p r o c e e d i n g s o f Ĩi-IL Six iti NATIONAL Co n f e r e n c e Oin MECHAIMCS Hanoi 3 - 5 December 1997 INTERACTION BETW EEN THE ELEMENTS WITH DIFFERENT DEGREES OF SMALLNESS IN NONLINEAR OSCILLATING SYSTEMS. Nguyen Van Dao Vietnam National University, Hanoi In trod u c tio n In dynamic svstem s there exist those elements characterizing friction, elasticity and excitations which ■vith different degrees of smallness in the differential equation of motion. These elements have no effect on illation in the first approximation, but thev interact one with another in the second approximation. The ation o f the interaction between these elements is of great interest. The asymptotic method of nonlinear ICS [1] is used to determine the equations for amplitudes and phases of oscillation. These equations are solved ligital computer. The following excitations and interactions will be considered : ratic n o n iin ea rity an d forced e x cita tio n, ratic nonỉinearity and parametric excitation, netnc and self - excitations, netnc and forced excitations. teraction between the elements characterizing the quadratic nonlinearity orced excitation. the dots indicate differentiation with respect to time, oc, q ,h and /? are constants, T = St and £ is a small ;ionless parameter characterizing the smallness of the terms behind it. The parameter £ is introduced ally and used as a book - keeping device and will be set equal to unity in the final solution. The quadratic term : due to curvature or and asymmetric material nonlineanty. The function (p{ỳ) is supposed to be a form Let us consider a nonlinear system governed bv the differential equation ( 1. 1) : v(r), T = St ( 1.2) (1.3) ■2 ( r ) x = íỊ c r t2 + q c o s 2 ộ9( r ) - £ 2 ( - A x + 2hx + fix2,) (1.4) A solution of this equation is sought by using the asymptotic method of nonlinear oscillation [1] ỈCOSỚ+ £U, (a , [f/,9) + £l u^(a.ỉự ,0) + £ 3 ,ớ = ạ> + ụ/, a , ụ/, Ờ) are periodic functions with penod 2n with respect to both variables Ụ/ and 6 and do not e first harmonics sin 9, COSỠ . The functions Aị ( a , ụ/), Bị (a , ự/) are periodic with respect to the variable : functions w ill be determ ined in the process of approximation calculations. ubstitutmg the expressions (1.5) into the equation (1.4) and comparing the coefficients of £ we obtain 4, Sin 6 - 2 a v ( r ) B ] COSỚ4- v 2( t ) tg the harmonics in ( 1.6) gives : = 0 . f d \ 0 6 2 + u . = aa] COS2 B + q c o s 2 (p ( r) . (1.6) r2 Ị (a a 2 -i-2 q c o s2 y /)cos26 r — sin 2 w sin 26 . ( r ) ỐV (v) ’ 3 v ( r ) (1.7) (1.8) 2 • Comparing the coefficients of £ in (1.4) we get- A , s i n < 9 - 2 a v ( r ) B : COSỚ+ v : ( r ) ÔG (1.9) I. COSỔ + Aacosớ T lh a vsm O - (kr COS3 6, I ves a q l A ~ 2 h v { z ) a —-<3s in 2 ^ , 3 v - ( r ) ( 5 c : 3 "i , aa T)BZ = Atf + — — — /? I< 3 a c o s2 ự / v 6 v ( r ) 4 J 3 v (r ) So, in the second approximation we have :o sớ + £ — — — ( c a r 4 - 2 q c o s lụ / ^ c o s lỡ - — S in 2 v /S in 2 ớ a ? 