Interaction between nonlinear parametric and forced oscillations

Vietnam Journal of Mechanics, NCNST of Vietnam T. XX, 1998, No 3 (16 - 23) IN T E R A C T I O N B E T W E E N N O N L IN E A R P A R A M E T R I C A N D FO R C E D O SCILLATIO N S N g u y e n V a n D a o , N g u y e n V a n D i n h , T r a n K im C hi The interaction of nonlinear oscillations is an important and interesting prob lem, which has attracted the attention of many researchers. Minorsky N. [5] has stated “Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction”. The interaction between the forced and “linear” parametric oscillations when the coefficient of the harmonic function of time is linear relative to the position has been studied in [l, 4], In this paper this kind of interaction is considered for “nonlinear” parametric oscillation with cubic nonlinearity of the modulation depth. The asymptotic method of nonlinear mechanics [ 1 ] is used. Our attention is focused on the stationary oscillations and their stability. Different resonance curves are obtained. 1. E quation of m otion and approxim ate solution Let us consider a nonlinear system governed by the differential equation where e > 0 is the small parameter; h > 0 is the damping coefficient; 7 > 0 , p > 0 , r > 0, u; > 0 are the constant parameters; eA = u 2 — 1 is the detuning parameter, where the natural frequency is equal to unity; and 6 > 0 is the phase shift between two excitations. The frequency of the forced excitation is nearly equal to the own frequency u, and the frequency of the nonlinear parametric excitation is nearly twice ELS large. So, both excitations are in fundamental resonance. They will interact one to another. Introducing new variables a and rp instead of X and X as follows, X + U 2X = E A x — h i — 7 1 3 + 2 p i 3 c o s2 w i + rcos[uit — (5) , ( 1 . 1) X = a cos 6. X = —acưsinớ, 9 = ut -H ĩp, ( 1.2 ) we have a system of two equations which is fully equivalent to (1.1) da e „ dxb 6 — = — — F sin Ớ, a - j- = — — F cos Ớ, at u at u (1.3 ) 16 ere F = A x — h i — 7 1 3 + 2px3 COS 2ut + r cos(wi — (5). ie equations (1.3) belong to the standard form, for which the asymptotic method applied [1]. Thus, in the first approximation we can replace the right hand sides (1.3) by their averaged values in time. We have the following averaged equations: le r e da e dtp £ ~dt= ~ 4 Z adt^~ 2 Ũ 90' /0 = 2h.ua + pa3 sin 20 + 2r s in ( 0 + 6), go = a E + p a 3 COS 2iỊ) + r cos(t/> + (5), 3 , E = A — - 7 a . 4 (1.4) (1.5) The stationary solution (ao^xpo) of the equations (1.4) are determined by da dĩp uations — = 0, ■— = 0 or dt dt / 0 = 2h u a 0 + pa.Q sin 2 0 0 + 2r sin (0 o + £) = 0, gQ = a0E 0 + p al C O S 2xị)0 + r cos(rjj0 + Ổ) = 0, H 6) 3 2 E 0 — A — -'ya.Q, 4 equivalently / l = fo COSTCO - <7o sin ĩpo 3 r = ~ rs in <5 + - sin(2i/>0 + <5) + 2h u a 0 COS 0 0 + (paổ — .EcOao sinV>0 = 0 , 1 2_ (1.7) = f Q s m i p o + (Jq co s xpQ = — ~ r COS 6 — - c o s (2t/>0 + <5) + 2hua0 sin 0 0 + (p °0 + Eo)ao COS 00 = 0. 