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Vietnam Journal of Mechanics, NCST of Vietnam Vol. ~1, 1999, No 2, (75 - 88) NONLINEAR OSCILLATORS UNDER DELAY CONTROL NGUYEN VAN DAO Vietnam National University, Hanoi 19 Le Thanh Tong, Hanoi, Vietnam ABSTRACT. In this paper, oscillations and stability of nonlinear oscillators with time delay are studied by means of the asymptotic method of nonlinear mechanics. Harmonic, superharmonic, subharmonic and parametric resonances of a Duffi.ng's oscillator are analyzed. Analytical method in combination with a computer is used. 1. Introduction The harmonically forced Duffing's oscillator with time delay state feedback has been investigated in .[1] by using the method of multiple scales [2]. Both primary and 1/3 subharmonic resonances of the Duffing's oscillator with weak nonlinearity and weak delay feedback have been e~amined. As shown in [1] the simplest model for various controlled nonlinear systems, e.g., active vehicle suspension systems when the nonlinearity in tires is taken into account, is described by a second order differential equation with time delay in the form d 2 x(t) dt 2 dx(t) + x(t) = -2€--;u---- JLX 3(t) + 2ux(t- ~) + 2v dx(t- ~) dt + 2pcos .Xt, (1.1) where €, JL, u, v and ~ are constants. To study all possible simple resonances in the dynamic system governed by equation (1.1}, in the present paper it is supposed that between the external frequency A and the natural frequency 1 there exists a relationship of the form A= n + cu, (1.2) where n = !?. is a rational number, p and q are integers. We suppose that paq rameters €, JL, u, v are small. The smallness of these parameters is insured by introducing small positive parameter c. 75 2. Harmonic Resonance f Assuming that n = 1 and equation (1.1) in the form is a small quantity of e - order, we can rewrite (2.1) where F dx(t) = -2€---;u--- J.Lx 3 (t) + 2ux(t- ~) + 2v dx(tdt ~) + 2pcos >..t. (2.2) The solution of equation (2.1) is found in the form x(t) a cos w(t), w(t) dx(t) . = = >..t + 0. "dt = -a>.. sm 'll(t), (2.3) where a and 0 are unknown functions. By substituting these expressions into (2.1) and solving for ~; ~: we obtain the following equations for a and 0: ~; = -~(2ux +F) sin w, dO a dt (2.4) e = - 1 (2ux +F) cos '11. In the first approximation we can replace the right hand sides of (2.4) by their averaged values: ~ = -~(aL + psinO), a d8 & where = (2.5) 2 ) + pcos o] -~>.. [a(M- ~J.La 8 , L = E>.. + u sin(>..~)- >..v cos(>..~), M = u + ucos(>..~) + >..vsin(>..~). The stationary solution of (2.5) is a satisfy the relationships: = a0 = const, (} aoL + psin60 a0 ( M- ~J.La6) + pcos 00 76 (2.6) = Oo = const which = 0, (2.7) = 0. From here we obtain: W(a~,,\) == a~[L 2 + (M- ~pa~) ]- p 2 = 2 tgDo = (2.8) 0, L (2.9) 3 M- Spa~ c The resonance curves are presented in Fig.1 for the parameters: p = 0.05, = 0.05, p = 1.5 and for various values of u, v: u = v = 0.05, ~ = 0 (curve 1), u = v = 0 (curve2), u = -v = 0.05 and~== 0 (curve3), ~ = 0.5 (curve4), ~ = 1 (curve5). Curve 1 corresponds to the case of an ordinary Duffing's oscillator without friction. Curve 2 corresponds to the well-known Duffi.ng's oscillator without time delay. Curve. 