Tool chatter in longitudinal turning is addressed with a new perspective using a complex machining model describing the coupled tool-workpiece dynamics subject to nonlinear regenerative cutting forces, instantaneous depth-of-cut (DOC) and workpiece whirling due to material imbalance. The workpiece is modeled as a system of three rotors: unmachined, being machined and machined, connected by a flexible shaft. The model enables workpiece motions relative to the tool and tool motions relative to the machining surface to be three-dimensionally established as functions of spindle speed, instantaneous DOC, rate of material removal and whirling. Excluding workpiece vibrations from the cutting model is found improper. A rich set of nonlinear behaviors of both the tool and the workpiece including period-doubling bifurcation and chaos signifying the extent of machining instability at various DOCs is observed. Presented numerical results agree favorably with physical experiments reported in the literature. It is found that whirling is non-negligible if the fundamental characteristics of machining dynamics are to be fully understood. The 3D model is explored along with its 1D counterpart, which considers only tool motions and disregards workpiece vibrations. Numerical simulations reveal diverse behaviors for the 3D coupled and 1D uncoupled equations of motion for the tool. Most notably, observations made with regard to the inconsistency in describing stability limits raise the concern for using 1D models to obtain stability charts. The nonlinear 3D model is linearized to investigate the implications of applying linear models to the understanding of machining dynamics. Taylor series expansion about the operating point where optimal machining conditions are desired is applied to linearize the model equations of motion. Modifications are also made to the nonlinear tool stiffness term to minimize linearization errors. Numerical experiments demonstrate inadmissible results for the linear model and good agreement with available physical data in describing machining stability and chatter for the nonlinear model. Effects of tool geometry, feed rate, and spindle speed on cutting dynamics are also explored. It is observed that critical DOC increases with increasing spindle speed and small DOCs can induce cutting instability -- two of the results that agree qualitatively well with published experimental data.

Tool chatter in longitudinal turning is addressed with a new perspective using a complex machining model describing the coupled tool-workpiece dynamics subject to nonlinear regenerative cutting forces, instantaneous depth-of-cut (DOC) and workpiece whirling due to material imbalance. The workpiece is modeled as a system of three rotors: unmachined, being machined and machined, connected by a flexible shaft. The model enables workpiece motions relative to the tool and tool motions relative to the machining surface to be three-dimensionally established as functions of spindle speed, instantaneous DOC, rate of material removal and whirling. Excluding workpiece vibrations from the cutting model is found improper. A rich set of nonlinear behaviors of both the tool and the workpiece including period-doubling bifurcation and chaos signifying the extent of machining instability at various DOCs is observed. Presented numerical results agree favorably with physical experiments reported in the literature. It is found that whirling is non-negligible if the fundamental characteristics of machining dynamics are to be fully understood. The 3D model is explored along with its 1D counterpart, which considers only tool motions and disregards workpiece vibrations. Numerical simulations reveal diverse behaviors for the 3D coupled and 1D uncoupled equations of motion for the tool. Most notably, observations made with regard to the inconsistency in describing stability limits raise the concern for using 1D models to obtain stability charts. The nonlinear 3D model is linearized to investigate the implications of applying linear models to the understanding of machining dynamics. Taylor series expansion about the operating point where optimal machining conditions are desired is applied to linearize the model equations of motion. Modifications are also made to the nonlinear tool stiffness term to minimize linearization errors. Numerical experiments demonstrate inadmissible results for the linear model and good agreement with available physical data in describing machining stability and chatter for the nonlinear model. Effects of tool geometry, feed rate, and spindle speed on cutting dynamics are also explored. It is observed that critical DOC increases with increasing spindle speed and small DOCs can induce cutting instability -- two of the results that agree qualitatively well with published experimental data.

No description available.

