On artifact solutions of semi-analytic methods in nonlinear dynamics

Nonlinear dynamics is a topic of permanent interest in mechanics since decades. The authors have recently published some results on a very classical topic, the dynamics of a softening Duffing oscillator under harmonic excitation focusing especially on low-frequency excitation (von Wagner in Arch Appl Mech 86(8):1383–1390, 2016). In this paper, it was shown that classical tools like harmonic balance and perturbation analysis may produce artificial solutions when applied without extra carefulness with respect to parameter ranges in the case of perturbation analysis or prior knowledge about the type of solution in case of harmonic balance. In the present paper these results are shortly summarized as they give the starting point for the additional investigations described herein. First, the method of slowly changing phase and amplitude is reviewed with respect to its capability of determining asymptotic stability of stationary solutions. It is shown that this method can also produce artifact results when applied without extra carefulness. As next example an extended Duffing oscillator is investigated, which shows, if harmonic balance is applied, “islands” of solutions. Using the error criterion in harmonic balance as described in von Wagner (2016) again artifact solutions can be identified

Non-Uniform Time Sampling for Multiple-Frequency Harmonic Balance Computations

A time-domain harmonic balance method for the analysis of almost-periodic (multi-harmonics) flows is presented. This method relies on Fourier analysis to derive an efficient alternative to classical time marching schemes for such flows. It has recently received significant attention, especially in the turbomachinery field where the flow spectrum is essentially a combination of the blade passing frequencies. Up to now, harmonic balance methods have used a uniform time sampling of the period of interest, but in the case of several frequencies, non-necessarily multiple of each other, harmonic balance methods can face stability issues due to a bad condition number of the Fourier operator. Two algorithms are derived to find a non-uniform time sampling in order to minimize this condition number. Their behavior is studied on a wide range of frequencies, and a model problem of a 1D flow with pulsating outlet pressure, which enables to prove their efficiency. Finally, the flow in a multi-stage axial compressor is analyzed with different frequency sets. It demonstrates the stability and robustness of the present non-uniform harmonic balance method regardless of the frequency set

On some aspects of the dynamic behavior of the softening Duffing oscillator under harmonic excitation

The Duffing oscillator is probably the most popular example of a nonlinear oscillator in dynamics. Considering the case of softening Duffing oscillator with weak damping and harmonic excitation and performing standard methods like harmonic balance or perturbation analysis, zero mean solutions with large amplitudes are found for small excitation frequencies. These solutions produce a ”nose-like” curve in the amplitude–frequency diagram and merge with the inclining resonance curve for decreasing (but non-vanishing) damping. These results are presented without any additional discussion in several textbooks. The present paper discusses the accurateness of these solutions by introducing an error estimation in the harmonic balance method showing large errors. Performing a modified perturbation analysis leads to solutions with non-vanishing mean value, showing very small errors in the harmonic balance error analysis

On Artifacts in Nonlinear Dynamics

Nonlinear oscillations are of permanent interest in the field of dynamics of mechanical and mechatronical systems. There exist several well-known semi-analytical methods like Harmonic Balance, perturbation analysis or multiple scales for such problems. We reconsider in our presentation the method of Harmonic Balance but add some additional steps in order to avoid artifacts and get information about the stability. The classical method of Harmonic Balance is therefore added by an error criterion, which considers the neglected terms. Looking on this error for increasing ansatz orders, it can be decided whether a solution exists or is an artifact of the method. For the low error solutions, a stability analysis is performed. As example, an extended Duffing oscillator with additional nonlinear damping and excitation is considered showing regions of separated island solutions. Also a nonlinear piezo-beam energy harvesting system is investigated. The described method enables to calculate solutions in a rapid manner with comparable low effort, to get an overview over regular responses of nonlinear systems.DFG, 253161314, Untersuchung des nichtlinearen dynamischen Verhaltens von stochastisch erregten Energy Harvesting Systemen mittels Lösung der Fokker-Planck-Gleichun

A frequency-domain approach to the analysis of stability and bifurcations in nonlinear systems described by differential-algebraic equations

A general numerical technique is proposed for the assessment of the stability of periodic solutions and the determination of bifurcations for limit cycles in autonomous nonlinear systems represented by ordinary differential equations in the differential-algebraic form. The method is based on the harmonic balance technique, and exploits the same Jacobian matrix of the nonlinear system used in the Newton iterative numerical solution of the harmonic balance equations for the determination of the periodic steady-state. To demonstrate the approach, it is applied to the determination of the bifurcation curves in the parameters' space of Chua's circuit with cubic nonlinearity, and to study the dynamics of the limit cycle of a Colpitts oscillato

A theoretical basis for the Harmonic Balance Method

The Harmonic Balance method provides a heuristic approach for finding truncated Fourier series as an approximation to the periodic solutions of ordinary differential equations. Another natural way for obtaining these type of approximations consists in applying numerical methods. In this paper we recover the pioneering results of Stokes and Urabe that provide a theoretical basis for proving that near these truncated series, whatever is the way they have been obtained, there are actual periodic solutions of the equation. We will restrict our attention to one-dimensional non-autonomous ordinary differential equations and we apply the results obtained to a couple of concrete examples coming from planar autonomous systems

Assessment of thermal instabilities and oscillations in multifinger heterojunction bipolar transistors through a harmonic-balance-based CAD-oriented dynamic stability analysis technique

We present a novel analysis of thermal instabilities and oscillations in multifinger heterojunction bipolar transistors (HBTs), based on a harmonic-balance computer-aided-design (CAD)-oriented approach to the dynamic stability assessment. The stability analysis is carried out in time-periodic dynamic conditions by calculating the Floquet multipliers of the limit cycle representing the HBT working point. Such a computation is performed directly in the frequency domain, on the basis of the Jacobian of the harmonic-balance problem yielding the limit cycle. The corresponding stability assessment is rigorous, and the efficient calculation method makes it readily implementable in CAD tools, thus allowing for circuit and device optimization. Results on three- and four-finger layouts are presented, including closed-form oscillation criteria for two-finger device