U.P.B. Sci. Bull., Series A, Vol. 74, Iss. 3, 2012
ISSN 1223-7027
APPLICATION OF MAX MIN APPROACH AND
AMPLITUDE FREQUENCY FORMULATION TO
NONLINEAR OSCILLATION SYSTEMS
Davood Domiri GANJI1, Mohammadreza AZIMI2
This paper applied the Max-Min Approach (MMA) and Amplitude
Frequency Formulation (AFF) to derive the approximate analytical solution
for motion of nonlinear free vibration of conservative, single degree of
freedom systems. In both nonlinear problems MMA and AFF yields the same
results. In comparison to forth-order runge-kutta method which is powerful
numerical solution, the results show that these methods are very convenient
for solving nonlinear equations and also can be used for the wide range of
time and boundary conditions for nonlinear oscillators.
Keywords: nonlinear oscillation, max min approach, amplitude frequency
formulation, analytical solution.
1. Introduction
This study has clarified the motion equation of two oscillators by
Max min approach and Amplitude frequency formulation to obtain the
relationship between Amplitude and angular frequency. As it can be
illustrated in Fig.1, m1 is mass of the block on the horizontal surface, m2 is the
mass of block which is just slipped in the vertical and is linked to m1 , L is length of
link, g is gravitational acceleration, and k is spring constant.
By assuming u =
x
, u << 1 , the equation of motion can be yield as
l
following terms:
⎛m
u + ⎜ 2
⎝ m1
⎞ 2 ⎛ m2 ⎞ 2 ⎛ k m2 g ⎞
m2 g 3
u =0
+
⎟ u u + ⎜
⎟ uu + ⎜
⎟u +
2lm1
⎠
⎝ m1 ⎠
⎝ m1 lm1 ⎠
(1)
In which u and t are generalized dimensionless displacement and time
variables.
1
Associated Prof., Dept.of Mechanical Engineering, Babol University of Technology, Iran
Bsc., Dept.of Mechanical Engineering, Babol University of Technology , Iran, e-mail:
[email protected]
2
132
Davood Domiri Ganji, Mohammadreza Azimi
Fig. 1. Geometry of first problem
The second problem is derived from the motion of simple pendulum
attached to a rotating rigid frame that is shown in fig.2 which has following
nonlinear differential equation:
θ + ( 1 − Λ cos (θ ) ) sin (θ ) = 0
(2)
In which θ and t are generalized dimensionless displacements and
Ω2 r
time variables, and Λ =
.
g
As it can be considered both of the problem are strongly nonlinear. In recent
years, many powerful methods such as Max Min Approach [1, 2],
Amplitude Frequency Formulation [3-6], Harmonic Balance Method (HBM)
[7], Homotopy Perturbation Method (HPM) [8-12], Variational Iteration
Method (VIM) [13, 14], Parameter Expansion Method (PEM) [15] and
Energy Balance Method (EBM) [16-19], are used to find approximate
solution to the nonlinear differential equations
This study has investigated two nonlinear oscillators by Max min
approach and Amplitude frequency formulations which are two methods
come from Chinese mathematics. The comparison between approximate
solutions and the forth-order runge kutta method assures us about accuracy
and validity of solution.
Fig. 2. Geometry of second problem
Application of max min approach and amplitude frequency […] oscillation systems
133
2. Application of MMA to the first problem:
In order to solve first problem with the max min approach, it can be
rewritten as:
⎛m ⎞
⎛m ⎞
m g
u + ⎜⎜ 2 ⎟⎟u 2 u + ⎜⎜ 2 ⎟⎟uu 2 + ω 02 u + 2 u 3 = 0, u (0 ) = A, u (0 ) = 0
2lm1
⎝ m1 ⎠
⎝ m1 ⎠
(3)
⎛ k m2 g ⎞
2
⎟⎟
ω 0 = ⎜⎜ +
m
lm
1 ⎠
⎝ 1
We can rewrite Eq. (3) in the following form:
u + f ( u ,u ,u,t ) u = 0
(4)
⎛⎛ m ⎞
⎛m ⎞
mg ⎞
f ( u ,u ,u,t ) = ⎜⎜ ⎜ 2 ⎟ uu + ⎜ 2 ⎟ u 2 + ω02 + 2 u 2 ⎟⎟ u .
