Proceedings of Vibration Problems, 14, 1, pp. 85-9 4, 1973 Institute o f Fundamental Technological Research, Polish Academy o f Sciences PARAMETRIC VIBRATIONS OF MECHANICAL SYSTEMS W ITH SEVERAL DEGREES O F FREEDOM UNDER THE ACTION OF ELECTROMAGNETIC FORCE NGUYEN VAN DAO (HANOI) 1. Systems with /I Degrees of Freedom Let us consider a vibrating system with n degrees of freedom which consists of a weight less cantilever beam carrying n concentrated masses mlf m2, m3, (Fig. 1). The elastic elements of the vibrating system have stiffness kLy k2y Fig. I. Supposing that some s -thmass is subjected to electromagnetic force, the differential equations of motion of the system considered can be written, in accordance with [1] in the form: ~dt ^ + c q = £sinv/’ mlxl+ki(xl-x2) = -híỉcl-p l(xl-x1):i, »iix2+l<i(xi - x i) + k1(x2 -xi) = - h ixi - p i(xI \l)3 - /32(x1- x ìy, (1-1) . mix,+ k t_l(x ,-x ,_ 1)+ k ,(x ,-x 1+1) = - h tx,-0,_ d x j -x ^ i ) 3 SI I ÕL PẢX, x,+ l) + 2 q ^ , mn XH + k„_ t (xn xn _ 1) -f kn xn — hnXH 86 N guyen V un Dao We assume that L = I(jrJ = z.0( l-»,*, + OTjX*), and that the friction forces and the non-linear terms in (1 .1) are small with respect to the remaining terms. Then, Eqs. (1.1) can be rewritten as: L-a'q + c = £sin vt — /j[L0q(— 2 , .Vj+ a2.v|) + ^L0( —ajij + 2a2 Vj.vJ], .V, ^ A , f-V, - V.,) = fjF\ , (1 2) »h.Xi+ x,)-ị k 2(.x2- x j) = ///",, »'íÀ:, + Ẳ ,_1(v., v.,_l) + Ắr,(.rs v.,+ 1) - - ^ q2 L.0 Z{+ uF,, ™»vj \„ - ,v„_ ,) 4 A,,. Y„ - ///■„. where / - / , - — // , .V, - /#, (a i - Vj)\ / ' r : - - /?, .V, - /?, (V; - V, )•’ - ,? ,( .v 2 - .Y j )\ (1.3) / r , = - //, vs vs —/?»_, (.V, - .r,_ ,)J -< f5(.v., - V, + , )3, 3 n •/'f». = ^/1 X ~ I ( ^ fi — \ It - I )^ — fttt •' We suppose that the characteristic equation of the homogeneous system m , V, +/r,(.r, - -V2) = 0, J v2 + * i (* 2 rj) + /:2(.v2 -A 3) = 0 , ^ ^ n - 1 (n . J ) -+■ k „ A „ — 0 > has no multiple roots and that its rootsw,, , 0)n are linearly independent. Then, to study the system (1 .2), wc shall analyze its particular solution corresponding to the one-frequency regime of vibrations [2]. To that end, we introduce the normal coordinates f i, by means of the formulae: (1.5) n X, = y ] c 'f ’t,. 5 = 1 , 2 , . . . ,« , Ơ -= 1 where cJ is algebraic supplement of the element placcd in the j-th column and the last line of the characteristic determinant of the system (1.4). We can easily verify that the normal coordinates f j , satisfy the following equa tions: Param etric vibrations o f mechanical system s with several degrees o f freed om 87 Here io(?> q, *3 , X,) = L0q(- 3 ^ , + 32.x2,) + L0qx,(-al + Ĩ7ĩot,) + Rq, 0 , = _ !_ ỳ ei»Ft- ĩ i h È L eo> 1 Mj Z j k IpMj ' ’ n Mi = £ /-1 In the first approximation, the investigation of one-frequency regime in the system considered can be reduced to a study of two equations: the first of (1 .6) and one of re maining n equations. The choice of the appropriate equation depends on the value of natural frequency CO in the neighbourhood of which the parametric vibrations are examined. Supposing that the frequency V of external force is near the natural frequency 0)j. Then we shall investigate the equations: ( 1 . 