Identification of physical parameters with memory in non-linear systems

In<../ Non-Lmear Meclurnics. Vol. 30, No. 6, pp. 841-860, 1995 Elsevier Science Ltd Printed in Great Britain w2cb7462/95 $9.50 + 0.00

Pergamon

0020-7462(95)00023-2

IDENTIFICATION OF PHYSICAL PARAMETERS MEMORY IN NON-LINEAR SYSTEMS

WITH

Julius S. Bendat J. S. Bendat Company, 833 Moraga Drive, Los Angeles, CA 90049, U.S.A.

Robert N. Coppolino Measurement Analysis Corporation, 23850 Madison St., Torrance, CA 90505, U.S.A.

and Paul A. Palo Naval Facilities Engineering Service Center, Port Hueneme, CA 93043, U.S.A. (Received 5 December 1994)

Abstract-A new technique called ‘Reverse MI/SO’ has been developed that greatly simplifies the identification of parameters in systems with amplitude non-linearities and frequency-dependent coefficients as described by non-linear integro-differential equations of motion. This paper illustrates the technique for single degree-of-freedom (SDOF) non-linear systems where linear and non-linear damping is described by memory functions of an exponential and exponential-cosine analytical form. Comparisons between analytical and numerical simulation results prove that the Reverse MI/SO technique is quite robust. A discussion is included outlining the importance of this new technique as applied to the (non-linear) dynamics of ships and stability studies.

INTRODUCTION

The theory and application of a new practical approach for non-linear system identification called Reverse MI/SO was described in [ 11. This technique was developed in work directed by the Naval Civil Engineering Laboratory (now the Naval Facilities Engineering Service Center) with sponsorship from the Office of Naval Research. In that first report, responses from various SDOF non-linear systems such as Duffing, Van der Pol, Mathieu and Deadband were simulated and used to numerically evaluate the technique. The system dynamics were described by non-linear second-order differential equations of motion with constant coefficients (i.e. zero memory). In all cases Reverse MI/SO successfully identified the (known) system coefficients. As outlined in [l], Reverse MI/SO is a very attractive identification (and analysis) technique. It is a frequency-domain technique that is applicable to almost-arbitrary nonlinearities (including non-analytical), requires only slightly more computations than an equivalent linear spectral analysis, and-most importantly-yields a coherence function (goodness-of-fit) for each linear and non-linear term. These coherence functions are vitally important because they allow for unambigous assessment of the accuracy of the fitted equation; anyone who has worked with stochastic data knows that such goodness-of-fit vector measures are far superior to simple scalar correlation coefficients for assessing the validity of proposed models. In essence, Reverse MI/SO’s most useful application is in finding the best-fit low-order non-linear equation of motion from measured or simulated dafa (‘best-fit’ acknowledges that it may not be possible to know if the chosen equation is in fact the ‘true’ system equation). There are practically no restrictions on the proposed form of the non-linear terms. Reverse MI/SO is also fully capable of identifying the frequency dependence of all linear and non-linear paths through a non-linear system. In fact, Reverse MI/SO automatically assumes this; by doing so, it automatically identifies the best integro-difirentiaI rather than simply difirential equation(s) describing the system dynamics. These features make Reverse MI/SO a natural technique for identifying phenomena such as rigid body dynamics of ships on a free surface, where the added mass and damping are 841

