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Sub-harmonic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method
Physics Letters A 371 (2007) 427–431 www.elsevier.com/locate/pla
Sub-harmonic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method Wen Jianmin a,∗ , Cao Zhengcai b a School of Automobile Engineering, Harbin Institute of Technology at Weihai, Weihai 264209, China b CIMS Research Center, Tongji University, Shanghai 200092, China
Received 31 January 2007; received in revised form 17 September 2007; accepted 24 September 2007 Available online 29 September 2007 Communicated by A.R. Bishop
Abstract An analytical technique, namely the homotopy analysis method (HAM), is applied to solve periodic solutions for sub-harmonic resonances of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter h. In this Letter, periodic analytic approximations for sub-harmonic resonances of nonlinear oscillations with parametric excitation are obtained by using the HAM for the first time, which agree well with numerical results. This Letter shows that the HAM is a powerful and effective technique for nonlinear dynamical systems. © 2007 Elsevier B.V. All rights reserved. Keywords: Sub-harmonic resonances; Nonlinear oscillation; Series solutions; Homotopy analysis method (HAM)
1. Introduction Exact solutions of differential equations are rare in many branches of fluid mechanics, solid mechanics, and physics because of nonlinearity, inhomogeneity, variable coefficients, and so on [1,2]. Hence, it is often necessary to determine approximate solutions of complicated nonlinear differential equations by using some analytical techniques. Foremost among the analytic techniques are the methods of perturbations (asymptotic expansions) in terms of a small/large parameter [3,4]. Because many nonlinear problems do not contain such a small/large parameter named perturbation quantity, the artificial small parameter method, the δ-expansion method and Adomian’s decomposition method were developed [5–7]. However, both of these perturbation and non-perturbation meth* Corresponding author.
E-mail addresses: [email protected] (J. Wen), [email protected] (Z. Cao). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.09.057
ods cannot present a simple way to adjust and control the convergence region of approximate solutions. In 1992, Liao [8] employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM does not need any small parameters. Different from all other analytic techniques, the HAM provides us with a simple way to adjust and control the convergence region of approximate series solutions [9]. HAM has been successfully applied to solve many types of nonlinear problems [10–21]. In this Letter, the basic idea of HAM is used to solve nonlinear oscillation systems with parametric excitations, governed by d 2x + 1 + ε cos(γ t) αx + βx 3 = 0, 2 dt
(1)
where x(t) is an unknown real function, ε, α, β, γ are known physical parameters. Here, we are interested in the periodic os-
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J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
cillations with the frequency γ (2) , n = 1, 2, 3, . . . . n In other words, given ε, α, β, γ and the frequency described above, the HAM is applied to find the corresponding unknown amplitude A, such that ω=
x (0) = 0.
x(0) = A,
(3)
Note that A is unknown in above equation, and is determined by an approach mentioned below. 2. Basic idea of HAM
(4)
and x(t) = AV (t),
(5)
Eq. (1) becomes V (t) + (1 + ε cos nωt) αV (t) + βA2 V 3 (t) = 0,
V (0) = 0.
+∞
ck cos kωt,
(8)
subject to the initial conditions (10)
According to the solution expression (8), u(τ ) can be expressed by u(τ ) =
+∞
ck cos kτ.
(11)
k=1
According to (11) and the initial conditions (10), it is obvious for us to choose such an initial guess u0 (τ ) = cos τ.
∂ 2 [Φ(τ, q)] ∂t 2 + (1 + ε cos nτ ) αΦ(τ, q) + βΛ2 (q)Φ 3 (τ, q) ,
(15)
where q ∈ [0, 1] is an embedding parameter, Φ(τ, q) is a kind of mapping of the unknown function u(τ ), and Λ(q) is a kind of mapping of the unknown amplitude A. Then, we construct the so-called zero-order deformation equation (1 − q)L Φ(τ, q) − u0 (τ ) = hqN Φ(τ, q), Λ(q) , (16)
where h = 0 is a nonzero auxiliary parameter. When q = 0, the solution of Eqs. (16) and (17) is obviously
where ck is a coefficient to be determined. This provides us with the so-called solution expression of V (t), which plays an important role in the frame of the HAM, as shown later. For the sake of simplicity, using the transformation τ = ωt and V (t) = u(τ ), Eqs. (6) and (7) are rewritten as ω2 u (τ ) + (1 + ε cos nτ ) αu(τ ) + βA2 u3 (τ ) = 0, (9)
u (0) = 0.
