The method of equivalent linearization is one of effective tools in solving random
vibration problems of mechanical systems. In this study, the modal responses of a simply
supported beam subjected to a space-wise and time-wise white noise loading are carried
out by the dual criterion of equivalent linearization. Our calculations are restricted in
two cases of single- and two-mode of beam vibrations. The exact solutions of the original
modal equation system obtained by Fokker-Planck equation are available for both cases.
A closed form of nonlinear algebraic system is obtained by the dual approach associated
with the frequency-response function method for the linearized modal system. In the
case of single-mode, the analytical solution of the first mode of the beam is easy to solve
explicitly for four methods considered (the exact solution, energy method, conventional
linearization and dual criterion method). Also, in the case of two-mode, the closed system is solved by the fixed-point iteration method. Numerical results show that the dual
criterion gives a good prediction on the random responses of the beam, especially in the
range of strong linearity of system parameters. Further investigations for random vibrations of other beam systems seem to be appropriate in order to verify the advantages of
the dual criterion.
14 trang |
Chia sẻ: huongthu9 | Lượt xem: 418 | Lượt tải: 0
Bạn đang xem nội dung tài liệu Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization, để tải tài liệu về máy bạn click vào nút DOWNLOAD ở trên
Vietnam Journal of Mechanics, VAST, Vol. 38, No. 1 (2016), pp. 49 – 62
DOI:10.15625/0866-7136/38/1/6629
VIBRATION ANALYSIS OF BEAMS SUBJECTED TO
RANDOM EXCITATION BY THE DUAL CRITERION OF
EQUIVALENT LINEARIZATION
Nguyen Nhu Hieu1,∗, Nguyen Dong Anh1, Ninh Quang Hai2
1Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
2Hanoi Architectural University, Vietnam
∗E-mail: nhuhieu1412@gmail.com
Received July 29, 2015
Abstract. In this paper responses of beams subjected to random loading are analyzed by
the dual approach of the equivalent linearization method. The external random loading is
assumed to be a space-wise and time-wise white noise in which the exact solutions of the
modal equations can be found. A system of nonlinear algebraic equations for linearization
coefficients of the modal linearized system is obtained in a closed form and is solved by
the fixed-point iteration method. Results obtained from the proposed dual criterion are
compared with the exact solution and those obtained from other approaches including
energy method, and conventional linearization method. It is observed that the solution
obtained by the dual criterion is in good agreement with the exact solution, especially, in
the case of strong nonlinearity of beam.
Keywords: Random vibration, equivalent linearization, dual criterion, modal response,
nonlinear beam.
1. INTRODUCTION
Over decades, the equivalent linearization (EQL) is one of the most extensively
used methods in investigating mechanical systems. The earliest researches on the EQL
method were carried out by Booton [1], Kazakov [2] and Caughey [3,4]. The fundamental
idea of the method lies on replacing the original nonlinear system under a random exter-
nal excitation by a linearized system under the same excitation in which linearization co-
efficients are found from a specified optimal criterion, for example, the mean-square error
criterion [4], spectral criterion [5]. This method has developed and been applied success-
fully to find approximate responses of nonlinear system subjected random excitation. For
discrete systems with single- and multi-degree-of-freedom, studies using the equivalent
c© 2016 Vietnam Academy of Science and Technology
50 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
linearization can be found in some works [6–11], review articles [12–16], and in mono-
graphs by Crandall and Mark [17], Lin [18], Roberts and Spanos [19], Socha [5] with ref-
erences therein. For continuous systems, the method of EQL is also applied [14, 20–26].
In vibration analysis of beam structures under random excitations, the EQL method is
studied by several authors. In Ref. [21], Herberts considered the effect of the mem-
brane force on the stresses in a simply supported Bernoulli-Euler beam by the method
of EQL. Seide [22] investigated nonlinear mean-square multimode responses of beams
subjected to uniform pressure uncorrelated in time. Using EQL method, he obtained
mean-square stresses and displacements of beams with arbitrary end conditions. Iwan
and Whirley [23] developed a version of the EQL that can be applied to continuous sys-
tems under non-stationary random excitations. Their technique allows the replacement
of the original nonlinear system with a time-varying linear continuous system. In [24,25],
a new technique of equivalent linearization method is proposed based on the energy ap-
proach. Unlike the traditional replacement of EQL method [3, 4], the energy method
requires that the mean-square error between the potential energy of the original nonlin-
ear system and that of corresponding linearized system must be minimum. In [26], Anh
et al. extended the approach of regulated equivalent linearization (RGEL) in studying
single-degree-of-freedom system to random vibration analysis of beams under random
loadings. The effective of the RGEL method is recorded by its excellent performance in
calculating approximate modal responses of the beam.
