The equivalent linearization method (ELM) is one of the most

commonly used methods and is an effective method for nonlinear

systems with weak nonlinear coefficients. For nonlinear systems

with larger nonlinear coefficients, the accuracy of this method is

significantly reduced. The dissertation focuses on researching and

developing EL method to improve errors when analyzing nonlinear

oscillations.

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covariance.
1.3 Some special stochastic processes
There are definitions of: Stationary random processes and
Ergodic process; Normal random process or Gaussian process; White
noise process; colored noise process; Wiener process and Markov
process.
1.4 Some approximately analytical methods for analyzing
random oscillation
Numerical methods, approximately analytical methods are very
popular methods. In detail, there are some useful methods in this
dissertation [29-31]:
- Perturbation technique.
- Fokker-Planck-Kolmogorov (FPK) equation technique.
- Stochastic averaging technique.
- Statistical linearization technique.
1.5 Fokker-Planck-Kolmogorov (FPK) equation technique and
Stochastic averaging technique
1.6 Overview of studies on random oscillations
The problem of random vibrations has been studied and
presented in many textbooks [26–33]. Oscillation analysis based on
nonlinear mathematical models requires appropriate methods. In the
theory of random oscillation, the stochastic equivalent linearization
method (ELM) replacing the nonlinear system by an equivalent
linear system is a common method because it preserves some
essential properties of the original nonlinear system. This method has
been described in many review papers [42, 43] and summarized in
monographs [29] and [44].
4
Although the accuracy of EL methods may not be high, this is
overcome by improved techniques [43]. Canor et al. [45] also wrote:
Thanks to its easy and fast implementation technique, the equivalent
linearization method has become a universal probability approach for
analyzing large nonlinear structures. The EL method has been used
in many research papers. A EL based on analytic method was
developed in [46, 47] to analyze nonlinear energy extraction
systems. The nonlinear oscillation system of the wing profile was
studied in [48, 49] by using EL method. Silva - Gonzlez et al. [52]
used the stochastic EL method to study elastic non-linear structure of
seismic load.
In Vietnam, the dissertation of Nguyen Ngoc Linh [4] analyzed
the nonlinear random oscillation of systems of 1 degree of freedom
by the stochastic EL method according to the weighted duality
criteria. The dissertation of Nguyen Nhu Hieu [5] has developed the
duality criterion in EL method for multi-degree-of-freedom nonlinear
systems with random excitation. Nguyen Minh Triet has conducted a
doctoral dissertation on the analysis of the response of the airplane
wing profile according to the duality approach, in which the study of
nonlinear periodic oscillation by EL method [6]. In his PhD
dissertation in 2002 [7] Luu Xuan Hung developed the Local Mean
Square Error Criterion (LOMSEC) based on the idea of replacing
integral over infinite domain (-∞ , + ∞) equals the integral over a
finite domain [-rx, + rx] where the system's response is
concentrated. Continuing development of this research direction, the
dissertation carried out by the PhD student will develop the Global-
Local Mean Square Error Criterion (GLOMSEC) of the EL method
to nonlinear MDOF systems subjected to random excitation. In this
development, a dual approach is used to address the finite domain of
integration [-rx, + rx].
Conclusion of chapter 1
Chapter 1 introduced some basic concepts and formulas for
probability theory and random processes, and some methods of
nonlinear random oscillation analysis. Some research results on
nonlinear random oscillations related to the dissertation have also
been reviewed and analyzed as the basis for the next chapters.
