Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire
The He’s Amplitude-Frequency Formulation is applied to study the periodic solutions of a strongly nonlinear system. This system corresponds to the motion of a mass attached to a stretched wire. The usefulness and effectiveness of the proposed technique is illustrated. The results are compared with e...
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Akbarzade, M. Farshidianfar, A. 2016-05-24T16:52:51Z 2016-05-24T16:52:51Z 2014 Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire / M. Akbarzade, A. Farshidianfar // Прикладная механика. — 2014. — Т. 50, № 4. — С. 137-144. — Бібліогр.: 19 назв. — англ. 0032-8243 https://nasplib.isofts.kiev.ua/handle/123456789/100638 The He’s Amplitude-Frequency Formulation is applied to study the periodic solutions of a strongly nonlinear system. This system corresponds to the motion of a mass attached to a stretched wire. The usefulness and effectiveness of the proposed technique is illustrated. The results are compared with exact solutions and those obtained by the harmonic balance show a good accuracy. Approximate frequencies are valid for the complete range of vibration amplitudes. Excellent ag Амплітудно-частотний підхід Хе застосовано до вивчення періодичних розв’язків сильно нелінійних систем. Проілюстровано корисність і ефективність запропонованої методики. Результати порівняно з точними розв‘язками і розв‘язками, отриманими на основі енергетичного балансу. Порівняння показало добру точність. Наближено обчислені частоти виявилися вірними у всьому діапазоні амплітуд коливань. Продемонстровано і обговорено узгодженість між наближеними та точними значеннями частот. en Інститут механіки ім. С.П. Тимошенка НАН України Прикладная механика Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire Применение амплитудно-частотного под- хода к нелинейным колебаниям системы в виде массы, присоединенной к растягиваемой проволоке Article published earlier |
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Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire |
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Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire Akbarzade, M. Farshidianfar, A. |
| title_short |
Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire |
| title_full |
Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire |
| title_fullStr |
Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire |
| title_full_unstemmed |
Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire |
| title_sort |
application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire |
| author |
Akbarzade, M. Farshidianfar, A. |
| author_facet |
Akbarzade, M. Farshidianfar, A. |
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2014 |
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English |
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Прикладная механика |
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Інститут механіки ім. С.П. Тимошенка НАН України |
| format |
Article |
| title_alt |
Применение амплитудно-частотного под- хода к нелинейным колебаниям системы в виде массы, присоединенной к растягиваемой проволоке |
| description |
The He’s Amplitude-Frequency Formulation is applied to study the periodic solutions of a strongly nonlinear system. This system corresponds to the motion of a mass attached to a stretched wire. The usefulness and effectiveness of the proposed technique is illustrated. The results are compared with exact solutions and those obtained by the harmonic balance show a good accuracy. Approximate frequencies are valid for the complete range of vibration amplitudes. Excellent ag
Амплітудно-частотний підхід Хе застосовано до вивчення періодичних розв’язків сильно нелінійних систем. Проілюстровано корисність і ефективність запропонованої методики. Результати порівняно з точними розв‘язками і розв‘язками, отриманими на основі енергетичного балансу. Порівняння показало добру точність. Наближено обчислені частоти виявилися вірними у всьому діапазоні амплітуд коливань. Продемонстровано і обговорено узгодженість між наближеними та точними значеннями частот.
|
| issn |
0032-8243 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/100638 |
| citation_txt |
Application of the amplitude-frequency formulation to a nonlinear vibration system typified by a mass attached to a stretched wire / M. Akbarzade, A. Farshidianfar // Прикладная механика. — 2014. — Т. 50, № 4. — С. 137-144. — Бібліогр.: 19 назв. — англ. |
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2025-11-25T22:42:42Z |
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| fulltext |
2014 ПРИКЛАДНАЯ МЕХАНИКА Том 50, № 4
ISSN0032–8243. Прикл. механика, 2014, 50, №4 137
M . A k b a r z a d e , A . F a r s h i d i a n f a r
APPLICATION OF THE AMPLITUDE-FREQUENCY FORMULATION
TO A NONLINEAR VIBRATION SYSTEM TYPIFIED
BY A MASS ATTACHED TO A STRETCHED WIRE
Department of Mechanical Engineering, Faculty of Engineering,
Ferdowsi University Of Mashhad, Mashhad, Iran
e-mail: mehdiakbarzade@yahoo.com
Abstract. The He’s Amplitude-Frequency Formulation is applied to study the periodic
solutions of a strongly nonlinear system. This system corresponds to the motion of a mass
attached to a stretched wire. The usefulness and effectiveness of the proposed technique is
illustrated. The results are compared with exact solutions and those obtained by the har-
monic balance show a good accuracy. Approximate frequencies are valid for the complete
range of vibration amplitudes. Excellent agreement of the approximate frequencies with the
exact one are demonstrated and discussed.
