Large-amplitude Cycles Research Articles

Bifurcation structures and bursting dynamics in a two degrees of freedom quasi-zero stiffness system with elastic constrain

In this manuscript, bifurcation structures and bursting patterns involving grazing phenomena are discussed based on a nonsmooth quasi-zero stiffness vibration isolation system. Within the selected parameter zone, analysis about stability and bifurcations of equilibria belonging to two sub-systems in phase space show that, equilibria of both sub-systems lose their stability through supercritical Hopf bifurcations. Numerical continuation performed on two sub-systems indicates that, the sub-system S− presents bi-stability consisting of small amplitude cycle branch circling around the S-shape equilibrium curve and large amplitude cycle branch, while the sub-system S+ presents mono-stability. Meanwhile, both cycle branches are terminated by fold bifurcations of cycle. Furthermore, as the quasi-static external force changes, periodic motions governed by the small amplitude branch of S− and cycle branch of S+ may touch the nonsmooth boundary and enter the other side of the boundary, which indicates the grazing bifurcations. By computing the bifurcation diagrams of the overall system, it is shown that grazing bifurcations result in complex distribution of cycle attractors as well as the occurrence of chaos. Based on these, multiple patterns of periodic and aperiodic bursting and the corresponding generation mechanisms are obtained. Besides, it is found that increasing the damping in coupling parts can improve the isolation efficiency.

Methods for editing random load spectra based on multi-parameter index preservation for accelerated durability testing

Herein, a multi-parameter index preservation-based (MPIP) load spectrum editing method is proposed for improving the efficiency of the accelerated durability testing of mechanical components. The editing process was conducted in the frequency domain by adopting a wavelet transform to decompose the load spectrum into different wavelet levels. The high-frequency wavelet levels were squared and summed at each time interval to obtain the wavelet energy distribution. The damage segments with large amplitude cycles were identified and extracted from the load spectrum based on the wavelet energy distribution. A shorter spectrum was obtained without significant changes in the indexes (the damage index, statistical index, and power spectrum density index). The effectiveness of the MPIP-based load spectrum editing method was demonstrated through the editing process of a coil spring load spectrum, along with a fatigue and damage assessment. The results show that the time duration reduction ratio of the load spectrum after editing was considerably lower than that of the load spectrum before editing. Further, the statistical indices (i.e., the mean, root mean square, and kurtosis) and power spectrum density Pearson correlation coefficient of the MPIP-based edited load spectrum were comparable to those of the original load spectrum. The results also demonstrate that the MPIP-based edited load spectrum indices and time-domain-based edited load spectrum have certain differences. However, the proposed method is more effective for accelerated durability tests. The results regarding the coil spring fatigue damage and lifetime assessment further validated this conclusion.

Accessing a broader range of energy states in metallic glasses by variable-amplitude oscillatory shear

The influence of variable-amplitude loading on the potential energy and mechanical properties of amorphous materials is investigated using molecular dynamics simulations. We study a binary mixture that is either rapidly or slowly cooled across the glass transition temperature and then subjected to a sequence of shear cycles with strain amplitudes above and below the yielding strain. It was found that well annealed glasses can be rejuvenated by small-amplitude loading if the strain amplitude is occasionally increased above the critical value. By contrast, poorly annealed glasses are relocated to progressively lower energy states when subyield cycles are alternated with large-amplitude cycles that facilitate exploration of the potential energy landscape. The analysis of nonaffine displacements revealed that in both cases, the typical size of plastically rearranged domains varies depending on the strain amplitude and number of cycles, but remains smaller than the system size, thus preserving structural integrity of amorphous samples.

