Academic literature on the topic 'Extender based forcing'
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Journal articles on the topic "Extender based forcing"
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Merimovich, Carmi. "Extender-based Radin forcing." Transactions of the American Mathematical Society 355, no. 5 (January 8, 2003): 1729–72. http://dx.doi.org/10.1090/s0002-9947-03-03202-1.
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SINAPOVA, DIMA, and SPENCER UNGER. "MODIFIED EXTENDER BASED FORCING." Journal of Symbolic Logic 81, no. 4 (December 2016): 1432–43. http://dx.doi.org/10.1017/jsl.2016.5.
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AbstractWe analyze the modified extender based forcing from Assaf Sharon’s PhD thesis. We show there is a bad scale in the extension and therefore weak square fails. We also present two metatheorems which give a rough characterization of when a diagonal Prikry-type forcing forces the failure of weak square.
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Gitik, Moti, and Menachem Magidor. "Extender based forcings." Journal of Symbolic Logic 59, no. 2 (June 1994): 445–60. http://dx.doi.org/10.2307/2275399.
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AbstractThe paper is a continuation of [The SCH revisited], In § 1 we define a forcing with countably many nice systems. It is used, for example, to construct a model “GCH below κ, c f κ = ℵ0, and 2κ > κ+ω” from 0(κ) = κ+ω. In §2 we define a triangle iteration and use it to construct a model satisfying “{μ ≤ λ∣c f μ = ℵ0 and pp(μ) > λ} is countable for some λ”. The question of whether this is possible was asked by S. Shelah. In §3 a forcing for blowing the power of a singular cardinal without collapsing cardinals or adding new bounded subsets is presented. Answering a question of H. Woodin, we show that it is consistent to have “c f κ = ℵ0. GCH below κ, 2κ > κ+, and ”. In §4 a variation of the forcing of [The SCH revisited, §1] is defined. It behaves nicely in iteration processes. As an application, we sketch a construction of a model satisfying:“κ is a measurable and 2κ ≥ κ+α for some α, κ < c f α < α” starting with 0(κ) = κ+α. This answers the question from Gitik's On measurable cardinals violating the continuum hypothesis.
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Merimovich, Carmi. "Extender-based Magidor-Radin forcing." Israel Journal of Mathematics 182, no. 1 (March 2011): 439–80. http://dx.doi.org/10.1007/s11856-011-0038-0.
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Merimovich, Carmi. "Supercompact extender based Prikry forcing." Archive for Mathematical Logic 50, no. 5-6 (April 10, 2011): 591–602. http://dx.doi.org/10.1007/s00153-011-0234-y.
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Merimovich, Carmi. "Supercompact extender based Magidor–Radin forcing." Annals of Pure and Applied Logic 168, no. 8 (August 2017): 1571–87. http://dx.doi.org/10.1016/j.apal.2017.02.006.
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Gitik, Moti. "Extender-based forcings with overlapping extenders and negations of the Shelah Weak Hypothesis." Journal of Mathematical Logic 20, no. 03 (January 30, 2020): 2050013. http://dx.doi.org/10.1142/s0219061320500130.
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Extender-based Prikry–Magidor forcing for overlapping extenders is introduced. As an application, models with strong forms of negations of the Shelah Weak Hypothesis for various cofinalities are constructed.
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Merimovich, Carmi. "Mathias like criterion for the extender based Prikry forcing." Annals of Pure and Applied Logic 172, no. 9 (October 2021): 102994. http://dx.doi.org/10.1016/j.apal.2021.102994.
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Merimovich, Carmi. "A power function with a fixed finite gap everywhere." Journal of Symbolic Logic 72, no. 2 (June 2007): 361–417. http://dx.doi.org/10.2178/jsl/1185803615.
