Quasilinear oscillations in systems with large static deflections

P roceed ings of the International Confluence on A pplied Dynam ics H anoi, 20-25/11/1995 QƯASELINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC DEFLECTIONS N g u y e n V a n D a o Vietnam National University, Hanoi A bstract In mechanical systems the static dcflcction of the clastic elements is usual not appeared in the equations of motion. The reason is that either a linear model of the clastic elements or their too small static dcflcction assumption was acccptcd. In the. present paper both nonlinear model of clastic elements and their large static dcficction arc considered, 30 that the nonlinear terms in the equation of motion appear with different degrees of smallness. In this case the nonlincarxty of the sys tem depends not only on the nonlinear characteristic of the clastic element but on its static itflcction. The distinguishing feature of the system under consideration is that if the clastic element had soft characteristic, the nonlinear system also be- longs to the soft OTIC. If the clastic element has hard characteristic, the system may be either soft or hard or neutral type, depending on the relationship between the parameters of the clastic element and its static deflection. The autonomous and non-autonomous system have been studied. Analytical meth ods in combination with Computer have bttn used. The problem of nonlinear oscillations of clastic structures with large static de flection in general, and beams, plates in particular, may be studied in a similar manner. PART 1 1. Introduction Let us consider the simplest oscillatory system which consists of a mass M and the spring as shown in the F ig .l. The spring supporting the mass is assumed to be nonlinear with the characteristic / ( u ) = C0U + /?0 U3 , (1 ) so that the spring force acting on the mass M is c . ( A - * ) + / » . ( A - x ) 3 , where c0 is positive constant and /?„ is either positive (hard characteristic) or negative (soft characteristic), A is the deformation of the spring at the s t a t i c equilibrium position. ThÌ3 position is chosen aa the reference position. W hen 1 = 0 , the spring force C0A + £0A3 is equal to the gravitational force Mg, that is -C0 A + /?0 A 3 = Mg. M umlreched poiifion > X M re ỉtren câ ị po sitio n — — x.fl F ig . 1 M easuring the displacement X from the static equilibrium position with I chosen to be positive in the upward direction, and applying N ew ton’s second law of motion to the mass M we obtain M i + c0 2 + 3 £ 0 A 2x - 3/?0 Ax2 + /?„ I 3 = 0. It is supposed that A is large and X is enough small, 30 that in comparison with linear term, /90x3 is a small quantity of second degree and jS0A i 3 is of the first degree of smallness: = 0 (c), £ox3 = 0(e3), ậữ&x2 = 0 (ff), 2 _ where e is a small positive parameter. In this case Ax2 is finite. Taking into account the viscous damping force h0x and exciting force P[t, z) which are both assumed to be small quantities of second degree and intro ducing the notation 2 c 0 + 3 & A 2 - ro u = — — , n = ~ r , M M = Ẽ2. M ' ( 2 ) we Can write the equation of motion of the mass M in the form: X + <jj2 x = e~ỊX2 — e2(hx + f i x 3 — / ( t , i ) ) . (3) In comparison with the classical Duffing equation, in the equation (3) the small terms appear with different degrees, most of them are of second degree of smallness. From the structure of the equation (3) one can predict that the influence of the forces on the motion of the mass M can be found in the second approximation of the solution. In the present paper a more general equation will be investigated. X + ui2 i = e-712 + c2F{r, < p (t),x , i), (4) 19 ìere r is a slow tim e T = et, F(r, <p[r), X, x) is the periodic function relatively ip w ith period 2rr which can be represented in the form N F[r,<plx,i) = ^2 n =-N ie coefficients of this expansion Fn[r, X, i) are polynoms of X, X. It is sum ed that the mom entary frequency i/(r) = is slowly changed over at e tim e and that Fn[T,x,x), u(r) have and enough number of derivatives latively to r for all finite values of r. We will be specially interested in e study of the resonance zone when w is near to - u, where p and q are tegers. A utonom ous system rst, we study a special case of the equation (4) when F[r, <p(r),x, x) does >t depend on time F{t,ip[t)ix,x) = Q{x,x). (5) )llowing to the a sy m p