Component mode mistuning (CMM) is a well-known, well documented reduced order modeling technique that effectively models small variations in blade-to-blade stiffness for bladed disks. In practice, bladed disks always have variations, referred to as mistuning, and are a focus of a large amount of research. One element that is commonly ignored from small mistuning implementations is the variation within the blade-to-blade damping values. This work seeks to better understand the effects of damping mistuning by utilizing both structural and proportional damping formulations. This work builds from previous work that implemented structural damping mistuning reduced order models formulated based on CMM. A similar derivation was used to create reduced order models with a proportional damping formulation. The damping and stiffness mistuning values investigated in this study were derived using measured blade natural frequencies and damping ratios from high-speed rotating experiments on freestanding blades. The two separate damping formulations that are presented give very similar results, enabling the user to select their preferred method for a given application. A key parameter investigated in this work is the significance of blade-to-blade coupling. The blade-to-blade coupling study investigates how the level of coupling impacts damping mistuning effects versus applying average damping to the bladed disk model. Also, the interaction of stiffness and damping mistuning is studied. Monte Carlo simulations were carried out to determine amplification factors, or the ratio of mistuned blade responses to tuned blade responses, for various mistuning levels and patterns.
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Abstract A wide range of mechanical systems have gaps, cracks, intermittent contact or other geometrical discontinuities while simultaneously experiencing Coulomb friction. A piecewise linear model with discontinuous force elements is discussed in this paper that has the capability to accurately emulate the behavior of such mechanical assemblies. The mathematical formulation of the model is standardized via a universal differential inclusion and its behavior, in different scenarios, is studied. In addition to the compatibility of the proposed model with numerous industrial systems, the model also bears significant scientific value since it can demonstrate a wide spectrum of motions, ranging from periodic to chaotic. Furthermore, it is demonstrated that this class of models can generate a rare type of motion, called weakly chaotic motion. After their detailed introduction and analysis, an efficient hybrid symbolic-numeric computational method is introduced that can accurately obtain the arbitrary response of this class of nonlinear models. The proposed method is capable of treating high dimensional systems and its proposition omits the need for utilizing model reduction techniques for a wide range of problems. In contrast to the existing literature focused on improving the computational performance when analyzing these systems when there is a periodic response, this method is able to capture transient and nonstationary dynamics and is not restricted to only steady-state periodic responses.
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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.
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null (Ed.)
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.
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Lacarbonara, Walter (Ed.)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.more » « lessFree, publicly-accessible full text available June 11, 2025
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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.
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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.
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Abstract In this paper, a new technique is presented for parametrically studying the steady-state dynamics of piecewise-linear nonsmooth oscillators. This new method can be used as an efficient computational tool for analyzing the nonlinear behavior of dynamic systems with piecewise-linear nonlinearity. The new technique modifies and generalizes the bilinear amplitude approximation method, which was created for analyzing proportionally damped structural systems, to more general systems governed by state-space models; thus, the applicability of the method is expanded to many engineering disciplines. The new method utilizes the analytical solutions of the linear subsystems of the nonsmooth oscillators and uses a numerical optimization tool to construct the nonlinear periodic response of the oscillators. The method is validated both numerically and experimentally in this work. The proposed computational framework is demonstrated on a mechanical oscillator with contacting elements and an analog circuit with nonlinear resistance to show its broad applicability.more » « less