OLLOQUIA MATHEMATICA SOCIETATIS JÁNOS BOLYAI 62. DIFFER ENT IAL EQUATIONS, B U D A PE S T (H U N G A R Y ), 1991 Nonlinear Differential Equation with Self-Excited and Parametric Excitations TRAN KIM CHI and NGUYEN VAN DAO In trodu ction In this paper, a single degree — o f— freedom cubic system subject sim ulta neously to a self-excited and parametric excitations is considered. Assuming that certain system param eters are small, the m ethod of multiple scales [1] is utilized to determine those frequency relationships which produce inter esting resonance phenomena. The various types of resonance exhibited by the system are studied and the stability analysis is presented. Exactly, this paper considers those dynamical systems whose motion is governed by the nonlinear differential equation (1) X + lj2x 4- e(3x3 — eR(x ) + eF{x , X) COS 'yt = 0, where differentiation with respect to the independent variable time (t) is denoted by a superdot, UJ, Ị3, 7 are constants, £ is a small positive param eter. The “negative” friction function R (x) is assumed to be of the form (2) , R(x) = h\x — h^x3, where hi are positive constants; and F (x,x ) is of form (3) F (x.x) — cxnx m, TRAN KIM CHI, NGUYEN VAN DAO :re c. n. rn are constants. If c = 0 equation (1) describes the self-excited cillator and if hi = h3 = 0 it presents a parametric oscillator. Each them considered separately has a define self-sustained oscillation. The lestion arises as to what will occur in the system represented by (1) when lese two oscillators are coupled. Does stationary solution (oscillation) of .) exist? Is it stable? The interaction of two autoperiodic oscillations of this type was first ivestigated bv Minorsky N. [2] for the system 4) X + e(x2 — l ) i + [1 + (a — cx2) COS 2t]x = 0 >y using stroboscopic method. He mentioned that the nonlinear oscillatory ystems exhibit always an interaction between the component oscillations md th at perhaps the whole theory of nonlinear oscillations could be formed )n the basis of interactions. In this paper we are interested in the resonant oscillations of equation ; l) and thus will concentrate only the stationary solutions of amplitude - phase equations. Three cases of equation (1) will now be analyzed: 1) rri, = n = 1, 2) m = 0, n = 2, 3) m = 0, n = 3. In order to determine critical frequency relationships leading to resonant oscillations in the foregoing equation, let where k is a rational number and eA is the detuning param eter. 1. M ethod of analysis. A rnplitude-phase equations We use the m ethod of multiple scales [1] to determine a first-order uniform expansion of the solution of (1). To this end, we let where, To — t and T\ = et. Substituting (6) into (1) and comparing the coefficients of equal powers of e, we obtain (5) CƯ2 = k27 2 + eA, (6) (7) (8) D ị x 0 + f i 2 rc0 = 0 , D qXi + ÍỶXị = —2 D qD\'Xq + /o, NONLINEAR DIFFERENTIAL EQUATION 67 /0 = — Axo - 0 xị + h \x 0 - h^xị - cx™x0 COS71. DQ — d/dTo, D ị—d/dTị, Í2 = Ảrỵ. The solution of (7) can be expressed as Xo — acos(fỉT0 + Ip), XQ — — a f 2 s i n ( Q ĩ b + ip), equation (8) yields D qX 1 + Q2x 1 = 2 £ìa' sin Ộ + 2Claip' COS ệ + / 0 , Ộ — f iT o + 'Ip, :re prime denotes differentiation with respect to T\. Case 1 : m — n = 1 In this case any particular solution of (11) contains secular terms if : 1 and k = 1/3. Elim inating the term that produce secular terms in X\, obtain For k = 1: 3 c 2a' = h\a — ^ /1 3 7 a — COS'?