2lm1 ⎠
⎝ m1 ⎠
⎝ ⎝ m1 ⎠
Inserting u = A cos (ωt ) an initial assumption and inserting this trial
function into Eq. (4), the maximum and minimum value of f ( u ,u ,u,t ) can
Where
be yield:
⎛m ⎞
mg
0 < ω 2 = f ( u ,u ,u,t ) < ω02 − ⎜ 2 ⎟ A2ω 2 + 2 A2
(5)
2lm1
⎝ m1 ⎠
Using He Chengtian's average [1], gives:
⎛
⎞
⎛m ⎞
mg
n ⎜⎜ ω02 − ⎜ 2 ⎟ A2ω 2 + 2 A2 ⎟⎟
2lm1 ⎠
⎛
⎞
⎛m ⎞
mg
⎝ m1 ⎠
(6)
= k ⎜⎜ ω02 − ⎜ 2 ⎟ A2ω 2 + 2 A2 ⎟⎟
ω2 = ⎝
m+n
2lm1 ⎠
⎝ m1 ⎠
⎝
Where m, n are weighting factors and k= n/(m+n). Substituting
approximate angular frequency into Eq. (4) yields:
u + ω 2 u = u + uf ( u , u ,u,t ) + ρ ( u ,u , u,t )
Where:
⎛m ⎞
⎛m ⎞
mg
ρ ( u ,u ,u,t ) = − ⎜ 2 ⎟ u 2u − ⎜ 2 ⎟ uu 2 − uω02 − 2 u 3
2lm1
⎝ m1 ⎠
⎝ m1 ⎠
⎛
⎞
⎛m ⎞
mg
+ ku ⎜⎜ ω02 − ⎜ 2 ⎟ A2ω 2 + 2 A2 ⎟⎟
2lm1
⎝ m1 ⎠
⎝
⎠
Substituting u (t ) = A cos(ωt ) into ρ , gives:
(7)
(8)
134
Davood Domiri Ganji, Mohammadreza Azimi
⎛m ⎞
⎛m⎞
ρ = ⎜⎜ 2 ⎟⎟ A 3 cos(ωt )3 ω 2 − ⎜ ⎟ A 3 cos(ωt )sin (ωt )2 ω 2 − Aω 02 cos(ωt )
⎝m⎠
⎝ m1 ⎠
⎛
⎛m ⎞
m g ⎞
m g
3
− 2 A 3 cos(ωt ) + kA cos(ωt )⎜⎜ ω 02 − ⎜⎜ 2 ⎟⎟ A 2ω 2 + 2 A ⎟⎟
2lm1
2lm1 ⎠
⎝ m1 ⎠
⎝
Using Fourier expansion series, secular term can be obtained:
(9)
∞
ρ (ωt ) = ∑ b2 n +1 cos[(2n + 1)ωt ] ≈ b1 cos(ωt )
n =0
⎛⎛ m ⎞
⎞
⎜ ⎜ 2 ⎟ A 3 cos(ϕ )3 ω 2 − ⎛⎜ m2 ⎞⎟ A 3 cos(ϕ )sin (ϕ )2 ω 2 ⎟
⎜m ⎟
⎜ ⎜ m1 ⎟⎠
⎟
⎝ 1⎠
π ⎝
⎜
⎟
m2 g 3
4 2⎜
⎟
3
2
A cos(ϕ )
b1 = ∫ ⎜ − Aω 0 cos(ωt ) −
⎟ cos(ϕ )dϕ
π 0
2lm1
⎜
⎟
⎜
⎟
⎛ 2 ⎛ m2 ⎞ 2 2 m2 g ⎞
⎟⎟ A ω +
A ⎟⎟
⎜ + kA cos(ϕ )⎜⎜ ω 0 − ⎜⎜
⎟
⎜
⎟
2lm1 ⎠
⎝ m1 ⎠
⎝
⎝
⎠
(10)
Avoiding secular term requires δ 1 = 0 , therefore angular frequency
can be yield:
ωMMA
1 8ω02lm1 + 3m2 gA2
=
2 l ( m2 A2 + 2 m1 )
(11)
3. Application of AFF to the first problem:
According to He's frequency formulation [3], we can assume for the
amplitude frequency formulation:
ω12 R2 ( t2 ) − ω22 R1 ( t1 )
2
(12)
ω =
R2 ( t2 ) − R1 ( t1 )
In order to solve Eq. (1), we first use two trial functions as follows:
u1 = A cos ( t )
