THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE
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1 Journal of Soun an Vibration (1996) 191(3), THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE E. M. WEINSTEIN Galaxy Scientific Corporation, 2500 English Creek Avenue, Builing 11, Pleasantville, NJ 08055, U.S.A. AND H. BENAROYA Department of Mechanical an Aerospace Engineering, Rutgers, The State University of New Jersey, P.O. Box 909, Piscataway, NJ 08855, U.S.A. (Receive 3 October 1994, an in final form 22 May 1995) This paper expans on previous work by Roríguez an van Kampen, an by Weinstein an Benaroya. The original work by Roríguez an van Kampen outline a metho of extracting information from the Fokker Planck equation without having to solve the equation itself. In the van Kampen expansion, the Fokker Planck equation is expane about the ranom component of the response. This expansion is use to erive a series of first orer ifferential equations in time for the moments of the response. These can then be solve for moments as a function of time. In the original work, this metho was emonstrate on the case of a Duffing oscillator excite by white noise. Weinstein an Benaroya evelope this metho for the case of a Duffing oscillator excite by colore noise, performe parametric stuies on the system parameters, an verifie their results by comparing them to Monte Carlo experiments. In this work, the van Kampen expansion is applie to systems of linke Duffing-linear oscillators excite by colore noise. Again, parametric stuies are performe on the system parameters an the results compare to those of Monte Carlo experiments. It is foun that the analytical results compare closely with the numerical results as long as the initial assumptions of the expansion are not violate. The closeness of the comparisons inicates that the van Kampen expansion is a vali technique for the stuy of this class of problem an is a useful tool in moelling offshore structures, plates in turbulent flows, an other flui structure interaction phenomena Acaemic Press Limite 1. INTRODUCTION The Fokker Planck equation has proven to be a useful tool in the analysis of simple non-linear oscillators excite by stochastic processes. As a partial ifferential equation for the probability ensity function of the response, its solution completely efines the solution of the problem. It can be use to analyze both a single oscillator of the form mx + (ẋ, x)x + k(x, x)x = F(t), (1) or a system of multiple, linke, oscillators of the form Corresponing author X/96/ $18.00/0 Mx + (x, x)x + K(x, x)x = F(t). (2) Acaemic Press Limite
2 398 E. M. WEINSTEIN AND H. BENAROYA In many cases, a physical system can be approximate by such a system of non-linear oscillators. The systems so moele can range from a Brownian particle to structures excite by von Kármán vortex sheing. Such moeling can be useful for gaining insight into a problem an the way in which the system will behave as certain parameters are varie. Once one has ecie on the system of oscillators to be use to represent the physical system, the erivation of the Fokker Planck equation is relatively straightforwar, although teious. The problem of how to solve it for the probability istribution of the response remains. In a very few cases, the Fokker Planck equation can be solve analytically, but in most cases no analytical solution exists an one usually must resort to a numerical solution. However, this can be computationally intensive an gives little insight into the larger problem. In their 1976 paper, Rorı guez an van Kampen [1] outline a metho of ealing with the case of an oscillator excite by weak Gaussian white noise. The Fokker Planck equation of the system is expane about the intensity,, of the riving function. This expansion is carrie to the orer O( 1/2 ). In this way the statistics of the fluctuations are obtaine irectly. This metho shows promise as a way to use the Fokker Planck equation to gain useful information about a wier variety of systems than was possible before. In previous work by Weinstein an Benaroya [2, 3], this metho was expane to oscillators excite by colore noise. Parametric stuies were performe on the intensity of the forcing function, the magnitue of the amping, an the correlation time of the colore noise forcing function. All results are valiate by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is applie to systems of couple Duffing linear oscillators excite by colore noise. This is one for two reasons. The first is to emonstrate that the expansion can be applie to a system of couple oscillators. The secon is that some physical systems are well moele by such couple oscillators. An example of such a system is an offshore structure uner the influence of ocean waves. These waves have a certain perioic nature, but are of ranom height an intensity. The structure itself can be moelle as a single oscillator or a series of oscillators. Wilson an Oa [4, 5] use the following oscillator to simulate the surge motion of moore offshore structures: mx + bx + b 1 x + b 2 x 2 + b 3 x 3 = w(t), where w(t) = c 1 y + c 2 y + n(t), n(t) = Gaussian white noise an y(t) = non-white ynamic input. Kanegaonkar an Halar [6] use the following moel for a guye tower platform: J + a + G + H 3, (3) where is the angle of rotation of the platform, G is a function of the platform geometry an H is a function of the vertical loa. The first of these two moels is similar in form to the moel iscusse in this paper, while the secon is similar in form to the moels examine in the previous papers cite above. There are several ways in which such couple oscillators can be formulate. The external force can rive either the linear or the Duffing oscillator. Here, it is ecie to have the external force rive the linear oscillator. The formulation of the solution woul be the same if the situation was reverse. There are also several ways in which the two oscillators can be couple: through a amping function, through a spring function, or through a combination of the two. In this analysis, the coupling chosen is of the form of a simple linear spring. Here, the system is formulate in such a way that the effect of the Duffing oscillator woul not fee back into the linear oscillator. Such a system woul more
3 DUFFING OSCILLATORS 399 accurately moel a system where the riving force is an overwhelming physical phenomenon impervious to the effect of the structure being moele. However, it appears that all of the above variations can be moele using the following technique. 2. DERIVATION AND EXPANSION OF THE FOKKER PLANCK EQUATION The first step in the erivation of the Fokker Planck equation for any system is the formulation of the governing equations. The governing equations for the system escribe in the Introuction are: where F(t) is efine by y (t) + 1 c y(t) = 1 c F(t), x 1(t) + 1 x 1(t) + x 1 (t) = y(t), (4, 5) x 2(t) + 2 x 2(t) + x 2 (t) + x 3 2(t) = k[x 1 (t) x 2 (t)], (6) F(t) = 0, F(t)F(t') = 2 (t t'), (7) where is the Dirac elta. That the solution of equation (4) is exponentially correlate colore noise is shown in several sources, such as Billah an Shinozuka [7]. Because the Fokker Planck equation requires that the governing equations be cast as a series of first orer ifferential equations, the following new variables are efine: v 1 = x 1, v 2 = x 2. (8, 9) Using these variables the governing equations can be rewritten in the following, equivalent, form: ẏ(t) = 1 c y(t) + 1 c F(t), x 1(t) = v 1 (t), (10, 11) v 1(t) = y(t) 1 v 1 (t) x 1 (t), x 2(t) = v 2 (t), (12, 13) v 2(t) = kx 1 (t) 2 v 2 (t) (1 + k)x 2 (t) x 3 2(t). (14) Denoting by f(x 1, v 1, x 2, v 2, y; t) the probability istribution function of the response at time t, the Fokker Planck equation of this system can be erive as t f = y (yf) v 1 f [(y x 1 v 1 v 1 x 1 )f ] v 2 f 1 x 2 v 2 [(kx 1 2 v 2 (1 + k)x 2 x 3 2)f ] y 2 f. (15) As was shown in Weinstein an Benaroya [2], the response of the oscillator can be separate into a eterministic component arising from the initial conitions, an a ranom component of magnitue O( ). However, by assuming the oscillator to be initially at rest, the eterministic component can be shown to be equal to zero. Therefore, the following substitutions are mae into equation (15): y =, x 1 = 1, v 1 = 1, x 2 = 2, v 2 = 2, (16 20) f(y, x 1, v 1, x 2, v 2 ; t) = 5/2 (, 1, 1, 2, 2 ; t). (21) (, 1, 1, 2, 2 ; t) is the joint probability ensity of the transforme variables. The factor 5/2 will be omitte from the efinition of. If carrie through the erivations, it woul be ivie out at a later stage.
4 400 E. M. WEINSTEIN AND H. BENAROYA The relationships between the partial erivatives of f an are easily obtaine as: f y =, f x 2 = 2, f =, x 1 1 f v 2 = 2, f v 1 = 1, (22 24) f t = t. (25 27) Equations (16) (20), as well as the equations above, equations (22) (27), can be substitute into the Fokker Planck equation, equation (15), yieling t = ( ) 1 1 [( ) ] [(k (1 + k) 2 3 2) ] 2 + 2,, (28) where the subscripts enote partial ifferentiation with respect to the variables in the subscripts. The left sie of equation (28) is transforme into partial erivatives of in the transforme variables,, 1, 1, 2, 2, yieling = ( ) 1 1 [( ) ] [(k (1 + k) 2 3 2) ] 2 + 2,. (29) One can now substitute the efinitions of, 1, 1, 2 an 2 into equations (11) (14) an multiply by 1/2. The resulting four equations are 1 = 1, 1 = 1 1 1, (30, 31) 2 = 2, 2 = k (1 + k) (32, 33) If one multiplies each of the above equations by an ifferentiates equation (30) with respect to 1, equation (31) with respect to 1, equation (32) with respect to 2 an equation (33) with respect to 2, each of the resulting equations can be ae to the transforme Fokker Planck equation, equation (29). The result is the following equation: This can be integrate with respect to to give = [ ] + 2,. (34) = + 2. (35) Thus, the Fokker Planck equation of the system of linke oscillators has been transforme to the following five equations: = + 2, 1 = 1, 1 = ( ), (36 38) 2 = 2, 2 = [k (1 + k) 2 3 2]. (39, 40) As shown in Weinstein an Benaroya [2], the time erivatives of the secon orer moments can be foun as follows: t 2 1 = 2 1 1, (41) t 2 1 = , (42) t 1 1 = , (43)
