MINISTRY OF EDUCTION AND TRAINING VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
GRADUATE UNIVERSITY OF SCIENCE AND TECHNOLOGY
-----------------------------
Nguyen Cao Thang
RESEARCH ON NON-LINEAR RANDOM VIBRATION BY THE GLOBAL – LOCAL MEAN SQUARE ERROR CRITERION
Major: Engineering Mechanics
Code: 9 52 01 01
SUMMARY OF MECHANICAL ENGINEERING AND ENGINEERING MECHANICS DOCTORAL DISSERTATION
HANOI - 2019
The dissertation has been completed at: Graduate University of Science and Technology – Vietnam Academy of Science and Technology
Supervisors : Dr. Luu Xuan Hung
Prof. Dr. Sc. Nguyen Dong Anh
Reviewer 1: …
Reviewer 2: …
Reviewer 3: ….
Dissertation is defended at Graduate University of Science and Technology – Vietnam Academy of Science and Technology at …, on date … month … 2019.
Hardcopy of the dissertation can be found at:
- Library of Graduate University of Science and Technology
- Vietnam National Library
1
INTRODUCTION
1. Rationale of the dissertation
The analysis, design and control of vibration play an important role in improving effectiveness and performance of structures, vehicles and engines. In recent years, the multi degree of freedom system is used for most of engineering applications. Accordingly, it is necessary to develop Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system, based on dual concept to develop Global local mean square error criterion (GLOMSEC) for MDOF nonlinear system.
2. Object of the dissertation
Apply dual concept to solve the limited area [-rx , + rx] in the local mean square error criterion (LOMSEC). By that way, propose the global – local mean square error criterion (GLOMSEC) for Gaussian equivalent linearization method (GEL) for randomly excited MDOF nonlinear system subjected to white noise or color noise excitation. The mean square of response solution will be concentrated to evaluate the accuracy of the proposed criterion by comparison with exact solution or other accepted solutions.
3. Research methodology
In the dissertation, analyse method, numerical method, Monte – Carlo Simulation method are considered. The analyse method is considered to create the error criterion: base on dual concept in analyse response of nonlinear systems (consider two different approaches to a problem) to obtain the linearized coefficients by close analysis method. The numerical method is considered to program by Matlab software to compute and simulate random nonlinear vibration MDOF systems. The Monte Carlo simulation is considered to find simulation solution for determination the accuracy of linearization method.
4. Scientific and practical application
- Develop Gaussian equivalent linearization method (GEL) – one of most popular method used in Random vibration. Particularly, the
2
Global Local Mean Square Error Criterion – GLOMSEC is generalized for MDOF random nonlinear system.
- Develop close equation system to determine mean square of responses. Investigate and evaluate the accuracy of proposed criterion for MDOF nonlinear random systems subjected to white noise or color process.
- The results of the dissertation are applied to analyse technical nonlinear random systems.
5. Structure of the dissertation
The structure of the dissertation includes: the introduction, 4 chapters, the conclusions, a list of publications, the references and the appendix.
CHAPTER 1. INTRODUCTION TO PROBABILITY THEORY AND SOME METHODS ANALYSING NON-LINEAR RANDOM VIBRATION
1.1. Random variable and its probabilistic properties
f M P M
)
(
)
lim ( n
Define probability of a random event [29], [69]: Perform n experiments, if the outcome M occurs m times, than probability of outcome M, denote P(M) is the limitation of frequence f(M) = m/n when the number of experiments n increases to infinity:
(1.1) Random variable X is a quantity that links each outcome r of an experiment with a real number X(r) satisfies:
a) Set X x is called an event M for each real number x,
b) probability of event X = equal zero:
PX = = 0 (1.2)
The cumulative distribution function (cfd) of the random variable X is defined for any real number x by:
3
F(x) = P[X x] (1.3)
1.2 Stochastic processes
There are definitions of: Probability density function; High order moment; Mathematical expectation; Mean square; Variance; Auto- correlation; 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]:
x
2
hx
x
( , ) g x x
t ( )
2 0
(2.10)
( , )
( )t
respectively; h is damping coefficient, Where, x , x and x are displacement, velocity and acceleration, g x x is a nonlinear
0h ,
is Gaussian white noise excitation with intensity
g x x
( , ) 0
0 is the natural frequency for
. The function, 2 ; equivalent linearization equation of (2.10) is as follows:
x
2
hx
x bx
kx
t ( )
2 0
(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]:
S
( , ) g x x
bx
kx
2
k d
min b k ,
(2.14)
Hence:
0;
0
S kd b
S kd k
xx
0
(2.15)
,
,
Assuming that the solution is a random process, the response x , x should be independent, that is , solving the system of equations (2.15) we obtain
b
k
2
2
xg x x x
xg x x x
(2.17) ,
6
2
2
x
,
x
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 .