for any integration constants C1 and C2 . According to Eq. (9), we define a nonlinear operator N Φ(τ, q), Λ(q)
(7)
k=1
u(0) = 1,
(14)
(6)
Here, we emphasize that the value of A in Eq. (6) is unknown. According to the initial conditions (7) and the nonlinear term in Eq. (6), the periodic solution of V (t) with the given frequency ω can be expressed by V (t) =
L(C1 sin τ + C2 cos τ ) = 0,
subject to the initial conditions ∂Φ(τ, q) Φ(0, q) = 1, = 0, ∂τ τ =0
subject to the initial conditions V (0) = 1,
which has the property
= ω2
Assuming that the solution of Eqs. (1) and (3) is periodic with the frequency ω defined by (2). Then, writing γ = nω
According to (11), obviously, we should choose the auxiliary linear operator 2 ∂ Φ(τ, q) + Φ(τ, q) , L Φ(τ, q) = ω2 (13) ∂t 2
(12)
Φ(τ, 0) = u0 (τ ).
(17)
(18)
When q = 1, the zero-order deformation equations (16) and (17) are equivalent to the original equations (9) and (10), provided Φ(τ, 1) = u(τ ),
(19)
Λ(1) = A.
(20)
Therefore, as the embedding parameter q increases from 0 to 1, Φ(τ, q) varies from the initial guess u0 (τ ) = cos τ to the unknown solution u(τ ) of Eqs. (9) and (10). Likewise, Λ(q) varies from the initial guess A0 to the unknown amplitude A. Note that the zero-order deformation equation (16) contains a nonzero auxiliary parameter h. Assuming that h is chosen so properly that the zero-order deformation equations (16) and (17) have solutions in the whole region q ∈ [0, 1], and besides, there exist 1 ∂ m Φ(τ, q) um (τ ) = (21) m! ∂q m q=0 and
1 d m Λ(q) Am = m! dq m q=0
(22)
for any m 1. Then, by means of Taylor series and using (18), we have Φ(τ, q) = u0 (τ ) +
+∞ m=1
um (τ )q m ,
(23)
J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
Λ(q) = A0 +
+∞
Am q m .
(24)
m=1
Assuming that h is so properly chosen that the power series (23) and (24) converge at q = 1, using (19) and (20), we have the series solutions +∞
u(τ ) = u0 (τ ) +
(25)
um (τ ),
m=1
A = A0 +
+∞
(26)
Am .
m=1
429
deformation equation (27) and (28) contains the so-called secular term τ cos τ , which however disobeys the solution expression (8). To avoid the secular terms, we must enforce bm,l (Am−1 ) = 0,
(31)
which provides us another algebraic equation for Am−1 . Then, the general solution of Eq. (27) reads μm bm,k (Am−1 ) h um (τ ) = χm um−1 (τ ) + 2 cos(kτ ) ω (1 − k 2 ) k=2
+ C1 cos τ + C2 sin τ,
(32)
For the sake of simplicity, define the vectors
um = u0 (τ ), u1 (τ ), u2 (τ ), . . . , um (τ ) , Am = {A0 , A1 , A2 , . . . , Am }.
where the integration constants C1 , C2 are determined by the initial conditions (28). The N th-order approximation is given by
Differentiating the zero-order deformation equations (16) and (17) m times with respect to the embedding parameter q, then dividing them by m!, and finally setting q = 0, we have the socalled mth-order deformation equation um−1 , Am−1 ) L um (τ ) − χm um−1 (τ ) = hRm ( (27)
u(τ ) ≈ u0 (τ ) +
subject to the initial conditions um (0) = 0,
um (0) = 0,
(28)
where ∂ m−1 N [Φ(τ, q), Λ(q)] 1 = (m − 1)! ∂q m−1 q=0
= ω2 um−1 (τ )
+ (1 + ε cos nτ ) αum−1 + β
m−1−k
uj (τ )
j =0
and χm =
um (τ ),
(33)
m=1
A ≈ A0 +
N
Am .
(34)
m=1
Note that γ is given, and thus ω = γn determined by the given positive integer n is known. For example, ω = γ when n = 1, ω = γ2 when n = 2, and so on. 3. Result analysis
Rm ( um−1 , Am−1 )
×
N
m−1−k−j
m−1
k
k=0
i=0
Ai Ak−i
ur (τ )um−1−k−j −r (τ )
(29)
r=0
0, m 1, 1, m > 1.
