A Second-Order Approximate Analytical Solution of a Simple

S. J. Liao1 Pendulum by the Process Analysis

Institute of Shipbuilding, Method University of Hamburg, Laemmerseith 90, In this paper, a new kind of analytical method of nonlinear problem called the D-2000 Hamburg 60, Federal Republic of Germany process analysis method (PAM) is described and used to give a second-order ap­ proximate solution of a simple pendulum. The PAM does not depend on the small parameter supposition and therefore can overcome the disadvantages and limitations of the perturbation expansion method. The analytical approximate results at the second-order of approximation are in good agreement with the numerical results. They are compared with perturbation solutions, and it appears that even the first- order solutions are more accurate than the perturbation solutions at second-order of approximation.

1 Introduction It is difficult to solve nonlinear problems, either numerically or theoretically. Traditionally, iterative techniques were used of swing, o>0 = /-; here, g is the gravity acceleration and / to find numerical solutions of nonlinear problems, but nearly all iterative methods are sensitive to initial solutions. Thus, it is the length of the simple pendulum. is not easy to obtain converged results in cases of strong non- It is easy to know that 10(01 ^(3. If the initial angle /3 is linearity. On the other side, as mentioned by Nayfeh (1980) small enough, then 0 is a small quantity and sin(0)« 0 is a good and O'Malley (1974), the perturbation expansion method is approximation; thus, the above equation has the solution widely used to analyze simple nonlinear problems. It is well known that the perturbation method is based on small or great 0(0 = (8 cos («oO (2) parameters. But, unfortunately, not every nonlinear problem with the period has such small or great parameters. And it also seems difficult to decide whether or not a parameter is small or great enough. T0 = —. (3) For example, it is well known that the motion of a simple O)0 pendulum is periodical, which can be described mathematically But, if /3 is not small, the above solutions are not accurate. as follows: For example, when (3 = 5TT/9, the numerical result of the dH_ period is \.232T0. Therefore, higher-order approximate so­ 2 + o>0 sin0 = 0 lutions should be given in this case. However, this is not easy, dt because Eq. (1) is nonlinear and has no small parameters. It 0(0 = 0 for t = Q (1) seems doubtful to be able to give a good perturbation ap­ proximation of 0(0 and its corresponding period, especially in 9'(0 = 0 for t = 0 the case of the great initial angle /3. where 0 is the angle of swing, t is the time, /3 is the initial angle As mentioned by Ortega and Rheinboldt (1970), the con­ tinuous mapping technique, or so-called homotopy method, has been generally used to widen the domain of convergence of a given method or as a procedure to obtain sufficiently close Present address: Institute of Underwater Engineering, Department of Naval starting points. The continuous mapping technique embeds a Architecture and Ocean Engineering, Shanghai Jiao Tong University, Shanghai parameter that typically ranges from zero to one. When the 200030, P. R. China. embedding parameter is zero, the equation is one of the linear Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY system. When it is one, the equation is the same as the original. OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Then, one can iterate, numerically along the solution path, by Discussion on this paper should be addressed to the Technical Editor, Prof. incrementing the imbedding parameter from zero to one; this Leon M. Keer, The Technological Institute, Northwestern University, Evanston, continuously maps the initial linear solutions into the solutions IL 60208, and will be accepted until four months after final publication of the of the original equation. Note that iterative techniques are paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, May 5, 1990; used at each step along the solution path if the equation is final revision, Sept. 30, 1991. Associate Technical Editor: D. J. Inman. nonlinear.

