VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY
INSTITUTE OF MECHANICS
o0o

NGUYEN NGOC LINH

ANALYSING NONLINEAR RANDOM VIBRATION BY
THE EQUIVALENT LINEARIZATION METHOD
Major: Engineering Mechanics
Code: 62 52 01 01
SUMMARY OF DOCTORAL THESIS HA NOI – 2015
Random vibrations are popular in engineering problems such
as structures subjected to wind or wave load, bearings or shafts in
vechicle components. Under such a loading case, the corresponding
vibration model is based on statistical and stochastic process theory.
The analysing of that model often leads to the construction and
solving the stochastic nonlinear differential equations. When the
excitation loading varies arbitrarily in time, the corresponding
response will also be arbitrary in time. Therefore, the response
process and its characteristic properties can be expressed by
averaging quantities in probability sense. The requirement of
analytical expression of response is very important because of two
main reasons, firstly it is allowed to confirm the accuracy of the
model in comparing with the measured data, secondly we can
estimate the adjustable and controlable paramaters in preliminary,
exact or check design stages. However, the exact analytical solutions
have been found for only a few nonlinear random vibration
problems. Although numerical methods make nonlinear problems to
be solvable, but a nonlinear system with numerical solution is
unlikely to satisfies the analysing requirements in practice. For
example, it takes a long time to construct exact calculating model not
only of a multi-degree-of-freedom system but also of single-degree-
of-freedom system with many input parameters. The approximate
analytical methods are therefore important in study of stochastic
nonlinear dynamical systems.
Among the approximate analytical methods, the equivalent
linearization method is one of the most popular one because of its 2
simplicity, it can be applied to both single-degree-of-freedom and

experiment. A stochastic process is a family of random variables
which depends on time. The main characteristics of a stochastic
process are evaluated by deterministic quatities such as probability
density function, mean and mean square values, variance, correlation
function and power spectral density.
The probability density function of a random process


x t
is







, , /
p x t F x t x t
  
(1.1)
where


,
F x t
is distribution function.


,

;
E x t x t x t p x t dx


 

(1.4)
The variance is defined as

   
 


   
2
2
2 2
x x x
D E x t m t x t x t

    
(1.5)
The covariance is



11 1 2 1 2 1 2
,
xx
D t t x x x x


cosr


(1.8) 4
In application, both the correlation and squared correlation
coefficients r và r
2
are used as a measure of the linear correlation
strenght between X and Y, they belong to effect size and are
classified into weak, medium and strong levels (Cohen, 1988).
1.2.2 Special random processes
Several special random processes are usually used in random
vibration such as stationary, Wiener, Markov processes are
introduced. The most frequently used process, Gaussian white noise,
has probability density function

 
2
2
1
exp
2
2
x
p x


 
(1.10)
where


x
i
a and


x
ij
K are drift and diffusion coefficients.
1.4 Random vibration subjected to Gaussian white noise
Consider single-degree-of-freedom illustrated as figure 1.1





,
tt tt pt
mx b x k x g x x u t
   
  
(1.11)
where m is mass, b
tt
is linear
damping coefficient, k

 
, when excitation 5
is Gaussian white noise then






/
f t u t m t

 

, (1.11) can be
expressed as bellow





2
2 ,
o
x hx x g x x t
 
   


  
(1.13)
where


g x
represents nonlinear restoring force, and




,
f H x x

is
the function of Hamiltonian total energy.
- when (1.12) has nonlinear restoring force and linear damping force





2
0
2
x hx x g x t
 
   


,
g x x

by coressponding linear ones,


,
g x x bx kx
 
 
, then one gets the linear system







2
2
o
x h b x k x t
 
    

 
(2.1)
where b, k are called the equivalent linearization coefficients.
Based on the equation error between (1.12) and (2.1)


,
xg x x xg x x
b k
x x
 
 

(2.4)
Conclusions of chapter 2
In this chapter the stochastic equivalent linearization method and
several developments of this method such as minimum potential
energy, regulated equivalent linearization, equivalent linearization
with non-Gauss distribution, partial equivalent linearization criteria
are introduced. 7
CHAPTER 3. THE DUAL CRITERION OF STOCHASTIC
EQUIVALENT LINEARIZATION METHOD
3.1. Basic idea of the general weighted dual criterion
Based on the dual approach, N.D.Anh (2010) suggested the criterion