6 v (r ) a s i n 2 ^ d w £ zA a —— = a + £' dt 2 v ( 1.10) y - ccq —— a J -f acos2ỉ// , v ( r ) 6 v ( r ) (1.12) (1.13) 8 12 Stationary Oscillation : Supposing that v ( r ) = CÙ = const and considering the staiionarv oscillation with constant amplitude a ase ụ/ we have : aq ^ A 2 n.2y/ = ha>, - J-co s2\j/ = ya , a ^ Q . (1.14) 6 2 laiing the phase Ụ/ we g e t: ■,<u) = 0 . (1.15) - - a V ’" )= 36 ya \ 2 J \ * r h Z(D From equation (1.15) it follows th at: , , 3 - 5a 2 £ A = Cù - 1 ~ 2(ũ) - 1), ỵ = — f i — — . (1.16) 8 12 a CÚ - 1 ± V a :<7 36 //:&r (1.17) 69 peiidcnce of the amplitude a on the external icy CO u presented in figure 1 for the :ters: e'~aq = 0,063 ,s : h = 0,01 ,£ zy = 0,08. The stability of nontrivial stationary solutions (a*0) equatioa (1.12) when CO is constant can be studied by the corresponding variational equations, which lead to adiiion [1] ■0 . (1.18) Fig. 1 jse function w ( 1. 16) is positive outside and negative inside the resonance curve, Lhe stable branch of the lance curve ìis the upper branch, which corresponds to the upper sign before the radical in (1.17). Thus, sen tiie rwo forms of oscillations corresponding to definite values of CO , the form with large amplitude is e and the forai with small amplitude is unstable. Following chapter 4 of [1], the trivial solution a=0 of the equation (1.12) is stable if the value CỦ does not 1 that iDtervad of the axis CO , from which the resonance curve is rising. In figure 1 the stable branches are m by heaw lines, while the unstable ones are shown by dotted lines. The passage of the system under consideration through resonance when v {r) is not a constant, but Lges bv the l&w : v (r ) = 1/ 0 T e ^ l , can be examined by integration of the differential equations (1.12). The meters lire chosen as /0 = 0, Qr) — 0.009 , [ị/ 0 = 0, £~h — 0.001 , S~Y = 0.01 , s~ CCCỊ = —0,024 , = 1, Li = 1 O' 5 (curve 1, Fig. 2); / i = 2.10 ~5 (curve 2, Fig. 2) ; /i = — 10 3 (curve 1, Fig.3) : = - 2 . 1 0 ' 5 (cnirve 2, Fig.3) From the expression (1.12) and (1.13) one can see that the quadratic noniineanty ( a )is always to eaize the system under consideration regardless of the sign o f a . Moreover, two elem ents characterizing dratic*' noniimearitv CDC2 and forced excitation q C Os2<p(r) combine together and act just like a parametric nation with an intensity ccq . H e system of equations ( 1.12) has a trivial solution a=0, which corresponds to a pure forced oscillation ỉer the actioni of an external excitation eq COS 2 cp : = -E — ZOSlCp 3 (1.19) 70. Fig. 3 action between the elements characterizing the quadratic nonlinearity rametric excitation. 