/;, ,N 4 4 3 c 2'^a'ip' = A a + ^/3a3 — ^ a 27 sin ip. For k = 1/3: 1 c 2a' — h \ d — /i 3 7 2a 3 + - a 2 COS 3-ỉ/>, 3) -• 12 ^ 7 c 4 27 a'lp1 — 3 A a + — - a 27 sin 3'0 . Case 2: m = 0, n = 2 The secular terms occur when /c = 1 and k = 1/3. The am plitude - phase equations now are of form: For Jfc = 1: TRAN KIM CHI, NGUYEN VAN DAO For k = 1/3: 15) Case 3: m = 0, n = 3 In this case the solution of equation (11) contains secular term s if 2. Stability analysis o f stationary solutions We now present the response curves and stability analysis for the constant solutions of am plitude - phase equations. Equations (12) and (13) have the same structure. Analogously for equations (14) and (15), also for equations (16) and (17). So, it is sufficient to study equations (12), (14) and (16). The response curves of equations (12) and (14) are ellipses. But the stability region of these curves is quite different (see Fig. 1, 2). Thus, consider system (12). Its nontrivial stationary solution: a — do, ip — 'Ipo is determined from ; = 1/2 and k = 1/4. For k = 1/2: (16) For k — 1/4: (17) (18) 4hi — 3/i372aồ — cao cos'lpo — 0, 4 A + 3 Ị3aị — cao7SÌn-0o = 0. NONLINEAR DIFFERENTIAL EQUATION 4 69 Fig. 1. Graph of response curve (19) for u = I, e(3 = 0.1, eh\ = 0.001, eh3 = 0.02, EC = 0.02. 'he response equation is readily found to be 19) Li = 0, lere 20) Li — 7 2(4/ii - 3/i37 2aỔ)2 + (4A + 3/5aồ)2 - c2Ý á ị In order to investigate the stability of a given stationary solution do, •po of (12) it is necessary to examine the characteristic roots of the linear variational system for this solution. The characteristic equation is of form: (21) here 7 2 A 2 - f e h i ^ X — £2 dWi 64 da2 = 0 , L = 0 (22) W i = a2[72(4/ii - 3/?,37 2a2)2 + (4A + 3pa2)2 - c27 2a 2]. The stability condition is (23) dWi da2 < 0 . L\ =0 The response curve (19) is shown in Fig. 1 where the steady-state amplitude ao is plotted against the non-dimensional frequency ratio 7] = Hereafter asymptotically stable solutions are represented as a solid line while unstable solutions are represented as a broken line. The stable part is the lower portion of the ellipse bounded by their vertical tangents, where the relation (23) is satisfied. For system (14) the response equation is '0 TRAN KIM CHI, NGUYEN VAN DAO ;24) (25) i 2 = 7 2 ( V j f t 3 7 2a ? ) 2 + ( f + and the characteristic equation has form: 7 2A2 + ^ 7 2(3/i372aị - 2/ii)A + (26) = 0 , 1/ 2= 0 W o — a 2„2 + = + j / f c ‘ - 1 6 The stability condition is then (27) 3 /? 3 7 2 a o - 2/ii > 0 (28) dW2 da2 > 0 1/2 = 0 These conditions are in contrast with (23). Only a part of the upper branch of the ellipse is stable when the two relations (27), (28) are simultaneously satisfied (Fig. 2). 2 t CJ Fig. 2. Graph of response curve (24) for CJ = 1, e/3 = 0.1, e/ll = 0.001, e/?3 = 0.015, EC = 0.02 For system (16) the response equation is 29) L 3 = 0, 30) L3 = J 2 (2hi - - h ^ á ù + ^2A + 2 ^ aoi ~ c2fl0 NONLINEAR DIFFERENTIAL EQUATION 71 and the characteristic equation will be 2 x 2 2 x I L 2 2 ^ , £ 2 d W 3 7 A + £ 7 2 A - / i i + a s'Y a 0 + V 8 4 ôa2 (31) = 0 , l 3 = 0 = a and the stability condition is (32) 7 2 f 2h\ — ^ 372a2^Ị +Í2A+^/5a2^Ị - c2a4 8 3 7 ^ 3 7 2aồ - / l i > 0, (33) Ỡ W 3 <9a2 > 0 . l 3 = 0 The response curve depends O I ! the value I = c — 3/i3Cj3. If / < 0. the response curve is an ellipse with the upper stability branch and lower unstability branch. If / = 0 the response curve is a parabola (Fig. 3). 