1 Introduction

The finite element method(FEM)has been widely applied in various fields of engineering and science,such as solid/ fluid mechanics,heat transfer, electromagnetics, and acoustics[1].The traditional FEM s are based on deterministic model,which can be called deterministic FEM s.However,in recent years,people have realized that many uncertain factors(e.g.,environmental loads,material parameters,geometries,and boundary conditions)exist in practical structures.Consequently,some non-deterministic FEMs have been developed.The probabilistic FEM(PFEM),which is based on the probability distribution of the input parameters[2–6]was first developed.However,in many cases,because of limited statistical information,obtaining an accurate probability density function(PDF)is difficult.Therefore,the PFEM suffers from many limitations in practical applications.Elishakoff[7]implemented a universal survey and systematically summarized the limitations of the probabilistic methods in engineering problems.

Recently,a number of non-probabilistic approaches for non-deterministic analysis have emerged[8].The fuzzy FEM(FFEM)has been studied by many researchers[9–11].FFEM is based on the possibility distribution of imprecise parameters,and the foundation is the subjective statistical data.In many cases,we cannot obtain sufficient statistical data from an objective test due to high cost and a long test cycle,but we can always obtain the possibility distribution function through careful planning.Thus,we can say that the fuzzy theory is an effective tool suitable for small-sample problems.

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As another non-probabilistic FEM,the interval FEM(IFEM),which is based on the interval concept to describe the non-deterministic model properties,has been studied on an academic level not only for static analysis[12–19],but also dynamic analysis[20,21]and reliability optimization problems[22–24].Furthermore,Yang and Sun[25]investigated the set-based eigenvalue problem using the Kriging model and particles warm optimization(PSO)algorithm.The IFEM and FFEM are not independent,and realization of the FFEM is based on the interval cut set of fuzzy parameters.

However,when the uncertainties of the imprecise parameters are described by convex set models such as the interval(hyper-cuboid),hyper-ellipsoid,Minkowski norm convex set,cumulative energy convex set,and multiple convex set models,their exact boundaries can hardly be determined because of limited statistical information.Therefore,for problems with scarce statistical data,the ordinary rigid convex set may cause an unexpected error.

In addition,existing research works about the FFEM mainly focus on the fuzzy properties or the possibility distributions of the structural responses.However,the fuzzy property or the possibility distribution cannot definitely reflect the rough scope of the structural response,and have weak comparability.Thus,they have significant limitations in engineering application.

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In the current study,the general convex set models are extended,and their boundaries are considered as fuzzy zones.Thus,a novel uncertainty model named fuzzy convex set(FCS)is presented.Then,the structural eigenvalue analysis methods based on FCS are investigated.The fuzzy properties and the definite indicators(conditional extremes)of the structural eigenvalues are studied and presented.The remainder of this paper is organized as follows.The FCS model is defined and its properties are investigated in Sect.2.Several typical FCS models are built in Sect.3.Section 4 presents a method for analysis of the fuzzy properties of the structural eigenvalues.In Sect.5,a non-fuzzy indicator,i.e.,the conditional extremes of the fuzzy eigenvalues,is presented,and two equivalent methods are provided.In Sect.6,several practical applications are provided to demonstrate the practicability and feasibility of the proposed methods.In Sect.7,some discussions are listed.In Sect.8,we complete this paper with some concluding remarks.

2 FCS models

The fuzzy properties of this system are analyzed according to the method shown in Fig.2.In order to ensure accuracy,the Monte Carlo method is applied for calculating the upper and lower bounds of every order eigenvalue.For every cut set level,the intervals values or the bounds of the structural eigenvalues are listed in Table 2.

In fact,probability and possibility are two uncertainty methods with different mathematical ideas.When obtaining the PDF is difficult,we can express the parameter uncertainty from the point of view of possibility. Equation(1)shows that the uncertainty measure of the possibility tends to be larger than that of the probability.Therefore,in engineering problems,when we cannot obtain accurate PDF,we can conservatively or safely apply the possibility distribution.