1 this paragraph the following equation = s{C0fc2 + pxcoscot) + £2 (ầx-2hx - px1) , id. where CÙ is the excitation frequency (2 .1) (2.2) y of frequencies, 1 is a natural frequency of the system under consideration, Ơ, P*h,p are constants, he solution of the equation (2.1) is also found in the form (1.5) with 0 = Củĩ + Ụ/ . By a similar 1 as in tlhe paragraph 1 we obtain : = 0 . — {a a 2 + p a COS y /) ^— (a a 2 + pa COS ụ/) cos2ớ - - ^ 7 sin ự/', sin 2 ớ 6ũ) 6 Cù n the setcond approximation we have S/9 + - - - 7 - 3(a a 2 + /7acosy/) - a a ' cos2G - p a c o s { 2 6 - lự) 61CÙ' 2( , 5 2 p 2 : n . 1 ? /ỉứ -f —r pan sin \ự + -L—ra sm 2 w , V 2 4 a r 8&> y / *) ^ *) (2.3) (2.4) (2.5) -£• A P ' D 2 P ' n ,„ a + —— - a - y a H r p a a c o s ^ -r — —aco slỉư 8a;' Vl2 á> 1 2&) 5a;: 8<y 1 2 ứ/)3 ■ rhe nonitrivial stationary solution of equations (2.5) is determined from Củ + 2 *5 ớ sin s in 2 ^ = 0 . + Ỏ Sacosụ / + /? c o s ly / = 0 . - — . s = — DCC . R = — . a ^ O . 6 24 ‘ 4 ilently r + / , cco s^ = 0 . ve :.6) / , cos y - / , sin ự = 0 . :osy/+-Ị £(./?-z + 2^a2) + 45:ứ2 sin ự/ = — , Z - 2 ; m 2) - 1 2 5 : a 2 c o s ^ / + IhcủR.smụ/ = 2 S a (Z - 2 ỵ a : ) - 4 S a i? . (2.7) .inaúơn o f y/ from equations (2.7) and the further discussion will be thesame as in the paragraph 4. teraction between the elements characterizing the parametric and xcita tions with different degrees of smallness. The noinlinear system under consideration is X = £p)XCQSú)t + S' A x - px- + D { 1 - á c 2 ) * (3.1) ủ)0,p, Ỗ)0%D)0,J3 are constants, £2A = CD" — 1 and 1 is a natural frequency. The term characterizing metric excitation is of the first order of smallness, while the terms characterizing the self-excued excitation f the seccond order of smallness. The structure of the equation (3.1) shows that in the first approximation [1] lents orn the right of (3.1) do not affect the oscillation. In the second approximation the terms mentioned and nevv nonlinear phenomena will occur. 71 le solution of the equation (3.1) will be found in the form (1.5), where 9 = (út 4- \ị/ . It is easy to find lation for determining Aị, Bị and Ui : n < 9 - 2 acủB, COSỚ + ỔT d o w, = p a COS 6 cos((9 - (ự). ị the harmonics sin Ỡ, COS G in (3.2) gives : 0 , 7 - COS lị/ — ^— T cos(2ớ — lị/). 6 Cú (3.2) (3.3) (3.4) Tie equation for A2, Ei and u2 is : , Ơ U-, ,\n ớ - la củ B -cosỡ + CỬ 7- - I ỚỠ 022 COS' 6)acủ sin <9. = pu, co s(ớ - Ụ/) + A a COS0 - /3a' COS3 <9- the coefficients of s in ớ and COSỚ in (3.5) yields — I -Daco(] -— az) + -^-^rSiĩ\2ụ/ ,aB1 = - 2(0 \ 4 40)- 1 2 <y r V 3/fc1 6 <U: J a Thus, in the second approximation we have Pa ! 1 X O S Ớ + £■ — — r C O S ụ / — -cos(2ớ ~ ụ/) . 2 ứJ' ' ^ L J and \ị/ satisfy the equations : •> r , 6 - Củt + ụ/ , (3.5) rCOs2 u/ 4ú) (3.6) (3.7) ỉ ; í n ố , p l a d\ụ s 2 \ p 2 3/? _3 p za — ~ DữCù{\ — — ữ ) -r —— sin 2^/ , a - — = - — (A + - ^ ~ ) ú f - — a ;-co s 2 ^ |_ 4 4a> J dt 2a>\_ 6a> 4 4<y (3.8) Equations (3.8) have a trivial solution a=0. The noa-trivial (a^O) stationary amplitude a 0 and phase \ị/ mined from the equations : ( ỹ \ ry f 2 \ . i 2 ụ /ữ = D d ) \ \ - - - a 0: l J L - co s 2ụ /0 = - A + — ^ ~ + i f a 2. (3.9) V 4 J 4 CŨ V 6(0 J 4 :nnined from the equations _ n i l * 2 in2ự/0 = a0 |, idng the phase