5
CHAPTER 2. EQUIVALENT LINEARIZATION METHOD
AND GLOBAL-LOCAL MEAN SQUARE ERROR
CRITERION
2.1. The classical criterion of equivalent linearization
We present the EL method for 1 DOF nonlinear random oscillator
systems of type [9, 29, 44]:
2
02 ( , ) ( )x hx x g x x t
(2.10)
Where, x , x and x are displacement, velocity and acceleration,
respectively; h is damping coefficient, ( , )g x x is a nonlinear
function, ( )t is Gaussian white noise excitation with intensity
2 ; 0 is the natural frequency for 0h , ( , ) 0g x x . The
equivalent linearization equation of (2.10) is as follows:
2
02 ( )x hx x bx kx t
(2.11)
where b, k are equivalent linearization coefficients. The error
between (2.10) và (2.11) satisfies the Mean Square Error Criterion
proposed by Caughey [10]:
2
d
,
( , ) mink
b k
S g x x bx kx (2.14)
Hence:
kd kd0; 0
S S
b k
(2.15)
Assuming that the solution is a random process, the response x , x
should be independent, that is 0xx , solving the system of
equations (2.15) we obtain
2
,xg x x
b
x
,
2
,xg x x
k
x
(2.17)
6
Equations (2.11) và (2.17) forms a system of equations for 3
unknowns x(t), b, k. Iterative algorithms are usually applied proposed
by Atalik and Utku [59] as follows:
a) Assign the initial value to the second order moments 2 2,x x .
b) Use (2.17) to determine the linear coefficients.
c) Solve equation (2.11) to find the new instantaneous second order
moments 2 2,x x .
d) Repeat b) and c) until the specified accuracy is reached.
We consider the nonlinear system with multi degrees of freedom
under random excitation
, ,Mx + Cx + Kx + Φ x,x x = Q t
(2.20)
where x , x , x are displacement, velocity and acceleration
vectors, respectively , ,M C Kij ij ijn n n n n nm c k
are
mass, damping and stiffness matrices; ,Φ x,x x - nonlinear function
vector, Q t is Gaussian white noise excitation vector with zero
intensity and spectral density matrix S ij n nS where ijS is
cross spectral density function of
iQ and jQ . The equivalent
linearization system is as follows:
,t
e e eM + M q C + C q K + K q Q
(2.21)
where e e eM , C , K are the equivalent matrices of mass, damping
and stiffness. In equation (2.21) we use the notation q t to show
that this is only an approximation to x t in the original nonlinear
equation (2.20). The error between the system (2.20) and the system
(2.21) is
. e e ee Φ q, q, q M q + C q + K q
(2.22)
7
The error e satisfies the Mean Square Error Criterion upon
e e eM , C , K :
min .E e e e
T
M ,C ,K
e e
(2.24)
Where the expectation in the left side of (2.24) is calculated by the
probability density function of (2.21). Atalik and Utku (1976) [59]
show that the criterion (2.24) leads to the following equation:
,
TT e e e Tzz M C K zΦ zE E
(2.25)
2.2. Some improved equivalent linearization criteria
For decades, many studies on equivalent linearization criteria have
been proposed to improve the accuracy of the equivalent
linearization method [11-24, 20-24, 67, 68].
2.3 Criterion of global-local mean square error
In this section, we propose a new equivalent linearization criterion
called the global-local mean square error criterion. We consider a
nonlinear random oscillation of one degree of freedom:
2
02 ( , ) ( )x hx x g x x t
(2.47)
Where, the notation is defined as above. The equivalent
linearization equation of (2.47) takes the form:
2
02 ( )x hx x x x t
(2.48)
where λ, μ are linearization coefficients. The error between
(2.47) và (2.48) is:
xxxxgxxe ,, (2.49)
The classical criterion gives [29, 44]
2
,
( , ) ( , ) mine x x P x x dxdx
(2.51)
Where ( , )P x x is the probability density function (PDF) of x and
x :
8
2 2
( , ) ( , )
, .
g x x x g x x x
x x
(2.53)
Since the integration domain in (2.51) is ( , ), the criterion
(2.51) is called the global mean square error criterion. With the
assumption that the integration should focus for a more accurate
solution, Anh and Di Paola proposed the local mean square error
criterion (LOMSEC) [15]:
0 0
0 0
2
,
( , ) ( , ) min
x x
x x
e x x P x x dxdx
(2.54)
Where, 00, xx are positive values. The integration (2.54) is
transformed into the non-dimension one by introducing
0 0,x xx r x r where r is a positive value, x and x are
standard deviations of x and x :
2 2
,
[ ( , )] ( , ) ( , ) min
x x
x x
r r
r r
e x x e x x P x x dxdx
(2.55)
where [.] denotes:
[ . ] ( . ) ( , )
x x
x x
r r
r r
P x x d xd x
(2.56)
We have analogously:
2 2
( , ) ( , )
( ) , ( ) .
g x x x g x x x
r r
x x
(2.57)
It is seen from (2.57) the local equivalent linearization
coefficients (LOMSEC) are functions of r, ( ), ( )r r .Using
the dual viewpoint, we can propose that r change across the non-
negative domain and the EL coefficients , can be chosen as the
following[24]:
9
0
0
1
( ) ( ) ,
1
( ) ( ) .
s
s
s
s
r Lim r dr
s
r Lim r dr
s
(2.60)
We have found from LOMSEC a new EL criterion called the
global-local mean square error criterion (GLOMSEC).
