excitation can be studied by means of the first-passage problem of probability theory. The system is considered to fail when the response process of interest passes out of specified region of safe operation for the first time at time T: for example, failures due to "bottoming" or "short-circuiting". Tis obviously a random variable and the solution to the problem is given by the probability distribution function P(T) of the first-passage time T. Much attention has been given to this problem, but because of the inherent analytical difficulties, little progress has been made with regard to the exact solution. However, many interesting approximations have been made. These involve approximations of the exclusion series or the multiple integral formulation of S. O. Rice [1, 2, 3], approximations to an integral equation describing the first-passage time density [4, 5], approximate and exact analyses of the envelope function [6, 7], series approximations for survival probabilities [8] and simulation [9]. Two recent papers by Yang and Shinozuka contain comparisons of some of the above methods along with several new ideas [10, 11]. This paper deals with obtaining the moments of the first-passage time T with the idea that a knowledge of the third, fourth and higher order moments will give additional insight into the behaviour of P(T). The first moment, or the mean first-passage time, is governed by the Pontriagin-Vitt equation. An equation describing the second moment can be deduced in terms of the first moment and so on for higher order moments. These partial differential equations are solved approximately by using a Galerkin technique. 449
450
a. SAHAY AND W. LENNOX
2. ANALYSIS The problem considered is the evaluation of the moments of the first-passage time, T. Tis the time required for the trajectory of the linear oscillator to leave a specified region in state space (see Figure I) for the first time, when started from some initial state. The oscillator is excited by wide-band noise and is assumed to be described by the stochastic differential equation
(l)
y + 2~to.)~ + to2.y = f i t ) ,
Y Figure 1. Sketch of the domain, R, and its boundary, F, in state space.
where ~ is the damping factor,~, is the undamped natural frequency of the system a n d f ( t ) is a stationary wide-band process with
E(f(t)) = O,
(2)
E(f(t)f(t+z))=Ks(O.
(3)
It is assumed that the correlation time, T,or, of the stochastic processf(t) is small compared with the time constant of the system:
flKA01dr 0
9
~
Ks(O)
1 < T~ = ~ 2~oJ,,"
This inequality implies that the trajectory of the response by a Markov vector process with
KAz) replaced by
where c is a measure of the intensity of the noise.
(4)
(y(t),p(t)) can be approximated
MOMENTS OF THE FIRST-PASSAGE TIME
451
Inside the "safe" domain, R, and on the boundary, F, the transition probability, w, satisfies the Fokker-Planck (forward) equation a~;'
a~ff,]
O
at
Ok,
Oj~
c a 2 ~, - co]y)~"1 + - ~
[(-2r
(6)
2 O~2
and the associated Kolmogorov (backward) equation Ow
Ow
Ow
~r-r = f,oa--7o- (2r
+ co~yo) ~-7-.-I oyo
c0 2 w
2
a.;,~
.
(7)
If y is the vector ( y , ) ) then w(y, Yo,z) is the joint probability that y(t - z) is in the interval Yo, Yo + dyo given that y(t) is in the interval y, y + dy. The initial and boundary conditions to be satisfied by w are w(y, Yo, 0) - 6(y - Yo),
(8)
which states that no changes of state can occur if the transition time is zero and w(y, Yo, "0 = 0,
Yo e F,
(9)
which implies that only those trajectories that have not crossed the boundary up to time interval z are taken into consideration. The expression
f.'(y, Yo,T) dy
W(z, Yo) =
(10)
R
represents the probability that a trajectory has not reached the boundary during a time interval ~. Initially, no realization has reached the boundary so that W(0, Yo) = 1
(! I)
and ultimately all realizations will reach the boundary with the result tV(oo, Yo) = 0.
(I 2)
The distribution function of the first-passage time P(z, yo) is then given by [12, 13]
1 - W(~, Yo). By integrating the Kolmogorov equation (7) as well as the boundary conditions (8) and (9) with respect to y over the region R, a differential equation describing Wis obtained: OtV OW -(2r Or = "YOayo
OIV c0 2 IV + 2 a~ '
(13)
with W(0, Yo) = I,
Yo E R,
(14)
IV(z, Yo) = 0,
Yo ~ F.
