Considerations to Rayleigh’s hypothesis

In 1907 Lord Rayleigh published a paper on the dynamic theory of gratings. In this paper he presented a rigorous approach for solving plane wave scattering on periodic surfaces. Moreover he derived explicit expressions for a perfectly conducting sinusoidal surface, and for perpendicular incidence of the electromagnetic plane wave. This paper was criticized by Lippmann in 1953 for he assumed Rayleigh’s approach to be incomplete. Since this time there have been published several arguments, proofs, and discussions concerning the correctness and the range of validity of Rayleigh’s approach not only for plane wave scattering on gratings but also for light scattering on nonspherical structures, in general. In the paper at hand we will discuss the different point of views on what is called “Rayleigh’s hypothesis” as well as the relevance of a found theoretical limit for its validity. Furthermore we present a numerical treatment of the original scattering problem of a p-polarized plane wave perpendicularly incident on a perfectly conducting sinusoidal surface (i.e., the scalar Dirichlet problem). In doing so we emphasizes the near-field solution especially within the grooves of the grating up to points on the surface, and below the surface. Two different Green’s function formulations of Huygens’ principle are used as starting points. One of this formulation results in the general T-matrix approach which is considered to be affected by Rayleigh’s hypothesis especially for near-field calculations. The other formulation provides a conventional boundary integral equation which is in accordance with Lippmann’s point of view and free of problems with Rayleigh’s hypothesis. But the obtained results show that Lippmann’s argumentation do not withstand a critical numerical analysis, and that the independence of least-squares approaches from Rayleigh’s hypothesis, as understood and proven by Millar, seems to hold also for certain methods which does not fit into such an approach.     

Dynamics of a Flexible Slider-Crank Mechanism Driven by a Non-Ideal Source of Energy

The dynamic response of a slider crank mechanism with a flexible connecting rod driven by an electric DC motor is examined. After formulating the governing nonlinear boundary value problem, it is reduced by a one-term truncation to a system of nonlinear ordinary differential equations representing the complete dynamics of the electromechanical system. First, the steady-state behavior is analyzed. It turns out that a constant crank rotational speed is not possible: fluctuations appear which can be limited by an appropriate choice of the system and the engine data. Then, the transient startup or rundown is dealt with by using a digital simulation of the model equations. It exhibits a rich variety of dynamic effects till the stationary speed is reached. Attention is focused on the influence of the flexibility which in both categories of motion yields typical features.     

Non-smooth and time-varying systems of geometrically nonlinear beams

Bending vibrations of geometrically nonlinear beams, which are connected with some clearance in their contact areas, are analyzed during dynamic extending and retracting motion of the different segments. For the physical model of a fork lifter, as an example of application, the governing system equations are derived by applying Hamilton's principle. Using a discretization procedure, based on admissible shape functions, a system of coupled, nonlinear, time-varying, ordinary differential equations is generated. Linearization and model reduction leads to a sequence of simple models. On the basis of these models, an adaptive state regulator and an adaptive full-state observer (Luenberger Observer) are designed for vibration suppression using the optimal linear quadratic regulator (LQR). The adaptive controller and observer are applied to the original, significantly more complicated, geometrically nonlinear and time-varying system with clearance so that the robustness of the controlled system can be studied during dynamic extending and retracting motions.     

Kinetische Stabilitätstheorie elastischer Stäbe unter Beachtung nichtlinearer Krümmungen und Torsion

Übersicht Unter Berücksichtigung nichtlinearer Terme sowohl in den Krümmungen als auch in der Torsion wird mit Hilfe des Prinzips von Hamilton das beschreibende Randwertproblem für kinetisches Kippen elastischer, querbelasteter Stäbe formuliert. Mittels der Methode der kleinen Schwingungen werden die zugehörigen Stabilitätsgleichungen hergeleitet, die bisher bekannte Beziehungen ergänzen. Das Beispiel des einfachsten Biege-Kipp-Problems illustriert die verbesserte Wirkung der vorgeschlagenen Gleichungen.

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Summary Considering non-linear terms in the curvatures as well as in the twist the governing boundary value problem for lateral bending of elastic, transverse loaded rods are formulated by means of Hamilton's principle. Using the method of small vibrations the associated linearized equations of stability are derived, which complete the currently accepted relations. The example of the simplest lateral bending problem illustrate the improved effect of the proposed equations.

     

Effects of Damping Mechanisms in Free and Forced Vibrations of Some Structural Members

For continuous vibrating systems, such as bars and beams, end-mounted in the environment, knowledge about the mass, damping and stiffness properties of the resonating environment is important for analyzing free and forced vibrations of such structural members which are also damped themselves. To finally get an identification of the clamping parameters, examinations of both vibrating structural members for various restraint conditions and dynamic interaction with viscoelastic halfspaces, etc., are required. As a first step, longitudinal bar vibrations are studied in detail. The method of separation of variables combined with the reformatted orthogonality relation, and numerical integration is applied for calculating the free and forced oscillations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Atomistic explanation of the Gough-Joule-effect

An interpretation of the Gough-Joule-Effect from an atomistic point of view is given by a special investigation of lattice vibrations using a non-interacting model. The calculations are restricted to crystals with ideal crystal structure built up of only one kind of particles. Approximate analytical formulas are derived. A comparison between the classical macroscopic theory and the here presented microscopic calculations is drawn. Received 5 April 2001 and Received in final form 16 August 2001