J. S. Bendat et al.

842

known to be frequency-dependent, and where at least one degree-of-freedom (roll) is known to be highly non-linear even for moderate excitations. Applications in this area are on-going and will be reported later. Reverse MI/SO is expected to play an integral supporting role in non-linear stability calculations. Of course, the validity of all such calculations are dependent on the accuracy of the underlying system equation. This article will demonstrate that Reverse MI/SO is a powerful new tool for identifying the ‘best’ low-order equation of motion directly from data, including equivalent representations of much higher-order systems. That in itself is noteworthy. However, the value of Reverse MI/SO is even greater because it features frequency-dependent measures of accuracy (coherence functions) for each linear and nonlinear path. By definitively establishing the validity of each term in the estimated system equations these functions provide important uncertainty and/or parametric bounds for the stability calculations. Thus, capabilities of Reverse MI/SO and the associated coherence functions can significantly improve the systems information available from real-world measurements. In the past it was engineering practice to design non-linear systems based solely on time domain simulations with, for example, finite element models with many degrees of freedom. As non-linear stability theory advances it is now understood that stability studies are an absolutely necessary complement to properly interpret and guide the finite element studies. The combination of low-order Reverse MI/SO and subsequent non-linear stability analyses, along with more accurate and detailed large degree-of-freedom finite element models, provides a very complete package for the analysis of non-linear systems. Finally, the fact that Reverse MI/SO naturally returns an integro-differential equation is significant because it allows for physical interpretations by engineers and scientists regarding specific phenomena and their behaviour, for example, that the coefficient associated with the first derivative of the response (wave-making damping for vessels at the free surface) is frequency dependent. This makes it useful in fundamental studies into the physics of various phenomena (such as wake vortices). This is not true for other higher-order statistical techniques such as Volterra series models. For all of these reasons, Reverse MI/SO is considered to be an important new tool for non-linear systems identification and stability studies, and one that offers significant advantages compared to existing techniques.

BRIEF

REVIEW

OF

REVERSE

MI/SO

This section reviews Reverse MI/SO (for further information see Cl]-[4]). To illustrate the technique, consider first a simple differential equation of motion for a non-linear SDOF system with non-linear viscous damping: mii(t) + cti(t) + ku(t) + dz(t) = F(t)

(1)

where the coefficients m, c, k and d are constants (frequency dependence is considered later) and F(t) = force input = physical excitation

@a)

u(t) = displacement output = physical response

(2b)

z(t) = ti(t)lii(t)l = non-linear viscous damping term

(2c)

m = mass coefficient

(2d)

c = linear damping coefficient

(2e)

k = linear stiffness coefficient

(2f)

d = non-linear viscous damping coefficient.

(2g)

Careful inspection of equation (1) allows for a very useful reinterpretation. First,*note that engineers have a bias because they know that F(t) caused u(t) (in essence, they associate a direction with the equality sign); they instinctively realize that the system is non-linear and

Physical parameters with memory in non-linear systems

843

is therefore difficult to identify. Consider instead what happens if the direction of the equality is interpreted mathematically as the opposite of the physical problem; in essence, reversed. Start with Fourier transforms of both sides of equation (1) and change variables to yield ~l(_f)xlu)

+

~2u-)X2(f)

=

yu-)

(3)

where Y(f) = Fourier transform of F(t)

(4)

X1 (f) = U(f) = Fourier transform of u(t)

(5)

X2(f) = Fourier transform of z(t) = li(t)lti(t)l

(6)

A,(f)

= k - (271f)2m + j(2rcf)c

(7)

.42(f)

=

(8)

d.

Equation (3) indicates that two excitations passed through two linear systems to produce one output; this is the concept of a reversed system. The original single input non-linear problem is now seen to be the same as a linear problem with multiple inputs. This is a reversed dynamic, multiple-input/single-output (MI/SO) system-hence, the technique is called ‘Reverse MI/SO’. This is not an equivalent linearization technique. This example is equivalent to the two-input/single-output linear model with correlated input records as shown in Fig. 1. These input records X1 (f) and X,(f) produce correlated output records that, together with an uncorrelated noise output record N(f), sum to the total output record Y(f). Practical procedures to identify the frequency response functions A1 (f) and A,(f) from measured input/output random data are utilized in [ l]-[4]. These procedures show how to replace any given set of correlated input records with a new set of mathematically computed uncorrelated input records U,(f) and V,(f) [not related to u(t) above] that are calculated using conditioned spectral density functions as shown in Fig. 2. Note also in Fig. 2 that the physical systems A1 (f) and A,(f) are replaced by different ‘conditioned’ systems L,(f) and L2(f). The Aj(f) physical frequency response functions are easily recovered algebraically from the Lj(f) operators.

Fig. 1. Two-input/single-output

Fig. 2. Two-input/single-output

linear model with correlated input records.

linear model with uncorrelated input records.