3 also represents the Duffing's oscillator without time delay and with a viscous friction 2(€- v)x two times larger than in the previous case. Hence, the maximum of the amplitudes strongly decreases. By increasing time delay (curves 4, 5), the resonance curves lean toward the right and the maximum of the amplitudes slightly decreases. L 3 0.2+ I / 4 / 0.19 / 014 2 0.50 O.IJ9 5 0.11+ 0.00 -t-.-.--.-.,---f--.--r-,.._,..-+-..,...--,-,--r-+ !.50 1.00 0.50 -0.01 +;::;::::;::;::r:;::;::;::;:;::;:::;::;::;::::;~~~1-:-- o.so f.DD 1.50 2.00 A Figure. 1b Figure1a Figure 1. Resonance curves in the case of harmonic resonance To study the stability of stationary solution ao, Do we use the variational equations obtained from {2.5) by letting a = ao + 6a, D = Do + 6D. Thus, we have dba = _:_ [L6a- a0 dt ,\ a 0 d6D dt = -x (M- ~pa~)6o] 8 ' c [ ( M- 9 pa;) 6a + aoL.6D] . 0 8 The characteristic equation for this system of equations is 2 [ 2 ( 3 2) ( 9 2) ] 2 2cao c a 0 p + -,\-Lp + ,\ 2 ao L + M- gpa0 M- 8pa0 = 0. 77 (2.10) Taking the expression (2.8) into account we can write this equation in the form: aop 2 aw c- 2 2c + -aoL · p + -ao ·= 0. A A2 oa~ Hence, the stability conditions will be 1) L >0 2) (2.11) aw - >O. (2.12) 8 ao It is easy to identify the stability zone by using the rule stated in [3]. In Figure 1 the stable branches are represented by solid lines, while the unstable branches by broken lines. It seems that time delay plays the same role as friction, decreasing the amplitudes and stabilizing the oscillations. 3. Superharmonic resonance of third order Supposing that n = d2 x(t) dt 2 ~'we have the equation (1.1} + 9A 2 • x(t) = E (6c-ux(t) in the form: + F 0 ] + 2pcos At, where dx(t) 3 F0 = -2edt- JLx (t) + 2ux(t- ~) + 2v dx(t- ~) dt · (3.1) (3.2) We transform equation (3.1) into a system of two equations of the first order relative to the amplitude a and phase (} as follows: x(t) = acos(3At + 0) + 2p* cos At, d~~t) = -3Aasin(3At + 8)- 2Ap* sin>.t, (3.3) I P* = 8A 2 • It is easy to find the equations for a and fJ : da c dt =- A(6uacos 'II+ F0 ) sin '11, 3 dO ca dt =- A(6uacos 'II+ F0 ) cos 'II, 3 (3.4) 'II= 3>.t + fJ. (3.5) where 78 In the first approximation we can replace the right hand sides of {3.4) by their averaged values. Hence, we have the following averaged equations: (3.6) where L1 = 3€>. + usin2>.Ll{ 3v>.cos3>.~, :~ = :a-~3~/~ + u cos(3M) + 3vAsin(3ALl.), (3.7) Stationary solution a = a0 , fJ = 60 of equations (3.6) satisfies relationships: aoL1 = JLP! sin 6o, 3 2) .3 ao (M1 - BJLa0 = JLP~ cos Do. (3.8} Eliminating the phase D0 we obtain the following equation for the resonance curves: (3.9) 4. Subharmonic resonance of order one third (1/3) Now, we consider the case when n = 3 in the relationship (1.2) and when the equation (1.1) has the form: . d2 x(t) >. 2 dt 2 + g-x(t) = c [2c] 3ux(t) + F + 2pcos >.t, 0 {4.1) where c-u = >.- 3 and F0 is the same as in {3.2). The solution of equation (4.1) is found in the form: x(t) =a cos (-it+ fJ) dx (t) { dt + 2ph cos >.t, a..\ Sin . ( -t ..\ + fJ ) - 2..\ph Sin . ..\t, = --. 3 3 (4.2) where a and fJ are new variables and Ph~- 79 9/ s.