Exact solutions for an oscillator with anti-symmetric quadratic nonlinearity

Abstract

Closed-form exact solutions for an oscillator with anti-symmetric quadratic nonlinearity are derived from the first integral of the nonlinear differential equation governing the behaviour of this oscillator. The mathematical model is an ordinary second order differential equation in which the sign of the quadratic nonlinear term changes. Two parameters characterize this oscillator: the coefficient of the linear term and the coefficient of the quadratic term. Not only the common case in which both coefficients are positive but also all possible combinations of positive and negative signs of these coefficients which provide periodic motions are considered, giving rise to four different cases. Three different periods and solutions are obtained, since the same result is valid in two of these cases. An interesting feature is that oscillatory motions whose equilibrium points are not at x = 0 are also considered. The periods are given in terms of an incomplete or complete elliptic integral of the first kind, and the exact solutions are expressed as functions including Jacobi elliptic cosine or sine functions.

Young’s modulus determination for a vibrated metal ruler using Arduino

“…If the mass is slightly depressed and released, the bar starts vibrating. Although the vibration of the cantilever is complicated and described by a nonlinear differential equation, based on the assumption of small deformation, observing the fundamental mode of its frequency, and regardless of the cantilever' s weight and deformation due to shear, a linear equation is acceptable [12,13,15].…”

Exact periodic solution of the undamped Helmholtz oscillator subject to a constant force and arbitrary initial conditions

“…Beléndez et al [32] derived an exact periodic solution for the unforced undamped Helmholtz oscillator that is characterized by antisymmetric quadratic nonlinearity and produces only symmetric vibrations, whereas Beléndez et al [31] derived an exact periodic solution for the unforced undamped Helmholtz oscillator that produces asymmetric vibrations. In both studies [31,32], the exact solutions were derived naturally from the first integral of the differential equation. However, the Helmholtz oscillators considered were unforced and had zero initial velocity.…”

“…A major shortcoming of the ansatz method is that it is not derived naturally from the differential equation, which is completely integrable [31,32]. Beléndez et al [32] derived an exact periodic solution for the unforced undamped Helmholtz oscillator that is characterized by antisymmetric quadratic nonlinearity and produces only symmetric vibrations, whereas Beléndez et al [31] derived an exact periodic solution for the unforced undamped Helmholtz oscillator that produces asymmetric vibrations. In both studies [31,32], the exact solutions were derived naturally from the first integral of the differential equation.…”

Closed-form solutions for the quadratic mixed-parity nonlinear oscillator

“…In this paper, we obtain the closed-form exact expressions for the period and the solution of a quadratic mixedparity nonlinear oscillator. A procedure similar to the one used in a previous paper when obtaining the exact solutions for an oscillator with anti-symmetric quadratic nonlinearity is considered [16]. Unlike the procedure considered by other authors [11,12], we do not assume an expression for the solution, but instead solve the nonlinear differential equation exactly.…”