WO2002041193A1 - Nonlinear systems - Google Patents

Abstract

The present invention provides techniques of frequency domain fatigue analysis which can be used to perform fatigue analysis for nonlinear structures or components subject to random loads. The techniques can be implemented in a way which is analogous to the fatigue analysis for linear structures or components and can readily be incorporated into a fatigue design routine. This allows design for fatigue to be readily implemented for nonlinear structures and components. The present invention also provides a technique which employs a link between the parameters or characteristics and the fatigue life or profile of the structure or components to be designed, and allows the design of the structure for fatigue to be implemented directly.

Description

Nonlinear Systems- .-.^'

The present invention relates to nonlinear systems and, more particularly, to fatigue analysis and design techniques in the frequency domain for nonlinear structures or components and associated computer programs and computer program products.

Traditional fatigue analysis methods, such as Osgood CC. : Fatigue Design; Wiley; 1970, for structures or components are based on time domain analysis methods. Such methods work well for defined deterministic loading patterns. However, many structures and components including, for example, aircraft and automotive components, offshore platforms, and many other systems are subject to random loading patterns. In such circumstances the traditional time domain fatigue analysis methods are difficult to apply and the stress or strain time histories have to be considerably simplified before results using these methods can be achieved- This can lead to inaccurate results and the fatigue analysis is often only performed as a checking procedure at the end of the design procedure due to computational complexity of the task. The design of a structure or component for fatigue using traditional fatigue analysis methods therefore typically involves a design, analysis and testing, redesign cycle to achieve the desired result.

Vibration fatigue analysis, which is also referred to as spectral fatigue analysis or as frequency based fatigue analysis, is an alternative to time domain fatigue analysis and can be used to estimate the fatigue life of structures and/or components in the frequency domain when the stress or strain histories associated with the structures and /or components are random in nature. ■ ;_..

Vibration fatigue analysis also involves determining the parameters that characterise the structures or components to be designed to achieve a desired or specified fatigue life. A computer model based fatigue design for structures or components which normally uses a finite element analysis (FEA) model of the structures or components allows design for fatigue without the need to make the structures or components. Therefore, for example, designers can accurately compute the life-span of a component as if it was made in different materials without having to manufacture and test prototypes. The design cycle based on this method is therefore much shorter and more cost effective.

Effective vibration fatigue or frequency based fatigue analysis and design techniques are currently limited to the case where the structure or component is linear or where a linear approximation is used to represent the behavior of the structure or component. However, almost all systems in the real world are nonlinear especially mechanical systems and composite materials. An offshore oil platform is, for example, ^' a nonlinear structural system that is subject to random loading. Fatigue analysis of such systems is currently limited to time domain fatigue analysis methods which involve assumptions and simplifications of the random stress or strain response patterns, or frequency based fatigue analysis based on a linearised approximation of the nonlinear dynamics . Both approaches involve approximations and are likely to lead to inaccurate fatigue calculations.

It is the object of the present invention to at least mitigate some of the problems of the prior art. Accordingly, a first aspect of the present invention provides a method for computing^" the fatigue of a nonlinear structure or component based on a predetermined loading pattern that represents a loading condition, the method comprises the steps of establishing a nonlinear dynamic model which represents the structure or component, producing a stress or strain response for the structure or component from the nonlinear dynamic model by performing a transient analysis of the structure or component when the structure or component is subject to the loading pattern; determining the power spectral density (PSD) of the stress or strain response of the structure or component to the predetermined loading pattern, computing the spectral moments from the PSD, constructing a probability density function (PDF) of the stress or strain ranges using the spectral moments and calculating the structure or component fatigue damage or fatigue life time using the obtained spectral moments and the constructed probability density function.

A second aspect of the present invention provides a method for computing the fatigue of a nonlinear structure or component based on a set of statistics of a predetermined loading pattern that represents a loading condition, the method comprises the steps of establishing a nonlinear dynamic model that represents the structure or component, evaluating the gain bounds of generalized frequency response functions of the nonlinear model either analytically or using an optimization procedure, determining a bound on the PSD response of the structure or component to the predetermined loading pattern, which is a function of a set of statistics of the predetermined loading pattern and gain bounds of generalized frequency response functions of the nonlinear model, computing the spectral moments from the bound on the PSD response, constructing a probability density function (PDF) of the stress or strain ranges, and calculating the fatigue damage or fatigue life time of the structure or component under the predetermined loading condition. Preferably, the fatigue damage or fatigue life time represents a worst case fatigue damage or shortest fatigue life time.

A third aspect of the present invention provides a method for the direct design of a nonlinear structure or component for fatigue when the structure or component is subject to random loads. Based on an explicit relationship between the variables associated with the fatigue of the structure or component to be designed and the parameters or characteristics of the structure or component, which could for example be obtained when the structure or component is subject to a specific random loading pattern or a class of random loading patterns, the method determines appropriate values of a parameter, parameters, or characteristics of the structure or component to achieve the required fatigue life time. The method comprises of the steps of expressing, for example, the stress or strain PSD response of the structure or component to a specific random loading history or the stress or strain PSD responses of the structure or component to a class of random loading histories in terms of the parameters or characteristics of the structure or component; and applying optimization or other procedures to determine the values of the structure or component parameter, parameters or characteristics which result in the specified fatigue life time. A fourth aspect of the present invention provides a method for the design of a nonlinear structure or component for fatigue when the structure or component is subject to random loads. The method follows the typical design, analysis, and redesign routine but uses the vibration fatigue analysis techniques in the present invention, that is, the first and second aspects of the present invention to perform the fatigue analysis and comprises the steps of designing a prototype of a nonlinear structure or component, performing the fatigue analysis for the designed structure or component using the fatigue analysis techniques in the present invention to check whether the fatigue^" of the structure or component satisfies the design requirements; and redesign the structure or component if the fatigue analysis indicates that the original design is not acceptable.

Advantageously, the embodiments of the present invention provide methods of fatigue analysis and design that can effectively be applied to nonlinear structures or components that are subject to random loads. Furthermore, embodiments provide methods that can relate the fatigue life or profile of a structure or component to be designed to the parameters that characterise the structure or component so as to implement a fatigue design directly.

Embodiments of the present invention will be described by way of example only with reference to the accompanying figures in which:

figure 1 shows a loading history which is the horizontal water particle velocity around an experimental offshore structure; .. ,

figure 2 shows the force response of the experimental offshore structure to the 'loading history in figure 1;

figure 3 shows the loading history representing a loading condition under investigation of the experimental offshore structure;

figure 4 illustrates the force response of the experimental offshore structure to the loading history shown in figure 3;

figure 5 illustrates the Power Spectrum Density (PSD) of the stress response of the experimental offshore structure to the loading history in figure 3 (in solid) and a bound on the stress response PSD (in dashed) ;

figure 6 depicts the probability density function (pdf) for stress ranges exerted on the experimental offshore structure when the structure is subject to the loading condition represented by the loading history of figure 3;

figure 7 shows a loading history representing a further loading condition under investigation for the experimental offshore structure;

figure 8 shows a loading history representing a still further loading condition under investigation for the experimental offshore structure;

figure 9 illustrates a force response of the experimental offshore structure when subjected to the loading history shown in figure 7; figure 10 illustrates a force^'"' response^' of the experimental offshore structure when subjected to the loading history shown in figure 8;

figure 11 illustrates the Power Spectrum Density

(PSD) of the stress response of the experimental offshore structure when subjected to the loading history of figure

7 (in solid) and a bound on the stress response PSD (in dashed) ;

figure 12 illustrates the Power Spectrum Density

(PSD) of the stress response of the experimental offshore structure when subjected to the loading history of figure 8 (in solid) and a bound on the stress response PSD (in dashed) ;

figure 13 depicts a probability density function

(pdf) of the stress ranges on the experimental offshore structure when the structure is subjected to the loading condition represented by the loading history shown m figure 7;

figure 14 depicts the probability density function (pdf) of the stress ranges on the experimental offshore structure when the structure is subjected to the loading condition represented by the loading history shown in figure 8 ;

figure 15 shows a time limited Fourier Transform result for the loading history shown in figure 3;

figure 16 shows a three-fold convolution integration result for the amplitude characteristic of a time limited Fourier Transform of the loading history shown in figure 3; figure 17 shows a calculation result for a^' statistic of the loading history shown in figure 3;

figure 18 shows a calculation result for a further statistic of the loading history shown in figure 3;

figure 19 shows a calculation result for a still further statistic of the loading history shown in figure 3;

figure 20 depicts the probability density function

(pdf) of the stress ranges on the experimental offshore structure which is evaluated using a bound on the PSD of stress response of the structure when subjected to the loading history shown in figure 3;

figure 21 depicts the probability density function

(pdf) of the stress ranges on the experimental offshore structure which is evaluated using a bound on the PSD of the stress response of the structure when subjected to the loading history shown in figure 7;

figure 22 depicts the probability density function (pdf) of the stress ranges on the experimental offshore structure which is evaluated using a bound on the PSD of the stress response of the structure when subjected to the loading history shown in figure 8;

figure 23 shows a specific loading history which is used to illustrate that there exists a very close relationship between the PSD of the stress response of the experimental offshore structure and its bound in some particular loading cases;

figure 24 illustrates the PSD of the stress response of the experimental offshore structure when subjected to the specific loading history shown ^''in figure 23 (in solid) and a bound on the PSD (in dashed) ;

figure 25 depicts the probability density functions

(pdfs) of the stress ranges on the experimental offshore structure evaluated using the PSD of the stress response of the structure when subjected to the specific loading history shown in figure 23 (in solid) and using a bound on the PSD (in dashed) respectively;

figure 26 shows a- simple vibration suspension system comprising a mass, a spring and a damper;

figure 27 depicts a loading history to be considered in implementing a fatigue design for the structural system shown in figure 26;

figure 28 shows a function of frequency in an expression for the stress response PSD of the structural system shown in figure 26 when subjected to the loading history shown in figure 27;

figure 29 shows a further function of frequency in an expression for the stress response PSD of the structural system shown in figure 26 when subjected to the loading history shown in figure 27; and

figure 30 shows a still further function of frequency in an expression for the stress response PSD of the structural system shown in figure 26 when subjected to the loading history shown in figure 27.

I Detailed descriptions

1.1 Nonlinear vibration fatigue analysis Given a time history of the^{** "}input loading on a structure or component, nonlinear vibration fatigue analysis assumes there is a general nonlinear relationship between the . stress or strain response of the structure or component and the corresponding input loading. As a result, the power spectrum density (PSD) of the stress or strain response under the input loading which is the basis of vibration fatigue analysis cannot be obtained using the well-known linear technique which evaluates the PSD of the stress or strain response simply by multiplying the -input loading PSD by the square of the modulus of the linear transfer function. Accordingly, the present invention provides techniques developed for nonlinear system analysis in both the time and frequency domains to solve the problem.

A vibration fatigue analysis is normally carried out for an engineering structure or component under the situation where the structure or component is subject to loading histories that represent working conditions. The present invention uses nonlinear system modelling, frequency domain analysis techniques and includes two techniques which can readily be used to implement the fatigue analysis when a general nonlinear relationship between the input loading and the output stress or strain is taken into account. The first approach uses loading histories that represent the operating conditions of the structure or component under investigation to implement the analysis, while the second approach uses the statistics of the representative loading histories to evaluate the worst case fatigue damage or shortest fatigue life of the structure or component under the considered conditions.