7 ) where L0q + ~ q = £ sin Vi - fiFZt ^ = k —— ẩm 1 ft > h c2in ỵ "s'- 3 > w?* = ^■-[Pic[ii(c[J'-citiì)3 + ậlc[j)(ct2n -c[J))3+ậìc,j ì(ct2n-c y ì)3+ + p, - 1 (cịJ> - c\L\y + p, c‘1>(c<'■> - <•<ị>,) + + /?„., c'B''(c'J1 - cii>!)3 + ft ’], ụFS = L0q (- ilcỳ>Ệj + <x2cV:)ỉỉJ) + L0qcíJ)Ệj{-31+2i1c‘»ỉJ). The remaining n- 1 normal coordinates fi, fy_j, £/+1, » fn arc far from the reso nance, their vibrations will be small in comparison with the resonant vibrations considered of the coordinate ỈJ, and in the first approximation they may be disregarded. Equations (1.7) describing the one-frequency regime of vibrations have the same structure as the equations of motion of the system with single degree of freedom [1]. This gives reason to expect that in each resonant region the same peculiarities of motion will be displayed as were found in the system with a single degree of freedom. Introducing the notations Wj’ r ~ Wj ’ 0 ~ loj ’ “* ■ 2Mị, ’ (18) « r = ~ I o , Ql= T~c> fJ = T T 5 - yj = 7T> A iJ CO j -*-'0 0 ^ j ^ j Eqs. (1.7) assume the form: q " + v>q = ejSinyjZ- (1.9) ' LoCOi + = -ụftỉ,j - / i ^ - a ĩ ^ 2 + fỉatq,2ỉj. 88 N guyen Van Dao Now, we transform the system (1.9) by means of the formulae: q = efsinyjT + Bsin(pt q' = yje* cosyjT + Qj Bcos<p, (1.10) ỈJ = - Ố - cos2y7T + /4ysinỡ; , where 2y b t ’j = Sin2y,r + /lyy;cos0; , e? = -^r— 2, b = — —^ 7 - , 9? = i2 ,r + 0 , 6j = yjT + y>j. J V ỉ-rỉ' 2 The transformed equations have the form: ịẲ Q J - 7 - = - - T - ^ / ' J C O S ? , dĩ LqCOj i/0 fj. (ỈT Lịj(uJ Gj B-j t = T ~ T = ^ 0 sinọ?, = —/íyU —yj)sinớjC0SỠj —//ScosOj-f- a j Yj ^j L = /4,(1-y * )sin 20,+ /^ sin0j4- S=hỉ'j + PỈ*-zĩq'2Ệ, where the non-written terms vanish when 5 = 0 . We suppose that yj is in the neighbourhood of 1 and that}/; and arc linearly independ ent. Then, in the first approximation the solution of the system (1.9) satisfies the equations obtained from (1.9) by averaging in time its right-hand part: [1 +00 0]-^-= -/< 4 * - (1.12) ^ = G ( B , 0 , A j , Vj), d.ij h ụ y‘ dt' = ^ 2 y^ f 2 Cl J' yjAllh = ị V -yj + + ị + -ị clAJcos2Vi+ ■■■ where c, = ỊL + 1 2 l-4 yỷ ’ /1 = q + ĩpb* +~2 p (I_4y2)a * Param etric vibrations ~vf mechanical system s with several degrees o f freedom 89 Since B -+ 0 when / -> 00, then below we shall take into account only the equations: dA Ị h . li . . - Yj ~ -t*ỶYjAJ + Ỷ CijS y)j’ (1.13) y jA j ~ár' = \ V - rJ + + -*2 CL AJcos2V>J> from which we obtain the amplitude Aj of vibrations: (1.14) and the phase (1.15) sin 2 yj = — yj, COS 2 yjj = + — Ị^cỉ - h 2 yj. c 1 C1 Equations (1.12)—(1.15) are different from the corresponding ones in the system with a single degree of freedom [1] only by the values of the constant coefficients. The method used enabled us to reduce the more complicated problem to the whole complex oĩn prob lems of the type considered earlier. In spite of this, in the first approximation each of such problems can be investigated independently of the others, because according to the con ditions of the problem, the resonant processes cannot be