Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1211.6626

A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities

Sami Karkar·Bruno Cochelin·Christophe Vergez
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we extend the method proposed by Cochelin and Vergez [A high order purely frequency-based harmonic balance formulation for continuation of periodic solutions, Journal of Sound and Vibration, 324 (2009) 243-262] to the case of non-polynomial nonlinearities. This extension allows for the computation of branches of periodic solutions of a broader class of nonlinear dynamical systems. The principle remains to transform the original ODE system into an extended polynomial quadratic system for an easy application of the harmonic balance method (HBM). The transformation of non-polynomial terms is based on the differentiation of state variables with respect to the time variable, shifting the nonlinear non-polynomial nonlinearity to a time-independent initial condition equation, not concerned with the HBM. The continuation of the resulting algebraic system is here performed by the asymptotic numerical method (high order Taylor series representation of the solution branch) using a further differentiation of the non-polynomial algebraic equation with respect to the path parameter. A one dof vibro-impact system is used to illustrate how an exponential nonlinearity is handled, showing that the method works at very high order, 1000 in that case. Various kinds of nonlinear functions are also treated, and finally the nonlinear free pendulum is addressed, showing that very accurate periodic solutions can be computed with the proposed method.

Explore related subjects

physics.class-phphysics.comp-ph

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sami Karkar, Bruno Cochelin, Christophe Vergez. 2012-11-28. A high-order, purely frequency based harmonic balance formulation for continuation of periodic solutions: The case of non-polynomial nonlinearities. https://doi.org/10.1016/j.jsv.2012.09.033

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↗