Parametric vibration of the prismatic shaft with hereditary and nonlinear geometry

Parametric vibration of the prismatic shaft with regard of physical and geometrical nonlinearity has been inves~igated in some publications (see for example). However, that vibration in the case of hereditary has not, to author's knowledge, been examined hitherto. In this paper it will be studied by means of the asymptotic method for high order systems. | T!'P chi CCacos6"{t- 6Cs"{sin6"{t] }y+ {P2Q 0 + (33 P 0 sin"{t+ + fJ• [co+ C2cos21t + b2 sin2"1t + C 4 cos4"{t + 9 ~4 sin4"{t + C6 cos6"ft]} ~~, () where fJ2, This equation will be solved by the asymptotic method [2]. Let's consider the case, when {31 , f3s, f34 are small quantities of first order so that fJ1 = e/31, fJ2 = e/32, f3s = e/33 , f34 = e/34• We shall deal with the oscillation in the resonance case, when there exists the following relation ·between the frequencies () p, q are integers, series o is detuning. The partial solution of the equation () is found in the form of y =a cos~+ eU1 (a,tj>,9) + e2 U2 (a,tj>, 9) + e 3 , here ••• , () ~ = ( ~'Yt + .p), 9 = 7t, a, .p are the functions satisfying the following differential equations d; = eA,(a,tj>) +e A (a,tj>) +e 2 ~~ = 2 3 , .•• , (w- ~'Y) + eB,(a, .P) + e B 2 () 2 (a, .P) + e3 •••• It is easy to prove that the resonance occurs when p 1 -=-·1·2·3 q 2 I I I ' () First of all, let's investigate the oscillation in the case p 1 - =- () 2 q In the first approximation we have y =a cos~= a cos 'Y ~: =a a7 (i-rt + t/>), () [h a-y- P cos2tj>], 1 1 ~~ = a [ (w2 - ·~) - S1 a 2 10 - w2 - 2h 1 R 1 + P 1 sin 2¢]. () where 6{i3Po p _ 4 1- R, = ' . 2 Stationary solution a0 , .Po of the system of the equations () is determined from relations ao(h,a1- P, cos 2t/>o) = 0, ao[ (w 2 - ~)- S,a5- 2h,w2 - R, + P, sin2t/>o] () = 0. Eliminating the phase in (), we get the equation of resonance curve for non-trivial stationary oscillation () From here we obtain A5 = where A2 = o S,a5 w2 (1- 2h 1 ) -~· - - D ± yC 2 4 ~· ' _ 12 - c• = w2 ' P{ - B2 a 2~ 2 , D _ R, w4 ' - () B 2 = hr . w2 ' w2 To study the stability of the stationary oscillation, we set in (} a= a0 +Sa, t/> = t/>o + St/>, where Sa, St/> are small pertubations. Substituting these expressions into equations (} .

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