Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties

Journal of Sound and Vibration 333 (2014) 2483–2503

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Stochastic interval analysis of natural frequency and mode shape of structures with uncertainties Chen Wang a, Wei Gao a,n, Chongmin Song a, Nong Zhang b a b

School of Civil and Environmental Engineering, University of New South Wales, Sydney, NSW 2052, Australia School of Electrical, Mechanical and Mechatronic Systems, University of Technology Sydney, Sydney, NSW 2007, Australia

a r t i c l e i n f o

abstract

Article history: Received 24 April 2013 Received in revised form 5 December 2013 Accepted 16 December 2013 Handling Editor: J. Lam Available online 21 January 2014

In this paper, natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach. Expressions for the mean value and variance of natural frequencies and mode shapes are derived by using perturbation method and random interval moment method. The bounds of these probabilistic characteristics are then determined by interval arithmetic. Two examples are given first to illustrate the feasibility of the presented method and the results are verified by Monte Carlo Simulations. The presented approach is also applicable to solve pure random and pure interval problems. This capability is demonstrated in the third and fourth examples through the comparisons with the peer research outcomes. & 2013 Elsevier Ltd. All rights reserved.

1. Introduction Realistically modeling uncertainties in determination of natural frequencies and mode shapes is crucial for studying characteristics of dynamic systems [1]. Stochastic methods have been well developed and become the most widely accepted ones for quantifying uncertainties when sufficient statistical information is available [2]. In these methods, uncertain inputs are modeled as random variables/fields/processes, and characteristics of uncertainties are described by mean value, variance and probability density function (PDF) etc. Probabilistic features of outputs are then obtained through uncertainty propagation [3]. Natural frequencies of dynamic systems with random parameters are often determined by solving the random eigenvalue problems [4]. Boyce [5] and Collins [6] did early research on this topic in sixties of last century. Scheidt and Purker [7] systematically studied random eigenvalue problems in their book. Various approaches [8–13] have been introduced to solve the random eigenvalue problems such as direct Monte Carlo Simulation method [8], perturbation method [9,13], random factor method [11] and asymptotic integral based method [12] etc. However, in the case of lack of trustworthy statistical information, selection of proper PDF can lead to subjective results [14] in stochastic approaches. In these circumstances, interval method was introduced as an alternative to quantify these uncertainties with possible value range between crisp bounds without additional information concerning variations in their intervals [15]. Interval eigenvalue problem [16] has attracted more and more attention in the past two decades. Hollot and Bartlett [17] investigated eigenvalues of interval matrices. Chen et al. [18] proposed perturbation method for computing the bounds of eigenvalues of vibration systems with interval parameters. In this paper, first-order and second-order perturbation methods were presented. Qiu et al. [19] introduced vertex theorem to compute eigenvalue bounds of structures with uncertain-but-bounded parameters. Gao [20] proposed interval factor method to conduct interval analysis

n

Corresponding author. Tel.: þ61 2 9385 4123. E-mail addresses: [email protected] (C. Wang), [email protected] (W. Gao), [email protected] (C. Song), [email protected] (N. Zhang).

0022-460X/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jsv.2013.12.015

2484

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

of natural frequency and mode shape of structures with interval parameters. Modares et al. [21] presented an element-byelement formulation to account for interval eigenvalue problem. Angeli et al. [22] investigated natural frequency intervals for systems with polytopic uncertainty. Fuzzy set theory [23] is another alternative method for quantifying uncertainties by using membership functions which describe the degree of possibility with which the uncertain quantities may take on the associated values [15]. Through α-level strategy, interval analysis indeed forms the core of fuzzy analysis [24]. In other words, a fuzzy analysis can be converted into a series of interval analyses on α-level levels. Numerous researches on uncertain problems in determining structural dynamic characteristics have been carried out by using single type of uncertain model. However, in real engineering problems, it is very common that a large number of design variables and parameters exist in a system simultaneously. Some of them may have sufficient statistical information while the others may not. In these circumstances, stochastic and interval models are required at the same time. Various attempts have been made on mixed uncertain problems in static problems [25–30]. Based on first-order perturbation method and moment method, Gao et al. [25,26] proposed random interval moment method to analyze static response of structures with mixed random and interval properties. Du et al. [27] investigated reliability based design of structures by considering a mixture of random and interval variables using optimization technique. Qiu et al. [28] introduced the interval arithmetic into conventional reliability theory. Gao et al. [29,30] incorporated random interval moment method with Monte Carlo Simulation and Quasi Monte Carlo Simulation to study static response and reliability of structures with mixed uncertainties. In the contrast to extensive research on static mixed uncertainty problems, dynamic problems of structures with mixed random and interval variables have not been well addressed. In this paper, natural frequencies and mode shapes of structures with mixed random and interval parameters are investigated by using a hybrid stochastic and interval approach. Expressions for the mean value and variance of natural frequencies and mode shapes are derived by using the first-order perturbation method and random interval moment method [25]. The bounds of these probabilistic characteristics are then determined by interval arithmetic. The presented method is also capable to solve pure random and pure interval problems. Four numerical examples are given to illustrate applications of the presented method. Two examples are presented first to demonstrate the feasibility of the presented method and the results are verified by Monte Carlo Simulations. The capabilities for solving pure random and pure interval problems are also illustrated through the comparison with other two published examples. 2. Random interval moment method Random interval moment method is briefly introduced here and more details can be found in Ref. [25]. Let XðRÞ be the set of all real random variables on a probability space Ω. xR is a random variable of XðRÞ. R denotes the set of all real numbers. xR , StdðxR Þ and VarðxR Þ are the mean value, standard deviation and variance of xR respectively. yI is an interval variable of IðRÞ which denotes the set of all the closed real intervals. A list of notations is attached in Appendix A for convenience: yI ¼ ½y; y ¼ ft; y rt r yjy; y A Rg yc ¼

y þy 2

;

Δy ¼

y y 2

;

ΔyI ¼ ½  Δy; þ Δy

(1) (2)

where y, y, yc , Δy and ΔyI are the lower bound, upper bound, midpoint value, maximum width and uncertain interval of yI respectively.
! ! Z RI is defined as a function of a random vector XR ¼ ðxR1 ; xR2 ; …; xRn Þ and an interval vector Y I ¼ ðyI1 ; yI2 ; …; yIm Þ. The mean
!
! !
!
!
! value of XR is XR ¼ ðxR1 ; xR2 ; …; xRn Þ and midpoint value of YI is Y c ¼ ðyc1 ; yc2 ; …; ycm Þ. Applying Taylor expansion at ( XR , Yc ), the following expression can be acquired:
! ! Z RI ¼ f ð XR ;YI Þ 9 8  =
!
! m < ∂f   R c ¼ f ð X ; Y Þþ ∑ 
!
! ΔyIj I  ; : ∂yj R j¼1 X ; Yc 9 9 8 8   = = n < ∂f  m < ∂2 f    I þ ∑ 
!
! þ ∑ 
!
! Δyj ðxRi  xRi Þ þ Re R R I   ; ; : : ∂xi ∂xi ∂yj R i¼1 j¼1 XR ; Y c X ; Yc

(3)

Ignoring the remainder term Re, the mean value and variance of the random interval function Z RI can be computed by using moments of linear function [31] as 9 8  = h i
!
! m < ∂f   RI RI R c ΔyIj (4) μðZ Þ ¼ E Z ¼ f ð X ; Y Þ þ ∑  I 
!
! ; : ∂y R c j¼1 j X ; Y h i VarðZ RI Þ ¼ E ðZ RI μðZ RI ÞÞ2

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

9 9 8  8  = = < ∂f  m < ∂2 f    I ¼ ∑ ∑ 
!
! þ ∑ 
!
! Δyj R R I ; i ¼ 1 k ¼ 1 :∂xi  XR ; Y c j ¼ 1 :∂xi ∂yj  XR ; Y c ; 9 9 8  8  = = < ∂f  m < ∂2 f    I  CovðxRi ; xRk Þ Δy þ ∑   j
!
! R I
! ; :∂xRk  R
! j ¼ 1 :∂xk ∂yj  XR ; Y c ; X ; Yc n

2485

n

(5)


! ! where E½ is the expectation operator, and CovðxRi ; xRk Þ denotes the covariance of random variables xRi and xRk . Z RI ¼ f ð XR ;Y I Þ is smooth and twice differentiable to random variable xRi and interval variable yIj . 3. Stochastic interval analysis of natural frequency Natural frequencies of a vibration system can be obtained through solving eigenvalue problem which can be expressed as
!
! ½K Φk ¼ λk ½M Φk

(6)
! where λk is the kth eigenvalue and Φk is the corresponding mode shape. ½K and ½M are stiffness matrix and mass matrix respectively. ωk is the kth natural frequency and λk ¼ ω2k . Normalized mode shape has the following important properties:
!
! ΦTk ½K Φk ¼ λk

(7)