844

J. S. Bendat Table

1. Notation

et al.

for SDOF

equations

m = mass coefficient

c k c(t) a b d d(r)

= = = = = = =

linear damping constant coefficient linear stiffness constant coefficient ca exp (- az)coshr = linear damping memory term linear damping exponential term Zrrf, = linear damping periodic term non-linear damping constant coefficient dp exp ( - p7) cos ~77 = non-linear damping memory term p = non-linear damping exponential term 4 = 2nf, = non-linear damping periodic term RM!$ = root-mean square value for the random excitation level of the force input data

Table 2. Simulated Parameter

4 RMS,

system numerical

parameters

Case 1

Case 2

Case 3

1.0 3.11 355.3 co 0 100 cc 0 5.0

1.0 3.71 355.3 40 0 100 20 0 5.0

1.0 3.71 355.3 40 10X 100 20 5n 5.0

Table 3. Digital

data processing

parameters

At = 0.01 s N = 1024 samples per subrecord T = NAt = 10.24 s per subrecord nd = 32 distinct subrecords N ,,,,a, = n,N = 32,768 samples T tOIP,= ndT = 327.68 s Nyquist cut-off frequency L = 1/2At = 50 Hz Bandwidth resolution B, = Af = 0.097 Hz

The final step of identifying the constant physical parameters k, m and c starts with Al(f) of equation (7). Note that Re[A1(f)]

MAr(

= k - (2~f)~rn

lim [Re{Al(f)/(2nf)‘}]

f-m

(9’4

= Wk

lim [Re{Al(f)}] /-‘O

(9a)

= k

= -m

v&4Im~l(27$) = c.

(9c) W)

(94

The constant physical parameter d is identified from direct inspection of the real part of the A,(f) calculated using equation (8), and for this example, it will be constant at all frequencies. Also, if the proposed equation of motion is correct, the imaginary part of A,(f) will be zero at all frequencies; the availability of such imaginary functions representing non-linear damping proportional to its derivative is another benefit of using Reverse MI/SO. As always, the analyst should compute appropriate coherence functions to assess the validity of each term and the fitted model,

845

Physical parameters with memory in non-linear systems

164

164.5

165

165.5

166

166.5

167

167.5

168

166.5

167

167.5

168

t (set)

164

164.5

165

165.5

166 t (set)

Fig. 3. Representative excitation (top) and response (bottom) time histories for all systems.

As discussed in [l], this approach is quite general in handling many other zero-memory non-linearities, including non-analytic functions, and even coupled systems. Furthermore, the a priori assumed zero-memory non-linearities can be considered as either parametric or non-parametric. Those and other important features are not pursued here.

EXTENSION

TO PHYSICAL

PARAMETERS

WITH

MEMORY

The present paper is directed towards analytical and numerical extensions of equation (1) to more general real-world cases where the physical parameters of the linear and non-linear damping terms have coefficients with memory-that is, they are frequency dependent. In a system with such behaviour, equation (1) is not a true equation of motion because the coefficient values are only valid for sinusoidal excitation at one particular frequency. A great deal of effort was spent by the naval architecture community in the 1960s to formally justify the existence of an equation of motion for systems with frequency-dependent coefficients that would be valid for any excitation (see [S]-[7]). This present work continues that same quest. For illustrative purposes, a similar question to equation (1) is used as described by the following general SDOF non-linear integro-differential equation of motion: t c(z)ti(t - t)dr + ku(t) +

mii(t) +

s0

- r)dz = F(t)

(10)

846

J. S. Bendat et al.

HZ

HZ Fig. 4. Representative excitation (top) and response (bottom) autospectra for all systems.

where F(t), u(t) and z(t) are the same as before in equations (2a)-(2c), but the previous constant damping coefficients c and d are replaced by memory terms c(r) and d(r), respectively. Thus, the linear and non-linear damping terms are now influenced by prior as well as by current motions. To be specific: c(z) = linear damping memory term

(11)

d(z) = non-linear damping memory term.