Recently, Anh et al. [27] have proposed a dual criterion of the EQL method for non-
linear single-degree-of-freedom systems under random excitations. The authors showed
that the accuracy of the mean-square response obtained by the dual criterion is signif-
icantly improved when the nonlinearity is increasing. This dual approach is then ex-
tended to cases of multi-degree-of-freedom systems [28].
Naturally, the dual approach may be extended to random vibration analysis of
many continuous systems. In this research, a version of dual criterion of the EQL method
is developed for analyzing modal responses of beams subjected to random loading. A
nonlinear algebraic system of linearization coefficients obtained in a closed form is solved
by an iteration method. To elucidate the dual approach, the obtained results are com-
pared with those of the conventional linearization and energy methods.
2. THE GOVERNING EQUATION OF BEAM
Consider the governing equation of a beam on elastic foundation, restrained at its
ends and subjected to a space-wise distributed time-dependent loading p (x, t) [24, 25]
EI
∂4w
∂x4
− N ∂
2w
∂x2
+ µA
∂2w
∂t2
+ β
∂w
∂t
+ K fw = p (x, t) , (1)
where the axial force N is given by
N =
EA
2L
L∫
0
(
∂w
∂x
)2
dx. (2)
Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 51
Here, A and I are the area and inertia moment of the cross-section, respectively; E
is the elastic modulus, µ the mass density, β the viscous damping coefficient, L the length
of the beam, K f the stiffness of the elastic foundation; w (x, t) is the deflection of the beam.
In this paper, a space-wise and time-wise white noise loading p (x, t) is considered.
In order to solve (1) one expands p (x, t)in series [24]
p (x, t) =
∞
∑
n=1
qn (t) φn (x), (3)
where qn (t) (n = 1, 2, . . .) are zero-mean Gaussian white noise stationary random pro-
cesses with corresponding correlations
E [qm (t) qn (t+ τ)] = 2piSmδmnδ (τ) , (m, n = 1, 2, . . .) (4)
in which δmn is the Kronecker-delta notation, δ (τ) is the Dirac-delta; the quantities Sn
(n = 1, 2, . . .) are constant spectral density values of random processes qn (t). The func-
tions φn (x) (n = 1, 2, . . .) in the series (3) are modal shapes satisfying the following rela-
tionships
d4φn
dx4
=
µA
EI
ω2nφn, (5)
1∫
0
φmφndξ = δmn, ξ =
x
L
(6)
where ωn (n = 1, 2, . . .) are natural frequencies associated with free vibration of the sys-
tem (1) without the axial force, viscous damping, elastic foundation and external excita-
tion for a simply supported boundary condition
ω2n =
n4EIpi4
µAL4
. (7)
Let the deflection function w (x, t) of the beam be expended in terms of an appro-
priate orthogonal set of modal shapes φn (x) as follows
w (x, t) =
∞
∑
n=1
wn (t) φn (x), (8)
where wn (t) is the modal contribution corresponding to nth-mode. Substituting Eq. (8)
into the expression of the axial force N in Eq. (2) yields
N =
EA
2L2
∞
∑
n=1
∞
∑
m=1
Knmwnwm, (9)
where it is denoted
Knm =
1∫
0
dφn
dξ
dφm
dξ
dξ = Kmn. (10)
52 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
In view of Eqs. (5) and (6), the governing equation (1) takes the form
∞
∑
n=1
(
µAw¨n + βw˙n + K fwn + µAω2nwn
)
φn − EA2L2
∞
∑
n=1
∞
∑
i=1
∞
∑
j=1
Kijwiwjwn
d2φn
dx2
= p (x, t) .
(11)
Multiplication of Eq. (11) by φm, integration over the length of the beam, and then
use of orthogonality conditions given in Eq. (6) yield a set of coupled nonlinear differen-
tial equations for modal amplitudes wm (t)
w¨m +
β
µA
w˙m +ω2mwm +
K f
µA
wm +
E
2µL4
∞
∑
n=1
∞
∑
i=1
∞
∑
j=1
KijKnmwiwjwn = bm (t) , (12)
where the random function bm (t) is given by
bm (t) =
1
µAL
L∫
0
p (x, t) φm (x) dx. (13)
Further in the expansion (8), it is assumed that only the first M modes of the beam
significantly contribute to formulate responses. Crandall and Yildiz [29] shown that if
the infinite series that are representing quantities such as displacement, mean-square
stresses, etc., converge, then the results can be made as accurate as desired by taking
sufficiently large M. For this reason, Eq. (12) can be taken by the following finite form
with the first M modes
w¨m +
β
µA
w˙m +ω2mwm +
K f
µA
wm + Gm (w1,w2, . . . ,wM) = bm (t) , (14)
in which nonlinear components Gm (m = 1, 2, . . . , M) are functions of M variables w1, . . . ,wM
Gm = Gm (w1,w2, . . . ,wM) =
E
2µL4
M
∑
n=1
M
∑
i=1
M
∑
j=1
KijKnmwiwjwn. (15)
Our objective is to find approximate mean-square modal responses of the beam
from the modal system (14). In the next section, the dual criterion of stochastic lineariza-
tion method will be applied to this nonlinear system.