!
! ΦTk ½M Φk ¼ 1

(8)

!
! I I I I Using the strategy of the random interval moment method [25], let vector aR ¼ ðaR1 ; aR2 ; …; aRn Þ and b ¼ ðb1 ; b2 ; …; bm Þ represent all random variables and interval variables respectively in a vibration system with S degrees of freedom. Stiffness !
! I matrix ½K and mass matrix ½M are functions of aR and b . Therefore the kth natural frequency ωk and its corresponding !
!
! I mode shape Φk are also functions of aR and b . Thus, Eq. (6) can be written as
! !I
! !I
! !I
!
! !I
!
! !I ½Kð aR ;b Þ U Φk ð aR ;b Þ ¼ ω2k ð aR ;b Þ U ½Mð aR ;b Þ U Φk ð aR ;b Þ

(9)


!
! Denoting mean value of aR ¼ ðaR1 ; aR2 ; …; aRn Þ by aR ¼ ðaR1 ; aR2 ; …; aRn Þ, midpoint values and uncertain interval of ! ! I I I I c c c c I I I I b ¼ ðb1 ; b2 ; …; bm Þ by b ¼ ðb1 ; b2 ; …; bm Þ and Δb ¼ ðΔb1 ; Δb2 ; …; Δbm Þ respectively, and expanding the stiffness and mass
! ! c matrices at ( aR ; b ) by Taylor expansion gives  
! !  "
! ! # m ∂½K   I c I R R Kð a ;b Þ ¼ Kð a ;b Þ þ ∑ Δbj  I
! ! j ¼ 1 ∂bj  aR ;bc   ! n m ∂2 ½K  ∂½K   I (10) þ ∑ þ ∑ Δbj ðaRi  aRi Þ þ Re   ! I
! ! R ∂aRi 
! c i¼1 j ¼ 1 ∂ai ∂bj  aR ;bc aR ;b  # 
! !  "
! m ∂½M  !  I c I R R Mð a ;b Þ ¼ Mð a ;b Þ þ ∑ Δbj  I
! ! j ¼ 1 ∂bj  aR ;bc   ! n m ∂2 ½M  ∂½M   I þ ∑ þ ∑ Δbj ðaRi  aRi Þ þ Re   ! I
! ! R ∂aRi 
! c i¼1 j ¼ 1 ∂ai ∂bj  aR ;bc aR ;b

(11)

where Re is the remainder term. Ignoring the remainder term, the random interval stiffness matrix can be expressed as
! !I
! !
! !I c ½Kð aR ;b Þ ¼ ½Kð aR ;b Þ þ½ΔKð aR ;b Þ
! !I
! !I where ½ΔKð aR ;b Þ represents a perturbation (small change) of ½Kð aR ;b Þ:    !  m ∂½K  n m ∂2 ½K 
! !I  ∂½K    I I R R ΔKð aR ;b Þ ¼ ∑ Δb þ ∑ Δb þ ∑    j j ðai ai Þ ! ! I
! ! R ∂bI 
! ∂aRi 
! c c R R ∂a j ¼ 1 ∂bj  aR ;bc i¼1 j ¼ 1 j i a ;b a ;b

(12)

(13)

Denoting ð∂½K=∂aRi Þj
! !, ð∂½K=∂bj Þj
! ! and ð∂2 ½K=∂aRi ∂bj Þj
! ! by K0aR , K0bI and K″aR bI , Eq. (13) can be rewritten as i j i j c c c aR ;b aR ;b aR ;b ( ) m n m
! !I I I ½ΔKð aR ;b Þ ¼ ∑ K0bI Δbj þ ∑ K0aR þ ∑ K″aR bI Δbj ðaRi  aRi Þ (14) I

I

j¼1

j

i¼1

i

j¼1

i

j

2486

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

Similarly, the mass matrix can also be expressed as
! !I
! !
! !I c ½Mð aR ;b Þ ¼ ½Mð aR ;b Þ þ½ΔMð aR ;b Þ

(15)


! !I
! !I where ½ΔMð aR ;b Þ represents a perturbation (small change) of ½Mð aR ;b Þ.    !  m ∂½M  n m ∂2 ½M 
! !I  ∂½M    I I R R Δb þ Δb ΔMð aR ;b Þ ¼ ∑ þ ∑ ∑    j j ðai  ai Þ ! ! I
! ! R ∂bI 
! ∂aRi 
! c c R R ∂a j ¼ 1 ∂bj  aR ;bc i¼1 j ¼ 1 j a ;b i a ;b

(16)

Representing ð∂½M=∂aRi Þj
! !, ð∂½M=∂bj Þj
! ! and ð∂2 ½M=∂aRi ∂bj Þj
! ! by M0aR , M0bI and M″aR bI , Eq. (16) can be i j i j c c c aR ;b aR ;b aR ;b rewritten as ( ) m n m
! !I I I (17) ½ΔMð aR ;b Þ ¼ ∑ M0bI Δbj þ ∑ M0aR þ ∑ M″aR bI Δbj ðaRi  aRi Þ I

I

j

j¼1

i¼1

i

j¼1

i

j


! Denoting perturbations (small changes) of the kth natural frequency and its corresponding mode shape by Δωk and Δ Φk , and
! !
! !I
! !
! !I
! !
!
!
! ! c c c c representing ½Kð aR ;b Þ, ½ΔKð aR ;b Þ, ½Mð aR ;b Þ, ½ΔMð aR ;b Þ, ωk ð aR ;b Þ, Φk ð aR ;b Þ by Kc , ΔK, Mc , ΔM, ωck and Φck respectively, the first-order perturbation of Eq. (9) can be expressed as
!
!
!
! ðKc þ ΔKÞð Φck þΔ Φk Þ ¼ ðωck þ Δωk Þ2 ðMc þ ΔMÞð Φck þΔ Φk Þ

(18)

Ignoring the higher-order terms, Δωk can be computed as Δωk ¼

h i
! 1
! T Φck ΔK  ωc2 ΔM Φck k c 2ωk

(19)

and consequently ωk ¼ ωck þ Δωk ¼ ωck þ

h i
! 1
! cT c c2 c Φk ΔK  ωk ΔM Φk 2ωk

(20)


!
! T 0 ω c Denoting Φck ½K0bI  ωc2 k UMbI  Φk by DbI , the mean value of the kth natural frequency can be evaluated by using moments of j

j

j

linear function [31] as   1 m I μðωk Þ ¼ E ωck þ Δωk ¼ ωck þ ∑ ðDωI Δbj Þ 2ωck j ¼ 1 bj

(21)

I

Due to existence of uncertain intervals Δbj , the mean value of the kth natural frequency is not a deterministic value but an interval value. As ωck and DωbI are deterministic values, μðωk Þ is equal to a linear combination of Δbj plus a deterministic value ωck . I

j

I

Using interval arithmetic, lower bound μðωk Þ and upper bound μðωk Þ can be achieved on bounds of Δbj as ( ) 1 m I ω μðωk Þ ¼ ωck þ ∑ ðD I Δbj Þ 2ωck j ¼ 1 bj min ( μðωk Þ ¼

ωck þ

1 m I ∑ ðDωI Δbj Þ 2ωck j ¼ 1 bj

(22)

) (23) max

The variance of the kth natural frequency can also be evaluated by using moments of linear function [31]. Denoting
!T
!
!
! 0 ω ω c cT c c2 Φck ½K0aR ωc2 k U MaR  Φk by DaR and Φk ½K″aR bI ωk U M″aR bI  Φk by DaR bI , the variance of the kth natural frequency can be i

i

i

i

j

computed as

n

n

j

i

j

h i Varðωk Þ ¼ E ðωk  μðωk ÞÞ2 !