d) Repeat b) and c) until the specified accuracy is reached.
We consider the nonlinear system with multi degrees of freedom under random excitation
,
,
(2.20)
Mx + Cx + Kx + Φ x, x x = Q t
vectors, respectively where x , x , x are displacement, velocity and acceleration are ,
M
K
m
C
k
,
ij
ij
c ij
n n
n n
n n
,
mass, damping and stiffness matrices;
Q t
vector,
intensity and spectral density matrix
cross spectral density function of
Φ x, x x - nonlinear function is Gaussian white noise excitation vector with zero S S ij iQ and
ijS is where n n jQ . The equivalent
e
e
,t
(2.21) linearization system is as follows: e M + M q C + C q K + K q Q
e
e
e
where
M , C , K are the equivalent matrices of mass, damping
x t
q t that this is only an approximation to in the original nonlinear equation (2.20). The error between the system (2.20) and the system (2.21) is
and stiffness. In equation (2.21) we use the notation to show
e
e
e
(2.22)
e Φ q, q, q M q + C q + K q
.
7
The error e satisfies the Mean Square Error Criterion upon
e
e
e M , C , K :
(2.24)
T e e
min .
e
e
e
E
M ,C , K
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:
T
T
e
e
E
(2.25)
e zz M C K
T zΦ z
E
, 2.2. Some improved equivalent linearization criteria
the accuracy of improve to For decades, many studies on equivalent linearization criteria have the equivalent been proposed 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:
x
2
hx
( , ) x g x x
t ( )
2 0
(2.47)
Where, the notation is defined as above. The equivalent linearization equation of (2.47) takes the form:
x
2
hx
x
x
x
t ( )
2 0
(2.48)
where λ, μ are linearization coefficients. The error between (2.47) và (2.48) is:
x
x
, xxe
, xxg
(2.49)
2
The classical criterion gives [29, 44]
e
x x P x x dxdx ( , ( ,
)
)
min ,
(2.51)
P x x is the probability density function (PDF) of x and ( , )
x :
Where
8
)
(
)
(
,
.
g x x x , 2
g x x x , 2
x
x
(2.53)
,
x
x
0
0
2
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]:
e
x x P x x dxdx ( , ( ,
)
)
min ,
x
x
0
0
0
(2.54)
x and
0, xx Where, into transformed x 0,x r r x x 0 standard deviations of x and x :
r
r
x
x
2
2
the non-dimension one by where r is a positive value, are positive values. The integration (2.54) is introducing x are
[
e
(
x x ,
)]
e
(
x x P x x dxdx ,
(
)
)
,
m in ,
r
r
x
x
(2.55)
r
r
x
x
where [.] denotes:
[ . ]
( . )
P x x d x d x )
(
,
r
r
x
x
(2.56)
)
We have analogously:
r ( )
,
r ( )
.
g x x x ( , 2
) g x x x ( , 2
x
x
(2.57)
( ), r
r ( )
It
is seen from (2.57) the local equivalent linearization coefficients (LOMSEC) are functions of r, .Using the dual viewpoint, we can propose that r change across the non- , can be chosen as the negative domain and the EL coefficients following[24]:
9
s
r ( )
r dr ( )
,
Lim s
1 s
0
s
r ( )
r dr ( )
.