Key words: Nonlinear vibration system; Amplitude-Frequency Formulation; Periodic
Solution; Angular frequencies.
1. Introduction.
The study of nonlinear problems is of crucial importance not only in all areas of physics
but also in engineering, since most phenomena in our world are essentially nonlinear and are
described by nonlinear equations recently many new approaches to nonlinear problems have
been proposed, for example, the variational iteration method [16], the homotopy perturba-
tion method [17 – 19], energy balance method [12 – 15] and the parameter-expanding
method [11].
To solve nonlinear problems, He proposed an amplitude–frequency formulation for
nonlinear oscillators, which was deduced using an ancient Chinese mathematics method and
it is now widely used by many authors [3 – 10]. In this paper He’s frequency-amplitude
formulation is used to solve nonlinear vibration system of conservative single degree of
freedom.
Consider the motion of a particle of mass m attached to the centre of a stretched elastic
wire [1, 2] and coefficient of stiffness of elastic wire equal to k . The length of the elastic
wire when no force is applied to it is 2a . We assume that the movement of the particle is
one-dimensional and this is constrained to move only in the horizontal x direction.
As we can see in Fig. 1, the ends of the wire are fixed a distance 2d a part. Length
d can be longer or equal to a . If d a , the wire is not stretched for 0x , and there is no
tension in each part of it. However, if d a , the wire is stretched for 0x , and the tension
in each part of the wire is ( - )k d a The equation of motion is given by the following nonlin-
ear differential equation [2]:
2
2 2 2
2
2 0
d x kax
m kx
dt d x
. (1)
138
Fig. 1
Mass attached to a stretched wire.
With the initial condition of
(0) , (0) 0
dx
x A
dt
.
Two dimensionless variables y and can be constructed as follows:
2
,
x k
y t
d m
. (2)
Substituting these dimensionless variables into Eq. (1) gives
2
2 2
0 , 0< 1
1
d y y
y
d y
. (3)
With the initial condition of
(0) , (0) 0
dy
y A
d
.
In Eq. (3) we have defined the following parameters:
0 ,
x a
A
d d
, (4)
as 0 a d it follows that 0 1 .
Eq. (3) is an example of a conservative nonlinear oscillatory system in which the restor-
ing force has an irrational form [1, 2] and this system and has the first integral.
2
1
( ) 0
2
dy
V y E
d
, (5)
where E is the ‘‘total energy’’ of the nonlinear oscillator and the potential function has the
irrational form [1]
2 21
( ) 1
2
V y y y . (6)
All the motions corresponding to Eq. (3) are periodic [1]; the system will oscillate
within symmetric bounds[ , ]A A , and the angular frequency and corresponding periodic
solution of the nonlinear oscillator are dependent on the amplitude A .
139
The main objective of this paper is to solve Eq. (3) by applying the first-order Ampli-
tude-Frequency Formulation, and to compare the approximate frequency obtained with the
exact one and with another approximate frequency obtained applying the harmonic balance
method.
2. Solution method.
Considers the following general nonlinear oscillators in the form:
( ) ( ), ( ), ( ) 0u t f u t u t u t . (7)
Oscillation systems contain two important physical parameters, i.e. the frequency and
the amplitude of oscillation, A . So let us consider such initial conditions
(0) , (0) 0.u A u
According to He’s amplitude-frequency formulation [8 – 10], we choose two trial func-
tions 1 cosu A t and 2 cos .u A t
Substituting 1u and 2u into Eq. (7), we obtain, respectively, the following residuals:
1 1 1 1 1( ) ( ), ( ), ( )R u t f u t u t u t (8)
and
2 2 2 2 2( ) ( ), ( ), ( )R u t f u t u t u t . (9)
In order to use He’s amplitude-frequency formulation [6 – 9], we set
1
4
11 1 1 0
1
4
cos( ) , 2
T
R R t dt T
T
, (10)
2
4
22 2 2 0
2
4 2
cos( ) ,
T
R R t dt T
T
. (11)
Applying He’s frequency-amplitude formulation [8 – 10] we have
2 2
2 1 22 2 11
22 11
R R
R R
, (12)
where
1 21 , . (13)
3. Results and discussion for small amplitudes (0 A<<1) .
Considering Eq. (3), for small values of A we can write
2
2
1 1
1 , 0 A<<1
2(1 )
y
y
. (14)
We can write Eq. (3) in the form of
2
2
2
1
1 0, 0< 1
2
d y
y y y
d
, (15)
with initial conditions of
(0) , (0) 0u A u .
