COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan,China
©2006 Tsinghua University Press & Springer
Interval Dynamic Analysis Using Interval Factor Method
Wei Gao*
School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, NSW 2052,
Australia
Email: [email protected]
Abstract A new method called the interval factor method (IFM) for the natural frequency and mode shape analysis of
truss structures with interval parameters is presented in this paper. Using the IFM, the structural physical parameters
and geometric dimensions can be considered as interval variables. The structural stiffness and mass matrices can then
be respectively divided into the product of two parts corresponding to the interval factors and the deterministic matrix.
The computational expressions for the midpoint value (mean value), lower bound, upper bound and interval change
ratio of the natural frequency and mode shape are derived from Rayleigh quotient by means of the interval operations.
The influences of the uncertainty of the structural parameters on the natural frequency and mode shape are
demonstrated by using truss structures.
Keywords: interval factor method, dynamic characteristics, interval analysis, truss sturcutres
INTRODUCTION
The static and dynamic analysis of structures with uncertain parameters is a very significant research field in
engineering [1-3]. In most practical engineering cases, structures have uncertainty in their parameters arising from
manufacturing tolerances, materials defects and variation in operating conditions. Uncertain structures may
correspond to the uncertainty of a single structure (for example, a bridge, offshore platform, antenna or space structure)
or a batch of nominally identical structures (eg. vehicles leaving the production line).
The most common approach to problems of uncertainty is to model the structural geometric and material parameters as
random variables. Under these circumstances, the mean value, variance, standard deviation of individual structural
parameters and the correlation between different structural parameters are provided by the probabilistic information
(probability density function and joint probability distribution function) of the structural parameters. Unfortunately,
the probabilistic approaches can not give reliable results unless sufficient experimental data or statistical information
are available to validate the assumptions about the probability densities of the random variables. In some cases, we can
only obtain the range or the lower bound and upper bound of the structural parameters. Therefore, the probabilistic
analysis method is not available if the probability density function can not be obtained, and the interval analysis
method will be very useful. In the interval analysis method, a bounded uncertain structural parameter can be described
as an interval variable and only its lower and upper bounds are required.
Since the mid-1960s, a new method called the interval analysis has appeared. Moore [4] and Alefeld [5] have done the
pioneering work, where they established the basic theory for the interval analysis and simply discussed the interval
operations and its application. Recently, Chen et al. [6-7] and Qiu et al.[8] have investigated the response and
eigenvalue problems of structures with uncertain parameters. In their studies, several important results have been
obtained by using interval analysis and matrix perturbation techniques. In perturbation methods, the uncertainty of all
the structural parameters are expressed as small parameters in the structural mass and stiffness matrices. These small
parameters are not interval variables, but simply small values. Therefore, it is very difficult to investigate the effect of
the uncertainty of the individual parameters on the structural natural frequency by their method.
In this paper, the interval dynamic characteristics analysis of interval structures is investigated, and a new method
called Interval Factor Method (IFM) is proposed. Truss structures are used to illustrate examples of this method, in
⎯ 1200 ⎯
which structural physical parameters (Young’s modulus and mass density) and geometry (length and cross-sectional
area of bar) are considered as interval variables.
The procedure of the IFM is as follows. Firstly, a structural parameter interval variable is expressed as an interval
factor multiplied by the mean value (midpoint value) of this structural parameter. Secondly, the structural mass and
stiffness matrices are expressed as interval factors of the structural parameters multiplied by their deterministic values,
respectively. Finally, the structural natural frequencies and mode shapes are expressed as the functions of these
interval factors. Therefore, the effect of the change of structural parameters on the structural natural frequencies and
mode shapes can be easily identified. This method allows for uncertainty in the material parameters and geometric
dimensions, and compared to other methods, the computational work to obtain the structural dynamical characteristics
is very small.