(15)
The closed form solution to this initial-boundary value problem is not known. In the absence of a complete solution for IV(r, Yo), a knowledge of the moments of the first-passage time would be useful.
452
n . S A H A Y A N D ~,V. L E N N O X
Let Mt(yo) be the mathematical expectation of the time required for the trajectory, with initial point Yo, to reach the boundary F o f the region R for the first time: ' r
Mt(yo)=-d
~++.o,v o
fWd+.
(16)
0r dZ=o
By integrating equation (13) with respect to z and using equation (14), the following equation is obtained for M~ 9
Ca2 3It 2 a~o~
aMt
+ Y'o-:--
a),o
aMt oyo
(2~o~o + o~.:o)-v:-. ~' = -l,
(17)
with M,(yo) = 0,
Yo ~ F.
(18)
Equation (17) was first derived by Pontriagin, Andronov and Vitt [14]. As in the procedure described by Pi [15], the nth order moment defined by OW M.CYo) = -- + f ~" a+ dr = n .I : - ' o
IV(z, Yo) dr
(I 9)
o
satisfies the equation c 32 ,_34.
2 a f'2 +Y'~
aM.
= -nM~_ x,
-(2~~176176176
n = 2, 3, 4,
(20)
with M~(yo) = O,
Yo ~ F.
(21)
3. CIRCULAR BOUNDARY For the problem of interest, R is the circle r = a in the phase plane and the following transformations are made:
y = ar cos O, 5'
ar sin O,
(.on
II = to~aZ/ c. e = 24, M * = to. Mr,
(22)
with the result that equation (17) becomes 1
/
02 M 1 cos 2 0 0 2 Mt /sin 2 0 + ~ Or2 r z O2 0
+sin2OOZM~l l r(r ~..l_ (re+2~)cos20] oM' r Or ~ ] - 2 L \ e - , : ~ , , / Or +
#o
-1, (23)
453
MOMENTS OF T H E FIRST-PASSAGE TIME
where the * has been dropped from M* for brevity. Now define the operator .ora as +
(24)
,
where sin 2 0 h- r --
2o=-
/l
cos 2 0 )
KO
1+
~
-
+e
-
Itr 2
sin 2 0 )
K'rO ~-
-
-
,
pr
sin 20 .
(25)
Equation (23) then reduces to a more compact form, ~[M,] =-1,
(26)
and the nth order moment satisfies the recurrence relation ~e[M.l = -riM._,.
(27)
Equation (26) and (27) provide a set of partial differential equations with zero boundary conditions for the moments MI, M2 . . . . . M., which must be solved successively in that order. It is extremely difficult to obtain close-form solutions for these partial differential equations. Bolotin [16] has suggested an approximate method based on Galerkin's method. The approximate solution is obtained in the form of a truncated series, k
(28)
M = ~ M (~ q~(r, 0),
where M t') are coefficients of the series and q~(r,0) are functions of (r,0) satisfying the conditions for M and complete in R. The coefficients xct-) can be found, by using equation (26), from the following relations: 2~
1
o~ ,/o {'C~a[~' M~(')q~'(r'O)]+ I} q~B(r,O)rdOdr=O,
(29)
where fl = I, 2 . . . . . k. It can be seen that the boundary F in the phase plane has been transformed to a unit circle. Thus, the boundary conditions now are given by M(I,0) = 0
on F.
(30)
A suggested set of functions, following Bolotin, is as follows: I ~ t j = ~ ( l --r2)r 2u-1) [1 + ( - - l ) J ] r J s i n ~ + [1 + (--l)Z-1]rJ-I cos (j
1)0
(31)
454
B. SAHAY AND W. LENNOX
The subset of functions q~=(r,O) can be selected from the set of functions described above. For example, consider the following subset. 9, = (1 -- r2),
~P2 = r2( l -- r 2) sin 0, q~3 = r2( 1 - r2) cos0, ~P. = r2(l - r2), ~Ps = r4( 1 -- r2)si n20, 96 = r4(l -- r 2)cos 20, ~7 = r4(l -- r2), (32) The following matrix equation is formed, by using equation (29): AM, = C,
(33)
where 2~" 1
A... = f f .~'{~p.(r, 0)} q~(r, 0) r dO dr, 0
(34a)
0 2~
I
Cm = - f f~p,.(r,O)rdOdr. 0
(34b)
0
By actual calculation of A's and C's, it turns out that the coefficients MCl~), which are associated with functions containing sinn0 and cosn0, have only the trivial solution zero, because they form a separate set of homogeneous equations, having no coupling with the equations containing coefficients Met~) which are associated with functions of r only. Thus, one can restrict the approximation to functions of the following type: tp,(r, O) -----r2tt-l)(l -- r2).