Next we develop the global-local mean square error criterion
(GLOMSEC) to MDOF systems:
tfzgz (2.61)
1 2, ,...,
T
nz z z z are vector of state variables, n is a natural
number, g is a nonlinear function of z, f(t) is a Gausian stochastic
process with zero mean. Denote:
tfzgzze (2.62)
Introduce new linearization terms into (2.62):
tfzgAzAzzze (2.64)
where ijaA is matrix n×n. Let y is a stationary solution of:
0 tfAyy (2.65)
From (2.64) and (2.65) we have:
ygAyye (2.66)
Denote p(y) probability density function (PDF) of y of (2.65).
According to the LOMSEC we have:
,min
0
0
1
0
1
1
0
1
22
ij
ynn
ynn
y
y a
y
y
i
y
y
i dyypyenye
i,j = 1,,n
It gives:
1
TT yyyygA (2.72)
10
The iterative algorithm is applied similarly to the one proposed by
Atalik and Utku [59]. According to the GLOMSEC, the EL
coefficients aij can be chosen as the following
00
1
0 0 0
1 2
0 0 0
1 2
0 0 0 0 0 0
1 2 1 20 0 0, ,..... 1 2 0 0
( , , ..... )
1
.... ( , , ..... ) .....
.....
n
n
ij ij n
yy
ij n n
y y y n
a a y y y
Lim a y y y dy dy dy
y y y
Conclusion of chapter 2
The second chapter deals with the development of the criterion of
Global – local mean squared error (GLOMSEC) for the systems of
one and multi degrees of freedom. The results in chapter 2 are
presented in the articles [1,6] of the List of publication of
dissertation.
CHAPTER 3. APPLICATION OF GLOMSEC IN ANALYSIS
OF RANDOM ONE DEGREE OF FREEDOM SYSTEMS
3.1. Analysis of domain of concentred response of nonlinear
systems
3.1.1. Duffing oscillation system subjected to white noise excitation
3.1.2 Nonlinear damped oscillation system subjected to white noise
excitation
We consider a nonlinear damped oscillation system subjected to
white noise excitation:
2 21x x x x x d t
(3.6)
The exact PDF two-dimensional probability density function of the
system [29, 44]:
22 2 2 2, exp 0.5p x x C x x x x
d
(3.7)
11
where C is the normalized constant. If Prob a x a is
chosen then the domain aa, will be detemined by:
Prob ,aaa x a p x x dx dx
(3.8)
Suppose we choose Prob 0.98a x a and consider the
parameter d = 2 while the nonlinear parameter changes. Then the
values a will be obtained (Table 3.2). From Table 3.2 we also find
that the finite domain [-a, a] in which the responses are concentrated
with probability 0.98. Observations show that the response domain
shrinks as the nonlinear parameter increases as shown in Table
3.2. and Figure 3.2 as follows:
Tab. 3.2. Values of a depending on
0.1 0.5 1 5 10 30 50 80 100
a 2.92 2.04 1.78 1.36 1.26 1.15 1.11 1.08 1.07
-4
-2
0
2
4
x -4
-2
0
2
4
x
0
0.02
0.04
p
-1
0
1
x -1
0
1
x
0
0.1
0.2
0.3
0.4
p
a. =0.1 b. =100
Fig.3.2. Graphics of PDF
of nonlinear damped system ( =0.1; 100)
12
3.2. Application examples of global-local mean square error
criterion (GLOMSEC)
3.2.1 Vibration with 3-order nonlinear damping
Consider a nonlinear damped oscillation system subjected to
white noise excitation:
3 22 ox h x x x t (3.11)
where , , ,oh are positive values. The corresponding
linearization system is as follows:
22 ox h b x x t (3.12)
where b is the linear coefficient. The mean square response of (3.12)
is:
2
2
22 2 o
x
h b
(3.13)
The coefficient b defined by the classical criterion is:
26b h x
(3.15)
By GLOMSEC is:
2,2 2
1,0
1
( ) 2 2.4119* 2
s
r
s
r
T
b b r h x Lim dr h x
s T
(3.22)
Substituting (3.22) into (3.13) gives:
2 2
2
2
2.4119
2*2.4119GL o
h h h
x
h
(3.23)
To evaluate approximate solutions we use the solution
2
ENL
x [29]. The relative errors of 2
GL
x ,
2
kd
x compared to the
solution 2
ENL
x are defined by (3.24):
13
2 2 2 2
( ) ( )2 2
*100%, *100%kd cx GL cxC GL
cx cx
x x x x
Err Err
x x
(3.24)
In Table 3.4, the results show that the solution 2
GL
x has better
accuracy than the solution 2
kd
x , in particular the largest error of
GLOMSEC is only 1.93%.
