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Improved stability analysis of the response of a duffing oscillator under filtered white noise
,,,r. J Non-Linear .Mrchmw Printed in Great Rnta~n
Vol. 28. No. 2, pp. 145 155, 1993
002&7462/93 36.00 + .JO T‘I 1993 Pergmon Press Lid
IMPROVED STABILITY ANALYSIS OF THE RESPONSE OF A DUFFING OSCILLATOR UNDER FILTERED WHITE NOISE P. K. KOLIOPULOS* and R. S. LANGLEY? *Department of Civil Engineering, ‘Department
of Aeronautics
University College, Cower Street, London WCIE 6BT, U.K. and and Astronautics, University of Southampton, Highheld, Southampton SO9 5NH, U.K. (Recked
3 September
1992)
Abstract The applicability of the equivalent linearisation method for the analysis of non-linear oscillators subjected to narrowband random excitation is discussed. It is shown that when jumps between competing response states occur, the choice of an appropriate value for the kurtosis of the response process is crucial for a reliable estimation of local statistical moments and for a more accurate stochastic stability analysis. Furthermore, a modification is proposed which takes into account the influence of certain non-linear terms in the variational equation for the cases of strong non-linearities. The theoretical predictions correlate well with the results based on extensive digital simulations.
1. INTRODUCTION
It is well known that if an oscillator with a hardening non-linear stiffness is subjected to sinusoidal excitation, it is possible for the oscillator response to exhibit the phenomenon of sharp jumps in amplitude. In recent years it has been realised that the same jumping phenomenon can occur if an oscillator with non-linear stiffness is subjected to narrowband random excitation, provided that the bandwidth of the excitation is sufficiently small [la]. The occurrence and persistence of jumps in the response process is reflected in non-unique temporal statistical moments and in multipeak probability density functions of the amplitude of the response. A limited number of approximate theoretical methods have been developed for the study of such systems with the aim of predicting regions (in the parameter space) of possible multiple response states, the estimation of the values of temporal response statistics and the influence of jumps on the response probability density. One method is the so-called quasi-harmonic (or quasi-static) method, which has been successfully implemented to tackle all the above-mentioned problems [S-7]. This simple, yet powerful, method suffers, however, from the fact that it is valid only when the input bandwidth is extremely small. Another technique is the popular equivalent linearisation method, which has been used for the first two problems [2,4,6, 8-l I]. There seems to be a disagreement, however, with regard to the significance of the equivalent linearisation predictions. In a series of publications lyengar 18, 123 has presented a stochastic stability analysis of the equation of motion of a hardening-type oscillator, which indicates that only one statistical moment is stable and, hence, the multivalued predictions of the equivalent linearisation analysis do not correspond to the real behaviour of the system. Other studies of non-linear softening systems driven by white noise have shown that the accuracy of the mean-square response predictions based on the equivalent linearisation method vary depending upon the nature and strength of the non-linearity of the system and the intensity of the excitation [9, lo]. Recent reported results based on extensive digital simulation studies on a hardening Duffing oscillator, however, have confirmed that the equivalent linearisation predictions correlate well with the observed behaviour of the oscillator provided that a suitable choice of the assumed response kurtosis is made [4]. In this study, the stability analysis presented by Iyengar [S, 121 is rederived and two restrictive assumptions are relaxed in order to take into account the influence of certain non-linear terms of the variational equation and to accommmodate for a non-Gaussian
Contributed NLM 28:2-B
by P. Spanos. 145
146
P. K. KOLIOPLILOS and R. S. LANGI.EY
assumption in the linearisation process. It is shown that for the case of relatively weak excitations the assumption of a quasi-harmonic (rather than Gaussian) response process is sufficient to improve the accuracy of the analysis while in cases of strong excitation the influence of the non-linear terms becomes significant and an inclusion of a further term in the variational equation becomes necessary. 7. EQUIVALENT
LINEARISATION
FORMULATION
Here the class of problems which involve a hardening Duffing oscillator with linear damping subjected to filtered white noise is considered. The equation of motion reads .;;_+ 2zs + ,&Y while the excitation
can be modelled as .. F + 25&C
+ i,.Y3) = F(t)
(1)
+ t0fL = W(t)
(2)
where the right-hand side of equation (2) represents a Gaussian white noise. The original non-linear system is replaced by an “equivalent” linear system. For a Duffing oscillator of the form of equation (1) the corresponding “equivalent” system can be written as .? + 2c(.;( + (0; - x = F(t) (3) where ~1)~ is defined so as to satisfy a certain criterion. The most commonly used criterion is the minimisation of the mean-square error of such an approximation, which leads to the following relation: (0: = (u;( 1 + ~(x”)/(.Y”), = w;( I + ‘r’ti_g:_) (4) where IC, is the unknown kurtosis of the response process. At this point, one has to make an assumption for the relationship between the fourth statistical moment of the response with its variance, which determines the required kurtosis. If the assumption is that the response is nearly Gaussian then X, = 3.0. In the case of a system with moderate or strong nonlinearities under narrow excitation, this assumption may lead to severe errors on the predictions of response moments and a more reasonable assumption is that the response is quasi-harmonic, leading to a value of kurtosis ti, = 1.50 141. If the degree of non-linearity is kept constant and the excitation bandwidth is varied, this value can be taken to be the lower bound for the kurtosis of the response of the system: as the input bandwidth increases, one would expect that the value of the kurtosis will also increase until it reaches an upper value which corresponds to a white-noise input. The value of this upper bound can be evaluated by first solving the FokkerrPlanckkKolmogorov (FPK) equation, which governs the probability density of the response, and then integrating the derived solution. For the Duffing oscillator described in equation (1) the solution of the associated FPK equation is of the form (5) where CJ~is the mean-square response of the system with ;’ = 0. The variation of the resulting response kurtosis (IC,) with the strength of non-linearity (~a;) is shown in Fig. 1. Employing the conventional input-output relations for the extended system of the shaping filter (2) followed by the linearised oscillator (3) an expression for the variance of the response is found as follows: r=
1+
((ti&S). jr(r
-
[Sr + EV2+ 4C6(&+ s)];
~1’)~ +
(6)
4r(s + 6)*(ar + 6~~))
where 1’= Qr/WJ,
b; = a/C&J,
C = v.&,
r = I + lix.y*a_;,
rl= i’ * CT&;:.
Numerical search of the roots of equation (6) defines regions in the parameter space (6, q, E, v) of multiple acceptable solutions [6, 111. The results of such an analysis in the (c, \I) plane, for a system with 6 = 0.08, three values of input intensity (q = 1.0, 0.5, 0.1). and
Improved
stability
analysis
147
2.9 -
. 2.6 -
Kx
2.7 . . .