(35)
The effect of this approximation is that the solutions will yield moments of the first-passage time of the envelope of the narrow-band process. For a well-behaved narrow-band process the crossing of the actual process can be estimated as being in the interval t ~ v < t W ~ < t~" +
where con is the centre frequency of the narrow-band process. For the mean first passage time, the matrix A,.. and the column vector C,. can be calculated by using the functions 9~ from equations (36) in equation9 (34), with the operator ~ defined as in equation (24). Upon making use of these functions, ~p,. ----r2"(l -- r2),
m = 0, 1, 2 . . . . .
~p,~= r2"(l -- r2),
n = 0, 1, 2 . . . . .
(36)
455
MOMENTS OF T H E FIRST-PASSAGE TIME
[A] is given by
~[,,(2,,__-L)
,,(2,,- 0+(,,+ o ( 2 . + I)
A(=+u'("+~) = It [ 2 n + 2m
2m + 2n + 2
r i ( n +2rr [21,[2n+2,n -- - .
2n + 1 2m+2n+2
n
(n + l)(2n + 1)]
' ~2+-~,,,+;~ j +
n+ I ] t 2m+gl"+4"Jn+l
2n + i
}]
-e~2n+2m+2 2m+2n+4+2m+2n+6
(37)
where m = 0, 1, 2 . . . . . n = 0, 1, 2 . . . . . with the condition that the terms n(2n - l)/(2n + 2m) and n/(2n + 2m) do not exist whenever m = 0, n = 0 simultaneously. The vector C= is given by 1 ' C,,,+, = -rr mq-
m +12.) '
m=0,1,2 .....
(38)
Once the numerical values of the elements of [A] and column vector C have been calculated, the vector M~ can be obtained from the non-homogeneous linear equations (34). M~, the mean first-passage time, can then be approximated from the series (28) as k
M, = ~ M(t=)~p,.
(39)
For higher order moments, M,, the matrix [A] remains unchanged, as can be seen from equations (27) and (34a), if the set of functions q~ in equation (28) remain the same as for 3/i. Since all the partial differential equations for other moments also have the same boundary conditions and q~ satisfies these for M1, this gives justification to retain the same set of functions for q~. However, from equation (27) it can be seen that the right-hand vector C in equation (33) changes for each moment. For example, for the nth order moment, elements of C will be given by (upon using a k term approximation)
~,=o
{
"-lk2m+2~+2
2m+2~+4
' 2 m + 2'~ + 6)'
(4o)
where m = 0, I, 2 . . . . . k.
4. RESULTS If the barrier is given in terms of c(o'y(i.e., a~ = C/4~o) 3 = xS(co,)/2r spectrum o f f ( t ) ) then 62 1./=~
44
3, where S(co) is the
456
B. SAtlAY AND W. LENNOX
and the results for the first ten moments (in cycles) using an eight term approximation are shown in Figure 2 for Yo = 0 and for various barrier levels c~. Also the mean and variance o f the first-passage time (in cycles) as functions o f initial position lyol, for three barrier levels ~ = 1, 2 and 3, are shown in Figures 3 and 4. It is of interest to note that these curves 108
107
106
105
104 A
k io ~
102
10
IO
0-1 I
2
3
4
5 Q2
6
"7
8
9
Figure 2. nth ozder moments (cycles as functions of barrier level for zero initial conditions. i
I
'
I
i
I
i
I
i
i
I 02
I
I 0.4
i
I 06
v
I 08
I
50
IlL
- - t'M I ~ ii n u
000 0
~
I0
IO
Fol Figure 3 . Mean first-passage time (cycles) as a function of initial position for barrier levels ~ = 1, 2, and 3.
