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An equivalent nonlinearization method for strongly nonlinear oscillations
MECHANICS RESEARCH COMMUNICATIONS
Mechanics Research Communications 32 (2005) 553–560 www.elsevier.com/locate/mechrescom
An equivalent nonlinearization method for strongly nonlinear oscillations Jianping Cai a
a,*
, Xiaofeng Wu b, Y.P. Li
c
Department of Mathematics, Zhangzhou Teachers College, Fujian 363000, China Department of Mathematics, Zhongshan University, Guangzhou 510275, China c Faculty of Science and Technology, University of Macau, Macau, China
b
Available online 29 October 2004
Abstract An equivalent nonlinearization method is proposed for the study of certain kinds of strongly nonlinear oscillators. This method is to express the nonlinear restored force of an oscillatory system by a polynomial of degree two or three such that the asymptotic solutions can be derived in terms of elliptic functions. The least squares method is used to determine the coefficients of approximate polynomials. The advantage of present method is that it is valid for relatively large oscillations. As an application, a strongly nonlinear oscillator with slowly varying parameters resulted from freeelectron laser is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method. 2004 Elsevier Ltd. All rights reserved. Keywords: Strongly nonlinear oscillator; Equivalent nonlinearization; Slowly varying parameter; Least squares method; Free-electron laser
1. Introduction In control engineering and oscillatory problems, we often meet with the following strongly nonlinear oscillator d2 y dy þ ekðy; ~tÞ þ gðy; ~tÞ ¼ 0 dt2 dt
*
Corresponding author. E-mail address: [email protected] (J. Cai).
0093-6413/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2004.10.004
ð1Þ
554
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
where ~t ¼ etð0 < e 1Þ is the slow scale. Methods of multiple scales and generalized KBM are effective to deal with such systems (for examples, Kuzmak, 1959; Kevorkian, 1987; Yuste, 1991; Dai and Zhuang, 1986; Cai and Li, 2004). However, these methods are based on the fact that the reduced equation (e = 0), although nonlinear, has a solution in terms of known functions. The solution can be expressed in terms of Jacobian elliptic functions when gðy; ~tÞ is a polynomial of degree two or three with respect to y. For generally nonlinear functions gðy; ~tÞ, Taylor series expansions (Nayfeh and Mook, 1979) and equivalent linearization methods (Krylov and Bogbliubov, 1943) are often used to approximate the nonlinear functions but the two methods are effective only for small amplitudes. Many efforts have been done to overcome this difficulty, such as Fourier series and approximate potential methods (Kevorkian and Li, 1988; Cai and Li, 2003), energy method (Li, 1995), generalized harmonic function (Xu and Cheung, 1994), perturbation-incremental method (Chan et al., 1996), combined equivalent linearization and averaging method (Mickens, 2003) and linearized perturbation technique (He, 2003). In this paper, an equivalent nonlinearization method is proposed to overcome the difficulty caused by certain kinds of nonlinear functions gðy; ~tÞ. This method is to approximate the nonlinear function by a polynomial of degree two or three such that the leading approximation is expressible in terms of elliptic functions. The least squares method is used to determined the coefficients of approximate polynomials. As an application, a strongly nonlinear oscillator with slowly varying parameters which models motion of free-electron laser is studied in detail. Compared with Taylor series expansions method, the advantage of present method is that it is valid for relatively large oscillations. Comparisons are also made with the numerical method and Taylor series expansions method to show the efficiency of the present method. 2. Basic idea of equivalent nonlinearization method For simplicity, we use the reduced equation of Eq. (1) d2 y þ gðyÞ ¼ 0 ð2Þ dt2 to illustrate the main idea of equivalent nonlinearization method. We may seek a polynomial p(y) to approximate the nonlinear function g(y) if the characteristic of g(y) is similar to a polynomial of degree two or three. The least squares method can be used to identify the coefficients of p(y), which requires minimizing the following expression Z y2 ðgðyÞ pðyÞÞ2 dy ! min : y1
where y1 and y2 are chosen by the concerned range of oscillation. The potential energy V ðyÞ ¼ can give useful information about this. The details are illustrated by some examples.