(12)

The extension of the simple products in equation (1) to the convolution integrals in equation (10) can be justified directly by the duality of the Fourier transform-that a product in one domain (here, frequency) is a convolution in the other domain (time). The previously analyzed system with constant linear and non-linear damping coefficients in equation (1) is a special case of equation (lo), where c(r) = J-’ {C(f)} = c 3-r {l} = &S(r)and similarly; d(z) = dd(z), where s(t) is the usual delta function and 3 is the Fourier transform. Just as before, take Fourier transforms of both sides of equation (10). This yields the result Al(f)XI(f)

+ Az(f)Xz(f)

= Y(f)

(13)

where X,(f) and X,(f) are the same as in the previous equations (5) and (6). However, instead of equations (7) and (8) AI

= k - CW12m + jGWfC(.f)

A*(f) = W-).

(14) (15)

847

Physical parameters with memory in non-linear systems

103

L :. : ,. :, .I

_

:..

:

““‘:“:‘_

HZ 200

100 t G g

0

iz -100

-200 i 10-l

100

HZ Fig. 5. Linear path frequency response function A,(f) for zero-memory system (0 = Reverse MI/SO estimate; line = exact).

The quantities C(f) and D(f) are C(j) = Fourier transform of c(t)

(16)

D(f) = Fourier transform of d(r).

(17)

Equations (14)-(17) are more general expressions for any forms of C(f) and D(f).

EXPONENTIAL

MEMORY

TERMS

Two analytical cases of physical damping with memory are treated in this paper, namely: (a) exponential memory, and (b) exponential-cosine memory. These were chosen as logical extensions to the ‘no memory’ delta functions of equation (1); the exponential memory represents a ‘short time interval’ positive memory, while the exponentialcosine memory extends the exponential memory to include both positive and negative memory. Exponential memory terms for equations (11) and (12) are defined by c(r) = ca exp (- UT)

(18)

d(z) = dp exp ( - pz)

(19)

848

al.

J. S. Bendat et

10-t

100

10’

HZ

10-t

100

10’

HZ Fig. 6. Non-linear path frequency response function A,(f) for zero-memory system (0 = Reverse MI/SO estimate; line = exact).

where c and d are linear and non-linear damping constants, respectively, and a = exponential memory term in c(r)

(204

p = exponential memory term in d(z).

Pb)

In general, a # p. The Fourier transforms of equations (18) and (19) are

C(f) = ca/Ca

(21)

W) = dp/Cp+iCW)l.

(22)

The non-memory case occurs when a and p approach infinity. For this exponential memory case, the integro-differential equation (10) can be expressed using the following simultaneous equations: mii(t) + Ii&)

+ h(t)

t&(r) + au,(r) =

+ U&) = F(t)

cati

t&(t) + au&) = dpz(t)

(23) (24) (25)

Physical parameters with memory in non-linear systems

849

0.6 -

10-l HZ Fig. 7. Cumulative coherence function for zero-memory system.

where equations (24) and (25) result from taking Fourier transforms of equations (26) and (27) and converting back to the time domain. The terms u,(t) and u,,(t) are given by I UC(t)=

ud(t)

=

ca[exp(-a@]ti(t

- z)dz

(26)

- z)dz

(27)

s0 ‘dp[exp(-pr)]z(t s0

with z(t) =

Cw)1’ i - [lqt)]”

i(t)

>0

i(t) < 0.

(28)

Equations (14) and (15) become A,(f)

= k - (27rj)‘m +j(27lf)ca/[u

A,(f)

=

WCP

+j(27cf)]

+~(wl.

(29)

(30)

Expanding, the real part of equation (29) is found to be Re[A,(f)]

= k - (2~f)~m + cu(27~j-)~/[u~ + (27~f)~] = k - (27~j”)~[m- a(f)]

(31)

where a(S) = ca/[u2 + (27~f)‘l.

(32)

This is an interesting observation, namely, that even for this simple trial function,jiequencyadded muss. In other words, this out-of-phase damping results in an in-phase effect. Naval architects have long “Ln 30-6-F dependent linear dumping creates a correspondingfrequency-dependent

850

J. S. Bendat et al.

10-l

10’

100

HZ

10-l

10’

100 HZ

Fig. 8. Comparison of linear path frequency response functions from non-linear Reverse MI/SO (0) and linear (:) techniques for zero-memory system.

known this to be true; added mass and linear damping are related through the ‘retardation function’. Also, the imaginary part of equation (29) can be shown to be Im[Ar(f)]

= ca2(27rj-)/[a2 + (2nf)23 = 427+(f).