A bioeconomic model of a fishery with saturated catch and variable price: Stabilizing effect of marine reserves on fishery dynamics

In this paper, we study a mathematical bio-economic model of a fishery with varying price. The three dimensional model describes the time evolution of the resource, the fishing effort and the price. The model is original because it considers a nonlinear harvesting function assumed to depend upon stock size and fishing effort with a saturation effect with respect to the resource as well as a price equation depending on demand and supply which is in addition proportional to price. Assuming that the price varies at a fast time scale, we are able to use ”aggregation of variables methods” in order to reduce the model in a two dimensional model at a slow time scale. This aggregated (reduced) model is analyzed. Several numerical simulations of the model are performed to substantiate our analytical findings. The existence of nonlinear harvesting makes the dynamics of the model more complicated, including multiple equilibria, bi-stability and limit cycle. Such large amplitude cycle variations are not desirable because they generate periods of overfishing at periods of very low activity. We then study the effects of marine reserves on the dynamics of the fishery, showing that for an adequate number of small reserves, limit cycle oscillations are switched off.

Multiple-S-Shaped Critical Manifold and Jump Phenomena in Low Frequency Forced Vibration with Amplitude Modulation

Motivated by the forced harmonic vibration of complex mechanical systems, we analyze the dynamics involving different waves in a double-well potential oscillator coupling amplitude modulation control of low frequency. The combination of amplitude modulation factor significantly enriches the dynamical behaviors on the formation of multiple-S-shaped manifold and multiple jumping phenomena that alternate between epochs of slow and fast motion. We can conduct bifurcation analysis to identify two harmonic vibrations. One is that the singular orbit makes multiple jumps to a fast trajectory segment from one attracting equilibrium to another as the expression of slow variable by using the DeMoivre formula. With the increase of tuning frequency, the system exhibits relaxation-type oscillations whose small amplitude oscillations are produced by nonlinear local cycles together with a distinct large amplitude cycle oscillation accounting for the Melnikov threshold values. The tuning frequency may not only affect the asymptotic expressions for the solution curves near fold singularities but also allow for the large amplitude orbit vibrations near fold-cycle singularities. Numerical analysis for computing critical manifolds and their intersections is used to detect the dynamical features in this paper.

Melnikov-threshold-triggered mixed-mode oscillations in a family of amplitude-modulated forced oscillator

This paper deals with the transitions through Melnikov thresholds and the corresponding fast–slow dynamics in a family of bi-parametric mechanical oscillators subjected to an amplitude modulation force both analytically and numerically. Applying Melnikov analytical method, the threshold condition for the occurrence of mixed-mode oscillations is obtained. First, the Melnikov threshold curves are drawn in different parameter space in the unperturbed system. Next, as the control variable crosses above the threshold, two states about mixed-mode oscillations can be identified: the rest state corresponding to local limit cycle from nontrivial equilibrium point and the spiking state corresponding to large amplitude cycle produced by saddle–node bifurcation of limit cycles. Such mixed-mode oscillations are created since the amplitude modulation force slowly and periodically switches between the rest states and spiking states. Furthermore, we elucidate the effect of modulation frequency on phase portraits and dynamical behaviors, and some numerical simulations are also included to illustrate the validity of our study.

Fatigue damage analysis of steel components subjected to earthquake loadings

PurposeExamination of steel moment resisting frames after the 1994 Northridge earthquake showed fatigue cracks presented in the beam–column connections of the frames. These observations indicate that fatigue failure may occur in the steel components of building structures in an earthquake event. To apply the fatigue design approach using the Palmgren–Miner’s rule for steel components of the moment resisting frames requires the knowledge regarding the damage index value at fatigue failure. The purpose of this paper is to perform fatigue tests to give the first damage values of steel components subjected to real earthquake-induced loadings.Design/methodology/approachThe added-damping-and-stiffness steel plates which are used in building structures for earthquake mitigation were fabricated and tested by constant amplitude, SAC block and earthquake-induced loadings to failure. The earthquake loadings were obtained from the dynamic analysis of a steel frame with the mentioned plates. The load cycles of the SAC block and the calculated earthquake loadings were counted using the rainflow-counting method, and the damage index value of each specimen were calculated using the Palmgren–Miner’s rule.FindingsReverse stiffness obtained from cyclic load-displacement loops is a robust and consistent parameter that can be used for determining fatigue failure of tested components. The Palmgren–Miner’s damage values at failure, caused by earthquake loadings, are smaller than 1, and in addition, are also smaller than those obtained from the tests of the SAC block loading. The large-amplitude cycles in the earthquake loading produce large damage on the specimens, and intermediate range cycles also produce damage that should not be neglected in the fatigue analysis.Originality/valueToday’s building design code allows large plastic deformation to occur in steel frames during an earthquake. However, the pre-Northridge earthquake steel frames showed fatigue cracks without the expected substantial plastic deformation at beam flanges. Proposed solutions to this problem were the reduced beam section neglecting the existence of the cracks at beam–column connections. This study considered the fatigue phenomenon in steel frames and provided the first set of tested fatigue damage values for steel components subjected to realistic earthquake loadings, which offered a possible method of dealing with fatigue cracks in the steel components of a building structure.