1

¼ ∑ ∑

i

DωaR i

i ¼ 1 l ¼ 1 4ωc2 k

þ

m

I ∑ ðDωaR bI Δbj Þ i j j¼1

DωaR l

m

þ ∑

j¼1

! I ðDωaR bI Δbj Þ l j

CovðaRi ; aRl Þ

     m m 1  ω I  ω I  ω ω ¼ ∑ ∑ DaR þ ∑ ðDaR bI Δbj ÞDaR þ ∑ ðDaR bI Δbj ÞCovðaRi ; aRl Þ i  l j ¼ 1 l j  i j i ¼ 1 l ¼ 1 4ωc2  j¼1 n

n

(24)

k

and the standard deviation of the kth natural frequency is Stdðωk Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðωk Þ

(25)

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

where CovðaRi ; aRl Þ is the covariance of random variables aRi and aRl . For mutually independent variables 0 !2 !2 n m 1 I ω ω R @ Varðωk Þ ¼ ∑ DaR þ ∑ ðDaR bI Δbj Þ Varðai Þ i i j i¼1 j¼1 4ωc2 k   !  m 1  ω I  ω R D R þ ∑ ðD R I Δbj ÞStdðai Þ  2ωck  ai j ¼ 1 ai bj

n

Stdðωk Þ ¼ ∑

i¼1

2487

(26)

(27)

I

Again, due to existence of uncertain intervals Δbj , variance and standard deviation of the kth natural frequency are not deterministic values but interval values. As DωaR and DωaR bI are deterministic values, Stdðωk Þ is equal to a linear combination of Δbj

I

i

i

j

I

plus a deterministic value. Consequently, lower bound Stdðωk Þ and upper bound Stdðωk Þ can be achieved on the bounds of Δbj by interval arithmetic as

( Stdðωk Þ ¼

i¼1

( Stdðωk Þ ¼

n

∑

n

∑

i¼1

  !)  m 1  ω I  ω R Stdða D þ ∑ ðD Δb Þ Þ   R I R j  i 2ωck  ai j ¼ 1 ai bj min

(28)

  !)  m 1  ω I  ω R Stdða D þ ∑ ðD Δb Þ Þ   R I R j  i 2ωck  ai j ¼ 1 ai bj max

(29)

and therefore lower bound Varðωk Þ and upper bound Varðωk Þ are acquired as Varðωk Þ ¼ ðStdðωk ÞÞ2

(30)

Varðωk Þ ¼ ðStdðωk ÞÞ2

(31)

4. Stochastic interval analysis of mode shape From Eq. (18), using eigenvalues instead of natural frequencies and ignoring higher-order terms, the following equation can also be obtained:
!
!
!
!
! Kc U Δ Φk þΔK U Φck ¼ Δλk UMc U Φck þ λck U ΔM U Φck þ λck U Mc U Δ Φk
! The change of the kth mode shape Δ Φk can be expressed as a superposition of deterministic mode shapes [18] S
!
! Δ Φk ¼ ∑ C q U Φcq

(32)

(33)

q¼1


! where C q is a constant with respect to Φcq .
! T Substituting Eq. (33) into Eq. (32), pre-multiplying the both sides by Φcq and using orthogonal properties, gives
!T
!T
!
! C q λcq þΦcq U ΔK U Φck ¼ λck UΦcq UΔM U Φck þ C q U λck

(34)

when q ak, from Eq. (34), C q can be computed as Cq ¼

h i
!
! T c Φcq ΔK λk U ΔM Φck λck λcq

ðq akÞ

(35)

when q ¼ k, first-order perturbation of Eq. (8) can be expressed as T
!
!
!
! ð Φck þ Δ Φk Þ ðMc þ ΔMÞð Φck þ Δ Φk Þ ¼ 1

(36)

Ignoring the higher-order terms, the following equation can be obtained:
!T
!
!
!T
!
! Φck UΔM U Φck þ Φck UMc U Δ Φk þΔ ΦTk U Mc U Φck ¼ 0

(37)


! T Pre-multiplying both sides of Eq. (33) by Φck U Mc and using the orthogonal properties gives
!
! T C q ¼ Φck U Mc UΔ Φk ðq ¼ kÞ

(38)

Substituting Eq. (38) and its transpose into Eq. (37) gives
!T
! C q ¼  12 Φck UΔM U Φck

(39)

2488

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

Therefore, S
!
!
!
!
! Φk ¼ Φck þ Δ Φk ¼ Φck þ ∑ C q U Φcq

(40)

q¼1

where

Cq ¼


!T
! Φcq ½ΔK  λck UΔM Φck λck  λcq

ðwhen q a kÞ

(41)


!T
! C q ¼  12 Φck UΔM U Φck ðwhen q ¼ kÞ

(42)

Consequently, the mean value of the kth mode shape can be evaluated by moments of linear function [31] as S
!
!
!
!
! μð Φk Þ ¼ E½ Φck þ Δ Φk  ¼ Φck þ ∑ E½C q  U Φcq

(43)

q¼1

where





E Cq ¼


! I 0 0
! cT c c ∑m j ¼ 1 ðΦq ½KbI  λk UMbI  Φk U Δbj Þ j

j

λck  λcq

ðwhen q a kÞ

(44)

m  
! 1
!T I E C q ¼  Φck U ∑ ðM0bI Δbj Þ U Φck ðwhen q ¼ kÞ j 2 j¼1

(45)


!
! T c Denoting Φcq ½K0bI  λk U M0bI  Φck by DϕI , the mean value of the kth mode shape can be expressed as j

bj

j

ϕ ∑m m j ¼ 1 ðDbI Δbj Þ
!
!
! 1
! T I j U Φcq  Φck U ∑ ðM0bI Δbj Þ U Φck U Φck j λ  λ 2 q k q¼1 j¼1 I

S
!
! μð Φk Þ ¼ Φck þ ∑

(46)

qa k

Eq. (46) can be rewritten as 1

20

3

DϕI 7 m 6B S
!
! 1
!
!
! C bj
! T C I7 6B c U Φcq  Φck UM0bI U Φck U Φck CΔbj 7 μð Φk Þ ¼ Φk þ ∑ 6B ∑ j A 5 2 j ¼ 1 4@ q ¼ 1 λk λq

(47)

qak

For the element on pth degree of freedom in the kth mode shape, the mean value of ϕpk can be evaluated as 1

20

3

DϕI C 7 m 6B S
! bj 1
!T C I7 6B Uϕcpk  Φck U M0bI U Φck U ϕcpk CΔbj 7 μðϕpk Þ ¼ ϕcpk þ ∑ 6B ∑ j A 5 2 j ¼ 1 4@ q ¼ 1 λk λq

(48)

qak

I

In Eq. (48), the uncertain intervals Δbj will result in a change range for the mean value of kth mode shape on pth degree of freedom. As S

DϕI

q¼1

λk λq

∑

qa k

bj

Uϕcpq 


! 1
! T Φc U M0bI U Φck U ϕcpk j 2 k

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2489

I

is a deterministic value, μðϕpk Þ is equal to a linear combination of Δbj plus a deterministic value ϕcpk . By means of the interval I

arithmetic, the lower bound μðϕpk Þ and upper bound μðϕpk Þ can be captured on the bounds of Δbj as 8 20 39 > > > > ϕ > D = < 6 7> B I m
! bj 1
! 6B S I7 0 cT c c c c μðϕpk Þ ¼ ϕpk þ ∑ 6B ∑ U ϕpq  Φk UMbI U Φk Uϕpk ÞΔbj 7 j > 5> 2 > > j ¼ 1 4@ q ¼ 1 λk  λq > > ; : q ak

(49)

min

8 20 39 > > > > > DϕI = < 7>
! m 6B S
! b 1 T 6 7 B I j 0 μðϕpk Þ ¼ ϕcpk þ ∑ 6B ∑ Uϕcpq  Φck U MbI U Φck U ϕcpk ÞΔbj 7 j > 5> 2 > > j ¼ 1 4@ q ¼ 1 λk λq > > ; : qa k

(50) max


!T
! c The variance of ϕpk can also be obtained by using moments of linear function [31]. Denoting Φcq ½K0aR λk U M0aR  Φck by DϕaR , i

i

i


!
!
!
!
!
! T T T Φcq ½K″aR bI  λck UM″aR bI  Φck by DϕR I , Φck M0aR Φck by M ϕaR and Φck M″aR bI Φck by M ϕR I , the variance of ϕpk can be expressed as ai bj ai bj i i j i j i j i h i Varðϕpk Þ ¼ E ðϕpk  μðϕpk ÞÞ2 0 !2 ! !1 1 s m m ϕcpq I I ϕ ϕ ϕ ϕ B ∑ @ DaR þ ∑ ðD R I Δbj Þ DaR þ ∑ ðD R I Δbj Þ A C C B ai bj al bj λk  λq i l C B q¼1 j¼1 j¼1 n n B C q ak CCovðaR ; aR Þ ¼ ∑ ∑ B i l C B ! ! c 2 C i¼1l¼1B m m C B ϕpk I I ϕ ϕ ϕ ϕ A @þ M aR þ ∑ ðM R I Δbj Þ M aR þ ∑ ðM R I Δbj Þ 2 0

i

j¼1

ai bj

l

j¼1

(51)

al bj

and standard deviation of ϕpk is Stdðϕpk Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Varðϕpk Þ

(52)

For mutually independent variables, 0 !2 !2 1 1 s m ϕcpq I ϕ ϕ B ∑ @ DaR þ ∑ ðD R I Δbj Þ A C C B ai bj λk λq i q¼1 j¼1 C B n B C q ak CVarðaR Þ B Varðϕpk Þ ¼ ∑ B i C ! C i¼1B 2 c 2 m C B ϕpk I ϕ ϕ A @þ MaR þ ∑ ðM R I Δbj Þ 2 0

i

j¼1

(53)

ai bj

and the standard deviation,  0 1 !2 !2 1   s m ϕcpq   I ϕ ϕ  ∑ @ C B DaR þ ∑ ðD R I Δbj Þ A   C B ai bj λk  λq i j¼1  C B n  q¼1   qa k C B R C   B Stdða Þ Stdðϕpk Þ ¼ SQRTB ∑  i  C !  C Bi ¼ 1  2 2 c m   C B I ϕ ϕ   þ ϕpk A @ M aR þ ∑ ðM R I Δbj Þ   2 ai bj i   j¼1 0

