WO2002041193A1 - Systemes non lineaires - Google Patents

Systemes non lineaires Download PDF

Info

Publication number
WO2002041193A1
WO2002041193A1 PCT/GB2001/005045 GB0105045W WO0241193A1 WO 2002041193 A1 WO2002041193 A1 WO 2002041193A1 GB 0105045 W GB0105045 W GB 0105045W WO 0241193 A1 WO0241193 A1 WO 0241193A1
Authority
WO
WIPO (PCT)
Prior art keywords
component
nonlinear
fatigue
evaluating
stress
Prior art date
Application number
PCT/GB2001/005045
Other languages
English (en)
Inventor
Stephen Alec Billings
Zi Qiang Lang
Original Assignee
The University Of Sheffield
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by The University Of Sheffield filed Critical The University Of Sheffield
Priority to AU2002215110A priority Critical patent/AU2002215110A1/en
Publication of WO2002041193A1 publication Critical patent/WO2002041193A1/fr

Links

Classifications

    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01M—TESTING STATIC OR DYNAMIC BALANCE OF MACHINES OR STRUCTURES; TESTING OF STRUCTURES OR APPARATUS, NOT OTHERWISE PROVIDED FOR
    • G01M7/00—Vibration-testing of structures; Shock-testing of structures
    • G01M7/02—Vibration-testing by means of a shake table
    • G01M7/025—Measuring arrangements
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/04—Ageing analysis or optimisation against ageing

Definitions

  • 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.
  • 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.
  • FFA finite element analysis
  • 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.
  • 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.
  • 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.
  • PSD power spectral density
  • PDF 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.
  • 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.
  • 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.
  • 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) ;
  • PSD Power Spectrum Density
  • 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
  • figure 12 illustrates the Power Spectrum Density
  • PSD 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
  • 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
  • figure 21 depicts the probability density function
  • 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;
  • PDF probability density function
  • 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
  • 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;
  • 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.
  • 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.
  • PSD power spectrum density
  • 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.
  • 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.
  • y nW is a ' nth-order output ' given by
  • K is the maximum lag , and y ( . ) , u ( . ) , and c carving radical ( . )
  • NARX Networkar Auto-Regression with exogenous input
  • y(k) 0.3 ⁇ (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 ⁇ y(k - 2)y ⁇ k - 3)
  • 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.
  • K f and M f are the parameters which define the fatigue characteristics of the associated materials using the so-called S-N relationship given by
  • BS 5400 Part 10: 1980
  • BS 5400 is incorporated herein in its entirety for all purposes.
  • 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
  • ⁇ * denotes the fatigue damage of the structure under the i-th loading situation.
  • 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.
  • 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.
  • K is the maximum lag
  • y ( . ) , u ( . ) , and c n beau( . ) pq are the system output, input, and model coefficients respectively.
  • NARX Networkar Auto-Regressive with exogenous input
  • 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)
  • NARX model 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)
  • 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 ⁇ + - + ⁇ - "
  • [a,b] is the interval which represents the frequency range of the possible loading histories.
  • L n c 10 (k 1 )exp(-jw dl k 1 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
  • H ⁇ B (w) C(wT s )L n H n B for n>2
  • H n (jw 1 ,--,jw n ) H d ⁇ (jT s w l ,...,jT s w n )
  • 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
  • Step (8) 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
  • 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.
  • 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 .
  • Dwi denotes the worst fatigue damage of the structure or component under the i-th loading situation.
  • 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
  • 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.
  • 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.
  • 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.
  • 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.
  • the Dirlik solution requires the stress response to be a stationary and Gaussion random process.
  • 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. )
  • 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.
  • 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.
  • 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.
  • a nonlinear fatigue design can be implemented using the following procedure.
  • 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.
  • 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.
  • 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.
  • time-limited Fourier transform ⁇ (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: -(" ' "
  • W 1 ⁇ • ••+ w denotes integration of ( . ) over the n-dimensional hyper- plane w - w x + ⁇ • -+w ⁇ , 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
  • Step (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
  • 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:
  • 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.
  • 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.
  • the design example mentioned above will be used later to illustrate this .
  • Example 1 Vibration fatigue analysis of an experimental offshore structure based on the loading histories representing three different loading conditions .
  • 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 .
  • 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 th ⁇ , .;• force response as below: • ⁇ - ⁇ - : * -,; '• " - *'
  • This model is of the general form of the NARX model with
  • 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
  • Tr. ls to yie Id
  • 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 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • the output PSD of a nonlinear system depends on more statistical information of the corresponding input rather than just the input PSD.
  • 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.
  • 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 whereas ??the second • •-technique of the present invention has been developed to address nonlinear vibration fatigue analysis problems.
  • Example 2 Vibration fatigue analysis of an experimental offshore structure based on statistics of the loading histories representing three different loading conditions.
  • Step (2) under the constraint w 1 + - -- + w n - w to yield first
  • Figure 15 shows one of the 30 time limited Fourier transform results.
  • c is a coefficient reflecting the relationship between the PSD of force and the PSD of stress and is taken to be
  • Example 1 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.
  • D A and T F reflect the real f tigue of the structure under the specific loading histories
  • 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.
  • D A and T F reflect the real f tigue of the structure under the specific loading histories
  • 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.
  • D A and T F reflect the real f tigue of the structure under the specific loading histories
  • 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.
  • 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.
  • 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.
  • 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
  • Figure 24 shows the PSD of the stress response of the structure to the specific loading history as a solid line.
  • 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.
  • 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.
  • 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.
  • Example 4 Implementation of a direct nonlinear vibration fatigue design for a mechanical system
  • 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 a 3 which is to be designed to achieve a desired fatigue life .
  • mapping from the time domain description of the suspension system to the frequency domain can be determined as
  • L(w, ⁇ ) denotes the conjugate . of (w,T)
  • C x (w), C 2 (w), and C 3 (w) are functions of the characteristics of the given ' "stochastic loading history and the fixed structure parameters ⁇ 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 a 3 from the expression of S yy (w) given above.
  • ⁇ 3 (n) [ f n C 3 (2 ⁇ f)df
  • J(a 3 ) ⁇ Q m Q (a 3 ) + ⁇ x m 1 (a 3 ) + ⁇ 2 m 2 (a 3 ) + ⁇ i m_ i (a 3 )
  • figure 27 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.- *'
  • 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
  • the optimal a 3 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