Table 3.4. The second moment of the response of the nonlinear
damping oscillator system 0.05, 1, 4oh h , and γ changes
γ
2
ENL
x 2
kd
x
( )
%
CErr
2
GL
x
( )
%
GLErr
1 0.4603 0.4342 5.61 0.4692 1.93
3 0.3058 0.2824 7.65 0.3090 1.05
5 0.2479 0.2270 8.32 0.2495 0.77
8 0.2025 0.1844 8.99 0.2032 0.35
10 0.1835 0.1667 9.16 0.1839 0.22
3.2.2. Van der Pol system to white noise
Consider Van der Pol system
2 2
ox x x x t
(3.25)
where , , , ,o are positive values, t is white noise with
unit intensity. We replace 2,g x x x x by the linear one bx ,
where b is the linear coefficient:
2ox b x x t (3.26)
14
The coefficient b defined by the classical criterion is:
2b x (3.29)
By GLOMSEC is:
1,2 2
0,0
1
( ) 0.8371
s
r
s
r
T
b b r x Lim dr x
s T
(3.34)
Mean square response 2
GL
x of Van der Pol system (3.25) by
GLOMSEC is:
2
2 2 2
2
1 1,6742
1,6742GL o
x
(3.36)
To evaluate approximate solutions, we use the Monte Carlo
simulation solution, [29]. The relative error between the approximate
solutions 2
GL
x ,
2
kd
x , compared to the simulation solutions
2
MC
x is calculated by the formula (3.24).
Table 3.5. Mean square responses of Van der Pol oscillator with
α*ε=0.2;
0 =1; =2; σ
2 changes
2
2
MC
x 2
kd
x
( )
%
CErr
2
GL
x
( )
%
GLErr
0.02 0.2081 0.1366 34.33 0.1574 24.32
0.20 0.3608 0.2791 22.46 0.3113 13.52
1.00 0.7325 0.5525 24.58 0.6095 16.79
2.00 1.0310 0.7589 26.40 0.8349 19.02
4.00 1.4540 1.0513 27.70 1.1544 20.61
15
In Table 3.5, the results 2
GL
x have better accuracy than 2
kd
x , in
which the largest error values respectively are 24.32% compared to
34.33%
3.2.3 Vibration in Duffing system to random excitation
Consider Duffing system subjected to white noise excitation:
2 32 ox hx x x t (3.37)
The notation is the same as in the previous example. The exact
solution is [29, 44]
2 2 2 4
2
2
x
2 2 4
2
4 1 1
exp
2 4
4 1 1
exp
2 4
o
c
o
h
x x x dx
x
h
x x dx
(3.39)
The equivalent liner system is:
22 ox hx x kx t (3.40)
The linearization coefficient by GLOMSEC is:
2, 2
1,0 0
2,2 2
1,0
1 1
( ) ( )
1
2.4119
s s
r
s s
r
s
r
s
r
T
k k r Lim k r dr Lim x dr
s s T
T
x Lim dr x
s T
(3.48)
Mean square response 2
GL
x of Duffing system (3.37) by
GLOMSEC:
2
2 2 41 2.4119
2* 2.4119
o oGL
x
h
(3.49)
16
The relative error between the approximate solutions 2
GL
x ,
2
kd
x with the exact one 2
xc
x defined by (3.24) and presented in
Tab. 3.6.