2.6
.
l.
2.4 ' 0
02
0.4
0.6
0.8
Y.-z
Fig. 1. Kurtosis
of the response of a Duffing oscillator strength of non-linearity
subjected to white noise as a function 0 < y. ui I 1.
of the
.. V
”
1.5
/,,.,. ,,,
,.
”
I
0.02
0.04
0.06
0.06
I
I
0.1
0.12
0.14
E
Fig. 2. Regions
of multiple
response states as predicted via the equivalent (equation (6)) with K, = 1.5.
linearisation
technique
a variation of input bandwidth parameter (0 s E I 0.15), are shown in Fig. 2. It is interesting to note that for a given system (6) and input intensity (a) the resulting domain of non-unique solutions is drastically reduced with the increase of the excitation bandwidth parameter (E) and eventually is completely eroded.
3. STABILITY
ANALYSIS
Following [8, 121, a stability analysis of equation (1) using the solution of the linearised equation (3) is carried out. Setting as xL the solution of equation (3) and z the difference between the solution x of the original system (1) and xr, we can write the difference equation corresponding to equations (1) and (3) as z + 2ai + Cf${z + y(z3 + 3XLZ2+ 3zxZ)) = yw&a:
- x2).
(7)
P. K. KOLIOPIJLOSand R. S. LANGLFV
148
For stability analysis the right-hand the variational equation becomes
side, which is independent
: + 226 + to; (2 + ;,(?
of z, can be neglected
+ 3.Q~2 + 3Zx:);
and
= 0.
(8)
At this point Iyengar has neglected all the terms which contain second and higher powers of Z, leading to (9a) Z + 2al + WjJZ(1 + 37.x:)’ = 0. The assumption that the difference z is small, however, may not be valid in cases of strong non-linearities and/or in regions of multiple solutions. Here it is proposed to take into account the influence of the non-linear terms by performing a further linearisation with respect to 2. It turns out that only the term containing ? results in a linearised component with stability significance (see Appendix A), which is evaluated as follows: -3 - _* _1
A = &Of
The parameter A depends on the product of the unknown variance and kurtosis of 2 and can be assumed to be equal to a fraction of the corresponding product of the variance and kurtosis of the solution of the linearised equation. Including this contribution to the variational equation, we obtain 2 + 2cti- + C&(1 The solution
xL of equation
+ A + 31’xZ) = 0.
(3) to a narrowband
excitation
(9b)
can be written
in the form
.‘cr = X, sin(toL,rf - 0)
(10)
where the response amplitude XL and phase H are slowly varying process and the linear effective frequency of the response wLef is defined [S], as the mean zero-crossing rate of the linearised response process .xL and is equal to
Setting for convenience rLef = (C)Lef/uOand using the parameters following relation holds:
defined in equation
(6). the
It is important to note that this relation depends on the assumption made for the value of the response kurtosis and, as will be shown later, a more realistic assumption (e.g. K, = I .S as opposed to K, = 3.0) can cause an enormous improvement on stability predictions. This observation, in fact, explains the discrepancy between the earlier published theoretical studies and digital simulation results for cases of weak excitation levels. Performing the transformation z(t) = u(t) exp( - rt) (12) equation (9b) yields ii + {C, + C, cos 2(Orerf - O)lu = 0 (13) where Cl = 1.5x;o;, co = c* + t&l + A - 62). Finally,
expressing
u(r) and its time derivative
u(t) = Uccos(~L,rt) it can be proved
+ U,sin(~~~rf),
after some manipulation
as C(t) = - U, sin(w,,,t)
+ CT,~~~(oL,rf)
(14)
that
ti, = OLef{ - Ii, sin(w Left) cos(WLeft) + Us cosZ(cOLeft) - U,j, - u
sin(w,,,
t)
(15a)
OLef
u. = c9Lef ( u, -
. U, sin2(WLrft) + U, cos(w Lrf t) sin(o),,, 2)) + -Lcos(0)Lef fl)Lef
t).
(15b)
149
Improved stability analysis
Making use of equation oscillation, we get
(13) and averaging
equations
(15a) and (15b) over the period
of
where Cl1 = cz2 = c;;;(fp),
Cl*,21 = y
_ &!L(_
Le
Assuming
that the solutions
of equations
of the integral
Us= U,,exp
should i =
where vLef is defined in equation e
= 1
(17) Lef
(16) are of the form
UC = UC, exp the argument
) +f~Jy).
Lef
(18)
satisfy the relation
Tjk: - 4(VLef -
6J2
(19)
(11) and
l.wc
f
co=el +
1+A-d2 VLef
VLef
The final solution based on equations (12), (14) and (18) is of an exponential form and, hence, the criterion for almost sure asymptotic stability is that the expected value of the argument of the exponential remains negative as time tends to infinity, i.e.
462
c: - 4(VL,f -
6,)” AX,) dXL
(21)
where XZ
XL ML)=~exp
-R (
. )
It should be emphasised here that the present formulation differs from the one found in Iyengar in two points. Firstly, the contribution of the higher-order non-linear terms of variational equation (8) is recognised through the parameter A. Secondly, the effect of a non-Gaussian assumption for the kurtosis of the process is taken into account in equation (11). As will be shown in the next session, these changes cause a substantial improvement on the theoretical predictions which now compare favourably with the simulation results.
4. SIMULATION
STUDIES
To test the validity of the theoretical developments, a series of numerical simulations were carried out in a digital computer based on equations (1) and (2). It may be noted from equations (l), (2) and (6) that specification of the four non-dimensional parameters E, v, 6 and q fully determine the non-dimensional response xof/o, = ~(y/r#‘~. If the parameter y is additionally specified then the dimensional response x may be deduced. In the present study, three basic configurations were considered, in which the system parameters were kept constant, 6 = 0.08, y = 0.3, while the input intensity factor assumed three values q = 0.1,0.5, 1.0. For each configuration parametric studies were performed varying the excitation bandwidth parameter E and the central normalised excitation frequency v. The equations of motion were solved using 4th order Runge-Kutta routine with a time step equal to l/80 of the mean period of excitation. The total length of each record was equal to 2000 forcing cycles, from which the global as well as the local values of variance were estimated and the probability histogram of the response amplitude was constructed. Typical graphs of the response amplitude and the corresponding probability histogram are shown in Figs 3 and 4. For clarity, only one point per mean excitation period is plotted which corresponds to the maximum observed response value. It is observed that for small excitation bandwidth (E = 0.01) the two response levels around which the system fluctuates are easily distinguished, resulting in a probabilty histogram with two sharp peaks. As the input bandwidth increases (E = 0.05), the distinction of the two states is less apparent.
