The influence of phase-locking on internal resonance from a nonlinear normal mode perspective

When a nonlinear system is expressed in terms of the modes of the equivalent linear system, the nonlinearity often leads to modal coupling terms between the linear modes. In this paper it is shown that, for a system to exhibit an internal resonance between modes, a particular type of nonlinear coupling term is required. Such terms impose a phase condition between linear modes, and hence are denoted phase-locking terms. The effect of additional modes that are not coupled via phase-locking terms is then investigated by considering the backbone curves of the system. Using the example of a two-mode model of a taut horizontal cable, the backbone curves are derived for both the case where phase-locked coupling terms exist, and where there are no phase-locked coupling terms. Following this, an analytical method for determining stability is used to show that phase-locking terms are required for internal resonance to occur. Finally, the effect of non-phase-locked modes is investigated and it is shown that they lead to a stiffening of the system. Using the cable example, a physical interpretation of this is provided

Identifying the significance of nonlinear normal modes

Nonlinear normal modes (NNMs) are widely used as a tool for understanding the forced responses of nonlinear systems. However, the contemporary definition of an NNM also encompasses a large number of dynamic behaviours which are not observed when a system is forced and damped. As such, only a few NNMs are required to understand the forced dynamics. This paper firstly demonstrates the complexity that may arise from the NNMs of a simple nonlinear system-highlighting the need for a method for identifying the significance of NNMs. An analytical investigation is used, alongside energy arguments, to develop an understanding of the mechanisms that relate the NNMs to the forced responses. This provides insight into which NNMs are pertinent to understanding the forced dynamics, and which may be disregarded. The NNMs are compared with simulated forced responses to verify these findings

LOWER MANTLE TEMPERATURE AND COMPOSITION

We evaluate the thermo-chemical state of the lower mantle by analysing the differences in the pattern of heterogeneity between shear and compressional velocity variations and the S−to−P heterogeneity ratio ( RS/P=δlnVS/δlnVP) as mapped in our model SPani and in alternative joint models. Robust structural differences between VP and VS evidence the presence of compositional heterogeneity within the two Large Low Shear Velocity Provinces (LLSVPs). We find also an increasing decorrelation with depth that can be associated with compositional layering of the LLSVPs. In addition, our model shows heterogeneity in the transition zone and mid mantle by complex morphology of subducting slabs and further differences between VP and VS that point to an unexpected heterogeneous lower mantle. Precise estimates of compositional heterogeneities are not yet affordable because of the difficulty to provide quantitative measure of RS/P, making it difficult to use this ratio to evaluate chemical heterogeneity. For instance, RS/P global median value ( ) drops from ∼2.8 to ∼1.9, at 2500 km depth when the VP component of SPani is replaced by a VP model resulting from a differently regularized inversion and obtaining an equally good data fit. An increase of 20% of the SPani VP anomalies also drastically reduces without significantly degrading the data fit. Noise in model parameters also leads to overestimate RS/P in the two LLSVPs as we show with synthetic tests. Additional mineral physics uncertainties for compositional effects on RS/P and for the conversion of δlnVS and δlnVP into density further complicates a precise chemical interpretation

An optimized tuned mass damper/harvester device

Much work has been conducted on vibration absorbers, such as tuned mass dampers (TMD), where significant energy is extracted from a structure. Traditionally, this energy is dissipated through the devices as heat. In this paper, the concept of recovering some of this energy electrically and reuse it for structural control or health monitoring is investigated. The energy-dissipating damper of a TMD is replaced with an electromagnetic device in order to transform mechanical vibration into electrical energy. That gives the possibility of controlled damping force whilst generating useful electrical energy. Both analytical and experimental results from an adaptive and a semi-active tuned mass damper/harvester are presented. The obtained results suggest that sufficient energy might be harvested for the device to tune itself to optimise vibration suppression

Fast Bayesian identification of a class of elastic weakly nonlinear systems using backbone curves

This paper introduces a method for the identification of the parameters of nonlinear structures using a probabilistic Bayesian framework, employing a Markov chain Monte Carlo algorithm. This approach uses analytical models to describe the unforced, undamped dynamic responses of structures in the frequency–amplitude domain, known as the backbone curves. The analytical models describing these backbone curves are then fitted to measured responses, found using the resonant-decay method. To investigate the proposed identification method, a nonlinear two-degree-of-freedom example structure is simulated numerically and analytical expressions describing the backbone curves are found. These expressions are then used, in conjunction with the backbone curve data found through simulated experiment, to estimate the system parameters. It is shown that the use of these computationally-cheap analytical expressions allows for an extremely efficient method for modelling the dynamic behaviour, providing an identification procedure that is both fast and accurate. Furthermore, for the example structure, it is shown that the estimated parameters may be used to accurately predict the existence of dynamic behaviours that are well-away from the backbone curve data provided; specifically the existence of an isola is predicted

Effect of Wind on Morphology and Mechanical Properties of Arabidopsis Thaliana

No abstract available