For given α, β, γ , ε and n, the periodic solution with the known frequency ω = γ /n and the corresponding unknown amplitude A can be determined by the analytic approach mentioned above. Note that there exists an auxiliary nonzero parameter h, which provides us with a simple way to ensure the convergence of solution series, as mentioned by Liao [9–17] and other authors [18–21]. For example, let us consider the case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 1 (i.e., ω = 10). Such kind of excitation is often called the primary parametric resonance. Obviously, the amplitude A contains the auxiliary nonzero parameter h. As suggested by Liao [9–17], one can plot the A–h curve to determine the so-called valid region of h, as shown in Fig. 1. Obviously, the series of A converges when 0 < h < 0.06. For instance, when h = 0.015,
Note that both um (τ ) and Am−1 are unknown, but we have now only one Eq. (27) for um (τ ). Thus, another algebraic equation must be given so as to determine Am−1 . It is found that the right-hand side of the mth-order deformation (27) is expressed by um−1 , Am−1 ) = bm,0 (Am−1 ) + hRm (
μm
bm,k (Am−1 ) cos(kτ ),
k=1
(30) where bm,k (Am−1 ) is a coefficient, μm is a positive integer dependent on the order m. According to the property (14) of the auxiliary linear operator L, when bm,1 (Am−1 ) = 0, the solution of the mth-order
Fig. 1. The 11th-order approximation of A versus h in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 1 (i.e., ω = 10).
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J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
Table 1 The analytic approximations of A by means of h Order n of approximation
A
Error: (An − An−1 )/An
1 2 3 4 5 6 7 8 9 10 11
5.76 5.77 5.79 5.80 5.81 5.82 5.83 5.83 5.84 5.85 5.85
0.00266 0.00243 0.00221 0.00200 0.00180 0.00162 0.00146 0.00131 0.00119 0.00107 0.000974 Fig. 4. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 3 (i.e., ω = 10/3). Solid line: HAM result; circle: numerical solution.
Fig. 2. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 1 (i.e., ω = 10). Solid line: HAM result; circle: numerical solution.
Fig. 5. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 4 (i.e., ω = 5/2). Solid line: HAM result; circle: numerical solution.
Fig. 3. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 2 (i.e., ω = 5). Solid line: HAM result; circle: numerical solution.
we have the convergent result A ≈ 5.85329, as shown in Table 1. The corresponding 2nd-order approximation of x(t) agrees well with the numerical result, as shown in Fig. 2, where the numerical result is obtained by substituting the convergent analytic result of the amplitude into the initial conditions x(0) = A, x(0) ˙ = 0 and using the Runge–Kutta’s numerical method. Similarly, in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 2 (i.e., ω = 5), we obtain the convergent amplitude A ≈
2.878075 by means of h = 0.04. And it is a little surprise that even the corresponding 2nd HAM approximation agrees well with the numerical ones, as shown in Fig. 3. In a similar way, in case of α = 1, β = 4, γ = 10, ε = 0.01, n = 3 and n = 4, we can always find a proper value of the auxiliary parameter h to ensure the convergence of the series A and x(t), which agree well with numerical results, as sown in Figs. 4 and 5, respectively. Note that when n = 2, 3, 4, the natural frequencies of oscillation are 1/2, 1/3 and 1/4 of the frequency γ of parametric excitation, respectively. Such cases are called subharmonic resonances, and are very important in engineering. Similarly, in case of α = 0.9, β = 4, γ = 1, ε = 0.1, and n = 1 (i.e., ω = 1), we obtain the convergent amplitude A ≈ 0.183846 by means of h = 0.01. One can plot the A–h curve to determine the so-called valid region of h, as shown in Fig. 6. Obviously, the series of A converges when −0.05 < h < 0.05. And it is a little surprise that even the corresponding 2nd HAM approximation agrees well with the numerical ones, as shown in Fig. 7.
J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
431
Indeed, the homotopy analysis method (HAM) has many advantages. Unlike perturbation methods, the HAM does not need any small parameters [8–21]. Different from all other analytic techniques, it provides us with a simple way to adjust and control the convergence region of approximate series solutions [9–21]. The current work illustrates that the HAM is indeed a powerful and versatile analytical technique for most types of nonlinear problems, and might have many applications in science and engineering. Acknowledgement
Fig. 6. The 11th-order approximation of A versus h in case of α = 0.9, β = 4, γ = 1, ε = 0.1, and n = 1 (i.e., ω = 1).