150
P. K. KOLIOPIJLOS and
R. S. LANGLEY
I
0.25
1 0.2
0.15
I i 01
0.05
1
25
50
75
Fig. 3. Time history
100
125
150
115
200 .I0
0
2.7
(left) and probability histogram (right) of the response amplitude with parameters ri = 0.08, q = 1.0, c = 0.01. \’ = 1.1.
0.12
3.6
4.5
for a system
r
0.1
0.08
0.06
0.04
0.02
I
25
50
75
100
125
150
115
200 UlP’
Fig. 4. Time hisory (left) and probability histogram (right) of the response amplitude with parameters ii = 0.08. q = 0.5, E = 0.05, v = 1.9.
for a system
5.4
151
Improved stability analysis
When two distinct states are exhibited by the response, it is possible to measure the kurtosis and variance of each, as well as the global values of these quantities. Such results are shown in Table 1 for the range of parameters covered by the simulation studies. Results are shown only for those cases in which two distinct response states could be clearly identifies in addition to the local (low and high) and global values of variance a2 and kurtosis K, the is also shown in the table-this represents the relative amount of the time ratio TkvlThigh spent in the two local states. It is clear from the table that the kurtosis of the high state is much nearer to the quasi-harmonic value of 1.5 than to the Gaussian value of 3, while the kurtosis of the low state is quite near to 3. What is initially surprising is that the global kurtosis is often much higher than that of either state-there is, in fact, a simple explanation
Table 1. v
lj
1.4
1.5
1.6
1.7
1.66
2.46
5.09.
3.45
2.60
3.05
=O.l
6 =
0.01 2.47
1.88
1.61
1.66
1.03
21.2
1.75
0.40
0.52
0.30
3.00
3.05
2.36
3.27
1.8
1.9
2.0
2.1
1.73 2.29
4.32
6.68
3.85
3.33
4.14
2.2
2.3
2.4
4.80
0.24
2.32 9 =O.l
1.66
E = 0.05
1.22 2.08
1.66
0.88
0.72 2.79 1.64 q = 0.5
1.61
1.62
1.62
E = 0.01
0.17
4.14
24.3
5.14
1.78
0.80
1.86
0.68
0.55
5.70
6.23
6.95
1.94
2.44
3.61
2.14
2.56
2.81
4.96
1.84 ‘1 = 0.5
1.66
1.66
1.67
E = 0.05
0.43
0.97
5.41
4.33
3.60
1.85
1.69
1.18
1.06
5.44
5.94
6.13
3.93
1.75
0.51
3.76
0.88
1.84
4.15
6.56
8.93
2.71
2.97
3.23
3.30
q = 1.0
1.62
1.64
1.71
1.65
E = 0.01
0.21
2.80
21.0
15.4
6.45
2.63
0.97
1.04
1.29
0.70
0.63
0.50
7.50
8.00
7.91
9.42
1.97
2.33
3.35
4.18
2.24
2.51
2.70
2.78
6.23
1.82 q= 1.0
1.70
1.67
1.70
1.75
E = 0.05
0.29
0.81
2.42
8.26
5.81
5.00
3.21
1.86
1.56
1.59
1.25
1.15
7.02
7.74
7.98
7.66
5.97
0.47
4.00
1.09
Fig. 5. Variance of the rcaponse for a s!,stcm wth parameters 6 = 0.0X. q 0.1. J. = 0.01 (MI) and ,i = 0.08, q = 0.1. c = 0.05 (right). Theorctlcal predictions based on equivalent linearisation with h-, = 1.5 (sun) and with K, 7 2.572 (wh.n) are compared with simulation results (sim). Instability regions predicted by equation (03) with h, = 1.5 (dark grey) and wth h, 2.575 (light grey).
n
i ____
1.5
25
2
\’
Fig. 6. Varlancc of the response (‘or a system wth parameters 0 = 0.0X. I/ == 0.5. i. = O.OI (lel’t) and d = 0.0X, q =- 0.5. z:: 0.05 (right). Thcorctical predictions hascd on equivalent linearisatlon with K, = I.5 (sin) and with I,, = 2.575 (v.h.n) arc compared with simulation rrsults (sim). Instabrlitj regons predicted by equation (%I) with K, : I.5 (light grcy) and hy equation (9h) with I,, = 1.5. / 5”,, (dark grey).
for this, as described in Appendix R, where it is shown that the global kurtosis is not a particularly meaningful quantity for the cases under consideration. The simulation results suggest that the quasi-harmonic value of K, = 1.50 provides a close lower bound on the kurtosis of the high state thigh. It is this kurtosis, rather than the one corresponding to the low state (K,“,, ) or to the global response kurtosis K~,~~,,~,, which is appropriate to the stability analysis of Section 3, as the resulting stability boundary affects the high state rather than the low one [8, 121. An upper bound on h-hiahmay be estimated by considering the limiting case of white-noise excitation. For the range of parameters con-
Improved
stability
analysis
a
= . .* t i a * * is * *
P
2 Ox _-._._~.__
6
-.__._._.___-._,_"_,_,
ox2 6
:
.-~-~~~~~+.-~~~~_-.~__._.~._~~.~.~~._ 3. -
lo.’ E-L 151”) E-L ,Wh “1 0 *
**
sim .
”
"
1.5
2
2.5
3
"
1.5
2
2.5
3
Fig. 7. Variance of the response for a system with parameters 6 = 0.08, n = 1.O, c = 0.01 (left) and 6 = 0.08, n = 1.0, c = 0.05 (right). Theoretical predictions based on equivalent linearisation with K, = 1.5 (sin) and with K, = 2.575 (whn) are compared with simulation results (sim). Instability regions predicted by equation (9b) with K, = 1.5, I. = 5% (light grey) and with K, = 1.5. i = 10% (dark grey).
