CSÁKI, F.

KÁRPÁTI, A.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

SOME REMARKS ON THE CONSTRUCTION OF PHASE TRAJECTORIES By F. CS~(KI and A. K~(RP~(TI Department of Automation, Poly technical University, Budapest (Received June 29, 1966) 1. Introduction In control engineering, mechanics, electrical engineering, electronics for the construction of the phase trajectories of the differential equation x + f(x) g(x) = 0 the PELL method, or for the simpler case ·when g(x) = x, the LrENARD method is used. A common disadvantage of both methods is the fact that for the con-struction of the phase trajectory only the tangent is precisely determined at each point, while an elementary section of the trajectory is fitted by an arc ,,-hose radius is only approximately true. The purpose of the present study is to clarify, on the one hand, how much the accuracy of both methods is effected by the parameters of the differential equations, and, on the other hand, to find an adequate method of construction for the exact determination of the center and radius of curvature of the phase trajectory at each point. 2. The Pell method [1] A differential equation to be solved is: :i: + f(x) g(x) = 0 Let it he then Thus x=v .. dv dx dv dv x=_·_= dx dt ·x=_·v dx dx dv dx 1(1;) + g(x) The PELL method [1] can he summarized as follows: a) The curves -x = f(v) and -v = g(x) are drawn. (2-1) (2-2) b) At the point F (Fig. 1) an angle of 4,5 0 is constructed. Thus the point 216 F. CSAKI and A. KARPATI L is determined. c) Drawing a parallel line to HL the point R is settled. d) The direction of RP is the direction of the normal of the trajectory. A small section of the curve beginning from the point P is approximated by an arc with centre R. Then from the ne'w point the 'whole construction is repeated. The beginning point P is fixed by the initial conditions x(O) andv(O). x -v=grx) Fig. 1 3. Determination of the center of curvature The accuracy of the PELL method can be greatly increased by the deter-mination of the center of curvature. The course of the proposed construction can be depicted from Figs. 2 and 3. Fig. 2 COSSTRUCTION OF PHASE TRAJECTORIES 217 The proposed method can be clarified by the following steps: a) The PELL construction is performed according to clause 2. b) Drawing in the point P a line at right angle to PR, point T results. c) The intersection of the continuation of QP and the parallel line from T to PR gives point Y. d) Drawing a perpendicular line to the x-axis in point R yields point Q. e) In point Q' the angle /3 is constructed in the sho·wn manner. The side of the angle determines point lV on the x-axis. A D o C' Fig. 3 f) The intersection of the continuation of PR and lVY finally gives the center of curvature C. The radius of curvature Q is given by CP. An auxiliary construction is necessary for the determination of the angle !) (Fig. 3). Let us introduce the following notations: df(v) =f'(v) = tan?:. dv dr dx = v'(x) = tan)' da(x) _CJ_ = g'(x) = tan q.. dx The steps of the auxiliary construction are a) At the point 0 angle er is constructed. b) The section UA is arbitrarily but adequately chosen. (3-1) (3-2) (3-3) c) At point A the angle e is constructed, the side of e determines point B. d) A perpendicular line at point B gives the point C. 218 F. CSAKI and A.. KARP.4Tl e) The latter point is turned around point B, thus, the point Cl is obtained. f) Constructing the angle ::;t; at point D the side yields point E. h) The angle OAE is a final result for obtaining the angle 13. The four special cases for the construction of angle 13 in the function of 0: and cp are summarized in Fig. 4. A D O,c«Ti/2 ~ o q!J < Ti/2 ~~ : . I I @ I WI ~r ~I 0 C' B [ A 0«::( < 17/2 T7/2,cf< IT CD r c Fig. 4 4. The verification of the proposed construction Thc solution of the differential equation x..L f(x) g(x) o n/2<cX<lT n/2 (cjJ(Ti is assumed in the form v = v(x), where r x. In other words, thc expression of the phase trajf'ctory is v v(x). The radius of curvature is given by the formula: Taking the relation d1' dx I} = -'---'---'---v" f(1') + g(x) v =1" (4,-1) CO;vSTRUCTIOZ', OF PHASE TRAJECTORIE.' into consideration v" is determined in the following way v f1 == _ d f(v) + g(X) dx Substituting (-! --------v [f'(V) '1" --7- g'(X)] f(v) + g(X) v' v If'(v)v' : g'(X)] :V'~ /: d where g'(x) - IZ(X)' - dx L and i'(v) = ~f(v) dv into (4-1); ·V g(x)] v' • L' Q= -f'(v). 'v' + g'(x) + t.'