
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], 0<z<o° and A kind of continuous mapping, 0(z)~0(z;p),X-~X(p) can be 2 there exist 6[k] (z;p) and \[k](p) inp € [0, 1], 0<z<oo for constructed as follows: A:>1 ^ 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.
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