(54)

where SQRT represents square root. I As DϕaR , DϕR I , M ϕaR and M ϕR I are deterministic values, Eq. (53) is a second-order polynomial function of Δbj . The bounds of ai bj ai bj i i Varðϕpk Þ can be achieved by using optimization methods, while more computational effort is required. Triangular Inequality I Theorem can also be easily applied here to give conservative results by taking bounds value of Δbj because  2 !2                m  ϕ  m  ϕ  ϕ  m I  I  I  ϕ ϕ ϕ 2 (55) ∑ r D þ ∑ ðD Δb Þ r D ðD Δb Þ  DaR    ∑ ðDaR bI Δbj Þ þ    j  j   aRi aRi aRi bIj aRi bIj  i  j¼1 i j    j¼1 j¼1  2 !2                m  ϕ  m  ϕ  ϕ  m I  I  I  ϕ ϕ ϕ 2 þ r M þ ðM Δb Þ r M ðM Δb Þ ∑ ∑  MaR    ∑ ðM R I Δbj Þ     R I R I j  j   ai ai aRi bj aRi bj  i   j ¼ 1 ai bj    j¼1 j¼1

(56)

2490

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

Consequently, the lower bound Varðϕpk Þ and upper bound Varðϕpk Þ can be expressed as 0 11 1 0 0 !2 s m ϕcpq I C B ∑ @ B ðDϕaR þ ∑ ðDϕR I Δbj ÞÞ2 A C C C B ai bj B λk  λq i j¼1 C C B n B q¼1 C C B B Varðϕpk Þ ¼ B ∑ B q ak CVarðaRi ÞC C C Bi ¼ 1 B 2 C C B B c m I 2 ϕ A A @ þ ϕpk ðM ϕ þ ∑ @ ðM R I Δbj ÞÞ 2 aR i

ai bj

min

1 !2 1 1 s m ϕcpq I ϕ ϕ A C C B ∑ @ B DaR þ ∑ ðD R I Δbj Þ C C B B ai bj λk λq i j¼1 C C B n B q¼1 C C B q ak B CVarðaR ÞC B Varðϕpk Þ ¼ B i C C B∑ B !2 C C Bi ¼ 1 B 2 c m C C B B I ϕ ϕ A A @ þ ϕpk @ M þ ðM Δb Þ ∑ j R I 2 aR 0

0

0

j¼1

(57)

!2

i

j¼1

(58)

ai bj

max

and therefore lower bound Stdðϕpk Þ and upper bound Stdðϕpk Þ are acquired as Stdðϕpk Þ ¼ ðVarðϕpk ÞÞ1=2

(59)

Stdðϕpk Þ ¼ ðVarðϕpk ÞÞ1=2

(60)

5. Numerical examples Four numerical examples are given to illustrate the applications of the presented method. First two examples are presented to demonstrate the feasibility of the presented method and the results are verified by Monte Carlo Simulations (MCS). The third and fourth examples are illustrations of its capabilities for solving pure random and pure interval problems. Published results of dynamic characteristics of structures with pure random or interval parameters are used to assess the accuracy of the presented method in these two examples. 5.1. Mixed random interval problem 5.1.1. Cantilever beam with hollow section Natural frequencies and mode shapes of a two dimensional cantilever beam with mixed random and interval parameters shown in Fig. 1 are investigated in this example. The cantilever beam made of aluminum is 2 m long with hollow rectangular section. The width and height of the cross-section are random variables with normal distribution, w ¼ 40 mm; sw ¼ 2 mm and h ¼ 60 mm; sh ¼ 3 mm. The thickness is an interval variable and t c ¼ 3 mm; Δt ¼ 0:15 mm. Elastic modulus E and mass density ρ of all elements are considered as deterministic values in this example, E ¼70 GPa and ρ ¼ 2700 kg=m3 . The beam is modeled as 10 beam elements with uniform length, and consistent mass is applied. Lower and upper bounds of the mean value and standard deviation of the first 10 natural frequencies acquired by the presented method and the MCS method are compared in Tables 1 and 2. To obtained convergent simulation results, 108 MCS which consists of 104 simulations of the interval variable and 104 simulations of the random variables are implemented. Relative errors with respect to the results of MCS are also computed and listed in the tables. The uncertain parameters in elements are considered as fully dependent, which means that values of same uncertain parameter in different elements are equal in each simulation. Consequently the computed relative errors of bounds appear to be same. Lower and upper bounds of the mean value of the 1st mode shape computed by the presented method and the mean value obtained by pure stochastic approach are plotted in Fig. 2. In the pure stochastic approach, the midpoint values of interval variables are taken. The results computed by the presented method are compared with the results generated by 108 MCS in Tables 3 and 4.

2000

60×40×3 Not in scale

Fig. 1. Front elevation (a) and cross section (b) of the aluminum cantilever beam (unit: mm).

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2491

Table 1 Bounds of mean value of the first 10 natural frequencies. Natural frequency (rad/s)

Mean value Presented method

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

98.3622 616.4455 1726.4476 3385.5103 5605.2570 8397.3124 11781.8368 15784.4133 20409.0502 25371.3938

0.1054 0.1054 0.1054 0.1054 0.1054 0.1054 0.1054 0.1054 0.1054 0.1054

98.8831 619.7096 1735.5891 3403.4364 5634.9365 8441.7757 11844.2210 15867.9910 20517.1152 25505.7340

 0.0672  0.0672  0.0672  0.0672  0.0672  0.0672  0.0672  0.0672  0.0672  0.0672

98.2587 615.7967 1724.6306 3381.9473 5599.3578 8388.4748 11769.4371 15767.8013 20387.5710 25344.6920

98.9496 620.1264 1736.7566 3405.7258 5638.7270 8447.4543 11852.1883 15878.6651 20530.9165 25522.8911

Table 2 Bounds of standard deviation of the first 10 natural frequencies. Natural frequency (rad/s)

Standard deviation Presented method

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

4.3495 27.2585 76.3414 149.7032 247.8577 371.3190 520.9786 697.9678 902.4636 1121.8925

 2.3074  2.3074  2.3074  2.3074  2.3074  2.3074  2.3074  2.3074  2.3074  2.3074

4.7295 29.6403 83.0119 162.7839 269.5149 403.7639 566.5004 758.9543 981.3185 1219.9204

1.6492 1.6492 1.6492 1.6492 1.6492 1.6492 1.6492 1.6492 1.6492 1.6492

4.4522 27.9023 78.1445 153.2390 253.7118 380.0892 533.2836 714.4530 923.7788 1148.3903

4.6528 29.1594 81.6651 160.1428 265.1422 397.2131 557.3093 746.6408 965.3973 1200.1281

Stochastic approach Lower bound of mean Upper bound of mean

1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

1st Mode Shape Fig. 2. Bounds of mean value of the 1st mode shape.

Lower and upper bounds of the mean values of the 2nd and 3rd mode shapes computed by the presented method and the mean values obtained by the pure stochastic approach are also plotted in Figs. B1 and C1 in Appendix B and C respectively. The bounds of the mean values and standard deviations of the 2nd and 3rd mode shapes acquired by the presented method are also verified by 108 MCS in Tables B1, B2, C1 and C2 in Appendix B and C. From Tables B1, B2, C1 and C2, it can be easily observed that the bounds of mean values and standard deviations computed by the presented method

2492

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

are in good agreement with the results generated by the 108 MCS. The presented method is much more efficient than MCS because it costs 0.3297 s while 108 MCS consumes 62962.8412 s on a computer with Intel Core [email protected] GHz. Due to the existence of the random and interval parameters in the system simultaneously, natural frequencies and corresponding mode shapes have both features of stochastic and interval variables. The mean values and standard deviations of natural frequencies and mode shapes are not deterministic values but interval values. The bounds of these probabilistic characteristics computed by the presented method can provide boundaries of possible PDF of natural frequencies and mode shapes of structures with mixed random and interval variables. The boundaries of PDF of the 1st and 2nd natural frequencies are shown in Figs. 3 and 4 (bold black continuous lines) by the combination of the lower and Table 3 Bounds of mean value of the 1st mode shape. Degree of freedom

Mean value Presented method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

1.8773E  02 1.8322E  01 7.1485E  02 3.3941E  01 1.5275E 01 4.6885E  01 2.5729E  01 5.7224E  01 3.8000E  01 6.5085E  01 5.1610E  01 7.0663E  01 6.6131E  01 7.4237E  01 8.1196E  01 7.6171E  01 9.6520E  01 7.6918E  01 1.1192E þ 00 7.7030E 01

