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Abstract 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. 

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. 

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. 

Abstract Coulomb friction has an influence on the behavior of numerous mechanical systems. Coulomb friction systems or dry friction systems are nonlinear in nature. This nonlinear behavior requires complex and time-demanding analysis tools to capture the dynamics of these systems. Recently, efforts have been made to develop efficient analysis tools able to approximate the forced response of systems with dry friction. The objective of this paper is to introduce a methodology that assists in these efforts. In this method, the piecewise linear nonlinear response is separated into individual linear responses that are coupled together through compatibility equations. The new method is demonstrated on a number of systems of varying complexity. The results obtained by the new method are validated through the comparison with results obtained by time integration. The computational savings of the new method are also discussed.