1.1.1 Nonlinear vibration fatigue analysis of structures based on loading histories which, ,,^* represent loading conditions " ^■■ ' ^''■^'- ^'' ^*" ■'■

When vibration fatigue analysis of a nonlinear structure or component , has to be considered, current techniques use the stress or strain response data under loading conditions to implement the analysis from either a practical test on the structure or a nonlinear finite element analysis (FEA) model transient analysis. The problems with these available techniques are twofold. Firstly, doing tests on the practical structure or performing nonlinear FEA model transient analyses repeatedly to cover all possible loading conditions can be very costly and time consuming. Secondly, because of time constraints and computational complexity, the available methods of fatigue analysis can not be effectively incorporated into a fatigue design routine when the system is nonlinear.

The first embodiment of nonlinear vibration fatigue analysis in the present invention can be applied to overcome these problems. The technique uses a nonlinear dynamic mathematical model of the structural system rather than the practical structure or a nonlinear FEA model to implement the transient analysis and involves a procedure that can therefore be readily incorporated into a fatigue design routine.

The nonlinear dynamic mathematical model is established using a nonlinear system identification technique from the system input and output data which can be obtained from a practical test on the structure or component or an FEA model simulation. A class of effective nonlinear system identification techniques can be used to establish the nonlinear dynamic mathematical model of a structural system directly from the system input and output data without using .any priori knowledge of the original structure. Such'-* -techniques ^'are well established and described in the NARMAX (Nonlinear Auto- Regression Moving Average with -exogenous input) methodology (Billings S.A., Chen S., Korenberg M.J., 1989, Identification of MIMO nonlinear systems using a forward regression orthogonal estimator, International Journal of Control, 49, pp2157-2189) .

The theory and method underlying this technique of the present invention will now be described in general terms in Steps (1) to (9) in the following.

(1) Collect data of a loading history and the corresponding stress response from a practical dynamic test on the structure or component to be analyzed or from a nonlinear FEA model transient analysis of the structure or component.

(2) Using the loading and stress response data and a nonlinear system identification method, establish a nonlinear dynamic mathematical model of the structure or component which could be, when the NARMAX model is applied, a nonlinear auto-regressive with exogenous input model of the form

y(k) = ∑ y_n(k) n = l where ^ynW is a ' nth-order output ' given by

_{Y n}W = ∑ ∑ _vq (l _{l f} . . . _f l_p+q)]] y tk - l_i) f[ u (k - 1 p=0 1_α . l _p+<,=l i=l i=p + l

with P^{+ (}3^{= n} where ^{n = 1}>^{' "}>^ND corresponds to various orders of equation nonlinearities,

K is the maximum lag , and y ( . ) , u ( . ) , and c„„ ( . )

• ^pq are the system output, input, and model coefficients respectively. • •

This model which is often referred to as the NARX (Nonlinear Auto-Regression with exogenous input) model is a polynomial form nonlinear difference equation model the sampling interval of which w ll be denoted as T^s .

A specific instance of the model such as

y(k) = 0.3υ(k -1)+ Q.ly(k -2) -0.02u(Jc - l)u(k -1) - 0.04u(k- l)u(k -2) - 0.06y(k - l) {k - 3)-0.0 βy(k - 2)y{k - 3)

may be obtained from the general form with

c₀₁(l) = 0.3, c₁₀(l) = 0.7, c₀₂ (1,1) = -0.02, c₀₂ (2,1) = -0.0 , c_u (1,3) = -0.06 c₂₀(2,3) = -0.08, else c_pq(.) = 0

It will be appreciated that the NARX model above is normally obtained from a NARMAX model by discarding the noise terms. The NARMAX model is the result obtained when using a nonlinear system identification technique based on the NARMAX methodology. The model includes terms which involve the system input, output, and noise. The noise terms ensure that the parameter estimates in the model are unbiased and that the deterministic part of the model reflects the dynamics of the underlying system. The above specific instance of the NARX model could, for example, be obtained from a NARMAX model such as y(k) = 0.3u(k-l) + ^Q.ly(k-2)-0.02u(k-l)u(k-l)^^Q.Q4u(k-l)u(k-2)_ι -0.06y (k-1) u (k-3) -0.08y (k-2) y^'(k-3) '^■¥ 0.2e (Jfc-i; u (k-1)^' -0.06e (k-2) y (k-1) u (k-3) + e(k) + 0.07e(ir - 2) - 0.04e(k - 3)

by discarding the terms which involve the noise e(k-i) where 1=0,1,2,3.

(3) Perform a transient analysis by simulating the model established in (2) above to evaluate the sampled stress response yk) _f k = 0,l,...,( -l)M/2-rM-l _{of the} structure or component when subject to a loading history representing a loading condition under investigation where M is chosen to be an even number and L is a positive integer which could typically be chosen, for example, to be greater than 20 in practice .

(4) Evaluate the PSD of the stress response from the sampled response obtained in Step (3) above using, for example, the Welch method where the data length for the Fast Fourier Transform (FFT) is M, the window used could be, for example, the Hanning window of length M, the sampling frequency is

^{3 "} ' ^s , and the number of the samples which are overlapped is M/2. The result obtained is

S_γv(w) w 2πi/MT_s, i = Q,l,...,.M/2

where ^Y d^w)>

Fourier transform of the stress response y(t) evaluated using the FFT from the windowed sampled data of the response

y[M(l -1)/2 (1) , ••^■ , y[(M -ϊ) + M(l-l)/2](M), 1 = 1, ... L

and h(i), i=l,...,M, represents the window function that was used.

(5) Evaluate the moments from the PSD of the stress as

m_n

for-^* n=0 , l , 2 , 4

(6) Construct a probability density function (Pdf) of the rainflow ranges (stress ranges) of the stress response directly from the moments from the PSD of the stress using, for example, the Dirlik solution

(Bishop N., 1997, Technique background notes relating to frequency life fatigue estimation software module. Ncode Inc. which is incorporated herein for all purposes.) as

^e^Q + ^{■ D}-^*Z _ΩΪ +D_^Ze ² p(S) = f (m₀,m₁,m₂,m₄) = Q R-^*

2^

where

_D, = ±∑Zz +&, ₀₃=1-0,-₀,, _{Q =} i-25⁽ -D3-D₂R⁾

1-R D_x

[1 ) Calculate the fatigue damage of the structure or component over a unit lifetime under the loading condition represented by the particular loading history which was used to evaluate the _, stress response in Step (3) above as

D= ^E(P)J S p(S)dS

where

, ^Kf and ^Mf are the parameters which define the fatigue characteristics of the associated materials using the so-called S-N relationship given by

N(S) =-^-

S ^f

The specific values of * and ^f .are . available from

British Standards (BS 5400: Part 10: 1980) given the materials by which the structure or component to be analyzed is constructed. BS 5400 is incorporated herein in its entirety for all purposes.

(8) Steps (3) to (7) can be repeated to evaluate the fatigue damage of the structure under other loading conditions represented by the corresponding loading histories to yield in total, for example, ^F fatigue damages

D_x , i= l, ... ,N_F

where ■* denotes the fatigue damage of the structure under the i-th loading situation.

(9) Evaluate the averaged fatigue damage ^DA of the structure from ^Dl' ^{1 = 1}' ^■■' ^{• ,NF} obtained above as

and then calculate the fatigue life of the structure as _ _1_

where ^Pj- represents the probability of the structure working under the i-th loading condition which could be obtained, for example, from observing historical loading data collected in practice.

In the case where the. loading histories representing loading conditions are deterministic signals rather than realizations of stochastic processes, it will be appreciated that Steps (4) to (7) above could be replaced by a conventional time domain fatigue analysis technique which evaluates the fatigue damage of the structure under each loading condition directly from the time domain stress response data y(k), k=0,l,....

1.1.2 Nonlinear vibration fatigue analysis of structures based on statistics of the loading histories which represent loading conditions

It is common practice in fatigue analysis for a structure or component to use only one loading history to represent a loading condition and to evaluate the fatigue of the structure or component under this condition based on the loading history data. When the structure or component under test is a linear system the PSD of the system output is uniquely determined by the simple product of the PSD of the system input and the square of the modulus of the linear system transfer function. So that, the fatigue analysis result obtained under a loading condition represented by one loading history can represent the results that would be obtained under a class of loading histories where the PSD' s of these loading histories are the same as the PSD of the loading history used for the analysis.

For nonlinear systems the PSD of the system output is generally determined by complex integral terms involving the generalised frequency response functions associated with the nonlinear component or structure and higher order moments of the corresponding input rather than the simple product of the PSD of the input with the square of the modulus of the system transfer function as in the linear system case. Therefore, the fatigue analysis result for a nonlinear system based on one loading history in the way described in Section 1.1.1 may, in many practical cases, represent only one result which is only representative for one particular loading history situation. Therefore, it would be preferable if fatigue analysis for a loading condition could accommodate not only one particular loading history situation but also other situations where the loading histories are statistically equivalent to_, the representative loading history. Statistically equivalent is defined hereafter.

A second embodiment of nonlinear vibration fatigue analysis of the present invention addresses this problem. The technique uses nonlinear system frequency domain analysis techniques to evaluate a bound on the PSD response of a structure or component when subjected to a loading history representing a loading condition. The result is a bound on the PSD responses of the structure or component to all loading histories which are statistically equivalent to the loading history used for the analysis in the sense of having the same statistics which reflect certain important statistical properties. This result, which is determined by the statistics of a representative loading history, can be used to evaluate the worst case fatigue damage of the structure or component under the considered loading condition.

The theory and method underlying this embodiment of the present invention will now be described in general terms in Steps (1) to (12) in the following.

(1) Collect data of a loading history and the corresponding stress response from a practical dynamic test on the structure or component to be analysed or from a nonlinear FEA model transient analysis of the structure or component

(2) Using the loading and stress response data and a nonlinear system identification method, establish a nonlinear dynamic mathematical model of the structure which could be, when the NARMAX (Nonlinear Auto-Regressive Moving Average with exogenous input) methodology is applied, a nonlinear auto-regressive with exogenous input model of the form.^"

.. . • . .. .1

where ^nW j__{s a} '.nth-order output' given by

with P^{+ C}5^{= n} where ^{n ~} x^{' " >N}D corresponds to various orders of equation nonlinearities,

K is the maximum lag, and y ( . ) , u ( . ) , and c_n„( . ) ^pq are the system output, input, and model coefficients respectively.

This model which is often referred to as the NARX (Nonlinear Auto-Regressive with exogenous input) model is a polynomial form nonlinear difference equation model the sampling interval of which will be denoted as T^s ■ .