sidered here, it has been found that the factor 70; which appears in Fig. 1 is generally less than 0.3 for the range of Q over which multiple states are obtained leading to values of kurtosis between 3.0 and 2.575. Noting that the value of 3.0 yields very loose stability bounds [12], it was decided that the value of 2.575 should be used to provide a suitable approximate upper bound on lChighin the present case. The recorded local values of response variance are shown in Fig. (5)(7) and are compared with two sets of predictions of the equivalent linearisation method (equation (6)) for the two cases K, = 1.50 and K, = 2.575. It is clear that the predictions based on the sinusoidal assumption serve as an upper bound and are superior to those based on the white-noise input assumption serving as a lower bound. As the bandwidth increases, however, the simulation results tend to deviate gradually from upper bound and fall between the two bounds. It is expected that for wideband excitations the lower bound will become more accurate except that in these cases no clear distinction between two response states is possible. Furthermore, results of the stability analysis described in Section 3 are presented. For the case of low intensity force (‘I = 0.1) it is shown that the non-linear terms of equation (8) are of a lesser importance and, hence, the use of equation (9a) is sufficient provided that the value of K, = 1.50 is used in equation (11). If the value of K, = 2.575 is used then the conclusion of the stability analysis (in accordance with Iyengar) is that all the high-amplitude response states are unstable; this is clearly in contrast with the simulation results which indicate that the system does indeed experience high-amplitude oscillations. The use of the more accurate value or the kurtosis extends the stability region and includes all the observed values. The cases of high-intensity force reveal that equation (9a) is not sufficient and one has to implement equation (9b), which takes into account the non-linear terms of equation (8). Here the kurtosis is set to the value of K, = 1.50 while the parameter A is set to be of the form A = iti,a:. The unknown coefficient 1. is assumed to be equal to 5% for the case v] = 0.5 and 10% for the case 9 = 1.0. These values are, in fact, consistent with the results of a study of the errors introduced by the equivalent linearisation technique in the limiting case of a white-noise excitation [13], where it is shown that the error in the variance is proportional to the excitation intensity and tends asymptotically (as the excitation intensity increases) to an upper limit of 17% of the value obtained via linearisation. It is clear from these two figures that the use of equation (9b) leads to a substantial improvement over the
P. K. KOLIOP~L~S and R. S. LAPGL~\,
154
predictions based on equation improved value for K, is used.
(9a), which
yields
results
of poor
accuracy
even
if the
5. CONCLUSIONS
In this study, the equivalent linearisation approach is implemented for the study ofjumps in the response of a Duffing oscillator subjected to filtered white-noise excitations. Parametric studies based on digital simulations have confirmed that under certain conditions the system can indeed jump between competing response states in a persistent way, with serious effects on the resulting probability functions. An improved version of a stochastic stability analysis is also presented and the main causes of errors in previously reported predictions are identified. The results based on the proposed analysis correlate well with the digital simulation studies.
REFERENCES I. K. Richard and G. V. Anand. Non-linear resonance in strings under narrow band random excitation. Part I. Planar response and stability. J. Sountl I’&. 86, X5-98 (I 983). 2. H. G. Davies and D. Nandlall, Phase plane for narrow band random excitation of a Dufing oscillator. J. Sound C’ih. 104, 277-283 (1986). 3. T. Fang and E. H. Dowel]. Numerical simulations of jump phenomena in stable Duffing systems. Int. J. Non-Linear Mech. 22, 267-214 (1987). 4. J. B. Roberts, Multiple solutions generated by statistical lineariaation and their physical significance. Int. J. Non-Linear Mrch. 26, 945-959 (1991). 5. W. C. Lennox and Y. C. Kuak. Narrow-band excitation of a non-linear oscillation. ASME J. oppl. Mrc,h. 43. 34&344 ( 1976). 6. M. F. Dimentberg. Sruristical Dynamic,\ of ,Yon-linrcw cmd Time- b’uriyny Swmrs. Research Studies Press. Somerset, England (1988). 7. P. K. Koliopulos and S. R. Bishop, Quasi-harmonic analysis of the behaviour of a hardening Duffing oscillator subjected to filtered white noise, Non-linear Dynumics (submitted) 8. R. N. lyengar, Stochastic response and stability of the Duffing oscillator under narrowband excitation. J. Sound Vih. 126. 255-263 (1986). 9. R. S. Langley, An investigation of multiple solutions yielded by the equivalent linearisation method. J. Sound Vih. 127. 271 281 (1988). IO. F. G. Fan and D. Ahmadi, On loss of accuracy and non-uniqueness of solutions generated by equivalent linearization and cumulant-neglect methods. J. Sound Vib. 137, 385401 (1990). I I. P. K. Koliopulos, S. R. Bishop and G. D. Stefanou, Response statistics of non-linear systems under variations of excitation bandwidth, in C’omputalional Stochastic Mechanics (Edited by P. D. Spanos and C. A. Brebbia), pp. 335.-348. Elsevier, London (I 99 1). 12. R. N. Iyengar, Multiple response moments and stochastic stability of a non-linear system. in Sfochos~rc Structurul Dynamics (Edited by S. T. Ariaratnam. G. 1. Schueller and 1. Elishakoff). pp. 159-172. Elsevier. London ( 1988). 13. W. F. Wu and Y. K. Lin. Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations. 1,~r. .I. Non-Linetrr Mech. 19. 349-362 (1984).
APPENDIX
A
Here we will prove that due to symmetry of equation (8) the linearisation (with respect of 3) of the second non-linear term appearing in equation (8) leads to a value equals to zero. The non-linear term under consideration is equal to 3u,? and, hence, its linearised version should be of the form (JU,:. The minimisation of the mean-square error determines the parameter i> as (\-;?) ,’ = __ (Al) ;_+> If we define two new variables Hence,
f = - 2 and ?, = ~ Y,,. it can be easily shown /I(_? u,.) = p(l. c,
The fact that the joint probability density function of Y, and z leads to the following
)=
p(
is an even function relations:
that they too satisfy equaton
2. ~ Y,).
(8). IA?)
while the numerator
of equation
(Al) IS an odd
tA3) Substituting
this result into equation
(Al) yields a value for the parameter
[J equal to zero.
Improved
stability
APPENDIX
analysis
155
B
Assume that a random process x(t) of total duration T,,,,, exhibiting a global variance u,&., and a fourth statistical moment F~,,~,~,, is a two-state process switching between a low-state process xL(t) with statistical Setting T,,,, the time parameters & , h. Ioyand a high-state process x”(t) with statistical parameters CT,&, Pi,+. spent in low state, and Thigh, the time spent in high state, the following relations hold:
(B2)
x’(t)dt
= T ,;,., { jo”“* x;(t)dt
+ j;
\-:,(r)dr}
(B4)
It is important to realise that under certain conditions, equation (B5) results in a value of global kurtosis much higher than the kurtosis of individual states. For example, consider the limiting case in which the low-state corresponds to extremely small statistical moments (i.e. CT& z 0, p(4 ,o~ z 0). In this case the global statistics are related to the local (high-state) statistics as follows:
Due to the fact that the global kurtosis is proportional to the inverse of the fraction of time spent in the higher state, the resulting global kurtosis may realise extremely high values when this fraction is small. This behaviour is reflected in some entries of Table 1, where the low state dominates and the resulting global variance is close to the low-state variance. The influence of the (rare) jumps in the high state on the fourth statistical moment, however, is much more significant, resulting in high values of the global kurtosis.