developed at the same time in more than on resonant region. The stability of stationary regimes of vibrations may be found by analysing Eqs. (1.12). The criteria of stability formed in [1] are: c w > 0 for Aj ^ 0, cAj w = Ị 'ị- fxpAj + 1 — y/ -fi2(cị-h2y}), and /Ầ2{h2y}-c2L) + (y} -l-ỊẲỔ)2 > 0 for Aj = 0. The study made in [1] concerning the stability of stationary regimes of motion will be suitable for the character of resonant processes described by Eqs. (1.12) in qualitative relation. This removes the necessity of analysis in detail the criteria of stability. Here we note only that for very slow change of frequency V in the system considered, n resonant peaks corresponding to the values V = (OỵÌYi = l),y = co2(y2 = 1) ••• arc observed (Fig. 2 ). Fio. 2. 7 Probleroy drgaA 90 N g uye n Van Dao 2. Parametric Resonance Id a System with Infinite Number of Degrees of Freedom W e i n v e s t ig a t e i n t h e C a r te s ia n c o o r d i n a t e s x9y,z a p r is m a t ic b e a m w i t h le n g t h / w h o s e c r o s s - s e c t io n i s sy m m e tr ic a l w i th r e s p e c t t o t w o m u t u a ll y p e r p e n d ic u la r a x e s . W e assumCị t h a t t h e a x is o f t h e b e a m in t h e u n d e fo r m e d s t a te c o i n c i d e s w ith t h e a x is X a n d t h a t th e . s y m m e tr ic a l a x e s a r e p a ra lle l t o t h e a x e s y a n d z ( F i g . 3 ). I Fig. 3. The beam UDder certain conditions of strengthening of its ends is subjected to the action of electromagnetic force which is ìị distant from the origin of the coordinates and directed to the axis y. We assume that the inductance L is a function of distance yị = yựị, t)j (2.1) L = I O',) = L0( 1 - + “2>'ỉ). and therefore the electromagnetic force depends on the location of the electromagnet 1 . , C'L and on the vibrations of the beam, and has intensity a —— . 2 oyl We -assume that the material of the beam follows the law [3] °x = A O = E ( l-d E 2el)ex, where ơx is the longitudinal force and €x is the longitudinal elongation. Then, the equation of motion of the beam is: (2.2) g2M d x 2 w h e r e Q is th e i n t e n s it y o f m a s s o f t h e b e a m , y = y(x, t)— th e d e fl e c t i o n , P(x, t)— t h e in t e n s it y of e x t e r n a l load, — t h e b e n d i n g m o m e n t : M - f í f ( y w ) > ’JyJỉ - Eỉ ỉ \ ' - dE'y ' [ ĩ? ) Substituting t h is e x p r e s si o n i n t o (2 .2 ) , w e o b t a i n : e - i f +EJ s - 3dE »/ St2 ÕX* \ c*y d2y 1 [ õx* ÕX2 w h e r e , (&)] Ji = / / y*dydz, J = / f y 2dydz. Param etric vibrations ọ/ mechanical system s w ith several degrees of free do m 91 We assume that the non-linear terms and the terms characterizing friction are small in comparison with the Unear terms. Then the equations of motion of the system considered can be represented in the form: (2.3) where (2.4) q + ũịq = eún v t+ nĩx ịy^ - f t - > > d2y ,J d*y _ õt2 dx* p 2t 02 L p - E L EJ ME>], ỉ c' Ĩ 9 /-* ’ p I * ' * - - T & +' [ & & +2( £ ) ì & +7 ' The external load P(x, r) has the form: (2.5) P(x, 0 = 2A * f o r /, + y , where A is the length of that element of the beam which is directly