~ Applying the notations of clause 3 f'(v). v' + g'(x) - I ~ (I I --·r cos:; Q == - as 1 -'- V':! ''-'' 1 tan x·tan'" + tan (p - I +--I cos2y 219 (4-2) After some algebraic manipulations and taking into accollnt t h(, relations cos ~) = -sin e and v/sin e = PR = r S111 y = cos e the following expression is ohtained: r 0==---- -.-----~-.~--- .. - I - sin e· cos 0 [tan x - tan e (tan (F - I)] Let us introduce the relation tan if = tan :x -- tan e (tan rp - I) (4-3) As a final result. the formula of the radius of curvature i" T (4-4) () I - sin e . cos e . tan fJ 220 F. CS.-iKI and A. KARPATI The validity of the construction can be verified on the basis of Fig. 2 by the following derivation: From triangle 1v.IYT Consequently Thus Hence PT = r·tan e PCN! <9:: = lklYT <r = c PT = r· tan c + 1.2 • tan c r . tan e = r· tan c 1.2 • tan c tane - tanc 1.2= T tan c: Let us introduce the notation thus tanc: = tan e - tan 6 1 + tan e . tan 6 and as from triangle NKY tan6 = consequently tan 0 v v 2· --+ v· tan e - v . tan f3 tane tane 2 - tan e . tan /3 tan2 e (4-5) (4-6) (4.-7) After substituting Eqs. (4,-6) and (4- 7) into (4-5) and peTforming some algebraic manipulations the final result (4,-4) is indeed obtained. Thus, the property of the construction is justified. No'w let us prove the truth of the auxiliary construction. On the basis of Fig. 3 OB =..., OA· tane BC =OB· tancp=OA· tane· tauq. OC' = OB - Cl B = OB - CB = OA (tan e tan e· tan cp) =OA· tan?: Thus that is COSSTRUCTIOiV OF PHASE TRAJECTORIES OE = OA [tan:x - tan 0 (tan er - 1)] tan{J= OE =tan:x-tan0(tan er- 1) OA 221 Eq. (4-8) is the very same as the definition (4-3), thus the auxiliary construction for obtaining the angle /J is indeed true. y e'\ \ N. 1 x Fig. 5 5. Special cases 5.1 . .iliodification of the LIENARD method [2] The LIENARD method is nothing but a special case of the PELL method, 'when g(x) = x. In the latter case our proposed method is also simplified significantly and we obtain the modified version of the LIENARD construction (see [2]). Taking x = tan er 1 into consideration it can be seen from Eq. (4-3) that in this case tan /3 = tan ex, thus the auxiliary construction can be left out. The modified LIENARD construction is shown for this case in Fig. 5. As T 0= -------------------- I-sin0·cos0·f'(v) if sin e . cos 0 . f'(v) ~ 0 then rp ~ T, consequently the LIENARD method gives a good approximation. This arrives in the vicinity of 0 = 0°, because of sin 0° = 0, and in the vicinity of 0 = 90°, because of cos 90° = 0 and other-wise if f'(v) is small enough. 222 F. CS.4KI and A. K.4.RP.-!n 5.2 Second special case Let us now assume that -X=V Q Q' V H f(x) = f(V) = v P(x,v) -v=g(x) Fig. 6 y Then the differential equation takes the following special form x + ;\; + g(x) = 0 The construction for this case is shown in Fig. 6. If and at the same time 5.3 Third special case f(x) =f(v) = 2 Cv g(x) = x x then the nonlinear differential equation is reduced to a linear one: The construction for the case of 2 C = 1 is shown in Fig. 7. CO".sTRUCTIO" OF PH.4.SE TRAJECTORIES 223 -x = fiv) y 5 T )( F -V=X Fig. 7 6. Conclnsions In the previous trcatment a construction was proposed for the detcrmina-tion of the centre and radius of curvature. The proposed method enables us to increase thc accuracy of the drawing of the phase trajectories in the general enough case of Eq. (2-1). This method can be regarded as an essential devel-opment of the well-kno,vn PELL method. The proposed method contains as a special case the modified LLENARD construction [2] in the same manner as the original LIENARD construction is a special case of the original PELL method. Summary A construction was proposed for obtaining the center and radius of curvature of the phase trajectories. The accuracy of drawing in the trajectories can thus be augmented. Some special cases are also given. References 1. PELL, W. H.: Graphical Solution of Single-Degree of Freedom Vibration Problem with Ar-bitrary Damping and Restoring Forces. Journal Applied Mechanics 24, 311-312 (1957). 2. SPEl\CE, R. - OZAL, Y. B.: Graphical Solution of a Free-Running Nonlinear Oscillator. Elektronics, Record, February 1966. pp. 292-293. , 3. Ln(;l\ARD, A.: Etude des oscillations entretenues. Revue Generale de l'Electricite 23, 901 - 946 (1928). Prof. Dr. Frigyes CS.iKI} Budapest XI., Attila K.iRP_iTI Hungary Egry J ozsef u. 18-20. 

SOME REMARKS ON THE CONSTRUCTION OF PHASE TRAJECTORIES

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SOME REMARKS ON THE CONSTRUCTION OF PHASE TRAJECTORIES

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Periodica Polytechnica (Budapest University of Technology and Economics)