In the difference between the quadratic non-linearity and the cubic one in a quasi-linear parametrically- excited system has been analyzed. In the present paper, the same question will be examined for a quasi-linear forced system and analogous resUlts as in will be obtained. | Journ:al of Mechanics, NCNST of Vietnam T. XVI, 1994, No 2 (20- 25) QUADRATIC AND CUBIC NON-LINEARITIES IN A QUASI-LINEAR FORCED SYSTEM NGUYEN VAN DINH Institute of Mechanics, NCNST of Vietnam SUMMARY. In !3L the difference between the quadratic non-linearity and the cubic one in a quasi-linear parametrically- excited system has been analyzed. In the present paper, the same question will be examined for a quasi-linear forced system and analogous resUlts as in [3] will be obtained. §1. SYSTEM UNDER CONSIDERATION AND DIFFERENT FORMS OF ITS DifFERENTIAL EQUATION Let us consider a quasi-linear forced system, described by the differential equation: () where q > 0, v > 0 are intensity and frequency of forced excitation, respectively; the signification of other symbols has been explained in [3]. Assuming that the order of smallness of h and q is e 2 , the differential equation {) can be written in the following forms, depending on the orders of smallness of {3, 1 and .6.: -if {3, /, .6. are of order e 2 , we have: X+ v 2 x = e 2 {.Bx 2 -1x 3 + . hi;+ qcos vt} () - if {3, /, .6. are of order e, we have: x + Vx = e{f3x2 - 3 "fX +} + e2 { - hx + qcosvt} () - at last, if {3 is of order e while 1 and .6. are of order e 2 , we have: x+ v2 x = e{f3x 2 } + e2 { -"jx3 + Ll,x- hx + q cos vt} () As in [3J, the case in which 1 and .6. are of different orders is rejected and, for the sake of simplicity, 1 is assumed to be positive. §2. SYSTEM WITH THE NON-LINEARITIES OF ORDER e2 First, we shall examine the case described by the differential equation (). As in [3], the asymptotic method is used and we obtain successively 20 I Al = o, Bl = 0, U! = - 2vA 2 = hva + qsinO, 3 - 2vaB2 =!::,.a- 41a 3 and o, (2. 1) + q cos e, a= -~{hv +~sinO}, 2v a iJ = _ () _!:_{ll- ~7a 2 +'!.cosO}. 2v a 4 Setting the right-hand sides of () equal to zero yields: hv + '!.sin 0 = 0, a 3 2 q t:l+ -cos 0 = 0, 4 a () -"'a and, after .