Combined with the condition that Ũ is a formal fuzzy set,we can learn thatŨ is an FCS according to Definition 1.The proof ends here.

In Sect.4,the possibility distribution of the structural eigenvalue was obtained according to the fuzzy properties of the input parameters.We assume that the possibility distribution of the structural eigenvalue is F(λ)and the upper and lower bounds of the zero-level cut set of F(λ)are andrespectively.The following fuzzy maximum and fuzzy minimum sets can be constructed:

In the following,the FCS model will be defined,and as its basis,the convex F-set and convex set are first introduced.

2.1 Convex F-set

Definition 1 We assume that C is a set in n-dimensional real space,i.e.,C⊂R n.If the relationship(1−t)x+t y∈C is always true as long as x,y∈C and t∈[0,1],C is a convex set.

Definition 2 We assume that R represents the real number if eld,F(R)represents all the fuzzy sets in R and is one of them,i.e.,.If Eq.(2)is always true as long as x1,x2,x3∈R and x1＞x2＞x3,Ã is a convex F-set.

whereis the membership function of and“∧”means taking small values.

According to Definition 2,we can obtain the following corollary:

Corollary1 The cut sets of convex F-set must be intervals;a fuzzy set with any cut set is an interval must be a convex fuzzy set.

2.2 FCS model

In this subsection,the FCS will be defined,and some properties will be proven.

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Definition 3 Let us assume that is a formal fuzzy set in R n and the membership function of isIf Eq.(3)is always true as long as x,y∈R n,t∈(0,1)and z= is called an FCS model.

According to Definition 3,we can obtain the following corollary:

The eigenvalue equation can be abbreviated as

Proof for Corollary 2

(1)We assume that is an FCS and x1,x2∈Uλ(λ∈[0,1],where Uλ is a λ-level cut set),i.e.,

According to Definition 3,Eq.(5)is always true as long as t∈(0,1)and x3=(1−t)x1+t x2.

Therefore,x3∈Uλ,and Uλis a convex set model.

(2)In contrast,we assume that is a normal fuzzy set in the domain R n.∀x1,x2∈R n,andThe following relationship can be obtained:

Let us assume that t∈(0,1)and x3=(1−t)x1+t x2.Because Uλis a convex set,x3∈ Uλ.The following relationship can then be obtained:

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We would like to point out that the FCS should be formal fuzzy sets because the limited statistical data can determine a convex set,which should be the kernel of the FCS.In other words,where Φ denotes the empty set.

3 Typical FCS models

In practice,people often choose the convex set models,which contain analytical expressions and are convenient to use.In the following,several FCS models will be built by introducing a new parameter—the fuzzy extending parameter—into some typical convex set models[28,29].

(1)Interval FCS model:

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where φ is the deviation of the interval—the kernel of the FCS—andis the fuzzy extending parameter.

(2)N-dimensional ellipsoidal FCS model:

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where W is a real symmetric positive definite matrix.(3)Minkowski norm FCS model:

where denotes the Minkowski norm and W is a real symmetric positive definite matrix.W 1/2=Λ1/2Q and W=Q TΛQ,where Q is an orthogonal matrix and Λ is a diagonal matrix.

(4)FCS model for cumulative energy:

(5)FCS model for slope tolerance:

(6)Multiple FCS model:

Let us assume that non-deterministic vector x=(x1,x2,…,xn)T can be expressed as,where k≤n and u i(i=1,2,...,k)are sub-vectors of x.The uncertainties of the sub-vectors can be quantified by the k FCS models,i.e.,(i=1,2,...,k).The FCS model for x can be obtained as follows

where∫denotes the collection of the correspondence relationships of x and its membership grades. is the nominal value vector of is the fuzzy vector that includes the fuzzy extending parameters.φ =(φ1,φ2,...,φk)is the scale parameter vector of the kernel of the multiple FCS model.