INTERVAL OPERATIONS AND INTERVAL FACTOR
Assume that I (R) , I ( R n ) and I ( R n×n ) denote the sets of all closed real interval numbers, n dimension real interval
vectors and n × n real interval matrices, respectively. R is the set of all real numbers. X I = [ x, x] is a member of
I (R) and can be usually written in the following form
X I = [ x c − Δx, x c + Δx]
(1)
xc =
x+x
2
(2)
Δx =
x−x
2
(3)
where x c and Δx denote the mean value (or midpoint value) of X I and the uncertainty (or the maximum width) in
X I , respectively. In this study, an interval ΔX I = [− Δx, Δx] is called an uncertain interval. An arbitrary interval
X I = [ x, x] can be written as the sum of its mean value and its uncertain interval ΔX I = [−Δx, Δx ] . Therefore
X I = x c + ΔX I
(4)
Eq. (4) can also be expressed as
X I = [ x c (1 −
x−x
x−x
Δx c
Δx
), x (1 + c )] = x c [1 −
,1+
]
c
c
2x
2xc
x
x
(5)
Here, we introduce X FI = [ x F , x F ] that is a member of I (R) and let
X FI = [ x F , x F ] = [1 −
x−x
x−x
,1+
]
c
2x
2xc
(6)
Substituting Eq. (6) into Eq. (5) yields
X I = X FI ⋅ x c
(7)
Because x c is a deterministic value and the uncertainty of X I is described by X FI , X FI is named as the interval factor
of X I in this study and it can be easily obtained
xF = 1 −
x−x
2
(8)
xF = 1 +
x−x
2xc
(9)
x Fc =
xF + xF
2
=1
(10)
⎯ 1201 ⎯
xF − xF
Δx F =
2
=
x − x Δx
= c
2xc
x
(11)
where x Fc and Δx F denote the mean value of X FI and the maximum width in X FI , respectively. In addition, Δx F can be
considered as the interval change ratio value (that is the maximum width of interval variable divided by its mean value)
to assess the dispersal degree of the interval [ x, x] .
NATURAL FREQUENCY AND MODE SHAPE ANALYSIS
Suppose that there are n elements in the analyzed truss structure. The stiffness matrix [K ] and mass matrix [M ] of
truss structure in global coordinate can be respectively expressed as
n
[M ] = ∑
e =1
[M ] = ∑ {12 ρ
n
(e)
(e)
A( e ) l ( e ) [I ]}
(12)
e =1
[K ] = ∑ [K
n
(e)
e =1
[
] = ∑{[T ]
n
(e) T
(e)
L
e =1
] [
E ( e ) A( e )
[G ][T ( e ) ]}
(13)
]
where M ( e ) , K ( e ) , E (e ) , A (e ) , L(e ) and ρ (e ) are the eth element’s mass matrix, stiffness matrix, Young’s
modulus, cross-sectional area, length and mass density, respectively. [I ] is a 6-order identity matrix. [G ] is a 6 × 6
[ ]
element to global coordinates, and [T ] is its transpose.
matrix ,where g11=g44=1，g14=g41=-1,other elements of [G ] are all equal to zero [9]. T (e ) is the transformation matrix
(e) T
that translates the local coordinates of the eth
In the following, we consider the structural physical parameters ( ρ (e ) , E (e ) ) and geometric dimensions ( A (e ) , L(e ) ) are
mumbers of I (R ) simultaneously, that is, they are all interval variables. Because there many structures that are
consisted of one kind of material or all elements are manufactured by the same means in engineering cases, we can
suppose that the interval change ratio of the same kind of interval structural parameter of each element is equal, for
C
C
I
C
example, ΔE (1) E (1) = ΔE ( 2 ) E ( 2 ) = ⋅ ⋅ ⋅ = ΔE ( n ) E ( n ) = ΔE E C . Then, the Young’s modulus E ( e ) , mass
density ρ ( e) , cross-sectional area A( e ) and length L( e ) can be respectively expressed as
I
I
c
I
c
I
E ( e ) = E FI ⋅ E ( e ) , ρ ( e ) = ρ FI ⋅ ρ ( e ) ,
I
c
I
I
A ( e ) = AFI ⋅ A ( e ) , L( e ) = LIF ⋅ L( e )
Where E FI , ρ FI , AFI and LIF are interval factors of E ( e )
E
(e) c
( e) c
,ρ
,A
(e) c
(e) c
and L
are mean values (or midpoint values) of E
I
(e) I
c
(14)
, ρ ( e)
,ρ
( e) I
I
, A( e )
,A
(e) I
I
and L( e )
(e) I
and L
I
, respectively;
, respectively.
Furthermore, the interval change ratio values of Young’s modulus, density, cross-sectional area and length can be
respectively expressed as
ΔE F = ΔE E c , Δρ F = Δρ ρ c , ΔAF = ΔA Ac , ΔLF = ΔL Lc
(15)
I
I
I
I
where ΔE F , Δρ F , ΔAF and ΔLF are interval change ratio values of E ( e ) , ρ ( e) , A( e ) and L( e ) , respectively.