140
According to He’s amplitude-frequency formulation [4 – 6]; we obtain, respectively, the
following residuals:
2 2
1
1
cos( ) 1 cos ( )
2
R A t A t
(16)
and
2 2 2
2
1
cos( ) cos( ) cos( ) 1 cos ( )
2
R A A A A
. (17)
In the equations, the frequency of oscillation is and the amplitude of oscillation is A .
In order to use He’s amplitude-frequency formulation [6 – 8], we set
1
2
4
11 1 1 0
1
8 34 1
cos( ) , 2
16
T A A
R R d T
T
(18)
and
2
2 2
4
22 2 2 0
2
8 3 8 84 1 2
cos( ) ,
16
T A A
R R d T
T
. (19)
Applying He’s frequency-amplitude formulation [6 – 8], we, therefore, obtain the first
order approximate solution for small amplitudes
23
1
8
A . (20)
For the first approaching Eq. (20) at 0A and for 1 we have
0
lim ( ) (1 )
A
A
.
4. Results and discussion for large amplitudes (A>>0) .
Eq. (3) can be rewritten in a form that does not contain the square-root expression [2]
22 2 2
2 2
0
(1 )
d y y
y
d y
, (21)
with initial condition of
(0) , (0) 0
dy
y A
d
.
According to He’s amplitude-frequency formulation [4 – 6]; we obtain, respectively, the
following residuals:
2 2 2
1 2 2
cos ( )
1 cos ( )
A
R
A
(22)
and
2 2 2
2 2
2 2 2
cos ( )
( cos( ) cos( ))
1 cos ( )
A
R A A
A
. (23)
In the equations, the frequency of oscillation is and the amplitude of oscillation is A .
In order to use He’s amplitude-frequency formulation [6 – 8], we set
1
2 2
2
4
11 1 1 0 2
1
2 1 arctan
4 1
cos( ) , 2
1
T
A
A A h
A
R R d T
T A A
(24)
141
and
2 3 2 44
22 2 0 2
2
4 2 1
cos( ) ( 2 1
3 1
T
R R d A A
T A A
2 3 2 2 3 2 2 2
22
2
3 arctan 4 1 2 1 3 1 ),
1
A
h A A A A A A T
A
. (25)
Applying He’s frequency-amplitude formulation [6 – 8], we, therefore, obtain the first
order approximate solution
3 2 6 2 4 2 3 2 2
2
2
2 1 6 6 6 1 arctan
2 1
2 1
A
A A A A A A h
A
A A A
. (26)
It is possible to solve Eq. (3) by applying the harmonic balance method. Following the
first-order harmonic balance method, a reasonable and simple initial approximation satisfy-
ing the conditions in Eq. (3) would be [2]
cosy A . (27)
Substitution of Eq. (27) into Eq. (3) gives
2
2
cos( )
cos( ) cos( ) 0
1 cos ( )
A
A A
A
. (28)
The power-series expansion of
21
y
y
is [2]
2 1
12
1
(2 1)!
( 1)
2 ! 1 !1
n n
n
n
y n
y y
n ny
(29)
Substituting Eq. (29) into Eq. (28) and taking into account Eq. (27) gives
2 2 2 1
1
1
(2 1)!
cos( ) cos( ) cos( ) ( 1) cos ( ) 0
2 ! 1 !
n n n
n
n
n
A
n n
. (30)
The formula that allows us to obtain the odd power of the cosine is
2 1
2
2 1 2 1 2 11
cos ( ) cos( ) cos(3 ) ... cos(2 1)( )
2 -1 0
n
n
n n n
n
n n
.(31
)
Substituting Eq. (31) into Eq. (30) gives
2 2
2 1
0
1 cos( ) (high order harmonics)=0n
n
n
c A
, (32)
where the coefficients 2 1nc are given by
1 1c
and
142
2 1 4 1 2
(2 1)!(2 1)!
( 1) , 1
2 ( !) 1 ! 1 !
n
n n
n n
c n
n n n
. (33)
For the first-order harmonic to be equal to zero, it is necessary to set the coefficient of
cos( ) equal to zero in Eq. (32), then
2
2 1
0
1 n
n
n
c A
. (34)
For the second-order harmonic we obtain [2]
1
2 4 62 3 3 13
1 ( ( )) , ( ) 1 ...