Fig.1 Typical distributions of.a Semi-rectangular distribution.b Sem i-trapezoidal distribution.c Parabolic distribution.d Normal distribution

The possibility distribution ofθ is biased to bes mall.Some typical distributions are shown in Fig.1,whereis the possibility function of the fuzzy extending parameter

4 Fuzzy properties of the structural eigenvalues

The eigenvalue problem of an undamped structure can be expressed as

where is the fuzzy vector of the structural parameters andandare the stiffness and mass matrices,respectively is the structural eigenvalue,which is equal to the square of the natural frequency,and is the corresponding eigenvector.

We assume that the possibility distribution function of fuzzy parameter is πãi(αi).We consider some cut sets under various levels;then,a series of general interval eigenvalue problems can be obtained,i.e.,

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The lower and upper bounds of the structural eigenvalues can form the interval vector as

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where

If the uncertainties of the structural parameters are quantified by FCS modela series of general eigenvalue problems constrained by convex set Cτcan be obtained by considering different cut-set levels.

This equation is the general eigenvalue problem with interval parameters.

Equations(15)and(19)are the set-based finite element(SFE)problems for structural eigenvalues.

According to the decomposition theorem of the fuzzy set,the i th-order fuzzy eigenvalue can be obtained.

Fig.2 Procedure for analysis of the structural eigenvalue responses under FCS constraint.A(α),A1(λ1),and A2(λ2)are the possibility degree of the input parameters,output parameter λ1,and output parameter λ2,respectively.SFEA denotes the SFE analysis

In practical problems,we can choose some discrete cutset levels and analyze the corresponding intervals of the eigenvalues.The approximate possibility distribution function can be further obtained based on these intervals and corresponding cut-set levels.When the structural parameters are constrained by the FCS model, the procedure for analysis of the fuzzy properties of the structural eigenvalues is shown in Fig.2.

5 Conditional extremes for structural fuzzy eigenvalues

The possibility distributions of the structural eigenvalues can be obtained according to Sect. 4. The conditional extremes of the structural eigenvalues under fuzzy constraints are further presented in this section,which would serve as a non-fuzzy quantification indicator.

We assume that the structural eigenvalue is λ(α).When α varies in the zero-level cut set,the upper and lower bounds ofλ(α)are denoted by M and m,respectively.The following fuzzy maximum and fuzzy minimum sets can be constructed:

We assume that the possibility distribution of α is A(α).According to the symmetric F-programming theory[27],if existing point α∗makes the following equation hold,point α∗ can be called the maximum element of λ(α)under the fuzzy constraint of A(α),and λ(α∗)is the conditional maximum value of the structural fuzzy eigenvalue.

Fig.3 Solution principle of the conditional extremes under fuzzy constraint.The solid line represents the possibility distribution of the structural eigenvalue λ.The dotted lines represent the membership function of the fuzzy maximum or minimum set for λ.The dot dash lines are auxiliary lines

Similarly,if existing point α∗∗ makes the following equation hold,point α∗∗ is called the minimum element of λ(α)under the fuzzy constraint of A(α),and λ(α∗∗)is the conditional minimum value of the structural eigenvalue under the constraint of A(α).

From the statistical cost perspective,although the possibility and probability distribution functions are all based on statistical data,the costs contain differences.The basis of the PDF is objective test data,and that of the possibility distribution function is subjective statistical data,which can be obtained using expert scoring or questionnaire.Therefore,generally speaking,the cost of the possibility statistics is lower than that of the probability statistics.In many cases,we cannot obtain sufficient statistical data from the objective test because of the limitations in terms of high cost and long test cycle,but can always obtain the possibility distribution function through careful planning.Thus,we can say that the fuzzy theory is another tool suitable for small sample-size problem,in addition to the convex set theory.

If existing point λ∗ makes the following equation hold

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λ∗is the conditional maximum value of the structural eigenvalue under the fuzzy constraint of A(α).