1. Interval mass matrix and stiffness matrix From Eq. (12), when the structural parameters are interval variables, it
can be easily obtained that
[M ]
(e) I
[
= ρ FI × AFI × l FI ⋅ ρ ( e ) ⋅ A ( e ) ⋅ L( e ) ⋅ [I ] = ρ FI × AFI × LIF ⋅ M (e )
c
[
Where M (e )
]
∗
c
c
[
]
∗
(16)
]
[
I
]
∗
is the deterministic part of mass matrix M ( e ) . It should be noted that M (e ) is not the midpoint value
[
]
I
(mean value) of M ( e ) except when only one of structural parameters is interval variable. The expression shows that
[
the mass matrix M ( e )
]
I
can be divided into the product of two parts, corresponding to the interval
[
]
∗
[
factors ρ FI , AFI , LIF and the deterministic matrix M (e ) . The uncertainty of M ( e )
⎯ 1202 ⎯
]
I
is dependent on the interval
[
factors ρ FI , AFI and LIF . Constructing the deterministic matrix M (e )
I
]
∗
is same as constructing the mass matrix in Eq.
c
I
c
I
c
(12) for the eth element, and taking the parameters as ρ ( e ) = ρ ( e ) , A ( e ) = A ( e ) and L( e ) = L( e ) . It should be note that
the element’s mass matrices should be expanded before assemblage. Applying the natural interval extension to Eq.
I
(12), [M ] can now be written as
[M ]I = ∑ [M (e) ]I = ∑ ( ρ FI ⋅ AFI ⋅ LIF [M (e) ]* ) = ρ FI × AFI × LIF ⋅ [M ]∗
n
n
e =1
e =1
(17)
where [M ] is the deterministic part in the mass matrix [M ] .
∗
I
Likewise, from Eq. (13), it can be easily obtained
[K ]I = ∑ [K (e) ]I = ∑ ( E F ×I AF
I
n
n
e=1
e =1
[ ]
∗
where K (e )
I
LF
[ ]
∗
⋅ K (e) ) =
E FI × AFI
∗
⋅ [K ]
I
LF
(18)
[ ]
and [K ] are the deterministic part of K ( e )
∗
I
and [K ] .
I
In perturbation method, interval stiffness and mass matrices can also be expressed respectively
I
C
I
I
C
I
∗
∗
∗
∗
as [M ] = [M ] + Δ[M ] or [M ] = [M ] + ε 1 [M ] , [K ] = [K ] + Δ[K ] or [K ] = [K ] + ε 2 [K ] . However, small
parameters ε 1 and ε 2 are not interval variables, Δ[M ] and Δ[K ] can not reflect the uncertainty of interval structural
parameters directly and obviously. In interval factor method, the uncertainty of interval structural parameter variable is
fully reflected by its interval factor. Hence, unlike perturbation method, the uncertainty of the individual parameters
can be examined.
I
I
2. Interval natural frequency Suppose that jth order natural frequency and modal shape of structure are
ω j I and {φ j }I , respectively. By using interval factor method, ω j I and {φ j }I can be written as
ω j I = ω j IF ⋅ ω j ∗
(19)
{φ j }I = φ j F ⋅{φ j }∗
(20)
I
where ω ∗j and {φ j }∗ are the deterministic part of ω j and {φ j }I , respectively.
I
By using the Rayleigh quotient expression, the jth natural frequency can be expressed as
{φ } [K ] {φ }
=
{φ } [M ] {φ }
IT
(ω j )
I 2
I
I
j
j
IT
I
(21)
I
j
j
Substituting Eqs. (17), (18), (19) and (20) into Eq. (21) yields
(ω j ) =
{φ } [K ] {φ }
⋅
{φ } [M ] {φ }
∗T
φ j IF × E FI × AFI × φ j IF
I 2
∗
j
φ j IF × ρ FI × AFI × LIF × LIF × φ j IF
∗
j
∗T
j
∗
∗
j
K ∗j
E FI
E FI
∗
=
(ω j ) 2
= I
⋅
I
I
I
∗
I
I
ρ F × LF × LF M j ρ F × LF × LF
(22)
K ∗j , M ∗j , ω j ∗ are all deterministic quantities, corresponding to the jth order stiffness, mass and natural frequency of the
I
c
I
c
I
c
I
c
structure when the parameters are E ( e ) = E ( e ) ， ρ ( e ) = ρ ( e ) , A ( e ) = A ( e ) and L( e ) = L( e ) . From the deterministic
stiffness and mass matrices [K ] and [M ] , the deterministic values of every jth natural frequency ω j can be obtained
∗
∗
∗
from the conventional finite element model.