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AbstractWe give an application of the extender based Radin forcing to cardinal arithmetic. Assuming κ is a large enough cardinal we construct a model satisfying 2κ = κ+n together with 2λ = λ+n for each cardinal λ < κ, where 0 < n < ω. The cofinality of κ can be set arbitrarily or κ can remain inaccessible.When κ remains an inaccessible, Vκ is a model of ZFC satisfying 2λ = λ+n for all cardinals λ.
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Poveda, Alejandro. "Contributions to the Theory of Large Cardinals through the Method of Forcing." Bulletin of Symbolic Logic 27, no. 2 (June 2021): 221–22. http://dx.doi.org/10.1017/bsl.2021.22.
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AbstractThe dissertation under comment is a contribution to the area of Set Theory concerned with the interactions between the method of Forcing and the so-called Large Cardinal axioms.The dissertation is divided into two thematic blocks. In Block I we analyze the large-cardinal hierarchy between the first supercompact cardinal and Vopěnka’s Principle (Part I). In turn, Block II is devoted to the investigation of some problems arising from Singular Cardinal Combinatorics (Part II and Part III).We commence Part I by investigating the Identity Crisis phenomenon in the region comprised between the first supercompact cardinal and Vopěnka’s Principle. As a result, we generalize Magidor’s classical theorems [2] to this higher region of the large-cardinal hierarchy. Also, our analysis allows to settle all the questions that were left open in [1]. Finally, we conclude Part I by presenting a general theory of preservation of $C^{(n)}$ -extendible cardinals under class forcing iterations. From this analysis we derive several applications. For instance, our arguments are used to show that an extendible cardinal is consistent with “ $(\lambda ^{+\omega })^{\mathrm {HOD}}<\lambda ^+$ , for every regular cardinal $\lambda $ .” In particular, if Woodin’s HOD Conjecture holds, and therefore it is provable in ZFC + “There exists an extendible cardinal” that above the first extendible cardinal every singular cardinal $\lambda $ is singular in HOD and $(\lambda ^+)^{\textrm {{HOD}}}=\lambda ^+$ , there may still be no agreement at all between V and HOD about successors of regular cardinals.In Part II and Part III we analyse the relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals. Two of these are the Tree Property and the Reflection of Stationary sets, which are central in Infinite Combinatorics.Specifically, Part II is devoted to prove the consistency of the Tree Property at both $\kappa ^+$ and $\kappa ^{++}$ , whenever $\kappa $ is a strong limit singular cardinal witnessing an arbitrary failure of the SCH. This generalizes the main result of [3] in two senses: it allows arbitrary cofinalities for $\kappa $ and arbitrary failures for the SCH.In the last part of the dissertation (Part III) we introduce the notion of $\Sigma $ -Prikry forcing. This new concept allows an abstract and uniform approach to the theory of Prikry-type forcings and encompasses several classical examples of Prikry-type forcing notions, such as the classical Prikry forcing, the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal. Specifically, we aim for a theorem asserting that every $\kappa ^{++}$ -length iteration with support of size $\leq \kappa $ has the $\kappa ^{++}$ -cc, provided the iterates belong to a relevant class of $\kappa ^{++}$ -cc forcings. While there are a myriad of works on this vein for regular cardinals, this contrasts with the dearth of investigations in the parallel context of singular cardinals. Our main contribution is the proof that such a result is available whenever the class of forcings under consideration is the family of $\Sigma $ -Prikry forcings. Finally, and as an application, we prove that it is consistent—modulo large cardinals—the existence of a strong limit cardinal $\kappa $ with countable cofinality such that $\mathrm {SCH}_\kappa $ fails and every finite family of stationary subsets of $\kappa ^+$ reflects simultaneously.
Books on the topic "Extender based forcing"
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Smiley, Will. From the Law of Ransom to the Law of Release. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198785415.003.0004.