Fig.4 The spring mass system

Table1 The cut sets of the physical parameters

Stiffness and mass parameters Cut set levels 0 0.05 0.1 0.2 0.3 0.5 0.7 0.9 1 k1(N/m)Upper 2010 2008.16 2007.77 2007.32 2007.00 2006.52 2006.09 2005.59 2005 Lower 2000 2001.84 2002.23 2002.68 2003.00 2003.48 2003.91 2004.41 2005 k2(N/m)Upper 1820 1816.32 1815.54 1814.63 1814.01 1813.04 1812.18 1811.19 1810 Lower 1800 1803.68 1804.46 1805.37 1805.99 1806.96 1807.82 1808.81 1810 k3(N/m)Upper 1610 1608.16 1607.77 1607.32 1607.00 1606.52 1606.09 1605.59 1605 Lower 1600 1601.84 1602.23 1602.68 1603.00 1603.48 1603.91 1604.41 1605 k4(N/m)Upper 1410 1408.16 1407.77 1407.32 1407.00 1406.52 1406.09 1405.59 1405 Lower 1400 1401.84 1402.23 1402.68 1403.00 1403.48 1403.91 1404.41 1405 k5(N/m)Upper 1210 1208.16 1207.77 1207.32 1207.00 1206.52 1206.09 1205.59 1205 Lower 1200 1201.84 1202.23 1202.68 1203.00 1203.48 1203.91 1204.41 1205 m1(kg)Upper 30.0 29.816 29.777 29.732 29.700 29.652 29.609 29.559 29.5 Lower 29.0 29.184 29.223 29.268 29.300 29.348 29.391 29.441 29.5 m2(kg)Upper 28.0 27.632 27.554 27.463 27.401 27.304 27.218 27.119 27.0 Lower 26.0 26.368 26.446 26.537 26.599 26.696 26.782 26.881 27.0 m3(kg)Upper 28.0 27.632 27.554 27.463 27.401 27.304 27.218 27.119 27.0 Lower 26.0 26.368 26.446 26.537 26.599 26.696 26.782 26.881 27.0 m4(kg)Upper 26.0 25.632 25.554 25.463 25.401 25.304 25.218 25.119 25.0 Lower 24.0 24.368 24.446 24.537 24.599 24.696 24.782 24.881 25.0 m5(kg)Upper 19.0 18.632 18.554 18.463 18.401 18.304 18.218 18.119 18.0 Lower 17.0 17.368 17.446 17.537 17.599 17.696 17.782 17.881 18.0

If existing point λ∗∗ makes the following equation hold

λ∗∗ is the conditional minimum value of the structural eigenvalue under the fuzzy constraint of A(α).

From this analysis,the following two relationships can be obtained:

The optimization problems in Eqs.(22)and(23)or Eqs.(25)and(26)can be resolved by the PSO algorithm.

Table2 The bounds of the structural eigenvalues under the corresponding cut set levels

Stiffness and mass parameters Cut set levels 0 0.05 0.1 0.2 0.3 0.5 0.7 0.9 1 λ1 Upper 6.4165 6.3109 6.2871 6.2631 6.2453 6.2204 6.1960 6.1701 6.1378 Lower 5.8824 5.9746 5.9920 6.0145 6.0327 6.0575 6.0797 6.1066 6.1378 λ2 Upper 45.9036 45.2102 45.0742 44.9067 44.7967 44.6215 44.4686 44.2965 44.0931 Lower 42.4420 43.0250 43.1603 43.3140 43.4125 43.5818 43.7179 43.8865 44.0931 λ3 Upper 107.6603 106.2108 105.9345 105.5541 105.3065 104.9604 104.6162 104.2430 103.8064 Lower 100.2015 101.5607 101.7894 102.0964 102.3438 102.6755 102.9967 103.3689 103.8064 λ4 Upper 172.1049 169.7428 169.1620 168.5651 168.1698 167.6126 167.0386 166.4414 165.7110 Lower 159.9541 161.9899 162.4605 162.9661 163.3220 163.8823 164.4105 164.9858 165.7110 λ5 Upper 226.6892 223.7540 223.1579 222.3846 221.9635 221.1740 220.5379 219.7907 218.8811 Lower 211.5602 214.2149 214.8112 215.4477 215.8500 216.6388 217.2253 217.9901 218.8811

The solution principle of the conditional extremes of the structural eigenvalue is shown in Fig.3.