A specific instance of the model such as

y(k)=0.3u(k-l)+0.7y(k-2)-0.02u(J-l)u(^-l)-0.04u(k-l)u(^-2) -0.06y(k - l)u(k - 3)- 0.08y(k - 2)y(k - 3)

may be obtained from the general form with

c₀₁(l) = 0.3, c₁₀(l)=0.7, c₀₂(l,l) =-0.02, c₀₂(2,l)--0.04, c_n(1,3) =-0.06 c₂₀(2,3) =-0.08, else c_pq(.) =0

It will be appreciated that the NARX model above is normally obtained from a NARMAX model by discarding the noise terms. The NARMAX model is the result obtained when using a nonlinear system identification technique based on the NARMAX methodology. ^"'^'"^' '"

The model includes terms which involve the system input, output, and noise.. The noise terms ensure that the parameter estimates in the model are unbiased and that the deterministic part of the model reflects the dynamics of the underlying system. The above specific instance of the NARX model could, for example, be obtained from a NAXMAX model such as

y (k) = 0.3u (k - V + Q.ly (k -2) -0.02u (k-1) u-(k -l) -0.Q4u (k -l) u (k -2) -0.06y (k-1) u (k-3) -0.08y (k-2) y (k-3) + 0.2e (k-1) u (k-1) -0.06e (k-2)y (k-1) u (k-3) + e(k) + 0.01e(k- 2)- 0.04e(k -3)

by discarding the terms which involve the noise e(k-i) where i=0,l,2,3.

(3) Evaluate the gain bounds of the generalized frequency response functions (GFRFs) of the nonlinear dynamic model from (2) above under the constraint for the frequency variables of w₁ + ^••• + w_n w_ _{τhis can be ac} ieved using two methods .

• Method 1 This is a relatively simple method. Evaluate, using a recursive algorithm, the gain bounds of the GFRFs of the identified NARX model under no constraints. Then using these results compute the gain bounds of the GFRFs under the constraint ^ + - + ^ - "

Denote the gain bounds of the GFRFs with no constraints as

H_{n f} n = 1,2,... and the gain bounds of the GFRFs under the constraint

H_a ^B.w), n = 1,2,

^a' ,^<-.,... _can IQQ determined using a recursive algorithm and the NARX model coefficients as follows

For n=l

where

∑c₀₁(*₁)exp(-_. w_dl.fc₁)

and [a,b] is the interval which represents the frequency range of the possible loading histories.

For n>2

where

L_n c₁₀(k₁)exp(-jw_dlk₁

and ⁿ represents the possible frequency range produced by the nth-order nonlinear output which can be determined from the input frequency range [a, b] using the following formula

[ ] means to take the integer part, na

1 - +1

(a + b)

I_k = [na - k(a + b),nb - k(a + b)] for k = 0,...,±^* - 1, I , = [0,nb - ι (a + b)]

Then determined as

H_π ^B (w) = C(wT_s)L_nH_n ^B for n>2

where

• Method 2

This method involves more calculations but can achieve more accurate and less conservative results. In this method, an optimisation procedure, such as, a Genetic Algorithm (GA) is applied directly to address the following optimization problem

to evaluate accurate gain bounds for the GFRF' s under the constraint *, +^•.. + w_Λ ^{= w} .

To implement this method, the GFRFs, ^Hn

_f n=l,2,... N, are first evaluated from the identified dynamic mathematical model. If a NARX model, for example, has been obtained using the NARMAX methodology in Step (2) above, the GFRFs of the system can be calculated using the following recursive algorithm:

H_n(jw₁,--,jw_n) = H^d _π(jT_sw_l,...,jT_sw_n)

where

K

{^~∑c₁₀(k₁)ex₁p[-j(w_d__l+,---,+w_dn)k_l]}H^d _n(jw_dl,...,jw_dn) =ι

2^c_0n(k₁,...,k_n)&ψ[-j(w_dlk_l+,---,+w_clnk_n_] =ι

n-ln-q K q=ip=_k_Jc_pl<fZ

⁺Σ Σ^Cpθ ^kl>--->^kp)^Hd">^p(J^Wil>---X^Wc_ln) p=2k₁,k_p=l where ^w«u =2 i i = l - n _and

rfXJ *&,... j *<&„) +^{• • •}+w_eu)j

with t?nl(j

*&,... J

X~+W_dn)k

(4) Using the (L-l)M/2+M samples ^UW , k=0, 1, ... , (L- l)M/2+M-l of the available loading history u(t) which represents the loading condition under analysis, evaluate the time limited Fourier transform of the loading history L times based on

u[-M(I-l)/2] , ^■•• , u[(M-l) + M(1- 1)/2]

for 1=1, ..., L respectively to yield

ϋ^{1)(jw,T), w = 2πi/MT_s, ± = -(M/2-Ϊ),--,M/2, 1 = 1,..., L

where the notation = M Ψ ^s m U ( ^\Jϊ w^m>j ψ-⁾ i •ndicates that

-j__{s a} time limited Fourier transform of u(t) obtained over the time duration ^~MX __ The time limited Fourier transform of a time signal u(t) over the time T is normally defined by

J f-'² u(t)e ^■jwt dt r/2

(5) Evaluate

n-1^/..^/N _defi_ned by

n=l,...,N using ^u d^wX⁾ for ^1~ ---X respectively. N is the maximum order of the dominant system nonlinearities expressed as the highest order of Volterra terms which could typically be taken as 3 or 4, or if necessary this can be determined using a method which determines an appropriate truncation of a Volterra series ^* '^'^expansion of the nonlinear system. ^{'"'" '}"

The calculation can be implemented using the algorithm

Conv(x- --x) where ° denotes the n-fold convolution of vector x and the notation T in ^.3^WX⁾ ±_s omitted here for simplification of expression.

(6) Calculate ⁵u_UO and ^s»ϊ ^w) . These are estimates of

S„_u(w which is defined by

where E[.] denotes expected value and ^su_UW which is defined by

The estimates are computed as

and

(7) Evaluate s w = (w) + - +

(2π) 2(1 X-1) -H-. » S 2(N-l)

(2*0 kw]^{2 w} _uw

÷Σ _iΣj_ 7 (^2πy^⁽^- ^{> ** '}>^{ffJ " '}^""^{' "}>> ^^MTs^'i-^0''^1'' ^{' M}2

to yield a bound on the PSD response of the structure or component to the loading history which represents the considered loading condition. It should be appreciated that the result obtained is determined by ^su \^w) _f i=l,,..,N, and ^su ^w) _f i=l,...,N, j=l, ...,N, i≠j, which are statistics of the representative loading history u(t). Therefore, the result is also the bound on the PSD responses of the system to loading histories, the statistics of which are the same as the statistics of the loading history used for the analysis.

S^B (w)

(8) Evaluate the moments from ^yy obtained in Step (7) above to yield

ⁱ⁼¹ ^MTs

for n=0,l,2,4

(9) Construct a probability density function (Pdf) of the rainflow ranges (stress ranges) of the stress response under the considered loading condition from the moments obtained in Step (8) above using, for example, the Dirlik solution (Bishop N., 1997, Technique Background notes relating to frequency life fatigue estimation software module. Ncode Inc.) as

where

l_ -__{D +D} ² 1.25(y-D, -D,R)

_D = L ¹ , D₃=l-D--D₂, Q= ^ 2 ^

1-R D_x

(10) Calculate the worst case fatigue damage over a unit lifetime on the structure under the considered loading condition to yield

where

^f and ^Mf are the parameters which define the fatigue characteristics of the associated materials using the so-called S-N relationship given by

N(S) _M S^Mf

The specific values of ^f and ^f can be obtained from British Standards (BS 5400: Part 10: 1980) given the materials by which the structure to be analysed is constructed.

(11) Steps (4) to (10) are preferably repeated to evaluate the worst fatigue damage of - the structure or component under other typical^"" loading conditions represented by the corresponding loading histories or predetermined loading patterns to yield in total, for example, ^Np worst fatigue damages .

D_m, i = l, ... ,N-

where ^Dwi denotes the worst fatigue damage of the structure or component under the i-th loading situation.

(12) Evaluate the averaged worst fatigue damage ^Df!A of the structure from ^Ni' -^{L = 1/ • ■} - '^NF _f obtained above as

where ^l represents the probability of the structure working under the i-th loading condition which could be obtained, for example, from observing historical loading data collected in practice. Then, if needed, evaluate the fatigue life of the structure from f« _as 1 ^υ»*

This represents the shortest or worst case fatigue life of the structure under the considered loading conditions .

Compared to the fatigue damage or life evaluated using the technique in Section I.1.1, the result obtained using this technique is relatively conservative because the fatigue calculations are based on a bound on the PSD of the stress or strain response to a loading history rather than the PSD itself. Normally the real fatigue damage will be less than *» and the real fatigue life will be greater than

However, it should be appreciated that the bound obtained in Step (7) above is a function of the statistics of the loading history used in the analysis. This means that the PSD responses of the structure to loading histories which have the same statistics are all bounded by this result. Thus, unlike the fatigue result obtained using the technique in Section 1.1.1 which may, generally speaking, represent only the situation under one particular loading history, the fatigue evaluated using the bound covers all situations where the loading histories are statistically equivalent to the representative loading history in the sense defined by the statistics which determine the bound. The result can therefore be used to reflect the worst case fatigue of the structure or component under the condition which is statistically represented by but not exactly the same as the particular loading history case.

Moreover the averaged worst fatigue damage ^m or the shortest fatigue life ^TFW obtained using this technique can be used as an important criterion for fatigue design. If a fatigue design results in an improvement on this criterion, the design can sufficiently guarantee that the structure would have a longer shortest fatigue life under the considered loading conditions. However, if a fatigue design brings up an improvement on the fatigue result evaluated using the technique in Section I.I.I, generally we may not be able to say that a better fatigue life can definitely be achieved by the design because the result may only represent one particular loading history situation for which the fatigue analysis is performed.

It can be observed from the equation in Step (7) above that when the structure or component under

investigation is linear, the result ^{yy }} obtained using this technique becomes

S*_y (w)

[_Hl (w)}² S (w)

where ^Ξuu^(.w) ±_{s an} estimation to the PSD of the input loading history. This is the well known relationship between the PSD response of a linear system and the PSD of the corresponding input which is used for linear vibration fatigue analysis to evaluate the PSD of the output stress or strain response of a system to an input loading. Therefore, the second technique of fatigue analysis in the present invention is an extension of the current linear vibration fatigue analysis technique to the nonlinear system case.

The basic idea of the nonlinear vibration fatigue analysis techniques in the present invention is to evaluate the PSD or a bound on the PSD of the stress or strain response of the system under investigation and then assess the fatigue using, for example, the Dirlik solution which relates the stress response PSD ^' to the fatigue damage. Theoretically, the Dirlik solution requires the stress response to be a stationary and Gaussion random process. However, it should be appreciated that a considerable divergence from this rigorous assumption can be tolerated in practice (Bishop N., 1997, Technique Background notes relating to frequency life fatigue estimation software module. Ncode Inc. )

In addition, it should be appreciated by one skilled in the art that the techniques in the present invention are not limited either to the NARX model representation or to the evaluation of the fatigue;^'- based on the so- called S-N curves. Any other^'" nonlinear ^' model representation either in discrete or continuous time can be used to describe the dynamic characteristics of the structure or component under investigation for the procedure in Section I.1.1. Similarly, any other nonlinear model representation either in discrete or continuous time can be used in the procedure of Section 1.1.2 provided that the GFRF' s or an equivalent frequency domain description can be computed from such a model. In some instances it may be ^* appropriate to compute the GFRF' s or an equivalent frequency domain description by other means that do not involve time domain modelling step. The GFRF' s or an equivalent frequency domain description can for example be computed directly from the recorded time domain data. The S-N curves can be replaced by other criteria such as, for example, damage tolerance which considers crack growth provided that these criteria can be related to the stress or strain response PSD of the structure or component under investigation.

