Newton- Raphson method and iteration methods

Transcription

1 DIEC NUMEICAL MEHOD FO CALCULAING HE PAAMEES OF HOIZONAL CICULA CUVES aga KHALIL Lecturer at Civil Engineering Department, Faculty of Engineering, Assiut University, Assiut, Egypt ملخص: ( ) ( ) ( ) ( ) ( ) ( ) ( ) Newton- aphson method and iteration methods ABSAC he horizontal circular curve can e descried y seven : (1) adius of the curve; ()deflection angle etween tangents; () tangent distance; (4) external distance; (5) middle ordinate; (6) long chord; and (7) length of the curve When the radius and deflection angle are given, the other five curve can e directly computed In some practical prolems, the radius and the deflection angle are unknown; two other must e known to solve the prolem Seven cases must e solved depending on the known curve, as mentioned y other authors his paper present three other cases and a direct method is proposed for two cases of the earlier seven KEY WODS: Circular curve; Layout; Direct method INODUCION he horizontal circular curve can e defined as shown in Fig (1) y seven : (1) adius of the curve, ; () deflection angle etween tangents, I; ()

2 tangent distance, ; (4) external distance, E; (5) middle ordinate, M; (6) long chord, Lc; and (7) length of the curve, L PC is the point of curvature, PI is the point of intersection and P is the point of tangent When the radius and deflection angle are given, the other five curve can e directly computed as presented in (Anderson et al 1985; Moffit and Bouchard 1987; Wolf and rinker 1989) PI I E L PC M Lc P I Fig (1) Principal of Horizontal Circular Curve Seven cases in which the curve radius and the deflection angle are unknown have een solved y Chen and Hwang (199), Li (199) and Easa (1994) he various cases they addressed are shown in tale (1) he solution of case (1) is direct, ut the other six cases must e iteratively solved ecause and I cannot e explicitly expressed in terms of the known as stated y Easa (1994) ale (1) he studied cases of the Circular Curve Case numer (1) Unknown () Given () O Formulas (4) 1, I, E ( ) + = + E, sin ( I / ) = / ( + E ), I, L o I = 180 L/ π, tan( I / ) = /, I L, E o I = 180 L/ π, cos( I / ) = /( + E) 4, I L, M o I = 180 L/ π, M = [ 1 cos( / ) ] 5, I L, Lc I = 180 L/ π, Lc = sin( I / ) 6, I, M tan( I / ) = /, M = [ 1 cos( / ) ] 7, I E, Lc cos ( I / ) = /( + E), sin( I / ) = Lc / Chen and Hwang (199) proposed using Newton-aphson (N) method for solving these cases heir method had some short comes as stated y Easa (199) that it needs an initial guessed value of the curve radius It requires computing the derivatives and may not converge for some initial values

3 Easa (1994) suggested a numerical method, called the iteration method, for finding the solution of the circular curve prolems in which the curve parameters cannot e determined directly He expressed the iteration function in terms of the radius of the circular curve he iteration method proposed y Easa is simpler than the Newton-aphson (N) method proposed y Chen and Hwang ecause it requires no derivatives and it usually converges for a larger set of initial values (Dueau 1995) Easa (1994) concluded that there are two solutions of case (6), his method similar to N method provides one solution and the other solution must e found using exhaustive search Li (199) also proposed a numerical iteration method, for solving the parameters of the horizontal circular curve He expressed the iteration function in terms of the deflection angle etween tangents, I and derived a formula for the initial estimates of I Li (1995) explain the compound iterative scheme which make the iterative process more efficient than the simple iterative scheme Easa (1995) said that similar to Chen and Hwang method, however, Li's method has not addressed the fact that case (6) has two solutions his paper presents three more cases never e mentioned efore for solving the prolem and proposed a direct solution for cases (6 and 7) DIEC MEHOD In this method a new direct formula was derived for case (6) and (7) and for a new three cases (8), (9) and (10) Also a simple form is derived for case (1) he formulas are shown in tale () he derived relationship of case (6) is presented and the relationships for other cases are listed Case (6) M cos ( I / ) = (1) sin ( I / ) ( M ) = () ise equation (1) and () to the power of and adding them together we get ( M ) ( M ) + 4 = 1 () After the needed areviations the final formula can e written as ( ) ( M + ) + ( M ) ( M ) = 0 M (4) he formulas for case (1), (8), (9) and (10) are simple while that for case (6) and (7) are in the form of the cuic polynomial

