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1 
gExpectations with application to risk measuresOffwood, Sonja Carina 05 March 2013 (has links)
Programme in Advanced Mathematics of Finance,
University of the Witwatersrand,
Johannesburg. / Peng introduced a typical ltration consistent nonlinear expectation, called a
gexpectation in [40]. It satis es all properties of the classical mathematical expectation
besides the linearity. Peng's conditional gexpectation is a solution to a
backward stochastic di erential equation (BSDE) within the classical framework of
It^o's calculus, with terminal condition given at some xed time T. In addition, this
gexpectation is uniquely speci ed by a real function g satisfying certain properties.
Many properties of the gexpectation, which will be presented, follow from the speci
cation of this function. Martingales, super and submartingales have been de ned
in the nonlinear setting of gexpectations. Consequently, a nonlinear DoobMeyer
decomposition theorem was proved.
Applications of gexpectations in the mathematical nancial world have also
been of great interest. gExpectations have been applied to the pricing of contingent
claims in the nancial market, as well as to risk measures. Risk measures were
introduced to quantify the riskiness of any nancial position. They also give an indication
as to which positions carry an acceptable amount of risk and which positions
do not. Coherent risk measures and convex risk measures will be examined. These
risk measures were extended into a nonlinear setting using the gexpectation. In
many cases due to intermediate cash
ows, we want to work with a multiperiod, dynamic
risk measure. Conditional gexpectations were then used to extend dynamic
risk measures into the nonlinear setting.
The Choquet expectation, introduced by Gustave Choquet, is another nonlinear
expectation. An interesting question which is examined, is whether there are
incidences when the gexpectation and the Choquet expectation coincide.

2 
Nonlinear output regulation of multivariable systems with some applications. / CUHK electronic theses & dissertations collectionJanuary 2012 (has links)
本文研究了多变量系统的非线性输出调节问题，并处理了机电和机械控制系统中的两个典型控制问题。 / 输出调节问题是控制中的核心问题之一。它同时处理轨迹跟踪和干扰抑制问题。参考输入和干扰是通过一定的动态系统产生，称为外部系统。因此输出调节问题和其它轨迹跟踪问题的区别在于它允许跟踪一类参考信号，而不需要精确的知道参考信号的值。而且，输出调节问题可应用于真实世界的很多控制问题，包括能源系统，交通和系统生物学。 / 众所周知，处理输出调节问题的基本框架包含两个步骤。首先把被控对象的鲁棒输 出调节问题转化成由被控对象和叫做内模的动态补偿器组成的增广系统的鲁棒镇定问题，然后鲁棒镇定增广系统。问题的关键在于如何设计一个合适的内模。这个内模不仅可以产生所有的稳态信息，而且使得增广系统可镇定。 / 虽然在过去的二十几年里关于非线性输出调节问题的研究取得了重大进展，大部分的结果只处理了单输入单输出的非线性系统。多变量非线性系统非线性输出调节问题有关的结果甚少。两大难点阻碍了多变量非线性系统非线性输出调节问题的进一步发展。一方面，由于被控对象的复杂结构，很难构造出一个合适的内模。这个内模既可以产生必需的稳态信息，又可以保证增广系统的镇定问题可解。另一方面，由于内模的引入，增广系统变成一个更加复杂的多输入多输出的非线性系统，其中既包含了动态不确定性，也包含了时变的静态不确定性。这个增广系统具有一定特殊的结构，并且从未出现在现有的文献中。增广系统的镇定问题很有挑战性。此外，关于实际系统非线性输出调节问题的研究甚少。因为输出调节方法对于未知的对象参数具有一定的鲁棒性，而且允许外部系统含有未知参数，通过输出调节方法我们可能得出更好的结果。基于以上的动机，我们将考虑具有特定结构的多变量系统的全局鲁棒输出调节问题，并解决两个典型控制问题，包括永磁同步电机的速度跟踪控制和球形倒立摆的轨迹跟踪控制。本文的贡献总结如下。 / 1. 本文考虑了一类多变量非线性系统的全周鲁棒输出调节问题。通过把内模附加到被控对象， 增广系统具有」定特殊的结构。它的镇定问题尚未有人处理过。我们把增广系统的镇定问题分解为多个单输入系统的镇定问题。然后利用changingsupply function 技术，我们通过法代的方法求解多个单输入控制问题，从而解决了这一镇定问题。理论结果可以解决面装式永磁同步电机的速度跟踪和负载干扰抑制问题。与现有的结果比较，我们的设计允许所有的电机参数未知。 / 2. 本文考虑了未知外部系统作用下的面装式永磁同步电机的速度跟踪和负载干扰抑制问题。由于外部系统未知，传统的鲁棒控制方法不能处理这一问题。而且，外部系统中的未知参数使得增广系统的镇定问题更具挑战性。因此我们需要把鲁棒控制和自适应控制相结合处理这一问题。利用changing supply function 技术和动态坐标变换技术相结合，我们用迭代的方法求解了两个单输入控制问题，进而解决了这一问题。 / 3. 本文考虑了用内模设计的方法处理永磁同步电机的速度跟踪和负载干扰抑制问题。当dq 轴电感相等时，这一问题己经得到了广泛的研究。然而dq 轴电感不等时，由于转速方程中的复杂非线性，这一问题的研究甚少。利用输出调节方法，电机参数可以具有更好的鲁棒性。由于增广系统不具有任何己知的特殊结构，我们研究出一种具体的工具来处理镇定问题。具体的说，我们研究出一种适用于nonISS (输入到状态稳定)系统的广义changing supply function 技术。通过把这个技术与一个特殊的非线性内模相结合，这一问题得到了很好的解决。 / 4. 本文给出了一种新的方法用于处理球形倒立摆的近似输出调节问题。这一方法基于神经网络近似调节器方程的解。求解调节器方程近似解的问题被转化成一个参数优化问题，进而有助于计算机实现的方便。而且，与现有结果中的基于三阶多项式的控制比较，对于参考输入信号的不同幅值的情况，这一方法的稳态跟踪误差保持了更加平稳的性能。 / 5. 本文给出了一种新的方法用于处理球形倒立摆的近似输出调节问题。这一方法是基于神经网络的增强型设计，并采用输出反馈控制。