MOMENTSOF THE FIRST-PASSAGETIME
457
I
I
I
I
I
I
i
I
I
I
I 02
I
I ~4
I
I 06
I
I 08
I
3.0
I I 7~,'? uuu
2-0
o
x
0
IO3
I~1 Figure 4. Variance of first-passage time as a function of initial position for barrier levels ~t= 1, 2 and 3. correspond exactly to those obtained by Ariaratnam for the envelope crossings o f a wideband excited linear oscillator [7]. 5. DISCUSSION The use of the set of functions given by equation (32) causes the problem to lose its dependency on 0 and implies that the angular position of Yo has no effect. This is obviously not true and, in fact, first passages from initial points in the first and third quadrants are more probable [3]. Also, inspection of the operator, equation (24), indicates that
~ , ( r , O) = M,(r, 0 + re), so that solutions will be identical in the first and third quadrants. However, if damping is small, then there will be very little difference in the times between envelope crossings and actual crossings. Thus the moments obtained here can be used as reasonable estimates for the first-passage moments of a two-sided barrier problem. ACKNOWLEDGMENT Financial aid for the study was provided by the National Research Council of Canada, Grant no. A3726. REFERENCES 1. K. L. CHANDIRAMANI1964 Ph.D. Thesis, Massachusetts htstitute of Technology. First-passage probabilities for a linear oscillator. 2. J. B. ROBERTS1968 Journal of Sound and Vibration 8, 301-328. An approach: to the first-passage problem in random vibration. 3. M. SmNOZUKA1965 Journal of ,4pplied Probability 2, 79-87. On the two sided barrier prob/em. 4. J. R. RICE 1964 Ph.D. Thesis, Lehigh University. Theoretical predictions of some statistical characteristics of random loading relevant to fatigue and fracture. 5. M. C. BERNARDand J. W. SHIPLEY1972 Journal ofSoundand Vibration 24, 121-132. The first passage problem for stationary random structural vibration.
458
B. SAHAY AND W. LENNOX
6. A. H. GREY 1966 Journal of Applied Mechanics 33, 187-191. First passage time in a random vibrational system. 7. S. T. ARIARATNAMand H. N. PI 1973 (to be published) International Journal of ControL On the first passage time for envelope crossing for a linear oscillator. 8. Y. K. LIN 1970 Journal of the American Institute of Aeronautics and Astronautics 8, 720-725. On the first-excursion failures for a randomly excited structure. 9. S. H. CRANDALL,K. L. CHANDmAMANIand R. G. COOK 1966 Journal of Applied Mechanics 33, 532-583. Some first passage problems in random vibration. 10. J. N. YANO and M. SmNOZUKA 1971 Journal of Applied Mechanics 38, 1017-1022. On the first-excursion probability in stationary narrow-band random vibration. 11. J. N. YANG and M. SHINOZUKA 1972 Journal of Applied Mechanics 39, 733-741. On the firstexcursion probability in stationary narrow-band random vibration. II. 12. R. L. STRA'rONOW'rCH1963 Topics in the Theory of Random Noise, VoL 1. New York: Gordon and Breach Science Publishers. 13. J. N. YANG and M. SHINOZUKA1970Journal of the Acoustical Society of America 47, 393-394. First-passage time problem. 14. L.S. PONTRIAGIN,A. A. ANDRONOVand A. A. Vrn- 1933 Journal of Experimental and Theoretical Physics 3, 165 (in German) 15. H. N. PI 1969 Ph.D. Thesis, University of Waterloo. The first-passage time for snap through o f shells. 16. V. V. BOLOTIN 1965 in Proceedings of the International Conference on Dynamic Stability of Structures. (Ed. G. Herrmann). Statistical aspects in the theory of structural stability. Oxford: Pergamon Press.
M O M E N T S O F T H E FIRST-PASSAGE TIME F O R A . NARROW-BAND
PROCESS
B. SAHAY
Department of Meckanical Engineering, Indian lnstitnte of Technology, Kanpur, htdia AND
W. LENNOX
Department of Civil Enghzeerhtg, University of Waterloo, Waterloo, Ontario, Canada (Received 10 April 1973, and in revisedform 25 September 1973) The moments of the first-passage time of a narrow-band stochastic process produced by a linear oscillator excited by wide-band noise are presented for the case of a "safe" region defined by a circular barrier in the phase plane. As long as damping is small the moments produced will be approximately the same as those for the two-sided barrier problem. An iterative set of partial differential equations governing the moments, the first being the Pontriagin-Vitt equation for the mean first-passage time, are solved by using a Galerkin approximation. No restrictions are required on the barrier size.