Large-amplitude consumer-resource cycles allow for the evolution of ontogenetic niche shifts in consumer life history

In many size-structured populations individuals change resources during the course of their ontogenetic development. Different resources often require different adaptations to be effectively exploited. This leads to a trade-off between small and large individuals in direct developing species. Specialization on the resource used later in life turns out to be hardly possible in case of equilibrium dynamics. However, size-structured populations often exhibit population cycles. Non-equilibrium dynamics can change evolutionary behavior when compared with equilibrium dynamics. Here, we study the evolution of specialization on a secondary resource that is available only to large individuals, using the framework of adaptive dynamics. We show that in case of small-amplitude cycles, specialization on a secondary resource is hardly possible. Specialization will either decrease the resource intake of large individuals or severely increase competition among small individuals such that they cannot mature. Specialization on a secondary resource is often possible in case the population exhibits large-amplitude cycles. Specialization in that case increases the resource intake of large individuals and therefore prevents starvation. While specialization on a secondary resource increases competition among small individuals, maturation is still possible in case of large-amplitude cycles. We furthermore show that there is ecological bistability where small- and large-amplitude cycles coexist, giving rise to evolutionary bistability.

Non-linear stability analysis of pressure drop oscillations in a heated channel

A study of non-linear stability analysis using co-dimension two (i.e., two free parameters being varied) bifurcations related to pressure drop oscillations (PDO) in a heated channel has been carried out in this paper. In the existing literature, in the context of PDO, mostly linear stability analysis is done. A few works on non-linear stability analysis are available; however, only co-dimension one bifurcation studies of PDO have been carried out in these works. However, in practice, since the inlet temperature of the coolant is also an independent operating parameter, both inlet temperature, and inlet mass flow rate need to be considered for stability analysis. It is also noted that the existing plethora of studies on PDO is limited to the prediction of supercritical Hopf bifurcation only. However, in the current study, two types of Hopf bifurcations have been identified namely subcritical and supercritical. A subcritical Hopf bifurcation exhibits unstable limit cycles, which is a signature of the existence of unstable solutions for slightly larger perturbations even in the linearly stable region. The existence of unstable solutions indicates that linear stability analysis is not sufficient to identify the overall stability behavior. Also, a generalized Hopf point on the stability boundary has been identified which denotes a boundary between subcritical and supercritical Hopf bifurcation. Furthermore, numerical simulations are carried out in different regions (both stable and unstable) of the parameter space to understand the non-linear phenomena of the system. Moreover, the bifurcations are explained in terms of the interaction of the external and internal characteristic curves of the system. The large and small amplitude cycles are shown in the characteristic curves of the system.

Identifying the Presence and Cause of Fashion Cycles in Data

Fashions and conspicuous consumption play an important role in marketing. In this article, the author presents a three-pronged framework to analyze fashion cycles in data composed of (1) algorithmic methods for identifying cycles, (2) a statistical framework for identifying cycles, and (3) methods for examining the drivers of such cycles. In the first module, the author identifies cycles by pattern-matching the amplitude and length of cycles observed to a user-specified definition. In the second module, the author defines the conditional monotonicity property, derives conditions under which a data-generating process satisfies it, and demonstrates its role in generating cycles. A key challenge in estimating this model is the presence of endogenous lagged dependent variables, which the author addresses using system generalized method of moments estimators. Third, the author presents methods that exploit the longitudinal and geographic variations in agents' economic and cultural capital to examine the different theories of fashion. The author applies her framework to data on given names for infants, shows the presence of large-amplitude cycles both algorithmically and statistically, and confirms that the adoption patterns are consistent with Bourdieu's theory of fashion as a signal of cultural capital.