2 2 elow , for sim p licity, w e con sid er only the case (5 = 0 . To elim ina te COS 2 t/)0 an d n2ĩịjo from (1.6) and (1.7), we use the combinations / = ~fo - Pao /i = rhuao - ịpa^ipaị - E0) - — 2phua,Q COS ipo = 0, sin xJjQ T T 3 9 = 2^0 + Paổỡi = 2 flo^o + 2 rpa° + 2p/iwao sin 00 ^2 + p a £(p a0 + £ o ) (1.8) r L~2 cos 00 = 0 17 The condition for equivalence of (1.6) and (1.8) is r2 Ỷ 4p2flo* As usual, equations (1.8) are considered as two linear algebraic equations relative to two unknowns u an(i V : u = sint/>o; V = COS 0 0 . T h e elim in a tio n o f th e p hase 00 can be d on e by using the relationship u2 + V 2 = 1. Two cases must be identified: 1. The “ordinary” case when the determinant D of the coefficients of u and V in (1.8) is different from zero, where D 2phujaị p aị{páị - Eo) - r 2 D — 4p2h 2u 2cLo + [r2 — pa^(pa 2. The “critical* case when D = 0 r2 2 + Pao(Pao + E o) 2phua,Q - E o )} [ V— + pa40{pal + E 0 )}. (1.9) 2. R esona nce curves in system w ithout damping Supposing that h = 0, the equations (1.6) become (paịcosx^o + r)sini/>0 = 0, d o { E o — p á ị) + r CO S rpo + 2 pflQ C O S 2 xpo r= 0. From the equations (2.1) it follows a) xpo = 0 w h ic h co rresp on d s to th e r eso n an ce cu rve c Ị 1^: £ 0 = - p a ồ - or A = ( - J - - p ) a ị - a n # 0. (2.2) ao ' 4 / do fr) 0 0 — ^ w h ich corresponds to th e resonance curve c j 2^: A = ( 'a 1 ~ p ) a 0 + ~ ’ a ° ^ °- (2 -3 ) \ 4 / clq c) t/>0 = ± a r c c o s ( — =• ) which corresponds to the resonance curve c 2: V p a g / r 2 ,2 \ y 2 E o ^ p á ị - — 4 or A = ( ^ 7 + p ) a 20 - — J (2 .4 ) pa l V 4 / pa* with lim itation: r2 < p2a6. The curves c [ 2^ and C 2 are presented in Fig. 1, where the curve C 2 is only the upper part of the curve (2 .4 ) ended at point I(r2 = p2clq). The parameters for Fig. 1 are chosen so that 4p > 7 . 18 ao = p2aồ°, <*1 = 2pr2a®, E 0 = A - a 2 = - 2 p4aồ4 + 8 p2h2u 2a l0° + p V a ® + r 4a£, (3.6) a 3 = p 6ŨQS - 8p4h 2u 2a l04 - 3 r 2p 4o ^2 + 16p 2/i4w 4aồ° _ o n J,2 L 2 , ,2 8 I 0-4 „2 6 I J l 2, ,2 2 _ _6 Ẩ ỈDp T tỉ UJ T p ^ 0 I ^ ' ỉ ' w 0 * It is easy to verify that the determinant D is different from zero along the reso nance curve (3.2). Since r 2 Ỷ 4p2a0 (equivalence condition of (1.6) and (1.8)) the equation (3.2) is equivalent to ĩỹ ( A ,a * )= 0 . (3.7) The resonance curves have three branches and are presented in Figs 2-3 for the parameters r = 0.01, p = 0.1, 7 = 0.25, and u 2 = 1.1. W ith increasing hj the upper branch 1 moves up and the two lower branches 2 and 3 are tied and then separated, as branches 4 and 5, see Fig. 2 for h = 0.01 and Fig. 3 for h = 0.027. Fig . 2 Fig. s 4. S ta b ility o f stationary oscillations Setting in (1.4) a = ao + <5a, 0 = 0 0 + <50 and neglecting the terms with higher than one degree relative to (5a, 6\Ị) we have the following equations in variation: a0 dỗa dt dỏ xị) dt e 4cư e 2u ( a r ) o Ểa + ( Ẽ ệ ) 60 \<9 V > /0 (4.1) 20 re the symbol ( )o denotes that a = do, 1p = ĩpQ. The characteristic equation lis system of equations is ( d fo ) A V da Jo 4 U) \ (d g 0\ £ /dgo\ V da Jo 2cj V dip ) 2 u The first stability condition will be a0 X = 0 . ổ a 0 ' The second stability condition is d / o d<7o = 4/iw a o > 0 . So = dfo dg0 > 0 d a 0 dxỊỉo dĩpo da0 From equations (1.8) / = ( 2 “ Pa0 cos V»o) 7o + (p«0 sin V’ojffoi <7 = (pag sin i/>o)7o + ( p a £ c o st/>0 + 2 / ^ ° ’ from / 0 = 0, g0 = 0, it follows: ẼẨ ỄJL - È L = ( r- l - 2 &) s da0 dxpo dxpo da0 V 4 p oj 2 Th<i second stability condition (4.4) is equivalent to s = d f d9 \ _ 1 d w Q T = — - 2 6 T \ d a 0 dxfro dxpodaQ/ 2 D T dao ’ 4 (4.2) (4.3) (4.4) (4.5) (4.6) dW dW __ ording to (3.4) we have = T ^ — along the resonance curve w = 0. da 0 do. 0 irefore the condition (4.6) takes the form: a0 dW 52 = n F 2 > 0 - £) d a ị (4.7) — r 2 It is noted that since w\ = —r6 < 0. £) = — > 0 one can easily l a 0 = 0 ’ la 0 = 0 2 itify the regions of the (ao, A) - plane where the functions w and D are positive 21 (4-) and negative (—) and therefore know the stability branches of the resonance curves. In Figures 2 - 3 these branches are presented by heavy lines, while the instability branches are shown by dotted lines. 