970 / Vol. 59, DECEMBER 1992 Transactions of the ASME Copyright © 1992 by ASME Downloaded 24 Apr 2009 to 202.120.2.30. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm If we derive the continuous mapping with respect to the imbedding parameter we will obtain the corresponding linear «^(0)*te0) = 0 equations of this kind of derivatives. This is an interesting property of continuous mapping and a pure mathematical proof 0(z;O) = /3 forz = 0 (9) can be given. By means of this property of continuous map­ 50(z;O) ping, a kind of general numerical method for nonlinear prob­ = 0 forz = 0. lems, called the finite process method (FPM), has been dz described by Liao (1991a, 1991b). The finite process method can avoid the use of iterative techniques to solve numerically nonlinear problems and it is insensitive to the initial solutions. Denote X0 = X(0) and 0o(z) = 0(z;0). For simplicity, select X0 Therefore, it can overcome the disadvantages and limitations = 1.0, called the initial solution of X(p). It is easy to know of iterative techniques. that the above linear Eq. (9) has the solution

Based on the same property of continuous mapping, an 0o(z) = /3cos(z). (10) analytical method for nonlinear problems, called the process analysis method (PAM), has been derived. It is interesting that this kind of analytical method does not depend on small or When p = 1.0, from the zero-order process Eq. (8), one great parameters and therefore can overcome the limitations has the final equation of perturbation techniques. d26(z;1.0) In this paper, the basic idea of the process analysis method 2 - + Xll.O)sin0(z;l.O) = O is described and examined by using a simple pendulum as an dz example. The main purpose of this paper is to give a kind of 0(z;l.O) = /3 for z = 0 (11) general analytical method for nonlinear problems, which is d0(z;1.0) independent on small or great parameters. = 0 for z = 0. dz

2 Basic Ideas of the Process Analysis Method Let Equations (11) are the same as (6). Denote that 0/z) = 0(z; 1.0) and X/= X(l .0), called the finalsolution. It is easy to understand 2TT (4) that 9j(z) and \f are just what we want to know. T From the above analysis, we can see that the zero-order process Eq. (8) gives a kind of relation between the initial and solutions 0O = |S cos(z), X0 = 1.0 and the final solutions 6f, Z = u>t (5) X/. But this kind of relation is nonlinear, because the zero- order process Eq. (8) is generally a nonlinear one. In the fol­ where T is the period and o) is the frequence of a simple lowing part of this section, a linear relation between 0O, X0, pendulum, respectively. and Of, Ay will be introduced and used to give a kind of solution Then Eq. (1) is transformed into at the second-order approximation. Define 2 T+A sin0 = O dk6(z;p) dz2 e[k\z;p)- (12) dp" 0(z) = /3 for ? = 0 (6)

0'(z) = O forz = 0. dk\(p) \lk](P)- (13) dpk Here,

o0 (7) as the Ath-order process derivative of d(z\p) and \{p), respec-

denotes the non-dimensional frequence or period of a simple Suppose: pendulum. 1 9(z;p), X(p) have definition inp € [0, 1], 01

^ AV)0tep) then, according to Taylor's theory, 60(z), X0 and 0/z), X/have + the following relations: +p\2(p) {sm[6(z;p)] - 6(z;p)} = 0 (8) (14) e(z;p) = P forz = 0 Kl k=\ p = 0 dd(z;p) = 0 forz = 0 dz (15) where/? € [0, 1], called the process-independent variable or k=\ Kl /) = 0 imbedding variable. For simplicity, call the continuous mapping 6(z;p) and \{p) process, or more precisely, zero-order process. Then, Eq. (8) where A:! = 1 x2x • • • x(k- l)xk is the factorial of k. could be called the zero-order process equation. 6{k\z;p),\m(jD)a.tp = 0 can be obtained in the following way. When p = 0, from zero-order process Eq. (8), one has the Deriving the zero-order process Eq. (8) with respect to p, initial equation: one can obtain the first-order process equation as follows:

Journal of Applied Mechanics DECEMBER 1992, Vol. 59 / 971

Downloaded 24 Apr 2009 to 202.120.2.30. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm ^^ + A2(p)0" W) + 2\(P)\[1](PMZ;P) a,„ = _ ,——— (AM>1). (25) 2AM(AM+1) + \2(p){sm[0(z;p)]-d(z;p)} Deriving the first-order process Eq. (16) with respect to p and then let p = 0, one can obtain the second-order process + 2p\(p)\ll](p) I sinWzw)] - e(z\p) )• equation at p = 0 as follows: (16) +p\2(p) I cos6(z;p) - 1) em(z\p) = 0

m [11 m e (z;p) = 0 forz = 0 = 4A (0) {0O - sin0o) + 2 {1 - cos0o)0 teO) n] d6 (z;p) [11 [l] [11 2 [21 - = 0 forz = 0. - 4A (O)0 (z;O) _ 2 {(A ) + A (0)) 0O (26) dz When p = 0.0, from first-order process Eq. (16), one has 0[21(z;O) = O forz = 0 ddl2\z;0) . . _ dz1 = 0 forz = 0. dz [11 [21 = Aijf0o(z)-sin0o)-2AoA (O)0o The above equation is linear with respect to 0 (z;O). Sub­ (17) [1] stituting (18), (19), (22), (23) in (26) and eliminating the secular 0 (z;O) = O forz = 0 terms, we have [11 2 30 (z;O) 21 1 MP) J2(0)J3(p) = 0 forz = 0. A0 = 3 dz 2 H 4)3 m It can be derived that 1 •ZiOT | JW-J(M y (-l) /2m + l«3) 00 2 -*—' AM(AM+1) 0 v 7 sin(0o) = 2^](- l)"72m + 1(/3)cos(2w + \)z (18) m = 1 m = 0 (27) and 2ml3 AM— 1 AM — 1

cos(0o) = MP) + 2 J] (- l)"72m(/3)cos(2wz) (19) and

[21 0 (z;O) = 2 bm cos (2AM + l)z where, /„((3) is first-sort of Bessel function denoted as m = 0 (-l)*(l3/2)2Ar+" Oo oo c cos [2( + w) + 1]z ^)=S kl(k+n)l ' (20) + ZJ ZJ ™ " m=\ n=\

Substituting (18) in (17), one has the first-order process \ equation at p = 0 as follows: d26l[\z;0) ZJ I ZJ C?„,„COS[2(M-AM)+1]Z (28) - + 0m(z;O) = !)3-2/ (/3)-2/3Ao11) cos(z) az2 1 m -2 2(-l) ^m+l(«COS(2AA7+l)z (21) 0[11(z;O) = O, forz = 0 where I 90"1fe;Q) , . n 00 OO 00 I 00 \ — = 0, forz = 0. b = b C d 29 dz ° - ZJ ">~ ZJ ZJ »-"~ ZJ ZJ »»> ( ) m=1 m=\ n=\ m~\ \ n=\ In order to let the above Eq. (21) have finite solution (i.e., the secular terms should be eliminated), it must be satisfied \n ^ m - 1/ that b J-ITJ^+M L 4/.C8) i /i08) JbOS) 11! 1 JM A (0) = 2AAZ(A«+1) (_ 0 m(m+X) 0 2 (-l)mflo[/2«(|8)-/2» 2C8)] + (AM>1) (30) l 2m(m+ 1) + + k\(k+\)\ (-l)'" V2m(/3)/2n + 1(ffl =S(-u* ' (AM>1,M>1) (31) '"" 4M(«+1)(M + AM)(« + AM+1) o2 + (-l)'" "./2m(/3)J2n + 1()3) (22) dm 16 24 \2j 4M(M + 1)(M - AK)(M - AM + 1) Thus, the /inear differential Eq. (21) has solution (AM>1;M>1, n^m, n^m- 1). (32)

[11 0 teO)=2 omcos(2AM+l)z, (23) 2.1 Approximation of Frequence w. According to where wo £»(P) = (33) UP)

«0 (24) and Taylor's theory, one has the first-order approximation of =- s«« frequence co as