3. A DUAL CRITERION FOR MODAL EQUATIONS OF BEAM
The dual criterion of stochastic linearization method appears from an idea that the
original nonlinear system can be replaced by an equivalent linearization system, and then
this equivalent system is replaced by another nonlinear system that belongs to the same
class of the original nonlinear system. Some obtained results using the dual criterion are
presented in works of Anh et al. [27,28] for single- and multi-degree-of-freedom systems
subjected to random excitations. Naturally, the dual criterion of stochastic linearization
needs to be developed in investigating continuous systems under random excitation. For
Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 53
the governing equation of the beam in the modal form (14), one can make a linearization
version as follows
w¨m +
β
µA
w˙m +ω2mwm +
K f
µA
wm +ω2mkeq,mwm = bm (t) , (16)
where keq,m (m = 1, 2, . . . , M) are non-dimensional linearization coefficients determined
from a specified criterion of stochastic linearization. We here utilize the dual criterion
[27, 28] for determining coefficients keq,m (m = 1, 2, . . . , M). In the first step, the original
nonlinear term Gm is replaced by a linearized one ω2mkeq,mwm, and then the linear term
ω2mkeq,mwm is replaced by another nonlinear quantity, λmGm, that can be considered as
a term belonging to the same class of the original function Gm, where the coefficients
keq,mand λm are determined from the following proposed criterion for beam vibration
e1 = E
[(
Gm −ω2mkeq,mwm
)2]
+ ρE
[(
ω2mkeq,mwm − λmGm
)2]→ min
keq,m,λm
, (17)
with the detuning parameter ρ taking two values 0 or 1. When the parameter ρ is equal
to zero, the criterion (17) becomes the conventional mean-square error criterion which
can be found in the literature. On the other hand, as the parameter ρ is taken to be
1, the criterion (19) is so-called dual one. In this criterion, the first expectation can be
understood as a component of the conventional replacement, whereas the second one
describes a dual replacement of the linearization problem. Similar to the conventional
linearization (see [6]), the criterion (17) leads to that partial derivatives of the expression
e1 with respect to variables keq,m and λm are equal to zero
∂e1
∂keq,m
= 0,
∂e1
∂λm
= 0, (m = 1, 2, . . . , M). (18)
The system (18) yields a set of algebraic equations of variables keq,m and λm,
(m = 1, 2, . . . , M) as follows(
(1+ ρ)ω2mE
[
w2m
])
keq,m = (E [wmGm]) (1+ ρλm) ,
λm =
(
ω2m
E [wmGm]
E [G2m]
)
keq,m.
(19)
Solving the system (19) for unknowns keq,m and λm, we arrive at
keq,m =
1
ω2m
1
1+ ρ− ρηm
E [wmGm]
E [w2m]
, (20)
λm =
ηm
1+ ρ− ρηm , (21)
where
ηm =
(E [wmGm])
2
E [w2m] E [G2m]
. (22)
It is observed that, using the dual criterion (17), the original nonlinear Eq. (14) of
modes of beam vibration is replaced by its linearization version (16), in which lineariza-
tion coefficients keq,m (m = 1, 2, . . . , M) are determined from expressions (20)-(22). In the
54 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
framework of this article, the dual criterion (17) is elucidated for random vibrations of a
simply supported beam under a space-wise and time-wise white noise loading.
4. RESPONSES OF A SIMPLY SUPPORTED BEAM AT BOTH ENDS
4.1. Equivalent linearization coefficients
For a simply supported beam, one has the following expression for the modal
shape φm [25]
φm (ξ) =
√
2 sin (mpiξ) . (23)
Using the expressions (10) and (23), one obtain
Knm = Kmn =
1∫
0
dφn
dξ
dφm
dξ
dξ =
{
pi2m2 if m = n,
0 if m 6= n. (24)
Eq. (14) becomes
w¨m +
β
µA
w˙m +ω2m
(
1+
α
m4
)
wm +
ω2m
2R2m2
M
∑
n=1
n2w2nwm =bm (t) , (25)
where
ω2m = ω
2
0m
4, ω20 =
EIpi4
µAL4
, α =
K f
µAω20
, R =
√
I
A
. (26)
The nonlinear functions Gm (m = 1, 2, . . . , M) in Eq. (25) take the form
Gm =
ω2m
2R2m2
M
∑
n=1
n2w2nwm. (27)
Substituting expressions Gm (m = 1, 2, . . . , M) from Eq. (27) into Eq. (20) yields
keq,m =
1
2R2m2
1
1+ ρ− ρηm
M
∑
n=1
n2E
[
w2nw2m
]
E [w2m]
, (28)
where
ηm =
M
∑
i=1
M
∑
j=1
i2 j2E
[
w2i w
2
m
]
E
[
w2jw
2
m
]
E [w2m]
M
∑
i=1
M
∑
j=1
i2 j2E
[
w2i w
2
jw
2
m
] . (29)
It is noted that, to calculate higher-order moments in Eqs. (28) and (29), we em-
ploy the following generalized formula expressed in terms of second-order moments of
Gaussian random processes with zero-mean [30]
E [z1z2 . . . z2m] = ∑
all independent pairs
(
∏
j 6=k
E
[
zjzk
])
, (30)
Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 55
where the number of independent pairs is equal to (2m)!