(13)
u2 = A cos (ωt )
(14)
Inserting Eqs. (12) and (13) into Eq. (1) , gives the following
Residuals:
Application of max min approach and amplitude frequency […] oscillation systems
⎛m ⎞
⎛m ⎞
3
2
R1 ( t1 ) = − A cos ( t ) − A3 ⎜ 2 ⎟ cos ( t ) + A3 ⎜ 2 ⎟ cos ( t ) sin ( t )
⎝ m1 ⎠
⎝ m1 ⎠
⎛mg⎞
3
+ Aω02 cos ( t ) + A3 ⎜ 2 ⎟ cos ( t )
⎝ 2lm1 ⎠
⎛m ⎞
3
R2 ( t ) = − A cos (ωt ) ω 2 − A3 ⎜ 2 ⎟ cos (ωt ) ω 2 + Aω02 cos (ωt )
⎝ m1 ⎠
⎛m ⎞
⎛mg⎞
2
3
+ A3 ⎜ 2 ⎟ cos (ωt ) sin (ωt ) ω 2 + A3 ⎜ 2 ⎟ cos (ωt )
⎝ m1 ⎠
⎝ 2lm1 ⎠
Weighted residuals can be introduced as follows:
4
R1 ( t1 ) =
T1
T1
4
∫ R ( t ) cos ( t ) dt ,T
1
=
1
0
T2
4
4
R 2 ( t2 ) =
T2
∫ R ( t ) cos (ωt ) ,T
2
2
2
2π
0
(15)
(16)
(17)
ω1
=
135
2π
(18)
ω2
Equating ω1 = 1 , ω2 = ω , we can obtain weighted residuals :
π
R1 ( t1 ) =
2
2
( −A cos ( t ) − A
π∫
3
0
⎛ m2 ⎞
3
2
3 ⎛ m2 ⎞
⎜
⎟ cos ( t ) + A ⎜
⎟ cos ( t ) sin ( t )
m
m
⎝ 1⎠
⎝ 1⎠
⎛mg⎞
+ Aω02 cos ( t ) + A3 ⎜ 2 ⎟ cos ( t ) 3 cos ( t ) dt
⎝ 2lm1 ⎠
1
3 ⎛ m g ⎞ 1⎛m ⎞
1
= Aω02 + A3 ⎜ 2 ⎟ − ⎜ 2 ⎟ A3 − A
2
8 ⎝ 2lm1 ⎠ 4 ⎝ m1 ⎠
2
)
(19)
π
⎛m ⎞
3
− A3 ⎜ 2 ⎟ cos (ωt ) ω 2 + Aω02 cos (ωt )
⎝ m1 ⎠
0
⎛m ⎞
⎛mg⎞
2
+ A3 ⎜ 2 ⎟ cos (ωt ) sin (ωt ) ω 2 + A3 ⎜ 2 ⎟ cos (ωt ) 3 cos (ϕ ) dϕ
⎝ m1 ⎠
⎝ 2lm1 ⎠
2
R 2 ( t2 ) =
2
( −A cos (ωt ) ω
π∫
2
)
(20)
1
3⎛ m g ⎞
1 ⎛m ⎞
1
Aω 2 + ⎜ 2 ⎟ A3 − A3 ⎜ 2 ⎟ ω 2 + ω02 A
2
8 ⎝ 2lm1 ⎠
4 ⎝ m1 ⎠
2
The substitution of Eqs. (19) and (20) into Eq. (12), angular
frequency can be yield:
=−
136
Davood Domiri Ganji, Mohammadreza Azimi
⎛ A A 3ω 2 (m1 m2 ) Aω 2 A 3 (m2 m1 ) ⎞
⎟ ×
ω = ⎜⎜ −
−
+
⎟
4
2
4
⎝2
⎠
(21)
2
2
3
2
3
2
⎛ ω 0 A 3m2 g A m2ω
⎞
⎞
⎛
ω A 3m2 g A m2 A ⎟
Aω
⎜
+
−
−
− ω 2 ⎜⎜ 0 +
−
− ⎟⎟
⎜ 2
lm
m
lm
m
16
4
2
2
16
4
2 ⎠ ⎟⎠
1
1
1
1
⎝
⎝
Solving Equation (21), Amplitude-frequency relationship can be
obtained:
−1
2
ω AFF =
1 8ω02 lm1 + 3m2 gA2
2 l m2 A2 + 2 m1
(
(22)
)
4. Application of MMA to the second problem:
Eq. (2) can be rewritten as follows:
1