We would like to express our sincere thanks to Professor Liao for his valuable comments and discussions. References
Fig. 7. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 0.9, β = 4, γ = 1, ε = 0.1, and n = 1 (i.e., ω = 1). Solid line: HAM result; circle: numerical solution.
4. Conclusions In this Letter, an analytical technique, namely the homotopy analysis method (HAM), is applied to obtain the periodic solutions for subharmonic resonances of nonlinear oscillations with parametric excitation. For given physical parameters and the frequency of sub-harmonic resonances, the convergent series solutions of the corresponding amplitude A and x(t) are explicitly obtained, which agree well with the numerical results. All of these verify that the HAM is valid for sub-harmonic resonances of nonlinear oscillation systems with parametric excitation.
[1] J. Zhou, Y. Zhu, Nonlinear Oscillations, Xi’an Jiaotong University Press, 1998, pp. 110–115. [2] A.H. Nayfeh, D.T. Mook, in: J. Song, W. Luo, S. Chen (Eds.), Nonlinear Oscillations, High Education Press, 1980, pp. 175–261. [3] N. Ali Hasan, Introduction to Perturbation Techniques, John Wiley & Sons, New York, 1981, pp. 1–24. [4] N. Ali Hasan, Problems in Perturbation, John Wiley & Sons, 1985, pp. 1–20. [5] A.M. Lyapunov, General Problems on Stability of Motion, Taylor & Francis, London, 1992. [6] A.V. Karmishin, A.I. Zhukov, V.G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Moscow, 1990. [7] D. Kaya, Appl. Math. Comput. 147 (2004) 69. [8] S.J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems. PhD thesis, Shanghai Jiaotong University, 1992. [9] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC, Boca Raton, 2003. [10] S.J. Liao, J. Fluid Mech. 385 (1999) 101. [11] S.J. Liao, A. Campo, J. Fluid Mech. 453 (2002) 411. [12] S.J. Liao, J. Fluid Mech. 488 (2003) 189. [13] S.J. Liao, Appl. Math. Comput. 147 (2) (2004) 499. [14] S.J. Liao, J. Su, A.T. Chwang, Int. J. Heat Mass Transfer 49 (2006) 2437. [15] S.J. Liao, Stud. Appl. Math. 117 (3) (2006) 239. [16] S.J. Liao, E. Magyari, Z. Angew. Math. Phys. (ZAMP) 57 (5) (2006) 777. [17] S.J. Liao, Int. J. Heat Mass Transfer 48 (2005) 2529. [18] T. Hayat, M. Sajid, Phys. Lett. A 361 (2007) 316. [19] S. Abbasbandy, Phys. Lett. A 361 (2007) 478. [20] S. Abbasbandy, Phys. Lett. A 360 (2006) 109. [21] F.M. Allan, M.I. Syam, J. Comput. Appl. Math. 182 (2005) 362.
Sub-harmonic resonances of nonlinear oscillations with parametric excitation by means of the homotopy analysis method Wen Jianmin a,∗ , Cao Zhengcai b a School of Automobile Engineering, Harbin Institute of Technology at Weihai, Weihai 264209, China b CIMS Research Center, Tongji University, Shanghai 200092, China
Received 31 January 2007; received in revised form 17 September 2007; accepted 24 September 2007 Available online 29 September 2007 Communicated by A.R. Bishop
Abstract An analytical technique, namely the homotopy analysis method (HAM), is applied to solve periodic solutions for sub-harmonic resonances of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM does not depend on any small physical parameters at all. Thus, it is valid for both weakly and strongly nonlinear problems. Besides, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter h. In this Letter, periodic analytic approximations for sub-harmonic resonances of nonlinear oscillations with parametric excitation are obtained by using the HAM for the first time, which agree well with numerical results. This Letter shows that the HAM is a powerful and effective technique for nonlinear dynamical systems. © 2007 Elsevier B.V. All rights reserved. Keywords: Sub-harmonic resonances; Nonlinear oscillation; Series solutions; Homotopy analysis method (HAM)
1. Introduction Exact solutions of differential equations are rare in many branches of fluid mechanics, solid mechanics, and physics because of nonlinearity, inhomogeneity, variable coefficients, and so on [1,2]. Hence, it is often necessary to determine approximate solutions of complicated nonlinear differential equations by using some analytical techniques. Foremost among the analytic techniques are the methods of perturbations (asymptotic expansions) in terms of a small/large parameter [3,4]. Because many nonlinear problems do not contain such a small/large parameter named perturbation quantity, the artificial small parameter method, the δ-expansion method and Adomian’s decomposition method were developed [5–7]. However, both of these perturbation and non-perturbation meth* Corresponding author.