1. INTRODUCTION Many problems in the reliability of systems subJected to random excitation can be studied by means of the first-passage problem of probability theory. The system is considered to fail when the response process of interest passes out of specified region of safe operation for the first time at time T: for example, failures due to "bottoming" or "short-circuiting". Tis obviously a random variable and the solution to the problem is given by the probability distribution function P(T) of the first-passage time T. Much attention has been given to this problem, but because of the inherent analytical difficulties, little progress has been made with regard to the exact solution. However, many interesting approximations have been made. These involve approximations of the exclusion series or the multiple integral formulation of S. O. Rice [1, 2, 3], approximations to an integral equation describing the first-passage time density [4, 5], approximate and exact analyses of the envelope function [6, 7], series approximations for survival probabilities [8] and simulation [9]. Two recent papers by Yang and Shinozuka contain comparisons of some of the above methods along with several new ideas [10, 11]. This paper deals with obtaining the moments of the first-passage time T with the idea that a knowledge of the third, fourth and higher order moments will give additional insight into the behaviour of P(T). The first moment, or the mean first-passage time, is governed by the Pontriagin-Vitt equation. An equation describing the second moment can be deduced in terms of the first moment and so on for higher order moments. These partial differential equations are solved approximately by using a Galerkin technique. 449
450
a. SAHAY AND W. LENNOX
2. ANALYSIS The problem considered is the evaluation of the moments of the first-passage time, T. Tis the time required for the trajectory of the linear oscillator to leave a specified region in state space (see Figure I) for the first time, when started from some initial state. The oscillator is excited by wide-band noise and is assumed to be described by the stochastic differential equation
(l)
y + 2~to.)~ + to2.y = f i t ) ,
Y Figure 1. Sketch of the domain, R, and its boundary, F, in state space.
where ~ is the damping factor,~, is the undamped natural frequency of the system a n d f ( t ) is a stationary wide-band process with
E(f(t)) = O,
(2)
E(f(t)f(t+z))=Ks(O.
(3)
It is assumed that the correlation time, T,or, of the stochastic processf(t) is small compared with the time constant of the system:
flKA01dr 0
9
~
Ks(O)
1 < T~ = ~ 2~oJ,,"
This inequality implies that the trajectory of the response by a Markov vector process with
KAz) replaced by
where c is a measure of the intensity of the noise.
(4)
(y(t),p(t)) can be approximated
MOMENTS OF THE FIRST-PASSAGE TIME
451
Inside the "safe" domain, R, and on the boundary, F, the transition probability, w, satisfies the Fokker-Planck (forward) equation a~;'
a~ff,]
O
at
Ok,
Oj~
c a 2 ~, - co]y)~"1 + - ~
[(-2r
(6)
2 O~2
and the associated Kolmogorov (backward) equation Ow
Ow
Ow
~r-r = f,oa--7o- (2r
+ co~yo) ~-7-.-I oyo
c0 2 w
2
a.;,~
.
(7)
If y is the vector ( y , ) ) then w(y, Yo,z) is the joint probability that y(t - z) is in the interval Yo, Yo + dyo given that y(t) is in the interval y, y + dy. The initial and boundary conditions to be satisfied by w are w(y, Yo, 0) - 6(y - Yo),
(8)
which states that no changes of state can occur if the transition time is zero and w(y, Yo, "0 = 0,
Yo e F,
(9)
which implies that only those trajectories that have not crossed the boundary up to time interval z are taken into consideration. The expression
f.'(y, Yo,T) dy
W(z, Yo) =
(10)
R
represents the probability that a trajectory has not reached the boundary during a time interval ~. Initially, no realization has reached the boundary so that W(0, Yo) = 1
(! I)
and ultimately all realizations will reach the boundary with the result tV(oo, Yo) = 0.