5. Conclusion The interaction between cubic nonlinear parametric and forced oscillations in a s y st e m g ov ern ed by th e d iffe r en tial e q u a tio n ( 1. 1) h as b een in v es tig a te d by the asymptotic method of nonlinear mechanics. The typical amplitude curves of stationary oscillations are presented in Figs 1-3. The amplitude curves in Fig 1-2 are sim ilar to that of the interaction between linear parametric and forced oscillations (see [l], Figs 94 and 98, page 275). The amplitude curves in Fig. 3 characterize the nonlinear system under consideration. For small values of a0 the forced component is dominated and the corresponding parts of resonance curves are sim ilar to those of forced oscillation. For large values of do the influence of the p a ra m etr ic c o m p o n e n t is cle a r, a n d as th e re su lt of th e in ter action b etw ee n tw o oscillations, the resonance curve has the form of an upward parabola. The stability of the stationary oscillations obtained is studied by using the variational equations. The stability criterion in the form (4.7) is convenient for g eo m e tric in te rp re ta tio n . T h e jum p p h en om en o n tak es p la ce on s om e b ran ch es o f the resonance curve. This work was financially supported by the Council for Natural Sciences of Vietnam. References 1. Mitropolskii Yu. A., Nguyen Van Dao. Applied asymptotic methods in non linear oscillations, Kluwer Publishers, 1997. 2. Zavodney L. D, Nayfeh A. H., Sanchez N. E. The response of a single - degree - of - freedom system with quadratic and cubic non - linearities to a principal parametric resonance. J. of Sound and Vibration, 1989, 129 (3), 417-442. 3. Zavodney L. D., Nayfeh A. H. The response of a single - degree - of - freedom system with quadratic and cubic non - linearities to a fundamental parametric resonance. J of Sound and Vibration (1988), 120 (1), 63-93. 4. Nguyen Van Dinh. Interaction between parametric and forced oscillations in fundamental resonance. Journal of Mechanics, Hanoi, Vietnam T .X V II, 1995, No. 3 (12-19) 5. Minorski N. Nonlinear oscillations. New York, 1962. Received October 5, 1998 22 TƯƠ NG TÁC GIỮA DAO ĐỘ NG CƯỠNG BỨC VÀ THÔNG SỐ PHI TUYẾN Sự tương tác cùa các dao động phi tuyến là một bài toán hay, quan trọng và ã thu hiít sự chú ý của nhiều nhà nghiên cứu. Minorsky N. đã phát biểu rằng: Toàn ìbộ lý thuyết dao động phi tuyến có thể được hình thành dựa trên ca sỏr ủa sự tương tác”. Sự tương tác giữa dao động cưỡng bức và dao động thông số “tuyến tính”, hi hệ :số của hàm điều hòa của thời gian là tuyến tính đối với thông số định vị iã đu’Ợ'c nghiên cứu trong các tài liệu ỊlỊ và [4]. Trong bài báo này xét sự tưcmg ác giữa dao đông thông sổ phi tuyến bậc ba với dao động cưỡng bức. Phưcmg iháp ti ệm cận của cơ học phi tuyến [1] đã đươc sứ dụng để nghiên cứu các dao lộng dììmg và sự ổn định của chúng. Cékc đường biên - tần điển hìn h của dao động dừng được biểu diễn trên hình -3. c a c đường cong trên hình 1-2 có dạng tưcmg tự như trường hợp tương tác ;iữa dsio động cưỡng bức và thông số “tuyến tính” (xem [1], hình 94 và 98 trang !75). C ác đường cộng hưổrng trên hình 3 rất đặc trưng cho hệ phi tuyến khảo át. V ứ i các giá trị ao nhổ thành phần cưỡng bức đóng vai trò áp đảo và phần lường cộng hường tương ứng có dạng tưcmg tự như trong trường hợp dao động ưỡng bức thuần túy. Với những giá trị lớn của ao, ảnh hưởng của thành phần hông ísố khá rõ. Kết quả của sự tương tác giữa hai dao động kể trên là đường :ông hitrờng có dạng parabôn. Sụr Ổn định của các dao động đừng được nghiên cứu bằng cách sử dụng ihưcrnig pháp biến phân. Tiêu chuẩn ổn định dưới dạng (4.7) rất thuận lợi cho áệc phiân định các nhánh ổn định. Hiiện tượng nhảy biên độ cũng xuất hiện trên một số nhánh của đường cộng ìưổrng . 