gives ,A ) = 0 , ,A) = , p r - T ^ o 2 + D 'ứ > £ 2 1 - —a V 4 16a; (3.10) , £ 2A = a r - 1. (3.11) From the equations (3.10), (3.11) we obtain approximately : : 1 - 3 : 2 Ip4 I 7 T Z ã ± S ^ - D { h 4 a° \ - (3.12) 72 a IS pioueu III ngure •+ lOf aic paiaiiiciciò 6 10”3 ,£ ' D = 10~3 ,Ổ = 40. — s l f5 — 0.01 (curve 1) and Ị5 — 0 (curve 2) 4 snoring the right hand sides of the equations (3.8) by R and Q, respectively, we have 1 2 ( a o - r Dổa° ’ l i r j 2 / £ = H a ứ) \ 6ú)~ 4 (3.13) s are ' c Q o f, Ờ' 0 = £ -a 0DỊ 1 - ^ - ư 0 j Fi^.4 <0. i r ' =^Tưf) \ô\ự ) \cu Jz 2củ~ r> 0 . (3.14) (3.15Ì 0 2, ỉn >- 0 , the stability conditions take the form A o ^ ( a 0; ,A ) (3.16) ổX2f gure 4 the heavy branch corresponds to the stability of stationary solution where the inequalities (3.16) are L It is easy to prove that the trivial solution 11=0 of the equations (3.8) is unstable. nteraction between the elements characterizing the forced and metric excitations. (4.10 Let us consider a nonlinear system described by the differential equation \T = s[q cos(2Cùi + z) + /7*cosứ)f] + £ : (Ax - I h x - /3xJ), Ú)2 - I, 1 is natural frequency. The terms containing q and p correspond to forced and parametric excitations, iveiv. It is easv to prove that in [he case under consideration B, - 0. * I4.2Ì 1 73 p a 1 " ' ’ Ũ " * * - ứ <7 c o s ( 2 ỵ / - x ) + — zos>y/ l c o s 2 ớ 3ứ) 1 [ pa . ~ \ q ^ ( 2 \ f / - ỵ ) + £- s m ụ / sin2ớ, and in the second approximation one has pa X = o . COS 6 - r e \ - t — r r C O S ự / — 2cù 3 CÙ gcos(2ụ/ - ỵ ) ~ — cosụ/ ícos2ớ (4.3) 3 Cú pa . q sm(2ợ/ - x ) + ^— s\nyj with a and ụ/ determined by the equations da _ e 2 di 2 cư £ 2 L ; _ p 2° ■ n . . pq X I — = 1 2/2a&> -L i — S i n 2 ụ / - - ^ s i r n y / - £ ) L 4 6 L adu/ £2 I p : 3 „ , P 'a pq — — = - —— (A - f — )a - —/5a — cos 2ụ/ -c o s ((// - £ ) I dt 2a) 6 4 4 6 -Ì ■p. . n z l r-_E 5_ B v puttmg A = - - - - - - - 4 6 we have the following equations for stationary solutions : fo = °. So - 0. f 0 = Icoha + Ra sin 2w + £ sin(ụ/ - j ) , g c = (A ~ — )a - — y3a3 + /to coslụ/ -r E cos(ự/ - x), (4.4) (4.5) (4.6) o r cq uiv a le n tlv / 0 cosy/ - g0 sin ^ = 0, / 0 sin ^ + g0 cost// = 0. From here we obtain : l c o h a s i n y / - l p a ' - - ( A - £ - + R) 4 6 a cosy/ + E C O S 7 = 0, 1 ^ - - (a + pL - R ) 4 6 a s i n [ ị/ + lcúha C O S ụ/ - E s i n X = 0 (4.7) (4.8) T h e conditions for reality of sin If/ and C O S ụ r are : > E 2 cos: X, 4ũ)2h 2 + l p a ' - - ( A + ^ - + R) 1 'ì Ẩ. > 4 6 (4.9) 74 4 ũJzh ị / 3 a 2 - ( & + £ - - R) > E l sin: ỵ. w, we consider two cases. The first case : Svstem without friction (h — 0). In this case we have 3az - ( A + ^ - + R) 6 a cos ụ/ = E c osỵ, -» - y f t r - (À + a sin ụ/ - E sin X- — p a 1 - (A -r —— h i?) ^ 0 and 4 6 - / f o r - ( À + — - / ? ) ^ 0 4 6 eliminating the phase ụ/ from (4.10) we obtain the equation of die resonance curve c f iT ,d 2 ) = 0 , re c y \ a : ) = —, ■) _ - / i r - ( A + — + /?)