Higher-Order Pseudoaveraging via Harmonic Balance for Strongly Nonlinear Oscillations

Abstract

Some strongly nonlinear conservative oscillators, on slight perturbation, can be studied via averaging of elliptic functions. These and many other oscillations allow harmonic balance-based averaging (HBBA), recently developed as an approximate first-order calculation. Here, we extend HBBA to higher orders. Unlike the usual higher-order averaging for weakly nonlinear oscillations, here both the dynamic variable and time are averaged with respect to an auxiliary variable. Since the harmonic balance approximations introduce technically O(1) errors at each order, the higher-order results are not strictly asymptotic. Nevertheless, as we show with examples, for reasonable values of the small expansion parameter, excellent approximations are obtained.

Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

“…The direct time solution of related equation systems is often too costly. Several approaches exist to study the steady-state response of nonlinear systems: methods of multiple scales aiming the research of consecutive approximations to the exact solutions [7], methods of harmonic balance [8][9][10][11][12][13] providing a projection of the sought solution on the harmonics of a basis sine function, and shooting-type methods [14] seeking the initial condition located on a steady state trajectory of the full system. Once the periodic solution is found, its stability is quite straightforward to obtain through the classical Floquet monodromy matrix [15].…”

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

“…Comparatively, little attention is granted to the determination of approximate solutions to autonomous damped equations. The few recent works (to our knowledge) in which analytical approximations to damped nonlinear oscillator equations are explicitly considered are by Liao [17], and Chatterjee and collaborators [18,19,20]. As a contribution to this effort, we have shown in a previous paper [21] how to merge the idea of expansion of constant [3] and the KBM method to derive better accurate slow flows for damped single degree of freedom oscillators.…”

Construction of approximate analytical solutions to strongly nonlinear damped oscillators

“…(ρ) = 15925248 − 125116416ρ 2 + 56615424ρ 4 − 1842624ρ 6 − 586120ρ 8 +59789ρ 10 − 2150ρ 12 + 28ρ14 (B.6) 2 (ρ) = −ρ 3 6186074112 − 20226834432ρ 2 + 6366156288ρ 4 − 527249088ρ 6 − 9586456ρ 8 + 3649821ρ 10 − 215839ρ 12 + 5498ρ 14 − 56ρ16 (B.7)…”