Interaction of subharmonic oscillations

Journal o f Technical Ph ysics. 16. 2, 227— 237. 1975. Polish Academ y o f Sciences. Institute of Fundamental Technological Research, Warszawa INTERACTION OF SUBHARMONIC OSCILLATIONS NGUYEN VAN DAO (HANOI) This paper deals with the interaction of subharmonic oscillations. As is generally known [1], the widely distributed species of non-linear systems which exhibit the sub- harmonic oscillations are: (0.1) 3í + tư2.r + ỖX + C3X3 = Z>0 + £cos(p/ + /?), and ( 0 .2 ) X+U)2X + ỗx + c2x 2 + c$x 3 — bcos(pt + P). These systems are characterized by the unsymmetry of either the elastic force or of the external force. We shall study the interaction of subharmonic oscillations in these systems by con sidering the combinative equation of the form (0 .3 ) X + U)2X + ỐX + C2X2 + C3X Ò = b0 + bcos(pt + P). This equation becomes the Eq. (0.1) when c2 = 0 and the Eq. (0.2) when b0 = 0. The interaction between the subharmonic and the parametric oscillations is investigated by considering the Eq. (2.1), which coincides with (0.3) if d = 0 and exhibits parametric oscillations when b0 = b2 = c2 = 0. 1. Interaction of Subhannooic Oscillations of the 1/2-Order Let us consider the case when p = 2v and V ~ 1, namely 1 = v2+fiơ and suppose that the friction force and the non-linear terms arc small of /i-order, so that we can write the Eq. (0.3) in the form: (1.1) X + V2X+ịi(ơx+ ỖX + c2x 2 + c3x 3) = b0 + b2 cos(2v/+£) b2 = b, fi is a small parameter. We shall find the periodic solution of the Eq. (1.1) in the form : bi (1.2) x0 = B0 + B2cos(2vt + £)-M 0sin(w + Y>0), B2 = - 3 2 • Only the most important parts of the solution are considered here: the constant term B0, the harmonic oscillation with a frequency 2v of the external force, and the subharmonic 228 Nguyen Van Dao oscillation A0sin(vt +¥>o), which occurs with a frequency twice smaller than that of the external force. Below we shall concentrate our attention on the latter oscillation. In order to obtain the unknown constants B0i A0, yj0, we substitute the expression (1.2) into ( 1 .1 ) and co m p a r e the coefficients at sin(vt + yĩ0) and COS(v/ + Y>o) and the free terms. We have: v2B0~b0 + /i |ơ50~C1 (bỉ + ~2 Bị + y /4oj + c3 B01 Bị + - - Bị + Y A2j — — B2Aq Ịơ + S + = 0, (1-3) -40|ơ + s+ -a-CòaỈJ = KA0cos(f}-2y0)y 4 ồvA 0 = KA0sin(P — 2ụ)0) where (1.4) K = (c2 + 3c3fi0)* 2 , 5 = 2f2£o + 3c3 From the two last equations of the system (1.3), we obtain a trivial stationary value A0 = 0 and non-trivial values A0 7* 0 determined by the equation: (1.5) w = 0, where w = (ơ+ s + 0 2)2 + <52v2-a:2, (1 .6 ) 3 e2 =-jc>Al The equation (1.5) gives the dependence of the frequency V on the amplitude A0 of sub harmonic oscillation (1.7) Q 2 = — (ơ + 5 ) ± ý K 2 — Ỗ2V2 . Following this formula, the amplitude curves are presented graphically in Fig. 1. The dependence of Q on c2 is