On the period of large amplitude free vibration of conservative autonomous oscillators with static and inertia type cubic non-linearities

The concern of this paper is the large amplitude free vibration of strongly non-linear oscillators u + mu + epsilon(1)u(2)u + epsilon(2)uu(2) + epsilon(2)u(3) = 0, where m = 1, 0, or -1, epsilon(1) and epsilon(2) are positive parameters which may be arbitrarily large, and u(t) may be of order unity. Approximate analytical solutions for the period of free motion are obtained, for comparison purposes, by using the single-term harmonic balance (SHB) method, the two-terms harmonic balance (2THB) method, and the two-term time transformation (2TT) method described in reference [1]. Parametric studies on the effects of m, epsilon(1) and epsilon(2) on the period-amplitude behaviour are presented as obtained by using the above three analytical methods. The results of these three methods are compared with each other and with those obtained numerically. For convenience, the results are displayed graphically. It is shown that for the case m = 1, a qualitative failure of the SHB method ocurs when epsilon(1) and epsilon(2) are in the range 1 . 5 < epsilon(1)/epsilon(2) < 1 . 8. It is also shown that for m = 0, or - 1, the period-amplitude behavior is of hardening type regardless of the value of epsilon(1) relative to epsilon(2). In all cases m = 1, 0, or - 1, the period becomes nearly constant independent of motion amplitude when this amplitude is relatively large. It is also shown that the period becomes a constant independent of motion amplitude and is equal to the linear period when epsilon(1) similar to 1 . 6 epsilon(2). (C) 1997 Academic Press Limited.