2 2 2
3 3 3 3
1 2

1 1
min
2 2
dn dn
dn dn dn dn
k
S A k B k B A


    
(3.3)
Using minimum condition in (3.3), one gets

2
1
2
dn
AB
k
B



(3.4)

2
dn






, then their correlation coefficient is 8

1/2 1/2
2 2
AB
r
A B
 (3.8)
Combining (3.6) and (3.8) yields

2
r


(3.9)
Hence µ is called the linear dependence level.
3.2.3 Geometric meaning of the dual criterion
Consider two dimension Hilbert space H of random functions




,
u x v x
with zero mean, finite second moment and the

 



(3.10)
where




,
u x v x
are represented by vectors u, v. Following
perpendicular projection theorem, there exists only projection vector
,
p
u

which sastifies

,
inf
p
u u u v

  
(3.11)
Vector
,
p

Two vector u and v are linearly independent if there exists a constant
c,
c R

such that
u cv

, in this case u and v fall in the same line
and


cos 1


. In case u and v fall on different lines, we only
express the linear relationship between v and the projection vector
p
u
of a projection from u onto v. In other words, vector u is replaced
by projection vector

p td
u k v

(3.14)
where
td
k
is equivalent coefficient. In equivalent linearization
replacement following mean square criterion, the squared distance

2 2
,
1 1
min
2 2
dn dn
dn dn dn dn
k
S A k B k B A


    
(3.17)

a)
0 / 2
 
  b) / 2
  
 

Figure 3.1 Vector projection of the dual criterion
As shown in (3.17), the forward replacement
dn
A k B

is vector
projection from A onto B and the backward one
dn dn
k B A

k B A A A r

 

   (3.18)
trong đó
1 1
2 2
tb dn
A A A

  (3.19)
Từ (3.40) ta có



cos
dn dn dn
A k B k B r
 
  (3.20)
3.2.4 Application of the dual criterion in second moment
analysing of random vibration
Consider random vibration (1.12) with the polynomial nonlinearity

     
1 1
, , ,
M N
i j

component of damping force. For linearizing
problem we can apply the superposition principle





   
1 1 1 1
, ; ,
, ; ,
i i j j
M M N N
i i j j
i i j j
g x x b x g x x k x
g x x bx b x g x x kx k x
   
 
   
   
  
   
(3.22)
The corresponding linear equation is







, ,
j j j
A g x x B x
 

determines

 
 
 
 
2
2
2 2 2 2
,
,
;
, ,
j
i
i j
i j
xg x x
xg x x
g x x x g x x x
 
 

 



2 2
1 1
,
,
1 1
;
2 2
M N
j
i
i j
i j
xg x x
xg x x
b k
x x
 
 
   
   
 
   
 
   
   
 

 





2
o
x b x x t
  
    

 
(3.28)
where b is linearization coefficient. The linear dependence level is

1/ 3


(3.29)
Table 3.1 Mean square response of Van der Pol oscillator with α=0.2; 

=1; =2; σ
2
varies
2


2
MC
x
2

, , ,
o
h
  
are real positive constant. The equivalent
linearization equation is





2
2
o
x h b x x t
 
   

 
(3.31)
where b is linearization coefficient. The linear dependence level is

3/ 5


(3.32)
Table 3.2 Mean square response of cubic damping oscillator with
0.05, 1, 4
o
h h

(3.33)
where
3
, ,
h c

are real positive constant. The equivalent linearization
equation is





1
2
x hx c k x t

   

 
(3.34)
where k is linearization coefficient. The linear dependence level is

3/ 5


(3.35)
Table 3.3 Response of Duffing oscillator with
1
1, 0.5, 2, 1

3.3.4 Oscillator with nonlinear damping and restoring





2 2 2 4 2 3
4 / 2 / 2 / 4
o o
x h x x x x x x t
    
     