6 Numerical examples and discussion

6.1 An analytic example

Let us take the following spring mass system shown in Fig.4 as our discussion object.

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The physical parameters such as the stiffness and mass parameters of this system are uncertain parameters and have fuzzy interval properties.Their possibility distributions are demonstrated as

To analyze the fuzzy property of the structural eigenvalues,the physical parameters are made some cut sets as listed in Table 1.

As we know, the elements of the stiffness matrix and mass matrix are functions about the stiffness and mass parameters.The generalized eigenvalue equation of this spring mass system is

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Corollary2(1)The cut sets of an FCS must be convex sets.(2)A normal fuzzy set with any one cut set is a convex set must be an FCS.

where K is the stiffness matrix,M is the mass matrix,X is the eigenvector and λ is structural eigenvalue.

According to fuzzy mathematics, membership function is the quantification of the compatibility of elements and a concept,or it represents the possibility that a certain element is F. For example, the possibility or the compatibility degree that Hans eats five eggs for breakfast is 0.8, but the probability is almost zero[26].In general,the probability and possibility have the following relationship[27]:

Fig.5 Possibility distributions and conditional extremes of the first-to fifth-order structural eigenvalues with fuzzy constraints.a First order.b Second order.c Third order.d Fourth order.e Fifth order

Table 3 The conditional extremes of the structural eigenvalues based on the fuzzy convex set and the symmetric F-programming theories

CM IV and CMAV denote the conditional minimum and conditional maximum values,respectively

Orders 1 2 3 4 5 CM IV 6.0743 43.6797 102.9037 164.2858 217.0463 CMAV 6.2069 44.5364 104.7541 167.2947 220.8313

Table 4 The interval eigenvalues of the spring mass system based on the interval matrix perturbation method(Ref.[30])

Orders 1 2 3 4 5 Lower 4.8128 41.653 99.744 160.15 211.50 Upper 7.4628 46.533 107.88 171.27 226.26

Table 5 The interval eigenvalues of the spring mass system based on the interval parameter perturbation method(Ref.[31])

Orders 1 2 3 4 5 Lower 5.3263 41.821 99.994 160.61 211.86 Upper 6.9493 46.365 107.63 170.81 225.90

The possibility or fuzzy distributions of the structural eigenvalue can be obtained based on the intervals listed in Table2according to the decomposition theorem and Eq.(20).The possibility distributions of the first-to fifth-order eigenvalues are shown in Fig.5.Then,the conditional maximum values and the conditional minimum values can be got and the results are shown in the same figure.The detail values are listed in Table 3.The interval eigenvalues based on the interval matrix perturbation method and interval parameter perturbation method are listed in Tables 4 and 5.

From this analytical example,the feasibility and scientific significance of the new methods proposed above have been verified.In addition,we can make some theoretical discussion and comparison as follows:

(1)The interval physical parameters under zero-cut set level are the same as that in Ref.[31].The interval matrix perturbation method and interval parameter perturbation method were used in that monographs to calculate the upper and lower bounds of the structural eigenvalues.However,the two methods are all low order perturbation method.They are effective only when the parameter uncertainty is small,in other word,the perturbation momentum of the stiffness and mass matrix is small.In addition,the correlation of the interval parameters is ignored in the interval matrix perturbation method.So,the accuracy of the two methods is limited.By comparison with them,the Monte Carlo method is more accurate,and the obtained results listed in Table 2 are more closed to the true values.Therefore,they lay a good foundation for obtaining the fuzzy distributions of the structural eigenvalues.

