New approach to approximate analytical solutions of a harmonic oscillator with weak to moderate nonlinear damping: Part I

TL;DR

This work develops an analytic framework for the free motion of a harmonic oscillator damped by nonlinear forces, introducing an ansatz $x(t)=A f(t)\cos(\omega_0 t+\varphi)$ that separates slow amplitude decay from rapid oscillations. By linking the energy-dissipation rate to the damping power and applying time averaging, it derives a master equation for $f(t)$, $\frac{df}{dt}=-(d_2 A f^2+d_1 f+d_0/A)$, and provides explicit equations to determine the initial amplitude $A$ and phase $\varphi$ from initial conditions. The authors obtain closed-form results for damping quadratic in velocity, while Coulomb damping and the combined quadratic+Coulomb damping are treated with quartic-in-$A$ solutions and, where needed, improved variants (e.g., $newX$, $newX2$) to push accuracy toward the motion’s end. Across weak-to-moderate damping, the new solutions show excellent agreement with numerical results, offering practical analytic tools for single-degree-of-freedom systems in physics and engineering, with Part II promising to extend to linear-nonlinear damping mixtures.

Abstract

We introduce a new approach to deriving approximate analytical solutions of a harmonic oscillator damped by purely nonlinear, or combinations of linear and nonlinear damping forces. Our approach is based on choosing a suitable trial solution, i.e. an ansatz, which is the product of the time-dependent amplitude and the oscillatory (trigonometric) function that has the same frequency but different initial phase, compared to the undamped case. We derive the equation for the amplitude decay using the connection of the energy dissipation rate with the power of the total damping force and the approximation that the amplitude changes slowly over time compared to the oscillating part of the ansatz. By matching our ansatz to the initial conditions, we obtain the equations for the corresponding initial amplitude and initial phase. Here we demonstrate the validity of our approach in the case of damping quadratic in velocity, Coulomb damping, and a combination of the two, i.e. in this paper we consider purely nonlinear damping, while the dynamics with combinations of damping linear in velocity and nonlinear damping will be analyzed in a follow-up paper. In the case of damping quadratic in velocity, by comparing our approximate analytical solutions with the corresponding numerical solutions, we find that our solutions excellently describe the dynamics of the oscillator in the regime of weak to moderately strong quadratic damping. In the case of Coulomb damping, as well as in the case of a combined Coulomb and quadratic damping, our approximate analytical solutions agree well with the corresponding numerical solutions until the last few half-periods of the motion. Therefore, for these two cases, we introduce improved variants of our approximate solutions which describe the dynamics well until the very end.

Figures (21)

• Figure 1: Schematic representation of a block-spring system with a restoring force $F_{res}(t)$ and a (total) damping force $F_d(t)$. Here we show the time instant at which $x(t)>0$ and $v(t)<0$, i.e. at which $F_{res}(t)<0$ and $F_d(t)>0$. See text for details.
• Figure 2: Solid red curves show the solutions $x(t)$ with initial conditions $(x_0=0.2 m,v_0=0)$ and various values of the ratio $d_2 A_0/\omega_0$. Solid blue curves show the solutions $\tilde{x}(t)$ and black dotted curves show the numerical solutions for the same initial conditions and the corresponding values of the ratio $d_2 A_0/\omega_0$. See text for details.
• Figure 3: Solid red curves show the energies $E(t)$ with initial conditions $(x_0=0.2 m,v_0=0)$ for the same values of the ratio $d_2 A_0/\omega_0$ as used in Fig. \ref{['slika1']}(a)-(d). Solid blue curves show the energies $\tilde{E}(t)$ and black dotted curves show the numerically obtained energies with the same initial conditions and for the corresponding values of the ratio $d_2 A_0/\omega_0$. See text for details.
• Figure 4: Solid red curves show the solutions $x(t)$ with initial conditions $(x_0=0,v_0=1.10m/s)$ for two values of the ratio $d_2 A_0/\omega_0$. Dotted black curves show the numerical solutions for the same initial conditions and the corresponding values of the ratio $d_2 A_0/\omega_0$. See text for details.
• Figure 5: Solid red curves show the energies $E(t)$ with initial conditions $(x_0=0,v_0=1.1m/s)$ for the same values of the ratio $d_2 A_0/\omega_0$ as used in Fig. \ref{['slika3']}(a) and (b). Solid blue curves show the energies $\tilde{E}(t)$ and black dotted curves show the numerically obtained energies with the same initial conditions and for the corresponding values of the ratio $d_2 A_0/\omega_0$. See text for details.