Lim s
1 s
0
(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:
z
zg
tf
T
z
z
,...,
2
z z , 1
n
(2.61)
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:
ze
zgz
tf
(2.62)
z
Az
Az
zg
ze
(2.64) Introduce new linearization terms into (2.62): tf
A
ija
where is matrix n×n. Let y is a stationary solution of:
y
Ay
0 tf
(2.65)
From (2.64) and (2.65) we have:
Ay
ye
yg
(2.66)
y
y
1
0 n
yn
0 y 1
Denote p(y) probability density function (PDF) of y of (2.65). According to the LOMSEC we have:
e
dyypy
i,j = 1,…,n
e
y
2 i
2 i
y
n
0 y 1 y
1
0 n
yn
,min a ij
T
T
It gives:
A
yy
yyg
1
(2.72)
10
a
a
(
,
y
,.....
y
)
ij
ij
0 y 1
0 2
0 n
0 n
....
a
(
,
y
,.....
y
)
.....
dy
ij
0 y 1
0 2
0 n
0 dy dy 1
0 2
0 n
Lim ,..... y
0 y y , 1
0 2
0 n
.....
y
1 0 0 y y 2 1
0 n
0 y 1 0
y 0
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
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
2
2
We consider a nonlinear damped oscillation system subjected to white noise excitation:
x
1
x
x
x
x
d
t
(3.6)
2
2
2
2
The exact PDF two-dimensional probability density function of the system [29, 44]:
C
exp
x
x
0.5
x
x
p x x ,
2
d
(3.7)
11
Prob
a x a
where C is the normalized constant. If is
aa,
a
will be detemined by: chosen then the domain
Prob
a
x
,
a
p x x dx dx
a
(3.8)
Prob
0.98
and consider the Suppose we choose a x a 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
4
0.04 p 0.02
1
2
0
0.4 0.3 p 0.2 0.1 0
0
0
x
-4 -4
x
-1 -1
-2 -2
-2
0 0
0 0
-1
x x
2 2
x x
-4
1 1
4 4
a 2.92 2.04 1.78 1.36 1.26 1.15 1.11 1.08 1.07
a. =0.1 b. =100
of nonlinear damped system (=0.1; 100) Fig.3.2. Graphics of PDF
12
3.2. Application examples of global-local mean square error criterion (GLOMSEC)
3.2.1 Vibration with 3-order nonlinear damping
3
Consider a nonlinear damped oscillation system subjected to white noise excitation:
x
2
x
x
t
2 o
(3.11)
h x
h are positive values. The corresponding
,
,
x
2
x
,o where linearization system is as follows:
h b x
t
2 o
(3.12)
2
x
2 h b
2 2
2 o
where b is the linear coefficient. The mean square response of (3.12) is:
(3.13)
The coefficient b defined by the classical criterion is:
b
x h
6
2
(3.15)
s
r
2
2
By GLOMSEC is:
b r ( )
h
x
b
dr
2.4119 * 2
h
x
Lim s
r
T 1 2, s T 1, 0
2 (3.22)
2
h
h
2
Substituting (3.22) into (3.13) gives:
x
GL
2 * 2.4119
2 h 2.4119 2 h o
(3.23)
2
2
2
To evaluate approximate solutions we use
x
x
x
kd
GL
2
ENL solution
[29]. The relative errors of , the solution compared to the
x
ENL
are defined by (3.24):
13
2
2
2
2
x
x
x
x
cx
cx
*100%,
*100%
Err (
C
)
Err (
GL
)
2
2
kd x
GL x
cx
cx
2
(3.24)
x
GL
2
In Table 3.4, the results show that the solution has better
x
kd
accuracy than the solution , in particular the largest error of
1,
0.05,
h 4
h
GLOMSEC is only 1.93%.