θ + sin (θ ) − Λ sin ( 2θ ) = 0,θ ( 0 ) = 0,θ ( 0 ) = A
(23)
2
Substitution of the relatively accurate approximations:
θ3 θ5
4θ 3 4θ 5
and sin ( 2θ ) ≈ 2θ −
into Eq. (23), yields:
sin (θ ) ≈ θ − +
+
6 120
3
15
2Λ ⎞ 5
⎛ 1 2Λ ⎞ 3 ⎛ 1
θ + ( 1 − Λ ) θ + ⎜ − +
−
(24)
⎟θ + ⎜
⎟θ = 0
3 ⎠
⎝ 6
⎝ 120 15 ⎠
To attack Eq. (24) by the max min approach, we rewrite it in the
following form:
θ = − f θ ,θ ,θ,t θ
(25)
(
Where
)
⎛ −1 2Λ ⎞
f (θ ,θ ,θ,t ) = ( 1 − Λ ) + ⎜ +
⎟θ
3 ⎠
⎝ 6
2
2Λ ⎞ 4
⎛ 1
+⎜
−
⎟ θ . Like
⎝ 120 15 ⎠
Section.2, It can be written:
(
)
1
2
1 4 2
1 − Λ < ω 2 = f θ ,θ ,θ,t < 1 − Λ + A2 + ΛA2 +
A − ΛA4
6
3
120
15
Similar of pervious example, angular frequency can be yields:
(26)
1
2
1 4 2
⎛
⎞
n ⎜ 1 − Λ + A2 + ΛA2 +
A − ΛA4 ⎟ + m ( 1 − Λ )
6
3
120
15
⎠
= ( 1 − k )( 1 − Λ )
ω2 = ⎝
(27)
m+n
1
2
1 4 2
⎛
⎞
A − ΛA4 ⎟
+ k ⎜ 1 − Λ + A2 + ΛA2 +
6
3
120
15
⎝
⎠
Where m, n are weighting factors and k= n/(m+n). Substituting
approximate angular frequency into Eq. (25) obtains:
Application of max min approach and amplitude frequency […] oscillation systems
θ + ω 2θ = θ + θ f (θ ,θ ,θ,t ) + ρ
137
(28)
Inserting u ( t ) = A cos (ωt ) as a trial function into ρ can be yield:
⎛
ρ = A cos(ωt )⎜⎜1 − b −
⎝
kA 2 2kΛA 2 kA 4 2kΛA 4
+
+
−
6
3
120
15
⎞
⎟⎟
⎠
2
4
⎛
⎞
A cos(ωt )
A 4 cos(ωt )
(29)
⎜1 −
+
−Λ⎟
6
120
⎟ A cos(ωt )
+ ⎜⎜
2
4 ⎟
2
4
⎜ + 4ΛA cos(ωt ) − 4ΛA cos(ωt ) ⎟
6
30
⎝
⎠
Using Fourier series and avoiding secular term, angular frequency
can be obtained:
1
1
1
1 4
ωMMA = 1 − A2 − ΛA4 + ΛA2 − Λ +
A
(30)
8
12
2
192
2
5. Application of AFF to the second problem:
Like pervious example, the trial functions θ1 (τ ) = A cos (τ ) and
θ 2 (τ ) = A cos (ωτ ) are inserted into Eq. (2) to yield the residuals:
1
1 5
3
5
R1 ( t ) = − A3 cos ( t ) +
A cos ( t )
6
120
4
4 5
Λ⎛
3
5⎞
− ⎜ 2 A cos ( t ) − A3 cos ( t ) +
A cos ( t ) ⎟
2⎝
3
15
⎠
(31)
1 3
1 5
3
5
A cos (ωt ) +
A cos (ωt )
6
120
(32)
Λ⎛
4 3
4 5
3
5⎞
− ⎜ 2 A cos (ωt ) − A cos (ωt ) +