972 / Vol. 59, DECEMBER 1992 Transactions of the ASME

Downloaded 24 Apr 2009 to 202.120.2.30. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm x'"(/?) although the zero-order process Eq. (8) is generally nonlinear. COi=CO0 Using process derivatives, a nonlinear problem with respect to x2(/?) UP) p = 0 6(z) can be transformed into an infinite number of linear prob­ lems with respect to 6lk\z;p) (k=\, 2, 3, • • •, oo) at/? = 0. But, [ = ^o(l-X^) (34) only a finite number of linear problems with respect to d \z;p){k = 1,2, • • •, n ) at p = 0 can be solved. It means that a and second-order approximation of frequence as p nonlinear problem can be approximated by a finite number of f_l_ Xm(p) , [Xm(/?)]2 Xpl(/?)> linear problems. One would say that a nonlinear problem could 0)2 = ^0 2 2 (X(p) 3 be discretized into np linear problems with respect to Arth-order x (/?) X (/?) "2X (p)j p = 0 process derivatives [k = 1,2, • • •, np). The greater np is, the 1 2 2 more exact this approximation is. These are the basic ideas of = CJ0 X0 ' + (A0'>) . (35) the Process Analysis Method.

3 Comparisons to the Numerical and Perturbation 2.2 Approximation of Period T. According to T/T0 = u>0/o>, one has the first-order approximation of period T as Results ]\ 1 In order to examine the solutions given by PAM, it is val­ uable to compare them to the numerical results and pertur­ 7-o-l-Xi" bation solutions at the same order of approximation. The original Eq. (1) can be solved numerically by means of /52 i M4 (36) Runge-Kutta's method. In numerical computation we select At = 0.0001 second and use double precision variables in the and the second-order approximation of period as computer program. On the other side, substitute T 1 2 (37) To I-W-V+M1)2 sin(0) = 0-^ + jy+... (40) in (6) and suppose that 0 be small enough and 6 and X could be described, respectively, as 2.3 Approximation of 8(t). Q(t) at first-order of approx­ (U, , o2/A(2(/,) _ imation is 6>=(3fe +/3e "+/3 e + (41) X=1+/3A(" + /32A(2)+--., (42) 'i(0 = (13 + a0) cosz + 2 am cos(2w + \)z then, we can obtain (a) perturbation solutions at first-order of approximation: = ((3 + a0) cosz - —-— cos(3z) + —— cos (5z) 12 0(/) = 0 cos (woO (43) MP) 1 //3 cos (7z) + • + COS (blif) (44) 24 192 90 V2,' 7-Q 5 (b) perturbation solutions at second-order of approximation: cos (3coi0 192 96 \2 w t 0(0= 0 + 0 (25) 1 192 193 + • • • cos (5o,0 + (38) ^1440 \2y 1+T6 and 6(t) at second-order of approximation is (46) a J T0 16 62(t)=U + a0 + -A cos(co20+2 \ m r

oo oo 3.1 Comparison of Nondimensional Period T/T0. The numerical and analytical results show that the nondimensional COS [(2/77 + l)w20 +25]^ COS[2(/7 + 777) + l](u2t) m = 1 /i = 1 period T/T0 of a simple pendulum is only dependent on the initial angle 13. The detailed comparison of numerical and an­ alytical results of T/TQ obtained, respectively, by PAM and the perturbation method is given in Table 1. It seems that PAM solutions, even at first-order of approximation, generally agree s s cos[2(«-/7j) + l](w20- (39)

\ii7±m- 1/

As the last part of this section, let us discuss simply the Table 1 Comparison of theoretical and numerical results of T/T0 expressions (14), and (15). The expressions (14), (15) describe a numerical PAM leruirbation method a kind of relation between the initial solution 0O =fi cos(z), X0 method first-orde second-order (second-order) = 1.0 and the final solution 8f, X/by means of/cth-order process 20° 1.008 1.008 1.008 1.008 [k] w 30° 1.017 1.017 1.017 1.017 derivatives 8 (z;p) and X (p) at/? = 0. The first-order process 40° 1.031 1.031 1.031 1.030 Eq. (17) and the second-order process Eq. (26) are linear with 50° 1.050 1.048 1.050 1.048 respect to 0[11(z;O) and 0[2,(z;O), respectively. One can prove 60° 1.073 1.070 1.073 1.069 70° 1.102 1.096 1.101 1.093 easily that the /tth-order process equation (k = 1, 2, 3,- • •) is 80° 1.138 1.127 1.136 1.122 lk [>] alway linear with respect to 9 '(z;0). Therefore, 9 (z;0), 90° 1.180 1.162 1.177 1.154 6l2](z;0),• • •, 6m (z;0),• • •, can be obtained without great dif­ 100° 1.232 1.201 1.225 1.190 110° 1.295 1.246 i 1.282 1.230 ficulties. It means that the expressions (14), (15) give a kind 120° 1.373 1.296 1.347 1.274 of linear relation between the initial solution and final solution, 130° 1.-171 1.351 1.424 1.322