Landscapes

  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Theoretical Computer Science (AREA)
  • Computer Hardware Design (AREA)
  • Evolutionary Computation (AREA)
  • Geometry (AREA)
  • General Engineering & Computer Science (AREA)
  • Investigating Strength Of Materials By Application Of Mechanical Stress (AREA)

Abstract

La présente invention concerne des techniques d'analyse de fatigue dans des domaines fréquentiels, pouvant être utilisées pour réaliser une analyse de fatigue pour des structures ou des composants non linéaires, soumis à des charges aléatoires. Ces techniques peuvent être mises en oeuvre d'une manière analogue à l'analyse de fatigue de structures ou de composants non linéaires et peuvent être aisément incorporées dans un sous-programme d'étude de fatigue. Cela permet une mise en oeuvre aisée de l'étude de la fatigue pour des structures et des composants non linéaires. La présente invention concerne également une technique qui utilise un lien entre les paramètres ou les caractéristiques et la résistance à la fatigue ou le profil de la structure ou des composants à concevoir, et qui permet de concevoir directement la structure pour résister à la fatigue.
PCT/GB2001/005045 2000-11-18 2001-11-16 Systemes non lineaires WO2002041193A1 (fr)

Priority Applications (1)

Application Number Priority Date Filing Date Title
AU2002215110A AU2002215110A1 (en) 2000-11-18 2001-11-16 Nonlinear systems

Applications Claiming Priority (2)

Application Number Priority Date Filing Date Title
GB0028768.0 2000-11-18
GB0028768A GB2369209A (en) 2000-11-18 2000-11-18 Fatigue analysis

Publications (1)

Publication Number Publication Date
WO2002041193A1 true WO2002041193A1 (fr) 2002-05-23

Family

ID=9903866

Family Applications (1)

Application Number Title Priority Date Filing Date
PCT/GB2001/005045 WO2002041193A1 (fr) 2000-11-18 2001-11-16 Systemes non lineaires

Country Status (3)

Country Link
AU (1) AU2002215110A1 (fr)
GB (1) GB2369209A (fr)
WO (1) WO2002041193A1 (fr)