(33)

For the exponential memory non-linear damping,

CW21

JWA2UX

= dp21Cp2 +

M42tf)l

= - dP(27wCP2

+

(344

cw21.

Wb)

These non-linear in- and out-of-phase components are not pursued here. The linear path scalars k, m, c and a can be identified from equations (31) and (33) by the relations lim Re(Ar(f)} J-0 lim [Re{A,(f))/(2nf)2] f-+m lim [Re{Ar(f)} f-0

= k

(35a)

= -m

- k + (2nf)zm]/(27r1f)2]

(35b) = c/a

(35c)

851

Physical parameters with memory in non-linear systems

104

h

103

2 B 2: 102

10’ 10-l

10’

100 HZ

HZ Fig. 9. Linear path frequency response function A,(f) for exponential memory system (0 = Reverse MI/SO estimate; line = exact).

lim [(27rf)Zm{A,(f)}] /-a

= ca’.

(35e)

The non-linear path parameters d and p can be identified from equations (34a) and (34b) by the relations

lim Re{Az(f)} = d J-0

lim J-0

C~~(Mf)hYW)l

lim [(2nf)Zm{A,(f)}]

= = -dp.

(3W

J--CC

This general case with exponential be numerically

illustrated

memory on both the linear and non-linear in a subsequent section.

damping

will

852

J. S. Bendat

10-l

et al.

101

100

HZ

10-l

10’

100

HZ Fig. 10. Non-linear path frequency response function AZ(f) for exponential memory system (0 = Reverse MI/SO estimate; line = exact).

EXPONENTIAL-COSINE

The exponential-cosine

MEMORY

TERMS

memory terms for the next illustrative case are defined by c(z) = ca exp ( - az) cos bz

(37)

d(z) = dp exp (- pz) cos qz.

(38)

Just as in the previous case, c and d are linear and non-linear damping constants, respectively, a and p are the linear and non-linear exponential memory terms, respectively, and b = 27rfb= cosine memory term in c(z)

(39a)

q = 271f, = cosine memory term in d(z).

W)

These results reduce to the previous exponential case when the cosine memory frequencies & and f4 are both zero. The Fourier transforms of equations (37) and (38) are C(f) = ca(a + j27rf)/[(a +j27# W)

= &(p +j2xf)/C(~

+ b*]

+j2M-)’ + q*l.

(40) (41)

853

Physical parameters with memory in non-linear systems

1

1

O.Q--

‘.

I

II,

F 1

.._

0.8 0.7

.

-

.._

,,_

0.6-.

0.5

_

0.4

-

0.3

-

.,..

.,,..

,..

._

,,-

.

:

..-

HZ Fig. 11. Cumulative

coherence function for exponential

memory system.

For this exponential-cosine memory case, the integro-differential expressed using the following simultaneous equations:

equation (10) can be

+ l&(t) + ku(t) + U&) = F(t)

h(t)

(42)

ii,(t) -+ 2ali,(t) + (a2 -I- b*)u,(t) = ca[i.i(t) -t ati(

(43)

i&(t) + 2pr&(t) + (p’ + 42)u&) = dp[i(t) + pz(t)]

(44)

where equations (43) and (44) result from taking Fourier transforms of equations (45) and (46) and converting back to the time domain, since y(t) and ud(r) are given here by

’ca[exp (-

uc(t) =

ar)] (cos br)ri(t - r) dr

s0 us(t)

=

t

dp[exp (- pr)] (cos qr)z(t - z) dz.

s0

(45)

(46)

The quantity i(t) is found by differentiating equation (28): (47) Equations (14) and (15) become here: A,(f)

= k -(27#m

+ j(2xf)ca(a

A(f)

= d&J +j2WC(p

+j2nI)”

+ j2$)/[(a + q21-

+ j2rrf)’ + b2]

(48) (49)

J. S. Bendat et al.

854

..................... ......