4 ale () he new formulas of the Circular Curve Case numer (1) Unknown () Given () Formulas (4) tan I / 4 = E /, = / tan 1, I, E ( ) ( I / ) 6, I, M ( M ) ( M + ) + ( M ) ( M ) tan ( I / ) = / 7, I E, Lc ( 8E) + ( 4E Lc ) ( E Lc ) ( Lc E ) cos ( I / ) = /( + E) = 0, = 0, 8, I, Lc cos( I / ) = Lc /( ), = Lc /( sin( I / ) ) 9, I E, M cos( I / ) = M / E, = ( E + M )/( tan( I / ) sin( I / ) ) 10, I M, Lc = Lc / 8M + M /, sin( I / ) = Lc / SOLVING HE CUBIC POLYNOMIAL o solve the general cuic polynomial ax + x + cx + d = 0, a 0 (5) First let ac p = 9a (6) 9ac 7a d q = 54a (7) D + = p q (8) Suppose p and q are real numers D > 0 : one real and two complex conjugate solutions D = 0 : three real solutions, at least two of which are equal D < 0 : three distinct real solutions Now let

5 α =q + D and (9) β =q D (10) he solutions are given y α + β (11) a ( α + β ) + ( ) α β i 1 (1) a ( α + β ) ( ) α β i 1 (1) a If D < 0, it is easier to let 1 q θ = cos (14) p and otain the solutions from the following formulas θ p cos a (15) θ π p cos + θ 4π p cos + a a (16) (17) APPLICAION he proposed method was applied to the numerical example of Chen and Hwang (199) he given for various cases are: L = m; = 795 m; E = 9990 m; M = 777 m; and Lc = m he results are shown in tale () he results of and I are the same for all cases except case (6) he proposed method showed that there are three solutions for case (6) he first solution is = m and I = (which is given in Chen and Hwang (199) he second solution is = 178 m and I = (which is given in Easa (1994) he third solution is = m and I = hese three solutions are shown graphically in figure ()

6 ale () he results of a numerical example Case numer (1) Given () (m) () 1, E I (degrees) (4) 6, M E, Lc , Lc E, M M, Lc CONCLUSIONS his paper presented a direct method for computing the radius of the horizontal circular curve and the deflection angle etween the tangents I using two of the other principal of the circular curve hree new direct cases (8), (9) and (10) and new direct formulas for cases (1), (6) and (7) are generated and applied to a numerical example of Chen and Hwang (199) he proposed formulas gave the same results as Chen and Hwang (199) and Easa (1994) for all cases except for case (6), the proposed formula gave three solutions Another research is needed to solve the rest cases in direct formulas I = I = I = = 795 M = 777 = 795 = 795 M = 777 = = 1788 = M = 777 I First solution Second solution hird solution Fig () Solutions for case (6)

7 EFEENCES 1 Anderson, J M, Mikhail, E M, and Davis E, Foote F S Introduction to surveying, McGraw-Hill yerson, New York, 1985 Chen, C, and Hwang, L Solving circular curve using Newton-aphson's method, Journal of Surveying Engineering, ASCE, 118(1), PP 4-, 199 Dueau, F Discussion of ' Simple numerical method for solving horizontal circular curves' y Easa, S M, Journal of Surveying Engineering, ASCE, 11(), PP 17-18, Easa, S M Discussion of ' Solving circular curve using Newton-aphson's method' y Chen, C, and Hwang, L, Journal of Surveying Engineering, ASCE, 119(), PP 10-11, Easa, S M Simple numerical method for solving horizontal circular curves, Journal of Surveying Engineering, ASCE, 10(1), PP 44-48, Easa, S M Closure of ' Simple numerical method for solving horizontal circular curves' y Easa, S M, Journal of Surveying Engineering, ASCE, 11(), PP , Li, K S A note on calculating parameters of circular curves, Survey ev, (July), PP , Li, K S Discussion of ' Simple numerical method for solving horizontal circular curves' y Easa, S M, Journal of Surveying Engineering, ASCE, 11(), PP , Moffitt F H and Bouchard H Surveying, Harper & ow pulishers, New York, Wolf P and Brinker C Elementary Surveying, Harper & ow pulishers, New York,