\2 · (4.3) Substituting expressions (4.2) into (4.1) and solving relative to the derivati~es of a and 8 we obtain: (2u 3c(2u ). ) da = -3c - --acoscp + Fo smcp dt .X 3 ' adO- = - - ~acoscp + Fo coscp, dt .X 3 - (4.4) .X where cp = 3t + 8. Averaging the right hand sides of (4.4) over time, we have the following averaged equations: (4.5) where, £2 M2 = ~ [€-X + 3usin ( .x:) - v.Xcos ( .x:)], = i1 [u- 9p,pi., The stationary solutions a relationships: . .· (.X~)] . + 3uco.s (.X~) 3 + v.Xsm 3 = ao, () = Oo a0 ( L 2 3 ao . - (M2- -p,a~8 (4.6) of equations (4.5) are determined by the ~ J.LaoPh sin 38o) = 0, 3 . -p,aoPh cos 30o) = 0. 4 (4.7) By eliminating 00 we obtain: (4.8) Using the last equation we can construct the resonance curves, giving the dependence of the amplitude ao on frequency .X of external force. In Figure 2 the resonance curves are drawn for the parameters Ph = 0;8, € = 0.01, J.t = 0.02 and u = v = 0 (curve 1), u = 0.01, v = -0.01 and~= 0 (curve 2), ~ = 0.05 (curve 3), ~ = 0.1 (curve 4). The abscissa- axis .X corresponds to the zero solution a= 0 of equations (4.5). Curve 1 is the resonance curve in the ordinary Duffing's oscillator without time delay. With the presence of delay elements (u, v) the resonance curve moves up. The larger the time delay~' the higher the resonance curve (see curves 2, 3, 4 for ~ = 0, ~ = 0.05 and ~ = 0.1 respectively). 80 .110 tlo 2.00 oot L - - - - - - - 1.10 0~+-~~~~~~~~~ J.lJ .J.IO .J.JJ A. auu +-----,-----....-lUO .J.2S J.JO }.. Figure2b Figure2a Figure 2. Resonance curves in the case of subharmonic resonance To study the stability of the stationary oscillations we use the variational equations: 3 d:ta = ~ : [- L 2 lia + a0 (-3M2+ ~Jta~ )tin], ao dliO dt 3e- [ = -T - ( 3 2) {4.9) ] M2 + 8~'a0 lia + 3aoL2li(} . The characteristic equation of this system of equations is of the form: (4.10) Hence, the stability conditions are 1) L2 > 0 2) aw2 > o. -a ao (4.11) From the expression (4.8) it is seen that for very small values of a 0 the function W2 (a5, A) is positive. Hence, in Figure 3, this function is positive outside of the "parabola" - resonance curve and is negative inside of this "parabola". If moving upwards along a straight line which paralells the ordinate axis ao and cuts the resonance curve, we go from the zone where W 2 is negative to the zone where W 2 is positive, then at the intersection point of the straight line with the resonance curve · t·1ve -a aw2 1s· post·t·1ve. rn t h e oppos1·te case aw-2 1s· negatiVe. · usmg · t h.1s th e d enva ao aao rule and taking the conditions (4.11) into consideration we can see that the upper (lower) branch of the resonance curve corresponds to the stability (instability) of stationary oscillation. In Figure 3a the stable branches are shown by solid lines and the unstable branches - by broken lines. Figure 3b shows the dependence of 81 L 2 on A. It is seen that for values interested of A(.\ = 3 + 3.5) the expression L2 is positive, and the first stability condition (4.11) is satified. It is easy to see that the zero solution a = 0 of equations (4.5) is stable, because the corressponding variational equation is: d6a = _ 3c L ha dt .\ 2 ' where L2 is positive in the interval interested of .\(3 + 3.5) {see Figure 2b). 