eiven for various values of B0 (which is taken approximately as Bq = b0/v2) B0 = 0 .5 , 1,2, and 5. The other parameters are Ô2 =0.01, 3c3 = 0.1, B2 = 1, V 2 = 1.4. From Fig. 1 it is clear that for every value of B0 there exist two sep arated branches of the amplitude curves. As the value of B0 increases the right branch moves to the left and the left branch rises. At first the maximum value of the amplitudes of the right branch somewhat decreases and then sharply increases. For stronger friction Ò2 = 0.032 the picture is rather different (sec Fig. 2, where 3c3 = 0.1, B2 = 1, V2 = 1.25). The right branch of the amplitude curves exists only for a sufficiently small value of B0. Figure 3 gives the dependence of o on B0 for various values of c2: 0, 0.1, 0.3, in the case Ô2 = 0.01, 3c3 = 0.1, B2 = 1, V2 = 1.4. The amplitude curves are symmetrical with rcspect to the axis B0 = -10c2 (in Fig. 3 the symmetric branches arc not presented). With the increase of the value c2, the amplitude curves move to the left. Interact ion o f subharmcnic oscillations 2? 9 Fig. 1. F ig . 2. The curves in Fig. 4 correspond to the systems with stronger friction: Ô2 = 0.032 (3C j = 0.1, B2 = l,v2 = 1.25). From this figure it will be seen that the friction (<5) strongly influences the location of the amplitude curves. In order to study the stability of the subhannonic oscillation obtained, we put in the Eq. (1.1) X = ' where x0 is expressed by (1.2 ). ĩn the first approximation, we have the following equation in variations | + /i<5£ + [v2 + ụ (ơ + 2c2x 0 + 3czxồ) ]ỉ = 0. 230 Nguyen Van Dao Fig. 3 Fig . 4. By using the transformation Ệ = 7/exp( — /1Ổ//2), we obtain fi2ô2 (1.8) 'Ì + V2 4- /i (ơ + 2c 1 x0 + 3c3 *o) — r,= 0 . Substituting here the expression x0 from (1.2) yields (1.9) where (1.10) ^ r + [ ậ - + 7 r ^ ớ»cos('ÍT- e-) b = °- 1 n-1 0o = v1+n(a+S)-¥^-ỊjLC3Al- ^(i2ỏ2, T = vt, eỉ= d l + dỉe, /. = 1,2, 3, 4, KẨ 3 dị, = -^-cosipo- cì B1Aữcos(JĨ-y)o), dlc = B1 siny>o-^-c3B1Aosin(0-y)o), 02, = -K sia ậ+ ^c3Aịsia2ip0, 9ỈC - KcoiỊÌ— -^-Cs^ồcoslyo, ớ3 = — Cị 3 2ẩ 0 , 0Ạ = — C jiij. Interaction o f subharmonic oscillations 231 The stability condition of the zcro-solution of Hill Eq. (1.9) is > 0 , /I = 1,2,3,4. For n = 1, 3, 4, the stability condition (1.11) has the form: (^-4 - ) V* + ° M > 0 ’ which is always satisfied with a sufficiently small parameter /4. For n = 2, we have (1.12) (0o- v 2)2+ y (0o+vJ)/xí <5ỉ - / i í0 i + - ^ - > 0. Substituting here ớ0, 62 from (1.10), we find Ịơ + S+yC3/ígj +<3V-Ị*j+ —cỈAt-ịciAỈKcosiP-lva) > 0. Taking into account the expressions (1.3), (1.6), we obtain 3c3 Ị ơ + s + c 3 > 0 or (1.13) - Ê " > 0 * ỚAq The solution A0 = 0 of the system (1.3) corresponds to the following solution of the system (1 .1 ) (1.14) x0 = B0 + B2cos(2vt + P). The equation in variations (1.8) in this case has the form: <“ 5> where I , 0S + 2 ^ fid*cos(2nvt — eH)\rj = 0, *-1 (1.16) ỠJ = v2+/i(ơ + s ) 0*e = * cos/ỉ, ỡf, = —Ksinfi, OỊ = y C jflf Then, the stability conditions of the solution (1.14) take the form: (1.17) (90—n2v2)2 + — (60 + n2v2)(i2ỗ2 —/x20ỉ + > 0, n = 1,2. In the ease of n = 2, this equality has the form v4 + 0 (/i) > 0 , ■which is always satisfied for a sufficiently small value of /i. 232 Nguyền Van Dao Substituting (1.16) into (1.17), we obtain for n = t the stability condition of the solu tion A0 = 0: (1.18) (Ơ + S)2 + Ỗ2V2- K 2 > 0, which denotes that the free term of the quadratic equation in relation to o2, (1 .5) mus be positive. By using the rule given in [2],we can find on the basis of inequalities (1.13) and (1.18' the stable and unstable branches of the amplitude curves. In Fig. 5 the stable branches are presented by heavy lines and the unstable ones by dotted lines. 2. Interaction Between Subharmonic and Parametric Oscillations Let us consider the equation (2.1) 3c+v2;t + fi(ơx+ <5i + 2*£iC0s2w + C2X2 + czx3) = b0 + b2cos(2vt + P). In the system described by this equation there exists, for b0 = b2 = c2 = 0, the para metric oscillation and for d = 0 the subharmonic oscillation. Although the manifestations of the subharmonic oscillation and the parametric one are different, both forms of oscillations have a common feature and they can coexist under certain conditions. The periodic solution of the Eq. (2.1) is found in the form: (2.2) Xq = B0+B2cos(2vt +(})+ys\nvt + zcosvtt Interaction o f subharmonic oscillations 233 where B2 = -b 2j3r2 and B0iy,z are constants. To determine the terms with frequency V equal to half the frequency of the external force, we substitute (2.2) into (2.1) and com pare the coefficients of sinrf, COSVi. We have: ( B 2 V2 4- z2 \ Bị H— — + — 2— I + /iCj | s 0 1 ^ + y B\ + y O'2 + r 2) | - -Ẹ- 5 2 [cos^(y2 —z2) + 2>’zsin^]| = b0> (2.3) Ơ + S + -^-c30 '2 + z2)\y = (Ảxos/? + £/Xy + (<5v + A!sin0)z, ơ + s+-^-CoO>2 + z 2) |z = (KsinP— 0v)y-(KcosP + d)z, where K = (c2 + 3c3 Bo) s 2 , 5 = 2c2 B0 + 3c3 Ị*ẳ Hr y * |Ị . From the two last equations of (2.3), we obtain after some simple transformations (2.4) ịa+s + ị c ^ ủ Ầị= (K2+d2 + 2dKcosậ)Aị + ỗ1v1Aị + 2ôvKs\nịì(z2—y Aị = (K2+d2 + 2dKcosp)Al, + 4 ỗvyz{K cos p + d), or |Ịơ+s + c 3 +<521 where A% = y2 + z2. Solving the Eq. (2.4) in relation to A0, we obtain 1) A 0 = 0 2) A0 # 0 determined from the relation (2.5) Ị ơ + 5 + i . C3/4ĩ | +Ô2V2 = K 2 + d 2 + 2dKcosp. 3 The dependence of Q = -'CyA\ on the parameter c2 is presented graphically in Fig. ( for various values of /?:/? = 0 , 7t/2y7i and for d = 0 .1 , Ỗ2 = 0 .0 1 , 3c3 = 0 .1 , i?2 = 1 V2 = 1.4 and B0 = 0.5. Fig. 7 presents the dependence of 0 on d in the case of Ổ2 = 0.01 3c3 = 0.1, B2 = 1, V2 = 1.4, /? = 71, B0 = 1, c2 = 0.2. From this figure it can be seer that the component of the oscillation with frequency V does not exist for all values of d but only for that lying outside a certain interval. In order to study the stability of the stationary oscillations, we write the equation ir variations (2.6) i? + U2Ỗ2 1 v2+ /i(ơ + 2</cos2r/ +2c2Xo + ĩcòxị)- \rj = 0. 