By Mikhail Yakovlevich Kushul’ (auth.)
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Additional info for The Self-Induced Oscillations of Rotors / Avtokolebaniya Rotorov / АВТОКОЛЕБАНИЯ à ОТОà ОВ
Example text
25) for the first approximation; they will contain q- r arbitrary constants, since they must satisfy the obvious identities '" 0 Al = m 1(t) 101 + ... ) ,r . In the original variables Xl' ... 16). 26). We note one feature of the resulting solutions. 28), even in the first approximation all the frequenices of the almost-periodic vibrations can in general be different from the frequenices of the free vibrations of the generating system Al' ... , Aq. In order to represent the cases possible here more precisely, we follow the usual procedure and separate all the'frequencies into the nonresonance frequencies Al, ...
X) are the characteristic functions of theshaft; the properties and methods for the determination of these functions were considered in the previous chapter. 2) O~ Here the Os and Ps are the generalized forces due to internal friction referred to the coordinates qs and Ps; and P~ are the generalized forces due to external friction referred to the same coordinates. The kinetic energy of the rotor system is I T ~ ~- r p [(' ~ 2 o~ r+ ( dt, ~) 2Jl dx + _~ 2} m, [ (' ~l~~__ l ~ -+ l' dt 2 at i I - " du ci dt ) 2] \,=".
If the operating speed is between the first- and second-order critical velocities and sufficiently far from the second-order critical velocity. then, as has been pointed out, we need retain only the first term of the series obtained by expanding in characteristic functions the purely forced vibrations due to imbalance. 24) in which, in contrast to the previous case. the coefficients cx, 0:1 .... will be ß= 44 I 6EI ~ X'X-' dx, o I E = a = ~ pE (x) X (x) dx 0 + E1hlpX (I). § 3. 14). Introducing the new variables xl' xz, ...