! V - P a 1 - ( A + £ - - / ? ) 4 6 ! 4 6 7 f — Ị3 a '- - (A -1- —— R) = 0 , i.e. if we have the resonance curve C; : 4 Ó - Ba~ = o r - I + -r i? , r 6 Q we have : ). a COS l ị/ = E c o s ỵ , 2Ra sin y / - E sin 7T 3 7T E i therefore COS y = 0 = 0 y = —, — ; sm 7 = ±1, w = ±arcsin —— 7 2 Z ifa f £ V ," £2 < 1 => ư > 4/?2 ư -> p - - - (3a~ - ( A -Ỉ /?) = 0 , i.e. if we have the resonance curve C 3 4 6 Ì n 2 _ 2 - B a -CO - 4 ỉn we have Oa sin (//■ = E sin -IR a COS ự/ = E c o sỵ, d therefore n i = 0 ==> £ = 0,;r.cos£ = =1. \ự = arccos ± £ 4 £ : (4.10) 4.11) 4.121 (4.13) (4.14) (4.15) (.4.1ÒÌ 751 r, 71 37Ĩ _ . n ủ7í so, ư 7 ^ 0, — ,71,— : the curves c^, C3 do not exist, lí X = — , — , then be 2, 2. 2 2 E 2 curve C] there is still semi-straight line c, in the plane ( a : , CÙ 2 ) , a 2 > —. If X = 0 , 7Z, then beside t 4Zc Cl th e re is s t i ll s e m i- s tr a ig h t lin e C3 in th e plane ( a 2 , CD2 ) , a : > 4 R 2 ' The second case : System with friction (h*0). W e go back to the equations (4.8) and denote D=4(D2h2 + -/5 a : - ( A + — ) - R 2 , 4 6 D,=E -CO S^f sin 7 - — Ba2 + A + — + R 4 6 2 Củh D;=E 2 ứ)/? .4' 6 a) If D ^ 0 . we have -C O S £ Sin J A A a sin t// = —r - acosu/ = —f D D 1 D }~ + D- a = ■ Z)2 (4.17) For the case ^ = 0 we obtain the following equation for the amplitude (a) and frequency ( CỦ Acù2h z + A + E l - I p a ' - R 6 4 - 2 " r- 77 2 2 f £ - a R 2 - - > W - ( A + 4 > 4(D2h'~ _ *■> o „ - b) If D = 0 ; we have — Ba~ — Cl)2 — 1 H ± *J R 2 — AcoJh2 A r * * A * (4.18) and sin ự/, cos^ exist only when D } = = 0 , or equivalently Z)1 COS^ - Z)2 sin 7 = 0, Z)1 sin x + D n COS7 = 0 From here we obtain : D 2 ry’ co. = —— sin 2 r , — B a.2 = CO.2 - 1-i i?c o s2r. 2h 4 6 (4.19) By the formulae (4.9), the amplitude is restricted as : E 2 a .2 > 4 R (420) 76' usion e interaction between the elements characterizing quadratic noniinearity and forced, parametric and seif- itations has been studied. The nonlinear systems under consideration belong to special types, where the rces have no effect on the first approximation. Theừ action and interaction appear only in the second ion. The amplitude and phase of oscillation are determined by using the asymptotic method o f nonlinear [1] a n d d ig it a l com p ute r. le role of each elem ent shown by the equations of the second approximation vanes from system to system, lie, In the equations (1.12) the terms characterizing quadratic nonlinearity ( a ) and forced excitation qual smallness (£■) always appeared as the product o f a and q . This means that each element { a ,q ) lone has no effect on the system under consideration and these elements have equal role. The equations that the quadratic noniinearity (a) gives effect only with the presence of parametric excitation (p), while ĩects the system even without the quadratic noniinearity. The equations (3.8) show that however the ation (D) is smaller than parametric excitation, they have an equal effect in