E-mail addresses: [email protected] (J. Wen), [email protected] (Z. Cao). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.09.057
ods cannot present a simple way to adjust and control the convergence region of approximate solutions. In 1992, Liao [8] employed the basic ideas of the homotopy in topology to propose a general analytic method for nonlinear problems, namely the homotopy analysis method (HAM). Unlike perturbation methods, the HAM does not need any small parameters. Different from all other analytic techniques, the HAM provides us with a simple way to adjust and control the convergence region of approximate series solutions [9]. HAM has been successfully applied to solve many types of nonlinear problems [10–21]. In this Letter, the basic idea of HAM is used to solve nonlinear oscillation systems with parametric excitations, governed by d 2x + 1 + ε cos(γ t) αx + βx 3 = 0, 2 dt
(1)
where x(t) is an unknown real function, ε, α, β, γ are known physical parameters. Here, we are interested in the periodic os-
428
J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
cillations with the frequency γ (2) , n = 1, 2, 3, . . . . n In other words, given ε, α, β, γ and the frequency described above, the HAM is applied to find the corresponding unknown amplitude A, such that ω=
x (0) = 0.
x(0) = A,
(3)
Note that A is unknown in above equation, and is determined by an approach mentioned below. 2. Basic idea of HAM
(4)
and x(t) = AV (t),
(5)
Eq. (1) becomes V (t) + (1 + ε cos nωt) αV (t) + βA2 V 3 (t) = 0,
V (0) = 0.
+∞
ck cos kωt,
(8)
subject to the initial conditions (10)
According to the solution expression (8), u(τ ) can be expressed by u(τ ) =
+∞
ck cos kτ.
(11)
k=1
According to (11) and the initial conditions (10), it is obvious for us to choose such an initial guess u0 (τ ) = cos τ.
∂ 2 [Φ(τ, q)] ∂t 2 + (1 + ε cos nτ ) αΦ(τ, q) + βΛ2 (q)Φ 3 (τ, q) ,
(15)
where q ∈ [0, 1] is an embedding parameter, Φ(τ, q) is a kind of mapping of the unknown function u(τ ), and Λ(q) is a kind of mapping of the unknown amplitude A. Then, we construct the so-called zero-order deformation equation (1 − q)L Φ(τ, q) − u0 (τ ) = hqN Φ(τ, q), Λ(q) , (16)
where h = 0 is a nonzero auxiliary parameter. When q = 0, the solution of Eqs. (16) and (17) is obviously
where ck is a coefficient to be determined. This provides us with the so-called solution expression of V (t), which plays an important role in the frame of the HAM, as shown later. For the sake of simplicity, using the transformation τ = ωt and V (t) = u(τ ), Eqs. (6) and (7) are rewritten as ω2 u (τ ) + (1 + ε cos nτ ) αu(τ ) + βA2 u3 (τ ) = 0, (9)
u (0) = 0.
for any integration constants C1 and C2 . According to Eq. (9), we define a nonlinear operator N Φ(τ, q), Λ(q)
(7)
k=1
u(0) = 1,
(14)
(6)
Here, we emphasize that the value of A in Eq. (6) is unknown. According to the initial conditions (7) and the nonlinear term in Eq. (6), the periodic solution of V (t) with the given frequency ω can be expressed by V (t) =
L(C1 sin τ + C2 cos τ ) = 0,
subject to the initial conditions ∂Φ(τ, q) Φ(0, q) = 1, = 0, ∂τ τ =0
subject to the initial conditions V (0) = 1,
which has the property
= ω2
Assuming that the solution of Eqs. (1) and (3) is periodic with the frequency ω defined by (2). Then, writing γ = nω
According to (11), obviously, we should choose the auxiliary linear operator 2 ∂ Φ(τ, q) + Φ(τ, q) , L Φ(τ, q) = ω2 (13) ∂t 2
(12)
Φ(τ, 0) = u0 (τ ).