Fig.6 Geometric model of the turbine blade

Table6 Nom inal values of various parameters

Elastic modulus E(×1011 Pa)Density ρ(×103 kg/m3)Angular velocity ω(×102 rad/s)Poisson’s ratio v 2.172 8.489 3.14 0.32

(2)The conditional extremes of the eigenvalues are based on the fuzzy interval description of the physical parameters and the symmetric F-programming theory,and they are defuzzy fication indicators for the fuzzy properties of the structural dynamic characteristics.It can be found that the conditional minimum value is higher than the lower bound value by IFEM under zero-cut set level,and the conditional maximum value is lower than the upper bound value by IFEM.Therefore,if the fuzzy property of the physical parameters is ignored, the interval eigenvalues will be much wider and have no practical reference value.

(3)The whole analysis process also shows that the procedure for analyzing the fuzzy distribution of structural eigenvalues shown in Fig.2 is correct and feasible.Besides,the conditional extremes are based on the F-programming theory and application of them is very image as shown in Fig.5 or Fig.3.The accuracy of the proposed method is also verified through this analytic example which has analytic eigenvalue equation.

Fig.7 Optimization procedures for the first- and second-order natural frequencies. a Conditional maximum value of the first-order natural frequency. b Conditional minimum value of the first-order natural frequency. c Conditional maximum value of the second-order natural frequency.d Conditional minimum value of the second-order natural frequency.The symbols“ps,”“w,”and “gbest”in every figure denote the population size,inertia weight factor,and global optimum value in the PSO algorithm.The “gbest”values are the possibility degrees of the conditional extremes

6.2 An engineering example

We consider a gas-turbine blade shown in Fig.6 as an illustration example.

The nominal values of the elastic modulus,density,angular velocity,and Poisson’s ratio are listed in Table 6.

We assume that the structural parameters have fuzzy properties.Now,let us analyze the fuzzy properties and theconditional extremes of the first-to fifth-order natural frequencies of the pre-stressed turbine blade with an angular velocity.Three cases are discussed below.

Table7 Conditional extremes of the natural frequencies and the corresponding parameters

Natural frequencies f(Hz) Possibility degree E(×1011 Pa) ρ(×103 kg/m3) ω(×102 rad/s)First order CM IV 50.71 0.9118 2.0814 8.8356 3.0151 CMAV 54.98 0.9028 2.2561 8.1444 3.2667 Second order CM IV 118.14 0.9145 2.0815 8.8357 3.0200 CMAV 128.21 0.9013 2.2589 8.1413 3.2589 Third order CM IV 165.80 0.9108 2.0812 8.8353 3.0804 CMAV 179.83 0.9045 2.2588 8.1417 3.2641 Fourth order CM IV 236.99 0.9124 2.0813 8.8354 3.0951 CMAV 256.91 0.9064 2.2588 8.1433 3.2652 Fifth order CM IV 452.80 0.9087 2.0813 8.8359 3.0181 CMAV 491.22 0.9035 2.2587 8.1419 3.1112

C a se 1

The uncertainties of the parameters are quantified by fuzzy intervals,i.e.,the exact boundaries of the structural parameters cannot be obtained.A fuzzy zone exists between uncertainty degrees of 4%and 5%of the nominal values.The optimization procedures for the conditional maximum and minimum values of the first-and second-order natural frequencies are shown in Fig.7.

The conditional maximum and minimum values of the first-to fifth-order natural frequencies and the corresponding parameters are listed in Table 7.

C a se 2

The uncertainties of the parameters are quantified by the following possibility distribution functions:

The possibility distribution and the conditional extremes of the first-to fifth-order natural frequencies are shown in Fig.8.

C a se 3

The uncertainties of the structural parameters are quantified by the FCS,i.e.,the hyper-ellipsoidal set has fuzzy boundaries.The fuzzy zone is located between the two hyperellipsoidal models with uncertainty degrees of 3%and 5%.

The conditional extremes obtained by the PSO method and the corresponding parameters are listed in Table 8.