02/41193

1.2 Nonlinear vibration fatigue designs

1.2.1 Nonlinear vibration fatigue design based on the nonlinear vibration fatigue analysis ^' techniques in the present invention

The nonlinear vibration fatigue analysis techniques in the present invention, hich are based on nonlinear system modelling and frequency domain analysis techniques can be readily implemented and incorporated into a typical vibration fatigue design routine. This overcomes the problem with existing techniques and allows fatigue design for nonlinear structures or components to be implemented in a way which is analogous to the design for linear systems.

Based on the new fatigue analysis techniques but using the traditional design, analysis, redesign procedure, a nonlinear fatigue design can be implemented using the following procedure.

(1) Construct a finite element model or build a prototype of the nonlinear structure or component according to a design based on given requirements. (2) Do a fatigue analysis for the nonlinear structure or component using one of the two techniques I.1.1 or 1.1.2 in the present invention or both of them to check whether the fatigue of the structure or component satisfies the design requirements. (3) If the fatigue analysis indicates that the design is acceptable, then the design process is completed. _Otherwise go back to Step (1) to redesign and build the finite element model or prototype of the structure and then to Step (2) to check the fatigue again until the analysis indicates a satisfactory fatigue life and/or an acceptable¹ worst case fatigue damage has been achieved by the design*.

This procedure follows the typical routine for a system design for fatigue but uses the new techniques in the present invention to assess the fatigue, in which the structure design and the fatigue analysis are still two separate processes.

However, if some parameters of the mathematical model of the structural system can be related directly to the fatigue and these parameters can be adjusted to achieve a specific fatigue requirement, the fatigue design could be implemented directly based on the direct link between the fatigue and the structure parameters to achieve a desired fatigue life. This problem can be addressed using the techniques in the present invention regarding direct vibration fatigue design of nonlinear structures or components.

1.2.2 Direct vibration fatigue design of nonlinear structures or components

The basic idea of the direct nonlinear vibration fatigue design technique of an embodiment of the present invention is to develop an explicit relationship between the parameters of the structure or component which is to be designed and the variables associated with the fatigue life of the structure or component and then to determine appropriate values for these parameters using this relationship to achieve a desired fatigue life.

For a nonlinear system subject to a stochastic or random loading history, the time-limited Fourier transform ϋ(jw,T) of an input loading and the time- limited Fourier transform Y(jw,T) of the corresponding output stress or strain are related^* by the following relationship: ^{-(" '}"

Y (jw,T) = ∑ INn JH_n (jw^-jw, )γ[ϋ(j _Wl ; T)dσ_w n=l (2π)^n~l ■ i -t-— +-/-,=!/ -Z=l where

is the nth-order GFRF of the system,

W₁ ^■ • ••+ =w denotes integration of ( . ) over the n-dimensional hyper- plane w - w_x +^■ • -+w_Ω , and N is the maximum order of the system nonlinearities .

The PSD of the stress or strain response can be expressed using the above expression for Y(jw,T) as

This is an expression from which an explicit link between the parameters of the structure or component to be designed and the PSD of the stress or strain response under the condition represented by an input loading can be derived and then used for direct fatigue design. The general steps to be followed to achieve this are

(1) Map the time domain dynamic mathematical model of the structure into the frequency domain to yield a description of the system' s GFRFs in terms of the parameters in the time domain model of the system.

(2) Substitute the mapping from the time domain model parameters to the frequency domain GFRFs into the above expression of S_yy(w) to yield an explicit relationship between the parameters of the structure or component and the PSD of the stress or strain response of the structure of^**' component , to a representative input loading.

(3) Use the relationship developed in Step (2) and the definition of the variables associated with the fatigue life of the structure or component which are normally functions of the stress or strain response

PSD, such as the moments based on the PSD of the stress response, to determine the parameters of the structure or component to implement the design.

When the original structural system can be described by a nonlinear differential equation model such that

where d is the differential operator, K is the maximum order of differential, p+q=n, and n=l,...,N_D corresponds to various orders of equation nonlinearity, the mapping of the system description from the time to the frequency domain is given by:

_0n(i₁,-" XJWι)^il---0^'*

where the recursive relation is given by

n-p+l

1=1 with

Notice that the above nonlinear differential equation is a general description of continuous time nonlinear systems. The differential equation

u = 0

would be, for example, represented by ^*the equation above with following definitions

c,_. c₀₂(0,0) = _l3 c_3Q(0,0,0) = c₂, c₂₁(2,2,0) = 1

When the original structural system is described by a nonlinear difference equation model such that

N„ y(k)=∑y_n(k) n-l where y_n(k) is a 'nth-order output' given by n ^κ p ρ+q

V_π(k) = ∑ ∑ c_pq(l₁„..,l_p+q)fjy(k-l_i)πu(k-l_i)

with p+ q=n where n = l,---,N_D corresponds to various orders of equation nonlinearities,

K K

Σ -Σ- Σ

K is the maximum lag and y ( . ) , u ( . ) , and c_pq( . ) are the output, input, and model coefficients respectively, the mapping of the system description from the time to the frequency domain is given by:

n-ln-q K

⁺ΣΣ Σ^⁽ »^⁾^-J^[^^*^I+:*₍^K^_*"-.J*{_M ⁾ g=l =l*₁,J^=l n ⁺Σ ∑ ^C O (^I — >k_p)Η_atP(jw_Λ,... ,jw_dn) p=2 k k_p^l and

with

H J^w _ctL>---X^w _dn) =** H_n(jw_Λ,..., jw_αh)exp[-j(w_dl+- ^• -1^)^]

The implementation of Step (2) above depends on the specific system description. A design example will be given later to illustrate how to express the PSD response of a structure to an input loading in terms of a parameter of the structure which is to be designed.

In step (3) above, when the moments based on the PSD of the stress response are used as the variables associated with the fatigue, these variables are functions of the parameters ^' of the structure to be designed such that m₀ = m₀ P₃

=^l(-^Ps) m₂ = m₂(P_s) where P_s denotes a vector of parameters to be designed. A criterion J(P_S) could be defined using these variables such that J(P_S) = β₀m_O(P_s) + β₁m₁ (P_s) + β_zm₂(P_s) + β^P_s) for the direct design where β_x ≥ Q, 1 = 0,1,2,4 and β₀ + β₁ + β₂ + β_i = 1

The basis of the design is to minimise the criterion J(P_S) in terms of P_s so as to achieve a desired fatigue life of the structure in the sense defined by the criterion under the considered loading condition. This method is based on the observation that larger values of the moments m i=0,l,2,4 normally correspond to a greater fatigue damage or a shorter fatigue life. The design example mentioned above will be used later to illustrate this .

Although the design procedures given here are only for the fatigue design of a structure or component under a specific loading history (see Section I.1.1), it should be appreciated that the principle of the method would be the same for fatigue design under many loading conditions represented by different loading histories. Similarly, the principle is also applicable to fatigue design based on the bounds on the PSD' s of the stress or strain responses to achieve a desired worst case fatigue damage or a desired shortest fatigue life (see Section 1.1.2). II Examples

II .1 Examples of nonlinear vibration fatigue analysis

Example 1: Vibration fatigue analysis of an experimental offshore structure based on the loading histories representing three different loading conditions .

(1) The velocity and force time histories from an experimental offshore structure which is a fixed cylinder were obtained from the University of

Salford. The cylinder was subjected to random waves.

The force was measured on a small cylindrical element and the input velocity is the ambient horizontal water particle velocity at the middle point of the element. The horizontal water particle velocity reflects the loading history on the experimental structure and the measured force is proportional _.to the stress response of the structure to the random waves. The velocity and force time histories which were used for identification of a nonlinear model of the system are shown in figures 1 and 2, respectively .

(2) A nonlinear model was fitted between the time histories of the inline force and the horizontal water particle velocity using the NARMAX methodology. The data shown in figures 1 and 2 was sampled under the sampling frequency of 25HZ for this nonlinear modelling. The NARMAX methodology includes effective nonlinear system modelling techniques which include methods for model structure selection, parameter estimation, and model validation. The application of these methods in this particular situation yields a NARX model representing the process between the velocity of random waves and thβ_, .;• force response as below: •^■-■ -^{: *}-,; ^'• " -^*'

y(k)=1.5593y(k~l)-0.4582y(k-2)-0.15585y(k-3)+1.2829u(k-l) -1.195u(k~3)+4.8262u(k-3)u(k-3)u(k-3)

This model is of the general form of the NARX model with

c₀₁( l )=1 . 2829 , c₀₁( 3 )=-1 . 195 , q₀( l )=l . 5593 , q₀(2 )=-0 . 4582 q₀( 3 )=-0 . 15585 , q,₃( 3 , 3 , 3 ) =4 . 8262 , else c_pq( . )=0

(3) A system transient analysis was performed based on the mathematical model from (2) above of the experimental offshore structure to evaluate the sampled inline force response y (k) , k=0, 1,..., (L- DM/2+M-1 where L=30 and M=4000 of the structure to the loading history which represents the first loading condition under investigation. Figures 3 and 4 show the first loading history and the corresponding force response which is obtained from. the system transient analysis based on the identified NARX model.

(4) The PSD of the stress response to the first loading history was evaluated using the sampled force response data obtained in Step (3) above. The result is shown by the solid line in figure 5. The PSD of the stress response is obtained from the PSD of the force response by multiplying by a proportional constant c which depends on the geometrical configuration of the structure and is taken to be

c=15x —ιe^Λ2

in this case. In calculating the PSD of the force, a Boxcar window of length M=4000 was used and T_s=l/25 s. (5) The moments m_n, n=0, 1, 2, 4 were eval.uated from the PSD of the stress response obtained ^■ in ^''Step _. (4) t& yield

m₀ =3.75744 m₁=2.0085 JΠ₂ =1.20636' m₄=5.53986

(6) Using the Dirlik Solution, construct p (S) , a Pdf of the rainflow ranges of the output stress, from the obtained values of m₀, - τa_ιr m₂, m₄. The result is shown in figure 6.

(7) Based on the Pdf obtained in Step (6) and the moments m_n, n=0, 1,2, 4 obtained in Step (5), calculate the fatigue damage of the structure under the first loading history condition over the unit life time

Tr. = ls to yie Id

E(P)

Di [- 3^ra' p ( s)

where and K_f and M_f are determined

from British Standards under the material Detail Class W as K_f =0.16χl0¹² and m_f=3.0. The result is 0^8.069138-10

(8) Repeat Steps (3) to (7) to, for example, evaluate the fatigue damage of the structure under two other loading conditions represented by the second and third loading histories which are shown in figures 7 and 8 respectively. The results obtained are

D₂ =9.98608e-ll D₃ =2.73610e-09

Figures 9 and 10 show the time domain force responses of the structure to the second and third loading histories respectively and the PSD' s of the corresponding stress responses are shown by solid lines in figures 11 and 12 respectively. The moments m_n , n =0, 1,2,4 , obtained using t,b.e second stress response PSD are ^•-':-^{■* ••}.^{; *}' -^■*

m₀ =0.91923 m₁ =0.49442 m₂ =0.29151 m₄ =0.60061 and the corresponding Pdf is shown in figure 13. The moments m_π, n=0, 1, 2, 4 obtained using the third stress response PSD are m₀=8.35732 m₁=4. 8713 m₂ = 2.64344 m₄ =5.63149 and the corresponding Pdf is shown in figure 14.

