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This work proposes a computational approach that has its roots in the early ideas of local Lyapunov exponents, yet, it offers new perspectives toward analyzing these problems. The method of interest, namely abstract dynamics, is an indirect quantitative measure of the variations of the governing vector fields based on the principles of linear systems. The examples in this work, ranging from simple limit cycles to chaotic attractors, are indicative of the new interpretation that this new perspective can offer. The presented results can be exploited in the structure of algorithms (most prominently machine learning algorithms) that are designed to estimate the complex behavior of nonlinear systems, even chaotic attractors, within their horizon of predictability. 

Vibration energy harvesting is increasingly being seen as a viable energy source to provide for our energy-dependent society. There has been great interest in scavenging previously unused or wasted energy in a large variety of systems including vibrating machinery, ocean waves and human motion. In this work, a bench-top system of a piecewise-linear nonlinear vibration energy harvester is studied. A similar idealized model of the system had previously been studied numerically, and in this work the method is adjusted to better account for the physical system. This new design is able to actively tune the system’s resonant frequency to match the current excitation through the adjustment of the gap size between the oscillator and mechanical stopper; thus maximizing the system response over a broad frequency range. This design shows an increased effective frequency bandwidth compared with traditional linear systems and improves upon current nonlinear designs that are less effective than linear harvesters at resonance. In this paper, the physical system is tested at various excitation conditions and gap sizes to showcase the new harvester design’s effectiveness. 

There has been an extensive amount of work developing reduced-order models (ROMs) for bladed disks using single-sector models and a cyclic analysis. Several ROMs currently exist to accurately model a bladed disk with under-platform dampers. To better predict the complex nonlinear response of a system with under-platform dampers, this work demonstrates how two linear models can determine bounds for the nonlinear response. The two cases explored are where the under-platform damper is completely stuck and also where the damper slides without friction. This work utilizes the component mode mistuning method to model small mistuning and a parametric ROM method to capture changes in properties due to rotational speed effects. Previously, these ROM methodologies have modeled freestanding bladed disk systems. To evaluate the ROM in predicting the bounds, blade tip amplitudes from the models are compared with high-speed rotating experiments conducted in a large, evacuated vacuum tank. The experimental data were collected during testing using strain gauges and laser blade tip timing probes. The blade amplitudes of the tip timing data, strain gauge data, and computational simulations are compared to determine the effectiveness of the simplified linear analysis in bounding the nonlinear response of the physical system. 

Abstract Recently, vibration energy harvesting has been seen as a viable energy source to provide for our energy dependent society. Researchers have studied systems ranging from civil structures like bridges to biomechanical systems including human motion as potential sources of vibration energy. In this work, a bench-top system of a piecewise-linear (PWL) nonlinear vibration harvester is studied. A similar idealized model of the harvester was previously looked at numerically, and in this work the method is adjusted to handle physical systems to construct a realistic harvester design. With the physically realizable harvester design, the resonant frequency of the system is able to be tuned by changing the gap size between the oscillator and mechanical stopper, ensuring optimal performance over a broad frequency range. Current nonlinear harvester designs show decreased performance at certain excitation conditions, but this design overcomes these issues while also still maintaining the performance of a linear harvester at resonance. In this investigation, the system is tested at various excitation conditions and gap sizes. The computational response of the resonance behavior of the PWL system is validated against the experiments. Additionally, the electromechanical response is also validated with the experiments by comparing the output power generated from the experiments with the computational prediction. 

Abstract In this paper, the forced response of a two degrees-of-freedom (DOF) bilinear oscillator with initial gaps involving inelastic collision is discussed. In particular, a focus is placed upon the experimental verification of the generalized bilinear amplitude approximation (BAA) method, which can be used for the accurate estimation of forced responses for bilinear systems with initial gaps. Both experimental and numerical investigations on the system have been carried out. An experimental setup that is capable of representing the dynamics of a 2DOF oscillator has been developed, and forced response tests have been conducted under swept-sine base excitation for different initial gap sizes. The steady-state response of the system under base excitation was computed by both traditional time integration and BAA. It is shown that the results of experiments and numerical predictions are in good agreement especially at resonance. However, slight differences in the responses obtained from both numerical methods are observed. It was found that the time duration where the DOFs are in contact with each other predicted by BAA is longer than that predicted by time integration. Spectral analyses have also been conducted on both experimental and numerical results. It was observed that in a frequency range where intermittent contact between the masses occurs, super-harmonic components of the excitation frequency are present in the spectra. Moreover, as the initial gap size increases, the frequency band where the super-harmonic components are observed decreases. 

A general formulation of piecewise linear systems with discontinuous force elements is provided in this paper. It has been demonstrated that this class of nonlinear systems is of great importance due to their ability to accurately model numerous scientific and engineering phenomena. Additionally, it is shown that this class of nonlinear systems can demonstrate a wide spectrum of nonlinear motions and in fact, the phenomenon of weak chaos is observed in a mechanical assembly for the first time. Despite such importance, efficient methods for fast and accurate evaluation of piecewise linear systems’ responses are lacking and the methods of the literature are either incompatible, very slow, very inaccurate, or bear a combination of the aforementioned deficiencies. To overcome this shortcoming, a novel symbolic-numeric method is presented in this paper that is able to obtain the analytical response of piecewise linear systems with discontinuous elements in an efficient manner. Contrary to other efficient methods that are based on stationary steady state dynamics, this method will not experience failure upon the occurrence of complex motion and is able to capture the entirety of the dynamics. 

Abstract In this paper, an experimental forced response analysis for a two degree of freedom piecewise-linear oscillator is discussed. First, a mathematical model of the piecewise linear oscillator is presented. Second, the experimental setup developed for the forced response study is presented. The experimental setup is capable of investigating a two degree of freedom piecewise linear oscillator model. The piecewise linearity is achieved by attaching mechanical stops between two masses that move along common shafts. Forced response tests have been conducted, and the results are presented. Discussion of characteristics of the oscillators are provided based on frequency response, spectrogram, time histories, phase portraits, and Poincaré sections. Period doubling bifurcation has been observed when the excitation frequency changes from a frequency with multiple contacts between the masses to a frequency with single contact between the masses occurs. 

Vibration energy is becoming a significant alternative solution for energy generation. Recently, a great deal of research has been conducted on how to harvest energy from vibration sources ranging from ocean waves to human motion to microsystems. In this paper, a theoretical model of a piecewise-linear (PWL) nonlinear vibration harvester that has potential applications in variety of fields is proposed and numerically investigated. This new technique enables automatic frequency tunability in the energy harvester by controlling the gap size in the PWL oscillator so that it is able to adapt to changes in excitations. To optimize the performance of the proposed system, a control method combining the response prediction, signal measurement and gap adjustment mechanism is proposed in this paper. This new energy harvester not only overcomes the limitation of traditional linear energy harvesters that can only provide the maximum power generation efficiency over a narrow frequency range but also improves the performance of current nonlinear energy harvesters that are not as efficient as linear energy harvesters at resonance. The proposed system is demonstrated in several case studies to illustrate its effectiveness for a number of different excitations.