/
(2mm!). Particularly, in view of
the present dual method for beam vibration, the following higher-order moment terms
will appear
E
[
w2mw
2
n
]
= E
[
w2m
]
E
[
w2n
]
+ 2 (E [wmwn])
2 = ymmynn + 2y2mn,
E
[
w2jw
2
jw
2
m
]
= E
[
w2i
]
E
[
w2j
]
E
[
w2m
]
+ 2
(
E
[
wiwj
])2 E [w2m]
+ 2
(
E
[
wjwm
])2 E [w2i ]+ 2 (E [wiwm])2 E [w2j ]
+ 8E
[
wiwj
]
E
[
wjwm
]
E [wiwm]
= yiiyjjymm + 2y2ijymm + 2y
2
jmyii + 2y
2
imyjj + 8yijyjmyim,
(31)
where the notation ymn for the second moment of wm is introduced
ymn = ynm = E [wmwn] . (32)
Substituting expressions (31) in Eq. (28), we arrive at
keq,m =
1
2R2m2
1
1+ ρ− ρηm
M
∑
n=1
n2
ynnymm + 2y2mn
ymm
, (33)
where
ηm =
M
∑
i=1
M
∑
j=1
i2 j2
(
yiiymm + 2y2im
) (
yjjymm + 2y2jm
)
ymm
M
∑
i=1
M
∑
j=1
i2 j2
(
yiiyjjymm + 2y2ijymm + 2y
2
jmyii + 2y
2
imyjj + 8yijyjmyim
) . (34)
For purpose of comparing the present dual criterion with other methods, in this
paper, we also present two known results of the conventional linearization [21, 22], and
energy method [24, 25]. From the linearized system (16), the following expressions of
the equivalent linearization coefficients keq,m (m = 1, 2, . . . , M) are obtained using the
conventional linearization method
keq,m conventional =
1
2R2m2
M
∑
n=1
n2
ynnymm + 2y2nm
ymm
. (35)
It is observed that the expression (35) is also obtained from (33) by taking the de-
tuning parameter ρ be zero. The linearization method based on energy criterion gives the
equivalent coefficients keq,m (m = 1, 2, . . . , M) obtained from the following system [24,25]{
1+ α+ k1,eq, 24
(
1+
α
24
+ k2,eq
)
, . . . , M4
(
1+
α
M4
+ kM,eq
)}T
=
2
ω20
A−1
{
E
[
w21U
]
E
[
w22U
]
. . . E
[
w2MU
]}T
,
(36)
56 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
where the matrix A and potential energy U of the system are determined as follows
A =
E
[
w21w
2
1
]
E
[
w21w
2
2
]
. . . E
[
w21w
2
M
]
E
[
w22w
2
1
]
E
[
w22w
2
2
]
. . . E
[
w22w
2
M
]
. . . . . . . . . . . .
E
[
w2Mw
2
1
]
E
[
w2Mw
2
2
]
. . . E
[
w2Mw
2
M
]
, (37)
U =
ω20
2
M∑
m=1
(
α+m4
)
w2m +
1
4R2
(
M
∑
m=1
m2w2m
)2 . (38)
To get a closed form of the equivalent linearization coefficients keq,m (m = 1, 2, . . . , M)
in Eq. (33), we utilize responses of the linearized (16) via spectral density of the external
excitations.
4.2. Responses of the linearized system
For the linearized system (16) under the random excitation b, one can obtain second-
order moments E [wmwn] of the responses wm as follows (see [19] for details)
E [wmwn] =
∞∫
−∞
Hm (−ω) Bmn (ω)Hn (ω) dω, (39)
where
Bmn (ω) =
Smδmn
(µA)2
, (40)
and the frequency-response function Hm (ω) is given by
Hm (ω) =
1(
1+ αm4 + keq,m
)
ω2m −ω2 + i βµAω
. (41)
Because Bmn (ω) given by (40) are constants, moments E [wmwn] can be rewritten as
E [wmwn] = Bmn
∞∫
−∞
Hm (−ω)Hn (ω) dω. (42)
Introducing (41) into the right-hand side of the expression (42) and employing
residual theorem in theory of complex variable functions, we get
ymn = E [wmwn] =
4piβBmn
µA
{[(
1+
α
m4
+ keq,m
)
ω2m −
(
1+
α
n4
+ keq,n
)
ω2n
] 2
+2
(
β
µA
)2 [(
1+
α
m4
+ keq,m
)
ω2m +
(
1+
α
n4
+ keq,n
)
ω2n
]}−1
.