to tic method of nonlinear oscillation [l, 2] the solu- an of the equation (4) in this case will be found in the form X — a COS 0 + ffUi (a, 6) + e 2u2 (a, 6) + . . . (6) here Ui(a, 0) are periodic functions of 6 with period 2ir which do not contain e first harmonics sinớ, C08Ổ and a, 9 satisfy the equations: ^ = cAi[a) + ff24 2(a) + i t (7) “37 = w + cBi (a) + ff2B 2(a) + at lbstituting these expressions into the equation (4) and comparing the •efficients of e and c2 we have: o2 w2 (-ggj- + Ui'j = 702 COS2 Ỡ + 2(UjjBi c os ỡ + 2uAi sin 6, q 2 Ú1 (^~0Q2 U a) = 2 ^ 7 ^ ! C08 6 + Q (a C O S 0 , —aai sin i) + + 2au1B2 COS 0 + 2u) A-Ì sin 6 + R{ A\, Bi), (8) here R[0, = iZ(A1(0) = 0. Comparing the coefficients of the harmonica I the first equation of (8 ) gives: Ax = 0, Bi - 0, Ui = ^cos2ớ). (9) Comparing the first harmonics sinớ and COS $ in the second equation of (8) where (/) is averaged valued on time of the function /. We consider now im portant examples: E xam p le 1. D uffing equation Supposing that Q(x, x) = -hi - fix3, we obtain The oscillations are damped with the frequency depending on the am pli- dd tude. W ith the grow of time the momentary frequency -J- either increases if a < 0 or decreases if a > 0 or is a constant if a = 0 . This is a, distin guishing feature of the system with large static deflection. The parameter a depends on the parameters c0, /?„ (spring) and A (static deflection). The considered Duffing equation is modeled [3] on the computer for a con crete case s = 0.25 in the system with hard characteristics p = 0.2 (Fig. 2). On the phase plan there exist three degenerated points X\ = 0, x2 = 5.52 and 13 = 14.47, where z2 is a saddle point while X\ and I 3 are stable focal ones. In the system with a soft characteristics p = - 0.2 (Fig. 3) the gener ated points are Xi = -2.4 , X2 = 0, 13 = 3.4 where x-2 is a stable focal point while the two other are saddle points. E xam p le 2 . Van-der-Pol equation It is assumed that Q(x, x) = —px3+ D { 1 —x2)i, where D is a positive constant. We have yields: Ai = — — (sin dQ[a cos Ớ, — a u sin 0)), (10) ( 11) T hus, in the second approximation we have where a and 6 are determined from the equations da dt d6 dt (13) 21 T 1 í I Ị 1 1 í 7 D uff ing e q u a t i o n (p =0.2) 2 0.0 000 X(t) duffi ng equati o n <p=-0.2> 2 5 . 0 0 0 0 2 5 .0 0 0 0 X(t) Fig. 3 id the equations of the second approximation are The oscillation is self-excited w ith a constant amplitude a0 = 2. The es sential difference in comparison with the classical Van-der-Pol oscillator 13 that the momentary frequency depends on the parameter Q which can be either positive or negative or zero. On the computer the Van-der-Pol equation has stable focal point I = 14.5 (Fig. 4) and a stable cycle with radius 2, which is independent from D > 0 . I ■ * I u / / V ' a n d erP o 1 / ( Q ■■■ ■! 1 1 V e q u a tio n \ \ \ V - _ \ A \ " v » . r s i w 1 I I 1 1 to • re net 1I 0 I \ v * \ V M \ V - f _ K W i . • 2 0 . 0 0 0 0 1 ! \j \j \ Fig. ị 3. N o n -s ta tỉo n a r y n o n -a u to n o m o u a s y ste m The approxim ate solution of the equation (4) in general case will be found in the form I = acos (-£ >+ + eui(r, a, VP, 0) + fi2u3 (r, a, ip,d) + , (15) where s = -<p + rp and u, (r, a, <p, Ớ) are periodic functions of ¥?, 9 with pe- 7 riod 2jt and do not contain the first harmonics COSỔ, sinớ. The unknown functions a and t/> satisfy the equations: ^ = e A i( r , a,4>) + c* A 2 [r, a, rp) + , d l _ _ (1 6 ) = u - -v[t) +cBi{rta,i>) + c2 B 3 (r, o, 4>) + at q By substituting the expressions (15) and (16) into the equation (4) and 23 mparing the coefficient of c and c2 we obtain: 2 / \ d 2 tiỵ d 2 U i n d 2 Ui 2 = ^a2 cos2 0 — — -y(r)) — 2awBi j COS 9 + Ị(w - ~u{r))a~Q~p' + 2w-Ai] «in o . « d^ll2 _ , . 3 2 U2 2 ^ 2 u 2 2 ^ W a ^ + 2“ ‘'(r> f ^ + u' a*r+ " “a = = 2 a 7 U i c o s ớ + F(r, <p, a COS Ớ, —aa; sin 0) - [(“ - f "(r)) a7 ■ 2au,B2 + ^ 7 + Bl 3^r . _ 3 5 i <9Bi 5 B i i + 2i4if?i + aA 1 —— + aBi — - + a —— sin 0 da ơr . ( 17) - {2"frf* + 2l/(r)fra; + 2l/M A > Ũ ỹ + 2uAiÊdẽ + 2‘/[T)BlỆó ĩ + ĩuBl^p' (18) + L - p- v M ) V q v V dyị) dd dyị> da J d<p dr J ?he unknown functions j4i, Bi and Uj will be determined from the equation 17). By comparing the coefficients of harmonics in (17) we obtain: Ai = 0 , Bi = 0, Ui = ( l - £ cos 2ớ) , 9 = -<p + v>. (19) \jialogously, we can find A i , B i and u2 from the equation (18) for the general form of the function F(rtip, x,x). However, we will concentrate itten tion on two important cases: Case 1 . T he p assage of the system throu gh the principal resonance zone it is supposed that the function F(r, (p, X, x) is of the form F(t,<p,x,x) = - h i - p x 3 + E sinip ự ), p = q = 1, (20) where E is a constant. In this case the equation (18) becomes: , , ><92 U2 „ . . 