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This chapter charts the “Law of Release,” a new system of rules that replaced the Law of Ransom. These rules were based on treaties signed from 1739 onward, but also on a variety of lesser agreements and unwritten understandings and the Islamic legal tradition. They were renewed frequently, and structured captivity as late as the 1850s. This chapter will explore the basic structures of the Law of Release—how captives were found, released, and sent home, and how slaveowners were convinced, coerced, or compensated to cooperate. I argue that while release was initially limited to Istanbul, and to the most visible captives, it extended both into elite households, and outward along the Ottoman corridors of power. This process tested the limits of the Ottoman state, forcing the state to cooperate with Russian officials for the benefit of both. They did so in the face of resistance from captors.
Book chapters on the topic "Extender based forcing"
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Rubin, Yoram. "Quantifying and Accounting for Uncertainty." In Applied Stochastic Hydrogeology. Oxford University Press, 2003. http://dx.doi.org/10.1093/oso/9780195138047.003.0018.
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This chapter deals with a wide range of issues with a common theme: coping with uncertainty. To this end, we look at the sources of uncertainty and the types of errors we need to deal with. We then explore methods for identifying these errors and for incorporating them into our predictions. This chapter extends our discussion on these topics in chapter 1, the discussion in chapter 3 on estimation under conditions of uncertainty, and image simulation using MC techniques. A comprehensive treatment of uncertainty needs to address two different types of errors. The first type is the model error, which arises from incorrect hypotheses and unmodeled processes (Gaganis and Smith, 2001), for example, from poor choice of governing equations, incorrect boundary conditions and zonation geometry, and inappropriate selection of forcing functions (Carrera and Neuman, 1986b). The second type of error is parameter error. The parameters of groundwater models are always in error because of measurement errors, heterogeneity, and scaling issues. Ignoring the effects of model and parameter errors is likely to lead to errors in model selection, in the estimation of prediction uncertainty, and in the assessment of risk. Parameter error is treated extensively in the literature: once a model is defined, it is common practice to quantify the errors associated with estimating its parameters (cf. Kitanidis and Vomvoris, 1983; Carrera and Neuman, 1986a, b; Rubin and Dagan, 1987a,b; McLaughlin and Townley, 1996; Poeter and Hill, 1997). Modeling error is well recognized, but is more difficult to quantify. Let us consider, for example, an aquifer which appears to be of uniform conductivity. Parameter error quantifies the error in estimating this conductivity. Modeling error, on the other hand, includes elusive factors such as missing a meandering channel somewhere in the aquifer. This, in essence, is the difficulty in determining modeling error; parameter error can be roughly quantified based on measurements if one assumes that the model is correct, but modeling error is expected to represent all that the measurements and/or the modeler fail to capture. To evaluate model error, the perfect model needs to be known, but this is never possible.
Conference papers on the topic "Extender based forcing"
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Rahmati, M. T., L. He, and R. G. Wells. "Interface Treatment for Harmonic Solution in Multi-Row Aeromechanic Analysis." In ASME Turbo Expo 2010: Power for Land, Sea, and Air. ASMEDC, 2010. http://dx.doi.org/10.1115/gt2010-23376.
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In the present work, an interface treatment method has been developed to extend a non-linear harmonic solution method for aeromechanic analysis in multiple blade row configurations. The main emphasis is two folds. Firstly the method will enable efficient and accurate analysis of blade aerodynamic forcing and damping characteristics under the influence of adjacent bladerows. Secondly, it will warrant that the same steady/mean base flow as defined by the aerothermal designers is used consistently in the aeromechanical analysis. For passing the harmonic disturbance across a rotor-stator interface, the present method employs the time series reconstruction in conjunction with a temporal Fourier transform. On the other hand, the consistency with a conventional steady multistage solution for the base flow is achieved by applying an extended mixing plane method. The validity and effectiveness of the method are examined and demonstrated.
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Giersch, Thomas, Peter Hönisch, Bernd Beirow, and Arnold Kühhorn. "Forced Response Analyses of Mistuned Radial Inflow Turbines." In ASME Turbo Expo 2012: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/gt2012-69556.