Long term simulation of wind-induced fatigue loadings

This paper deals with the fatigue analysis of slender structures subjected to the wind action. A 50-year conditioned Monte Carlo simulation of the bending moment at the base of a steel lighting pole is performed. Long-term simulations of the static part of the bending moment, related to the mean wind speed, are combined with short-term fluctuations due to the turbulence. Multiple time series of the stress at the base of the pole, with length ranging between 10min and 3days, are generated and processed by means of the Rainflow counting algorithm. The resulting large size data set of the fatigue damage is analyzed in order to clarify some cloudy issues related to the fatigue analysis: the contribution to the damage of the low-frequency, large-amplitude cycles in the stress-time series; the contribution of the transition cycles between consecutive wind states; the length of the stress-time series necessary to get a reliable stress cycle histogram in the framework of the classical state approach; the variability of the damage resulting from a given mean wind speed; the variability of the annual damage.

Approximation of a physiologically structured population model with seasonal reproduction by a stage-structured biomass model

Seasonal reproduction causes, due to the periodic inflow of young small individuals in the population, seasonal fluctuations in population size distributions. Seasonal reproduction furthermore implies that the energetic body condition of reproducing individuals varies over time. Through these mechanisms, seasonal reproduction likely affects population and community dynamics. While seasonal reproduction is often incorporated in population models using discrete time equations, these are not suitable for size-structured populations in which individuals grow continuously between reproductive events. Size-structured population models that consider seasonal reproduction, an explicit growing season and individual-level energetic processes exist in the form of physiologically structured population models. However, modeling large species ensembles with these models is virtually impossible. In this study, we therefore develop a simpler model framework by approximating a cohort-based size-structured population model with seasonal reproduction to a stage-structured biomass model of four ODEs. The model translates individual-level assumptions about food ingestion, bioenergetics, growth, investment in reproduction, storage of reproductive energy, and seasonal reproduction in stage-based processes at the population level. Numerical analysis of the two models shows similar values for the average biomass of juveniles, adults, and resource unless large-amplitude cycles with a single cohort dominating the population occur. The model framework can be extended by adding species or multiple juvenile and/or adult stages. This opens up possibilities to investigate population dynamics of interacting species while incorporating ontogenetic development and complex life histories in combination with seasonal reproduction.

The DeAngelis–Beddington functional response and the evolution of timidity of the prey

We study the evolution of “timidity” of the prey (i.e., its readiness to seek refuge) in a predator–prey model with the DeAngelis–Beddington functional response. Using the theory of adaptive dynamics, we find that a predator–prey population at equilibrium always favours less timidity. Low levels of timidity, however, may destabilise the population and lead to cycles. Large-amplitude cycles favour a positive level of timidity, but if such cycles do not occur, timidity will evolve all the way to zero, where the prey no longer responds to the predator by seeking refuge, and in which case the DeAngelis–Beddington functional response has become identical to the Holling type–II functional response.

Predicting Population Dynamics from the Properties of Individuals: A Cross-Level Test of Dynamic Energy Budget Theory

Individual-based models (IBMs) are increasingly used to link the dynamics of individuals to higher levels of biological organization. Still, many IBMs are data hungry, species specific, and time-consuming to develop and analyze. Many of these issues would be resolved by using general theories of individual dynamics as the basis for IBMs. While such theories have frequently been examined at the individual level, few cross-level tests exist that also try to predict population dynamics. Here we performed a cross-level test of dynamic energy budget (DEB) theory by parameterizing an individual-based model using individual-level data of the water flea, Daphnia magna, and comparing the emerging population dynamics to independent data from population experiments. We found that DEB theory successfully predicted population growth rates and peak densities but failed to capture the decline phase. Further assumptions on food-dependent mortality of juveniles were needed to capture the population dynamics after the initial population peak. The resulting model then predicted, without further calibration, characteristic switches between small- and large-amplitude cycles, which have been observed for Daphnia. We conclude that cross-level tests help detect gaps in current individual-level theories and ultimately will lead to theory development and the establishment of a generic basis for individual-based models and ecology.