(9) Assuming the probabilities of the structure working as predicted under each loading condition to be P₁=0.6 F₂=0.2 P₃=0.2 and evaluating the averaged fatigue damage D_A of the structure under this assumption yield D_A = P_lD_l + P₂D₂ + P₃D₃ = 1.05134e - 09

The fatigue life of the structure is then obtained as

1 ^ψ = = 30.16133 years

31536000xD_A

The result obtained in this example reflects the fatigue damage or life of the experimental offshore structure under the loading conditions represented by the three loading histories shown in figure 3, figure 7, and figure 8. The identified nonlinear dynamic model given in Step (2) above is used to implement the structure transient analysis in Step (3) . This allows the fatigue analysis to be easily performed and to be incorporated into a fatigue design routine.

But in many practical cases the nonlinear vibration fatigue analysis performed in this way, which is based on a specific loading history for the analysis under each loading condition, may not yield a result which covers all loading history situations within the range of the considered loading condition. For example, the output PSD' s of a nonlinear system will only-,.be the same' under inputs with the same PSD' s when the input is a real, stationary process with zero mean , and a Gaussion distribution. When the loading histories of a structure or component do not satisfy these conditions, the fatigue analysis result obtained in this way based on one loading history could be different from the result obtained in the same way but based on another loading history. This is because the PSD' s of the output stress responses to the two input loading histories may be different even if the two loading histories are the same in as much as they are categorised under the same loading condition in terms of the PSD. In other words, the fatigue analysis result obtained based on one loading history could be different from the result obtained based on another loading history although the two loading histories are under the same loading condition because they have the same PSD.

Generally, the output PSD of a nonlinear system depends on more statistical information of the corresponding input rather than just the input PSD. Take a simple case such as, for example, a nonlinear system with only a third order nonlinearity in the input, the statistical information defined by the sixth-order moment of the system input will be required to characterise the corresponding output PSD. Therefore, in general when the system is nonlinear the class of input loadings which can produce the same stress or strain response PSD could be very small. This means that any fatigue analysis based on one specific loading history should not be assumed to cover all loading histories which could reasonably be considered to be under the same loading condition. Therefore, the fatigue analysis result for a nonlinear system obtained based on one loading history may, in some cases, represent only the result for that particular loading history alone. This is why .....the second ^••-technique of the present invention has been developed to address nonlinear vibration fatigue analysis problems.

In the following an example is used to illustrate how to use the second technique (see Section 1.1.2) to compute the fatigue. This example uses the same loading histories as in the first example in order to compare the results obtained using the two different techniques.

Example 2: Vibration fatigue analysis of an experimental offshore structure based on statistics of the loading histories representing three different loading conditions.

(1) The same as Step (1) in Example 1

(2) The same as Step (2) in Example 1.

(3) Using the first method, evaluate the gain bounds of the GFRFs of the nonlinear dynamic model obtained in

Step (2) under the constraint w₁ + - -- + w_n - w to yield first

H₂=0, H„=0, n>4 and ^c03 ⁺∑^cu^H2 + c₂₀HιH2 +∑c₁₂Hi +∑c₂₀H₂H ∑c_Q3 4 . 8262

+∑C₂₁(H?)² +∑C₃₀(H₁ ^B)³ , and then

1 . 2829exp f jw/25)-l . 195exp fr-3 jw/25)

|l-l . 5593exρ f jw/25)+0 . 4582exp (-2 jw/25) + 0 . 15585exp(-3 jw/25)

iϊ|(w)= C (jw^*];) L₃H = 4.8262 C(w/25)

4 . 8262 l-1 . 5593exp fjw^*/'25)+0 . 4582exp f2jw/25)+0 . 15585exp(-3jw/25j (4) Evaluate the time limited Fourier transform of the loading history shown in figure, 3,_Λ which represents the first loading condition, L=30 times based on the sampled series of the loading signal u[ 4000(I-l)/2 ], ••^■ ,u[ (4000-l)+ 4000(l-l)/2 ] I =l,--,30 to yield

[7⁽ⁱ⁾ ( !^*v-,:r = 4000/25 = 160s) w = 2Λi/l 60 i = - (4000/2 - 1 ) ,- - - , 4000/2

1 = 1, — ,30

Figure 15 shows one of the 30 time limited Fourier transform results.

(5) Evaluate ϋ ^](w,T = 160s), n=l,3, from U^{I) (w,T = 160s) obtained in Step (4) above for l =l,---,30 using the algorithm in Step (5) on page 26 to yield.

ϋ[¹ (2πi/l 60,T

30 and

Figures 15 and 16 show one of the 30 results of

i=0,—,4000/2, respectively.

(6) Calculate the estimation results for S_]_u(w) , s"(w) , and ^s _uu ^w) using U„^{{ )}(w,τ = 160s), n =l,3 , 1=1, ...,30, obtained in Step (5) to yield S_ _u(w) , s£jj(w) , and S^_u(w) where

The results are shown in figures 17-19. (7) Evaluate a bound on the PSD of the stress response of the structure to the loading history representing the first loading condition as (w)

w = 2πi/l 60 i = 0 , — , 4000/2 where c is a coefficient reflecting the relationship between the PSD of force and the PSD of stress and is taken to be

16 c=15x π here. The result is shown by the dashed line in figure 5.

(8) Evaluate the moments m_n , n =0, 1, 2, 4, from S_yy(w) obtained in Step (7) to yield

JΠ₀=7.36010 =3.90780 m₂=2.32209 m₄ =1.36707

(9) Using the Dirlik Solution, construct p(S), a Pdf of the rainflow ranges of the output stress, from the obtained values of m₀, m_x, m₂, m₄. The result is shown in figure 20.

(10) Based on the Pdf obtained in Step (9) and the moments m_a , n =0,l,2,4 obtained in Step (8), calculate a worst fatigue damage of the structure under the first loading condition over the unit life time τ_L = ls to yield

where E(p)=. ^■0.6974, and K_f and M_f are determined

from British Standards under the material Detail Class as K_f=0.16xl0¹² and m_f=3.0. The result is

„=2.00834e-09 (11) Repeat Steps (3) to (10) but. ^'* evaluate the.*^' worst fatigue damage of the structure under two other loading conditions represented by the second and third loading histories which are shown in figures 7 and 8 respectively. The results obtained are

D_W2=1.16820e-10 D_m =1.65991e-08

The bounds on the PΞD's of the stress responses of the structure to the second and third loading histories are shown by dashed lines in figures 11 and

12 respectively. The moments m_n, n =0, 1, 2, 4 , obtained using the bound on the second stress response PSD are m₀ =1.09709 m₁ ~ 0.59146 m₂=0.34581 m₄=0.17555 and the corresponding Pdf is shown in figure 21. The moments m_n, n =0,l,2,4 obtained using the bound on the third stress response PSD are m₀ =30.08130 m₁ =15.96680 m₂ =9.80847 m =6.35577 and the corresponding Pdf is shown in figure 22.

(12) Assuming that probability of the structure working as predicted under each loading condition to be

P₁=0.6 P₂=0.2 P₃ = 0.2 and evaluating the averaged worst fatigue damage D_WA of the structure under this as sumption yield

D_WA = P_lD_wl + P₂D_ff2 + P₃D_W3 = 4.54B18e - 09

The shortest fatigue life of the structure can then be obtained as 1 FW — ^" ■ = 6.97197 years

31536000x 0, WA

Compared* to the fatigue analysis results D_A or T_F obtained in Example 1, D_m or r_Fff obtained in Example 2 is obviously more conservative. This is natural because (i) The same loading histories, representing three different loading-- -conditions are used in the two examples.

(ii) The result obtained in. Example 1 is the real ^• fatigue damage or life of the experimental offshore structure under the specific loading conditions represented by the three loading histories. However, the result may only be correct for these three specific loading history cases.

(iii) The result obtained in example 2 reflects the worst fatigue damage or shortest fatigue life of the structure under the loading conditions which are represented by the three loading histories but the analysis also accommodates all possible loading history cases which are statistically equivalent -to the representative loading histories in the sense of having the same statistics S (w), S_u ³ _u (w), and S™(w).

D_A and T_F reflect the real f tigue of the structure under the specific loading histories, whereas D_WA and T_FW are the worst fatigue damage and shortest fatigue life of the structure under the loading conditions represented by the loading histories in terms of a set of statistics. In practice it would be informative to evaluate both D_A (or T_F) and D_m (or τ_Ff/) to provide a complete profile of the fatigue of the structure under investigation.

Figures 5, 11, and 12 show comparisons between the PSD responses of the structure to the specific loading histories and the corresponding bounds. The former was used for the fatigue analysis in Example 1 and the latter was the basis when performing the worst fatigue calculations in Example 2. It can be,;,, observed from the figures that, apart from the second loading'' history situation where the amplitude of the system input is relatively small and the structure - can actually be approximately regarded to be linear, there exists a significant difference between the PSD responses and the bounds. This is the general situation regarding the PSD of a practical nonlinear system output and its bound and a direct reason why a more conservative fatigue analysis result was obtained in Example 2.

However, in some specific input loading cases, the PSD of a nonlinear system output could be very close to its bound and the fatigue analysis result obtained using the second nonlinear vibration fatigue analysis technique can also be used to represent the real fatigue damage of the structure in these cases. An example of this is given in the following.

Example 3: Vibration fatigue analysis of an experimental offshore structure under a specific loading history.

In this example, the experimental offshore structure is the same as the structure analysed in Examples 1 and 2. Therefore, the dynamic model description of the system and the bounds on the GFRFs of the system model which are needed for the fatigue analysis are all the same as in the previous examples. The specific loading history under investigation is a signal as _.shown in figure 23 whose analytical description is given by

where a₀ = 0.3239 x 2 x π^* , b_Q = 0.6002x 2 x ;r , and M_u =3.2 . • Apply the first nonlinear vibration fatigue analysis technique (see Section 1.1.1) of ^' 'an^'" embodiment ^"of the present invention to evaluate the structure fatigue under this specific loading history.

Figure 24 shows the PSD of the stress response of the structure to the specific loading history as a solid line. The moments m_n , n = 0,1,2,4, evaluated from the PSD of stress are

m₀=1.0e + 03x4.72545 ^=1.06 + 03x2.23605 m₂ =l^*.0e + 03xl.15460 m_i =1.0e + 03x0. 9064

and the corresponding Pdf is shown by the solid line in figure 25. Based on the moments and the Pdf, the fatigue damage of the structure under the given specific loading history can be evaluated to give the result

£=2.90547e-05

• Apply the second nonlinear vibration fatigue analysis technique (see Section 1.1.2) of an embodiment of the present invention to evaluate the worst structure fatigue under the loading condition represented by this specific loading history.

Figure 24 shows the obtained bound on the PSD of the stress response of the structure to the specific loading history as a dashed line. The moments _n , n =0,1,2,4, evaluated from the bound on the PSD of stress are

+ 03xl.47194 m_Λ =1.0e +03x0.59128 and the corresponding Pdf is shown as. a dashed line in figure 25. Based on the moments <^:--an<d Pdf, ^* the^' worst fatigue damage of the structure under the loading condition represented by the specific loading history is obtained as

D_H =4.20140e-05

Comparing D and D_w shows that the two nonlinear vibration fatigue analysis techniques yield similar results. The reason for this can be explained from figure 24 which indicates that the^* PSD of the stress response to the particular loading history is very close to the evaluated bound on this PSD.