Ry 0
gðuÞdu
Example 1. Consider the following nonlinear oscillator d2 y þ a sin y by ¼ 0 dt2 The energy integral is 2 1 dy þ V ðyÞ ¼ 0 2 dt where 1 V ðyÞ ¼ a cos y by 2 þ a 2
ð3Þ
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
555
is the potential (for simplicity we chose V(0) = 0). When a P 1 and 0 < b < 1, V(y) has a minimum point at y = 0 and two maximum points at y = ±ys, 0 < ys 6 p. There are periodic solutions around oscillatory center y = 0 (see Cai and Li, 2003 for details). According to the characteristic of function a sin y by, we seek a polynomial of the form cy + dy3 satisfying Z y2 2 F ðc; dÞ ¼ ða sin y by cy dy 3 Þ dy ! min : y1
¼ 0 and oF ¼ 0. y1 and y2 can be chosen within the interval which requires coefficients c and d satisfying oF oc od [ys, ys]. For example, y1 and y2 can be chosen as p2 and p2 respectively if ba 6 p2. Then c and d can be worked out as c ¼ 0:988792a b;
d ¼ 0:145062a
So the equivalent nonlinear equation of Eq. (3) is d2 y þ ð0:988792a bÞy 0:145062ay 3 ¼ 0 dt2
ð4Þ
The solutions of Eq. (4) can be expressed in terms of Jacobian elliptic function (see Chen and Cheung, 1996 for details). Comparisons of the numerical solution with asymptotic solutions of Eq. (3) with a = 1, and y 0 (0) = 0 are shown in Fig. 1. In this paper the symbolic language Mathematica is used b ¼ p2, yð0Þ ¼ 2p 5 to implement the asymptotic and numerical solutions. Example 2. Consider a mass slides on a smooth surface while restrained by a linear spring modeled as (Ex. 2.18 in Nayfeh and Mook, 1979) d2 y 2ay þ 2y pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 dt2 1 þ y2
ð5Þ
1
0.5
10
20
30
40
50
-0.5
-1
numerical solution, - - - equivalent nonlinearization method ____ third-order Taylor expansions method Fig. 1. Solution and approximations of Eq. (3).
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J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
Its potential energy V ðyÞ ¼ y 2 2a
pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y 2 þ 2a
is ‘‘U-shaped’’ and there are periodic solutions around oscillatory center y = 0 when 0 < a < 1. For the amplitude y 2 [2.5, 2.5] (2.5 is chosen according to the concerned range of amplitude, another value is available), we seek a polynomial of the form cy + dy3 satisfying !2 Z 2:5 2ay 2y pffiffiffiffiffiffiffiffiffiffiffiffiffi cy dy 3 dy ! min : ð6Þ 1 þ y2 2:5 which gives c = 21.47163a, d = 0.132121a. So the equivalent nonlinear equation of Eq. (5) becomes d2 y þ ð2 1:47163aÞy þ 0:132121ay 3 ¼ 0 dt2 and Comparisons of the numerical solution with asymptotic solutions of Eq. (5) with a = 0.5, yð0Þ ¼ 2p 3 y 0 (0) = 0 are shown inpFig. 2. When a > 1, the potential V has one minimum point at y = 0 and two maxffiffiffiffiffiffiffiffiffiffiffiffiffi imum points at y ¼ pffiffiffiffiffiffiffiffiffiffiffiffi a2 1ffi . The potential is ‘‘W-shaped’’ and there are two families of periodic solutions centered about y ¼ a2 1. Here we consider only the oscillation around the right hand side center. We look for a polynomial of the form cy + dy3 satisfying !2 Z y2 2ay 3 2y pffiffiffiffiffiffiffiffiffiffiffiffiffi cy dy dy ! min : 1 þ y2 y1 By the potential function V, we can approximate the range of amplitude, which can be used to choose the values of y1 and y2. For a = 1.1, y1 and y2 can be chosen as 0 and 0.6 respectively. Then c and d can be worked out as c = 21.9835a, d = 0.766629a. So the equivalent nonlinear equation of Eq. (5) is
2
1
5
10
15
20
25
30
-1
-2
numerical solution, - - - equivalent nonlinearization method ____ third-order Taylor expansions method Fig. 2. Solution and approximations of Eq. (5) with a = 0.5.