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Radial turbine wheels designed as blade integrated disks (blisk) are widely used in various industrial applications. However, related to the introduction of exhaust gas turbochargers in the field of small and medium sized engines a sustainable demand for radial turbine wheels has come along. Despite those blisks are state of the art, a number of fundamental problems, mainly referred to fluid-structure-interaction and therefore to the vibration behavior, have been reported. Aiming to achieve an enhanced understanding of fluid-structure-interaction in radial turbine wheels a numerical method, able to predict forced responses of mistuned blisks due to aerodynamic excitation, is presented. In a first step the unsteady aerodynamic forcing is determined by modeling the spiral casing, the stator vanes and the rotor blades of the entire turbine stage. In a second step the aerodynamic damping induced by blade vibration is computed using a harmonic balance technique. The structure itself is represented by a reduced order model being extended by aerodynamic damping effects and aerodynamic forcings. Mistuning is introduced by adjusting the modal stiffness matrix based on results of blade by blade measurements that have been performed at rest. In order to verify the numerical method, the results are compared with strain-gauge data obtained during rig-tests. As a result a measured low engine order excitation was found by modeling the spiral casing. Furthermore a localization phenomenon due to frequency mistuning could be proven. The predicted amplitudes are close to measured data.
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Bredmose, Henrik, Signe Schløer, and Bo Terp Paulsen. "Higher-Harmonic Response of a Slender Cantilever Beam to Fully Nonlinear Regular Wave Forcing." In ASME 2012 31st International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2012. http://dx.doi.org/10.1115/omae2012-83475.
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The higher-harmonic response of a vertical cantilever beam to fully nonlinear wave loads is investigated. Such responses are also known as ‘ringing’ and is of practical interest in the context of offshore wind turbine foundations which, in contrast to the classical incidents of ringing at deep water, are placed at intermediate or shallow water. The purpose of the study is to provide generic results which can be of later use for the interpretation of more complex cases of irregular wave forcing. To this end, the problem parameters are defined and reduced by dimensional analysis. A simple numerical model is proposed, based on linear beam theory and fully nonlinear regular stream function waves. The hydrodynamic forces are determined from the extended Morison equation. Parametric studies of the response dependence to relative forcing period, wave height and depth are presented and discussed. A central finding of the paper is that for waves of 85% maximum height, the third-harmonic response increases substantially when the depth is reduced from deep-water conditions into intermediate depth.
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Tammisola, Outi, and Matthew P. Juniper. "Adjoint Sensitivity Analysis of Hydrodynamic Stability in a Gas Turbine Fuel Injector." In ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/gt2015-42736.
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Hydrodynamic oscillations in gas turbine fuel injectors help to mix the fuel and air but can also contribute to thermoacoustic instability. Small changes to some parts of a fuel injector greatly affect the frequency and amplitude of these oscillations. These regions can be identified efficiently with adjoint-based sensitivity analysis. This is a linear technique that identifies the region of the flow that causes the oscillation, the regions of the flow that are most sensitive to external forcing, and the regions of the flow that, when altered, have most influence on the oscillation. In this paper, we extend this to the flow from a gas turbine’s single stream radial swirler, which has been extensively studied experimentally (GT2008-50278) [8]. The swirling annular flow enters the combustion chamber and expands to the chamber walls, forming a conical recirculation zone along the centreline and an annular recirculation zone in the upstream corner. In this study, the steady base flow and the stability analysis are calculated at Re 200–3800 based on the mean flow velocity and inlet diameter. The velocity field is similar to that found from experiments and LES, and the local stability results are close to those at higher Re (GT2012-68253) [11]. All the analyses (experiments, LES, uRANS, local stability, and the global stability in this paper) show that a helical motion develops around the central recirculation zone. This develops into a precessing vortex core. The adjoint-based sensitivity analysis reveals that the frequency and growth rate of the oscillation is dictated by conditions just upstream of the central recirculation zone (the wavemaker region). It also reveals that this oscillation is very receptive to forcing at the sharp edges of the injector. In practical situations, this forcing could arise from an impinging acoustic wave, showing that these edges could be influential in the feedback mechanism that causes thermoacoustic instability. The analysis also shows how the growth rate and frequency of the oscillation change with either small shape changes of the nozzle, or additional suction or blowing at the walls of the injector. It reveals that the oscillations originate in a very localized region at the entry to the combustion chamber, which lies near the separation point at the outer inlet, and extends to the outlet of the inner pipe. Any scheme designed to control the frequency and amplitude of the oscillation only needs to change the flow in this localized region.