Routes to bursting in a periodically driven oscillator

Abstract The dynamical behaviors of a periodic excited oscillator with multiple time scales in the form that order gap exists between the frequency of the excitation and the natural frequency, are investigated in this Letter. By regarding the whole excitation term as a parameter, bifurcation sets are derived, which divide the generalized parameter space into several regions corresponding to different kinds of dynamics. Different types of bursting phenomena, such as fold/Hopf bursting, fold/Hopf/homoclinic bursting and Hopf/homoclinic bursting, are presented, the mechanism of which is obtained based on the bifurcations of the generalized autonomous system as well as the introduction of the so-called transformed phase portraits. Furthermore, the evolution of the bursting is discussed in details, in which one may find that when the two limit cycles caused by the Hopf bifurcations of the two related equilibrium points interact with each other, homoclinic bifurcation may occur, leading to the merge of the two cycles to form a large amplitude cycle. The homoclinic bifurcation may cause the two asymmetric bursters to merge into a symmetric enlarged burster, in which the large amplitude of the spiking state agrees well with the amplitude of the cycle caused by the homoclinic bifurcation.

Nitinol Fatigue Life for Variable Strain Amplitude Fatigue

Nitinol fatigue testing results are presented for variable strain amplitude cycling. The results indicate that cycles smaller than the constant amplitude fatigue limit may contribute to significant fatigue damage when they occur in a repeating sequence of large and small amplitude cycles. The testing utilized two specimen types: stent-like diamond specimens and Z-shaped wire specimens. The diamond specimens were made from nitinol tubing with stent-like manufacturing processes and the Z-shaped wire specimens were made from heat set nitinol wire. The study explored the hypothesis that duty cycling can have an effect on nitinol fatigue life. Stent-like structures were subjected to different in vivo loadings in order to create more complex strain amplitudes. The main focus in this study was to determine whether a combination of small and large amplitudes causes additional damage that alters the fatigue life of a component.

Identifying the Presence and Cause of Fashion Cycles in the Choice of Given Names

Fashions and conspicuous consumption play an important role in marketing. In this paper, we present a three-pronged framework to analyze fashion cycles in data -- a) algorithmic methods for identifying cycles, b) statistical framework for identifying cycles, and c) methods for examining the drivers of such cycles. In the first module, we identify cycles based on pattern-matching the amplitude and length of cycles observed to a user-specified definition. In the second module, we define the Conditional Monotonicity Property, derive conditions under which a data generating process satisfies it, and demonstrate its role in generating cycles. A key challenge that we face in estimating this model is the presence of endogenous lagged dependent variables, which we address using system GMM estimators. Third, we present methods that exploit the longitudinal and geographic variations in agents' economic and cultural capital to examine the different theories of fashion. We apply our framework to data on given names for infants, show the presence of large amplitude cycles both algorithmically and statistically, and confirm that the adoption patterns are consistent with Bourdieu's theory of fashion as a signal of cultural capital.