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
557
0.6 0.5 0.4 0.3 0.2 0.1
10
20
30
40
50
numerical solution, - - - equivalent nonlinearization method third-order Taylor expansions method Fig. 3. Solution and approximations of Eq. (5) with a = 1.1.
d2 y þ ð2 1:9835aÞy þ 0:766629ay 3 ¼ 0 dt2
ð7Þ
Comparisons of the numerical solution with asymptotic solutions of Eq. (5) with a = 1.1, y(0) = 0.1 and y 0 (0) = 0 are shown in Fig. 3.
3. Application to strongly nonlinear oscillators with slowly varying parameters We now apply the equivalent nonlinearization method to a pendulum with a slowly varying ‘‘tangential force’’ (Eq. (4.14) in Kevorkian and Li, 1988) d2 y dy ð8Þ þ ekðy; ~tÞ þ cð~tÞ sin y dð~tÞ ¼ 0 dt2 dt ~tÞ where cð~tÞ > 0, dð~tÞ > 0, dð < 1 and ~t ¼ etð0 < e 1Þ is the slow scale. Kevorkian and Li (1988) used cð~tÞ respectively approximate potential and Fourier series methods to obtain the asymptotic solutions of Eq. (8) but their results have evident errors compared with numerical results. From the potential of system ~tÞ (8), we know that the system oscillates around the resonance center y r ¼ arcsin dð and a saddle point is cð~tÞ located at y s ð~tÞ ¼ p y r ð~tÞ. With the nonlinear function cð~tÞ sin y dð~tÞ, the classical perturbation methods are invalid because the solution of the reduced equation of Eq. (8) cannot be expressed in terms of any elementary or known transcendental functions. Approximate approaches must be used. According to the characteristic of function cð~tÞ sin y dð~tÞ, we look for a polynomial of the form a0 ð~tÞ þ a1 ð~tÞy þ a2 ð~tÞy 2 satisfying Z y2 ðcð~tÞ sin y dð~tÞ a0 ð~tÞ a1 ð~tÞy a2 ð~tÞy 2 Þ2 dy ! min : y1
where y1 and y2 can be chosen around the oscillatory center yr. Then Eq. (8) is equivalent to
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J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
d2 y dy þ ekðy; ~tÞ þ a0 ð~tÞ þ a1 ð~tÞy þ a2 ð~tÞy 2 ¼ 0 dt2 dt
ð9Þ
After taking a transformation Y = y yc (yc is the oscillatory center of system (9) and is determined by a0 ð~tÞ þ a1 ð~tÞy c þ a2 ð~tÞy 2c ¼ 0Þ, Eq. (9) becomes d2 Y dY þ að~tÞY þ bð~tÞY 2 ¼ 0 þ ekðY ; ~tÞ ð10Þ 2 dt dt where að~tÞ ¼ a1 ð~tÞ þ 2a2 ð~tÞy c ; bð~tÞ ¼ a2 ð~tÞ. Now the multiple scales method can be used to obtain the asymptotic solutions of Eq. (10). Suppose that the solution of Eq. (10) can be developed in the multiple scales form Y ðt; eÞ ¼ Y 0 ðtþ ; ~tÞ þ eY 1 ðtþ ; ~tÞ þ e2 Y 2 ðtþ ; ~tÞ þ +
ð11Þ dtþ dt
where ~t ¼ et is the slow scale. The fast scale t , following Kuzmak (1959), is defined as ¼ xð~tÞ with an unknown xð~tÞ to be determined by the periodicity of the solution of Eq. (11). Y0, Y1, . . . , are periodic functions of t+. Substituting Eq. (11) into Eq. (10) and equating powers of e gives the leading order equation x2 ð~tÞ
o2 Y 0 þ að~tÞY 0 þ bð~tÞY 20 ¼ 0 otþ2
ð12Þ
The solution of Eq. (12) can be expressed in terms of Jacobian elliptic function (see Cai and Li, 2004 for details) Y 0 ¼ A0 ð~tÞcn2 ½KðvÞu; vð~tÞ þ B0 ð~tÞ
ð13Þ p ffiffi ffi where u = t+ + u0, K(v) is the complete elliptic integral of the first kind associated with the modulus v and 3av a 2v 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 A0 ¼ B0 ¼ 2b 2b v2 v þ 1 v2 v þ 1 x4
¼
16K
4
a2 v þ 1Þ
ðv2
The equation governing v is v2 J ðvÞ 5
ðv2 v þ 1Þ4
¼
2cb2 5
9a2
Z
!