.

.

h

2

3

.....

.....,..

.I .....

...

... ... . ..

....

:

._.

.......

10-t

100

101

HZ

HZ Fig. 12. Comparison of linear path frequency response functions from non-linear Reverse MI/SO (0) and linear (*) techniques for exponential memory system.

The real and imaginary parts of equation (48) are iWW31

= k - (27~O~Cm- B(f)1

(504

where

B(f) = (a2 +

- b2 + 4n2f2) b2 - 47~~j-~)~ + 16a27r2fz

ca(a”

(5W where r(f) =

ca(a’ + b2 + 47r2f2) (a2 + b2 - 47c2f2)’ + 16a27r2f2’

(504

From equation (49):

W) (514

855

Physical parameters with memory in non-linear systems

100

10-t

10’

HZ

HZ Fig. 13. Linear path frequency response function A,(f) for exponential-cosine memory system (0 = Reverse MI/SO estimate; line = exact).

The linear path scalars k, M, c, a and b can be identified as in the previous case, except that equation (35d) replaced by:

(52) The non-linear path parameters d, p and q can be identified from equations (5la) and (51b) by the relations

lim CRe{~2t_f>>l = &zltP2+ 4’)

(534

f-0

limCW42(fWW)l f-0

= - 402 - q2Mp2+ q2J2

lim [(27rf)Zm{A2(f)}] = - dp. f-m

(53c)

Equations (37~(53) represent the second numerical example in this paper, namely, linear and non-linear damping, each with exponential-cosine memory. ‘In actual practice, the form of the calculated frequency response functions is unrestricted and arbitrary. The two analytic functions illustrated in this report were chosen simply for ease of illustration of converting between the frequency and time domains and they should not be considered as limiting forms for Reverse MI/SO.

856

J. S. Bendat et al. COMPUTER

SIMULATION

RESULTS

Extensive studies on these matters have been conducted on a wide variety of systems. The remainder of this paper illustrates the application of Reverse MI/SO to systems with the following characteristics: Case I. Linear and non-linear damping, each without memory (baseline reference case). Case 2. Linear and non-linear damping, each with exponential memory. Case 3. Linear and non-linear damping, each with exponential-cosine memory. In all cases a random wideband excitation was artificially created, and the equations of motion were numerically integrated using the Adams-Bashford predictor-corrector procedure [S] to find the corresponding responses. The excitation signals were defined using (a) the random number generator in MATLAB (normal distribution option), and (b) an g-pole Butterworth digital filter set to yield random signals with roughly uniform spectral content from 0 to 25 Hz. Required MI/SO spectral analyses were performed on the simulated excitation and response time-history records to estimate the reverse dynamic system frequency response functions. The notation for the various SDOF linear and non-linear equations with memory coefficients is given in Table 1. Simulated system parameters for the three cases shown in Table 2 are consistent with values used previously in [l]. The digital data processing param-eters used for all cases are listed in Table 3. A Hanning window was applied to reduce leakage effects together with overlap processing of 50% to recover ‘lost’ data.

l.,_

[1

L

HZ

10-l

100

101

HZ Fig. 14. Non-linear path frequency response function AZ(f) for exponential-cosine memory system (0 = Reverse MI/SO estimate; line = exact).

Physical

parameters

with memory

in non-linear

857

systems

Discussion of simulation results

Random excitation and response time histories were analyzed for all three cases using MI/SO spectral analysis for a reverse dynamic system. As shown in Fig. 1, the two reverse dynamic mathematical inputs consist of the displacement response, u(t), and the non-linear viscous damping term, ti(t)lti(t)). The single reverse dynamic mathematical output is the excitation force, F(t). Results for each MI/SO analysis case are summarized below. Case 1. Linear and non-linear damping, each without memory. Time history segments for F(t) and u(t) are illustrated in Fig. 3 and the corresponding