o
(
)
Err (
GL
)
2
2
2
x
x
x
Table 3.4. The second moment of the response of the nonlinear damping oscillator system , and γ changes
ENL
kd
GL
CErr %
%
γ
0.4603 0.4342 5.61 0.4692 1.93 1
0.3058 0.2824 7.65 0.3090 1.05 3
0.2479 0.2270 8.32 0.2495 0.77 5
0.2025 0.1844 8.99 0.2032 0.35 8
10 0.1835 0.1667 9.16 0.1839 0.22
3.2.2. Van der Pol system to white noise
2
Consider Van der Pol system
x
x
x
t
2 o
x
(3.25)
are positive values,
,
,
,
,o
2
t by the linear one bx ,
g x x
x x
where is white noise with
unit intensity. We replace , where b is the linear coefficient:
x
x
b x
t
2 o
(3.26)
14
2
b
x
(3.29)
The coefficient b defined by the classical criterion is:
s
r
2
2
By GLOMSEC is:
b
b r ( )
x
dr
0.8371
x
r
T 1 1, s T 0, 0
2
(3.34) Lim s
x
GL
Mean square response of Van der Pol system (3.25) by
1
2
2
x
GLOMSEC is:
GL
1,6742
2 1,6742 2 o
2
2
2
(3.36)
x
x
kd
GL
2
To evaluate approximate solutions, we use the Monte Carlo simulation solution, [29]. The relative error between the approximate , compared to the simulation solutions solutions ,
x
MC
is calculated by the formula (3.24).
0 =1;=2; σ2 changes
(
)
Err (
GL
)
2
2
2
x
x
x
2
MC
kd
GL
CErr %
%
Table 3.5. Mean square responses of Van der Pol oscillator with α*ε=0.2;
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
2
2
x
x
GL
kd
In Table 3.5, the results have better accuracy than , in
which the largest error values respectively are 24.32% compared to 34.33%
3.2.3 Vibration in Duffing system to random excitation
3
Consider Duffing system subjected to white noise excitation:
x
2
hx
x
x
t
2 o
(3.37)
2
2
4
x
exp
x
x
dx
2 o
h 4 2
1 2
1 4
2
x
The notation is the same as in the previous example. The exact solution is [29, 44]
c x
2
4
exp
x
x
dx
2 o
4 h 2
1 2
1 4
(3.39)
x
2
hx
x
kx
The equivalent liner system is:
t
2 o
(3.40)
s
s
r
2
k
k r ( )
k r dr ( )
x
dr
Lim s
Lim s
1 s
1 s
T 2, T 1,
r
0
0
s
r
2
2
x
dr
2.4119
x
Lim s
r
T 1 2, s T 1, 0
The linearization coefficient by GLOMSEC is:
2
(3.48)
x
GL
Mean square response of Duffing system (3.37) by
2
x
2.4119
GLOMSEC:
4 o
2 o
GL
1 2 * 2.4119
2 h
(3.49)
16
2
x
GL
2
2
The relative error between the approximate solutions ,
x
x
xc
kd Tab. 3.6.
with the exact one defined by (3.24) and presented in
0.25,
o
; changes 1
(
)
Err (
GL
)
2
2
2
x
x
x
xc
kd
GL
CErr %
%
responses of Duffing system, Table 3.6 Mean square h 1,
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
2
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]
( )D t
2
(3.63)
The system (3.63) is replaced by the linear one
ec
D t ( )
2
(3.66)
17
ec by GLOMSEC is:
s
s
r
e
2 1/ 2
c
e c r ( )
{ } E
e c r dr ( )
dr
1.49705
E {
2 1/ 2 }
Lim s
Lim s
1 s
r
0
T 1 3 , t s T 1, 0
Mean square response by GLOMSEC is:
2/3
E
E
0.76415
2 1/ 2
2
2
GL
GL
D e c
D { } E
1.49705
D
The linearization coefficient
2/3
E
E
0.7323
2 1/2
2
2
C
C
D e c
D { } E
1.5958
D
Mean square response by the classical criterion is:
2/3
E
E
0.765
2
2
ENL
ENL
D
2
Mean square response by the nonlinear equivalent linearization method is:
x
GL
2
The relative error between the approximate solutions ,
x
C
, compared to the nonlinear equivalent linearization method, is
4.314%;
0.130%
Err (
C
)
Err (
GL
)
calculated by the formula (3.24). We have:
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
3
3 x 1 1
x 2
3
1 0
1 0 0 1
x 1 x 2
0 1 2
x 1 x 2
x 1 x 2
w t ( ) 1 w t ( ) 2
a 2 2
2 1 a
3 x 2 22
b x 1 b x 2
x 1
(4.1)
,
a b , ,
(i=1, 2) are constants.