~t
exp
kð0; sÞds 0
where constant c can be determined by initial values of the system, and Z K 1 J ðvÞ ¼ sn2 ðu; vÞcn2 ðu; vÞdn2 ðu; vÞdu ¼ ½ð1 vÞðv 2ÞKðvÞ þ 2ðv2 v þ 1ÞEðvÞ 15v2 0
pffiffiffi Here, E(v) is the complete elliptic integral of the second kind associated with the modulus v. This paper just concerns applications of leading order approximations. More details of higher order solutions, readers can refer to Kevorkian and Li (1988) or Li (1987). Example 3. Consider an equation of motion of free-electron laser (Eq. (2.23) in Kevorkian and Li (1988)) d2 / d/ 1 ¼0 þ cð~zÞ sin / dð~zÞ þ e2 ðeð~zÞ sin / þ dð~zÞÞ 2 d^z d^z 1
ð14Þ
9:376:31~z 4:983:36~z 14:19:47~z where ~z ¼ e2^z, cð~zÞ ¼ 7:554:98~ , dð~zÞ ¼ 7:554:98~ and eð~zÞ ¼ 7:554:98~ . The system oscillates zþ1:68~z2 zþ1:68~z2 zþ1:68~z2 4:983:36~z around the oscillatory center / ¼ arcsin 9:376:31~z, which is about 0.56. For the amplitude / 2 [0.4, 1.9], the equivalent equation of Eq. (14) is
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
559
1.5
1
0.5
10
20
30
40
50
numerical solution, - - - equivalent nonlinearization method third-order Taylor expansions method Fig. 4. Solution and approximations of Eq. (14).
d2 / d/ 1 þ 0:01113cð~zÞ dð~zÞ þ 1:12631cð~zÞ/ 0:314788cð~zÞ/2 ¼ 0 þ e2 ðeð~zÞ sin / þ dð~zÞÞ 2 d^z d^z Following the above procedure, we can carry out the approximate solution of leading order. Comparisons of the numerical solution with asymptotic solutions of Eq. (14) with e = 3.36 · 104, /(0) = 1.5 and / 0 (0) = 0 are shown in Fig. 4. It is worth to point out that the present result is more accurate than the results of approximate potential and Fourier series methods (see Figs. 3 and 4 in Kevorkian and Li (1988)).
4. Conclusions (1) The equivalent nonlinearization method presented in this paper is effective for certain strongly nonlinear oscillators whose characteristic of nonlinear restored force is similar to polynomials of degree two or three. The least squares method can be used to determine the coefficients of approximate polynomials. The method works not only for small oscillations but also for relatively large oscillations. (2) The comparisons show that the results of the present method are in good agreement with the numerical results, while the results of Taylor series expansions have large errors when the amplitudes are not small. (3) The application of the present method to the equation of motion of free-electron laser gives better result than that of Kevorkian and Li (1988).
Acknowledgment The authors are grateful to an anonymous referee for helpful comments.
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References Cai, J., Li, Y.P., 2003. Approximate potentials with applications to strongly nonlinear oscillators with slowly varying parameters. Shock and Vibration 10 (5–6), 379–386. Cai, J., Li, Y.P., 2004. Strongly non-linear oscillations with slowly varying parameters. Journal of Sound and Vibration 275 (1–2), 241– 248. Chan, H.S.Y., Chung, K.W., Xu, Z., 1996. A perturbation-incremental method for strongly nonlinear oscillators. International Journal of Non-Linear Mechanics 31 (1), 59–72. Chen, S.H., Cheung, Y.K., 1996. Perturbation method for certain strongly nonlinear oscillations. Journal of Sound and Vibration 192 (2), 453–464. Dai, S.Q., Zhuang, F.Q., 1986. Asymptotic solutions to a class of nonlinear oscillation systems. Scientia Sinica (Science in China) Series A 29 (7), 737–744. He, J.H., 2003. Linearized perturbation technique and its applications to strongly nonlinear oscillations. Computers and mathematics with Applications 45, 1–8. Kevorkian, J., 1987. Perturbation techniques for oscillatory systems with slowly varying coefficients. SIAM Review 29, 391–461. Kevorkian, J., Li, Y.P., 1988. Explicit approximations for strictly nonlinear oscillators with slowly varying parameters with applications to free-electron lasers. Studies in Applied Mathematics 78 (2), 111–165. Krylov, R.E., Bogbliubov, N., 1943. Introduction to Nonlinear Mechanics. Princeton University Press, Princeton. Kuzmak, G.Z., 1959. Asymptotic solutions of nonlinear second order differential equations with variable coefficients. Pure Math Manuscript 23, 515–526. Li, L., 1995. Energy method for computing periodic solutions of strongly nonlinear systems (1)-autonomous systems. Nonlinear Dynamics 9 (3), 223–247. Li, Y.P., 1987. Free electron lasers with variable parameter wigglers, a strictly nonlinear oscillator with slowly varying parameters. PhD dissertation. University of Washington, Seattle. Mickens, R.E., 2003. A combined equivalent linearization and averaging perturbation method for nonlinear oscillator equations. Journal of Sound and Vibration 264, 1195–1200. Nayfeh, A.H., Mook, D.T., 1979. Nonlinear oscillations. Wiley, New York. Xu, Z., Cheung, Y.K., 1994. Averaging method using generalized harmonic functions for strongly nonlinear oscillators. Journal of Sound and Vibration 174 (4), 563–576. Yuste, S.B., 1991. On Duffing oscillators with slowly varying parameters. International Journal of Non-Linear Mechanism 26 (5), 671– 677.