Table 3.6 Mean square responses of Duffing system,
1, 0.25, 1o h ; changes
2
xc
x 2
kd
x
( )
%
CErr
2
GL
x
( )
%
GLErr
0.1 0.8176 0.8054 1.49 0.8327 1.857
1.0 0.4680 0.4343 7.194 0.4692 0.263
10 0.1889 0.1667 11.768 0.1839 2.626
100 0.0650 0.0561 13.704 0.0624 4.076
The results show that the approximation determined by the
classical criterion has good accuracy with small nonlinear elastic
coefficient , the error increases to over 13% as the nonlinear elastic
coefficient increases. Accuracy of GLOMSEC criterion is better with
maximum error of 4.1%.
3.2.4. Duffing system with nonlinear damping to white noise
3.2.5. Vibration of ship
The rolling motion of the ship in random waves has been
considered by [55], [56], [57]. The equation of the ship's motion is of
the form [56-57]
2 2 ( )D t
(3.63)
The system (3.63) is replaced by the linear one
2 ( )
ec D t (3.66)
17
The linearization coefficient ec by GLOMSEC is:
3 ,2 1/ 2 2 1/2
1,0 0
1 1
( ) ( ) { } 1.49705 { }
s s
t re e e
s s
r
T
c c r Lim c r dr E Lim dr E
s s T
Mean square response by GLOMSEC is:
2/3
2 2
2 1/2
0.76415
1.49705 { }eGL GL
D D D
E E
c E
Mean square response by the classical criterion is:
2/3
2 2
2 1/2
0.7323
1.5958 { }eC C
D D D
E E
c E
Mean square response by the nonlinear equivalent linearization
method is:
2/3
2 2 0.765
ENL ENL
D
E E
The relative error between the approximate solutions 2
GL
x ,
2
C
x , compared to the nonlinear equivalent linearization method, is
calculated by the formula (3.24). We have:
( ) ( )4.314%; 0.130%C GLErr Err
The results show that GLOMSEC gives the good agreement with
the ENL solution and GLOMSEC improves the accuracy of the
classical criterion.
Conclusion of chapter 3
In Chapter 3, the GLOMSEC was applied to analyze the mean
square responses for a number of 1-order-freedom random
oscillating systems. The examples applied confirmed the outstanding
advantages proposed in the GLOMSEC. The results are presented in
[1,3,5] of List of publications of the dissertation.
18
CHAPTER 4. APPLICATION OF GLOMSEC TO THE
ANALYSIS OF RANDOM MDOF SYSTEMS
4.1. Two-degree-freedom nonlinear oscillation system
Consider the two-degree-freedom nonlinear oscillation system
described by:
332
1 1 1 21 1 1 1 11
2 33
2 1 2 2 2 22 2 2 2 1
0 ( )1 0
0 ( )0 1
x b x xx x x w ta
x x x w ta x b x x
(4.1)
where: , , , ,i i ia b (i=1, 2) are constants. 1 2( ), ( )w t w t are white
noise processes with zero mean and ( ) ( ) 2 ( )i i iE w t w t S (i=1,
2), ( ) is Delta Dirac function,
1 2,S S = const. The equivalent linear
system is:
2
1 1 1 11 11 12 1 11 12
2
2 2 2 221 1 2 22 21 2 22
( )1 0
( )0 1
e e e e
e e e e
x x x w tc c k a k
x x x w tc c a k k
(4.4)
where , ; ( , 1,2)e eij ijc k i j
are equivalent linear coefficients. The
equation error is:
( , )
e eC X K Xx x (4.5)
33
1 1 1 21
33
2 2 2 2 1
( , )
x b x x
x x
x b x x
111 12 11 12
221 22 21 22
; ; ;e eC X K
e e e e
e e e e
xc c k k
xc c k k
1
2
;X
x
x
(4.6)
To simplify the calculation we suppose that 1 2,x x
are
independent. Using the Appendix of dissertation and noting
2 1 2 1 0 ( )n mi jE x x i j
GLOMSEC gives:
19
2,211 11 1 1
1,0
1
( )
s
re e
s
r
T
c c r E x Lim dr
s T
,
2,222 22 2 2
1,0
1
( )
s
re e
s
r
T
c c r E x Lim dr
s T
2, 1,2 211 11 1 2
1, 0,0 0
1 1
( ) 3 .