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Downloaded 24 Apr 2009 to 202.120.2.30. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm better with the numerical results than perturbation solutions According to (38) and (39), the PAM solutions, at both first at second-order of approximation. Even in the case of a great and second-order of approximation, can be written in the form initial angle 0 = 130 deg, the second-order PAM solution can of also give a good enough approximate value of a period, but 00 the perturbation solution at the same-order of approximation 0(O=XM2*-iCOs[(2£-lM] (*=1,2,---). (47) has about a ten percent relative error. 3.2 Spectrum Analysis. It is well known that the solution Numerically, we have 0(0 of the original Eq. (1) is a periodical function. So, 0(0 can ,7/2 be expressed in the form of a Fourier series. *-H 0(0 cos(nwt)dt (n=l,2,---) (48) Table 2 Spectrum analysis: Numerical result where the numerical solution of 0(0 is used, and thus a nu­ merical integral over [0,772] is needed. 13 30° 60° 90° 120° AJP 1.001 1.006 1.015 1.030 The numerical values Ak(k = 1, 2,•••) and their corre­ A2//3 3.29E-6 -2.21E-7 4.77E-S 3.38E-6 sponding analytical values at first and second-order of ap­ A3lf3 -1.45E-3 -6.14E-3 -1.53E-2 -3.20E-2 proximation given by PAM are shown as Table 2, Table 3, A, IP 5.90E-7 -8.39E-8 -4.24E-8 1.43E-7 and Table 4, respectively. These results show that the second- A5//3 3.43E-6 6.62E-5 3.96E-4 1.65E-3 order PAM solutions are in better agreement with the nu­ Ae//3 2.25E-7 -6.85E-8 -4.86E-8 1.41E-7 Ail/3 -1.61E-7 -7.84E-7 -1.22E-5 -1.01E-4 merical solutions than the first-order PAM results. Note that the analytical values of A2/<;(k = 1,_2, •—) are zero and the corresponding numerical values of A2, AA, and A6 given in Table 3 Spectrum analysis: First-order PAM solutions Table 2 are so small that they can be as the numerical errors 13 30° 60° 90° 120° in the integral. It appears that PAM captures the leading non­ A, 113 1.001 1.005 1.011 1.017 linear harmonics found in the numerical solutions. A3/0 -1.40E-3 -5.33E-3 -1.10E-2 -1.72E-2 The numerical results and analytical solutions at second- A, IH 1.61E-6 2.49E-5 1.19E-4 3.47E-4 Ar/P -1.32E-9 -8.23E-8 -8.95E-7 -4.70E-6 order of approximation given, respectively, by the perturbation method and PAM, in case of j3 = 57r/9, are shown Fig. 1. The analytical and numerical values of Ak{k> 1) are so much Table 4 Spectrum analysis: Second-order PAM solutions smaller than A\ that

13 30° 60° 90° 120° 0(O = (3cos(o)2O (49) A, IP 1.001 1.006 1.014 1.025 A3/P -1.45E-3 -6.03E-3 -1.41E-2 -2.60E-2 can give a good enough approximation of the original Eq. (1). A„IP 1.96E-6 4.57E-5 3.29E-4 1.34E-3 This simplified expression gives essentially the same results as A-,10 -1.88E-9 -2.22E-7 -4.27E-6 -3.57E-5 the second-order PAM solutions (39).