与我们之前工作中的神经网络设计比较，我们给出了一种新的方法用于近似前馈控制，而不是调节器方程的解。求解前馈控制近似解的问题被转化成一个参数优化问题。我们采用HookeJ eeeves 方法和Powell 方法解决了参数优化问题。当所有的状态可用的时候，我们的设计对于被控对象参数的变化具有一定的鲁棒性。 / In this thesis, we investigate nonlinear output regulation problem of multivariable systems and their applications to two typical control problems in electromechanical and mechanical control systems. / The output regulation problem is one of the central problems in control. It deals with trajectory tracking and disturbance rejection problem simultaneously. Both the reference inputs and disturbances are generated by some dynamical system called exosystem. Thus it is in contrast with some other traj ectory tracking problems in that it allows tracking of a family of reference signals which does not have to be known exactly. Moreover, it can be applied to many control problems in real world including energy systems, transportation and system biology. / It is well known that the general framework for tackling the output regulation problem consists of two steps. The first step is to convert the robust output regulation problem for the given plant into a robust stabilization problem for the socalled augmented system composed of the given plant and a dynamic compensator called internal model, and the second step aims to robustly stabilize the augmented system. The key issue lies in how to design a suitable internal model which can generate all the steadystate information and render the augmented system to be stabilizable. / Although the research on the nonlinear output regulation has made significant progress for over two decades, most of the results only deals with singleinput, singleoutput nonlinear systems. There are few results on nonlinear output regulation of multivariable nonlinear systems. Two challenges hinder the further development of nonlinear output regulation of multivariable nonlinear systems. On the one hand, due to the complicated structure of the system, it is difficult to construct a suitable internal model which can generate the necessary steadystate information while ensuring the solvability of stabilization problem of the augmented system. On the other hand, due to the introduction of the internal model, the augmented system becomes a more complicated multiinput, multioutput nonlinear system containing both dynamic uncertainty and timevarying static uncertainty. Such augmented system takes certain special structure which has never been encountered before. The stabilization problem of such augmented system is very challenging. Moreover, t he research on the nonlinear output regulation of practical systems is very rare. Because output regulation approach offers certain robust property to unknown plant parameters and also allows unknown parameters in exosystem, better results may be achieved by using output regulation approach. Therefore we are motivated to consider global robust output regulation problem of multivariable systems with certain structures and solve two typical control problems including speed tracking control of permanent magnet synchronous motor and trajectory tracking control of spherical invert d pendulum. The main contributions of the thesis are summarized as follows. / 1. The global robust output regulation problem for a class of multivariable nonlinear systems is considered. By attaching the internal model to the given plant, the augmented system takes certain special structure and the stabilization problem of such a system has never been handled. We decompose the stabilization problem of the augmented system into the stabilization problem of several singleinput systems. Then we solve the problem by solving several singleinput control problems via a recursive approach utilizing the changing supply function technique. The theoretical result is applied to the speed tracking control and load torque disturbance rejection problem of a surfacemounted