Mechanics Research Communications 32 (2005) 553–560 www.elsevier.com/locate/mechrescom
An equivalent nonlinearization method for strongly nonlinear oscillations Jianping Cai a
a,*
, Xiaofeng Wu b, Y.P. Li
c
Department of Mathematics, Zhangzhou Teachers College, Fujian 363000, China Department of Mathematics, Zhongshan University, Guangzhou 510275, China c Faculty of Science and Technology, University of Macau, Macau, China
b
Available online 29 October 2004
Abstract An equivalent nonlinearization method is proposed for the study of certain kinds of strongly nonlinear oscillators. This method is to express the nonlinear restored force of an oscillatory system by a polynomial of degree two or three such that the asymptotic solutions can be derived in terms of elliptic functions. The least squares method is used to determine the coefficients of approximate polynomials. The advantage of present method is that it is valid for relatively large oscillations. As an application, a strongly nonlinear oscillator with slowly varying parameters resulted from freeelectron laser is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method. 2004 Elsevier Ltd. All rights reserved. Keywords: Strongly nonlinear oscillator; Equivalent nonlinearization; Slowly varying parameter; Least squares method; Free-electron laser
1. Introduction In control engineering and oscillatory problems, we often meet with the following strongly nonlinear oscillator d2 y dy þ ekðy; ~tÞ þ gðy; ~tÞ ¼ 0 dt2 dt
*
Corresponding author. E-mail address: [email protected] (J. Cai).
0093-6413/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2004.10.004
ð1Þ
554
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
where ~t ¼ etð0 < e 1Þ is the slow scale. Methods of multiple scales and generalized KBM are effective to deal with such systems (for examples, Kuzmak, 1959; Kevorkian, 1987; Yuste, 1991; Dai and Zhuang, 1986; Cai and Li, 2004). However, these methods are based on the fact that the reduced equation (e = 0), although nonlinear, has a solution in terms of known functions. The solution can be expressed in terms of Jacobian elliptic functions when gðy; ~tÞ is a polynomial of degree two or three with respect to y. For generally nonlinear functions gðy; ~tÞ, Taylor series expansions (Nayfeh and Mook, 1979) and equivalent linearization methods (Krylov and Bogbliubov, 1943) are often used to approximate the nonlinear functions but the two methods are effective only for small amplitudes. Many efforts have been done to overcome this difficulty, such as Fourier series and approximate potential methods (Kevorkian and Li, 1988; Cai and Li, 2003), energy method (Li, 1995), generalized harmonic function (Xu and Cheung, 1994), perturbation-incremental method (Chan et al., 1996), combined equivalent linearization and averaging method (Mickens, 2003) and linearized perturbation technique (He, 2003). In this paper, an equivalent nonlinearization method is proposed to overcome the difficulty caused by certain kinds of nonlinear functions gðy; ~tÞ. This method is to approximate the nonlinear function by a polynomial of degree two or three such that the leading approximation is expressible in terms of elliptic functions. The least squares method is used to determined the coefficients of approximate polynomials. As an application, a strongly nonlinear oscillator with slowly varying parameters which models motion of free-electron laser is studied in detail. Compared with Taylor series expansions method, the advantage of present method is that it is valid for relatively large oscillations. Comparisons are also made with the numerical method and Taylor series expansions method to show the efficiency of the present method. 2. Basic idea of equivalent nonlinearization method For simplicity, we use the reduced equation of Eq. (1) d2 y þ gðyÞ ¼ 0 ð2Þ dt2 to illustrate the main idea of equivalent nonlinearization method. We may seek a polynomial p(y) to approximate the nonlinear function g(y) if the characteristic of g(y) is similar to a polynomial of degree two or three. The least squares method can be used to identify the coefficients of p(y), which requires minimizing the following expression Z y2 ðgðyÞ pðyÞÞ2 dy ! min : y1
where y1 and y2 are chosen by the concerned range of oscillation. The potential energy V ðyÞ ¼ can give useful information about this. The details are illustrated by some examples.