i
i
i
1 E w t w t ( ) ( i
2 2 ( ) S i
i
( ), w t w t ( ) are white (i=1, ) ,S S = const. The equivalent linear
1
2
, where: noise processes with zero mean and 2), ( ) is Delta Dirac function, system is:
1 0
e c 11
x 1 x
0 1
2
x 1 x 2
w t ( ) 1 w t ( ) 2
1 e c 21
e c 12 2
1
e c 22
e k 11 e 21
a k 2 2
e 12 e k 22
2 1 a k
x 1 x 2
(4.4)
,
k
;
i j ( ,
1,2)
Consider the two-degree-freedom nonlinear oscillation system described by:
e ij
e c ij equation error is:
where are equivalent linear coefficients. The
e
e
x x ( ,
)
(4.5)
C X K X
3
3 x 1 1
x 2
( , ) x x
3
2
1
3 x 2 22
b x 1 b x 2
x 1
e
e
(4.6)
X
C
;
X
;
K
;
x 1 x
k
x 1 x 2
2
e c 11 e c 21
e c 12 e c 22
e k 11 e 21
e k 12 e k 22
; ,x x 1 2
1
m
n
i 0 (
)
j
1 2 x j
To simplify that
are the calculation we suppose independent. Using the Appendix of dissertation and noting GLOMSEC gives: 2 E x i
19
s
r
r ( )
dr
e c 11
e c 11
1
2 E x Lim 1
s
r
T 1 2, s T 1, 0
s
r
c
c
r ( )
dr
e 22
e 22
2
2 E x Lim 2
s
r
T 1 2, s T 1, 0
s
s
r
r
r ( )
dr
3
dr
.
e k 11
e k 11
2 b E x Lim 1
s
2 E x Lim 2
s
r
r
T 1 2, s T 1, 0
T 1 1, s T 0, 0
s
s
r
r
r ( )
dr
3
dr
.
e k 12
e k 12
2 b E x Lim 2
s
2 E x Lim 1
s
r
r
T 1 2, s T 1, 0
T 1 1, s T 0, 0
s
s
r
r
k
k
r ( )
dr
3
dr
.
e 21
e 21
2 b E x Lim 1
s
2 E x Lim 2
s
r
r
T 1 2, s T 1, 0
T 1 1, s T 0, 0
s
s
r
r
k
k
r ( )
dr
3
dr
.
e 22
e 22
2 b E x Lim 2
s
2 E x Lim 1
s
r
r
T 1 2, s T 1, 0
T 1 1, s T 0, 0
,
(4.11)
s
s
r
r
dr
0.83706
dr
2.41189
lim s
lim s
r
r
T 1 2, s T 1, 0
T 1 1, s T 0 0
The limits in (4.11) equal to:
(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
với
2
1
,x x follow 1
2
a b S
. 0 1
2
1
)
(
)
)
(
Err (
GL
)
2 E x
2 E x
2 E x 2 C
2 ENL
1 ENL
2 E x 1 GL
2 E x 2 GL
CErr %
CErr ( %
GLErr %
2 E x 1 C
%
, 1 2
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
Table 4.1. The mean square responses of 1 2
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
3
z
z
2 (
z
z
)
f
Consider Duffing system to random color noise excitation:
(4.41)
f
f
w
f
where f is the color noise random process
2 2 f f
cx
kx
f
. (4.22)
The nonlinear system is replaced by the linear one x (4.27)
21
The linear coefficients by GLOMSEC are:
k
2
2.41189
,
c
.
2 2 x
(4.45)
By the classical criterion:
k
2
3
2 2
c
,x
(4.46)
2
By the Energy criterion:
k
2.5
c
2 2 ,x
2
2
(4.50)
2
The relative errors between approximate solutions ,x GL ,
2
2
,x C ,x E are presented in Table 4.3. The results show the ,x C , namely
,x GL is much better accurate then the solution solution with the bigest errors 2.392% compared to 11.398%, respectively.
2
, ,S,
,
1
compared to
2 f
changes.
2
2
2
,x C
,x E
%GLErr
,x GL
%CErr
Tab. 4.3. Mean square response with
0.1 1.86038 1.75024 5.920 1.88195 1.159
0.66376 0.60015 9.583 0.67688 1 1.977
0.16687 0.14855 10.979 0.17072 10 2.307
100 0.03720 0.03296 11.398 0.03809 2.392
4.2.3. Duffing system with nonlinear damping to color noise excitation
3
3
z
z z
z
f
Consider Duffing system with third order nonlinear damping:
(4.51)
where f is the color noise defined by (4.22). The nonlinear system is replaced by the equivalent linear one (4.27).