  
(3.36) 13
where
, , ,
o
h
  
are real positive constant. The equivalent
linearization equation is






2
kd
x
error (%)
2
dn
x
error (%)
0.1 0.7677 0.6659 13.26 0.8176 6.50
1 0.4652 0.3816 17.98 0.4808 3.35
10 0.1924 0.1509 21.55 0.1928 0.20
100 0.0666 0.0513 22.97 0.0659 1.16
3.3.5 Lutes – Sarkani oscillator

 
sgn ( )
a
x x x f t

 

(3.39)
where a, γ are real positive constant,
( )
f t
is Gaussian white noise
with
o
S const


1; 1
S

 
and a varies
a
2
x
c
x
2
kd
x
error (%)
2
dn
x
error (%) µ
0.039 1.901 1.537 19.18 2.661 39.94 0.67
0.222 1.564 1.395 10.81 1.880 20.22 0.80
0.424 1.332 1.265 5.03 1.446 8.57 0.90
1.000 1.000 1.000 0.00 1.000 0.00 1.00 14
1.775 0.810 0.779 3.83 0.835 3.00 0.90
2.200 0.751 0.695 7.43 0.779 3.74 0.80
2.715 0.699 0.614 12.06 0.716 2.54 0.67
3.000 0.676 0.577 14.59 0.683 1.06 0.60
3.437 0.647 0.528 18.34 0.634 1.96 0.50

and
2/3 1

 
.
CHƯƠNG 4. THE WEIGHTED DUAL CRITERION OF
STOCHASTIC EQUIVALENT LINEARIZATION METHOD
4.1 The weighted dual criterion có weighting parameter
4.1.1 Concept of the weighted dual criterion
For evaluating the different influence of replacements, consider the
criterion

     
2 2
,
1 min
ts ts
ts ts ts ts
k
S p A k B p k B A


     
(4.1) 15
where p is normalised weighting parameter with property

0 1

1
ts
p
p






(4.4)
where the linear dependence level µ has the same form as in (3.6)

2
2 2
AB
A B


(4.5)
Those above calculations are implemented under the assumptions
that
2 2
0, 0
A B
 
and

1
p



1
tb ts
A p A p A

   (4.9)



cos
ts ts
A k B
 
 (4.10)
The main geometric properties of the weighted dual criterion are
(a). Vector k
ts
B is projection of a vector projection A
tb
onto vector B,
with the length of vector A
tb
equals weighted mean of lenght of
vectors A and λ
ts
A.
(b). Vector λ
ts
A is projection of perpendicular vector projection k

2
cos 0


,
0


(the weakest linear dependence level),
then A and B are perpendicular,
0
ts


,
0
ts
k

(Figure 4.1b).

a)
1, 0
 
 
and
 

b)
0, / 2

p

. The criterion (4.1) now has only

 
 
2
1
,
min
ts ts
ts ts
ts p
k
S k B A



   (4.11)
From the minimum condition in (4.11), one has
0
ts ts
k

 
from
which
0
AB



 

1 1
p
   
 
(4.12)



0 1
p

(4.13)
- Medium linear dependence level,
1/ 3 2 / 3

 

1/ 2
p

 (4.14)

     
(4.18)
Next, we consider the interpolation problem applied to nonlinear
system which has exact solution to find
1

and
2

. The Lutes
Sarkani oscillator governed by (3.39) can be chosen due to following
reasons: It represents a class of nonlinear system and has exact
solution; It has continuous linear dependence level. The result is

1 2
1.162 6 / 5; 1.514 3/ 2
 
    
 
(4.19)
At discontinuity point µ = 1/3, the equivalent linearization
coefficient is considered the arithmetic mean

 
1/3
(1/3) (1/3)
2 2
1 1 1 3 11
2 2 2 5 20
AB AB

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
m
p
(m)
p =1/2
(m)
p =
p =0
p =27/49
(m)
(m)
(m)
p =
(m) -6m/5+1
-3m/2+3/2

Figure 4.3 The graph of the weighting coefficient p(µ)
Table 4.1 Weighting parameter and linearization coefficient of the weighted dual criterion
N
Linear
dependence

2 Weak
1/ 3


27
49
p 

1/3
2
11
20
AB
k
B

3 Medium


1/ 3, 2 / 3



1
2
p


2
1

 