The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von K\'rm\'{a}n-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented. We used two approaches to study the responses of shells to a primary resonance: a $direct$ approach and a $discretization$ approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency- and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared. For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating $2n+(2n)^2$ nonlinear first-order ordinary-differential equations simultaneously, where $n$ is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results. The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown. Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo symmetry-breaking, saddle node, and period-doubling bifurcations. / Ph. D.

<p> Nonlinear flexural vibrations of simply supported rectangular plates with thermal coupling  are studied for the case when the plate is harmonically excited by the force acting normal to the  midplane of the plate. The coupled thermo-mechanical equations are derived by applying the  Galerkin procedure on the von-Karman equation and the energy equation for an element of the  plate. The thermo-mechanical equations are second order in transverse displacement and first order  in thermal dynamics. In our first study, we represent the transverse displacement, bending moment  and membrane force due to temperature by one mode approximation, and study the response of  thermoelastic plate in time and frequency domain. The analysis of forced vibration to a transverse  harmonic excitation is carried out using harmonic balance as well as direct time integration coupled  to a Fourier analysis for a range of excitation frequencies. The effects of thermal coupling, material  nonlinearity and different amplitudes of excitation on the thermoelastic plate’s transverse  displacement and thermoelastic variables are investigated. The method of averaging is applied to the one mode case to transform the nonlinear modal  equations into sets of two-dimensional dynamical systems which govern the amplitudes and phases  of the two modes. The averaged system is studied in detail by using pseudo arc-length continuation  schemes implemented in MATCONT. The physical phenomena of interest in this study arise when a plate exhibits two distinct  linear modes of vibration with nearly the same natural frequency. To analyze the dynamics of the  thermoelastic plate in this scenario, we utilize a two-mode approximation. The response of the  plate, as a function of excitation frequency, is determined for the two-mode model using  MATCONT, and several bifurcation points are identified. Our analysis reveals two types of  solutions: single-mode and coupled-mode solutions. We find that stable single-mode and coupled mode solutions can coexist over a wide range of amplitudes and excitation frequencies. Under the influence of thermal coupling, our analysis using MATCONT reveals the  identification of Neimark-Sacker bifurcation points. After a detailed study of the Neimark-Sacker  region using Fourier spectrum and Poincare section, we conclude that a pitchfork bifurcation  occurs, resulting in stable asymmetric solutions. We further investigate the effect of in-plane forces  or mechanical precompression on the thermoelastic plate, using MATCONT for a fixed value of  force, damping, and excitation frequency. We find that the in-plane forces lead to buckling, which  12 is identified as a branch point cycle (pitchfork bifurcation) in MATCONT. Consequently, the  bifurcation diagram of transverse displacement as a function of in-plane forces can be divided into  prebuckling and post buckling regions, with multistable solutions in each region. To validate our one mode model, we use ANSYS software to verify the transverse  displacement and temperature results. We validate the frequency and time domain results for both  the linear and nonlinear cases, and plot contours using ANSYS to observe the variation of  displacement and temperature over the surface of the plate. Our one mode model results closely  match with the ANSYS results, leading us to conclude that our one mode approximation is accurate  and that the coupled thermo-mechanical equations we derived are correct.  </p>

No description available.

Nous considérons un système mécanique en vibration non linéaire, pour lequel nous fournissons une solution approchée par l'utilisation des développements multiples échelles; nous proposons d'abord une étude avec double échelles puis avec triple échelles où nous comparons les deux approches. Une preuve rigoureuse de ces développements a été faite. L'étude de la stabilité de la solution est nécessaire pour montrer la convergence au voisinage de la résonance. Un lien entre l'amplitude de la réponse vibratoire et la fréquence du système en vibration libre a été mis en évidence. / We consider small solutions of a vibrating mechanical system with smooth non-linearities for which we provide an approximate solution by using multiple scale analysis; we first use a double scale analysis; in order to improve the approximation, then we perform a triple scale analysis; a rigorous proof of convergence of the triple scale method is included; for the forced response, a stability result is needed in order to prove convergence in a neighborhood of a primary resonance. The amplitude of the response with respect to the frequency forcing is described and it is related to the frequency of a free periodic vibration.

