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

Karlo Lelas; Robert Pezer

Home/Archive/arXiv

arXiv · 2512.23590

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

Karlo Lelas·Robert Pezer

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

physics.class-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Karlo Lelas, Robert Pezer. 2025-12-29. New approach to approximate analytical solutions of a harmonic oscillator with weak to moderate nonlinear damping: Part I. https://arxiv.org/abs/2512.23590

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unifying pendulum-like dynamical regimes via complex time

We present an alternative derivation of exact solutions for pendulum-like systems across all dynamical regimes using a standard technique in undergraduate mathematical physics course: the Cauchy residue theorem, entirely independent from the traditional Jacobi elliptic framework. The solutions are exact in both the time and frequency domains, providing continuous trajectories and precise frequency decompositions. Pendulum-like dynamics is a foundational model across many areas of physics, underlying systems ranging from classical nonlinear oscillators to superconducting qubits and cold-atom tunneling platforms. While its time-domain solutions are well-known in terms of Jacobi elliptic functions, its obscured frequency-domain solutions have led a large body of theoretical and experimental work to develop and apply approximate methods when studying its spectral behavior. We discover that all regimes arise from a single spectral kernel, with parity selection distinguishing the periodic motions and the separatrix representing their discrete-to-continuum limit. Regime changes thus correspond to symmetry-driven reorganizations in frequency space rather than changes in the underlying spectral structure, with the stopping trajectory representing the continuum limit. By completely bypassing the traditional Jacobi elliptic functions, and relying on the symmetrical structure of the complex time plane, the derivation shows that the symmetrical spectral structure is not merely algebraic artifacts of elliptic integrals, but fundamentally connected to the dynamical symmetries of the system. The derivation presented here not only serves as a powerful pedagogical exercise for this long-standing physics problem, but also reveals a highly symmetrical spectral organization in nonlinear dynamics.

physics.class-ph↗

arXiv2026

Spectral taxonomy for quartic systems: fundamental clock, parity, and continuum

A symmetric quartic potential is a physics model with expansive applications, ranging from broadband energy harvesters, quantum tunneling in molecules and the early universe, to torque-free spacecraft rotation. Its rich dynamics have been traditionally classified into regimes and expressed as disjointed time-domain solutions. Here we build a taxonomy for this broad class of motions and discover that their regimes exhibit a symmetrical spectral structure: they share a fundamental clock, obey parity selection, and dissolve into the separatrix through a discrete-to-continuum transition. Applied to the famous Dzhanibekov effect where a rotating body periodically undergoes rapid 180-degree flips in its attitude, the taxonomy reveals its spectral anatomy. The three principal-axis rotations share a common clock while occupying distinct parity channels, with stable-axis branches exchanging DC bias across the separatrix. We discuss a case study where the three spectral pillars: clock, parity, and continuum, survive the Wick rotation from real-time into imaginary-time kinematics.

physics.class-ph↗

arXiv2026

Bypass transition under wall cooling over a flat plate with an elliptical leading edge

This study uses wall-resolved large-eddy simulations to investigate bypass transition in a flat-plate boundary layer subjected to free-stream turbulence. Two thermal configurations are considered: an adiabatic wall and a uniformly cooled wall. The plate geometry includes an elliptic leading edge and is designed for direct experimental reproduction, providing a framework for future numerical-experimental comparisons. The resolved leading edge allows direct examination of the early stages of bypass transition. Both simulations recover the classical sequence of receptivity, vortex tilting, lift-up, streak amplification, secondary instability and turbulent-spot growth. The wall-normal transport term is identified as a precursor of streak formation, while shear sheltering is quantified and linked to the frequency-dependent penetration of free-stream disturbances into the boundary layer. Wall cooling does not modify the bypass-transition mechanisms nor the onset location of transition. Instead, it generates thermal streaks alongside the velocity streaks and shifts the latter slightly closer to the wall, consistent with optimal-perturbation predictions. The velocity streaks retain similar amplitudes, growth rates and spanwise spacings in both thermal conditions. A conditional analysis reveals a pronounced asymmetry between high- and low-velocity streaks. Although breakdown is systematically observed within low-velocity streaks, the strongest pre-transitional evolution occurs within the high-velocity streak population, which undergoes significant amplification and a progressive displacement towards the wall before the onset of intermittency. These observations suggest that high-velocity streaks may actively contribute to the reorganisation of the streak field leading to secondary instability and breakdown.

physics.class-ph↗