(43)
It is seen that a closed system of nonlinear algebraic equations for unknowns keq,m
(m = 1, 2, . . . , M) is obtained by substituting (43) in to the right hand side of Eq. (33). As
noted before, because the contribution of the first modes of the system is significant, we
here restrict our calculations for modal responses of the beam in two cases: single-, and
Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 57
two-mode using three approaches: the conventional linearization, energy method, and
present dual criterion.
5. NUMERICAL RESULTS AND DISCUSSIONS
It is seen that, in Eq. (25) for vibrational modes, as R tends to infinity, the effect of
nonlinear terms Gm(m = 1, 2, . . . , M) disappear. Therefore, one can view the magnitude
of the quantity 1/R as the parameter related to the magnitude of nonlinearity of the
original nonlinear system (25). Assume that spectral densities (m = 1, 2, . . . , M) of the
stochastic processes qm (t) have the same value, i.e. S1 = S2 = . . . = SM = S0. For this
assumption, from the Fokker-Planck equation corresponding to the system (25), the exact
expression of the probability density function can be obtained [24]
P (w1,w2, . . . ,wM) =
1
C
exp
−βµAω202piS0
M∑
m=1
(
α+m4
)
w2m +
1
4R2
(
M
∑
m=1
m2w2m
)2
=
1
C
exp
{
−βµA
piS0
U (w1,w2, . . . ,wM)
}
,
(44)
where U = U (w1,w2, . . . ,wM) is the potential energy of the system (25) given by (38),
and C is the normalization constant
C =
∞∫
−∞
. . .
∞∫
−∞︸ ︷︷ ︸
M− f old
exp
−βµAω202piS0
M∑
m=1
(
α+m4
)
w2m +
1
4R2
(
M
∑
m=1
m2w2m
)2 dw1 . . . dwM.
(45)
The exact modal mean-square response of Eq. (25) is evaluated by the following
multiple integral with M-fold
E
[
w2m
]
exact
=
1
C
∞∫
−∞
. . .
∞∫
−∞︸ ︷︷ ︸
M−fold
w2mexp
− βµAω202piS0
M∑
m=1
(
α+m4
)
w2m+
1
4R2
(
M
∑
m=1
m2w2m
)2 dw1 . . . dwM.
(46)
In general, the multiple integral (46) must be calculated using a numerical method.
In the following computation, we use the exact solution (46) in the case of single-mode
(M = 1) and of two-mode (M = 2) to elucidate the accuracy of the proposed dual
criterion method (33), and other methods for comparison purpose.
5.1. The case of single-mode
For the single-mode, M = 1, the governing equation of the simply supported beam
(25) takes the form
w¨1 +
β
µA
w˙1 +ω20 (1+ α)w1 +
ω20
2R2
w31 =
1
µA
q1 (t) . (47)
58 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
This is the well-known Duffing oscillator subjected to random excitation [4, 5].
From Eq. (46), one can get an exact solution of mean-square response of w1 in Eq. (47) as
follows (see also [22, 26])
E
[
w21
]
exact, 1 =
∞∫
−∞
w21 exp
{
− βµApiS0
(
1
2ω
2
0 (1+ α)w
2
1 +
ω20
8R2 w
4
1
)}
dw1
∞∫
−∞
exp
{
− βµApiS0
(
1
2ω
2
0 (1+ α)w
2
1 +
ω20
8R2 w
4
1
)}
dw1
= (1+α)R2
[
K3/4
(
(1+α)2
4
R2
R201
)
−K1/4
(
(1+α)2
4
R2
R201
)]/
K1/4
(
(1+α)2
4
R2
R201
)
,
(48)
where Kν (y) is the modified Bessel function of the second kind of order ν of the variable
y, and the quantity R01 is given by
R01 =
√
piS0
β (µA)ω20
. (49)
Using Eqs. (43) and (28) for M = 1, we get the following single-mode approxi-
mate mean-square response E
[
w21
]
depending upon the nonlinear parameter 1/R in the
following form
E
[
w21
]
dual,1 =
2R201
1+ α+
√
(1+ α)2 + 307
R201
R2
. (50)
Similarly, approximate mean-square responses of w1 corresponding to the conven-
tional linearization (35) and energy method (36) in the case of single-mode are obtained,
respectively,
E
[
w21
]
conventional,1 =
2R201
1+ α+
√
(1+ α)2 + 6R
2
01
R2
, (51)
E
[
w21
]
energy,1 =
2R201
1+ α+
√
(1+ α)2 + 5R
2
01
R2
. (52)
Numerical results for the first mode of beam vibration in the case of single-mode
using four methods, including the exact solution (48), conventional linearization (51), en-