The linear coefficients by GLOMSEC are:
22
k
2
2.41189
,
c
2.41189
.
2 x
2 x
(4.54)
By the classical criterion:
k
2
,
c
.
2 3 x
2 3 x
2
2
(4.55)
2
The relative errors between approximate solutions ,x GL ,
2
compared to Monte Carlo solution
,x C ,x MC are presented in Tab. 4.4. ,x GL is much better accurate then the
2
The results show the solution
,x C obtained by the classical criterion.
, , S,
1
solution
2 f
, ,
Table 4.4. Mean square response with
2
2
2
,x C
,x MC
%GLErr
,x GL
%CErr
changes.
0.1 2.62060 2.14097 18.302 2.46218 6.045
1 0.82554 0.65974 20.084 0.76111 7.805
10 0.22073 0.16413 25.642 0.19043 13.727
100 0.05138 0.03502 30.841 0.04067 20.845
Conclusion chapter 4
In chapter 4, the global-local mean square error criterion (GLOMSEC) was applied to analyze the mean square responses for a number of random nonlinear MDOF oscillation systems and non- oscillation systems subjected to random narrow band color noise. The results in chapter 4 are presented in [1,2,4,6] in the List of publications of dissertation.
23
CONCLUSIONS AND RECOMMENDATIONS
1. Conclusion of the dissertation
General conclusions
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.
New contributions of the dissertation
- Application of the above criterion to analyze mean square responses for a number of one- and multi-degree-of-freedom nonlinear oscillation systems subjected to white noise random excitation
- Developed the global-local mean square error criterion (GLOMSEC) for one- and multi-degree-of-freedom nonlinear oscillation systems subjected to white noise random excitation
- Developed the global-local mean square error criterion (GLOMSEC) for nonlinear oscillation systems subjected to narrow- band random color noise.
- The applied examples have confirmed the outstanding advantage of GLOMSEC and that GLOMSEC gives approximate solutions with small errors when analyzing the mean square responses of nonlinear random oscillating systems with medium and large nonlinearities.
2. Further research directions
The research results of the dissertation can be developed for multi-degree-of-freedom nonlinear systems subjected to color noise excitation, and/with simultaneous harmonic and parametric excitation, and for mechatronic systems.
LIST OF PUBLICATIONS OF DISSERTATION
1. Luu Xuan Hung, Nguyen Cao Thang, Performance analysis of global-local mean square error criterion of stochastic linearization for nonlinear oscillators, Vietnam Journal of Mechanics, 2019, Vol. 41, No. 1, pp.1-15, DOI: https://doi.org/10.15625/0866-7136/12015.
2. Luu Xuan Hung, Nguyen Cao Thang, A new stochastic linearization technique for nonlinear oscillators under colored noise excitation, 10th National Conference on Mechanics, 2017, Vol. 1, pp.211-220, Hanoi.
3. Luu Xuan Hung, Nguyen Cao Thang, Analysis of randomly excited nonlinear oscillators by the global-local mean square error criterion, 4th International Conference on Engineering Mechanics and Automation (ICEMA 4), 2016, pp.197-204, Hanoi.
4. Luu Xuan Hung, Nguyen Cao Thang, Extension of Global-local mean square error criterion to nonlinear oscillators under narrow band excitation, J. of Multidisciplinary Engineering Science Technology, 2016¸ 3, Iss. 11, pp.6000-6005 (International journal).
linearization for
5. Luu Xuan Hung, Nguyen Cao Thang, A new improvement of Gaussian equivalent stochastic nonlinear oscillators, 2nd National Conference on Mechanical Engineering and Automation, 2016, pp.274-280, Hanoi.
6. N.D. Anh, L.X. Hung, L.D. Viet, N.C. Thang, Global-local mean square error criterion for equivalent linearization of nonlinear systems under random excitation, Acta Mechanica, 2015, 226, N9, pp.3011-3029, DOI: 10.1007/s00707-015-1332-4 (SCI Journal).