7 - 4C0 for I < 0, CƯ = 1, e/3 = 0.1, eh\ = 0.001, e / i 3 = 0.01, ec = 0.03 If I > 0 the response curve is a hyperbola (Fig. 4, 5). It is seen from these figures th at increasing the am plitude of the parametric excitation the stable stationary oscillation of system considered disappears. 72 TRAN KIM CHỊ NGUYEN VAN DAO 2 Ĩ r * Fig. 4■ Graph of response curve (32) for / > 0, UJ — 1, e/3 = 0.1, eh\ = 0.001, Eh3 = 0.01, EC 4 CO' 0.04 for I > 0, U) — 1, £0 — 0.1, e h\ — 0.001, £/13 = 0.01, EC — 0.2 3. C oncluding rem arks Summing up, the above analysis shows the following peculiarities of the considered nonlinear equations: 1. Stable stationary solutions (oscillations) exist only in certain regions defined by inequalities like (23), (27), (28), (32) and (33). Outside these regions there are no stationary solutions. NONLINEAR DIFFERENTIAL EQUATION 73 2. As we know, the simple self-excited system (c = 0) exhibits always :he stable stationary oscillation. W ith the presence of the parametric excita tion (c Ỷ 0) depending on the am plitude (m, n) of excitation the stationary oscillation m aintains only with certain excitation frequencies (7 ). The sta bility region of stationary oscillation also depends on the excitation form. Sometimes the lower branch is stable (Fig. 1), sometimes the upper branch is stable (Fig. 2, 3, 4) and sometimes no branch is stable (Fig. 5). 3. The stability conditions like (27), (32) indicate the lower energy level below which oscillations cannot maintain themselves (since a2 = X 2 + Q ~ 2X 2 is a measure of the energy stored in system). As on the other hand, 3/i372aồ = 4hỵ for the simple self-excited oscillator (c = 0) corresponding to (14) and 3/ì372aồ = 16^1, for the simple self-excited oscillator corresponding to system (16), it follows from (27) and (32) that, the oscillator represented by (1) with m = 0, n = 2 or m = 0, n = 3 develops less energy than a simple self-excited oscillator. Everything happens as if the param etric part of the nonlinear oscillator absorbed the energy developed by the simple self-excited oscillator. R eferences 1. A. H. N ayfeh, D. T. M oo k , Nonlinear Oscillations. New York, Wiley-Interscience, 1979. 2. N. Minorsky, Nonlinear Oscillations. D. Van. Nostrand Company, Inc., Princeton, New York-London-Toronto, 1962. Tran Kim Chi, Nguyen Van Dao Institute of Mechanics National Centre for Scientific Research of Vietnam 224 DOI CAN, HANOI. . BOLYAI 62. DIFFER ENT IAL EQUATIONS, B U D A PE S T (H U N G A R Y ), 1991 Nonlinear Differential Equation with Self-Excited and Parametric Excitations TRAN KIM CHI and NGUYEN VAN DAO In trodu. curves and stability analysis for the constant solutions of am plitude - phase equations. Equations (12) and (13) have the same structure. Analogously for equations (14) and (15), also for equations. 0.02 For system (16) the response equation is 29) L 3 = 0, 30) L3 = J 2 (2hi - - h ^ á ù + ^2A + 2 ^ aoi ~ c2fl0 NONLINEAR DIFFERENTIAL EQUATION 71 and the characteristic equation will be 2 x 2 2 x