Example 3 reflects an important phenomenon concerning' the PSD of a system stress or strain response which is the basis of the first technique of fatigue analysis in this invention and a bound on the PSD which is the basis of the second technique of fatigue analysis in this invention. Although, generally, there may be a considerable difference between the PSD and its bound so that a relatively conservative result is obtained using the second technique to yield a worst case estimation of fatigue under the considered loading condition, . under some specific loading histories such as, for instance, the above loading history for Example 3, the PSD and its bound are very close so that the worst case fatigue damage obtained using the second technique is no longer a conservative result; it can also represent the real fatigue damage of the structure under the specific loading history which is used for the analysis. II .2 An example of direct nonlinear, vibration fatigue design "" "

Example 4. Implementation of a direct nonlinear vibration fatigue design for a mechanical system

Consider a fatigue design problem for a simple vibration suspension system composed of a mass, a spring and a damper as shown in figure 26 where u(t) is the base displacement representing the input loading, and y_d(t) is the displacement of the mass which is assumed to be proportional to the stress response y(t) of the structure such that y(t) = κ_sYd(t) Under the assumption that the damper characteristic f_d(.) can be represented as a third order polynomial but without the second order term, the relationship between the input loading and the stress response for this specific structure can be described as

[ y(t) = κ_sy_d(t)

[m_by_d (t) = k_s [u(t) - y _d (t)] + a_λ [ύ(t) - y_d (t)] + a₃ [ύ(t) - y_d (t) j³

The vibration fatigue design of the mechanical structure is implemented in the following, assuming that the structure is subject to a stochastic loading history and all the parameters of the system are fixed except for a₃ which is to be designed to achieve a desired fatigue life .

(1) Map the time domain description of the system into the frequency domain

The mapping from the time domain description of the suspension system to the frequency domain can be determined as

(2) Express the PSD of the 'stress response in terms of the parameter to be designed

Assume N=3 and substitute the expressions above for ffiOi),

^{snd H}. J^wvJ^wzX^w_) into the general expression of S_yy(w) to yield an expression of S_yy(w) in terms of the parameter a₃ as

S_yy (w) = (w)a₃ ² + 2C₂ (w)a₃ + C₃ (w)

where

E{Re[N(w,T)L(w,T §

C₂ (w) = lim

in which L(w,T) and N(w,T) are defined by

L(w,T) = H₁(j w)ϋ(jw,T)

where L(w,τ) denotes the conjugate . of (w,T) Note that for any w, C_x(w), C₂(w), and C₃(w) are functions of the characteristics of the given ^'"stochastic loading history and the fixed structure parameters κ_s , k_s, m_b, and a_x . The PSD response of the structure to the given loading history can therefore be determined from the parameter a₃ from the expression of S_yy(w) given above.

(3) Relate the parameter of the structure to be designed to the fatigue and implement the fatigue design

The moments based on the PSD of the stress response can be expressed using the result in Step (2) above in terms of a₃ as m_n(a₃) = α_x(n)al + α₂(n)a₃ + α₃(n) n=0 ,1,2,4

where α_x(ή) = [fⁿC,(2πf)df

α₃(n) = [ fⁿC₃(2πf)df

Because the fatigue can be directly determined from the moments m_n, n=0,l,2,4, using, for example, the Dirlik solution, which defines the relationship between the fatigue damage D of the structure and the moments based on the PSD of the stress response, the above relationship between the moments and the parameter a₃ is an important link between the fatigue and the parameter to be designed and can therefore be employed in fatigue design.

The design can be implemented in many different ways based on the relationship between the moments ^ 1=0,1,2,4 and a₃. One of these methods could be, for example, to define a criterion J(a₃) using the bments m_i i=0,l,2,4 such that

J(a₃) = β_Qm_Q(a₃) + β_xm₁(a₃) + β₂m₂(a₃) + β_im__i(a₃)

where β. ≥O, i=0, 1,2,4 and β_Q + β_x + β₂ + β₄ =ι , which can then be minimised in terms of a₃ to achieve a desired fatigue life associated with the weights β₀, β_x, β₂, β_i in the criterion. This method is based on the fact that larger values for the moments _i 1=0,1,2,4 normally correspond to a greater fatigue damage or a shorter fatigue life. The optimisation is straightforward in this specific case since J(a₃) can be further written as

^*J(a₃) = ∑β_nm_n (a ) = ∑β_n [α (ή)al + ₂(n)a₃ + α₃ (ή ] = γ₁a + γ₂a₃ + γ₃ n=0,n≠3 n=0,-n≠3 where

4 4 4

7ι = ∑β_α<*ι(n)> ϊz = β ifl), ^and 7_ ^•= ∑β_nα₃(^ή) n=0,π≠3 Π=0,Λ≠3 n=0,-π≠3

This i s a s imple second order polynomial in terms of a₃ . Since , theoretically,

γ_x = ∑β_nα_x(n) > 0 n=0,n≠3 the optimal a₃ which produces a minimal J(a₃) can be obtained as

Now consider a specific case where K_s = \MPαlm, m_b =240 kg, k_s =16000N/m and _\_x =2960 and the specific loading history to be investigated is as shown in figure 27 which is a random signal generated by passing a zero mean Gaussian distributed random process with a standard deviation 0.005- through a Butterworth bandpass filter the passband of whi"ch--^:*i.s [1,10] 'HZ.-^*'

The mapping from the time domain description of the system to the frequency domain can be directly obtained by substituting these specific parameter values into the above analytical expressions of

H_x(jw_x), H₂(j w_x,jw₂), and H₃ jw_x ,jw₂,jw₃) .

The stress response PSD of the structure to the specific loading history in terms of the parameter a₃

S_yy (w) = q (w)al + 2C₂ (w)a₃ + C₃ (w) can be obtained by evaluating C_x(w), C₂(w), and C₃(w) practically from the PSD responses of the structure to the given loading history under three different values of a₃. The results of C_x(w), C₂(w), and C₃(w) obtained for this specific case are shown in figures 28, 29, and 30, respectively.

The moments based on the PSD of the stress response can then be expressed in terms of parameter a₃ as

m_n (a₃) = a_x(n)a₃ + ₂(ή)a₃ + α₃(n) n=0, 1,2,4

where _x(n), α₂(n), and α₃(n), n=0, 1, 2, 4, are the results evaluated from C_x(w), C₂(w), and C₃(w) and are given in the table below.

Define a criterion for the fatigue, design as

The weights β₀, β_x, β₂, β_ are taken as /J₀=0.65, /J-=0.34, ?₂=0.01, and β_t =0 and

4 4 4

Υi - ∑βn^{α n} > 72 = ∑^_π«ϊ(^ή), 7₃ = ∑β_nα₃ ⁿ) n=Q,n≠3 n=0,n≠3 n=0,n≠3 where _x(ή), α₂(n), and ₃(n), n =0, 1, 2, 4, are the values given in the table above. The optimal a₃ , which causes the criterion to reach a minimum so as to achieve a desired fatigue life in the sense defined by this criterion, can then be obtained as

This example illustrates a very simple case of direct nonlinear vibration fatigue design. It will be appreciated that similar design principles can readily be extended to very general situations. Note that the key point of this technique is to relate the structure parameters to be designed to the PSD of the stress or strain response, this can then lead to a criterion associated with the fatigue in terms of these structure parameters, and finally the structure design for fatigue can be implemented by determining these parameters using a ^' optimisation routine to achieve a minimum or maximum for the criterion.

Claims

1. A method for computing a fatigue characteristic of a nonlinear structure or component based on a predetermined loading pattern that represents a loading condition, the method comprises the steps of establishing a nonlinear dynamic model which represents the structure or component, producing a stress or strain response for the structure or component from the nonlinear dynamic model by performing a transient analysis of the structure or component when the structure or component is subject to the predetermined loading pattern; determining a power spectral density (PSD) of the stress or strain response of the structure or component to the predetermined loading pattern, computing the spectral moments from the PSD, constructing a probability density function (Pdf) of the stress or strain ranges from the spectral moments, and calculating the fatigue characteristic of the structure or component using the spectral moments and the constructed probability density function.

2. A method as claimed in claim 1 in which the step of establishing a nonlinear dynamic model which represents the structure or component comprises the steps of analysing known loading data and a corresponding stress or strain response of the structure or component using a nonlinear system identification technique to produce the nonlinear dynamic model.

3. A method as claimed in claim 2 in which the nonlinear system identification technique is at least one of the NARMAX methodology, parametric regression methods, non- parametric methods , nonlinear - estimations , training or learning , in either discrete time /-or-. -continuous' time .

A method as claimed in claim 3 in which the NARMAX methodology comprises establishing

N „ y (k) = ∑ y _n ( k) n = l

where ^ynW f_{s a} ' nth-order 'output ' given by

with ^{p + q n} where ^{n > ' " > ND} corresponds to various orders of equation nonlinearities,

K K K

Σ -Σ- Σ

K is the maximum lag, and y(.)_f u(.)_f and c_π„(.) ^pq are the system output, input, and model coefficients respectively.

5. A method as claimed in any preceding claim in which the step of performing the transient analysis comprises the steps of applying the predetermined loading pattern to the established nonlinear dynamic model to produce the stress or strain response.

6. A method as claimed in any preceding claim in which the step of determining a power spectrum density (PSD) of, the stress or strain response of the structure or component to the predetermined loading pattern comprises the step of evaluating

where ^γ ^w), 1=1,...,L, _{is the Four}j_e_r transform of the stress or strain response y(t) evaluated using an FFT from the windowed sampled data of the response

y[M(l -l)/2]ft(l) , ^•- , y[(M - l) + M(l ~ l)/2]h(M), 1 = 1, ...L

and h(i), i=l,...,M, represents a window function.

7. A method as claimed in any preceding claim in which the step of constructing a probability density function for the stress or strain ranges comprises the step of computing the spectral moments from the power spectrum density.

A method as claimed in claim 7 where the step of computing the spectral moments from the power spectrum density comprises the step of evaluating

for n=0,l,2,4

9. A method as claimed in claim 7 in which the step of constructing the probability density function comprises the step of evaluating

D, ~ D_?Z ~ —

-Xe + X-e²*² + D₃Ze ² p ( S ) = f (m₀,m₁,m₂, ₄) = Q R² !___

where

1-y- ^■Ό_X + Ό_X 1.2S(y-O₃-D^' ₂Rr

D₂ = ^D ₃ =1-D₁-D₂, Q =

1-R

R = - r-^χ _m-^π l_ __{Dι +D}2

10. A method as claimed in any preceding claim in which the step of calculating the structure or component fatigue damage or fatigue life time using the probability density function comprises evaluating

_O__ M s p(S)dS κ_f J

where

, ^f and f are the parameters which define the fatigue characteristics of the associated materials using the S-N relationship given by

K_f S^Mf

11. A system for computing a fatigue characteristic of a nonlinear structure or component based on a predetermined loading pattern that represents a loading condition, the system comprises means for establishing a nonlinear dynamic model which represents the structure or component, means for producing a stress or strain response for the structure or component from the nonlinear dynamic model by performing a transient analysis of the structure or component when the structure or component is subject to the predetermined loading pattern; means for determining a power spectral density (PSD) of the stress or strain response of the structure or component to the predetermined loading pattern, computing the spectral moments from the PSD, constructing a probability density function (Pdf.)^". of the stress or strain ranges, and calculating the ^' fatigue^* characteristic of the structure or component using the spectral moments and the probability density function.-

12. A system as claimed in claim 11 in which the means for establishing a nonlinear dynamic model which represents the structure or component comprises means for analysing known loading data and a corresponding stress or strain response of the structure or component using a nonlinear system identification technique to produce the nonlinear dynamic model .

