A cube-law air track oscillator

Abstract

A mechanical oscillator has been constructed for which the restoring force is closely proportional to the cube of the displacement. The simple air track system allows the properties of this nonlinear oscillator to be experimentally confirmed. In particular, the period-amplitude product TA is observed to be constant and the amplitude of the third harmonic component in the oscillation matches that predicted. The apparatus allows students to expand their studies of linear oscillators into nonlinear phenomena, a topic of considerable interest.

Anharmonic oscillations of a conical buoy

“…For small displacements y ≪ 1, P 3 (y) ≈ 3(1 − α) 2/3 y, the equation is approximately linear and so the movement is quasi harmonic with period T SO = 2π α/6 . 3 1/(1 − α). In other physical situations the oscillator exhibits anharmonic displacements and the period will depend on the amplitude (or total energy).…”

“…Anharmonic oscillators are useful examples of nonlinear phenomena. Many vibrating systems found in the real world are nonlinear whether they be macroscopic mechanical oscillators [1,2,3,4] or microscopic atomic oscillators [5]. The pendulum at high angles is a classic example of an anharmonic oscillator [6,7].…”

“…of amplitude [a, b] ≡ [−y C , y 1 ] for α ≤ 3/4 and [y 1 , −y V ] for α > 3/4 given by Eqs. (2) and(3). It means that the procedure to initiate the movement should be as follows: for α < 3/4 the base of the cone should be raised near the free surface of the liquid; for α > 3/4 the vertex of the cone should be lowered until near complete immersion; for α = 3/4 the choice of the base or vertex to initiate the movement is irrelevant since the amplitude of the oscillation, that has a maximum (equal to h) corresponds to a displacement between the base and the vertex i.e.…”

Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators

“…It is clear that none of the standard methods for constructing a perturbation solution to this equation apply since these procedures assume that when β = 0 the resulting differential equation is that for the harmonic oscillator [24]. One example of this nonlinear oscillator is a cube-law air track oscillator [38]. The motion of a ball-bearing oscillating in a glass tube that is bent into a curve [23] is another example, as well as the motion of a mass attached to two identical stretched elastic wires for small amplitudes when the length of each wire without tension is the same as half the distance between the ends of the wires [39].…”

Physical pendulum model: Fractional differential equation and memory effects

“…Other experiments that provide insight into pendular motion include the air-track [53,54], the drinking straw [55,56], and the free-fall [57].…”