Vol. 28. No. 2, pp. 145 155, 1993
002&7462/93 36.00 + .JO T‘I 1993 Pergmon Press Lid
IMPROVED STABILITY ANALYSIS OF THE RESPONSE OF A DUFFING OSCILLATOR UNDER FILTERED WHITE NOISE P. K. KOLIOPULOS* and R. S. LANGLEY? *Department of Civil Engineering, ‘Department
of Aeronautics
University College, Cower Street, London WCIE 6BT, U.K. and and Astronautics, University of Southampton, Highheld, Southampton SO9 5NH, U.K. (Recked
3 September
1992)
Abstract The applicability of the equivalent linearisation method for the analysis of non-linear oscillators subjected to narrowband random excitation is discussed. It is shown that when jumps between competing response states occur, the choice of an appropriate value for the kurtosis of the response process is crucial for a reliable estimation of local statistical moments and for a more accurate stochastic stability analysis. Furthermore, a modification is proposed which takes into account the influence of certain non-linear terms in the variational equation for the cases of strong non-linearities. The theoretical predictions correlate well with the results based on extensive digital simulations.
1. INTRODUCTION
It is well known that if an oscillator with a hardening non-linear stiffness is subjected to sinusoidal excitation, it is possible for the oscillator response to exhibit the phenomenon of sharp jumps in amplitude. In recent years it has been realised that the same jumping phenomenon can occur if an oscillator with non-linear stiffness is subjected to narrowband random excitation, provided that the bandwidth of the excitation is sufficiently small [la]. The occurrence and persistence of jumps in the response process is reflected in non-unique temporal statistical moments and in multipeak probability density functions of the amplitude of the response. A limited number of approximate theoretical methods have been developed for the study of such systems with the aim of predicting regions (in the parameter space) of possible multiple response states, the estimation of the values of temporal response statistics and the influence of jumps on the response probability density. One method is the so-called quasi-harmonic (or quasi-static) method, which has been successfully implemented to tackle all the above-mentioned problems [S-7]. This simple, yet powerful, method suffers, however, from the fact that it is valid only when the input bandwidth is extremely small. Another technique is the popular equivalent linearisation method, which has been used for the first two problems [2,4,6, 8-l I]. There seems to be a disagreement, however, with regard to the significance of the equivalent linearisation predictions. In a series of publications lyengar 18, 123 has presented a stochastic stability analysis of the equation of motion of a hardening-type oscillator, which indicates that only one statistical moment is stable and, hence, the multivalued predictions of the equivalent linearisation analysis do not correspond to the real behaviour of the system. Other studies of non-linear softening systems driven by white noise have shown that the accuracy of the mean-square response predictions based on the equivalent linearisation method vary depending upon the nature and strength of the non-linearity of the system and the intensity of the excitation [9, lo]. Recent reported results based on extensive digital simulation studies on a hardening Duffing oscillator, however, have confirmed that the equivalent linearisation predictions correlate well with the observed behaviour of the oscillator provided that a suitable choice of the assumed response kurtosis is made [4]. In this study, the stability analysis presented by Iyengar [S, 121 is rederived and two restrictive assumptions are relaxed in order to take into account the influence of certain non-linear terms of the variational equation and to accommmodate for a non-Gaussian
Contributed NLM 28:2-B
by P. Spanos. 145
146
P. K. KOLIOPLILOS and R. S. LANGI.EY
assumption in the linearisation process. It is shown that for the case of relatively weak excitations the assumption of a quasi-harmonic (rather than Gaussian) response process is sufficient to improve the accuracy of the analysis while in cases of strong excitation the influence of the non-linear terms becomes significant and an inclusion of a further term in the variational equation becomes necessary. 7. EQUIVALENT
LINEARISATION
FORMULATION
Here the class of problems which involve a hardening Duffing oscillator with linear damping subjected to filtered white noise is considered. The equation of motion reads .;;_+ 2zs + ,&Y while the excitation
can be modelled as .. F + 25&C
+ i,.Y3) = F(t)
(1)
+ t0fL = W(t)
(2)
where the right-hand side of equation (2) represents a Gaussian white noise. The original non-linear system is replaced by an “equivalent” linear system. For a Duffing oscillator of the form of equation (1) the corresponding “equivalent” system can be written as .? + 2c(.;( + (0; - x = F(t) (3) where ~1)~ is defined so as to satisfy a certain criterion. The most commonly used criterion is the minimisation of the mean-square error of such an approximation, which leads to the following relation: (0: = (u;( 1 + ~(x”)/(.Y”), = w;( I + ‘r’ti_g:_) (4) where IC, is the unknown kurtosis of the response process. At this point, one has to make an assumption for the relationship between the fourth statistical moment of the response with its variance, which determines the required kurtosis. If the assumption is that the response is nearly Gaussian then X, = 3.0. In the case of a system with moderate or strong nonlinearities under narrow excitation, this assumption may lead to severe errors on the predictions of response moments and a more reasonable assumption is that the response is quasi-harmonic, leading to a value of kurtosis ti, = 1.50 141. If the degree of non-linearity is kept constant and the excitation bandwidth is varied, this value can be taken to be the lower bound for the kurtosis of the response of the system: as the input bandwidth increases, one would expect that the value of the kurtosis will also increase until it reaches an upper value which corresponds to a white-noise input. The value of this upper bound can be evaluated by first solving the FokkerrPlanckkKolmogorov (FPK) equation, which governs the probability density of the response, and then integrating the derived solution. For the Duffing oscillator described in equation (1) the solution of the associated FPK equation is of the form (5) where CJ~is the mean-square response of the system with ;’ = 0. The variation of the resulting response kurtosis (IC,) with the strength of non-linearity (~a;) is shown in Fig. 1. Employing the conventional input-output relations for the extended system of the shaping filter (2) followed by the linearised oscillator (3) an expression for the variance of the response is found as follows: r=
1+
((ti&S). jr(r
-
[Sr + EV2+ 4C6(&+ s)];
~1’)~ +
(6)
4r(s + 6)*(ar + 6~~))
where 1’= Qr/WJ,
b; = a/C&J,
C = v.&,
r = I + lix.y*a_;,
rl= i’ * CT&;:.
Numerical search of the roots of equation (6) defines regions in the parameter space (6, q, E, v) of multiple acceptable solutions [6, 111. The results of such an analysis in the (c, \I) plane, for a system with 6 = 0.08, three values of input intensity (q = 1.0, 0.5, 0.1). and
Improved
stability
analysis
147
2.9 -
. 2.6 -
Kx
2.7 . . .
2.6
.
l.