13. A system as claimed in claim 12 in which the nonlinear system identification technique is at least one of NARMAX methodology, parametric regression methods, non- parametric methods, nonlinear estimations, training or learning, in either discrete time or continuous time.

14. A system as claimed in claim 13 in which the NARMAX methodology comprises means for establishing

Nn y(k) = ∑ y_n(k) n-l

where ^y»^ is a 'nth-order output' given by

p+q y _n ( k) = ∑ ∑ ^C _pq ( l 1 ' p=0 1₁ ,l_{p+ q}=l * r i=[l ^{y (k} -ⁱi⁾ ii=pi+l^{u (k} -¹!-

with P^{+ c}3-ⁿ where ^{n •*}->^'">-^Nβ corresponds to various orders of equation nonlinearities,

K K K

Σ -Σ- Σ K is the maximum lag, and y(.), u(.), and- c- j- t . )

, ^pq are „ the system output, input, and model coefficients respectively.

15. A system as claimed in any of claims 11 to 14 in which the means for performing the transient analysis comprises means for applying the predetermined loading pattern to the established nonlinear dynamic model to produce the stress or strain response.'

16. A system as claimed in any of claims 11 to 15 in which the means for determining a power spectrum density (PSD) of the stress or strain response of the structure or component -to the predetermined loading pattern comprises means for evaluating

0,l,...,.N/2

y(-1) ( -j _w -I _ -] T where ^r---^ru _{r 1S} ^he Fourier transform of the stress or strain response y(t) evaluated using an FFT from the windowed sampled data of. the response

y[M(l-l)/2]i(l) , ••• , y[(M - l) + M(l - l)/2]h(M), 1 = 1, ...L

and h(i), i=l,...,M, represents a window function.

17. A system as claimed in any of claims 11 to 16 in which the means for constructing a probability density function for the stress or strain ranges comprises means for computing the spectral moments from the power spectrum density.

18. A system as claimed in claim 17, /.-where the means for computing the spectral moments " 'f om the -power'' spectrum density comprises means for evaluating

for n=0,l,2,4.

19. A system as claimed in claim 17 in which means for constructing the probability density function comprises means for evaluating

p(S) =f (m₀,m₁,m₂,m₄) =

where

m, m, m,

Z= - _™ = ^' 2 ^(χ„ -r⁾

2,/m_π f m_n 'in, ^Dι = i + r²

l-γ - O_x + O² _x _1.25(^* -D₃-D₂R)

D₂-.= . i ± ±., D₃=l-D₁-D₂, Q

1-R O_χ

R= l^y_j,^χ_-_D^_{ι + D} ²

20. A system as claimed in any of claims 11 to 19 in which the means for calculating the structure or component fatigue damage or fatigue life time using the probability density function comprises means for evaluating E(p)

D=- J S^M*p(S)dS

K_f where E

, f and f are the parameters which define the fatigue characteristics of the associated materials using the so-called S-N relationship given by

N(S) = ^ S ^£

21. A method for computing a fatigue characteristic of a nonlinear structure or component based on a set of statistics of a predetermined loading pattern that represents a loading condition, the method comprises the steps of establishing a nonlinear dynamic model that represents the structure or component, evaluating gain bounds of generalized frequency response functions (GFRF) of the nonlinear model, determining a bound on the power spectral density of the stress or strain response of the structure or component to the predetermined loading pattern, the bound on this power spectral density being a function of the set of statistics of the predetermined loading condition and the gain bounds of the generalized frequency response functions of the nonlinear model, computing the spectral moments from the bound on the power spectral density; constructing a probability density function (PDF) of the stress or strain ranges, from the spectral moments, and calculating the fatigue characteristic under the predetermined loading condition using the spectral moments and the probability density function.

22. A method as claimed in claim 21 in which the fatigue characteristic is at least one of fatigue damage or fatigue life representing worst cas.e fatigue or shortest fatigue life time. ' •'<^*:---. ^' "^** " ^■•"

23. A method as claimed in any of claims 21 to 22 in which the step of establishing a nonlinear dynamic model which represents the structure or component comprises the_. steps of analysing known loading data and a corresponding stress or strain response of the structure or component using a nonlinear system identification technique to produce the nonlinear dynamic model.

^' 24. A method as claimed in claim 23 in which the nonlinear system identification technique is at least one of NARMAX methodology, parametric regression methods, non- parametric methods, nonlinear estimations, training or learning, in either discrete time or continuous time.

25. A method as claimed in claim 24 in which the NARMAX methodology comprises establishing

ND yW = ∑ y a(k⁾ "-I where ^y^ -¹ is a 'nth-order output' given by

with P ^{+ c}3 ⁿ where ⁿ X ^{' " > N}D corresponds to various orders of equation nonlinearities,

K K K

Σ -Σ- Σ

K is the maximum lag, and y(.), u(.), and c_nα(.) ,, , ^{a J pq} are the system output, input, and model coefficients respectively.

26. A method as claimed in any of claims 21 to 25 in which the step of determining the ouri'd on power ^{" '} spectral density of the stress or strain response of the structure or component to the predetermined loading pattern comprises the step of evaluating gain bounds of generalised frequency response functions of the nonlinear dynamic model under a predetermined frequency variable constraint.

27. A method as claimed in claim 26 in which the step of evaluating the gain bounds of the generalised frequency response functions under the predetermined frequency variable constraint comprises the steps of evaluating estimates for the gain bounds of the generalised frequency response functions of the nonlinear dynamic model under no constraints and using the estimates to compute the gain bounds of the generalised frequency response functions under the constraint ^w> ^{'" w}" ^{~ w} .

28. A method as claimed in either of claims 26 and 27 in which the gain bounds of the generalised frequency response functions of the nonlinear dynamic model under the predetermined frequency variable constraint are evaluated as follows: denote the gain bounds of the GFRFs with no constraints as

and the gain bounds of the GFRFs under the constraint w, + ^■■• + w_n = w _as

^a _π , _^.,_.,... _can ^_e determined.:-^1, using a recursive algorithm and the NARX model coefficients as follows

For n=l

where

Σ c₀i(.*i)exp⁽rjv, dl^kl.

H^dι jw_dl) =

^{1 ~} ∑^co(*ι)^exp(-jw_Λ*!)

*₁₌1

and [a,b] is the interval which represents the frequency range of the possible loading histories.

For n≥2

where

L_n = min ^1_Σ^cιo(^kι)^e PHw_{dl 1}) k₁₌l

and ^n represents the possible frequency range produced by the nth-order^' nonlinear output which can be determined from the input frequency range [a, ,b_ using the following formula ' ■ ■^•• ^{' •}-• " " ^*'

[] means to take the integer part, na i = + 1

(a + b)

I_k = [na - k(a + b),nb - k(a + b)) for k = 0,...,±^* - 1, I.„ = [Q,nb - i^*(a + b)]

Then determined as

where

C(wT_s)=

1- ∑c_1Q(k₁) exp [-jτ_swk ) k,=l

29. A method as claimed in claim 26 in which the step of determining the bound on power spectral density of the stress or strain response of the structure or component to the predetermined loading pattern comprises the step of evaluating gain bounds of generalised frequency response functions of the nonlinear dynamic model under a predetermined frequency variable constraint using an optimisation procedure or a genetic algorithm.

30. A method as claimed in claim '29- ^; in which^{' '}"the*'^' step of evaluating the gain bounds using an optimisation procedure or a genetic algorithm under the predetermined frequency variable constraint comprises the step of optimising

31. A method as claimed in either of claims 29 and 30 in which the step of evaluating the gain bounds using an optimisation procedure or a genetic algorithm under a predetermined frequency variable constraint comprises calculating the values of the generalised frequency response functions using the following recursive algorithm:

H_π (jw_x, -,j w_n ) = H^d _n (j T_sw_x,...,j T_sw_n )

where

{l ' ∑ c_l0(k₁)Qxp[-j(w_dl+,- --,+w_d__n)k₁]}H^d„(jw_dX,..., jw_dn)

JW_dx,..,X W_dll)

where ^{W di} = T„W 1 = 1,- rn and

with

32. A method as claimed in any of claims 21 to 31 in which the step of determining the bound of the power spectral density comprises the step of evaluating

where ^uu ' , i=l, ..., N, 3=1, ..., N, l≠j , are a set of statistics of the predetermined loading pattern.

33. A method as claimed in claim 32 in which the step of evaluating

^i≠J' comprises the steps of evaluating a time limited Fourier transform of the predetermined loading pattern u(t) L times based on the samples

U[M(1 - 1)/2] , ■ • • , u[(M - 1) + M{1 - 1)/2]

for 1=1, . . . , L respectively to yield ϋ^w(jw,T), w = 2πi/MT_s, ± = -(M/2-l),---,M/2,^>'" 1=1, ... ,L

where the notation ^{τ MTs} in ^{ϋ (Jw}>^τ) indicates that ' jX>(^j-iw™,- T^>-⁾~) j__{s a} time limited Fouri ■er transform of u₍t) obtained over the time duration ^T~MT _Sf th._e time limited Fourier transform of a time signal u(t) over the time T being defined by

£ ^)e -jwt dt

v_alu_atin_g ^ϋ" for), n-1, ..,N _{defined y}

\ϋ^w_j(w-w_l,...-w_n__x,τj_idw_x.,d.w_n_₁ n=l,...,N for ¹ = ¹,---, respectively, where N is a predetermined, preferably maximum, order of dominant system nonlinearities expressed as the highest nonlinear order in the Volterra series expansion of the system, using the following algorithm

Conv(x---x) < tl. where " denotes the n-fold,-..convolution*, -of-*- vector x and the notation T in ^ϋ v^w>τ) ±_s omitted here for simplification of expression, and calculating

^suu ^w _and ^fo, which are estimates of ^s (^w^ which i< defined as

where E[.] denotes expected value and ^suυ ^(wJ which i. defined by

, using

and

i =l, ... ,N, j = l, ... ,N i≠j

34. A method as claimed in any of claims 21 to 33 in which the step of computing the spectral moments from the bound on the power spectrum density comprises the step of evaluating

for n=0, 1,2,

35. A method as claimed in any of claims 21 to 34 which comprises the step of constructing the probability density function of the stress or strain ranges using the spectral moments.