7 Discussion

On the basis of the proposed theories,given examples,and results,some discussions can be listed as follows:

(1)The FCS model presented in this paper developed the theory of convex F-set and can meet practical engineering requirements.When an FCS is single dimensional,it is reduced to a normal convex F-set,i.e.,a fuzzy number.The FCS model can effectively overcome the limitations of the traditional rigid convex set model,e.g.,the rigid boundary is difficult to determine and the potential error is difficult to control.Therefore,FCS can better reduce the project risk.

(2)The ideas of analyzing the fuzzy properties of structural eigenvalues based on FCS model have been verified to be feasible.The further obtained conditional maximum and minimum values based on fuzzy programming theory are quantitative “defuzzification”indicators and have a significant value in engineering application.

Fig.8 Possibility distributions and conditional extremes of the first-to fifth-order natural frequencies with fuzzy constraints.a First order.b Second order.c Third order.d Fourth order.e Fifth order

Table8 Conditional extremes of the first to fifth natural frequencies and the corresponding parameters

Natural frequencies f(Hz) Possibility E(×1011 Pa) ρ(×103 kg/m3) ω(×102 rad/s)First order CM IV 51.6980 0.8428 2.1167 8.6812 3.1240 CMAV 53.9589 0.8305 2.2232 8.2890 3.1429 Second order CM IV 120.2095 0.8428 2.1176 8.6906 3.1375 CMAV 125.9723 0.8334 2.2158 8.2641 3.1534 Third order CM IV 168.6258 0.8426 2.1142 8.6759 3.1477 CMAV 176.7117 0.8304 2.2225 8.2907 3.1396 Fourth order CM IV 241.0274 0.8404 2.1099 8.6545 3.1369 CMAV 252.4831 0.8286 2.2098 8.2435 3.1444 Fifth order CM IV 460.8577 0.8403 2.1224 8.7081 3.1336 CMAV 483.1880 0.8331 2.2174 8.2666 3.1342

(3)This sample analysis shows that the presented two methods for solving the conditional extremes are all feasible and applicable in practice.If we can obtain the possibility distribution of the structural eigenvalues based on a series of cut sets of the input parameters and corresponding SFEA,the graphical solution method shown in Fig.3 is more convenient.If we construct the fuzzy maximum and minimum sets using input parameters as independent variables(i.e.,Eq.(21)),we can directly solve the optimization problems(i.e.,Eqs.(22)and(23)).In theory,the two methods are equivalent,but they have their own advantages and disadvantages.The graphical solution method can show the results in a more intuitive and concise manner but needs some SFEA times,which can reduce efficiency.In addition,an unavoidable error will occur in the drawing and reading processes.The optimization method only needs a single time of SFEA under a zero-level cut set and a single optimization time.Relatively speaking,the latter method is more efficient and accurate.However,this approach is relatively abstract in its application,and the accuracy is closely related to the optimization parameters and contains certain randomness.Generally speaking,we can choose any method depending only on the required accuracy,our habits,and even our preferences.

8 Conclusions

This paper has proposed a novel uncertainty model called the FCS model,and the problems of structural eigenvalues with fuzzy constraints are studied.The fuzzy theory(or in other words,the possibility theory)is another effective method for resolving small sample size-problems,in addition to the convex set models.A method is developed to analyze the fuzzy properties of structural eigenvalues with FCS constraint.According to the symmetric F-programming theory,the conditional maximum and minimum values of the structural eigenvalues are presented,which can serve as a non-fuzzy indicator for fuzzy eigenvalue problems.A practical application is provided,which indicates the effectiveness and practicability of the presented method.Although this study focuses on the structural eigenvalue problem,the presented methodology can be easily extended to solve other engineering problems with uncertainties,such as static response analysis,lifetime prediction,buckling,post-buckling and structural reliability.Moreover,the structural model is not limited to FEM.The method proposed in this paper can also be easily combined with the boundary element method and other novel methods such as the mesh less local Petrov–Galerkin method[32,33].These can be possible interesting study subjects in the future.

Acknowledgements This work was supported by the National Natural Science Foundation of China(Grant 51509254).

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