(17)
(18)
When q = 1, the zero-order deformation equations (16) and (17) are equivalent to the original equations (9) and (10), provided Φ(τ, 1) = u(τ ),
(19)
Λ(1) = A.
(20)
Therefore, as the embedding parameter q increases from 0 to 1, Φ(τ, q) varies from the initial guess u0 (τ ) = cos τ to the unknown solution u(τ ) of Eqs. (9) and (10). Likewise, Λ(q) varies from the initial guess A0 to the unknown amplitude A. Note that the zero-order deformation equation (16) contains a nonzero auxiliary parameter h. Assuming that h is chosen so properly that the zero-order deformation equations (16) and (17) have solutions in the whole region q ∈ [0, 1], and besides, there exist 1 ∂ m Φ(τ, q) um (τ ) = (21) m! ∂q m q=0 and
1 d m Λ(q) Am = m! dq m q=0
(22)
for any m 1. Then, by means of Taylor series and using (18), we have Φ(τ, q) = u0 (τ ) +
+∞ m=1
um (τ )q m ,
(23)
J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
Λ(q) = A0 +
+∞
Am q m .
(24)
m=1
Assuming that h is so properly chosen that the power series (23) and (24) converge at q = 1, using (19) and (20), we have the series solutions +∞
u(τ ) = u0 (τ ) +
(25)
um (τ ),
m=1
A = A0 +
+∞
(26)
Am .
m=1
429
deformation equation (27) and (28) contains the so-called secular term τ cos τ , which however disobeys the solution expression (8). To avoid the secular terms, we must enforce bm,l (Am−1 ) = 0,
(31)
which provides us another algebraic equation for Am−1 . Then, the general solution of Eq. (27) reads μm bm,k (Am−1 ) h um (τ ) = χm um−1 (τ ) + 2 cos(kτ ) ω (1 − k 2 ) k=2
+ C1 cos τ + C2 sin τ,
(32)
For the sake of simplicity, define the vectors
um = u0 (τ ), u1 (τ ), u2 (τ ), . . . , um (τ ) , Am = {A0 , A1 , A2 , . . . , Am }.
where the integration constants C1 , C2 are determined by the initial conditions (28). The N th-order approximation is given by
Differentiating the zero-order deformation equations (16) and (17) m times with respect to the embedding parameter q, then dividing them by m!, and finally setting q = 0, we have the socalled mth-order deformation equation um−1 , Am−1 ) L um (τ ) − χm um−1 (τ ) = hRm ( (27)
u(τ ) ≈ u0 (τ ) +
subject to the initial conditions um (0) = 0,
um (0) = 0,
(28)
where ∂ m−1 N [Φ(τ, q), Λ(q)] 1 = (m − 1)! ∂q m−1 q=0
= ω2 um−1 (τ )
+ (1 + ε cos nτ ) αum−1 + β
m−1−k
uj (τ )
j =0
and χm =
um (τ ),
(33)
m=1
A ≈ A0 +
N
Am .
(34)
m=1
Note that γ is given, and thus ω = γn determined by the given positive integer n is known. For example, ω = γ when n = 1, ω = γ2 when n = 2, and so on. 3. Result analysis
Rm ( um−1 , Am−1 )
×
N
m−1−k−j
m−1
k
k=0
i=0
Ai Ak−i
ur (τ )um−1−k−j −r (τ )
(29)
r=0
0, m 1, 1, m > 1.
For given α, β, γ , ε and n, the periodic solution with the known frequency ω = γ /n and the corresponding unknown amplitude A can be determined by the analytic approach mentioned above. Note that there exists an auxiliary nonzero parameter h, which provides us with a simple way to ensure the convergence of solution series, as mentioned by Liao [9–17] and other authors [18–21]. For example, let us consider the case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 1 (i.e., ω = 10). Such kind of excitation is often called the primary parametric resonance. Obviously, the amplitude A contains the auxiliary nonzero parameter h. As suggested by Liao [9–17], one can plot the A–h curve to determine the so-called valid region of h, as shown in Fig. 1. Obviously, the series of A converges when 0 < h < 0.06. For instance, when h = 0.015,
Note that both um (τ ) and Am−1 are unknown, but we have now only one Eq. (27) for um (τ ). Thus, another algebraic equation must be given so as to determine Am−1 . It is found that the right-hand side of the mth-order deformation (27) is expressed by um−1 , Am−1 ) = bm,0 (Am−1 ) + hRm (
μm
bm,k (Am−1 ) cos(kτ ),
k=1
(30) where bm,k (Am−1 ) is a coefficient, μm is a positive integer dependent on the order m. According to the property (14) of the auxiliary linear operator L, when bm,1 (Am−1 ) = 0, the solution of the mth-order
Fig. 1. The 11th-order approximation of A versus h in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 1 (i.e., ω = 10).