 0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131

1.9673E  02 1.9200E  01 7.4911E  02 3.5568E  01 1.6007E  01 4.9132E  01 2.6962E  01 5.9967E 01 3.9821E  01 6.8204E  01 5.4084E  01 7.4050E  01 6.9301E  01 7.7796E 01 8.5087E  01 7.9821E  01 1.0115E þ00 8.0605E  01 1.1728E þ00 8.0722E  01

 0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483

1.8794E  02 1.8342E  01 7.1566E  02 3.3979E  01 1.5293E  01 4.6938E  01 2.5758E  01 5.7289E  01 3.8043E  01 6.5158E  01 5.1669E  01 7.0743E  01 6.6206E  01 7.4321E  01 8.1288E  01 7.6257E  01 9.6629E  01 7.7005E  01 1.1205E þ00 7.7117E  01

1.9702E  02 1.9228E  01 7.5022E  02 3.5620E  01 1.6031E  01 4.9205E  01 2.7002E  01 6.0056E  01 3.9880E  01 6.8306E 01 5.4164E  01 7.4160E  01 6.9404E 01 7.7911E  01 8.5214E  01 7.9940E  01 1.0130E þ00 8.0725E  01 1.1746E þ00 8.0841E  01

Table 4 Bounds of standard deviation of the 1st mode shape. Degree of freedom

Standard deviation Presented method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

3.5023E  04 3.4181E  03 1.3336E 03 6.3321E  03 2.8498E  03 8.7470E  03 4.8000E  03 1.0676E  02 7.0893E  03 1.2142E  02 9.6285E  03 1.3183E  02 1.2338E 02 1.3850E  02 1.5148E  02 1.4211E  02 1.8007E  02 1.4350E 02 2.0880E  02 1.4371E  02

 1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226

3.8710E  04 3.7779E  03 1.4740E  03 6.9986E  03 3.1497E 03 9.6677E  03 5.3053E  03 1.1800E  02 7.8355E 03 1.3420E  02 1.0642E  02 1.4571E  02 1.3636E  02 1.5308E  02 1.6743E  02 1.5706E  02 1.9902E  02 1.5861E  02 2.3078E 02 1.5883E  02

0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916

3.5710E  04 3.4851E  03 1.3598E 03 6.4562E  03 2.9056E  03 8.9185E  03 4.8941E  03 1.0885E  02 7.2282E  03 1.2380E  02 9.8173E  03 1.3442E  02 1.2579E  02 1.4121E  02 1.5445E  02 1.4489E  02 1.8360E  02 1.4631E  02 2.1289E  02 1.4652E  02

3.8406E  04 3.7483E  03 1.4624E 03 6.9436E  03 3.1250E  03 9.5918E  03 5.2636E  03 1.1707E 02 7.7740E  03 1.3315E  02 1.0558E  02 1.4456E  02 1.3529E 02 1.5188E  02 1.6611E  02 1.5583E  02 1.9746E  02 1.5736E  02 2.2897E  02 1.5759E  02

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2493

PDF of the 1st natural frequency

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 75

80

85

90

95

100

105

110

115

120

natural frequency ω (rad/s) Fig. 3. PDF of the 1st natural frequency obtained by presented method and 108 MCS. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

PDF of the 2nd natural frequency 0.016

0.014

PDF obtained by 10 MCS

0.012

0.01

0.008

0.006

0.004

0.002

0 450

500

550

600

650

700

750

natural frequency ω (rad/s) Fig. 4. PDF of the 2nd natural frequency obtained by presented method and 108 MCS. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

upper bounds of the mean value and standard deviation acquired from the presented method. The PDF obtained by 108 MCS are also plotted in these figures (red point and dash line). Figs. 3 and 4 indicate that the presented method is capable of giving approximate boundaries of possible PDF of natural frequencies of structures with mixed random and interval parameters efficiently. 5.1.2. Three dimensional steel frame structure A model of 3D steel frame structure shown in Fig. 5 with mixed random and interval parameter is investigated in this example to demonstrate the feasibility of presented method to complex structures. The effects of random and interval parameters on structural natural frequencies are also discussed in this example. The structure has 20 levels and each level is 0.36 m in height. The layout of columns consists of 4  3 bays in size of 0.5 m  0.5 m. The elevations are shown in Fig. 6. The structure is modeled as 1020 3D beam elements and has total 2400

2494

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

Fig. 5. 3D view of steel frame structure (unit: m).

Fig. 6. Front (a), side (b) and top (c) elevations of steel frame structure (unit: m).

degrees of freedom. Columns and beams of the structure have square hollow sections. The outer dimension of column is 40 mm  40 mm and thickness is 3 mm. The outer dimension of beam is 25 mm  25 mm and thickness is 2 mm. In this example, the outer dimensions of column and beam are considered as random variables. Mean value and standard deviation of the side length of column are ac ¼ 40 mm and sac ¼ 2 mm. Mean value and standard deviation of the side length of beam are ab ¼ 25 mm and sab ¼ 1 mm. The thicknesses of hollow sections of column and beam are interval variables. The midpoint values are t c ¼ 3 mm and t b ¼ 2 mm, and uncertain intervals are Δt Ic ¼ ½  0:2; 0:2 mm and Δt Ib ¼ ½  0:1; 0:1 mm respectively. Elastic modulus E, shear modulus G and mass density ρ are deterministic values, E ¼ 200 GPa, G ¼ 77 GPa and ρ ¼ 7850 kg=m3 . Consistent mass is applied. Bounds of mean values and standard deviations of the first 10 natural frequencies obtained by the presented method are shown in Table 5. The effects of the random and interval parameters on natural frequencies are indicated in Figs. 7–11 by using Dispersal Degree (DD) [25] which is defined by coefficient of variation νxR for random variables and interval change ratio μyI for interval variables: νxR ¼

sxR xR

(61)

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2495

Table 5 First 10 natural frequencies of 3D steel frame. Natural frequency (rad/s)

Mean value

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10

Standard deviation

Lower bound

Upper bound

Lower bound

Upper bound

36.9988 38.3316 43.0689 100.1579 103.0970 112.8396 180.1622 184.7953 201.3243 254.8576

37.3006 38.6435 43.3913 100.7232 103.6800 113.3866 181.4009 186.0719 202.6000 256.5315

1.3124 1.3894 1.5870 3.8373 4.0034 4.4624 6.9342 7.1514 7.7754 9.7362

1.4467 1.5319 1.7515 4.2450 4.4292 4.9434 7.6606 7.9007 8.5980 10.7542

1.8

33.6

Mean value of 1st natural frequency

Standard deviation of 1st natural frequency

Lower bound. Upper bound.

33.5

33.4

33.3

33.2

33.1

33

32.9

Lower bound. Upper bound.

1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

0.01

0.02

0.03

0.04

0.05

0

0.01

Dispersal Degree (DD)

0.02

0.03

0.04

0.05

Dispersal Degree (DD)

Fig. 7. Mean value (a) and standard deviation (b) of the 1st natural frequency when DD ¼ νac ¼ νab ¼ μtIc ¼ μtI . b

34.2 0.045

Standard deviation of 1st natural frequency

Mean value of 1st natural frequency

34 33.8 33.6 33.4 33.2 33 32.8 32.6 32.4

0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0

0

0.01

0.02

0.03

Dispersal Degree (DD)

0.04

0.05

0

0.01

0.02

0.03

0.04

0.05

Dispersal Degree (DD)

Fig. 8. Mean value (a) and standard deviation (b) of the 1st natural frequency when DD ¼ νac ; μtIc ¼ μtI ¼ νab ¼ 0. b

2496

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

34.2

Standard deviation of 1st natural frequency

Mean value of 1st natural frequency

1.6

Lower bound. Upper bound.

34 33.8 33.6 33.4 33.2 33 32.8 32.6 32.4

Lower bound. Upper bound.

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0

0.01

0.02

0.03

0.04

0.05

0

0.01

Dispersal Degree (DD)

0.02

0.03

0.04

0.05

Dispersal Degree (DD)

Fig. 9. Mean value (a) and standard deviation (b) of the 1st natural frequency when DD ¼ νab ; μt Ic ¼ μtI ¼ νac ¼ 0. b

1 0.8 Standard deviation of 1st natural frequency

Mean value of 1st natural frequency

33.3

33.25

33.2

33.15

33.1

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

0

0.01

0.02

0.03

0.04

0.05

0

0.01

Dispersal Degree (DD)

0.02

0.03

0.04

0.05

Dispersal Degree (DD)

Fig. 10. Mean value (a) and standard deviation (b) of the 1st natural frequency when DD ¼ μtI ; μt Ic ¼ νac ¼ νab ¼ 0. b

where sxR and xR are standard deviation and mean value of random variable xR : μyI ¼

Δy ; yc

(62)

where Δy and yc are maximum width and midpoint value of interval variable yI . Fig. 7 shows that the mean value and standard deviation of the 1st natural frequency obtained by the presented method are not deterministic values but intervals when structures have a mixture of random and interval parameters. The widths of intervals increase with the dispersal degrees of random and interval parameters. In the case that structures have random parameters only (Figs. 8 and 9), the mean value of the 1st natural frequency is deterministic value. The standard deviation of the 1st natural frequency increases with dispersal degrees of random parameters linearly. When structures have interval parameters only (Figs. 10 and 11), the 1st natural frequency obtained by the presented method is an interval variable and the interval increases with dispersal degrees of interval parameters linearly, however, the standard deviation of the 1st natural frequency is zero.