5. Parametric resonance Let us consider a dynamic system described by the differential equation: (5.1) where F1 dx(t) = -2e---;u- - J.Lx 3 (t) + 2ux(t- A)+ 2v dx(t- A) dt + 2px(t) cos >..t. (5.2) Different from (1.1), here the external force appears as parametric excitation in the form 2/x(t) ·cos At. It is supposed that >.. = 2 + c-u, so that the equation (5.1) can be written in the form: (5.3} Introducing the amplitude a and the phase fJ as new variables, associated with x and :i; by the formulae: >.. = a cos( -t + 8), 2 >.. (>..-t + fJ ) -dx = --asin dt 2 2 ' x {5.4) we have the following equations which are equivalent to {5.3): da 2c d(J 2c dt = -T(uacos 71 { a dt = -T(uacos71 . + Fl) sm71, + Fl) cos71, (5.5) In the first approximation, equations (5.5) can he replaced by the following averaged equations: da ga dO ca + psin28), dt = -T(La { a-= dt (5.6) 3 --(M .>t 3 - -JJ,a 4 2 + pcos28) ' where La = . (.AA) E.>t + 2u sm 2 - >.v cos (.>tA) 2 , (5.7) Ma = a + 2u cos ( ).: ) + >.v sin ( ).: ) . The stationary solution a = a0 =!= O, fJ = 80 of equations (5.6) is determined by the, relationships: La + psin 28 0 = 0, (5.8} 3 2 Ma - 4JJ.a0 + pcos 28 0 = 0. Eliminating 80 we get: Wa ( a02 , .>t) 3 2) = L 32 + ( Ma- 4pa 0 2 - p2 = 0. (5.9) The resonance curves are plotted in Figure 3 for the parameters p = 0.42, € = 0.1, JJ. = 0.3, u = v = 0 (curve 1}, u = 0.05, v = -0.1 and 6 = 1, (curve 2), A = 0.95 (curve 3). By decreasing the delay parameter, the amplitude of oscillation decreases. The maximum of the amplitudes is very sensitive to a change_in the delay parameter. ao f.OO aso 0.24 0.1"1 +-..--...........---r---r--r-.---r--,----r--. 2.SO A 2.00 1.50 0.00+-~~~--~~~~ {.50 200 2.50 A Figgure 9a Figure 9b Figure 9. Resonance curves for parametric oscillations 83 The stabilility of the stationary oscillations obtained is examined by using the corresponding variational equations: dDa 2e= --paocos2fJ 0 · 6(} dt .X ' do(} 3e- 2 2e. ao dt = 2.X J.Lao. oa + --rpao sm20o. ofJ. - where, 6a =a- a0 , is 6(} = (}- 00 . (5.10) The characteristic equation of equations (5.10) (5.11) It is easy to see that awa aa~ = 3J.L (~J.La2- Ma). 2 ° 4 Hence, the stability conditions of stationary solution (ao =/= 0) are 1) La> 0 2) awa > o. -a ao (5.12) Since L 3 is positive in the interval interested of .X(1.5 -;-2.5), we then consider only the second stability condition (5.12). It is easy to identify the sign of the function W3 in the plane (.X, a0 ), because for ao = 0 and for very large values of .X the function W 3 (5.9) is positive. This function vanishes on the resonance curve and changes sign when crossing the resonance curve. According to the well-known rule [3] we can see that the upper branch of the resonance curve is stable and the lower branch is unstable. In order to study the stabilility of the trivial solution a = 0 of the equations (5.6) it is convenient to use the cartesian coordinate (u, v) instead of the polar coordinates a and (} as follows: (5.13) z = asin(J, y =a cos(), which gives dy dt = - ~; = - ~ [L 3 y+ (Ma - ~J.La 2 - P) z] .X i ((Ma - ~ J.La 4 2 ' + p) y + Laz]. (5.14) The characteristic equation of this system is 2 P 2e- c-2 2 + TLap + .X 2 W 3 (a 0 , .X) = 0. (5.15) Hence, the abcissa axis .X (a= 0 or y = z = 0) is stable where La > 0 and W3 (a~, .X) > 0. 84 (5.16)