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Beuth, Jan Paul, Jakob G. R. von Saldern, Thomas Ludwig Kaiser, Thoralf G. Reichel, Christian Oliver Paschereit, Bernhard Ćosić, and Kilian Oberleithner. "Flow Response of an Industrial Gas Turbine Combustor To Acoustic Forcing Extracted From Unforced Data." In ASME Turbo Expo 2021: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/gt2021-59718.
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Abstract Gas turbine combustors are commonly operated with lean premix flames, allowing for high efficiencies and low emissions. These operating conditions are susceptible to thermoacoustic pulsations, originating from acoustic-flame coupling. To reveal this coupling, experiments or simulations of acoustically forced combustion systems are necessary, which are very challenging for real-scale applications. In this work we investigate the possibility to determine the flame response to acoustic forcing from snapshots of the unforced flow. This approach is based on three central hypothesis: first, the flame response is driven by flow fluctuations, second, these flow fluctuations are dominated by coherent structures driven by hydrodynamic instabilities, and third, these instabilities are driven by stochastic forcing of the background turbulence. As a consequence the dynamics in the natural flow should be low-rank and very similar to those of the acoustically forced system. In this work, the methodology is applied to experimental data of an industry-scale swirl combustor. A resolvent analysis is conducted based on the linearized Navier-Stokes equations to assure analytically the low-rank behavior of the flow dynamics. Then, these dynamics are extracted from flow snapshots using spectral proper orthogonal decomposition (SPOD). The extended SPOD is applied to determine the heat release rate fluctuations that are correlated with the flow dynamics. The low-rank flow and flame dynamics determined from the analytic and data-driven approach are then compared to the flow response determined from a classic phase average of the acoustically forced flow, which allow the research hypothesis to be evaluated. It is concluded that for the present combustor, the flow and flame dynamics are low-rank for a wider frequency range and the response to harmonic forcing can be determined quite accurately from unforced snapshots. The methodology further allows to isolate the frequency range where the flame response is predominantly driven by hydrodynamic instabilites.
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Redkar, Sangram, and S. C. Sinha. "Order Reduction of Nonlinear Systems With Periodic-Quasiperiodic Coefficients." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-85306.
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In this work, some techniques for order reduction of nonlinear systems involving periodic/quasiperiodic coefficients are presented. The periodicity of the linear terms is assumed non-commensurate with the periodicity of either the nonlinear terms or the forcing vector. The dynamical evolution equations are transformed using the Lyapunov-Floquet (L-F) transformation such that the linear parts of the resulting equations become time-invariant while the nonlinear parts and forcing take the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-modulated functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states. Three methods are proposed to carry out this model order reduction (MOR). First type of MOR technique is a linear method similar to the ‘Guyan reduction’, the second technique is a nonlinear projection method based on singular perturbation while the third method utilizes the concept of ‘quasiperiodic invariant manifold’. Order reduction approach based on invariant manifold technique yields a unique ‘generalized reducibility condition’. If this ‘reducibility condition’ is satisfied only then an accurate order reduction via invariant manifold is possible. Next, the proposed methodologies are extended to solve the forced problem. All order reduction approaches except the invariant manifold technique can be applied in a straightforward way. The invariant manifold formulation is modified to take into account the effects of forcing and nonlinear coupling. This approach not only yields accurate reduced order models but also explains the consequences of various ‘primary’ and ‘secondary resonances’ present in the system. One can also recover all ‘resonance conditions’ obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handing systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. These methodologies are applied to some typical problems and results for large-scale and reduced order models are compared. It is anticipated that these techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems.