5 t 2 2 = 2 2 2, DUFFING OSCILLATORS 401 t 1 2 = , (44, 45) t 1 2 = , (46) t 2 2 = 2k 1 2 2(1 + k) , (47) t 1 2 = k 2 1 (1 + k) , (48) t 1 2 = k 1 1 (1 + k) ( ) , (49) t 2 2 = (1 + k) k , (50) t 2 = , t 1 = 1 + 1, (51, 52) t 1 = 2 1 ( 1 + ) 1, (53) t 2 = 2 + 2, (54) t 2 = k 1 (1 + k) 2 ( 2 + ) 2. (55) The previous 15 equations, equations (41) (55), can be cast as a single matrix equation, where x = Ax + b, (56) t x = 1 2, b = 0,
6 402 E. M. WEINSTEIN AND H. BENAROYA an A = k 0 2 2k k k k k k k k 0 1 k 2
7 DUFFING OSCILLATORS 403 Equation (56) can be solve by stanar techniques to yiel all the time evolving secon orer moments of the system. 3. RESULTS In Figure 1 are shown the results of the analysis of the case of 1 = 1 0, 2 = 1 0, c = 0 01 an coupling spring constant k = 1 0. The Monte Carlo results for the same system are shown in Figure 2. Agreement between the two sets of results is generally very goo, with no major points of ifference. In general, the analytical results show slightly e-emphasize local maxima an minima as compare to the Monte Carlo results. This effect is most pronounce in the cross-correlation curves, such as 1 1 (Figures 1(c, e, f, j) an 2(c, e, f, j)) that have local maxima at about 5 s. A comparison of Figures 1(c) an 2(c) shows this effect most clearly. It is also seen that, in general, the steay state values of the Monte Carlo results are somewhat lower than those of the analytical results. This was shown to be a numerical artifact in the previous two papers [2, 3]. It comes from the fact that in truncating the expansion at a particular orer of, the largest terms neglecte are negative in the perio that the moments are evloping, 0 t 4. Four stuies on the effects of varying each of the several parameters, while keeping all others constant, are performe: (1) 1 is varie from 0 7 to 10, (2) 2 is varie from 0 7 to 10 0, (3) k, the constant of the coupling spring, is varie from 0 1 to 10 0, an (4) c is increase from to The results of the first of these stuies are shown in Figures 3 an 4. The results are as woul be expecte from the previous work. The responses of the linear oscillator o not change shape signficantly with changing values of 1 ; however, the magnitue oes ecrease with increasing values of 1. This is a irect result of the physics of the problem: increase amping of an oscillator shoul ecrease the excursions. The ecrease in the magnitue of the response of the Duffing oscillator is a irect result of the ecrease magnitue of the response of the linear oscillator. The Duffing oscillator is excite by the isplacement of the linear oscillator; it is reasonable that a ecrease in the magnitue of the isplacement of the linear oscillator will be reflecte in a ecrease in the response of the Duffing oscillator. The agreement between the analytical an numerical ata is goo. The analytical ata show slightly e-emphasize local maxima an minima, although the ifference is not marke. The largest ifference is seen in Figures 3(j) an 4(j). A comparison of these two figures shows the Monte Carlo traces becoming negative at 5 s an 10 s, while the analytical traces merely reach local minima of about zero. The results of the stuy of the effect of varying 2 is shown in Figures 5 an 6. These figures show no change in the response of the linear oscillator with changing 2. This is because, while the Duffing oscillator is excite by the isplacement of the linear oscillator, the linear oscillator is inepenent of the Duffing oscillator; i.e., neither x 2 nor v 2 appear in the equations of motion of the linear oscillator. Therefore, one woul not expect any variation of the Duffing oscillator to affect the linear oscillator. However, the effect on the Duffing oscillator of varying 2 is quite pronounce an similar to the effect on the linear oscillator of varying 1. All response curves of the Duffing oscillator ecrease in magnitue as 2 increases. This is for the reasons state above. A comparison of Figures 5 an 6 shows the same similarities an contrasts as for the 1 stuy. In Figures 7 an 8 are shown the results of varying the spring constant, k, of the coupling spring. Again it is seen that varying the nature of the coupling has no effect on the linear oscillator: there is no change in the moments of the linear oscillator, 2 1,
8 404 E. M. WEINSTEIN AND H. BENAROYA Figure 1. The magnitue of moments of response of couple oscillators versus time, 1 = 1 0, 2 = 1 0, c = 0 01, k = 1 0, calculate analytically. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) an 2 1. The moments of the Duffing oscillator, 2 2, 2 2 an 2 2, o increase with increasing values of k. This is again because the riving force of the Duffing oscillator is k( 1 2 ). Therefore, the larger values of k will lea to larger values of the forcing function, an so larger values of the moments. The cross-correlations, 1 2, 1 2 (Figures 7(f, h) an 8(f, h)) show something unique to this stuy. Looking at the