autospectra are given in Fig. 4. Similar results occur for Cases 2 and 3. The identified frequency response functions associated with the linear and non-linear system components in Case 1 are provided in Figs 5 and 6, respectively. In each of these figures, the MI/SO spectral analysis result is depicted by the ‘0’ symbol at discrete frequencies, and the exact analytical reference result is shown as a continuous solid line. This convention is used for all three example cases to compare the MI/SO and exact reference results. The estimate of the non-linear frequency response function, AZ(f), illustrated in Fig. 6, gives the correct simulated value for d in agreement with equation (8). The close comparison of MI/SO functions with the exact frequency response functions is confirmed here by the cumulative coherence functions shown in Fig. 7. The near-unity value of the lower coherence component indicates that most of the output autospectrum can be accounted for by the linear path approximation. The reader is cautioned that an optimum linear analysis would return a coherence even closer to unity, leading to the mistaken conclusion that the system was essentially linear. This would have dire consequences for the common, realworld situation where the data correspond to ‘measurements of opportunity’ with a moderate excitation (say, a recent storm or earthquake), but the identified system parameters are to be used with much larger survival excitations for the final design. In these cases of extrapolation to larger excitations, the identification of small non-linearities are extremely important.

1

0.9

0.8

0.6 E 3 E

0.5

0.4

10-l

100

HZ Fig. 15. Cumulative

coherence

function

for exponential-cosine

memory

system.

858

J. S. Bendat et al.

Figure 8 shows the estimates for A,(f) from the appropriate MI/SO-based reverse dynamic system analysis compared to an incorrect SI/SO analysis, where it is implicitly assumed that the system is strictly linear. The ‘0’ symbol represents the properly computed Reverse MI/SO estimate, and the ‘*’ symbol represents this erroneously computed SI/SO estimate. A key observation to note here is the substantial difference in the two estimates, even though the linear coherence function in Fig. 7 mistakenly indicates that the system non-linearity is small. Case 2. Linear and non-linear damping, each with exponential memory. The identified frequency response functions associated with the linear and non-linear system components in Case 2 are provided in Figs 9 and 10, respectively. Figure 9 is quite similar in appearance to Fig. 5, but Fig. 10 now indicates a frequency-dependent non-linear damping. The close comparison between the Reverse MI/SO and the exact frequency response functions is confirmed here by the cumulative coherence functions shown in Fig. 11 that sum to near-unity values at all frequencies. The estimated non-linear frequency response function, A,(f), illustrated in Fig. 10, has negligible bias error and indicates non-linear damping with exponential memory, in agreement with equation (22). Figure 12 compares the estimates for A,(f) from the appropriate MI/SO-based reverse dynamic system analysis and from the wrong SI/SO analysis, where it is implicitly assumed that the subject system is strictly linear (same key as Fig. 8). Note the substantial difference

I

-200 10-l

!

I

I

I

I I I 1

1

,

100

HZ Fig. 16. Comparison of linear path frequency response functions from non-linear Reverse MI/SO (0) and linear (*) techniques for exponential-cosine memory system.

Physical parameters with memory in non-linear systems

859

in the two estimates, even though the coherence functions shown in Fig. 11 indicates that the system non-linearity is not strong. Case 3. Linear and non-linear damping, each with exponential-cosine memory. The identified frequency response functions associated with the linear and non-linear system components are provided in Figs 13 and 14, respectively. Figure 13 is quite similar in appearance to the previous figures, but Fig. 14 is quite different than the previous Figs 6 and 10. The close comparison between the Reverse MI/SO and the exact frequency response functions is confirmed here by the cumulative coherence functions shown in Fig. 15 that sum to near-unity values. The non-linear frequency response function, AZ(f), illustrated in Fig. 14, has negligible bias error and correctly indicates non-linear damping with exponential-cosine memory in agreement with equation (41). Figure 16 compares estimates for Al(f) from the Reverse Ml/SO-based dynamic system analysis and from a simple SI/SO analysis, where it is implicitly assumed that the subject system is strictly linear (same key as Fig. 8). Once again, note the substantial difference in the two estimates, even though the coherence functions in Fig. 15 show that the system non-linearity is not strong. Statistical

