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Journal of Mechanics, NCNST of Vietnam T. XVI, 1994, No 3 (5 - 10) WEAK SELF-SUSTAINED SYSTEM UNDER THE ACTIONS OF LESS WEAK EXCITATIONS NGUYEN VAN DINH Institute of Mechanics, NCNST of v~·etnam SUMMARY. It has been known that, in several cases, to study quasi~linear oscillating system, the degrees of smallness of various factors must be distinguished in detail [2-7]. To affirm again this interesting remark, we shall examine a weak (of order e 2 ) self-sustained system subjected to less weak (of order e:) excitations in resonance cases. It will be seen that the system considered is enhanced. §1. SYSTEM UNDER CONSIDERATION AND ITS APPROXIMATE SOLUTION Let us consider a quasi-linear oscillating system described by the following differential equation: (1.1) f(x,wt) = { + 3p cos 2wt j, = (3x 2 fz = 2pxcoswt (1.2) where: x - an oscillatory variabb; e > 0 - small parameter; overdots denote differentiation with respect to timet; {3, 1- coefficients of the quadratic and cubic non-linearities, respectively; ho > 0 - damping viscous coefficient; h > 0, k > 0, p 2 0- constants; 1 - the natural frequency, e: 2 D. - the de tuning parameter assumed to be of order e 2 • If p = 0, we have a "pure~ self-sustained system with the positive friction force (hX- kX 3 ). If p > 0, the mentioned system is subjected to the external excitation 3p cos 2wt in subharmonic resonance of order one-half or to the parametric one 2px cos wt in principal resonance (by fundamental we mean the cases where the natural frequency is near that of the external excitation or one-half of that of the parametric one). . The damping (negative frictiori) force is introduced to facilite the analyses andj as it will be shown below, the quadratic non_-)inearity {Jx 2 is necessary in the case of external excitation. Using the asymptotic method fl], the solution of the differential equation (1.1) will be found in the form: x= acos,P+w,(a,O,¢) +s2u2(a,O,,P), a= eA1(a,O) +<2 A2(a,O), ' 2 0= 0 Bao aBo B6o Baa -+-<0 (1.12a) (1.12b) The first condition (1.12a) leads to the inequality: (1.13a) The second condition ( 1.12b) can be written as: BW -->0 BA6 6 (1.13b) §2. SELF-SUSTAINED SYSTEM UNDER EXTERNAL EXCITATION For the case f = fl (extern;tl excitation) we have: 2 2 {fJa- - c fJao s 21/;- pcos(21/;- 20) } u 1 = ·1t.~-' 2 2 6 and: 2 {w (!.kw a >-) - (3p sin ze} 4 w ae = e2a{[/:;.(315fJ2)a2]- fJp cosze} 2w 4 6w w a=- c (2.1) 2 a 2w 2 - 2 2 (2.2) 2 The trigonometrical equation { 1. 7) becomes: fJp 6.- 2cos20"' =0 (2.3) w therefore, the stability conditions (1.6) for the equilibrium regime are: (2.4a) or approximatively: (2.4b) The relationship (1. 9) is: from which, we obtain: (2.6) where e=±(s1_5fJ 2 ) (2.7) D, = (k2 + e)fJ2p2 - (k!:;.- U)2 So (2.8) 3 4 6 in the interval: (2.9) Let us analyse the stability conditions ( 1.13). It is easy to verify that is the "amplitude" of the pure self-sustained system. Indeed, if A; p = 0, from (2.8) we deduce /:;. = f: , then, from (2.6), we obtain A6 = ~ =A:. Thus, the first stability condition (1.13a) requires that the amplitude of the stable stationary oscillation must be large enough (greater than one-half of that of the pure self-sustained system). The second stability condition (1.13b) is: aw, aA 2o > 0 (2.10) or Comparing (2.10) and (2.6) we conclude that only the oscillation whose amplitude corresponds to the sign + in (2.6) may be stable. 7 To estimate the influence of the external excitation, let us compare the resonance curves A5(A) of the pure self-sustained when p = 0 with those of the combined one when p > 0. First, we suppose that A > 0 i.e. h > ho (the positive friction is greater than the negative one). 5 2 31 . . . a Iways unsta ble. For ..,< = 0 1.e. · · neutra1·1zed , 1 num regime 1s ~ = -(3- t h e system 1s Th e equ1'l'b .· . 4 6 the resOnance curve is an ellipse of center Co (A = 0 1 A 2 = A;) and its backbone curve CoCb is the abscissa line A 2 = AZ. Increasing (decreasing) €, th.e ellipse is deformed, its center C A5 = (A = ~k), , A 2 = A:) moves to the right (left) along C0 Cb, the backbone curve CC' has positive ~ ~. If p = 0 (pure self-sustained system), these ellipses degenerate to their k2 + corresponding centers {00 or 0). Figure 1 shows the resonance curves for the case ). = 0.00012 > 0; k = 0.0024; [P = 0.0036; p = 0.03 and ~ = O(a); ~ = 0.0024(b, c). Heavy (dashed) curves correspond to stable (unstable) regimes. Obviously, the influence of the external excitation is significant: the maximum amplitude of the "combined" system is greater enough in comparison with that of the pure self-sustained one. (negative) slope c i 1 I \ \ '' , I I ' ' / / '' ' / ' '' ' ', \ \ ...._ ..._ '- \ I / ' "'_, / ' -----... .