the second approximation and eparatelv act on the system. For the equation (4.5) the terms characterizing forced (q ) and parametric itations are not in equality. The effect of forced excitation exists only with the presence of parametric , while the effect of parametric excitation will exist even with the absence o f forced one. Some related can be found in publications [2-6] listed below. wledgments Fhe author is srateful to Dr. Tran Thi Kim Chi for numerical calculations on the digital computer. This work iCially supported by the Council for natural Sciences of Vietnam. References x>iskii Y u.A ., Nguyen Van Dao. Applied Asymptotic methods in nonlinear Oscillations. Kluwer Academic shers, 1997. tney L.Ù., N ayfeh A.H. The response of a singie-degree-of-freedom system with quadratic and Cubic non- rides to a fundamental parametric resonance .J. of sound and vibration (1988) 120 (1), 63-93. en Van Dinh. Interaction between parametric and forced oscillations in fundamental resonance. J. of lanic*' NCNST of Vietnam, TOM 17, N°3, 1995. en Van Dinh. Non-linearities in a quasi-linear system subjected to external and parametric excitations of rent orders. J. of Mechanics, NCNST o f Vietnam, TOM 18, N° u 1996. en Van Dirih, Tran Kim Chi. Fundamental resonance in one generalized system of Vanderpol type. J. of hanics, NCNST of Vietnam, TOM 18, N° 3,1996. Kim Chit N guyen Van Dinh. On the interaction between forced and parametric oscillations in a system with degrees of freedom. J. of Mechanics, NCNST of Vietnam, TOM 19, N° 1, 1997. TƯƠNG TÁC GIỮA NHỮNG PHAN TỊỬ CÓ BẬC BÉ KHÁC NHAU TRONG CÁC HỆ DAO ĐỘNG PHI TUYEN. Nguyên Văn Đạo Đại học Quốc gia Hà Nội Trong các hê động lực có những phần tử đặc trưng cho ma sát, tính đàn hổi và kích động. Chúng xuát hiên c bé khác nhau trong phương trình vi phân của chuyén động. Những phần từ này tuy không có ảnh hường dến >ng trong xáp xi thứ nhất, song chúng tác động qua lại vói nhau trong xáp xi thứ hai và tạo nèn những hiệu ứng /ến mới. Phương pháp tiêm cận của cơ hoc phi tuyến được sử dụng dô lập phương trình xác định bièn đô và pha dao sau dó các phương trình này được giải trèn máy tính. Những kích đông và tương tác sau đáy đã dươc khảo sát. 1. Phi tuyến bâc hai và kích động cưởng bức. 2. Phi tuyến bậc hai và kích động thông só. 3. Kích động thông số và tự chấn. 4. Kích động thông số và cưởng bức. 77 . motion. These elements have no effect on illation in the first approximation, but thev interact one with another in the second approximation. The ation o f the interaction between these elements. the further discussion will be thesame as in the paragraph 4. teraction between the elements characterizing the parametric and xcita tions with different degrees of smallness. The noinlinear. NATIONAL Co n f e r e n c e Oin MECHAIMCS Hanoi 3 - 5 December 1997 INTERACTION BETW EEN THE ELEMENTS WITH DIFFERENT DEGREES OF SMALLNESS IN NONLINEAR OSCILLATING SYSTEMS. Nguyen Van Dao Vietnam