9 DUFFING OSCILLATORS 405 Figure 2. The magnitue of moments of response of couple oscillators versus time, 1 = 1 0, 2 = 1 0, c = 0 01, k = 1 0, calculate via a Monte Carlo experiment. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2. progression for traces as k is varie shows markely ifferent results than any other presente in this paper. In all other figures presente, as a parameter is varie, the curves change monotonically in magnitue, but o not change significantly in shape. In these figures, the trace corresponing to k = 10 0, the largest value of the parameter, has the meian value. Also, the traces corresponing to the two lowest values of the parameter,
10 406 E. M. WEINSTEIN AND H. BENAROYA Figure 3. The effect of increasing 1 on the moments of response of couple oscillators, 2 = 1 0, k = 1 0, c = 0 01, calculate analytically. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., 1=0 7;, 1 = 1 0;, 1 = 3 0;, 1 = k = 0 1 an k = 0 3, o not show any iscernible local maxima in the first 2 5 s; they increase monotonically from zero to their maxima. The other three traces show increasing local maxima with the local maximum of the k = 10 0 trace of roughly three times as great as the steay state value. The ecrease in the magnitue of these traces, after each reaches its local maximum, represents a shift in the phase angle between the two oscillators.
11 DUFFING OSCILLATORS 407 Figure 4. The effect of increasing 1 on the moments of response of couple oscillators, 2 = 1 0, k = 1 0, c = 0 01, calculate via a Monte Carlo experiment. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., 1=0 7;, 1 = 1 0;, 1 = 3 0;, 1 = This can be seen by examining two linke linear oscillators, with both masses equal to one, the constant of the spring linking the first mass to groun equal to one an the linking spring equal to k. If one consiers a system with k very large, then the two masses will move virtually as one; i.e., the two masses will move in unison with a phase angle of zero. As k becomes of orer unity the masses will begin to move inepenently, an
12 408 E. M. WEINSTEIN AND H. BENAROYA Figure 5. The effect of increasing 2 on the moments of response of couple oscillators, 1 = 1 0, k = 1 0, c = 0 01, calculate analytically. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., 1=0 7;, 1 = 1 0;, 1 = 3 0;, 1 = as k ecreases further, m 1 an m 2 will become 180 egrees out of phase. This is what is reflecte in these curves. In Figures 9 an 10 are shown the effects on the response of the oscillators of varying the correlation time of the noise. As in the previous papers, it is seen that there is negligible effect in varying c from an 0 01, an very small effect in changing
13 DUFFING OSCILLATORS 409 Figure 6. The effect of increasing 2 on the moments of response of couple oscillators, 1 = 1 0, k = 1 0, c = 0 01, calculate via a Monte Carlo experiment. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., 2=0 7;, 2 = 1 0;, 2 = 3 0;, 2 = c to 0 1. For this oscillator, noise of correlation time c 0 01 can be consiere as white without any loss of information. For many systems, the white noise approximation is acceptable for the case of colore noise with correlation time c = 0 1. As in the other stuies, the agreement between the analytical an experimental results was quite goo.
14 410 E. M. WEINSTEIN AND H. BENAROYA Figure 7. The effect of varying k on the moments of response of couple oscillators, 1 = 1 0, k = 1 0, c = 0 01, calculate analytically. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., k=0 1;, k = 0 3;, k = 1 0;, k = 3 0;, k = In summary, the ifferences between Figures 1 an 2 persist through all the parametric stuies: the expansion results show slightly e-emphasize local extrema an slightly exaggerate large time solutions. In general, the first effect is more pronounce than the secon. If the magnitue of two ajacent local extrema is taken
15 DUFFING OSCILLATORS 411 Figure 8. The effect of varying k on the moments of response of couple oscillators, 1 = 1 0, 2 = 1 0, c = 0 01, calculate via a Monte Carlo experiment. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., k=0 1;, k = 0 3;, k = 1 0;, k = 3 0;, k = as the absolute value of the ifference of the magnitue, then the ifference between these two methos ranges from 0 to 50 percent. However, they are always in evience an always occur at about the same time in the results of the two methos. The ifference in the steay state solutions is uner 20 percent. Qualitatively, however, the results are
16 412 E. M. WEINSTEIN AND H. BENAROYA Figure 9. The effect of varying c on the response of couple oscillators, 1 = 1 0, 2 = 1 0, k = 1 0, calculate analytically. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., c=0 001;, c = 0 01;, c = 0 1;, c = 1 0;, c = quite similar. Both sets of results share the same shape, local extrema, time scales, an inicate similar behavior of the system. Both show ientical trens for the variation of the parameters as long as the assumption is vali that the intensity of the noise is small compare to unity an the amping is at least of orer unity.
