tailieunhanh - Identifying the resonance curve of a system subjected to linear and quadratic parametric excitations

In the present paper, we examil}e the case in which the linear and quadratic parametric excitations are present. The asymptotic method [ 1] is applied. We are interested in the method of identifying the resonance curve. The results obtained show that the "associated" equations can be used. | Vietnam Journal of Mechanics, NCST of Vietnam Vol. 21, 1999, No 3 (137- 146) IDENTIFYING THE RESONANCE CURVE OF A SYSTEM SUBJECTED TO LINEAR AND QUADRATIC PARAMETRIC EXCITATIONS (case with damping) NGUYEN VAN ,DINH - TRAN KIM CHI Institute of Me~hanics In [3], a quasi linear oscillating system simultaneously subjected to linear an;d cubic parametric excitations has been studied. In the present paper, we examil}e the case in which the linear and quadratic parametric excitations are present . The asymptotic method [1] is applied. · We are interested in the method of identifying the resonance curve. The results obtained show that the ."associated" equations can be used. § System under consideration - Original and Associated equatioi1-s Let u~ consider a quasilinear oscillating system described by the equation: · . x + w 2 x = e-{ ~x- h:i;- ryx3 + 2pxcos2wt + 2qx 2 coswt} differenti~l · (). where 2p > 0, 2q > 0 and 2w, w are intensities and frequencies of the linear ~d the quadratic parametric excitations; other notations have been explained in [2). Introducing slowly varying amplitude a and dephase fJ by means of formul 0}, it is necessary to distinguish two domains: the equivalence domain and the non-equivalence one. The equivalence domain satisfy T = 4p2 - q2 a 2 f. 0 i,e. a2 f. 4 2 a2 = P * q2 • (} Obviously, in the equivalence domain, the original and associated equations are equivalent; consequently, corresponding parts of Co and C coincide each with another. 138 The non-equivalence domain is the line . 2 2 T = 0 . a =a •. Alon~ the non-equivalence line, the equa~ions () (10 , g0 ) and(!, g) .are not equivalen~, C 0 differs from C. However, from (), It follows that all ·solutions of (/0 , g0 ) al~o satisfy (/,g). Hence Co C C, . along the non-equivalence line the elemen~s (representative point and dephase angle) of the original resonance curve C 0 must be and may be found among those of the associated resonance .