23 Vietnaim Journal of Mechanics, NCST of Vietnam Vol. 21, 1999, No 2, (75 - 88) N O N L IN E A R O SCILLATO R S U N D E R D E L A Y C O N T R O L N g u y e n V a n D a o Vietnam National University, Hanoi 19 Le Thanh Tong, Hanoi, Vietnam A B S T RACT. In this paper, oscillations and stability of nonlinear oscillators with time delay a.re studied by means of the asymptotic m ethod of nonlinear mechanics. Harmon ic, sup'erharmonic, subharmonic and parametric resonances of a Duffing’s oscillator are analyzed. Analytical m ethod in combination with a computer is used. ltro d m c tio n T he hatrrnon ically forced D u ffin g ’s o sc illa to r w ith tim e d elay s ta te feed ba c k has in v e st ig a te d in [l| by u sing th e m e th o d o f m u ltip le sc ale s [2]. B o th p rim ary L/3 su lb ha rm o nic r e so nan ce s o f th e D u ffin g ’s o sc illa to r w ith w eak n on lin ea rity vveak dlelay feed b ack h a ve b een e xa m in e d. A s sho w n in [ l| th e sim p les t m od el a rio us c on tr olled n on lin ea r s y ste m s, e .g ., a ctiv e v eh icle su s pe n sio n s ys te m s 1 th e riionlin ea rity in tires is ta ken in to a cco un t, is d escrib e d by a sec on d ord er rentia l e q u atio n w ith tim e d ela y in th e form ' - ị p - f- .xịt) = — — — ỊẦX3(t) + 2u x(t - A ) + 2 v — - + 2 p c o s \t, ( l . l ) ( CỈ/ c d V re £, ịii, u, V a n d A are c on sta n ts. To Jitiudy all p os s ib le s im p le reso na nce s in th e d yn am ic s y ste m g ove rn ed by itio n (*1.1), in th e p re sen t pa p er it is su p p osed th a t b etw e en th e e x ter n a l u en cy A and th e n atu ral freq uen c y 1 the r e e x ists a r e la tio n sh ip o f th e form À = n + 6TƠ, (1-2) p re n == - is a rational number, p and q are integers. We suppose that pa- Q eter.^ £, /X, Uy V are sm a ll. T h e s m a lln es s of th ese p ara m e ter s is in su red by o du cin ig sm a ll p o sitiv e pa ra m eter e. 75 / Assuming that n = 1 and / is a small quantity of E - order, we can rewrite equation (1.1) in the form 2. Harm onic Resonance dzx{t) dt2 wheire + A x(t) = e[2ơx 4- F), (2-1) F = - ụ,x3(t) + 2ux(t - A) + — + 2pcos At. (2.2) at (it The solution of equation (2.1) is found in the form x(t) = a COS #(<)? 'P (í ) = + Ớ. dx(t) dt aAsin'I'^), (2.3) wheire a and 0 are unknown functions. By substituting these expressions into (2.1) da do an d solving for — — we obtain the following equations for a and 0: at at da E . . — = — Y [2ơx + F ) sin ị i , ' I I (M a — = h 2 ơ £ + F) cos V&. dt A v ' III the first approximation we can replace the right hand sides of (2.4) by their averaged values: da £ — = - - { a L + p sin Ớ), dt Ả (2 5 ) a (^ M n a2^j + pcosO do E a wheire dt A L — (X + u sin(AA) - At; cos(AA), ^ M = Ơ + ucos(AA) + Ausin(AA). The stationary solution of (2.5) is a = a0 = const, 6 — 90 = const which satissfy the relationships: do L 4- psind 0 = 0, ( 3 2\ (2.7) a0 - gMaoJ + P cos 00 = 0. 76 re we obtain: tgớo W(aị,\)=aị L 2 + [M - ị i i a ỉ 2 p2 = 0, (2.9) (2.8) e resonance curves are presented in Fig. 1 for the parameters: p = 0.05, £ — = 1.5 an d for v ario us v alu es of u, t>: u = V = 0.05, A = 0 (cu rve 1), u = V = e 2 ) , u = —V = 0.0 5 and A = 0 (c u rv e 3 ) , A = 0 .5 (c u r v e 4 ) , A = 1 ( c u r v e 5 ). 