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

Incidence and Growth of Patent Thickets - The Impact of Technological Opportunities and Complexity

We investigate incidence and evolution of patent thickets. Our empirical analysis is based on a theoretical model of patenting in complex and discrete technologies. The model captures how competition for patent portfolios and complementarity of patents affect patenting incentives. We show that lower technological opportunities increase patenting incentives in complex technologies while they decrease incentives in discrete technologies. Also, more competitors increase patenting incentives in complex technologies and reduce them in discrete technologies. To test these predictions a new measure of the density of patent thickets is introduced. European patent citations are used to construct measures of fragmentation and technological opportunity. Our empirical analysis is based on a panel capturing patenting behavior of 2074 firms in 30 technology areas over 15 years. GMM estimation results confirm the predictions of our theoretical model. The results show that patent thickets exist in 9 out of 30 technology areas. We find that decreased technological opportunities are a surprisingly strong driver of patent thicket growth

Fast Variations of Gamma-Ray Emission in Blazars

The largest group of sources identified by EGRET are Blazars. This sub-class of AGN is well known to vary in flux in all energy bands on time-scales ranging from a few minutes (in the optical and X-ray bands) up to decades (radio and optical regimes). In addition to variations of the gamma-ray flux between different viewing periods, the brightest of these sources showed a few remarkable gamma-ray flares on time-scales of about one day, confirming the extension of the ``Intraday-Variability (IDV)'' phenomenon into the GeV range. We present first results of a systematic approach to study fast variability with EGRET data. This statistical approach confirms the existence of IDV even during epochs when no strong flares are detected. This provides additional constraints on the site of the gamma-ray emission and allows cross-correlation analyses with light curves obtained at other frequencies even during periods of low flux. We also find that some stronger sources have fluxes systematically above threshold even during quiescent states. Despite the low count rates this allows explicit comparisons of flare amplitudes with other energy bands.Comment: 5 pages including figures, LaTex, uses aipproc.sty, to appear in the proceedings of the 4th Compton Symposium at Williamsburg, V

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

How to measure patent thickets – a novel approach

The existing literature identifies patent thickets indirectly. In this paper we propose a novel measure based on patent citations which allows us to measure the density of patent thickets directly. We discuss the algorithm which generates the measure and present descriptive results validating it. Moreover, we identify technology areas which are particularly impacted by patent thickets.patenting; patent thickets; patent portfolio races; complexity

The strategic use of patents and its implications for enterprise and competition policies

This report was commissioned as a study into the strategic use of patents. In the course of its case investigations and legislative reviews the European Commission became aware of changes in the use of intellectual property, in particular the use of patents. It was noted that firms’ uses of intellectual property are becoming increasingly strategic. This raised concerns about the implications of firms’ patenting behaviour for enterprise and competition policy. The following report contains a comprehensive review of patenting behaviour, the extent to which patenting is becoming more strategic and the implications this has for competition and enterprise policies

On the influence of external stochastic excitation on linear oscillators with subcritical self-excitation applied to brake squeal

A characteristic of linear systems with self-excitation is the occurrence of non-normal modes. Because of this non-normality, there may be a significant growth in the vibration amplitude at the beginning of the transient process even in the case of solely negative real parts of the eigenvalues, i.e. asymptotic stability of the trivial solution. If such a system is excited additionally with white noise, this process is continually restarted and a stationary vibration with dominating frequencies and comparably large amplitudes can be observed. Similar observations can be made during brake squeal, a high-frequency noise resulting from self-excitation due to the frictional disk-pad contact. Although commonly brake squeal is considered as a stable limit cycle with the necessity of corresponding nonlinearities, comparable noise phenomena can in the described model even observed in a pure linear case when the trivial solution is asymptotically stable