Ry 0
gðuÞdu
Example 1. Consider the following nonlinear oscillator d2 y þ a sin y by ¼ 0 dt2 The energy integral is 2 1 dy þ V ðyÞ ¼ 0 2 dt where 1 V ðyÞ ¼ a cos y by 2 þ a 2
ð3Þ
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
555
is the potential (for simplicity we chose V(0) = 0). When a P 1 and 0 < b < 1, V(y) has a minimum point at y = 0 and two maximum points at y = ±ys, 0 < ys 6 p. There are periodic solutions around oscillatory center y = 0 (see Cai and Li, 2003 for details). According to the characteristic of function a sin y by, we seek a polynomial of the form cy + dy3 satisfying Z y2 2 F ðc; dÞ ¼ ða sin y by cy dy 3 Þ dy ! min : y1
¼ 0 and oF ¼ 0. y1 and y2 can be chosen within the interval which requires coefficients c and d satisfying oF oc od [ys, ys]. For example, y1 and y2 can be chosen as p2 and p2 respectively if ba 6 p2. Then c and d can be worked out as c ¼ 0:988792a b;
d ¼ 0:145062a
So the equivalent nonlinear equation of Eq. (3) is d2 y þ ð0:988792a bÞy 0:145062ay 3 ¼ 0 dt2
ð4Þ
The solutions of Eq. (4) can be expressed in terms of Jacobian elliptic function (see Chen and Cheung, 1996 for details). Comparisons of the numerical solution with asymptotic solutions of Eq. (3) with a = 1, and y 0 (0) = 0 are shown in Fig. 1. In this paper the symbolic language Mathematica is used b ¼ p2, yð0Þ ¼ 2p 5 to implement the asymptotic and numerical solutions. Example 2. Consider a mass slides on a smooth surface while restrained by a linear spring modeled as (Ex. 2.18 in Nayfeh and Mook, 1979) d2 y 2ay þ 2y pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 dt2 1 þ y2
ð5Þ
1
0.5
10
20
30
40
50
-0.5
-1
numerical solution, - - - equivalent nonlinearization method ____ third-order Taylor expansions method Fig. 1. Solution and approximations of Eq. (3).
556
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
Its potential energy V ðyÞ ¼ y 2 2a
pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ y 2 þ 2a
is ‘‘U-shaped’’ and there are periodic solutions around oscillatory center y = 0 when 0 < a < 1. For the amplitude y 2 [2.5, 2.5] (2.5 is chosen according to the concerned range of amplitude, another value is available), we seek a polynomial of the form cy + dy3 satisfying !2 Z 2:5 2ay 2y pffiffiffiffiffiffiffiffiffiffiffiffiffi cy dy 3 dy ! min : ð6Þ 1 þ y2 2:5 which gives c = 21.47163a, d = 0.132121a. So the equivalent nonlinear equation of Eq. (5) becomes d2 y þ ð2 1:47163aÞy þ 0:132121ay 3 ¼ 0 dt2 and Comparisons of the numerical solution with asymptotic solutions of Eq. (5) with a = 0.5, yð0Þ ¼ 2p 3 y 0 (0) = 0 are shown inpFig. 2. When a > 1, the potential V has one minimum point at y = 0 and two maxffiffiffiffiffiffiffiffiffiffiffiffiffi imum points at y ¼ pffiffiffiffiffiffiffiffiffiffiffiffi a2 1ffi . The potential is ‘‘W-shaped’’ and there are two families of periodic solutions centered about y ¼ a2 1. Here we consider only the oscillation around the right hand side center. We look for a polynomial of the form cy + dy3 satisfying !2 Z y2 2ay 3 2y pffiffiffiffiffiffiffiffiffiffiffiffiffi cy dy dy ! min : 1 þ y2 y1 By the potential function V, we can approximate the range of amplitude, which can be used to choose the values of y1 and y2. For a = 1.1, y1 and y2 can be chosen as 0 and 0.6 respectively. Then c and d can be worked out as c = 21.9835a, d = 0.766629a. So the equivalent nonlinear equation of Eq. (5) is
2
1
5
10
15
20
25
30
-1
-2
numerical solution, - - - equivalent nonlinearization method ____ third-order Taylor expansions method Fig. 2. Solution and approximations of Eq. (5) with a = 0.5.