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Table 1 The analytic approximations of A by means of h Order n of approximation
A
Error: (An − An−1 )/An
1 2 3 4 5 6 7 8 9 10 11
5.76 5.77 5.79 5.80 5.81 5.82 5.83 5.83 5.84 5.85 5.85
0.00266 0.00243 0.00221 0.00200 0.00180 0.00162 0.00146 0.00131 0.00119 0.00107 0.000974 Fig. 4. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 3 (i.e., ω = 10/3). Solid line: HAM result; circle: numerical solution.
Fig. 2. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 1 (i.e., ω = 10). Solid line: HAM result; circle: numerical solution.
Fig. 5. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 4 (i.e., ω = 5/2). Solid line: HAM result; circle: numerical solution.
Fig. 3. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 2 (i.e., ω = 5). Solid line: HAM result; circle: numerical solution.
we have the convergent result A ≈ 5.85329, as shown in Table 1. The corresponding 2nd-order approximation of x(t) agrees well with the numerical result, as shown in Fig. 2, where the numerical result is obtained by substituting the convergent analytic result of the amplitude into the initial conditions x(0) = A, x(0) ˙ = 0 and using the Runge–Kutta’s numerical method. Similarly, in case of α = 1, β = 4, γ = 10, ε = 0.01, and n = 2 (i.e., ω = 5), we obtain the convergent amplitude A ≈
2.878075 by means of h = 0.04. And it is a little surprise that even the corresponding 2nd HAM approximation agrees well with the numerical ones, as shown in Fig. 3. In a similar way, in case of α = 1, β = 4, γ = 10, ε = 0.01, n = 3 and n = 4, we can always find a proper value of the auxiliary parameter h to ensure the convergence of the series A and x(t), which agree well with numerical results, as sown in Figs. 4 and 5, respectively. Note that when n = 2, 3, 4, the natural frequencies of oscillation are 1/2, 1/3 and 1/4 of the frequency γ of parametric excitation, respectively. Such cases are called subharmonic resonances, and are very important in engineering. Similarly, in case of α = 0.9, β = 4, γ = 1, ε = 0.1, and n = 1 (i.e., ω = 1), we obtain the convergent amplitude A ≈ 0.183846 by means of h = 0.01. One can plot the A–h curve to determine the so-called valid region of h, as shown in Fig. 6. Obviously, the series of A converges when −0.05 < h < 0.05. And it is a little surprise that even the corresponding 2nd HAM approximation agrees well with the numerical ones, as shown in Fig. 7.
J. Wen, Z. Cao / Physics Letters A 371 (2007) 427–431
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Indeed, the homotopy analysis method (HAM) has many advantages. Unlike perturbation methods, the HAM does not need any small parameters [8–21]. Different from all other analytic techniques, it provides us with a simple way to adjust and control the convergence region of approximate series solutions [9–21]. The current work illustrates that the HAM is indeed a powerful and versatile analytical technique for most types of nonlinear problems, and might have many applications in science and engineering. Acknowledgement
Fig. 6. The 11th-order approximation of A versus h in case of α = 0.9, β = 4, γ = 1, ε = 0.1, and n = 1 (i.e., ω = 1).
We would like to express our sincere thanks to Professor Liao for his valuable comments and discussions. References
Fig. 7. Comparison of the 2nd-order HAM approximation of x(t) with the numerical solution in case of α = 0.9, β = 4, γ = 1, ε = 0.1, and n = 1 (i.e., ω = 1). Solid line: HAM result; circle: numerical solution.
4. Conclusions In this Letter, an analytical technique, namely the homotopy analysis method (HAM), is applied to obtain the periodic solutions for subharmonic resonances of nonlinear oscillations with parametric excitation. For given physical parameters and the frequency of sub-harmonic resonances, the convergent series solutions of the corresponding amplitude A and x(t) are explicitly obtained, which agree well with the numerical results. All of these verify that the HAM is valid for sub-harmonic resonances of nonlinear oscillation systems with parametric excitation.
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