4 64 512
f A f A A A A
. (35)
Table 1. Comparison of He’s frequency-amplitude formulation (Eq. (20)) with harmonic
balance frequency (Eq. (35)) for small values of A . ( =0.5)
Table 1
A He’s frequency-amplitude formulation harmonic balance frequency
0.01 0.7071 0.7071
0.05 0.7074 0.7075
0.1 0.7084 0.7085
0.2 0.7124 0.7126
0.3 0.7189 0.7190
0.5 0.7395 0.7369
Table 2. Comparison of He’s frequency-amplitude formulation (Eq. (20)) with harmonic
balance frequency (Eq. (35)) for small values of A . ( =1.0)
Table2
A He’s frequency-amplitude formulation harmonic balance frequency
0.01 0.0061 0.0063
0.05 0.0306 0.0316
0.1 0.0612 0.0630
0.2 0.1225 0.1249
0.3 0.1837 0.1844
0.5 0.3062 0.2935
The exact frequency can then be derived as follows [2]:
1
1
2 2 2 2 202 (1 ) 2 ( 1 1 )
exact
Adu
A u A A u
. (36)
Now we are going to obtain an asymptotic representation for large amplitudes. We con-
sider the expression for the exact frequency exact Eq. (36) and we do the change 1/A B .
For large amplitudes A we have 0B . Taking this into account, and doing the
power-series expansion of the result for small values of B , we obtain [2]
1
1
2 2 2 202 1 2 ( 1 )
exact
du
u B B B u
(37)
and
143
1
1 2 2
2 2 2 2
0
1 3
...
2 1 (1 ) 1 2(1 ) 1
exact
B B
du
u u u u u
. (38)
The power-series expansion for the exact frequency for small values of B (large values
of A ) is:
1 2
2 2
2 2
2 2( 2)
... 1 ...
2 2exact B B
A A
=
2
2
0.63662 0.23134
1 ...
A A
. (39)
Table 3
Comparison of He’s frequency-amplitude formulation frequency (Eq. (26)) with exact
frequency (Eq. (39)) for large amplitude ( = 0.5).
A He’s frequency-amplitude
formulation
Exact frequency Error (%)
5 0.9398 0.9340 0.6176
6 0.9495 0.9453 0.4362
7 0.9565 0.9533 0.3305
8 0.9898 0.9593 0.2631
9 0.9618 0.9639 0.2171
10 0.9694 0.9676 0.1842
Table 4
Comparison of He’s frequency-amplitude formulation frequency (Eq. (26)) with exact
frequency (Eq. (39)) for large amplitude ( = 1).
A He’s frequency-amplitude
formulation
Exact frequency Error (%)
5 0.8755 0.8634 1.3937
6 0.8961 0.8875 0.9706
7 0.9109 0.9043 0.0715
8 0.9221 0.9168 0.7283
9 0.9308 0.9264 0.5755
10 0.9377 0.9340 0.3984
The method of He’s frequency-amplitude formulation is capable of producing analytical
approximation to the solution to the nonlinear system, valid even for the case where the am-
plitude are not small but we see that harmonic balance solution is valid only for small ampli-
tude.
5. Conclusions.
He’s frequency-amplitude formulation has been used to solve nonlinear vibration sys-
tem typified by a mass attached to a stretched wire. With the procedure, the analytical ap-
proximate frequency and the corresponding periodic solution, valid for small as well as
large amplitudes of oscillation, can be obtained. The method, which is proved to be a power-
ful mathematical tool to the search for angular frequencies of nonlinear vibration systems,
can be easily extended to any nonlinear equation, and the present letter can be used as para-
digms for many other applications in searching for periodic solutions, limit cycles or other
approximate solutions for real-life physics problems. We think that the method have great
potential which still needs further development.
144
Р Е ЗЮМ Е . Амплітудно-частотний підхід Хе застосовано до вивчення періодичних
розв’язків сильно нелінійних систем. Проілюстровано корисність і ефективність запропонованої
методики. Результати порівняно з точними розв‘язками і розв‘язками, отриманими на основі енерге-
тичного балансу. Порівняння показало добру точність. Наближено обчислені частоти виявилися вір-
ними у всьому діапазоні амплітуд коливань. Продемонстровано і обговорено узгодженість між на-
ближеними та точними значеннями частот.
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*From the Editorial Board: The article corresponds completely to submitted manuscript.
Поступила 26.01.2011 Утверждена в печать 03.12.2013
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