1 corresponds to the case of an ordinary Duffing’s oscillator without friction. 2 corresponds to the well-known Duffing’s oscillator without time delay. 3 also* re presen ts th e D u ffin g’s o scillato r w ith o u t tim e d ela y a nd w ith a 5 fricti'on 2(£ — v)x two times larger than in the previous case. Hence, the lu m o f th e a m p litu d e s stro n gly de cr e ases. B y in creasing tim e delay (c urv es 4, : reson ance curves lean toward the right and the maximum of the amplitudes y decreases. Figure la Figure, lb Figure 1. Resonance curves in the case of harmonic resonance To study the stability of stationary solution ao,0o we use the variational atioris obtained from (2.5) by letting a = aQ + (5a, 0 = 0Q + 69. Thus, we have ao dSa dt d60 It Lỏa — a o ( M 9 ị ạ a ị y e (^ M fia^j6 a + doL.i (2.10) e characteristic equation for this system of equations is 2 2ea0 e2 flop + —~—Lp + — a0 L 1 Hr M = 0. 77 [...]... com puter and for generating the figures T h is work was supported by the Council for Natural Science of Vietnam REFEREN CES 1 23- H aiyan H u, E a rl H Dowell, Lawrence N Virgin Resonances of a harm onically forced Duffing oscillator with time delay state feedback Nonlinear Dynam ics 15 : 3 1 1 - 327 K lu w e r A c a d e m ic P u b lish e r s , 1998 Nayfeh A H and Mook D T Nonlinear oscillations, ... the relationship (1.2) and when the uation (1.1) has the form: d 2x(t) A2 \2 e + ~ x [ t ) = c J "*(() + Fo + 2p cos Ai, (4.1) ìere ẽơr = A — 3 and Fo is the same as in (3.2) The solution of equation (4.1) is md in the form: x (i) = a COS ị^ —t + O^j + 2 p lt cos Ai, /A \ v , — — = —— sin ( —t + 9 I — 2A P it sin At — (it 3 \3 / dx(t) aX (4.2) iere a and 0 are new variables and ÌL 8A2 Pi 79 (4.3)... sign of the function w 3 in the plane (A, do), because for a0 = 0 and for very large values of A the function w 3 (5.9) is positive T h is 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 stab ilility... e solution of equation (7.2) will be found in the form (2.3) In the first approxim ation, the amplitude a and the phase 9 are determined by the equations: da ea Í ra 2 Ea A + q - ~ sin 2Ớ), dt A v 4 ’ do —ea a— = (m 4- < 7y dt A = 86 (7.5) cos 20^ , L and M are of the form (2.6) The amplitude and phase of stationary itions sa tisfy the equations: 4 L + 0 and H^3(aỔ,A) > 0 84 (5.16) ler word, on the abscissa... word, on the abscissa - axis A the segments lying outside the interval, which the resonance curve is growing up, are stable and the interval lying the resonance curve is unstable (Figure 3a) a r a r n e tr ic e x c ita tio n of th e se co n d d eg ree Changing the structure of the parametric excitation, we consider the system ibed b y the differential equation of the form (5.1) with the function F in . stated “Perhaps the whole theory of nonlinear oscillations could be formed on the basis of interaction . The interaction between the forced and “linear” parametric oscillations when the coefficient. linearities to a fundamental parametric resonance. J of Sound and Vibration (1988), 120 (1), 63-93. 4. Nguyen Van Dinh. Interaction between parametric and forced oscillations in fundamental. linear parametric and forced oscillations (see [l], Figs 94 and 98, page 275). The amplitude curves in Fig. 3 characterize the nonlinear system under consideration. For small values of a0 the forced