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
557
0.6 0.5 0.4 0.3 0.2 0.1
10
20
30
40
50
numerical solution, - - - equivalent nonlinearization method third-order Taylor expansions method Fig. 3. Solution and approximations of Eq. (5) with a = 1.1.
d2 y þ ð2 1:9835aÞy þ 0:766629ay 3 ¼ 0 dt2
ð7Þ
Comparisons of the numerical solution with asymptotic solutions of Eq. (5) with a = 1.1, y(0) = 0.1 and y 0 (0) = 0 are shown in Fig. 3.
3. Application to strongly nonlinear oscillators with slowly varying parameters We now apply the equivalent nonlinearization method to a pendulum with a slowly varying ‘‘tangential force’’ (Eq. (4.14) in Kevorkian and Li, 1988) d2 y dy ð8Þ þ ekðy; ~tÞ þ cð~tÞ sin y dð~tÞ ¼ 0 dt2 dt ~tÞ where cð~tÞ > 0, dð~tÞ > 0, dð < 1 and ~t ¼ etð0 < e 1Þ is the slow scale. Kevorkian and Li (1988) used cð~tÞ respectively approximate potential and Fourier series methods to obtain the asymptotic solutions of Eq. (8) but their results have evident errors compared with numerical results. From the potential of system ~tÞ (8), we know that the system oscillates around the resonance center y r ¼ arcsin dð and a saddle point is cð~tÞ located at y s ð~tÞ ¼ p y r ð~tÞ. With the nonlinear function cð~tÞ sin y dð~tÞ, the classical perturbation methods are invalid because the solution of the reduced equation of Eq. (8) cannot be expressed in terms of any elementary or known transcendental functions. Approximate approaches must be used. According to the characteristic of function cð~tÞ sin y dð~tÞ, we look for a polynomial of the form a0 ð~tÞ þ a1 ð~tÞy þ a2 ð~tÞy 2 satisfying Z y2 ðcð~tÞ sin y dð~tÞ a0 ð~tÞ a1 ð~tÞy a2 ð~tÞy 2 Þ2 dy ! min : y1
where y1 and y2 can be chosen around the oscillatory center yr. Then Eq. (8) is equivalent to
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J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
d2 y dy þ ekðy; ~tÞ þ a0 ð~tÞ þ a1 ð~tÞy þ a2 ð~tÞy 2 ¼ 0 dt2 dt
ð9Þ
After taking a transformation Y = y yc (yc is the oscillatory center of system (9) and is determined by a0 ð~tÞ þ a1 ð~tÞy c þ a2 ð~tÞy 2c ¼ 0Þ, Eq. (9) becomes d2 Y dY þ að~tÞY þ bð~tÞY 2 ¼ 0 þ ekðY ; ~tÞ ð10Þ 2 dt dt where að~tÞ ¼ a1 ð~tÞ þ 2a2 ð~tÞy c ; bð~tÞ ¼ a2 ð~tÞ. Now the multiple scales method can be used to obtain the asymptotic solutions of Eq. (10). Suppose that the solution of Eq. (10) can be developed in the multiple scales form Y ðt; eÞ ¼ Y 0 ðtþ ; ~tÞ þ eY 1 ðtþ ; ~tÞ þ e2 Y 2 ðtþ ; ~tÞ þ +
ð11Þ dtþ dt
where ~t ¼ et is the slow scale. The fast scale t , following Kuzmak (1959), is defined as ¼ xð~tÞ with an unknown xð~tÞ to be determined by the periodicity of the solution of Eq. (11). Y0, Y1, . . . , are periodic functions of t+. Substituting Eq. (11) into Eq. (10) and equating powers of e gives the leading order equation x2 ð~tÞ
o2 Y 0 þ að~tÞY 0 þ bð~tÞY 20 ¼ 0 otþ2
ð12Þ
The solution of Eq. (12) can be expressed in terms of Jacobian elliptic function (see Cai and Li, 2004 for details) Y 0 ¼ A0 ð~tÞcn2 ½KðvÞu; vð~tÞ þ B0 ð~tÞ
ð13Þ p ffiffi ffi where u = t+ + u0, K(v) is the complete elliptic integral of the first kind associated with the modulus v and 3av a 2v 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1 A0 ¼ B0 ¼ 2b 2b v2 v þ 1 v2 v þ 1 x4
¼
16K
4
a2 v þ 1Þ
ðv2
The equation governing v is v2 J ðvÞ 5
ðv2 v þ 1Þ4
¼
2cb2 5
9a2
Z
!