errors

Throughout all the simulations done to date, modest to moderate statistical bias and random errors occur in the estimates for the linear damping functions. The source of this bias has not yet been identified. Relatively small statistical bias and random errors occur in the estimates for the non-linear damping functions. These statistical errors do not compromise the fidelity of the overall identified linear and non-linear systems, as indicated by the high values for the cumulative coherence functions in Figs 7, 11 and 15. Repeating comments made throughout this report, a key feature of this technique is the availability of coherence functions for each non-linear path. These play a crucial and unambiguous role in the proper interpretation of the estimated functions and their errors-as a function of frequency and for each individual system path. In the cases reported here and in all other cases, the cumulative coherence quantitatively establishes how well and where the proposed model fits the data, thereby assessing also the effect of bias and random errors on the response.

CONCLUSIONS

This analytical and numerical study shows that a methodology now exists to practically and accurately identify a wide range of non-linear systems. In particular, this article shows how to identify physical parameters in SDOF linear and non-linear systems when the linear and non-linear damping terms have coefficients with memory of an exponential or exponential-cosine form. Formulas are also stated on how to identify the physical parameters in SDOF non-linear systems when the coefficients have other forms of memory. Three conclusions emphasized in this report are: (1) For SDOF non-linear systems, the reverse dynamics system MI/SO linear method can correctly identify frequency dependence/memory functions in SDOF non-linear integro-differential equations of motion for both linear and non-linear damping terms with exponential or exponential-cosine memory. The ability to correctly identify nonlinearities as well as frequency dependences (or equivalently, memory) will provide a more realistic starting point for non-linear stability studies. (2) the Reverse MI/SO technique is robust, is not limited to any special forms for the excitation, can accommodate a wide range of non-linearities and memory functions, and requires the same order of computing resources as a linear analysis. (3) For SDOF non-linear systems, SI/SO-based optimum linear frequency response function-estimates (as done in conventional modal analysis work) gives erroneous results (due to the misleading high coherence) compared to the appropriate Reverse MI/SObased frequency response function estimates. This is especially important if the system parameters are to be used with larger excitations in design applications.

860

J. S. Bendat et al.

This leads to the recommendation that Reverse MI/SO be used in all systems identification applications, even where the system is ‘assumed’ linear. Even if non-linearities are not suspected, applying Reverse MI/SO non-parametrically with two additional signals given by the response squared and cubed (representing the first even and odd non-linearities, respectively) can be readily implemented with many existing multiple input software packages, does not significantly increase the analysis time, and-most importantly-quantitatively establishes whether the system is linear or not; as shown in this study, a high coherence from a linear analysis does not reliably establish that the system is truly linear. Routine use of Reverse MI/SO would eliminate the danger of misinterpreting high coherences from such linear analyses. Future reports are planned regarding on-going applications to experimental data; analysis rather than system identification uses; bias and random errors; and studies of systems where coefficients exhibit (for example) Reynolds number dependences.

REFERENCES 1. J. S. Bendat, P. A. Palo and R. N. Coppolino, A general identification technique for nonlinear differential equations of motion. Prob. Engng Mech. 7. 43 (1992). 2. J. S. Bendat and A. G. Piersol. Enyinwr~rrq ,Ipp/l( urions of Correlation and Spectra/ Analysis (2nd Edn). Wiley-lnterscience, New Y’ork (1993). 3. J. S. Bendat. Nonlinear System .4nal~srs and Idenfifcutionfrom Random Data. Wiley-lnterscience, New York ( 1990). 4. J. S. Bendat and A. G. Piersol, RANDOM DATA: Analysis and Measurement Procedures (2nd Edn). WileyInterscience, New York (1986). 5. W. Cummins, The Impulse Response Function and Ship Motions. Report 1661, David Taylor Model Basin, Carderock, MD (1962). 6. J. Hooft, Advanced Dynamics of Marine Structures. Wiley-Interscience, New York (1982). 7. H. Bingham, F. Korsmeyer and J. Newman, Prediction of the seakeeping characteristics of ships. 20th Naval Hydrodynamics Symposium, Santa Barbara, CA (1994). 8. F. Hildebrand, Aduanced Calcufusfor Applications. Prentice-Hall, Englewood Cliffs, NJ (1962).