(I 2)
The distribution function of the first-passage time P(z, yo) is then given by [12, 13]
1 - W(~, Yo). By integrating the Kolmogorov equation (7) as well as the boundary conditions (8) and (9) with respect to y over the region R, a differential equation describing Wis obtained: OtV OW -(2r Or = "YOayo
OIV c0 2 IV + 2 a~ '
(13)
with W(0, Yo) = I,
Yo E R,
(14)
IV(z, Yo) = 0,
Yo ~ F.
(15)
The closed form solution to this initial-boundary value problem is not known. In the absence of a complete solution for IV(r, Yo), a knowledge of the moments of the first-passage time would be useful.
452
n . S A H A Y A N D ~,V. L E N N O X
Let Mt(yo) be the mathematical expectation of the time required for the trajectory, with initial point Yo, to reach the boundary F o f the region R for the first time: ' r
Mt(yo)=-d
~++.o,v o
fWd+.
(16)
0r dZ=o
By integrating equation (13) with respect to z and using equation (14), the following equation is obtained for M~ 9
Ca2 3It 2 a~o~
aMt
+ Y'o-:--
a),o
aMt oyo
(2~o~o + o~.:o)-v:-. ~' = -l,
(17)
with M,(yo) = 0,
Yo ~ F.
(18)
Equation (17) was first derived by Pontriagin, Andronov and Vitt [14]. As in the procedure described by Pi [15], the nth order moment defined by OW M.CYo) = -- + f ~" a+ dr = n .I : - ' o
IV(z, Yo) dr
(I 9)
o
satisfies the equation c 32 ,_34.
2 a f'2 +Y'~
aM.
= -nM~_ x,
-(2~~176176176
n = 2, 3, 4,
(20)
with M~(yo) = O,
Yo ~ F.
(21)
3. CIRCULAR BOUNDARY For the problem of interest, R is the circle r = a in the phase plane and the following transformations are made:
y = ar cos O, 5'
ar sin O,
(.on
II = to~aZ/ c. e = 24, M * = to. Mr,
(22)
with the result that equation (17) becomes 1
/
02 M 1 cos 2 0 0 2 Mt /sin 2 0 + ~ Or2 r z O2 0
+sin2OOZM~l l r(r ~..l_ (re+2~)cos20] oM' r Or ~ ] - 2 L \ e - , : ~ , , / Or +
#o
-1, (23)
453
MOMENTS OF T H E FIRST-PASSAGE TIME
where the * has been dropped from M* for brevity. Now define the operator .ora as +
(24)
,
where sin 2 0 h- r --
2o=-
/l
cos 2 0 )
KO
1+
~
-
+e
-
Itr 2
sin 2 0 )
K'rO ~-
-
-
,
pr
sin 20 .
(25)
Equation (23) then reduces to a more compact form, ~[M,] =-1,
(26)
and the nth order moment satisfies the recurrence relation ~e[M.l = -riM._,.
(27)
Equation (26) and (27) provide a set of partial differential equations with zero boundary conditions for the moments MI, M2 . . . . . M., which must be solved successively in that order. It is extremely difficult to obtain close-form solutions for these partial differential equations. Bolotin [16] has suggested an approximate method based on Galerkin's method. The approximate solution is obtained in the form of a truncated series, k
(28)
M = ~ M (~ q~(r, 0),
where M t') are coefficients of the series and q~(r,0) are functions of (r,0) satisfying the conditions for M and complete in R. The coefficients xct-) can be found, by using equation (26), from the following relations: 2~
1
o~ ,/o {'C~a[~' M~(')q~'(r'O)]+ I} q~B(r,O)rdOdr=O,
(29)
where fl = I, 2 . . . . . k. It can be seen that the boundary F in the phase plane has been transformed to a unit circle. Thus, the boundary conditions now are given by M(I,0) = 0
on F.