2.4 ' 0
02
0.4
0.6
0.8
Y.-z
Fig. 1. Kurtosis
of the response of a Duffing oscillator strength of non-linearity
subjected to white noise as a function 0 < y. ui I 1.
of the
.. V
”
1.5
/,,.,. ,,,
,.
”
I
0.02
0.04
0.06
0.06
I
I
0.1
0.12
0.14
E
Fig. 2. Regions
of multiple
response states as predicted via the equivalent (equation (6)) with K, = 1.5.
linearisation
technique
a variation of input bandwidth parameter (0 s E I 0.15), are shown in Fig. 2. It is interesting to note that for a given system (6) and input intensity (a) the resulting domain of non-unique solutions is drastically reduced with the increase of the excitation bandwidth parameter (E) and eventually is completely eroded.
3. STABILITY
ANALYSIS
Following [8, 121, a stability analysis of equation (1) using the solution of the linearised equation (3) is carried out. Setting as xL the solution of equation (3) and z the difference between the solution x of the original system (1) and xr, we can write the difference equation corresponding to equations (1) and (3) as z + 2ai + Cf${z + y(z3 + 3XLZ2+ 3zxZ)) = yw&a:
- x2).
(7)
P. K. KOLIOPIJLOSand R. S. LANGLFV
148
For stability analysis the right-hand the variational equation becomes
side, which is independent
: + 226 + to; (2 + ;,(?
of z, can be neglected
+ 3.Q~2 + 3Zx:);
and
= 0.
(8)
At this point Iyengar has neglected all the terms which contain second and higher powers of Z, leading to (9a) Z + 2al + WjJZ(1 + 37.x:)’ = 0. The assumption that the difference z is small, however, may not be valid in cases of strong non-linearities and/or in regions of multiple solutions. Here it is proposed to take into account the influence of the non-linear terms by performing a further linearisation with respect to 2. It turns out that only the term containing ? results in a linearised component with stability significance (see Appendix A), which is evaluated as follows: -3 - _* _1
A = &Of
The parameter A depends on the product of the unknown variance and kurtosis of 2 and can be assumed to be equal to a fraction of the corresponding product of the variance and kurtosis of the solution of the linearised equation. Including this contribution to the variational equation, we obtain 2 + 2cti- + C&(1 The solution
xL of equation
+ A + 31’xZ) = 0.
(3) to a narrowband
excitation
(9b)
can be written
in the form
.‘cr = X, sin(toL,rf - 0)
(10)
where the response amplitude XL and phase H are slowly varying process and the linear effective frequency of the response wLef is defined [S], as the mean zero-crossing rate of the linearised response process .xL and is equal to
Setting for convenience rLef = (C)Lef/uOand using the parameters following relation holds:
defined in equation
(6). the
It is important to note that this relation depends on the assumption made for the value of the response kurtosis and, as will be shown later, a more realistic assumption (e.g. K, = I .S as opposed to K, = 3.0) can cause an enormous improvement on stability predictions. This observation, in fact, explains the discrepancy between the earlier published theoretical studies and digital simulation results for cases of weak excitation levels. Performing the transformation z(t) = u(t) exp( - rt) (12) equation (9b) yields ii + {C, + C, cos 2(Orerf - O)lu = 0 (13) where Cl = 1.5x;o;, co = c* + t&l + A - 62). Finally,
expressing
u(r) and its time derivative
u(t) = Uccos(~L,rt) it can be proved
+ U,sin(~~~rf),
after some manipulation
as C(t) = - U, sin(w,,,t)
+ CT,~~~(oL,rf)
(14)
that
ti, = OLef{ - Ii, sin(w Left) cos(WLeft) + Us cosZ(cOLeft) - U,j, - u
sin(w,,,
t)
(15a)
OLef
u. = c9Lef ( u, -
. U, sin2(WLrft) + U, cos(w Lrf t) sin(o),,, 2)) + -Lcos(0)Lef fl)Lef
t).
(15b)
149
Improved stability analysis
Making use of equation oscillation, we get
(13) and averaging
equations
(15a) and (15b) over the period
of
where Cl1 = cz2 = c;;;(fp),
Cl*,21 = y
_ &!L(_
Le
Assuming
that the solutions
of equations
of the integral
Us= U,,exp
should i =
where vLef is defined in equation e
= 1
(17) Lef
(16) are of the form
UC = UC, exp the argument
) +f~Jy).
Lef
(18)
satisfy the relation
Tjk: - 4(VLef -
6J2
(19)
(11) and
l.wc
f
co=el +
1+A-d2 VLef
VLef
The final solution based on equations (12), (14) and (18) is of an exponential form and, hence, the criterion for almost sure asymptotic stability is that the expected value of the argument of the exponential remains negative as time tends to infinity, i.e.
462
c: - 4(VL,f -
6,)” AX,) dXL
(21)
where XZ
XL ML)=~exp
-R (
. )
It should be emphasised here that the present formulation differs from the one found in Iyengar in two points. Firstly, the contribution of the higher-order non-linear terms of variational equation (8) is recognised through the parameter A. Secondly, the effect of a non-Gaussian assumption for the kurtosis of the process is taken into account in equation (11). As will be shown in the next session, these changes cause a substantial improvement on the theoretical predictions which now compare favourably with the simulation results.
4. SIMULATION
STUDIES
To test the validity of the theoretical developments, a series of numerical simulations were carried out in a digital computer based on equations (1) and (2). It may be noted from equations (l), (2) and (6) that specification of the four non-dimensional parameters E, v, 6 and q fully determine the non-dimensional response xof/o, = ~(y/r#‘~. If the parameter y is additionally specified then the dimensional response x may be deduced. In the present study, three basic configurations were considered, in which the system parameters were kept constant, 6 = 0.08, y = 0.3, while the input intensity factor assumed three values q = 0.1,0.5, 1.0. For each configuration parametric studies were performed varying the excitation bandwidth parameter E and the central normalised excitation frequency v. The equations of motion were solved using 4th order Runge-Kutta routine with a time step equal to l/80 of the mean period of excitation. The total length of each record was equal to 2000 forcing cycles, from which the global as well as the local values of variance were estimated and the probability histogram of the response amplitude was constructed. Typical graphs of the response amplitude and the corresponding probability histogram are shown in Figs 3 and 4. For clarity, only one point per mean excitation period is plotted which corresponds to the maximum observed response value. It is observed that for small excitation bandwidth (E = 0.01) the two response levels around which the system fluctuates are easily distinguished, resulting in a probabilty histogram with two sharp peaks. As the input bandwidth increases (E = 0.05), the distinction of the two states is less apparent.