PM synchronous motor. Compared with existing results, our design allows all the motor parameters to be uncertain. / 2. A speed tracking and load torque disturbance rejection problem of surfacemounted PM synchronous motor subject to an unknown exosystem is considered. Because the exosystem is unknown, traditional robust control approach cannot be used to tackle this problem. Moreover, the unknown parameters in the exosystem make the stabilization problem of augmented system even more challenging. Therefore we need to incorporate robust control with adaptive control to t ackle the problem. We solve the problem by solving two singleinput control problems via a recursive approach utilizing the changing supply function and dynamic coordinate transformation techniques. / 3. A speed tracking and load torque disturbance rejection problem of PM synchronous motor by internal model design is considered. When dq axes inductances are equal, the problem has been extensively studied. However, when dq axes inductances are not equal, such problem has received little attention for the complicated nonlinearity in speed equation. By using output regulation approach, a better robust property with respect to motor parameters can be achieved. As the augmented system does not take any known special form, we develop a specific tool to deal with the stabilization problem. In particular, a generalized changing supply function technique applicable to nonISS (inputtostate stable) systems is developed. This technique, in conjunction with a particular nonlinear internal model leads to an effective solution to the problem. / 4. An alternative approach to the approximate output regulation problem of spherical inverted pendulum is considered. This method is based on a neural network approximation of the solution of the regulator equations. The problem of seeking approximate solution of the regulator equations has been converted into a parameter optimization problem which has lent itself to a convenient implementation in computer. Moreover, compared with a third order polynomial based control in existing results, the steadystate tracking errors of the method maintains a more uniform performance over different amplitudes of the reference input signals. / 5. An alternative approach to the approximate output regulation problem of spherical inverted pendulum is considered. This method has been based on the neural network enhanced design with output feedb ack control. Different from the neural network design in our previous work, we present a novel method to approximate the feedforward control instead of the solution of regulator equations. The problem of seeking approximate solutions of feedforward control is converted into a parameter optimization problem. We solve the parameter optimization problem by both HookeJeeves method and Powell method. When all the states are available, our design also offers certain robustness to plant parameter variations. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Ping, Zhaowu. / "October 2011." / Thesis (Ph.D.)Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 160170). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese. / Abstract  p.i / Acknowledgement  p.VI / Chapter 1  Introduction  p.1 / Chapter 1.1  Literature Overview  p.1 / Chapter 1.2  Contribution of Thesis  p.7 / Chapter 1.3  Organization of Thesis  p.9 / Chapter 2  Background and Preliminaries  p.10 / Chapter 2.1  Fundamentals of Nonlinear Systems  p.10 / Chapter 2.1.1  Lyapunov Stability  p.11 / Chapter 2.1.2  Inputtostate Stability  p.13 / Chapter 2.2  Framework of Robust Output Regulation  p.15 / Chapter 2.3  Stabilization of Nonlinear Systems  p.19 / Chapter 2.3.1  Technical Lemmas for ISS Systems  p.20 / Chapter 2.3.2  Technical Lemmas for NonISS Systems  p.23 / Chapter 2.4  Feedforward Control of Nonlinear Output Regulation  p.28 / Chapter 2.4.1  Polynomial Based Approximation Method  p.31 / Chapter 2.4.2  Neural Network Based Approximation Method  p.32 / Chapter 3  Global Robust Output Regulation for a Class of Multivariable Systems  p.39 / Chapter 3.1  Introduction  p.40 / Chapter 3.2  Preliminaries  p.42 / Chapter 3.3  Main Results  p.48 / Chapter 3.4  Application to Speed Control of SurfaceMounted PM Synchronous Motor  p.56 / Chapter 3.5  Conclusion  p.71 / Chapter 4  Speed Tracking Control of SurfaceMounted PM Synchronous Motor Subject to an Unknown Exosystem  p.72 / Chapter 4.1  Problem Formulation and Preliminaries  p.73 / Chapter 4.2  Solvability of The Motor Control Problem  p.79 / Chapter 4.3  Evaluation of The Control Law  p.87 / Chapter 4.4  Conclusion  p.99 / Chapter 5  Speed Tracking Control of PM Synchronous Motor by Internal Model Design  p.100 / Chapter 5.1  Problem Formulation and Preliminaries  p.101 / Chapter 5.2  Solvability of the Motor Control Problem  p.107 / Chapter 5.3  Evaluation of the Control Law  p.110 / Chapter 5.4  Conclusion  p.119 / Chapter 6  Approximate Output Regulation of Spherical Inverted Pendulum by Neural Network Control  p.120 / Chapter 6.1  Introduction  p.121 / Chapter 6.2  Neural Network Based Approximate Solution of the Regulator Equations  p.124 / Chapter 6.3  Approximate Control Law  p.131 / Chapter 6.4  Conclusion  p.135 / Chapter 7  Neural Network Enhanced Output Regulation of Spherical Inverted Pendulum  p.136 / Chapter 7.1  Neural Network Based Output Feedback Control Law  p.137 / Chapter 7.1.1  HookeJ eeves Method  p.139 / Chapter 7.1.2  Powell Method  p.143 / Chapter 7.2  Robustness Analysis  p.149 / Chapter 7.2.1  HookeJeeves Method  p.149 / Chapter 7.2.2  Powell Method  p.153 / Chapter 7.3  Conclusion  p.156 / Chapter 8  Conclusions  p.157 / Bibliography  p.160 / Biography  p.171

3 
Degree theory in nonlinear functional analysis.Pillay, Paranjothi. 21 October 2013 (has links)
The objective of this dissertation is to expand on the proofs and concepts of Degree Theory, dealt with in chapters 1 and 2 of Deimling [28], to make it more readable and
accessible to anyone who is interested in the field. Chapter 1 is an introduction and contains the basic requirements for the subsequent chapters. The remaining chapters aim at defining a llvalued map D (the degree) on the set M = {(F, Ω, y) / Ω C X open, F : Ὠ → X, y ɇ F(∂Ω)} (each time, the elements of M satisfying extra conditions) that satisfies :
(D1) D(I, Ω, y) = 1 if y Є Ω.
(D2) D(F, Ω, y) = D(F, Ω1 , y) + D(F, Ω2, y) if Ω1 and Ω2 are disjoint open subsets of Ω o such that y ɇ F(Ὠ \ Ω1 U Ω2 ).
(D3) D(I  H(t, .), Ω, y(t)) is independent of t if H : J x Ὠ →X and y : J → X.
An important property that follows from these three properties is (D4) F1(y) ≠ Ø if D(F, Ω, y) ≠ 0.
This property ensures that equations of the form Fx = y have solutions if D(F, Ω, y) ≠ 0.
Another property that features in these chapters is the Borsuk property which gives us conditions under which the degree is odd and hence nonzero. / Thesis (M.Sc.)University of DurbanWestville, 1989.

4 
Nonlinear dynamics of composite plates and other physical systems /Nayfeh, Jamal Faris, January 1990 (has links)
Thesis (Ph. D.)Virginia Polytechnic Institute and State University, 1990. / Vita. Abstract. Includes bibliographical references (leaves 188200). Also available via the Internet.

5 
Investigations in structural optimization of nonlinear problems using the finite element methodSedaghati, Ramin 01 March 2018 (has links)
Structural optimization is an important field in engineering with a strong foundation on continuum mechanics, structural finite element analysis, computational techniques and optimization methods. Research in structural optimization of linear and geometrically nonlinear problems using the force method has not received appropriate attention by the research community.