17 DUFFING OSCILLATORS 413 Figure 10. The effect of varying c on the response of couple oscillators, 1 = 1 0, 2 = 1 0, k = 1 0, calculate via a Monte Carlo experiment. (a) 2 1 ; (b) 2 1 ; (c) 1 1 ; () 2 2 ; (e) 1 2 ; (f) 1 2 ; (g) 2 2 ; (h) 1 2 ; (i) 1 2 ; (j) 2 2., c=0 001;, c = 0 01;, c = 0 1;, c = 1 0;, c = CONCLUSIONS In general, the quantitative ifferences between the results of the two methos is small, an the qualitative ifferences are almost negligible. Quantitatively, the ifferences inclue an unerrepresentation of the local extrema an a small overstatement of the steay state magnitues. The significant similarity between the results given by the analytical an
18 414 E. M. WEINSTEIN AND H. BENAROYA computational methos implies that this aaptation of the van Kampen expansion is an accurate tool for preicting the statistics of the response of a couple Duffing linear oscillator excite by colore noise. Although not irectly shown here, it seems reasonable to assume that this metho woul give goo results for an arbitrary combination of linear an Duffing oscillators. It can be expecte that the magnitue of the response will ecrease as the ranom isturbance is propagate through the cascae of oscillators. However, it is still reasonable to assume that the energy resient in any oscillator will still be proportional to the energy input to the system. Therefore, the ranom part of the response will remain proportional to the square root of the forcing function intensity, although it may not be of the same orer. It was also seen that, epening on the magnitue of the correlation time as compare to the natural perio of the oscillator, simplifying assumptions can be mae for certain correlation times which permit using the white noise approximation. It is certainly not propose that this metho replace more traitional analysis tools such as finite elements. However, it coul have important applications in etermining the relative importance of the variables in the problem an the egree to which the larger structure can be moele as as a series of smaller structures as is common in structural analysis. ACKNOWLEDGMENTS This work was performe uner the support of the Feeral Aviation Aministration Technical Center. The authors woul like to thank Lawrence Neri of the Technical Center for his interest an support, Harry Kemp, also of the Technical Center for his support an expertise, an also the Department of Mechanical an Aerospace Engineering at Rutgers, the State University of New Jersey. The secon author woul also like to thank the Office of Naval Research an scientific officer Thomas F. Swean for support uner grant number N REFERENCES 1. R. RODRI GUEZ an N. G. VAN KAMPEN 1976 Physica 85A, Systematic treatment of fluctuations in a nonlinear oscillator. 2. E. M. WEINSTEIN an H. BENAROYA 1994 Journal of Statistical Physics 77(3/4), Nov. The van Kampen expansion for the Fokker Planck equation of a Duffing oscillator. 3. E. M. WEINSTEIN an H. BENAROYA 1994 Journal of Statistical Physics 77(3/4), Nov. The van Kampen expansion for the Fokker Planck equation of a Duffing oscillator excite by colore noise. 4. J. F. WILSON 1984 Dynamics of Offshore Structures. New York: John Wiley. 5. H. ODA, T. OZAKI an Y. YANAMOUCHI 1987 In Nonlinear Stochastic Dynamics Engineering Systems (F. Ziegler an G. I. Schuëller, eitors), New York: IUTAM, Springer-Verlag. A nonlinear system ientification in the analysis of offshore structure ynamics in ranom waves. 6. H. KANEGAONKAR an A. HALDAR 1987 In Nonlinear Stochastic Dynamics Engineering Systems (F. Ziegler an G. I. Schuëller, eitors), New York: IUTAM, Springer-Verlag. Nonlinear ranom vibrations of compliant offshore platforms. 7. K. Y. R. BILLAH an M. SHINOZUKA 1990 Physical Review 12(42), Dec. Numerical metho for colore noise generation an its application to a bistable system.
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