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2497

1 33.45 Lower bound. Upper bound.

Standard deviation of 1st natural frequency

Mean value of 1st natural frequency

Lower bound. Upper bound.

0.8

33.4 33.35 33.3 33.25 33.2 33.15 33.1 33.05 33

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

32.95

-1 0

0.01

0.02

0.03

0.04

0.05

0

0.01

Dispersal Degree (DD)

0.02

0.03

0.04

0.05

Dispersal Degree (DD)

Fig. 11. Mean value (a) and standard deviation (b) of the 1st natural frequency when DD ¼ μtIc ; μtI ¼ νac ¼ νab ¼ 0. b

k2

k1 m1

k20 m2 ...

m20

Fig. 12. Random spring-mass oscillators.

Table 6 First 10 natural frequencies obtained by the presented method and Adhikari [12]. Natural frequency

ω1 ω2 ω3 ω4 ω5 ω6 ω7 ω8 ω9 ω10

Mean value

Standard deviation

Presented method

Firstorder perturb.

Secondorder Perturb.

MCS

Presented method

Firstorder perturb.

Secondorder perturb.

MCS

3.8303 2.1141% 11.4683 2.0838% 19.0391 2.0715% 26.4982 2.0524% 33.8017 2.0398% 40.9069 1.9998% 47.772 1.9830% 54.3568 1.9074% 60.6225 1.8299% 66.5326 1.7558%

3.8303 2.1141% 11.4683 2.0838% 19.0391 2.0715% 26.4982 2.0524% 33.8017 2.0398% 40.9069 1.9998% 47.772 1.9830% 54.3568 1.9074% 60.6225 1.8299% 66.5326 1.7558%

3.7588 0.2079% 11.2544 0.1798% 18.6845 0.1705% 26.0058 0.1560% 33.1757 0.1500% 40.1527 0.1192% 46.8964 0.1138% 53.3682 0.0540% 59.5312  0.0032% 65.3511  0.0512%

3.7510

0.1295  7.3014% 0.3878  8.0825% 0.6438  6.9653% 0.896  6.1682% 1.1429  6.3350% 1.3832  6.4900% 1.6153  5.1943% 1.8379  5.5501% 2.0498  4.5228% 2.2496  4.0858%

0.1295  7.3014% 0.3878  8.0825% 0.6438  6.9653% 0.896  6.1682% 1.1429  6.3350% 1.3832  6.4900% 1.6153  5.1943% 1.8379  5.5501% 2.0498  4.5228% 2.2496  4.0858%

0.1327  5.0107% 0.3982  5.6174% 0.6643  4.0029% 0.9315  2.4505% 1.2006  1.6063% 1.4727  0.4394% 1.7487 2.6353% 2.3062 18.5159% 2.3324 8.6404% 2.6462 12.8237%

0.1397

11.2342 18.6527 25.9653 33.1260 40.1049 46.8431 53.3394 59.5331 65.3846

0.4219 0.6920 0.9549 1.2202 1.4792 1.7038 1.9459 2.1469 2.3454

5.2. Pure random problem The spring-mass system shown in Fig. 12 is cited from Ref. [12] by Adhikari. In the spring-mass system, 20 masses are connected by springs. The mass and stiffness are random variables with Gaussian distribution and uncorrelated. Mean values of masses and stiffness are 1 kg and 2500 N/m, and randomness (coefficient of variation) associated with mass and stiffness is 15 and 20 percent respectively.

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503 4

9 10

5

1

8

3 6

13 11

7

2

14

12

1000

2498

15

4×1000

Fig. 13. Plane steel truss structure (unit: mm).

Table 7 First 10 eigenvalues obtained by presented method and Modares et al. [21]. Eigen-value

λ1 λ2 λ3 λ4 λ5 λ6 λ7 λ8 λ9 λ10

Lower bound

Upper bound

Presented method

Modares et al.

Relative error (%)

Presented method

Modares et al.

Relative error (%)

410,339 1,592,996 3,380,741 9,437,097 11,958,015 17,255,290 20,547,745 23,940,339 27,703,022 33,178,730

410,330 1,592,959 3,380,649 9,436,747 11,957,569 17,254,949 20,547,852 23,940,622 27,701,932 33,176,699

0.0022 0.0023 0.0027 0.0037 0.0037 0.0020 0.0005 0.0012 0.0039 0.0061

418,109 1,621,683 3,446,562 9,516,362 12,068,310 17,325,237 20,683,124 24,062,324 27,896,207 33,465,517

418,099 1,621,646 3,446,470 9,516,020 12,067,867 17,324,898 20,683,225 24,062,601 27,895,173 33,463,457

0.0024 0.0023 0.0027 0.0036 0.0037 0.0020 0.0005 0.0012 0.0037 0.0062

Natural frequencies of the spring-mass system are calculated by using the presented method and the first 10 natural frequencies are shown in Table 6. The results computed by the first-order perturbation method, second-order perturbation method and MCS provided in Adhikari0 s paper [12] are also listed in Table 6 for comparison. The numbers of percentage under the natural frequencies represent relative errors with respect to the results generated by MCS given in Ref. [12]. The figures in Table 6 indicate that the results obtained by the presented method are identical to the results calculated by the first-order perturbation method [12] because the presented method is developed based on the first-order perturbation technique. The accuracy of the presented method is lower than second-order perturbation method which requires more computational efforts. 5.3. Pure interval problem The plane steel truss structure shown in Fig. 13 is cited from the papers by Modares et al. [21]. In this example, the elastic modulus of elements 1, 2, 7, 12, 14 and 15 are uncertain but bounded E ¼ ½0:205  1012 ; 0:210  1012  Pa. The mass density ρ and cross section area A of each element is ρ ¼ 7800 kg=m3 and A ¼ 0:0012 m2 respectively. Lumped mass is applied. As eigenvalues were computed in paper [21], for comparison, expressions for mean value and variance of eigenvalues (instead of natural frequencies) are derived by using the same methodology as the presented method. Pure interval eigenvalue problem can be solved by using the equation for mean value as m
!
! T I μðλk Þ ¼ λck þ ∑ fΦck ½K0bI  λck UM0bI  Φck UΔbj g j¼1

j

(63)

j

The bounds of the first 10 eigenvalues computed by the presented method are listed in Table 7, and compared to the results acquired by Modares et al. [21]. The figures in Table 7 indicate that the results obtained by the presented method are in good agreement with the results computed by Modares et al. [21]. 6. Conclusions Natural frequencies and mode shapes of structures with mixed random and interval uncertain parameters are investigated in this paper by using a hybrid stochastic and interval approach. Expressions for the mean value, standard deviation and variance of natural frequencies and mode shapes are derived by using first-order perturbation method and random interval moment method. Due to the existence of random and interval variables simultaneously, the mean value, standard deviation and variance of natural frequency and mode shape are not deterministic values but interval values. The bounds of these probabilistic characteristics acquired by the presented method are able to provide good approximate boundaries of possible PDF of natural frequencies and mode shapes of structures with mixed random and interval variables.

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2499

The applications of the presented method on mixed random interval problems are illustrated in the first two numerical examples. The computational results of the first example are verified by 108 MCS, while the presented method is much more efficient. The presented method is also capable of solving the pure random and pure interval problems. These capabilities are demonstrated in the last two examples. As the presented method is developed based on first-order perturbation technique, it has same limitations of first-order perturbation method, such as lower accuracy and applicable to small interval change range. To obtain more accurate results, higher-order perturbation methods and optimization methods can be employed but more computational efforts are required. How to improve accuracy of the present method efficiently will be studied in the future research.

Acknowledgments The authors would like to thank the reviewers for constructive comments on the earlier version of the paper. This research was supported by the Australian Research Council through ARC Discovery Grants.