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Yao, Ming-Hui, Wei Zhang, Xiang-Ying Guo, and Dong-Xing Cao. "Multi-Pulse Orbits With a Melnikov Method and Chaotic Dynamics in Motion of Composite Laminated Rectangular Thin Plate." In ASME 2008 Conference on Smart Materials, Adaptive Structures and Intelligent Systems. ASMEDC, 2008. http://dx.doi.org/10.1115/smasis2008-559.
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This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s three-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial differential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization approach, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the extended Melnikov method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation. The results of numerical simulation also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.
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Straatman, A. G., N. C. Gallego, Q. Yu, and B. E. Thompson. "Characterization of Porous Carbon Foam as a Material for Compact Recuperators." In ASME Turbo Expo 2006: Power for Land, Sea, and Air. ASMEDC, 2006. http://dx.doi.org/10.1115/gt2006-90598.
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Experiments are presented to quantify the convective heat transfer and the hydrodynamic loss that is obtained by forcing water through blocks of porous carbon foam (PCF) heated from one side. The experiments were conducted in a small-scale water tunnel instrumented to measure the pressure drop and the temperature rise of the water passing through the blocks and the base temperature and heat flux into the foam block. In comparison to similar porosity aluminum foam, the present results indicate that the pressure drop across the porous carbon foam is higher due to the large hydrodynamic loss associated with the cell windows connecting the pores, but the heat transfer performance suggests that there may be a significant advantage to using PCF over aluminum foam for extended surface convection elements in recuperators and electronic cooling devices.
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Iklódi, Zsolt, Xavier Beudaert, and Zoltan Dombovari. "On the Modelling Bases of In-Motion Dynamic Characterization of Flexible Structures Subject to Friction and Position Control Delay." In ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/detc2022-90924.
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Abstract This article presents a characterisation technique of in-motion machine dynamics based on the principles of numerical continuation. A linear two degree of freedom mechanical model is considered, representing e.g. a flexible moving column of a machine tool, and is subjected to a non-smooth friction and a delayed feedback drive control force, resulting in a model governed by a system of piecewise-smooth delay differential equations. By applying harmonic forcing to the system, periodic solutions can be found, through the continuation of which, an accurate vibratory characterisation of in-motion machine dynamics can be acquired. In the continuation routine, spectral collocation techniques were employed to formulate the discretized boundary value problem of piecewise-smooth periodic orbits, and the pseudo-arclength method was implemented. Special care was attributed to the detection of grazing and sliding bifurcations, and the continuation routine was also extended to allow continuation of these critical points.
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Pesheck, E., and C. Pierre. "A Global Methodology for the Modal Reduction of Large Nonlinear Systems Containing Quadratic and Cubic Nonlinearities." In ASME 1997 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1997. http://dx.doi.org/10.1115/detc97/vib-3952.
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Abstract A methodology is presented for the systematic modal reduction of structural systems which contain quadratic and cubic nonlinearities in displacement. The procedure is based on the center manifold approach for describing individual nonlinear modes, but it has been extended to account for simultaneous motion within several chosen modal coordinates. Motions of the reduced system are constrained to lie on high-dimensional manifolds within the phase space of the original system. Polynomial approximations of these manifolds are obtained through third order for arbitrary system parameters. Algorithms have been developed for automation of this procedure, and they are applied to an example system. Free and forced responses of the reduced system are discussed and compared to responses reduced through simple modal truncation. A more rigorous treatment of harmonic forcing is proposed, which will allow for the production of high-dimensional, time-dependent manifolds through a simple adaptation of the unforced procedure.