TWO PATHS TO ISOCHRONICITY IN A CLASS OF ONE DEGREE OF FREEDOM OSCILLATORS

摘要

We propose a class of problems in nonlinear vibrations related to avoiding undesirable hysteresis and jump phenomena by designing an oscillator for which the backbone curve is a straight vertical line. In particular we consider the class of conservative oscillators of the form: (xe) + xx~2+f(x) =0 and we choose f(x) so that the frequency of oscillation is independent of amplitude. We do this in two ways: 1) by expanding f(x) in a power series and using a perturbation method to compute the coefficients, and 2) by starting with a simple harmonic oscillator (which has a straight line backbone curve), and choosing a transformation which puts the resulting equation in the above form. We show that both methods result in a closed form expression for f(x) which involves the imaginary error function erfi(z).