--;'/f - _\t_--::.-.- -0,0020 \\ I ----------~ ..,../ ~..::::.--_ .-- _. I ..-- _. / I _.,.. A ,' Fig. 1 Suppose now that A < 0 (h < h0 ) but >. 2 < (3 2 p 2 (the resulting damping force is small enough). The equilibrium regime is unstable in the interval [A[ < (3 2 p 2 - >. 2 and stable if 2 2 1~1 > JfJ2p - A • The center Co lies below the abscissa axis Ob.., the center C moves along the abscissa line CoCb, on the left (right) if > 0 (~ < o). In figure 2, the resonance CUrVeS correspond to the case where A = -0.0006, other coefficients remain unchanged. In this case, the oscillation appears under the action of the external excitation. At last, if A < 0 and A2 > f3 2 p 2 , the equilibrium regi~e is always stable, the system is not excited. It is noted that, if j3 = 0, the intensity p of the external excitation is absent in (2.2) and the forced oscillation eu1 = -epcos(21/J- 20) is of order e. So withOut quadratic non-linearity, the interaction between external and self-sustained excitations can be neglected. y e i ! 8 l ! c' I I \ \ I / / / .I / I I '' I '' .... _____ - / '' ' I I ' 0.0020 / '' '/ / / I' / / ' '' c~ \ I I I / I / ' / / Fig. 2 §3. WEAK SELF-SUSTAINED SYSTEM UNDER PARAMETRIC EXCITATION For the case f = f2 (parametric excitation), we have: A 1 = 0, B1 = o, pa u 1 = 1- { pacos8-w2 3 cos(2,P ~e) } (3.1) and: . cz a { 3 2 2 w(-kw a 2w 4 a=-- pz . } -.\)+-sm28 w2 2 aB=-.:.O:{[(Ll.+ Zp 2 ) 2w 3w 31 a2 ] +i_cosze} 4 (3.2) w2 The staPility condition of the equilibriuill: regime are: (3.3) The relationship (1.9) is of the form: or (3.5) where (3.6) 9 in the intervt,t] (3.7) The first stability condition {1.12a) of the stationary oscillation is given by the same inequality (1.13a). The second stability condition (1.12b) gives: awz -- > 0 a(A5) 2p2 A 02 > or k>. k2+'l + ey(t. + 3) kz+ 1 z (3.8) Comparing (3.3)-(3.7) and (2.4)-(2.10) we find that for k = 0.0024; A = 0.0012; p 2 = 0.0018 and 1 = 0, 1 = ±0.0024 the resonance curves are given in the same figure 1 with a little modifica- . 2 2 tion: the center C0 deplaces on the left, its new abscissa is 6 = - ...£_ 3 CONCLUSION We have examined an_ oscillating system subjected simultaneously to weak (of order e 2 ) positive friction force and to less weak (of order e) external or parametric excitations in the resonance cases. The results obtained show that these force and excitations renforce their actions together so that the oscillation of large amplitude can be observed. The resonance curves are the ellipes whose centers correspond to the oscillatory regime of the pure self-sustained system. In the case of ' external excitation, the quadratic non-linearity (of order e) is necessary. This publication is completed with financial support from the National Basis Research Program in Naturnal Sciences. REFERENCES 1. Eororrro6oB H. H., MHTporroJihCKHtt IO. A. AcHMTITOTH'l£eCKHe MeTO,!l,l:.I B TeopHH HeJUIHett- KoJie6_aHH:tt, MocKBa, 1963. Nguyen. Van Dinh. Stability of a parametrically-excited, system under two excitations. Journal of MechaniCs, No 2, 1993. Nguyen Van Dinh. Resonance in a parametrically- excited system ~nder two excitations of different ord_ets. Proceedings of the yth National Conference on Mechanics, Vall, 1993. Nguyen Van Dinh. Quadratic and cubic non-linearities in a quasi-linear parametrically-excited system. Journal of Mechanics No 3, 1993. Nguyen Van Dinh. Quadratic and cubic non-linearities in a quasi-linear forced system. Journal of Mechanics No 1, 1994. Nguyen Van Dao. Non-linear oscillations in· systein with large static deflection of elastic elements. Journal of Mechanics No 4, 1993. Nguyen Van Dao. Some problem ·of nonlinear oscillations in system with large static deflection of elastic elements, Proceedings of the NCNST No 2, 1993 Nguyen Van Dinh. On the a·eph_ase angle. in a variational system of the equilibrium regime. Journal of Mechanics No 1, 1994. HbiX 2. 3. 4. 5. 6. 7. 8. Received November 2, 1999 Ht TV CHAN YEU DUG! TAC DQNG CUA NRUNG KICH DQNG iT YEU H

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