From Eq. (22), the interval value of ω j can be obtained
I
ωj = (
EF
ρ F ⋅ LF ⋅ LF
)
1
2
∗
⋅ω j =(
1
1 − ΔE F
∗
) 2 ⋅ω j
(1 + Δρ F ) ⋅ (1 + ΔLF ) ⋅ (1 + ΔLF )
⎯ 1203 ⎯
(23)
ωj = (
1
1
1 + ΔE F
EF
∗
∗
) 2 ⋅ω j = (
) 2 ⋅ω j
(1 − Δρ F ) ⋅ (1 − ΔLF ) ⋅ (1 − ΔLF )
ρ F ⋅ L F ⋅ LF
(24)
From Eqs. (23) and (24), the mean value, maximum width and interval change ratio of jth natural frequency can be
obtained
ω jc =
(ω j + ω j )
2
(ω j − ω j )
, Δω j =
2
, Δω j F =
Δω j
ω jc
=
(ω j − ω j ) 2
(25)
(ω j + ω j ) 2
3. Interval modal shape From the modal analysis theory, the modal matrix [φ ] has the following orthogonal property
[φ ]T [M ][φ ] = [I ]
(26)
[φ ]T [K ][φ ] = [Ω] = diag [ω 2 ]
(27)
Eqs. (26) and (27) can be written as
φ FI × ρ FI × AFI × LIF × LIF × φ FI ⋅ ([φ ]∗ [M ]∗ [φ ]∗ ) = [I ]
T
φ FI × E FI × AFI × φ FI
LIF
⋅ ([φ ]
∗T
(28)
[ ]
E FI
[K ] [φ ] ) = I I I ⋅ diag ω ∗ 2
ρ F × LF × LF
∗
∗
(29)
From Eqs. (28) and (29), we can obtain the same result
φ j IF = φ FI = (
1
1
) 2
I
I
ρ × AF × LF
(30)
I
F
For i = 1 to n degrees of freedom in structures, [φ ] and {φ j } can be written as
⎡φ11
⎢ M
⎢
[φ ] = φ L φ j = ⎢φi1
⎢
⎢ M
⎢φ n1
⎣
[
]
K φ1 j
L
M
L φij
L M
L φ nj
L φ1n ⎤
L M ⎥⎥
L φin ⎥
⎥
L M ⎥
L φ nn ⎥⎦
(31)
From Eq. (30), we can obtain that the interval change ratio of each element of modal matrix are equal. The interval
value of any elements can be obtained according to the interval operations
φij = [1 (1 + Δρ F ) ⋅ (1 + ΔAF ) ⋅ (1 + ΔLF )]1 2 ⋅ φij ∗
(32)
φij = [1 (1 − Δρ F ) ⋅ (1 − ΔAF ) ⋅ (1 − ΔLF )]1 2 ⋅ φij ∗
(33)
From Eqs. (32) and (33), the mean value, maximum width and interval change ratio of mode shape can be obtained
φij c =
(φij + φij )
2
, Δφij =
(φij − φij )
2
, Δφij F =
Δφij
φij c
=
(φij − φij ) 2
(φij + φij ) 2
(34)
EXAMPLES
According to the preceding computing formulas and the solving method, the corresponding computational program is
designed. The conventional dynamic characteristics of deterministic truss structures can be obtained by any FEM
analysis software, such as ANSYS, NASTRAN and ALGOR.
In the following simulation, an 8-meter caliber antenna shown in Figure 1 is used as an example. This is a 96-node and
336-element space truss structure, in which there are 12 kinds of elements whose mean values of cross-sectional areas
⎯ 1204 ⎯
are
different
(see
Table
1).
The
structural
c
5
c
3
3
and E = 2.058 × 10 ( Mpa ) , ρ = 7.65 × 10 (kg m ) .
parameters
are
all
interval
variables
Table 1 The mean values of element cross-sectional areas
Kinds of elements
Mean value (×10-4 m2)
A1
A2
A3
A4
A5
A6
A7
A8
A9 A10 A11 A12
3.0 4.0 6.0 2.0 3.0 3.0 6.0 2.0 3.0 4.0 6.0 2.0
In order to investigate the effect of the change (uncertainty) of interval variables E, ρ , A and L on the structural
dynamic characteristics, the values of interval change ratio ΔE F , Δρ F , ΔAF and ΔLF of interval structural parameters
E , ρ , A and L are respectively taken as different groups. The computational results of natural frequency and natural
modal shape are given in Table 2 and Table 3, respectively.