234 Nguyen Van Dao F ig . 6. Substituting here the expression x0 from (2.2), we can write the Eq. (2.6) in the form: 4 (2.7) 7ị+ịoo + 2ụ ^ ỡ .c o s í m - í - e .) ] ^ = 0, 1 •where ỡ0 = v2 + ^ (ơ + 5 ) + - |/ i C 3^(ẳ - 9Ỉ = ® i + 0 « , ® I* = K y - ^ c 3B2(y cos p + z sin ậ ), 0lc = Ar— -y c3i?2Osin/?-zcos/?)> ớ2, = - A T s i n / 3 + -^ C j> > z , (2.8) ỡ2e = d + K c o s ậ - - Ị Cì(y2- Z 2), Interaction o f subharmonic oscillations 235 = J c3B2Cycosậ-zsinậ), 0JC = 4 c3B2Cysinfi + zcosfi), 0 * = ị c 3BỈ. Now, the stability conditions are (2.9, [9o- ( = ) 7 + f + ( ^ ] ^ - l + ^ > , = « . , . 1 , 2 , 3 , 4 . It will be seen that for sufficiently small value of ỊẮ the conditions (2.9) are satisfied for n 7* 2. In the case of n = 2, the inequality (2.9) yields (2.10) Ịơ + — c3-í4ổ + sj ■+■ Ỗ2V2 > AT2-fi/2 + 2/^/cos/3 + -y^r CỈ Aq — — c$d(y2 — z2) — y Ac3 [2yzsin/? + (.ỵ2 —z2)cos/?]. Taking into account the formula (2.5) and Ịơ + S+ ^ czAq^Aq = 2Kyzsiũậ+(Kcos(ỉ + d) (y2-z 2)f obtained from the two last equations of the system (2.3), the condition (2.10) can be rewritten in the form: (2.11) c3ịơ+ S+ ~c3Aề^ > 0 or <2-12> 4 S r > ° ’ where (2.13) W1 « 0 is the equation of the amplitude curve, w l = ị a + S + j C i A ị j + Ỗ2V2 — K 2—d2 — 2 d K c o s ộ . When A0 = 0 the solution (2.2) takes the form (2.14) x0 = B0 + B2 cos(2vf4-/?). The stability (or instability) of the stationary value A0 = 0 (x0 = y0 = 0) corresponds to the stability (or instability) of the solution (2.14). The equation (2.6) in this case has the form: (2.15) rị + (00 + 2/1 [02 cos(2rf — e2) + 04 cos(4vf — £4)]) 7] = 0, 236 Nguyen Van Dao where 08 =v2 + n(cr + S ) - 8 a2!=-Ksinp, 6°u = d+ Kcosạ , 0° = I c3 fl|. As the right part of the Eq. (2.15) contains only cosinus of the even /, then the stability condition is (2.16) <fl8-/»V)i + j (0S+iiM ) ^ ốí- ^ 5 ỉ + > 0. This condition is always satisfied for /7 = 2 . For n = 1 it takes the form (2.17) (ơ + S)2 + ỏ2v2- K 2- d 2-2Kdcosp > 0. The inequality (2.17) shows that the free term in the Eq. (2.13) must be positive. The geometric interpretation of the stability conditions (2.12) and (2.17) is similar to that presented in Section 1. References 1. c . H ay ash i, Non-linear oscillations in physical systems. Me Graw-Hill, New York, 1964. 2. Nguyen van DaO, On the phenomenon o f para m etric resonance o f a non-linear vibrator under the action of electromagnetic force, Proc. Vibr. Probl., 13, 3, 1972. 3. N. M inorsky , Nonlinear oscillations, D. Van N ostrand C om pany, Inc, New Y ork 1962. 