The present thesis constitutes a comprehensive study in the area of structural optimization. Development of new methodologies for analysis and optimization and their integration in finite element computer programs for analysis and design of linear and nonlinear structural problems are among the most important contributions.
For linear problems, a force method formulation based on the complementary energy is proposed. Using this formulation, the element forces are obtained without the direct generation of the compatibility matrix. Application of the proposed method in structural size optimization under stress, displacement and frequency constraints has been investigated and its efficiency is compared with the conventional displacement formulation. Moreover, an efficient methodology based on the integrated force method is developed for topology optimization of adaptive structures under static and dynamic loads. It has been demonstrated that structural optimization based on the force method is computationally more efficient.
For nonlinear problems, an efficient methodology has been developed for structural optimization of geometrical nonlinear problems under system stability constraints. The technique combines the nonlinear finite element method based on the displacement control technique for analysis and optimality criterion methods for optimization. Application of the proposed methodology has been investigated for shallow structures. The efficiency of the proposed optimization algorithms are compared with the mathematical programming method based on the Sequential Quadratic Programming technique. It is shown that structural design optimization based on the linear analysis for structures with intrinsic geometric nonlinearites may lead to structural failure.
Finally, application of the group theoretic approach in structural optimization of geometrical nonlinear symmetric structures under system stability constraint has been investigated. It has been demonstrated that structural optimization of nonlinear symmetric structures using the group theoretic approach is computationally efficient and excellent agreement exists between the full space and the reduced subspace optimal solutions. / Graduate

6 
Nonlinear output regulation with timevarying or nonlinear exosystems. / CUHK electronic theses & dissertations collectionJanuary 2011 (has links)
In this thesis, we investigate the global robust output regulation problem for nonlinear systems subject to timevarying or nonlinear exosystems. / One of the crucial issues in output regulation problem is the design of the appropriate internal model. Internal model is a dynamical compensator which possesses an essential ability of generating all possible steadystate input information asymptotically, and it should not only lead to a welldefined augmented system but also ensure the stabilizability of the augmented system. Besides, stabilization techniques for the augmented system should also be carefully chosen to meet the needs in different scenarios, e.g. the timevarying settings. Efforts are put on both sides throughout the thesis. / Output regulation problem, also known as servomechanism problem, is one of the central topics in control theory. The control objective is to design a feedback control law for the given plant so as to achieve asymptotic tracking for a class of reference signals and asymptotic rejection for a class of disturbance signals while maintaining the stability of closedloop system. The reference or the disturbance signals are assumed to be generated from a dynamical system called the exosystem. Normally, the exosystem is a linear autonomous system, e.g. a harmonic oscillator, and the exogenous signals represent step or ramp signals, or sinusoidal signals contains finite number of harmonics. The extensions of the exosystem, from linear to nonlinear, autonomous to nonautonomous, significantly enlarge the categories of the exogenous signals, and more importantly, such extensions motivate the development of the output regulation theory in both scientific research and practical application. / Paying special attention to the appearance of timevarying or nonlinear exosystems, our research is mainly conducted under the general framework for tackling the output regulation problem. In general, first we convert the output regulation problem of the original plant into the stabilization problem of the augmented system which is composed of the plant and the designed internal model. Second, we achieve the global stabilization of the augmented system by robust and adaptive control approaches, according to both parameter uncertainty and dynamic uncertainty in either plant or the exosystem. / The main contributions of the thesis are outlined as follows. 1. A framework for handling the robust output regulation problem for general timevarying nonlinear systems subject to timevarying exosystem is proposed. Especially, certain existence conditions of a timevarying internal model is given, and problem conversion can be achieved. As an application of this framework, we give the solvability conditions of the output regulation problem for the timevarying nonlinear systems in output feedback form. Further, when parameter uncertainties occurred in the timevarying exosystem, we solve the corresponding adaptive robust output regulation problem resorting to some adaptive control methods. These results can also be applied to the timevarying nonlinear systems in lower triangular form. 2. The global robust output regulation problem for nonlinear systems subject to nonlinear exosystem is considered. A new class of internal models is introduced which relaxes the existence conditions of the former one. Also, this class of internal models has the merit that it is zero input globally asymptotically stable which greatly facilitates the global stabilization of the augmented system. Compared with the existing results, the new method solves the global robust output regulation problem without restrictions on the initial conditions or trajectory bounds of the exosystems, and the bound of the parameter uncertainties of the plant is not necessarily known. Moreover, utilizing the Nussbaum gain technique, the unknown control direction case can also be handled by modifying the control law. 3. The theoretical results have been applied to several practical control problems, such as the global disturbance rejection problem for FitzHughNagumo model with Mathieu equation, the synchoniztion of periodicallyforced pendulum with Rayleigh equation, etc .. / Yang, Xi. / Adviser: Jie Huang. / Source: Dissertation Abstracts International, Volume: 7304, Section: B, page: . / Thesis (Ph.D.)Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 123134). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [201] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

7 
Visualization and analysis of electrodynamic behavior during cardiac arrhythmiasBray, MarkAnthony P. January 2003 (has links)
Thesis (Ph. D. in Biomedical Engineering)Vanderbilt University, 2003. / Title from PDF title screen. Includes bibliographical references.