(30)
A suggested set of functions, following Bolotin, is as follows: I ~ t j = ~ ( l --r2)r 2u-1) [1 + ( - - l ) J ] r J s i n ~ + [1 + (--l)Z-1]rJ-I cos (j
1)0
(31)
454
B. SAHAY AND W. LENNOX
The subset of functions q~=(r,O) can be selected from the set of functions described above. For example, consider the following subset. 9, = (1 -- r2),
~P2 = r2( l -- r 2) sin 0, q~3 = r2( 1 - r2) cos0, ~P. = r2(l - r2), ~Ps = r4( 1 -- r2)si n20, 96 = r4(l -- r 2)cos 20, ~7 = r4(l -- r2), (32) The following matrix equation is formed, by using equation (29): AM, = C,
(33)
where 2~" 1
A... = f f .~'{~p.(r, 0)} q~(r, 0) r dO dr, 0
(34a)
0 2~
I
Cm = - f f~p,.(r,O)rdOdr. 0
(34b)
0
By actual calculation of A's and C's, it turns out that the coefficients MCl~), which are associated with functions containing sinn0 and cosn0, have only the trivial solution zero, because they form a separate set of homogeneous equations, having no coupling with the equations containing coefficients Met~) which are associated with functions of r only. Thus, one can restrict the approximation to functions of the following type: tp,(r, O) -----r2tt-l)(l -- r2).
(35)
The effect of this approximation is that the solutions will yield moments of the first-passage time of the envelope of the narrow-band process. For a well-behaved narrow-band process the crossing of the actual process can be estimated as being in the interval t ~ v < t W ~ < t~" +
where con is the centre frequency of the narrow-band process. For the mean first passage time, the matrix A,.. and the column vector C,. can be calculated by using the functions 9~ from equations (36) in equation9 (34), with the operator ~ defined as in equation (24). Upon making use of these functions, ~p,. ----r2"(l -- r2),
m = 0, 1, 2 . . . . .
~p,~= r2"(l -- r2),
n = 0, 1, 2 . . . . .
(36)
455
MOMENTS OF T H E FIRST-PASSAGE TIME
[A] is given by
~[,,(2,,__-L)
,,(2,,- 0+(,,+ o ( 2 . + I)
A(=+u'("+~) = It [ 2 n + 2m
2m + 2n + 2
r i ( n +2rr [21,[2n+2,n -- - .
2n + 1 2m+2n+2
n
(n + l)(2n + 1)]
' ~2+-~,,,+;~ j +
n+ I ] t 2m+gl"+4"Jn+l
2n + i
}]
-e~2n+2m+2 2m+2n+4+2m+2n+6
(37)
where m = 0, 1, 2 . . . . . n = 0, 1, 2 . . . . . with the condition that the terms n(2n - l)/(2n + 2m) and n/(2n + 2m) do not exist whenever m = 0, n = 0 simultaneously. The vector C= is given by 1 ' C,,,+, = -rr mq-
m +12.) '
m=0,1,2 .....
(38)
Once the numerical values of the elements of [A] and column vector C have been calculated, the vector M~ can be obtained from the non-homogeneous linear equations (34). M~, the mean first-passage time, can then be approximated from the series (28) as k
M, = ~ M(t=)~p,.
(39)
For higher order moments, M,, the matrix [A] remains unchanged, as can be seen from equations (27) and (34a), if the set of functions q~ in equation (28) remain the same as for 3/i. Since all the partial differential equations for other moments also have the same boundary conditions and q~ satisfies these for M1, this gives justification to retain the same set of functions for q~. However, from equation (27) it can be seen that the right-hand vector C in equation (33) changes for each moment. For example, for the nth order moment, elements of C will be given by (upon using a k term approximation)
~,=o
{
"-lk2m+2~+2
2m+2~+4
' 2 m + 2'~ + 6)'
(4o)
where m = 0, I, 2 . . . . . k.
4. RESULTS If the barrier is given in terms of c(o'y(i.e., a~ = C/4~o) 3 = xS(co,)/2r spectrum o f f ( t ) ) then 62 1./=~
44
3, where S(co) is the
456
B. SAtlAY AND W. LENNOX
and the results for the first ten moments (in cycles) using an eight term approximation are shown in Figure 2 for Yo = 0 and for various barrier levels c~. Also the mean and variance o f the first-passage time (in cycles) as functions o f initial position lyol, for three barrier levels ~ = 1, 2 and 3, are shown in Figures 3 and 4. It is of interest to note that these curves 108
107
106
105
104 A
k io ~
102
10
IO
0-1 I
2
3
4
5 Q2
6
"7
8
9
Figure 2. nth ozder moments (cycles as functions of barrier level for zero initial conditions. i
I
'
I
i
I
i
I
i
i
I 02
I
I 0.4
i
I 06
v
I 08
I
50
IlL
- - t'M I ~ ii n u
000 0
~
I0
IO
Fol Figure 3 . Mean first-passage time (cycles) as a function of initial position for barrier levels ~ = 1, 2, and 3.