Desenvolve uma pesquisa em dinâmica estrutural não-linear com enfoque teórico e experimental direcionada para uma importante classe de estruturas flexíveis. É motivada pelos novos requerimentos das indústrias em termos de inovações, das agências certificadoras em termos de segurança e conforto e, por restrições relativas ao meio ambiente cada vez mais severas. Como conseqüência, o cenário atual e os desafios da engenharia moderna são bem diferentes daqueles encontrados antigamente. Atualmente as estruturas são mais flexíveis e operam sob condições cada vez mais severas. O aumento da flexibilidade torna as não-linearidades mais ativas e, juntamente com a ação de diferentes formas de excitação, produzem um cenário dinâmico complexo. Neste cenário, diversos fenômenos dinâmicos intrinsicamente não-lineares podem se desenvolver e conseqüêntemente comprometer a integridade estrutural, prejudicar a operação e incrementar os problemas de ruído. Tais fenômenos são altamente perigosos, principalmente por não serem previstos e nem ao menos conhecidos pela teoria dinâmica linear. Dentre estes fenômenos, a pesquisa se propõe a abordar dois: vibrações ressonantes paramétricas e autoparamétricas. Especificamente, a pesquisa investiga a influência da viscosidade do meio de operação e da presença de excitações combinadas nos fenômenos de ressonância paramétrica. No caso das ressonâncias autoparamétricas o objetivo específico é avaliar técnicas experimentais na caracterização do fenômeno, assim como, promover entendimentos mais profundos sobre suas características. Para atingir os objetivos propostos, são construídas duas estruturas de laboratório com características aeronáuticas. A primeira faz alusão a um estabilizador vertical. Nesta estrutura foram desenvolvidos os trabalhos relativos à vibração ressonante paramétrica. A segunda é uma simplificação de um sistema estrutural asa-pilone-turbina. Nesta segunda estrutura foram avaliadas técnicas experimentais para a identificação, caracterização e análise da vibração ressonante autoparamétrica. Os resultados teóricos e experimentais demonstram que a viscosidade do meio de operação age positivamente na dinâmica da estrutura, reduzindo níveis máximos de vibração em regime permanente, simplificando a dinâmica em respostas transientes e facilitando as relações de estabilidade/instabilidade. Por fim, apresenta resultados experimentais demonstrando que a energia vibratória da asa pode ser transferida por intermédio de uma ressonância autoparamétrica principal para a sub-estrutura pilone-turbina resultando em níveis extremamente elevados de vibração. / This document presents results of theoretical and experimental investigations on the non-linear vibration characteristics of an important class of flexible structures. The motivation for such a study arises mainly from the increasing need for lightweight structural members. The weight reduction associated to the use of novel materials contribute to the increase of flexibility what can cause the appearance of nonlinear effects not previously observed. These nonlinear phenomena associated to the fact that, in field conditions the structure is frequently subjected to complex dynamic loads of different nature, results in a complex dynamic environment when estimation of the structure's dynamic response is concerned. Moreover, these nonlinear effects potentially may cause undesired vibration level, in some cases causing bad function and failure of the entire structure. The research is focused on studying the effects of medium viscosity as well as combined excitations on parametrically resonant vibrating structures. It is speciffically aimed characterize the phenomenons either analytically and experimentally by constructing laboratory test specimens that resemble aircraft structures. For that purpose a vertical fin is built in order to conduct experiments on the principal parametric resonance phenomenon. An analytical single degree of freedom model of this structure including nonlinear terms is derived and the results of numerically simulated results through perturbation technique are compared to experimental results obtained in the laboratory. A second structure is built that resembles a typical wing-pylon-engine substructure and it is used to study autoparametric resonance vibrations. In this case the structure is considered with multiple degrees of freedom and the results of a finite element model is correlated with experimentally obtained results. Theoretical and experimental results show that the environment viscosity affects in a significant manner the dynamic response of the structures under test, decreasing the maximum vibration levels in steady-state regime, simplifying the dynamics in transient responses and facilitating the relationship between instability/stability. At the end, it is shown experimental results demonstrating that vibratory energy from the wing substructure can be transferred by an autoparametric resonance to the substructure pylon-engine. All the experimental results do not found linear theory counterparts.