Evidence for Chaos in Mitochondrial Dynamics

The triggering of post-ischemic arrhythmias and heart contractile dysfunction at acute levels of oxidative stress and energetic failure represents a new paradigm in pathophysiology. Mitochondrial network oscillations are able to drive the excitability of the heart cell setting the stage of cardiac fibrillation. Not known is whether chaotic mitochondrial dynamics can give rise to fibrillation, considered as a form of spatio-temporal chaos in electrical activity. Thus, herein we seek for evidence of the existence of mitochondrial chaos. An upgraded computational model of mitochondrial redox-energetics including reactive oxygen species (ROS) production and ROS scavenging in both mitochondrial and cytoplasmic compartments was utilized to search for chaotic dynamics under oxidative stress. Stability analysis of the model showed that mitochondrial oscillations can be observed when the superoxide, O2.−, scavenging capacity is compromised in energized mitochondria. The transition between stable steady states and oscillatory dynamics is characterized by a series of Hopf bifurcations (HB), from which subcritical ones are observed with large positive maximum Lyapunov exponents. This behavior is a signature of sensitivity to initial conditions, a hallmark of chaos. Return plots show that the oscillatory regime comprises large amplitude cycles of membrane potential, NADH, O2.−, and H2O2 with complex wave forms. A three dimensional phase plane plot of mitochondrial ADP, NADH, and O2.− at the subcritical HB shows a strange attractor. The power spectrum of the corresponding time series exhibits a power law distribution, characterized by a broad spectrum of frequencies (spectral exponent = −2.3), while other mitochondrial state variables such as O2.− showed exponential decay. Detrended fluctuation analysis of the time series indicated that the signals do not correspond to colored noise. Taken together our results suggest that mitochondria can exhibit chaotic dynamics under conditions of reduced O2.− scavenging capacity.

Deformation Characteristics of Dry Hostun Sand with Principal Stress Axes Rotation

Coaxiality between the principal directions of the stress tensor and the principal directions of the plastic strain increment tensor is assumed in conventional plasticity models. In order to investigate coaxiality, or non-coaxiality, between these two principal directions, a series of drained tests on dry Hostun sand was carried out using a precision Hollow Cylinder Apparatus (HCA). The applied stress path includes large Principal Stress Axes Rotation (PSAR). Two of the three principal stresses are kept constant. Therefore, among the three principal stresses, only the intermediate principal stress, which is confining pressure (same pressure outside the hollow cylinder for internal and external lateral surfaces), changes during loading. During these tests, at different stress levels, elastic (or quasi-elastic) properties are also investigated, using small amplitude quasi-static cycles. These small cycles are performed in two different directions by successively changing only the axial stress σzz or the shear stress σθz. Elastic experimental properties are well simulated using the DI Bendetto-Geoffroy-Sauzeat (DBGS) hypo-elastic model, which takes into account PSAR. For each test, the elastic part of deformation is calculated using the DBGS model and removed from global strain so that it is possible to to focus only on the irreversible part (plastic part). Then, the principal directions of stress and plastic strain increment are compared. Experimental results show that there is no coaxiality between these directions. This observation attests to the existence of a non-coaxial plasticity. In addition, the coupling between tge coaxial and non-coaxial part is clearly shown. Experimental results reveal that the plastic strain part is very important for the first large amplitude cycles and remains greater than the elastic part even after 20 cycles.

Unique plastic and recovery behavior of nanofilled elastomers and thermoplastic elastomers (Payne and Mullins effects)

AbstractWe have proposed recently that the mechanical properties of nano‐filled elastomers are governed by the kinetics of rupture and re‐birth of glassy bridges which link neighboring nanoparticles and allow for building large rigid clusters of finite life‐times. The latter depend on parameters such as the temperature, the nanoparticle‐matrix interaction, and the distance between neighboring fillers. Most importantly these life‐times depend on the history of deformation of the samples. We show that this death and re‐birth process allows for predicting unusual non‐linear and plastic behavior for these systems. We study in particular the behavior after large deformation amplitude cycles. At some point we put the systems at rest under large deformation, and let the stress relax in this new deformed state. During this relaxation process the life‐time of glassy bridges increases progressively, even for large deformation states. The systems thus acquire a new reference state, which corresponds to a plastic deformation. The stretching energy of the polymer strands of the rubbery matrix is larger than in the initial undeformed state, but this effect is compensated by a new configuration of glassy bridges, which are much stiffer. For plastic deformations of less than about 10%, the new system acquires mechanical properties around this new reference state which are very close to those of the initial system. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 1495–1508, 2010