s s
r re e
s s
r r
T T
k k r b E x Lim dr E x Lim dr
s T s T
2, 1,2 212 12 2 1
1, 0,0 0
1 1
( ) 3 .
s s
r re e
s s
r r
T T
k k r b E x Lim dr E x Lim dr
s T s T
2, 1,2 221 21 1 2
1, 0,0 0
1 1
( ) 3 .
s s
r re e
s s
r r
T T
k k r b E x Lim dr E x Lim dr
s T s T
2, 1,2 222 22 2 1
1, 0,0 0
1 1
( ) 3 .
s s
r re e
s s
r r
T T
k k r b E x Lim dr E x Lim dr
s T s T
(4.11)
The limits in (4.11) equal to:
2,
1,0
1
lim 2.41189
s
r
s
r
T
dr
s T
,
1,
00
1
lim 0.83706
s
r
s
r
T
dr
s T
(4.12)
To evaluate the approximate solution while the original nonlinear
system does not have an exact solution, we use the approximate
probability density function by the equivalent nonlinear method
(ENL) [77]. Table 4.1 presents the approximate mean square
responses and their relative errors compared to ENL solutions.
20
Table 4.1. The mean square responses of 1 2,x x follow 1 2 với
1 2 1 2 0 1a b S .
1
2
,
21 ENLE x
21 CE x
( )
%
CErr
21 GLE x
( )
%
GLErr
2
2 ENL
E x
22 CE x
( )
%
CErr
22 GLE x
( )
%
GLErr
0.1 1.573 1.216 22.68 1.407 10.54 1.573 1.151 26.83 1.327 15.64
1 0.496 0.422 15.07 0.488 1.59 0.496 0.370 25.51 0.419 15.50
5 0.253 0.220 13.19 0.254 0.268 0.253 0.205 19.19 0.234 7.573
10 0.194 0.171 12.07 0.197 1.533 0.194 0.162 16.48 0.186 4.178
It is seen that GLOMSEC gives good improvements for the
accuracy of approximate solutions when the nonlinearity increases.
4.2. Nonlinear oscillation systems subjected to color noise
The introduction of the 1-DOF system subjected to color noise
excitation in Chapter 4 is because the color noise random process is
described as a white noise process passing through a second-order
differential filter. The oscillation equation is solved with the filter
equation so it can be considered as a system of multi-degrees-of-
freedom.
4.2.1. Extend GLOMSEC to the case of random color noise
excitation
4.2.2. Duffing system to color noise excitation
Consider Duffing system to random color noise excitation:
2 3( )z z z z f (4.41)
where f is the color noise random process
2 2
f ff f f w
. (4.22)
The nonlinear system is replaced by the linear one
x cx kx f (4.27)
21
The linear coefficients by GLOMSEC are:
2 2 22.41189 , .xk c (4.45)
By the classical criterion:
2 2 23 ,xk c
(4.46)
By the Energy criterion:
2 2 22.5 ,xk c
(4.50)
The relative errors between approximate solutions 2,x GL ,
2
,x C
compared to 2,x E are presented in Table 4.3. The results show the
solution 2,x GL is much better accurate then the solution
2
,x C , namely
with the bigest errors 2.392% compared to 11.398%, respectively.
Tab. 4.3. Mean square response with 2 2f, ,S, , 1
changes.
2
,x E
2
,x C %CErr
2
,x GL %GLErr
0.1
1
10
100
1.86038
0.66376
0.16687
0.03720
1.75024
0.60015
0.14855
0.03296
5.920
9.583
10.979
11.398
1.88195
0.67688
0.17072
0.03809
1.159
1.977
2.307
2.392
4.2.3. Duffing system with nonlinear damping to color noise
excitation
Consider Duffing syste

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