150
P. K. KOLIOPIJLOS and
R. S. LANGLEY
I
0.25
1 0.2
0.15
I i 01
0.05
1
25
50
75
Fig. 3. Time history
100
125
150
115
200 .I0
0
2.7
(left) and probability histogram (right) of the response amplitude with parameters ri = 0.08, q = 1.0, c = 0.01. \’ = 1.1.
0.12
3.6
4.5
for a system
r
0.1
0.08
0.06
0.04
0.02
I
25
50
75
100
125
150
115
200 UlP’
Fig. 4. Time hisory (left) and probability histogram (right) of the response amplitude with parameters ii = 0.08. q = 0.5, E = 0.05, v = 1.9.
for a system
5.4
151
Improved stability analysis
When two distinct states are exhibited by the response, it is possible to measure the kurtosis and variance of each, as well as the global values of these quantities. Such results are shown in Table 1 for the range of parameters covered by the simulation studies. Results are shown only for those cases in which two distinct response states could be clearly identifies in addition to the local (low and high) and global values of variance a2 and kurtosis K, the is also shown in the table-this represents the relative amount of the time ratio TkvlThigh spent in the two local states. It is clear from the table that the kurtosis of the high state is much nearer to the quasi-harmonic value of 1.5 than to the Gaussian value of 3, while the kurtosis of the low state is quite near to 3. What is initially surprising is that the global kurtosis is often much higher than that of either state-there is, in fact, a simple explanation
Table 1. v
lj
1.4
1.5
1.6
1.7
1.66
2.46
5.09.
3.45
2.60
3.05
=O.l
6 =
0.01 2.47
1.88
1.61
1.66
1.03
21.2
1.75
0.40
0.52
0.30
3.00
3.05
2.36
3.27
1.8
1.9
2.0
2.1
1.73 2.29
4.32
6.68
3.85
3.33
4.14
2.2
2.3
2.4
4.80
0.24
2.32 9 =O.l
1.66
E = 0.05
1.22 2.08
1.66
0.88
0.72 2.79 1.64 q = 0.5
1.61
1.62
1.62
E = 0.01
0.17
4.14
24.3
5.14
1.78
0.80
1.86
0.68
0.55
5.70
6.23
6.95
1.94
2.44
3.61
2.14
2.56
2.81
4.96
1.84 ‘1 = 0.5
1.66
1.66
1.67
E = 0.05
0.43
0.97
5.41
4.33
3.60
1.85
1.69
1.18
1.06
5.44
5.94
6.13
3.93
1.75
0.51
3.76
0.88
1.84
4.15
6.56
8.93
2.71
2.97
3.23
3.30
q = 1.0
1.62
1.64
1.71
1.65
E = 0.01
0.21
2.80
21.0
15.4
6.45
2.63
0.97
1.04
1.29
0.70
0.63
0.50
7.50
8.00
7.91
9.42
1.97
2.33
3.35
4.18
2.24
2.51
2.70
2.78
6.23
1.82 q= 1.0
1.70
1.67
1.70
1.75
E = 0.05
0.29
0.81
2.42
8.26
5.81
5.00
3.21
1.86
1.56
1.59
1.25
1.15
7.02
7.74
7.98
7.66
5.97
0.47
4.00
1.09
Fig. 5. Variance of the rcaponse for a s!,stcm wth parameters 6 = 0.0X. q 0.1. J. = 0.01 (MI) and ,i = 0.08, q = 0.1. c = 0.05 (right). Theorctlcal predictions based on equivalent linearisation with h-, = 1.5 (sun) and with K, 7 2.572 (wh.n) are compared with simulation results (sim). Instability regions predicted by equation (03) with h, = 1.5 (dark grey) and wth h, 2.575 (light grey).
n
i ____
1.5
25
2
\’
Fig. 6. Varlancc of the response (‘or a system wth parameters 0 = 0.0X. I/ == 0.5. i. = O.OI (lel’t) and d = 0.0X, q =- 0.5. z:: 0.05 (right). Thcorctical predictions hascd on equivalent linearisatlon with K, = I.5 (sin) and with I,, = 2.575 (v.h.n) arc compared with simulation rrsults (sim). Instabrlitj regons predicted by equation (%I) with K, : I.5 (light grcy) and hy equation (9h) with I,, = 1.5. / 5”,, (dark grey).
for this, as described in Appendix R, where it is shown that the global kurtosis is not a particularly meaningful quantity for the cases under consideration. The simulation results suggest that the quasi-harmonic value of K, = 1.50 provides a close lower bound on the kurtosis of the high state thigh. It is this kurtosis, rather than the one corresponding to the low state (K,“,, ) or to the global response kurtosis K~,~~,,~,, which is appropriate to the stability analysis of Section 3, as the resulting stability boundary affects the high state rather than the low one [8, 121. An upper bound on h-hiahmay be estimated by considering the limiting case of white-noise excitation. For the range of parameters con-
Improved
stability
analysis
a
= . .* t i a * * is * *
P
2 Ox _-._._~.__
6
-.__._._.___-._,_"_,_,
ox2 6
:
.-~-~~~~~+.-~~~~_-.~__._.~._~~.~.~~._ 3. -
lo.’ E-L 151”) E-L ,Wh “1 0 *
**
sim .
”
"
1.5
2
2.5
3
"
1.5
2
2.5
3
Fig. 7. Variance of the response for a system with parameters 6 = 0.08, n = 1.O, c = 0.01 (left) and 6 = 0.08, n = 1.0, c = 0.05 (right). Theoretical predictions based on equivalent linearisation with K, = 1.5 (sin) and with K, = 2.575 (whn) are compared with simulation results (sim). Instability regions predicted by equation (9b) with K, = 1.5, I. = 5% (light grey) and with K, = 1.5. i = 10% (dark grey).