Cited By (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US7369966B1 (en) 2007-02-12 2008-05-06 Honeywell International Inc. Methods and apparatus to design a wheel of a multiple-axle vehicle
CN101615215B (zh) * 2009-08-05 2012-05-09 中国海洋石油总公司 一种半潜式平台结构简化疲劳设计方法
CN105651478A (zh) * 2015-12-15 2016-06-08 西安交通大学青岛研究院 一种基于振动信号测试零部件疲劳寿命的分析方法
CN107194050A (zh) * 2017-05-11 2017-09-22 电子科技大学 随机载荷作用下涡轮盘结构的概率疲劳寿命预测方法
WO2018180880A1 (fr) * 2017-03-31 2018-10-04 日本電気株式会社 Dispositif d'analyse, dispositif de diagnostic, procédé d'analyse et support d'enregistrement lisible par ordinateur
WO2019110957A1 (fr) * 2017-12-04 2019-06-13 Bae Systems Plc Estimation d'endommagement par fatigue dans une structure
CN110879912A (zh) * 2018-09-05 2020-03-13 西门子股份公司 疲劳分析方法与装置
CN114112633A (zh) * 2021-11-26 2022-03-01 山东大学 一种基于非线性超声的金属早期疲劳损伤检测方法及系统

Families Citing this family (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US6704664B2 (en) * 2001-12-18 2004-03-09 Visteon Global Technologies, Inc. Fatigue sensitivity determination procedure
US10890499B2 (en) 2017-12-21 2021-01-12 Caterpillar Inc. System and method for predicting strain power spectral densities of light machine structure

Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US4858146A (en) * 1986-08-13 1989-08-15 The Babcock & Wilcox Company Automated design of structures using a finite element database
WO1998014765A1 (fr) * 1996-09-30 1998-04-09 Ford Motor Company Procede pour definir des tests de vibrations aleatoires pour la validation de la durabilite de produits
US5930155A (en) * 1997-03-25 1999-07-27 Hitachi Metals, Ltd. Method of evaluating endurance of vehicle wheel by computer simulation
US6212486B1 (en) * 1998-09-17 2001-04-03 Ford Global Technologies, Inc. Method of identifying critical elements in fatigue analysis with von mises stress bounding and filtering modal displacement history using dynamic windowing

Family Cites Families (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US5847259A (en) * 1995-12-01 1998-12-08 Ford Motor Company Computer program, system and method to specify sinusoidal vibration tests for product durability validation

Patent Citations (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US4858146A (en) * 1986-08-13 1989-08-15 The Babcock & Wilcox Company Automated design of structures using a finite element database
WO1998014765A1 (fr) * 1996-09-30 1998-04-09 Ford Motor Company Procede pour definir des tests de vibrations aleatoires pour la validation de la durabilite de produits
US5930155A (en) * 1997-03-25 1999-07-27 Hitachi Metals, Ltd. Method of evaluating endurance of vehicle wheel by computer simulation
US6212486B1 (en) * 1998-09-17 2001-04-03 Ford Global Technologies, Inc. Method of identifying critical elements in fatigue analysis with von mises stress bounding and filtering modal displacement history using dynamic windowing

Cited By (13)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
US7369966B1 (en) 2007-02-12 2008-05-06 Honeywell International Inc. Methods and apparatus to design a wheel of a multiple-axle vehicle
CN101615215B (zh) * 2009-08-05 2012-05-09 中国海洋石油总公司 一种半潜式平台结构简化疲劳设计方法
CN105651478A (zh) * 2015-12-15 2016-06-08 西安交通大学青岛研究院 一种基于振动信号测试零部件疲劳寿命的分析方法
JP7014223B2 (ja) 2017-03-31 2022-02-15 日本電気株式会社 分析装置、診断装置、分析方法及びプログラム
WO2018180880A1 (fr) * 2017-03-31 2018-10-04 日本電気株式会社 Dispositif d'analyse, dispositif de diagnostic, procédé d'analyse et support d'enregistrement lisible par ordinateur
JPWO2018180880A1 (ja) * 2017-03-31 2020-02-06 日本電気株式会社 分析装置、診断装置、分析方法及びコンピュータ読み取り可能記録媒体
CN107194050B (zh) * 2017-05-11 2021-03-02 电子科技大学 随机载荷作用下涡轮盘结构的概率疲劳寿命预测方法
CN107194050A (zh) * 2017-05-11 2017-09-22 电子科技大学 随机载荷作用下涡轮盘结构的概率疲劳寿命预测方法
WO2019110957A1 (fr) * 2017-12-04 2019-06-13 Bae Systems Plc Estimation d'endommagement par fatigue dans une structure
GB2568964B (en) * 2017-12-04 2022-05-25 Bae Systems Plc Estimating fatigue damage in a structure
US11772823B2 (en) 2017-12-04 2023-10-03 Bae Systems Plc Estimating fatigue damage in a structure
CN110879912A (zh) * 2018-09-05 2020-03-13 西门子股份公司 疲劳分析方法与装置
CN114112633A (zh) * 2021-11-26 2022-03-01 山东大学 一种基于非线性超声的金属早期疲劳损伤检测方法及系统