ergy method (52), and dual criterion method (50) are illustrated in Tabs. 1 and 2. The sys-
tem parameters are ω0 = 1, β = 0.1, µA = 1. Tab. 1 shows a comparison of relative errors
between results obtained from approximate and exact solution methods. The stiffness pa-
rameter α of the system is fixed at 1, whereas the nonlinearity parameter 1/R varies from
0.01 to 10.0. It is seen that, for small values of 1/R, for example 1/R = 0.01, 0.02, 0.05,
the conventional linearization yields quite small errors, about 0.05%, whereas errors of
the energy and dual criterion methods are larger. In the range [1, 10] of 1/R, the error
of conventional linearization becomes larger 10% while that of the energy method and
Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 59
Table 1. Mean-square response of w1 of the simply supported beam in case of single-mode with
ω0 = 1, α = 1, β = 0.1, µA = 1, S0 = 1 and various values of 1/R (CL: Conventional
Linearization; EM: Energy Method; DM: Dual Criterion Method)
1/R E
[
w21
]
exact,1 E
[
w21
]
CL,1 Error (%) E
[
w21
]
EM,1 Error (%) E
[
w21
]
DM,1 Error (%)
0.01 15.6895 15.6895 0.0001 15.6926 0.0195 15.6948 0.0335
0.02 15.6349 15.6346 0.0014 15.6468 0.0761 15.6554 0.1317
0.05 15.2778 15.2707 0.0466 15.3403 0.4086 15.3907 0.7388
0.10 14.2613 14.1964 0.4549 14.4101 1.0437 14.5706 2.1690
0.20 11.9200 11.6419 2.3324 12.0674 1.2370 12.4086 4.0991
0.50 7.3952 6.8668 7.1448 7.3248 0.9518 7.7220 4.4191
1.00 4.4131 3.9581 10.3102 4.2767 3.0909 4.5614 3.3618
2.00 2.4261 2.1276 12.3034 2.3146 4.5968 2.4842 2.3928
5.00 1.0294 0.8890 13.6400 0.9712 5.6599 1.0463 1.6388
10.00 0.5251 0.4510 14.1093 0.4934 6.0421 0.5322 1.3564
Table 2. Mean-square response of w1 of the simply supported beam in case of single-mode with
ω0 = 1, β = 0.1, µA = 1, S0 = 1, R = 1 and various values of α
α E
[
w21
]
exact,1 E
[
w21
]
CL,1 Error (%) E
[
w21
]
EM,1 Error (%) E
[
w21
]
DM,1 Error (%)
1.0 4.4131 3.9581 10.3102 4.2767 3.0909 4.5614 3.3618
2.0 4.0310 3.6844 8.5978 3.9549 1.8889 4.1930 4.0181
3.0 3.6976 3.4334 7.1448 3.6624 0.9518 3.8610 4.4191
4.0 3.4056 3.2038 5.9244 3.3975 0.2382 3.5629 4.6202
5.0 3.1489 2.9944 4.9075 3.1581 0.2921 3.2960 4.6714
6.0 2.9224 2.8036 4.0656 2.9422 0.6756 3.0573 4.6147
7.0 2.7218 2.6300 3.3715 2.7475 0.9442 2.8438 4.4840
8.0 2.5433 2.4721 2.8008 2.5719 1.1241 2.6528 4.3056
9.0 2.3840 2.3284 2.3324 2.4135 1.2370 2.4817 4.0991
10.0 2.2412 2.1975 1.9480 2.2703 1.3000 2.3281 3.8785
dual criterion are remaining about 6%. For large values of 1/R, for instance 1/R = 5,
1/R = 10, however, the error of the dual criterion method is smallest (about 2%).
In Tab. 2, the parameter 1/R is taken to be 1, the stiffness parameter α varies from
1.0 to 10.0, other parameters have the same values as in Tab. 1. It is observed that, the
dual criterion method gives a good prediction on response errors (about 5%) as α varies,
and the error of energy method is smallest.
5.2. The case of two-mode
Numerical computations used the fixed-point iteration method [25, 26] to find ap-
proximate mean-square response of the first mode of beam vibration are carried out for
60 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
the nonlinear algebraic system (33) and (43) with unknowns km,eq in the case of two-mode.
The exact solution is obtained from the multiple integral (46) with M = 2. The obtained
numerical results are presented in Tabs. 3 and 4. Tab. 3 shows that, the error of dual cri-
terion method is in good agreement with that of the energy method. In Tab. 4, in general,
the dual criterion and energy method yield values that are close to the exact solutions
with different value of the stiffness parameter α.