subjected to the action of electromagnetic force. To solve the system (2.3), we note first that the generative equations (/1 = 0) (2.6) have the solution: (2.7) <7 +-0ẳ<7 = e sin V/, C/2y <?r2 ■ " ÕX* 7 +b r- = 0 q = e* sinvr + Ssin 9>, oo y = y .x M c v * ( j 2 -bt+y*]’ Hm. 1 where Bt 0 , f„, arc arbitrary constants, xn are the eigenfunctions which define the natural modes of vibrations of the beam and depend on the boundary conditions. Equations (2.3) are different from the corresponding ones of the system (2.6) only by small terms fiF1, /iF2. Consequently, it is natural to propose the following form of solution of the system (2.3): (2.8) CO q = e*sinv/ + 5sin9>, y = JT X,(x)s„(t), *-1 where (p = i?o/ + 0 and By 0 , sn arc functions of time. 92 N g uyen V a n Dao Now, instead of determining the functions q and y, we determine the functions B, SH. To find the differential equations for these variables, we represent F2 in the form of a series: To seek the functions of time Vn, we multiply both sides of the equality (2.9) by xl9 and integrate the result over the total length of the beam; due to the orthogonality of the eigen functions there remains only one term on the right-hand side which corresponds to the number n, so that where Kn are polynomials of third degree, relatively of slts2 , We consider now parametric resonance when the frequency of the electric circuit V is in the neighbourhood of 0) j assuming that the natural frequencies (0lf0)2 are inde pendent. Then we retain in (2.11) only the coordinate Sj. The remaining coordinates Sỵ Sj_l9 Sj+ị are far from the resonance and their values will be small in comparison with S j and in the first approximation we can disregard them. Thus, following the expres sions (2.4), (2.8), we have: 00 (2.9) I I (2.10) Substituting (2.8), (2.9) into (2.3) and equating the coefficients ỵ„, we arrive at: (2.11) q + Qịq = esin vt+pFlt snic>2ns„ = /.trnL0*1q2Sll + /xK„(sl, s 2 , j , , i 2l ,uF2 = - ~ Sj AO+p [xyx]'1+ 2XỊ"2 xyisf + Ĩ - . Therefore, from (2.10), (2.11) we obtain the equations for q, sn: (2.12) q + ũịq = e si n w + /ifh Jijr S j+ c o jS j = - — S j + PajSj+bjq2Sj-cqzf where (2 .1 3 ) Parametric vibrations of mechanical sy ste m s with several decrees o f freedom 93’ It is easily seen that the system of Eqs. (2.12) is the complete analogy of the differen tial equations of vibrations of a system with single degree of freedom. To avoid repeti tion, we shall refer below to the paper [1], where the problem of construction of a solu tion of the system of equations of the form (2.12) is considered in detail. Thus, following the results of [1], we conclude that when the frequency V of an electric circuit is near to CO I , then the beam co nsidered vibrates strongly w ith frequency V (p a ra  metric resonance). T his type o f resonance takes place also when the frequency V is near to w2 ,a >3 However, it must be emphasized that in the system with distributed para meters the vibrations with ‘the lowest frequency (i'jj) play the main role. Some experiments were performed with beams and systems of several degrees of free dom. The