sidered here, it has been found that the factor 70; which appears in Fig. 1 is generally less than 0.3 for the range of Q over which multiple states are obtained leading to values of kurtosis between 3.0 and 2.575. Noting that the value of 3.0 yields very loose stability bounds [12], it was decided that the value of 2.575 should be used to provide a suitable approximate upper bound on lChighin the present case. The recorded local values of response variance are shown in Fig. (5)(7) and are compared with two sets of predictions of the equivalent linearisation method (equation (6)) for the two cases K, = 1.50 and K, = 2.575. It is clear that the predictions based on the sinusoidal assumption serve as an upper bound and are superior to those based on the white-noise input assumption serving as a lower bound. As the bandwidth increases, however, the simulation results tend to deviate gradually from upper bound and fall between the two bounds. It is expected that for wideband excitations the lower bound will become more accurate except that in these cases no clear distinction between two response states is possible. Furthermore, results of the stability analysis described in Section 3 are presented. For the case of low intensity force (‘I = 0.1) it is shown that the non-linear terms of equation (8) are of a lesser importance and, hence, the use of equation (9a) is sufficient provided that the value of K, = 1.50 is used in equation (11). If the value of K, = 2.575 is used then the conclusion of the stability analysis (in accordance with Iyengar) is that all the high-amplitude response states are unstable; this is clearly in contrast with the simulation results which indicate that the system does indeed experience high-amplitude oscillations. The use of the more accurate value or the kurtosis extends the stability region and includes all the observed values. The cases of high-intensity force reveal that equation (9a) is not sufficient and one has to implement equation (9b), which takes into account the non-linear terms of equation (8). Here the kurtosis is set to the value of K, = 1.50 while the parameter A is set to be of the form A = iti,a:. The unknown coefficient 1. is assumed to be equal to 5% for the case v] = 0.5 and 10% for the case 9 = 1.0. These values are, in fact, consistent with the results of a study of the errors introduced by the equivalent linearisation technique in the limiting case of a white-noise excitation [13], where it is shown that the error in the variance is proportional to the excitation intensity and tends asymptotically (as the excitation intensity increases) to an upper limit of 17% of the value obtained via linearisation. It is clear from these two figures that the use of equation (9b) leads to a substantial improvement over the
P. K. KOLIOP~L~S and R. S. LAPGL~\,
154
predictions based on equation improved value for K, is used.
(9a), which
yields
results
of poor
accuracy
even
if the
5. CONCLUSIONS
In this study, the equivalent linearisation approach is implemented for the study ofjumps in the response of a Duffing oscillator subjected to filtered white-noise excitations. Parametric studies based on digital simulations have confirmed that under certain conditions the system can indeed jump between competing response states in a persistent way, with serious effects on the resulting probability functions. An improved version of a stochastic stability analysis is also presented and the main causes of errors in previously reported predictions are identified. The results based on the proposed analysis correlate well with the digital simulation studies.
REFERENCES I. K. Richard and G. V. Anand. Non-linear resonance in strings under narrow band random excitation. Part I. Planar response and stability. J. Sountl I’&. 86, X5-98 (I 983). 2. H. G. Davies and D. Nandlall, Phase plane for narrow band random excitation of a Dufing oscillator. J. Sound C’ih. 104, 277-283 (1986). 3. T. Fang and E. H. Dowel]. Numerical simulations of jump phenomena in stable Duffing systems. Int. J. Non-Linear Mech. 22, 267-214 (1987). 4. J. B. Roberts, Multiple solutions generated by statistical lineariaation and their physical significance. Int. J. Non-Linear Mrch. 26, 945-959 (1991). 5. W. C. Lennox and Y. C. Kuak. Narrow-band excitation of a non-linear oscillation. ASME J. oppl. Mrc,h. 43. 34&344 ( 1976). 6. M. F. Dimentberg. Sruristical Dynamic,\ of ,Yon-linrcw cmd Time- b’uriyny Swmrs. Research Studies Press. Somerset, England (1988). 7. P. K. Koliopulos and S. R. Bishop, Quasi-harmonic analysis of the behaviour of a hardening Duffing oscillator subjected to filtered white noise, Non-linear Dynumics (submitted) 8. R. N. lyengar, Stochastic response and stability of the Duffing oscillator under narrowband excitation. J. Sound Vih. 126. 255-263 (1986). 9. R. S. Langley, An investigation of multiple solutions yielded by the equivalent linearisation method. J. Sound Vih. 127. 271 281 (1988). IO. F. G. Fan and D. Ahmadi, On loss of accuracy and non-uniqueness of solutions generated by equivalent linearization and cumulant-neglect methods. J. Sound Vib. 137, 385401 (1990). I I. P. K. Koliopulos, S. R. Bishop and G. D. Stefanou, Response statistics of non-linear systems under variations of excitation bandwidth, in C’omputalional Stochastic Mechanics (Edited by P. D. Spanos and C. A. Brebbia), pp. 335.-348. Elsevier, London (I 99 1). 12. R. N. Iyengar, Multiple response moments and stochastic stability of a non-linear system. in Sfochos~rc Structurul Dynamics (Edited by S. T. Ariaratnam. G. 1. Schueller and 1. Elishakoff). pp. 159-172. Elsevier. London ( 1988). 13. W. F. Wu and Y. K. Lin. Cumulant-neglect closure for non-linear oscillators under random parametric and external excitations. 1,~r. .I. Non-Linetrr Mech. 19. 349-362 (1984).
APPENDIX
A
Here we will prove that due to symmetry of equation (8) the linearisation (with respect of 3) of the second non-linear term appearing in equation (8) leads to a value equals to zero. The non-linear term under consideration is equal to 3u,? and, hence, its linearised version should be of the form (JU,:. The minimisation of the mean-square error determines the parameter i> as (\-;?) ,’ = __ (Al) ;_+> If we define two new variables Hence,
f = - 2 and ?, = ~ Y,,. it can be easily shown /I(_? u,.) = p(l. c,
The fact that the joint probability density function of Y, and z leads to the following
)=
p(
is an even function relations:
that they too satisfy equaton
2. ~ Y,).
(8). IA?)
while the numerator
of equation
(Al) IS an odd
tA3) Substituting
this result into equation
(Al) yields a value for the parameter
[J equal to zero.
Improved
stability
APPENDIX
analysis
155
B
Assume that a random process x(t) of total duration T,,,,, exhibiting a global variance u,&., and a fourth statistical moment F~,,~,~,, is a two-state process switching between a low-state process xL(t) with statistical Setting T,,,, the time parameters & , h. Ioyand a high-state process x”(t) with statistical parameters CT,&, Pi,+. spent in low state, and Thigh, the time spent in high state, the following relations hold:
(B2)
x’(t)dt
= T ,;,., { jo”“* x;(t)dt
+ j;
\-:,(r)dr}
(B4)
It is important to realise that under certain conditions, equation (B5) results in a value of global kurtosis much higher than the kurtosis of individual states. For example, consider the limiting case in which the low-state corresponds to extremely small statistical moments (i.e. CT& z 0, p(4 ,o~ z 0). In this case the global statistics are related to the local (high-state) statistics as follows:
Due to the fact that the global kurtosis is proportional to the inverse of the fraction of time spent in the higher state, the resulting global kurtosis may realise extremely high values when this fraction is small. This behaviour is reflected in some entries of Table 1, where the low state dominates and the resulting global variance is close to the low-state variance. The influence of the (rare) jumps in the high state on the fourth statistical moment, however, is much more significant, resulting in high values of the global kurtosis.