Figure 1: Quarter of 8-meter caliber antenna(unit: mm)
In addition, in order to verify the effectiveness of the Interval Factor Method, the simulation results that obtained by the
conventional finite element method are also given in Table 2 and Table 3. In simulations, two different values of any
structural parameter are selected, one is its lower bound value and the other one is its upper bound value. Then, 16
different combinations of the lower bound and upper bound values of structural parameters E , ρ , A and L are used to
compute the structural natural frequency and mode shape by FEM analysis software ANSYS.
It can be seen from Table 2 and Table 3 that: 1) The analytic results by the method (IFM) proposed in this paper agreed
very well with that of simulation method, by which the effectiveness of our method is verified.
Table 2 The computational results of natural frequency ( ∗ Simulation Method)
Model
Deterministic model
ΔE F = Δρ F = ΔAF = ΔLF =0
ΔE F =0.03 Δρ F = ΔAF = ΔLF =0
ω1
ω1c
ω1
Δω1 F = Δω1 ω1
22.818
22.818
22.818
0
22.133
22.818
23.502
0.030000
Δρ F =0.03 ΔE F = ΔAF = ΔLF =0
22.153
22.838
23.523
0.030001
ΔAF =0.03 ΔE F = Δρ F = ΔLF =0
22.818
22.818
22.818
0
ΔLF =0.03 ΔE F = Δρ F = ΔAF =0
21.569
22.879
24.251
0.059948
ΔE F = Δρ F =0.03 ΔAF = ΔLF =0
21.488
22.858
24.229
0.059947
ΔE F = Δρ F =0 ΔAF = ΔLF =0.03
21.569
22.879
24.251
0.059948
ΔE F = Δρ F = ΔAF = ΔLF =0.03
20.255
23.003
25.751
0.119465
ΔE F = Δρ F = ΔAF = ΔLF =0.05
18.725
23.334
27.944
∗
∗
∗
18.725
⎯ 1205 ⎯
23.334
27.944
0.197535
∗
0.197535
c
2) The change (uncertainty) of the cross-sectional area does not does not have any effect on the structural natural
frequency; however, the change (uncertainty) of the Young’s modulus does not have any effect on the structural natural
modal shape. 3) When the interval change ratio of physical parameters is equal to that of geometric dimensions, the
uncertainty of geometric dimensions will produce greater effect on the uncertainty of structural mode shape. 4) It is
shown that the change or error of structural parameters will produce considerable effect on the computational results of
structural natural frequency and modal shape. Along with the increase of the interval change ratio (uncertainty) of
physical parameter and geometric dimensions, the interval change ratio (dispersal degree) of structural natural
frequency and modal shape will notably increase.
Table 3 The computational results of natural mode shape ( ∗ Simulation Method)
Model
Deterministic model
ΔE F = Δρ F = ΔAF = ΔLF =0
ΔE F =0.03 Δρ F = ΔAF = ΔLF =0
φ11
φ11c
φ11
Δφ11 F = Δφ11 φ11
2.1190
2.1190
2.1190
0
2.1190
2.1190
2.1190
0
Δρ F =0.03 ΔE F = ΔAF = ΔLF =0
2.0572
2.0891
2.1845
0.030001
ΔAF =0.03 ΔE F = Δρ F = ΔLF =0
2.0572
2.0891
2.1845
0.030001
ΔLF =0.03 ΔE F = Δρ F = ΔAF =0
2.0572
2.0891
2.1845
0.030001
ΔE F = Δρ F =0.03 ΔAF = ΔLF =0
2.0572
2.0891
2.1845
0.030001
ΔE F = Δρ F =0 ΔAF = ΔLF =0.03
1.9973
2.1247
2.2520
0.059948
ΔE F = Δρ F = ΔAF = ΔLF =0.03
1.9391
2.1304
2.3217
0.089784
ΔE F = Δρ F = ΔAF = ΔLF =0.05
1.8304
2.1509
2.4714
∗
∗
∗
1.8304
2.1509
2.4714
c
0.149011
∗
0.149011
CONCLUSIONS
(1) The effect of the change of E , ρ , A and L on the change of the structural natural frequency and modal shape
are different; the uncertainty of bar’s length L produce the greatest effect on the structural natural frequency,
however, the uncertainty of ρ , A and L produce the same effect on the change of the structural modal shape.
(2) Comparing with the case that only one of the uncertainty of E, ρ , A and L is taken into account, the change of the
structural dynamic characteristics are greater when their uncertainty are considered simultaneously.
(3) The examples show that the dynamic characteristics analysis results of the structure with interval parameters can
be obtained expediently and the model and method presented in this paper are rational and feasible.
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⎯ 1206 ⎯
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⎯ 1207 ⎯