Appendix A
! aR ¼ ðaR1 ; aR2 ; …; aRn Þ random vector of design variables
!
! aR ¼ ðaR1 ; aR2 ; …; aRn Þ mean value of random vector of aR ! I I I I b ¼ ðb1 ; b2 ; …; bm Þ interval vector of design variables ! ! c c c c I b ¼ ðb1 ; b2 ; …; bm Þ midpoint value of interval vector b

! I I I I I Δb ¼ ðΔb1 ; Δb2 ; …; Δbm Þ Uncertain interval of interval vector b R x random variable xR mean value of xR StdðxR Þ standard deviation of xR VarðxR Þ variance of xR yI ¼ ½y; y interval variable of yI y,y lower and upper bounds of yI y þy 2 y y Δy ¼ 2 I

yc ¼

midpoint value of yI maximum width of yI

Δy ¼ ½  Δy; þΔy uncertain interval of yI
! !
! ! Z RI ¼ f ð XR ;YI Þ function of XR and Y I ½K global stiffness matrix
! !
! ! c c c K ¼ ½Kð aR ;b Þ global stiffness matrix at ð aR ;b Þ


! !I ΔK ¼ ΔKð aR ;b Þ uncertain interval of ½K
! ! c partial derivative of ½K with respect to aRi at ð aR ;b Þ ! c a ;b
! ! I c 0 ∂bKc KbI ¼ I j
! ! partial derivative of ½K with respect to bj at ð aR ;b Þ ∂bj j c aR ;b K0aR ¼ i

∂bKc j ∂aRi
! R


! ! I c j
! ! second partial derivative of ½K with respect to aRi and bj at ð aR ;b Þ c R a ;b ½M global mass matrix $
! !
! ! c c c M ¼ Mð aR ;b Þc global mass matrix at ð aR ;b Þ K″aR bI ¼ j

i

∂2 ½K ∂aRi ∂bIj


! !I ΔM ¼ ΔMð aR ;b Þ uncertain interval of ½M
! ! c partial derivative of ½M with respect to aRi at ð aR ;b Þ ! c a ;b
! ! I c M0bI ¼ ∂bMI c j
! ! partial derivative of ½M with respect to bj at ð aR ;b Þ ∂bj j c R a ;b

M0aR ¼ i

∂bMc j ∂aRi
! R

2500

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

M″aR bI ¼ j

i


! ! I c j
! ! second partial derivative of ½M with respect to aRi and bj at ð aR ;b Þ c aR ;b

∂2 ½M I ∂aRi ∂bj


! XR ¼ ðxR1 ; xR2 ; …; xRn Þ random vector
!
! XR ¼ ðxR1 ; xR2 ; …; xRn Þ mean value of random vector of XR ! YI ¼ ðyI1 ; yI2 ; …; yIm Þ interval vector !
! Y c ¼ ðyc1 ; yc2 ; …; ycm Þ midpoint value of interval vector Y I kth natural frequency ωk
! !
! ! c c c ωk ¼ ωk ð aR ;b Þ kth natural frequency at ð aR ;b Þ λk kth eigenvalue
! !
! ! c c c λk ¼ ωk ð aR ;b Þ kth eigenvalue at ð aR ;b Þ
! kth mode shape Φk
! ! !
! !
! c c c Φk ¼ Φ k ð aR ;b Þ kth mode shape at ð aR ;b Þ element on pth degree of freedom in kth mode shape ϕpk
! !
! ! c c c ϕpk ¼ ϕpk ð aR ;b Þ element on pth degree of freedom in kth mode shape at ð aR ;b Þ
!
! T 0 c DωaR ¼ Φck ½K0aR  ωc2 k U MaR  Φk i

i

DωaR bI j

i

i


!
! T c ¼ Φck ½K″aR bI ωc2 k UM″aR bI  Φk j

i

i

j


!
! T c DϕI ¼ Φcq ½K0bI  λk U M0bI  Φck bj

j

j


!
! T c ¼ Φcq ½K0aR  λk UM0aR  Φck

DϕaR

i

i

i


!
! T c DϕR I ¼ Φcq ½K″aR bI  λk UM″aR bI  Φck ai bj

M ϕaR i

MϕR

j

i

i

j


!
! T ¼ Φck M0aR Φck i

I

ai bj


!
! T ¼ Φck M″aR bI Φck i

j

Appendix B See Fig. B1, Tables B1 and B2. 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

Stochastic approach

-0.8

Lower bound of mean Upper bound of mean

-1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2nd Mode Shape Fig. B1. Bounds of mean value of the 2nd mode shape obtained by presented method and MCS.

2

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2501

Table B1 Bounds of mean value of the 2nd mode shape obtained by presented method and MCS. Degree of freedom

Mean value Presented method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

1.0368E  01 9.3883E  01 3.3696E  01 1.3006E þ00 5.8889E  01 1.1389E þ00 7.6499E  01 5.6603E  01 7.9879E  01  2.6575E  01 6.5979E  01  1.1843E þ00 3.5487E  01  1.9769E þ 00  8.2146E  02  2.5145E þ 00  6.1432E  01  2.7620E þ00  1.1729E þ00  2.8037E þ 00

 0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1483  0.1131  0.1483  0.1131  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483

1.0865E  01 9.8382E  01 3.5311E  01 1.3630E þ00 6.1711E  01 1.1935Eþ00 8.0166E  01 5.9316E  01 8.3708E 01  2.5359E 01 6.9141E  01  1.1301E þ 00 3.7188E  01  1.8865E þ00  7.8389E  02  2.3995E þ00  5.8622E  01  2.6356Eþ 00  1.1193E þ00  2.6755Eþ 00

 0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1483  0.1131  0.1483  0.1131  0.1483  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131  0.1131

1.0380E  01 9.3989E  01 3.3735E  01 1.3021E þ00 5.8955E  01 1.1402E þ00 7.6586E  01 5.6667E  01 7.9969E  01  2.6614E  01 6.6053E  01  1.1861E þ00 3.5527E  01  1.9799E þ 00  8.2268E  02  2.5183E þ00  6.1523E  01  2.7661E þ 00  1.1747E þ00  2.8079Eþ 00

1.0881E  01 9.8529E  01 3.5364E  01 1.3650E þ00 6.1803E  01 1.1953E þ00 8.0285E  01 5.9404E  01 8.3832E  01  2.5388E  01 6.9244E 01  1.1314E þ00 3.7243E 01  1.8886E þ00  7.8478E  02  2.4022E þ00  5.8689E  01  2.6386E þ00  1.1205E þ 00  2.6785Eþ 00

Table B2 Bounds of standard deviation of the 2nd mode shape obtained by presented method and MCS. Degree of freedom

Standard deviation Presented method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

1.9342E  03 1.7515E  02 6.2865E  03 2.4265E  02 1.0986E  02 2.1247E  02 1.4272E  02 1.0560E  02 1.4902E  02 4.7311E  03 1.2309E  02 2.1084E  02 6.6205E  03 3.5195E  02 1.4624E 03 4.4766E  02 1.0937E  02 4.9171E  02 2.0881E  02 4.9915E  02

 1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226

2.1378E  03 1.9359E 02 6.9482E  03 2.6819E  02 1.2143E  02 2.3484E  02 1.5774E  02 1.1672E  02 1.6471E  02 5.2291E  03 1.3605E  02 2.3303E 02 7.3174E  03 3.8900E  02 1.6164E  03 4.9478E  02 1.2088E  02 5.4347E  02 2.3079E 02 5.5169E  02

0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916

1.9721E  03 1.7858E  02 6.4097E 03 2.4740E  02 1.1202E  02 2.1664E  02 1.4552E  02 1.0767E 02 1.5194E  02 4.8239E  03 1.2550E  02 2.1497E 02 6.7503E  03 3.5885E  02 1.4911E  03 4.5644E  02 1.1151E  02 5.0135E 02 2.1291E  02 5.0893E  02

2.1210E  03 1.9207E 02 6.8936E  03 2.6608E  02 1.2047E  02 2.3299E  02 1.5650E  02 1.1580E  02 1.6342E  02 5.1880E  03 1.3498E  02 2.3120E  02 7.2599E  03 3.8594E  02 1.6037E  03 4.9090E  02 1.1993E  02 5.3920E  02 2.2898E  02 5.4736E  02

2502

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

Appendix C See Fig. C1, Tables C1 and C2.

0.6 Stochastic approach Lower bound of mean Upper bound of mean

0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

3rd Mode Shape Fig. C1. Bounds of mean value of the 3rd mode shape obtained by presented method and MCS.