Table 3. Mean-square response of w1of the simply supported beam in case of two-mode with
ω0 = 1, α = 1, β = 0.1, µA = 1, S0 = 1 and various values of 1/R
1/R E
[
w21
]
exact,1 E
[
w21
]
CL,1 Error (%) E
[
w21
]
EM,1 Error (%) E
[
w21
]
DM,1 Error (%)
0.01 15.6865 15.6866 0.0007 15.6901 0.0232 15.6921 0.0358
0.02 15.6233 15.6232 0.0006 15.6372 0.0887 15.6449 0.1385
0.05 15.2125 15.2050 0.0494 15.2847 0.4744 15.3290 0.7658
0.10 14.0574 13.9915 0.4688 14.2343 1.2587 14.3672 2.2039
0.20 11.4744 11.2121 2.2857 11.6951 1.9235 11.9358 4.0207
0.50 6.7777 6.3387 6.4772 6.8873 1.6175 7.0652 4.2423
1.00 3.9033 3.5576 8.8571 3.9627 1.5219 4.0385 3.4636
2.00 2.0923 1.8799 10.1519 2.1284 1.7243 2.1529 2.8951
5.00 0.8714 0.7764 10.9030 0.8891 2.0276 0.8936 2.5492
10.00 0.4414 0.3923 11.1312 0.4510 2.1828 0.4522 2.4529
Table 4. Mean-square response of w1of the simply supported beam in case of two-mode with
ω0 = 1, β = 0.1, µA = 1, S0 = 1, R = 1 and various values of α
α E
[
w21
]
exact,1 E
[
w21
]
CL,1 Error (%) E
[
w21
]
EM,1 Error (%) E
[
w21
]
DM,1 Error (%)
1.0 3.9033 3.5576 8.8571 3.9627 1.5219 4.0385 3.4636
2.0 3.5878 3.3203 7.4557 3.6862 2.7440 3.7299 3.9603
3.0 3.3112 3.1035 6.2725 3.4338 3.7025 3.4526 4.2703
4.0 3.0677 2.9058 5.2784 3.2038 4.4375 3.2038 4.4375
5.0 2.8526 2.7256 4.4511 2.9947 4.9812 2.9808 4.4929
6.0 2.6616 2.5616 3.7582 2.8047 5.3755 2.7807 4.4741
7.0 2.4915 2.4121 3.1849 2.6321 5.6419 2.6010 4.3960
8.0 2.3393 2.2760 2.7071 2.4752 5.8105 2.4394 4.2811
9.0 2.2026 2.1517 2.3090 2.3326 5.9016 2.2938 4.1422
10.0 2.0793 2.0383 1.9738 2.2027 5.9355 2.1623 3.9922
Vibration analysis of beams subjected to random excitation by the dual criterion of equivalent linearization 61
6. CONCLUSIONS
The method of equivalent linearization is one of effective tools in solving random
vibration problems of mechanical systems. In this study, the modal responses of a simply
supported beam subjected to a space-wise and time-wise white noise loading are carried
out by the dual criterion of equivalent linearization. Our calculations are restricted in
two cases of single- and two-mode of beam vibrations. The exact solutions of the original
modal equation system obtained by Fokker-Planck equation are available for both cases.
A closed form of nonlinear algebraic system is obtained by the dual approach associated
with the frequency-response function method for the linearized modal system. In the
case of single-mode, the analytical solution of the first mode of the beam is easy to solve
explicitly for four methods considered (the exact solution, energy method, conventional
linearization and dual criterion method). Also, in the case of two-mode, the closed sys-
tem is solved by the fixed-point iteration method. Numerical results show that the dual
criterion gives a good prediction on the random responses of the beam, especially in the
range of strong linearity of system parameters. Further investigations for random vibra-
tions of other beam systems seem to be appropriate in order to verify the advantages of
the dual criterion.
ACKNOWLEDGEMENTS
This research is funded by Vietnam National Foundation for Science and Technol-
ogy Development (NAFOSTED) under grant number: “107.04-2015.36”.
REFERENCES
[1] R. C. Booton. Nonlinear control systems with random inputs. Circuit Theory, IRE Transactions
on, 1, (1), (1954), pp. 9–18.
[2] I. E. Kazakov. An approximate method for the statistical investigation of nonlinear systems.
Trudy VVIA im Prof. NE Zhukovskogo, 394, (1954), pp. 1–52. (in Russian).
[3] T. K. Caughey. Equivalent linearization techniques. The Journal of the Acoustical Society of
America, 35, (11), (1963), pp. 1706–1711.
[4] T. K. Caughey. Nonlinear theory of random vibrations. Advances in Applied Mechanics, 11,
(1971), pp. 209–253.
[5] L. Socha. Linearization methods for stochastic dynamic systems, Vol. 730. Springer Science & Busi-
ness Media, (2007).