4. B. o. Kohohehko, n . c. KoBAJifewyK, o ỜUHdMUHecKOM 63auM0Òeủcmeuu MỂXãHU3M06 ỉeHepupoỗaHux KO/ieôaHuủ 8 He.iuHeũHbỉx cucmeM QXy M exaHHK a T B c pao ro Tc.ia, M ocK B a, 4 , 1973. Strcszczcnic ODDZIALVW ANIE DRG A N SUBHARMONICZNYCH W srod wielu drgaủ nieliniowych nalezy wyrỏzni£ jak o najbardziej istotne: drgania harmoniczne, subharmoniczne, ultraharmoniczne, paramctrycznc i samowzbudnc. Drgania harmoniczne S 4 to drgania z czẹsto$cÌ4 siìy zewnẹtrzncj (przy zalozeniu, ze siỉẹ zcw nẹtrznạ mozna przedstawié ja k o harm onicznạ funkcjẹ trygonom etrycznạ sin(cư/ + x) lub cos(a>/ + a)). D rgan ia subharmoniczne majạ okrcs rỏwny cal- kowitej wiclokrotnosci okrcsu sily zcwnẹtrzncj. N a odwrot, najmniejszy okrcs drgart ultraharm onicznych je s t rố w ny 1 Ịn o k re su siỉy z e w nẹ trzn ẹ j. Drgania p a r a m ctry c z n c ch a ra k tc ry z u jạ sic ty m , 2C od by vvạịạ sic z cz ẹsto s ciq róvv n ạ polovvie czẹ sto sc i p a ra m c try c z n e g o p o b u d z c n ia . D rg a n ia s a m o w z b u d n e m a jạ czẹstos(í bliskạ czẹstosci wlasnej. Tc drgania sạ podtrzym ywanc przcz staỉe zrốdla encrgii nicdrganiovvcgo cha- rakteru. Oddziafywaniu nicliniowych drgarí byty poswiẹconc pew ne prace [1, 3, 4J. Jcdnakzc oddziaiywaniu s u b h a r m o n ic z n y c h d rg ari z in n ym i ro d z a ja m i d rg a rt n ie lin io w yc h nic poẳv viẹcon o d o tcj p o ry nic zbẹd - n cj uw agi. N in ic jszy a rty k u l p rz ed s ta w ia b a d a n ic od d z ia ly w a n ia s u b h a rm o n ic z n y c h d rg ar t z inn ym i ro d z a ja m i (Jrgart nie liniow ych , k tó rc d o ty c h cz a s z n a jd o w a ly sir p o z a p o lem b a d ah . Oddzialywanic drgart subharmonicznych bada siẹ przez rozpatrzcnic kombinowanego ro w na nia p o sta ci (0 .3 ), a od d ziaty w an ie d rg a r t s u b h a r m o n ic z n y c h i p a ra m c try c z n y c h b a d a siẹ za p o m o c ^ ro w n an ia (2.1). Do analizy rỏvvnaứ typu (0.3) i (2.1) mogạ byà sprowadzonc zagadnicnia drgart cle k tr o m e c h a n icz - ncgo ukỉadu [2) z uwzglẹdnicnicm statyczncj deformacji nicliniowcgo clem entu sprxzystcgo. 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AeộopMauHH HeJiHHeHHoro ynpyroro 3JieMCHT3 BbiJiH Haâ^eH bi cTauHOHapHJbie cyốrapMOHimecKHe KOJie6aHHH c MacTOTOÌí, paBHOÌ* noJioBHHC Ma- CTOTbl BHCIiIHerO B03Ốy>KfleHHH H HX yCTOHMHBOCTb DEPARTM ENT OF MATHEMATICS AND PHYSICS POLYTECHNICAL INSTITUTE, HANOI Received O ctober 14, 1974 . Sciences. Institute of Fundamental Technological Research, Warszawa INTERACTION OF SUBHARMONIC OSCILLATIONS NGUYEN VAN DAO (HANOI) This paper deals with the interaction of subharmonic oscillations. . unsymmetry of either the elastic force or of the external force. We shall study the interaction of subharmonic oscillations in these systems by con sidering the combinative equation of the form (0. KcoiỊÌ— -^-Cs^ồcoslyo, ớ3 = — Cị 3 2ẩ 0 , 0Ạ = — C jiij. Interaction o f subharmonic oscillations 231 The stability condition of the zcro-solution of Hill Eq. (1.9) is > 0 , /I = 1,2,3,4. For n