36. A method as claimed in claim 35 in which the step of constructing the probability density function comprises the step of evaluating

p(S) = f (m₀,m₁,m₂,m₄)

where

_ l -γ - O_x + Ό 1.25 (y - D_j -D₂R)

, D₃=l-D₁-D₂, Q =

^~ 1-R r-*_ >2

^■ Ό;

R^*=- l-^*-D₁ + Ό_X

37. A method as claimed in any of claims 21 to 36 in which the step of calculating the fatigue characteristic comprises the step of evaluating

E(P)

D_w = J^" S^Mrp(S)dS

K, where E(p)

, f and ^M* are the parameters which define the fatigue characteristics of the associated materials using the S-N relationship given by

»<« _s

38. A method as claimed in . any of claims 21 to 37 further comprises the steps of calculating a plural of values for the fatigue characteristic each for respective predetermined loading patterns and calculating an average value for the fatigue characteristic.

39. A method as claimed in claim 38 in which the step of evaluating the average value of the fatigue characteristic comprises the step of evaluating

where ^DmX^{χ i r} - - - ^r^_{P are} ^he plurality of values for the fatigue characteristic and represents the probability of the structure or component working under the i-th predetermined loading pattern.

40. A method as claimed in claim 39 further comprises the step of evaluating the fatigue life time of the structure or component using the calculated fatigue characteristic using at least one of T=l/D_wi or T=1/D_WA.

41. A system for computing a fatigue characteristic of a nonlinear structure or component based on a set of statistics of a predetermined loading pattern that represent a loading condition, the system comprising means for establishing a nonlinear ^' dynamic model that represents the structure or^" -component, ^''"^■'' "^''• means for evaluating gain bounds of generalized frequency response functions (GFRF) of the nonlinear model, means for determining a bound on the power spectral density of the stress or strain response of the structure or component to the predetermined loading pattern, the bound on the power spectrum density being a function of the set of statistics of the loading condition and the gain bounds of the generalized frequency response functions of the nonlinear model, means for computing the spectral moments from the bound on the PSD, constructing a probability density function (PDF) of the stress or strain ranges from the spectral moments, and calculating the fatigue characteristic under the predetermined loading condition using the spectral moments and the probability density function.

42. A system as claimed in claim 41 in which the fatigue characteristic is at least one of fatigue damage or fatigue life representing worst case fatigue or shortest fatigue life time respectively.

43. A system as claimed in any of claims 41 to 42 in which the means for establishing a nonlinear dynamic model which represents the structure or component comprises means for analysing known loading data and a corresponding stress or strain response of the structure or component using a nonlinear system identification technique to produce the nonlinear dynamic model.

44. A system as claimed in claim^' 43, .in which the nonlinear system identification technique ^" i"s^{': '}a^;t least one "of NARMAX methodology, parametric regression methods, non- parametric methods, nonlinear estimations, training or learning, in either discrete time or continuous time.

45. A system as claimed in claim 44 in which the NARMAX methodology comprises establishing

y(k) = ∑ y_n(k) n = l

where ^' -¹ is a 'nth-order output' given by

p+q

_Yn(k) =∑ ∑ c_pq(l ,,..., l_p+q) y(k-l_i)π ( -1,) p=01_l,l_p+q=l i=p+l

with P^{+ (}3 ⁿ where ⁿ '^'"'' ^D corresponds to various orders of equation nonlinearities,

K is the maximum lag, and y(-), u(.), and c_nrτ(.) ^pq are the system output, input, and model coefficients respectively.

46. A system as claimed in any of claims 41 to 45 in which the step of determining the bound on the power spectral density of the stress or strain response of the structure or component to the predetermined loading pattern comprises means for evaluating gain bounds of generalised frequency response functions of the nonlinear dynamic model under a predetermined frequency variable constraints.

47. A system as claimed in claim 46 in. which^* the means for evaluating the gain bounds of -the -.generalis'ed -frequency response functions under the predetermined frequency variable constraint comprises means for evaluating estimates of the gain bounds of the generalised frequency response functions of the nonlinear dynamic model under no constraints and means for using the estimates to compute the gain bounds of the generalised frequency response functions under^* the constraint wl+ ...wn=w.

48. A system as claimed in either of claims 46 and 47 in which the gain bounds of the generalised frequency response functions of the nonlinear dynamic model under the predetermined frequency variable constraint are evaluated as follows denote the gain bounds of the GFRFs with no constraints as

H_n , n = 1,2,...

and the gain bounds of the GFRFs under the constraint ^ ^{+ ■" + w« = w} as

H_n ^B(w) , n = 1,2,...

^XH„ / n ¹,²,... _can τ_De determined using a recursive algorithm and the NARX model coefficients as follows

For n=l

where ∑c₀₁(k_x)exp(-jw_dXk_x) k_=l

H lCfodl)

¹ - ∑ ^cιo ( ι ) ^exp(-J ^d A ) k_=l

and [a,b] is the interval which represents the frequency range of the possible loading histories.

For n≥2

where

K

- mm !- ^cιo( ι)^eχPHw_dlk₁ k,=l

and ^π represents the possible frequency range produced by the nth-order nonlinear output which can be determined from the input frequency range [a, b] using the following formula

[] means to take the integer part, na i = + 1

(a + b)

I_k = [na - k(a + b), nb - k(a + b)] for k = 0,...,i -1,

then ^Hn (^w)> ^{n 1}J2,... _{can be} approximately determined as

H^B (w) = C(wT_s)L_nH_n ^B for n>2

where

49. A system as claimed in claim 46 in which the means for determining the bound on power spectral density of the stress or strain response of the structure or component to the predetermined loading pattern comprises means for evaluating gain bounds of generalised frequency response functions of the nonlinear dynamic model under a predetermined frequency variable constraint using an optimisation procedure or a genetic algorithm.

50. A system as claimed in claim 49 in which the means for evaluating the gain bounds using an optimisation procedure or a genetic algorithm under a predetermined frequency variable constraint comprises means for optimising

1. A system as claimed in either of,;claims 49 and 50 in which the means for evaluating ' the^•-.gain bounds'^' using an optimisation procedure or a genetic algorithm under a predetermined frequency variable .constraint comprises means for calculating the values of the generalised frequency response functions using the following recursive algorithm:

H_n(jw_x,-,jw_n) H^d _n(jT₃w_x,...,jT_sw_n)

where

i¹ ~ ∑ ^cιo (^kι ) expf- fodi +> ^{• •} ' >+™_dn )^kJ) H^dn(jw_d ,...,jw_dn) k,=l

= Σ^_{n l}j---,^k _n)^Q -3(w_dχk_χ+,---,+W_dπk_n)] kτ,k„=l

n K ⁺ Σ^C _pθ(^k _lτ-_> ^k _p ^Hdn,p W_dl,...,jW_dn) p=2k₁,k_p=l where ^wdi=^τ _s ^wι i = l/-/n _and

with

tfati %,-J »&)

_

52. A system as claimed in any of claims^* 41 to 51 in which the means for determining the bound of the power spectral density comprises means for evaluating

where ™w , i=l,...,N, j=l, ...,N, i≠j, are a set of statistics of the predetermined loading pattern.

53. A system as claimed in claim 52 in which the means for evaluating

comprises means for evaluating a time limited Fourier transform of the predetermined loading pattern L times based on the samples

u[M(l -I)/ 2} , ^■•• , u[(M-l) + M(l -l)/2]

for 1=1,..., L respectively to yield

U^m(jw,T), w = 2πi/MT₃, i = -(M/2-l),---,M/2, 1 = 1,..., L

where the notation ^{= MT}s __n ^U (^jw,τ) n icates that

^j ' ' is a time limited Fourier transform of u(t) obtained over the time duration ^T:=M'I's_r the time limited Fourier transform of a time signal u(t) over the time T being defined by f¹² u(t)e ~^~~jj"^wtdt evaluating ^u° for), n-1, .. ,^N defined'by^"

n=l,...,N for 3^. = 1,- --,L respectively, where N is a predetermined, preferably maximum, order of dominant system nonlinearities expressed as the highest nonlinear order in the Volterra series expansion of .the system, using the following algorithm

Conv(x---x) where « denotes the n-fold convolution of vector x and the notation T in ^ϋ tJw.r) simplification of expression, and calculating

^suufo) and ^suv(^w _r which are estimates of ^s™fo) which defined as

where E [ . ] denotes expected value^{: *' *}and ^s^ ^w) which is defined by

l,...,N, i ≠j

, using

and

i =l, ... ,N, j= l, ...,N i≠j

54. A system as claimed in any of claims 41 to 53 in which the means for computing the spectral moments . from the bound on the power spectral density comprises means for evaluating

M/2

2τι± „ - rfⁿS^B _yy(27rf)df=∑fⁿS^Byy(2πf)_<5^' t = ∑ -^MTs yy M s MTs_y

for n=0,l,2,4

55. A system as claimed in any of claims 41 to 54 which comprises means for constructing the probability density function of the stress or strain ranges using the spectral moments .

56. A system as claimed in claim 55 in which the means for constructing the probability density function comprises means for evaluating

where

57. A' system as claimed in any of claims 41 to 56 in which the means for calculating the fatigue characteristic comprises means for evaluating

where

, ^f and f are the parameters which define the fatigue characteristics of the associated materials using an S-N relationship given by _t •

N(S) = - - S ^£

58. A system as claimed in any of claims 41 to 57 further ^■ comprises means for calculating a plurality of values for the fatigue characteristic each for respective predetermined loading patterns and calculating an average value for the fatigue characteristic.

59. A system as claimed in claim 58 ,άn which the means of evaluating the average value-,, of ^* the •^'' fatigue characteristic comprises means for evaluating

where ^D _Wi, ± ^^l , . . . ,^N _{F are the p}ι_uraii y of values for the

, fatigue characteristics, * represents the probability of the structure or component working under the i-th predetermined loading pattern.

60. A system as claimed in claim 59 further comprising means for evaluating the fatigue life time of the structure or component using the calculated fatigue characteristic using at least one of T=l/D_wi or T=1/D_WA.

61. A^' method for the design of a structure or component for fatigue when the structure or component is subject to predetermined loading patterns using a relationship between a selected variable or variables associated with the fatigue of the structure or component to be designed and parameters or characteristics of the structure or component, the method comprising the steps of expressing the stress or strain PSD response of the structure or component to the predetermined loading pattern or the stress or strain PSD responses of the structure or component to a class of predetermined loading patterns in terms of the parameters or characteristics of the structure or component; establishing the relationship between a selected variable or variables associates with the fatigue of the structure or component and the parameters or characteristics of the structure or component using the expression for the stress or strain PSD response; and applying an optimization or,^" other procedure to determine values of the structure or- component^' parameter, parameters or characteristics which result in the specified fatigue life time.

62. A method for the design of a nonlinear structure or component for fatigue characteristic when the structure or component is subjected to predetermined loading patterns, the method comprising the steps of: designing a prototype of a nonlinear structure or component, performing the fatigue analysis for the designed structure or component using a method or system as claimed in any preceding claim to determine whether the fatigue characteristic of the structure or component satisfies a design requirement; and redesign the structure or component if the fatigue analysis indicates that the prototype is not acceptable.

63. A method of manufacturing a nonlinear structure or component comprising the steps of designing the nonlinear structure or component using a method or system as claimed in any preceding claim; and materially producing the nonlinear structure or component so designed.

64. A method for designing a nonlinear structure or component substantially as described herein with reference to and/or as illustrated in any of the accompanying drawings.

65. A system arranged to implement a method as claimed in any preceding method claim.

66. A computer program element comprising computer code means to implement a method or system -as claimed' in any preceding claim.

67. A computer program product comprising a storage medium having stored thereon a computer program element as claimed in claim 66.