MOMENTSOF THE FIRST-PASSAGETIME
457
I
I
I
I
I
I
i
I
I
I
I 02
I
I ~4
I
I 06
I
I 08
I
3.0
I I 7~,'? uuu
2-0
o
x
0
IO3
I~1 Figure 4. Variance of first-passage time as a function of initial position for barrier levels ~t= 1, 2 and 3. correspond exactly to those obtained by Ariaratnam for the envelope crossings o f a wideband excited linear oscillator [7]. 5. DISCUSSION The use of the set of functions given by equation (32) causes the problem to lose its dependency on 0 and implies that the angular position of Yo has no effect. This is obviously not true and, in fact, first passages from initial points in the first and third quadrants are more probable [3]. Also, inspection of the operator, equation (24), indicates that
~ , ( r , O) = M,(r, 0 + re), so that solutions will be identical in the first and third quadrants. However, if damping is small, then there will be very little difference in the times between envelope crossings and actual crossings. Thus the moments obtained here can be used as reasonable estimates for the first-passage moments of a two-sided barrier problem. ACKNOWLEDGMENT Financial aid for the study was provided by the National Research Council of Canada, Grant no. A3726. REFERENCES 1. K. L. CHANDIRAMANI1964 Ph.D. Thesis, Massachusetts htstitute of Technology. First-passage probabilities for a linear oscillator. 2. J. B. ROBERTS1968 Journal of Sound and Vibration 8, 301-328. An approach: to the first-passage problem in random vibration. 3. M. SmNOZUKA1965 Journal of ,4pplied Probability 2, 79-87. On the two sided barrier prob/em. 4. J. R. RICE 1964 Ph.D. Thesis, Lehigh University. Theoretical predictions of some statistical characteristics of random loading relevant to fatigue and fracture. 5. M. C. BERNARDand J. W. SHIPLEY1972 Journal ofSoundand Vibration 24, 121-132. The first passage problem for stationary random structural vibration.
458
B. SAHAY AND W. LENNOX
6. A. H. GREY 1966 Journal of Applied Mechanics 33, 187-191. First passage time in a random vibrational system. 7. S. T. ARIARATNAMand H. N. PI 1973 (to be published) International Journal of ControL On the first passage time for envelope crossing for a linear oscillator. 8. Y. K. LIN 1970 Journal of the American Institute of Aeronautics and Astronautics 8, 720-725. On the first-excursion failures for a randomly excited structure. 9. S. H. CRANDALL,K. L. CHANDmAMANIand R. G. COOK 1966 Journal of Applied Mechanics 33, 532-583. Some first passage problems in random vibration. 10. J. N. YANO and M. SmNOZUKA 1971 Journal of Applied Mechanics 38, 1017-1022. On the first-excursion probability in stationary narrow-band random vibration. 11. J. N. YANG and M. SHINOZUKA 1972 Journal of Applied Mechanics 39, 733-741. On the firstexcursion probability in stationary narrow-band random vibration. II. 12. R. L. STRA'rONOW'rCH1963 Topics in the Theory of Random Noise, VoL 1. New York: Gordon and Breach Science Publishers. 13. J. N. YANG and M. SHINOZUKA1970Journal of the Acoustical Society of America 47, 393-394. First-passage time problem. 14. L.S. PONTRIAGIN,A. A. ANDRONOVand A. A. Vrn- 1933 Journal of Experimental and Theoretical Physics 3, 165 (in German) 15. H. N. PI 1969 Ph.D. Thesis, University of Waterloo. The first-passage time for snap through o f shells. 16. V. V. BOLOTIN 1965 in Proceedings of the International Conference on Dynamic Stability of Structures. (Ed. G. Herrmann). Statistical aspects in the theory of structural stability. Oxford: Pergamon Press.