experimental results in the cases considered were in good agreement with the theoretical results. This fact testifies to the acceptability of the limitations used in the prob lem and shows that the approximate solutions found by using the assumption concerning the one-frequency regime of vibrations in the regions of resonance can be adopted for practical purpose. For the cantilever beam with parameters E = 2 • 107 N/cm2, J = 16' 10“3cm4, rp = = 10 4N • s2/cm2, I = 46cm; therefore, 0)i = 14.8, 0)2 = 93.7, strong vibrations with frequency of electric circuit V when V is in the region 13.5- 14.3 H z were very small. For the same beam, but when / = 58cm and therefore C0L = 26.3, substantial parametric resonance when V is in the region 27.1-29.4Hz was observed. R e fere n ce s 1. N ci;yen v an D a o, On the phm onunon of parametric resonance o f a non-lỉnơur vibrator under the ac tion of electromagnetic force, P ro c . V ibr. P r o b l., 1 3 , 3, 1972. 2 . I I . I I . Eoro/iiOBO B , KD. A . M iiTPono/ibC K HÌi, AcuMWtiotnuuecKue Memoòbt 8 meopuu ỉte.iuneũiibix KO.ICỔCI- ftuii, M ocK Ba 19 6 3. 3. H . K a u d e r e r , a ĩichtíinear Mechanik, B e rlin 19 5 8. Streszczenie PARAMETRYCZNE DRGANIA UKLAD 0W MECHAN1CZNYCH o WIELU STOPN1ACH SWOBODY, WYWOLANE DZIA LANIEM ELEKTROM AGNETYCZNEJ S1LY Niniejsza praca jest kontynuacịạ poprzedniej pracy [1], Z badano w nicj drgania mechanicznego ukia- du o n stopniach sw obody oraz bclki przy zalozeniu, ze sạ one obci^zonc elek.tromagnetycznạ siỉạ para- metrycznie pobudzajạcạ drgania. Podobnic ja k w [1] rozpatiyw ane drgania param etryczne majạ cz$sto$ố rósvnạ czcstoắci đrgaií w elektrycznym obwodzie. W yznaczono am plitudy drgan i zbadano ich statecznosé. 94 N guyen Van D ao PC3WMC riAPAM ETPIM ECK HE KOJIEBAHRH M E X A H H ^H C K H X CH C TEM CO MHOrHMH CTEriERHMH CBOBOJltl BBI3BAHHWE JXEfitCTBHEM 3JIE K T P 0M A T H H TH 0fl C H JIM H a cT o am an p aố o ia HBJIHCTCH npOAOJi>KCHMCM n peA b inym eìí p aổ oT b i [1 ]. HccJicAOBaHbi B HCH K0- /IC ỒaK HH MCXaHlWCCKOft CHCTCMbX c n CTCnCHHMH CBOÕOAbl H ỐaJIK H npH npCOTOJIO>KCH IUI, MTO OH11 Ha- rpy>K eH bi 3JICKTpoMarHHTH0 A cm io ii napaMeTpHMccKH B03ốy>K juiK)ineìi KOJieỐaHHH. AHaaorM MHo KOK B [1 ] paCCMaTpHBaCMBIC napaMCTpHMCCKHC KO^cõaHHH HMCIOT *iacTOTy paB iryio MaCTOTC K0/ic6aHHií B 3J1CK- TpjfMeCKOM KOHType. O npcA C ^C RU aM iuurry/Ịbi KOiieốaHHÌi H HCcrceAOBaiia HX ycTOH'iHBocTb. d e p a r t m e n t o f m a t h e m a t i c s a n d p h y s ic s PO LY TEC H NIC INSTITUTE, HANOI Received March 8, 1972 . F FREEDOM UNDER THE ACTION OF ELECTROMAGNETIC FORCE NGUYEN VAN DAO (HANOI) 1. Systems with /I Degrees of Freedom Let us consider a vibrating system with n degrees of freedom which consists of. several decrees o f freedom 93’ It is easily seen that the system of Eqs. (2.12) is the complete analogy of the differen tial equations of vibrations of a system with single degree of freedom. To avoid. 3. The beam UDder certain conditions of strengthening of its ends is subjected to the action of electromagnetic force which is ìị distant from the origin of the coordinates and directed to the