-- NUMERICAL RESULT

--0TH ORDER APPROXIMATION --1TH ORDER APPROXIMATION --2TH ORDER APPROXIMATION

T INITIAL ANGLE 100 DEGREF ) .1

TIME Fig. 1 Comparison of the analytical solutions at second-order of ap­ proximation to the numerical results

974 / Vol. 59, DECEMBER 1992 Transactions of the AS ME

Downloaded 24 Apr 2009 to 202.120.2.30. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm With comparison of expressions (36), (38), to expressions The process analysis method is also based on suppositions, (46) and (45), it appears that the first-order PAM solutions i.e., there should exist zero-order process and &th-order process include the terms of the perturbation solutions at second-order derivatives, and the corresponding Taylor's series should con­ of approximation. verge to the solution of the original equations. Therefore, PAM is also dependent on suppositions, but these suppositions are 4 Conclusion and Discussion not small-parameter suppositions. Owing to this reason, one In this paper, the basic idea of a new kind of analytical can use it to solve more nonlinear problems, especially those method for nonlinear problems, called PAM, is described and without small or great parameters. used to give a second-order approximate solutions of a simple However, it should be pointed out that Taylor's series has pendulum. These solutions are compared to the numerical and generally a. finite radius of convergence. (Only few functions perturbation solutions. The comparison shows that even the have converged Taylor's series with infinite radius of conver­ first-order PAM solutions agree better with the numerical so­ gence; for example, sin(x), cos(x), and so on.) PAM requires lutions than the second-order perturbation solutions. So, we the itth-order process derivatives (k = 1, 2,- • •) to exist, and have reason to believe that PAM can indeed give more accurate to be well behaved and be bounded (i.e., they must not have analytical results than the perturbation method. Note that small turning points or diverge to infinity inp € [0, 1]), and converge. or great parameter supposition is not needed for PAM. This Up to now, it is unknown whether or not the corresponding is an advantage of PAM. Taylor's series, given by PAM, have always a larger radius of The perturbation method seems like a kind of art. Especially convergence than an asymptotic series based on the pertur­ in the case of singular perturbation problems one must use bation method. These are the limitations of PAM. different techniques, for example, the methods of multiple- scale expansions, the method of matched asymptotic expan­ Acknowledgments sions and so on, to solve different problems. Therefore, ex­ Professors W. H. Isay and H. Soeding are acknowledged periences seem important for the perturbation method. But, for their suggestions. Thanks to the DAAD for the financial contrary to perturbation techniques, PAM has the simplicity support. in logic. This is another advantage of PAM. The process analysis method is based on the two concepts References of process and process derivatives. Process is a kind of con­ Liao, S. J., 1991a, "A General Numerical Method for The Solution of Gravity tinuous mapping which connects the initial solutions with the Wave Problems. Part 1: 2D Steep Gravity Waves in Shallow Water," Inter­ solutions of the original nonlinear problem. But this contin­ national Journal for Numerical Methods in Fluids, Vol. 12, pp. 727-745. uous mapping described by the zero-order process equation is Liao, S. J., 1991b, "The numerical limiting form of 2D nonlinear gravity waves past a submerged vortex by Finite Process Method (FPM)," The Sixth generally nonlinear for a nonlinear problem. It is interesting International Workshop on Water Waves and Floating Bodies, Woods Hole, and important that the £th-order process equations are linear Mass., pp. 139-143. with respect to the &th-order process derivatives. Then, ac­ Nayfeh, A. H., 1980, International to Perturbation Techniques, John Wiley cording to Taylor's theory, the final solution and initial so­ and Sons, New York. Ortega, J. M., and Rheinboldt, W. C, 1970, Iterative Solution of Nonlinear lution can be connected by the Ath-order process derivatives Equations in Several Variables, New York, Academic Press, pp. 230-235. (k = 1, 2, • • •)• That is why the process derivatives should be O'Malley, R. E., 1974, Introduction to Singular Perturbations, Academic, used. New York.

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