[6] T. S. Atalik and S. Utku. Stochastic linearization of multi-degree-of-freedom non-linear sys-
tems. Earthquake Engineering & Structural Dynamics, 4, (4), (1976), pp. 411–420.
[7] E. T. Foster. Semilinear random vibrations in discrete systems. Journal of Applied Mechanics,
35, (3), (1968), pp. 560–564.
[8] W. D. Iwan and I.-M. Yang. Application of statistical linearization techniques to nonlinear
multidegree-of-freedom systems. Journal of Applied Mechanics, 39, (2), (1972), pp. 545–550.
[9] R. C. Micaletti, A. S¸. C¸akmak, S. R. K. Nielsen, and H. U. Ko¨ylu¨olu. Error analysis of sta-
tistical linearization with Gaussian closure for large-degree-of-freedom systems. Probabilistic
Engineering Mechanics, 13, (2), (1998), pp. 77–84.
[10] I. Elishakoff, L. Andriamasy, and M. Dolley. Application and extension of the stochastic lin-
earization by Anh and Di Paola. Acta Mechanica, 204, (1-2), (2009), pp. 89–98.
62 Nguyen Nhu Hieu, Nguyen Dong Anh, Ninh Quang Hai
[11] N. D. Anh and L. X. Hung. An improved criterion of Gaussian equivalent linearization
for analysis of non-linear stochastic systems. Journal of Sound and Vibration, 268, (1), (2003),
pp. 177–200.
[12] L. Socha. Linearization in analysis of nonlinear stochastic systems. AppliedMechanics Reviews,
44, (1991), pp. 99–422.
[13] L. Socha. Linearization in analysis of nonlinear stochastic systems: recent results-part I: The-
ory. Applied Mechanics Reviews, 58, (3), (2005), pp. 178–205.
[14] L. Socha. Linearization in analysis of nonlinear stochastic systems, recent results-part II: Ap-
plications. Applied Mechanics Reviews, 58, (5), (2005), pp. 303–315.
[15] S. H. Crandall. A half-century of stochastic equivalent linearization. Structural Control and
Health Monitoring, 13, (1), (2006), pp. 27–40.
[16] I. Elishakoff and L. Andriamasy. The tale of stochastic linearization technique: Over half a century
of progress. Springer, (2012).
[17] S. H. Crandall and W. D. Mark. Random vibration in mechanical systems. Academic Press,
(2014).
[18] Y.-K. Lin. Probabilistic theory of structural dynamics. New York, McGraw-Hill, (1973).
[19] J. B. Roberts and P. D. Spanos. Random vibration and statistical linearization. New York, Wiley,
(1990).
[20] R. E. Herbert. Random vibrations of a nonlinear elastic beam. The Journal of the Acoustical
Society of America, 36, (11), (1964), pp. 2090–2094.
[21] R. E. Herbert. On the stresses in a nonlinear beam subject to random excitation. International
Journal of Solids and Structures, 1, (2), (1965), pp. 235–242.
[22] P. Seide. Nonlinear stresses and deflections of beams subjected to random time dependent
uniform pressure. Journal of Engineering for Industry, 98, (3), (1976), pp. 1014–1020.
[23] W. D. Iwan and R. G. Whirley. Nonstationary equivalent linearization of nonlinear continu-
ous systems. Probabilistic Engineering Mechanics, 8, (3), (1993), pp. 273–280.
[24] I. Elishakoff, J. Fang, and R. Caimi. Random vibration of a nonlinearly deformed beam by a
new stochastic linearization technique. International Journal of Solids and Structures, 32, (11),
(1995), pp. 1571–1584.
[25] J. Fang, I. Elishakoff, and R. Caimi. Nonlinear response of a beam under stationary random
excitation by improved stochastic linearization method. Applied mathematical modelling, 19,
(2), (1995), pp. 106–111.
[26] N. D. Anh, I. Elishakoff, and N. N. Hieu. Extension of the regulated stochastic linearization
to beam vibrations. Probabilistic Engineering Mechanics, 35, (2014), pp. 2–10.
[27] N. D. Anh, N. N. Hieu, and N. N. Linh. A dual criterion of equivalent linearization method
for nonlinear systems subjected to random excitation. Acta Mechanica, 223, (3), (2012),
pp. 645–654.
[28] N. D. Anh, V. L. Zakovorotny, N. N. Hieu, and D. V. Diep. A dual criterion of stochastic
linearization method for multi-degree-of-freedom systems subjected to random excitation.
Acta Mechanica, 223, (12), (2012), pp. 2667–2684.
[29] S. H. Crandall and A. Yildiz. Random vibration of beams. Journal of Applied Mechanics, 29, (2),
(1962), pp. 267–275.
[30] D. Middleton. An introduction to statistical communication theory. McGraw-Hill New York,
(1960).
Các file đính kèm theo tài liệu này:
- vibration_analysis_of_beams_subjected_to_random_excitation_b.pdf