Esta dissertação tem como objetivo principal realizar uma investigação sobre técnicas de identificação e caracterização de não linearidades em dinâmica estrutural. Em particular, busca-se aplicar as transformadas de Hilbert e Hilbert-Huang na identificação e caracterização de um sistema estrutural composto de uma viga metálica em balanço com uma massa concentrada em sua extremidade livre e apresentando uma não linearidade cúbica. Este efeito não linear é obtido através de um fio metálico tracionado que é fixo à extremidade livre da viga em balanço. O sistema é modelado através de ferramentas analíticas convencionais e o modelo do sistema é usado em simulações numéricas. Ensaios experimentais de vibração livre são realizados e os sinais medidos são usados como dados de entradas nas transformadas afim de se investigar o fenômeno não linear presente na estrutura. Os resultados simulados numéricamente apresentaram uma excelente correlação com os resultados experimentais no que se refere à identificação e caracterização do efeito não linear cubico presente no sistema. Algumas distorções entre resultados simulados e experimentais foram observados no que se refere à quantificação do parâmetro não linear, mas de forma geral, as técnicas empregadas produziram resultados bastante satisfatórios. / The major goal of this dissertation is to investigate currently available techniques to the identification and characterization of nonlinearities in structural dynamics. Particular attention is paid to the application of the Hilbert and Hilbert-Huang transforms in the nonlinearity identification and characterization process. For that purpose, a structural system composed of a steel cantilever beam carrying a lumped mass at its free end is used. The nonlinear effect is introduced by attaching the lumped mass to a thin steel wire that is positioned transversely to the beam\'s longitudinal axis. By varying the traction force on this wire different levels of nonlinearity can be observed on the beam\'s bending motion when it undergoes free vibrations. The system analytical model is obtained by employing standard modeling techniques and this model is used in numerical simulations. An experimental survey is carried out on an actual prototype in order to provide a comparison basis for the numerically simulated results. The output vibration signals resulting from either the numerical simulations and experimental tests are then used with the transform methods studied and the results are compared. A good correlation is observed between numerical and experimental data, what is a clear indication of the robustness of the Hilbert and Hilbert-Huang transforms.

High cycle fatigue (HCF) failure has been studied extensively over the last two decades. Its impact on jet engines is severe enough that may result in engine losses and even life losses. The main requirement for fatigue life predictions is the stress caused by mechanical vibrations. One of the factors which have major impact on the vibratory stresses of bladed disk assemblies is a phenomenon called &ldquo / mistuning&rdquo / which is defined as the vibration localization caused by the loss of cyclic periodicity which is a consequence of inter&amp / #8208 / blade variations in structural properties. In this thesis, component mode synthesis method (CMSM) is combined with nonlinear forced response analysis in modal domain. Newton&amp / #8208 / Raphson and arc length continuation procedures are implemented for the solution. The component mode synthesis method introduces the capability of imposing mistuning on the modal properties of each blade in the assembly. Forced response analysis in modal domain reduces the problem size via mode truncation. The main advantage of the proposed method is that it is capable of calculating nonlinear forced response for all the degrees&amp / #8208 / of&amp / #8208 / freedom at each blade with less computational effort. This makes it possible to make a stress analysis at resonance conditions. The case studies presented in this thesis emphasize the importance of number of modes retained in the reduced order model for both CMSM and nonlinear forced response analysis. Furthermore, the results of the case studies have shown that both nonlinearity and mistuning can cause shifts in resonance frequencies and changes in resonance amplitudes. Despite the changes in resonance conditions, the shape of the blade motion may not be affected.