~t
exp
kð0; sÞds 0
where constant c can be determined by initial values of the system, and Z K 1 J ðvÞ ¼ sn2 ðu; vÞcn2 ðu; vÞdn2 ðu; vÞdu ¼ ½ð1 vÞðv 2ÞKðvÞ þ 2ðv2 v þ 1ÞEðvÞ 15v2 0
pffiffiffi Here, E(v) is the complete elliptic integral of the second kind associated with the modulus v. This paper just concerns applications of leading order approximations. More details of higher order solutions, readers can refer to Kevorkian and Li (1988) or Li (1987). Example 3. Consider an equation of motion of free-electron laser (Eq. (2.23) in Kevorkian and Li (1988)) d2 / d/ 1 ¼0 þ cð~zÞ sin / dð~zÞ þ e2 ðeð~zÞ sin / þ dð~zÞÞ 2 d^z d^z 1
ð14Þ
9:376:31~z 4:983:36~z 14:19:47~z where ~z ¼ e2^z, cð~zÞ ¼ 7:554:98~ , dð~zÞ ¼ 7:554:98~ and eð~zÞ ¼ 7:554:98~ . The system oscillates zþ1:68~z2 zþ1:68~z2 zþ1:68~z2 4:983:36~z around the oscillatory center / ¼ arcsin 9:376:31~z, which is about 0.56. For the amplitude / 2 [0.4, 1.9], the equivalent equation of Eq. (14) is
J. Cai et al. / Mechanics Research Communications 32 (2005) 553–560
559
1.5
1
0.5
10
20
30
40
50
numerical solution, - - - equivalent nonlinearization method third-order Taylor expansions method Fig. 4. Solution and approximations of Eq. (14).
d2 / d/ 1 þ 0:01113cð~zÞ dð~zÞ þ 1:12631cð~zÞ/ 0:314788cð~zÞ/2 ¼ 0 þ e2 ðeð~zÞ sin / þ dð~zÞÞ 2 d^z d^z Following the above procedure, we can carry out the approximate solution of leading order. Comparisons of the numerical solution with asymptotic solutions of Eq. (14) with e = 3.36 · 104, /(0) = 1.5 and / 0 (0) = 0 are shown in Fig. 4. It is worth to point out that the present result is more accurate than the results of approximate potential and Fourier series methods (see Figs. 3 and 4 in Kevorkian and Li (1988)).
4. Conclusions (1) The equivalent nonlinearization method presented in this paper is effective for certain strongly nonlinear oscillators whose characteristic of nonlinear restored force is similar to polynomials of degree two or three. The least squares method can be used to determine the coefficients of approximate polynomials. The method works not only for small oscillations but also for relatively large oscillations. (2) The comparisons show that the results of the present method are in good agreement with the numerical results, while the results of Taylor series expansions have large errors when the amplitudes are not small. (3) The application of the present method to the equation of motion of free-electron laser gives better result than that of Kevorkian and Li (1988).
Acknowledgment The authors are grateful to an anonymous referee for helpful comments.
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