3 2 ti2 2 3 2 ti2 3 " ( r ) f ^ r + ỉ " ( ’ ) 3 ^ + " a i ? + w U ỉ = = 2a~iui COS 9 + haw sin 9 — f3az COS3 9 + E sin <p(t) 3 A 2 — — v(r)) — 2aw.Ỡ2j C08 9 + £(ui — v {T))a ~Q^ + 2cưAaj s in 9. (21) dip Comparing the coefficients of sinớ and C03Ớ in (21) we obtain (w — v(r)) ~ I cluB i = - a a 3 — E ú n Ip, d B (u — v [t ) )cl— — + 2c jA 2 = — h a u — E COS t/>, " d r p a V ~ * Z ' Solving these equations we have ha E /. A 2 = ■ ■ C03 t/>, 2 CJ + i/( r ) _ a 2 £7 i?2 = — a + — - — — sint/;. 2cj a UI + y ( r ) (22) Comparing the coefficients of the other harmonics in (21) and solving the equation obtained we get Uj = ĩ è ĩ ( ắ + f ) a3cos39- (23) Thus, in the second approximation we have: X = a COS Ớ + y r ( x ~ 3 cos 2d) ’ ^ = SP(Ế) -t- V-'(0 ) (2 4 ) where a and v> satisfy the equations: 4 d a 2 ■/i £ — a + r-r C08 d t 12 w + ^ (t) -I drp e2a 2 e2£ - 7 - = UI - i/(r) + —— a"4 + — — sin 1p. dt 2u a u + v [t)} (25) These equations are solved on the personal computer by using the finite C^h c E difference method for the parameters —— = 0.5 • 103, —— = 0.158 • 10~3 U) UI 2 5 1 = + 0.1 (F ig .5), - - 0.1 (F ig.6) with the initial values: t = 0 , OJ w 3 = 10~ 6, v>0 = 0 . The parameter T) = — for Fig. 5 is rj = 0.97 + 10-6 i r v e 1 , A t = 0 .0 4 ) , n = 0 .9 7 + 1 0 " 6t ( c u r v e 2 , A t = 0 . 4 ) , TỊ = 1 .0 3 - 1 0 _ 6 i rve 3), r\ = 1.03 - 10“6i (curve 4) and for Fig. 6 is »7 = 1.02 - 10-6 t rve 1), TJ = 1.02 - 10-5 i (curve 2), T) = 0.97 + 10-6 i (curve 3, A t = 0.04), : 0.97 + 10-5 i (curve 4, At = 0.4). e stationary am plitudes corresponding to the constant values of the fre- ỉncy V are presented in the F ig .7 for the values mentioned above of / , and — + 0.1 (curve 1), a = 0 (curve 2), = - 0.1 (curve 3). The ivy (dashed) lines in this figure correspond to the stability (instability) oscillations. mparing the Figs 5, 6 and Fig. 7 it is seen that increasing the velocity of ssing through the resonance, the m axim um of the amplitude decrease and s peak appear after the resonance peak. The maximum of the amplitudes stationary oscillations is biggest. Fig. 5 Fig. 6 I Fig. 7 27 [...]... s e x c it in g th e s e lf-su sta in e d o scilla tio n and p a ra m etric one coexist in one system Following the assum ptions in [41, these oscillations are w eak T h e y a p p ea r on ly in th e seco n d a p p r o x im a tio n of the so lu tio n and their in te r a c tio n is w eak , also In comparison with the classical problem on the interaction between self­ excited and parametric oscillations. .. rig in y = z = 0 (a = 0 ) of this system is alw ays unstable, because the c h a ra c te ristic equation of the lin e ar term s of (82) has the roots w ith p o sitiv e real p art F ig l ị III w eak in t e r a c t io n o f s e l f -e x c i t e d o s c il l a t io n w it h forced one in N O N L IN E A R SYSTEMS WITH LARGE STATIC DEFLECTION In th is section the attention is concentrated on stu d yin g... p aid s p e c ia l t t e n tio n NO N L IN E A R OSCILLATIONS OF THE SYSTEM WITH LARGE STATIC DEFLECTION F THE e l a s t ic e l e m e n t s a n d l im it e d p o w e r su p p l y 1 th is s e c tio n th e n o n -lin ea r o sc illa tio n s o f a m a c h in e w it h r o ta tin g unbal- nce a nd large s t a t ic deflection o f th e n o n -lin ea r spring and lim ite d pow er apply are c o n sid e re d... s give I< = mrx sin (p + mgr sin Taking into account the driving moment £,( . P roceed ings of the International Confluence on A pplied Dynam ics H anoi, 20-25/11/1995 QƯASELINEAR OSCILLATIONS IN SYSTEMS WITH LARGE STATIC DEFLECTIONS N g u y e n V a. combination with Computer have bttn used. The problem of nonlinear oscillations of clastic structures with large static de flection in general, and beams, plates in particular, may be studied in a. following problems have been examined: I The non-linear oscillations of electrom echanical systems with limited Dwer supply and large static deflection of the elastic elements. I The interaction