A cos (ωt ) ⎟
2⎝
3
15
⎠
Weighted residuals can be yield as:
1 3 1
1
1
1 5
R1 ( t1 ) = −
A − ΛA5 + ΛA3 − AΛ +
A
(33)
16
24
4
2
384
1
1
1 5 1
1
1
1
~
R2 (t 2 ) = A − A 3 +
A − AΛ − Aω 2 − ΛA 5 + ΛA 3
(34)
2
16
384
2
2
24
4
Inserting Eq. (33) and Eq. (34) into Eq. (12), angular frequency can
be yield:
1
1
1
1 4
ω AFF = 1 − A2 − ΛA4 + ΛA2 − Λ +
A
(35)
8
12
2
192
R2 ( t ) = − A cos (ωt ) ω 2 + A cos (ωt ) −
138
Davood Domiri Ganji, Mohammadreza Azimi
6. Results and Disscussions
In this section, the obtained results will be studied in some numerical
cases. In Fig.3 the comparison between Analytical solutions and runge-kutta
result is shown. As we see, amplitude frequency formulation and Max min
approach obtain same results and have a high validity in comparison with
runge-kutta method.
Fig.3. Comparison between AFF & MMA & Runge-kutta forth order in first problem for
π
A = , g = 9.81 m s 2 , k = 100 N m 2 , m1 = 5 kg , m2 = 1kg ,l = 1m
6
Fig.4 shows the comparison between forth-order runge kutta and
analitycal solutions in second problem for A =
π
3
, Λ = 0.25 . Good agreement
can be illustrated in Fig.4.
Fig.4. Comparison between AFF & MMA & Runge-kutta forth order in second problem for
A=
π
3
, Λ = 0.25
Application of max min approach and amplitude frequency […] oscillation systems
139
6. Conclusions
In this paper, Max min approach and Amplitude frequency
formulation which are two powerful methods and derived from Chinese
mathematics are applied to the motion equations of two nonlinear
oscillators. At first the equation of motion was derived for both problems.
Max min approach and Amplitude frequency formulation was applied to the
nonlinear ordinary differential equations and finally the results compared
with forth order runge kutta method. Simple procedure and high accuracy
and validity are the advantages of these methods.
.
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