Also Published As

Publication number Publication date
GB0028768D0 (en) 2001-01-10
AU2002215110A1 (en) 2002-05-27
GB2369209A (en) 2002-05-22

Similar Documents

Publication Publication Date Title
Cheng et al. Volterra-series-based nonlinear system modeling and its engineering applications: A state-of-the-art review
Avci et al. Self-organizing maps for structural damage detection: a novel unsupervised vibration-based algorithm
US20050273296A1 (en) Neural network model for electric submersible pump system
US20070244678A1 (en) Design Optimization System and Method
CN111950868B (zh) 一种基于生成对抗网络的综合能源系统负荷场景生成方法
WO2002041193A1 (fr) Systemes non lineaires
CN110334865B (zh) 一种基于卷积神经网络的电力设备故障率预测方法及系统
CN114282309B (zh) 一种基于多目标代理模型的静叶调节机构系统可靠性分析方法
CN113051838A (zh) 一种基于数字孪生技术的空间在轨制造结构可靠寿命预测方法
Mao et al. Highly efficient parameter estimation algorithms for Hammerstein non‐linear systems
Hu et al. Accelerated life testing (ALT) design based on computational reliability analysis
CN109543894B (zh) 一种核电站松脱部件事前预测系统及预测方法
Bussetta et al. Nonlinear updating method: a review
Chen et al. A Mahalanobis Distance Cumulant‐Based Structural Damage Identification Method with IMFs and Fitting Residual of SHM Measurements
CN113740671A (zh) 一种基于vmd和elm的故障电弧识别方法
Kim et al. A study toward appropriate architecture of system-level prognostics: physics-based and data-driven approaches
Randall et al. Updating modal models from response measurements
Van de Wouw et al. A Volterra series approach to the approximation of stochastic nonlinear dynamics
Mohammadpour et al. Dispersion reduction of the analyses data for more reliable fragility curves of selected electric substations equipment
Dackermann et al. Application of cepstrum analysis and artificial neural networks for the damage identification of a two-storey framed structure based on response-only measurements
CN114741946B (zh) 一种航空发动机典型机构多构件失效相关的可靠性仿真方法
Hunter Jr Bilinear system characteristics from nonlinear time series analysis
JP2006195542A (ja) モデル同定装置およびモデル同定プログラム
Junior et al. Influence of sampling rate and discretization methods in the parameter identification of systems with hysteresis
CN117473302B (zh) 汽车配件的结构可靠性测试方法、装置、设备及存储介质

Legal Events

Date Code Title Description
AK Designated states

Kind code of ref document: A1

Designated state(s): AE AG AL AM AT AU AZ BA BB BG BR BY BZ CA CH CN CO CR CU CZ DE DK DM DZ EC EE ES FI GB GD GE GH GM HR HU ID IL IN IS JP KE KG KP KR KZ LC LK LR LS LT LU LV MA MD MG MK MN MW MX MZ NO NZ PH PL PT RO RU SD SE SG SI SK SL TJ TM TR TT TZ UA UG US UZ VN YU ZA ZW

AL Designated countries for regional patents

Kind code of ref document: A1

Designated state(s): GH GM KE LS MW MZ SD SL SZ TZ UG ZM ZW AM AZ BY KG KZ MD RU TJ TM AT BE CH CY DE DK ES FI FR GB GR IE IT LU MC NL PT SE TR BF BJ CF CG CI CM GA GN GQ GW ML MR NE SN TD TG

121 Ep: the epo has been informed by wipo that ep was designated in this application
DFPE Request for preliminary examination filed prior to expiration of 19th month from priority date (pct application filed before 20040101)
REG Reference to national code

Ref country code: DE

Ref legal event code: 8642

122 Ep: pct application non-entry in european phase
NENP Non-entry into the national phase

Ref country code: JP

WWW Wipo information: withdrawn in national office

Country of ref document: JP