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

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

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

Energy Harvesting From Bistable Systems Under Random Excitation

The transformation of otherwise unused vibrational energy into electric energy through the use of piezoelectric energy harvesting devices has been the subject of numerous investigations. The mechanical part of such a device is often constructed as a cantilever beam with applied piezo patches. If the harvester is designed as a linear resonator the power output relies strongly on the matching of the natural frequency of the beam and the frequency of the harvested vibration which restricts the applicability since most vibrations which are found in built environments are broad-banded or stochastic in nature. A possible approach to overcome this restriction is the use of permanent magnets to impose a nonlinear restoring force on the beam that leads to a broader operating range due to large amplitude motions over a large range of excitation frequencies. In this paper such a system is considered introducing a refined modeling with a modal expansion that incorporates two modal functions and a refined modeling of the magnet beam interaction. The corresponding probability density function in case of random excitation is calculated by the solution of the corresponding Fokker-Planck equation and compared with results from Monte Carlo simulations. Finally some measurements of ambient excitations are discussed.DFG, 253161314, Untersuchung des nichtlinearen dynamischen Verhaltens von stochastisch erregten Energy Harvesting Systemen mittels Lösung der Fokker-Planck-Gleichun

On the discretization of a bistable cantilever beam with application to energy harvesting

A typical setup for energy harvesting is that of a cantilever beam with piezoceramics excited by ambient base vibrations. In order to get higher energy output for a wide range of excitation frequencies, often a nonlinearity is introduced by intention in that way, that two magnets are fixed close to the free tip of the beam. Depending on strength and position of the magnets, this can either result in a mono-, bi- or tristable system. In our study, we focus on a bistable system. Such systems have been investigated thoroughly in literature while in almost all cases the beam has been discretized by a single shape function, in general the first eigenshape of the linear beam with undeflected stable equilibrium position. There can be some doubts about the suitability of a discretization by a single shape function mainly due to two reasons. First: In case of stochastic broadband excitations a discretization, taking into consideration just the first vibration shape seems not to be reasonable. Second: as the undeflected position of the considered system is unstable and the system significantly nonlinear, the question arises, if using just one eigenshape of the linear beam is a suitable approximation of the operation shapes during excited oscillations even in the case of harmonic excitation. Are there other, e.g. amplitude dependent, possibilities and/or should multiple ansatz functions be considered instead? In this paper, we focus mainly on the second point. Therefore, a bistable cantilever beam with harmonic base excitation is considered and experimental investigations of operation shapes are performed using a high-speed camera. The observed operation shapes are expanded in a similar way as it is done in a theoretical analysis by a corresponding mixed Ritz ansatz. The results show the existence of distinct superharmonics (as one can expect for a nonlinear system) but additionally the necessity to use more than one shape function in the discretization, covering also the amplitude dependence of the observed operation shapes

Zipping Segment Trees

Stabbing queries in sets of intervals are usually answered using segment trees. A dynamic variant of segment trees has been presented by van Kreveld and Overmars, which uses red-black trees to do rebalancing operations. This paper presents zipping segment trees - dynamic segment trees based on zip trees, which were recently introduced by Tarjan et al. To facilitate zipping segment trees, we show how to uphold certain segment tree properties during the operations of a zip tree. We present an in-depth experimental evaluation and comparison of dynamic segment trees based on red-black trees, weight-balanced trees and several variants of the novel zipping segment trees. Our results indicate that zipping segment trees perform better than rotation-based alternatives

Engineering Top-Down Weight-Balanced Trees

Weight-balanced trees are a popular form of self-balancing binary search trees. Their popularity is due to desirable guarantees, for example regarding the required work to balance annotated trees. While usual weight-balanced trees perform their balancing operations in a bottom-up fashion after a modification to the tree is completed, there exists a top-down variant which performs these balancing operations during descend. This variant has so far received only little attention. We provide an in-depth analysis and engineering of these top-down weight-balanced trees, demonstrating their superior performance. We also gaining insights into how the balancing parameters necessary for a weight-balanced tree should be chosen - with the surprising observation that it is often beneficial to choose parameters which are not feasible in the sense of the correctness proofs for the rebalancing algorithm.Comment: Accepted for publication at ALENEX 202

On the detection of artifacts in Harmonic Balance solutions of nonlinear oscillators

Harmonic Balance is a very popular semi-analytic method in nonlinear dynamics. It is easy to apply and is known to produce good results for numerous examples. Adding an error criterion taking into account the neglected terms allows an evaluation of the results. Looking on the therefore determined error for increasing ansatz orders, it can be evaluated whether a solution really exists or is an artifact. For the low-error solutions additionally a stability analysis is performed which allows the classification of the solutions in three types, namely in large error solutions, low error stable solutions and low error unstable solution. Examples considered in this paper are the classical Duffing oscillator and an extended Duffing oscillator with nonlinear damping and excitation. Compared to numerical integration, the proposed procedure offers a faster calculation of existing multiple solutions and their character