8 
An idempotentanalytic ISS small gain theorem with applications to complex process modelsPotrykus, Henry George 28 August 2008 (has links)
Not available / text

9 
An idempotentanalytic ISS small gain theorem with applications to complex process modelsPotrykus, Henry George. January 2002 (has links) (PDF)
Thesis (Ph. D.)University of Texas at Austin, 2002. / Vita. Includes bibliographical references. Available also from UMI Company.

10 
Dynamics of geometrically nonlinear sliding beamsBehdinan, Kamran 31 July 2018 (has links)
The elastodynamics of flexible frame structures is of interest in many areas of engineering. In certain structural systems the deflections can be large enough to warrant a nonlinear analysis. For example, offshore structures, long suspension bridges and other relatively slender structures used in space applications require a geometrically nonlinear analysis. In addition, if the structure has deployable elements, as in some space structures, the required analysis becomes even more complex. Typical examples are spacecraft antennae, radio telescopes, solar panels and spacebased manipulators with deployable elements.
The main objective of the present work is to formulate the problem of sliding beams undergoing large rotations and small strains. Further we aim to develop efficient finite element technique for analysis of such complex systems. Finally we wish to examine the nature of the motion of sliding beams and point out its salient features.
We start with two well known approaches in the nonlinear finite element static analysis of highly flexible structures, namely, the updated Lagrangian and the consistent corotational methods and extend these techniques to dynamic analysis of geometrically nonlinear beam structures. We analyse several examples by the same methods and compare the performance of each for efficiency and accuracy.
Next, using McIver's extension of Hamilton's principle, we formulate the problem of geometrically flexible sliding beams by two different approaches. In the first the beam slides through a fixed rigid channel with a prescribed sliding motion. In this formulation which we refer to as the sliding beam formulation, the material points on the beam slide relative to a fixed channel. In the second formulation the material points on the fixed beam are observed by a moving observer on a sliding channel and the beam is axially at rest. The governing equations of motion for the two formulations describe the same physical problem and by mapping both to a fixed domain, using proper transformations, we show that the two sets of governing equations become identical.
It is not, possible to find analytical solutions to our problem and we choose the Galerkin numerical method to obtain the transient response of the problem for the special case axially rigid beam. Next we follow a more elegant approach wherein we use the developed incremental nonlinear finite element approaches (the updated Lagrangian and the consistent corotational method) in conjunction with a variable time domain beam finite elements (where the number of elements is fixed and as mass enters the domain of interest, but the sizes of elements change in a prescribed manner in the undeformed configuration).
To verify the formulation and its computational implementation we analyse many examples and compare our findings with those reported in the literature when possible. We also use these illustrative examples to identify the importance of various terms such as axial flexibility and foreshortening effects. Finally we look into the problem of parametric resonance for the beam with periodically varying length and we show that the regions of stability obtained in the literature, using a linear analysis, do not hold when a more realistic nonlinear analysis is undertaken. / Graduate

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