Table C1 Bounds of mean value of the 2nd mode shape obtained by presented method and MCS. Degree of freedom

Mean value Presented method

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

 2.6763E  01  2.2093Eþ 00  7.0935E 01  1.8295Eþ 00  8.8741E  01 1.9878E  01  6.1715E 01 2.2731E þ00  2.3113E  02 3.1085E þ00 5.3050E  01 2.1227E þ00 7.3617E  01  2.0929E  01 4.4218E  01  2.7783Eþ 00  2.6813E  01  4.3056Eþ 00  1.1734Eþ 00  4.6050Eþ 00

 0.1483  0.1483  0.1483  0.1483  0.1483  0.1131  0.1483  0.1131  0.1483  0.1131  0.1131  0.1131  0.1131  0.1483  0.1131  0.1483  0.1483  0.1483  0.1483  0.1483

 2.5538E  01  2.1083E þ 00  6.7691E  01  1.7458E þ 00  8.4682E  01 2.0831E  01  5.8892E  01 2.3820E þ00  2.2056E  02 3.2575E þ 00 5.5593E  01 2.2244E þ 00 7.7145E  01  1.9972E  01 4.6337E  01  2.6513E þ 00  2.5586E  01  4.1087E þ 00  1.1198E þ00  4.3944E þ 00

 0.1131  0.1131  0.1131  0.1131  0.1131  0.1483  0.1131  0.1483  0.1131  0.1483  0.1483  0.1483  0.1483  0.1131  0.1483  0.1131  0.1131  0.1131  0.1131  0.1131

 2.6802E  01  2.2126E þ00  7.1041E  01  1.8322E þ00  8.8873E  01 1.9900E  01  6.1807E  01 2.2756E þ 00  2.3148E  02 3.1120E þ 00 5.3110E  01 2.1251E þ00 7.3700E  01  2.0960E  01 4.4268E  01  2.7825E þ00  2.6852E  01  4.3120E þ00  1.1752E þ 00  4.6119Eþ 00

 2.5567E 01  2.1106E þ 00  6.7767E 01  1.7478E þ00  8.4778E 01 2.0861E  01  5.8959E  01 2.3855E þ 00  2.2081E  02 3.2623E þ 00 5.5675E 01 2.2277E þ 00 7.7260E 01  1.9994E  01 4.6406E  01  2.6543E þ 00  2.5615E  01  4.1134E þ 00  1.1210E þ 00  4.3994E þ 00

Table C2 Bounds of standard deviation of the 2nd mode shape obtained by presented method and MCS. Degree of freedom

Standard deviation Presented method

1 2 3 4 5

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

4.7645E  03 3.9332E 02 1.2629E  02 3.2570E 02 1.5798E  02

 1.9226  1.9226  1.9226  1.9226  1.9226

5.2660E  03 4.3472E  02 1.3958E  02 3.5998E  02 1.7461E  02

0.7916 0.7916 0.7916 0.7916 0.7916

4.8579E  03 4.0103E  02 1.2876E  02 3.3208E  02 1.6108E  02

5.2247E  03 4.3131E  02 1.3848E  02 3.5716E  02 1.7324E  02

C. Wang et al. / Journal of Sound and Vibration 333 (2014) 2483–2503

2503

Table C2 (continued ) Degree of freedom

Standard deviation Presented method

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

MCS

Lower bound

Relative error (%)

Upper bound

Relative error (%)

Lower bound

Upper bound

3.7084E 03 1.0987E  02 4.2407E  02 4.1148E  04 5.7992E  02 9.8971E  03 3.9601E  02 1.3734E  02 3.7260E  03 8.2494E  03 4.9462E  02 4.7734E  03 7.6653E  02 2.0891E  02 8.1983E  02

 1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226  1.9226

4.0988E  03 1.2144E  02 4.6870E 02 4.5480E  04 6.4097E  02 1.0939E  02 4.3769E  02 1.5180E  02 4.1182E  03 9.1177E 03 5.4669E  02 5.2759E  03 8.4721E  02 2.3090E  02 9.0612E  02

0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916 0.7916

3.7811E  03 1.1202E  02 4.3238E  02 4.1955E  04 5.9129E  02 1.0091E  02 4.0377E  02 1.4003E  02 3.7990E  03 8.4111E  03 5.0432E  02 4.8670E  03 7.8155E  02 2.1300E  02 8.3590E 02

4.0666E  03 1.2048E  02 4.6502E  02 4.5123E  04 6.3593E 02 1.0853E  02 4.3426E  02 1.5061E  02 4.0858E  03 9.0461E  03 5.4239E  02 5.2344E  03 8.4056E  02 2.2908E  02 8.9901E  02

References [1] G.S. Szekely, G.I. Schueller, Computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties, Computer Methods in Applied Mechanics and Engineering 191 (2001) 799–816. [2] G.I. Schueller: The effects of uncertainties in structural analysis, Proceedings of the Third International Conference on Advances in Structural Engineering and Mechanics, 2004, pp. 1275–1282. [3] S.H. Lee, W. Chen, A comparative study of uncertainty propagation methods for black-box-type problems, Structural and Multidisciplinary Optimization 37 (2009) 239–253. [4] H. Benaroya, Random eigenvalues, algebraic methods and structural dynamic models, Applied Mathematics and Computation 52 (1992) 37–66. [5] W.E. Boyce, Random Eigenvalue Problems. Probabilistic Methods in Applied Mathematics, Academic Press, New York, 1968. [6] J.D. Collins, W.T. Thomson, The eigenvalue problem for structural systems with statistical properties, AIAA Journal 7 (1969) 642–648. [7] J.V. Scheidt, W. Purkert, Random Eigenvalue Problems, North Holland, New York, 1983. [8] R.A. Ibrahim, Structural dynamics with parameter uncertainties, Applied Mechanics Reviews 40 (1987) 309–328. [9] D. Song, S.H. Chen, Z.P. Qiu, Stochastic sensitivity analysis of eigenvalues and eigenvectors, Computers and Structures 54 (1995) 891–896. [10] P.B. Nair, A.J. Keane, An approximate solution scheme for the algebraic random eigenvalue problem, Journal of Sound and Vibration 260 (2003) 45–65. [11] W. Gao, Natural frequency and mode shape analysis of structures with uncertainty, Mechanical Systems and Signal Processing 21 (2007) 24–39. [12] S. Adhikari, Joint statistics of natural frequencies of stochastic dynamic systems, Computational Mechanics 40 (2007) 739–752. [13] M.M. Kaminski, J. Szafran, Random eigenvibrations of elastic structures by the response function method and the generalized stochastic perturbation technique, Archives of Civil and Mechanical Engineering 9 (2009) 5–32. [14] H.D. Gersem, D. Moens, W. Desmet, D. Vandepitte, Interval and fuzzy dynamic analysis of finite element models with super elements, Computers and Structures 85 (2007) 304–319. [15] B. Moller, M. Beer, Engineering computation under uncertainty – capabilities of non-traditional models, Computers and Structures 86 (2008) 1024–1041. [16] A. Dief, The interval eigenvalue problem, Journal of Applied Mathematics and Mechanics 71 (1991) 61–64. [17] C.V. Hollot, A.C. Bartlett: On the eigenvalue of interval matrices, Proceedings of the 26th IEEE Conference on Decision and Control, 1987, pp. 794–799. [18] S.H. Chen, Z.P. Qiu, Z.S. Liu, Perturbation method for computing eigenvalue bounds in structural vibration systems with interval parameters, Communications in Numerical Methods in Engineering 10 (1994) 121–134. [19] Z.P. Qiu, X.J. Wang, M.I. Friswell, Eigenvalue bounds of structures with uncertain-but-bounded parameters, Journal of Sound and Vibration 282 (2005) 297–312. [20] W. Gao, Interval natural frequency and mode shape analysis for truss structures with interval parameters, Finite Elements in Analysis and Design 42 (2006) 471–477. [21] M. Modares, R.L. Mullen, R.L. Muhanna, Natural frequencies of a structure with bounded uncertainty, Journal of Engineering Mechanics 132 (2006) 1363–1371. [22] P. Angeli, F. Barazza, F. Blanchini, Natural frequency interval for vibration systems with polytopic uncertainty, Journal of Sound and Vibration 329 (2010) 994–995. [23] L. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353. [24] D. Moens, D. Vandepitte, A survey of non-probabilistic uncertainty treatment in finite element, Computer Methods in Applied Mechanics and Engineering 194 (2005) 1527–1555. [25] W. Gao, C.M. Song, F. Tin-Loi, Probabilistic interval analysis for structures with uncertainty, Structural Safety 32 (2010) 191–199. [26] W. Gao, C.M. Song, F. Tin-Loi, Probabilistic interval response and reliability analysis of structures with a mixture of random and interval properties, Computer Modeling in Engineering and Sciences 46 (2009) 151–189. [27] X.P. Du, A. Sudjianto, B. Huang, Reliability-based design with the mixture of random and interval variables, Journal of Mechanical Design 127 (2005) 1068–1076. [28] Z.P. Qiu, D. Yang, I. Elishakoff, Probabilistic interval reliability of structural systems, International Journal of Solids and Structures 45 (2008) 2850–2860. [29] W. Gao, D. Wu, C.M. Song, F. Tin-Loi, X.J. Li, Hybrid probabilistic interval analysis of bar structures with uncertainty using a mixed perturbation MonteCarlo method, Finite Elements in Analysis and Design 47 (2011) 643–652. [30] C. Wang, W. Gao, C.W. Yang, C.M. Song, Non-deterministic structural response and reliability analysis using a hybrid perturbation-based stochastic finite element and quasi-Monte Carlo method, Computers, Materials & Continua 25 (2011) 19–46. [31] J.R. Benjamin, C.A. Cornell, Probability, Statistics and Decision for Civil Engineers, McGraw-Hill Inc., New York, 1970.