 
19
4.1.3 Other properties of the weighted dual criterion
0 0.1 0.2 1/3 0.4 0.5 0.6 2/3 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1d
k
(m)
m

Figure 4.4 Ratio


 
 

 
  
(4.21)





2 2
,
,
1
1
;
1 1
j
i
j
i
i j
i i j j
xg x x
xg x x
p
p
b k
p p


 

 

 
 

 

(4.23)



2
1
,
1
1
N
j
j
j
j j
xg x x
p
k
p
x

20
equation is





1
x c k x t

  


(4.26)
where k is linearization coefficient. The linear dependence level is

0.85


(4.27)
Table 4.2 Mean square response of quadratic restoring oscillator;
1, 2
 
 
,

varies


a
o
x hx x x t
  
   

 
(4.28)
where
, , , ,
o
h a
  
are real positive constant,
a
x

is odd function.
The equivalent linearization equation is





2
0
2
x hx k x t
 
   



,a = 7 and γ varies


2
x
c
x
2
kd
x
error (%)

2
dn
x
error (%)

2
ts
x

error (%)

µ
0.1 0.6170 0.4733 23.30 0.5388 12.68 0.7131 15.57 0.082
1.0 0.4156 0.2871 30.92 0.3323 20.03 0.4679 12.61 0.082
10 0.2595 0.1678 35.35 0.1958 24.56 0.2835 9.23 0.082
100 0.1549 0.0963 37.82 0.1128 27.16 0.1656 6.90 0.082


   

 
(4.32)
with k is linearization coefficient. The component linear dependence
level are

1 2
3/ 5; 4 / 35
 
 
(4.33)
Table 4.4 Mean square response of cubic-quintic restoring oscillator;
1 3
2 1
h c c
  
,
2


and
5
c
varies
5
c

2

a
x x x x x f t
  
   
  
(4.34)
where
0
,
 
, a are real positive constant,


f t
is Gaussian white
noise with
o
S const
 . The equivalent linearization equation is



2
0
x bx x t
 
  

 
(4.35)

(%)


ts
h a

error
(%)
µ p
ts

1 1 1 0 1 0 1 0 1 0
0.7008 0.8329 0.7741 7.06 0.8617 3.46 0.8125 2.45 0.800 0.300
1.0000 0.7979 0.7071 11.38 0.8165 2.33 0.8165 2.33 0.667 0.500
1.3955 0.7644 0.6351 16.92 0.7522 1.60 0.7522 1.60 0.500 0.500
1.8820 0.7349 0.5650 23.12 0.6746 8.21 0.6952 5.40 0.333 0.551
2.0000 0.7290 0.5503 24.51 0.6568 9.91 0.7205 1.16 0.300 0.640
2.4353 0.7104 0.5024 29.27 0.5962 16.07 0.7255 2.14 0.200 0.760
3.1302 0.6876 0.4416 35.78 0.5158 24.98 0.7215 4.94 0.100 0.880
5.2277 0.6453 0.3245 49.72 0.3624 43.85 0.6590 2.11 0.010 0.988
7.1107 0.6230 0.2627 57.84 0.2861 54.08 0.5956 4.40 0.001 0.999 22
4.2.5 Free vibration

2 1
0
n
x x

2 2
n n
t dt t dt
   
 
  
 

 
   

   
   
 
(4.40)
Table 4.6 Frequency of free vibration with n varies
n
cx


kd


sai số
(%)
dn


error


at
0


and
1


are defined. Based
on boundary conditions, the analytical expressions of


p

for the 23
weak and strong linear dependence level are constructed from the
interpolation problem using exact data of Lutes- Sarkani oscillator.
CONCLUSIONS AND RECOMMENDATIONS
New results of this thesis are:
1. The dual criterion and the weighted dual criterion in terms of least
square condition are constructed based on the dual approach for
equivalent linearization. The expression of the weighted dual
criterion is general form of the dual and classical criteria, applied
to stationary Gaussian processes with zero mean.
2. Combining the meaning of linear dependence and influence of the
forward and backward replacements leads to the classification of
of linear dependence level between nonlinear function and