Helmut J. Holl

The equations of Lagrange written for a non-material volume

SummaryThe Lagrange equations are extended with respect to a non-material volume which instantaneously coincides with some material volume of a continuous body. The surface of the non-material volume is allowed to move at a velocity which is different from the velocity of the material surface. The non-material volume thus represents an arbitrarily moving control volume in the terminology of fluid mechanics. The extension of the Lagrange equations to a control volume is derived by using the method of fictitious particles. Within a continuum mechanics based framework, it is assumed that, the instantaneous positions of both, the original particles included in the material volume, and the fictitious particles included in the control volume, are given as function of their positions in the respective reference configurations, of a set of time-dependent generalized coordinates, and of time. The corresonding spatial formulations are also assumed to be available. Imagining that the fictitious particles do transport the density of kinetic energy of the original particles, the partial derivatives of the total kinetic energy included in the material volume with respect to generalized coordinates and velocities are related to the respective partial derivatives of the total kinetic energy contained in the control volume. Hence follow the Lagrange equations for a control volume by substituting the above relations into the classical formulations for a material volume. In the present paper, holonomic problems are considered. The correction terms in the newly derived version of the Lagrange equations contain the flux of kinetic energy appearing to be transported through the surface of the control volume. This flux comes into the play in the form of properly formulated partial derivatives. Our version of the Lagrange equations is tested using the rocket equation and a folded falling string as illustrative examples.

The Rayleigh-Ritz Technique and the Lagrange Equations in Continuum Mechanics: Formulations for Material and Non-Material Volumes

In the present Lecture, we use the Rayleigh-Ritz technique in connection with the Lagrange equations in order to approximate the partial differential equations of continuum mechanics by means of a system of ordinary differential equations in time. We start from the local form of the equation of balance of momentum, from which we proceed to an extended Hamilton principle, eventually ending up with the Lagrange equations. We lay special emphasis upon the functional dependencies to be considered in the Rayleigh-Ritz technique with respect to the spatial and the material descriptions of continuum mechanics. In analogy to the notion of a material-time derivative, we define material variations and material partial derivatives of the quantities and entities that enter the Lagrange equations, and we relate these derivatives to local partial derivatives, the latter being particularly feasible with respect to the spatial formulation of continuum mechanics. Having formulated the Lagrange equations for the case of a material volume, we present a re-formulation for problems that are posed with respect to non-material volumes. As an example for such a problem, we shortly treat the coiling of a strip. The presented re-formulation of the Lagrange equations for non-material volumes represents an alternative derivation of recent results by Irschik and Holl (2002).

A Modal-Based Substructure Method Applied to Nonlinear Rotordynamic Systems

The discretisation of rotordynamic systems usually results in a high number of coordinates, so the computation of the solution of the equations of motion is very time consuming. An efficient semianalytic time-integration method combined with a substructure technique is given, which accounts for nonsymmetric matrices and local nonlinearities. The partitioning of the equation of motion into two substructures is performed. Symmetric and linear background systems are defined for each substructure. The excitation of the substructure comes from the given excitation force, the nonlinear restoring force, the induced force due to the gyroscopic and circulatory effects of the substructure under consideration and the coupling force of the substructures. The high effort for the analysis with complex numbers, which is necessary for nonsymmetric systems, is omitted. The solution is computed by means of an integral formulation. A suitable approximation for the unknown coordinates, which are involved in the coupling forces, has to be introduced and the integration results in Greens functions of the considered substructures. Modal analysis is performed for each linear and symmetric background system of the substructure. Modal reduction can be easily incorporated and the solution is calculated iteratively. The numerical behaviour of the algorithm is discussed and compared to other approximate methods of nonlinear structural dynamics for a benchmark problem and a representative example.

Eine effiziente Substrukturmethode für transiente Probleme der nichtlinearen Rotordynamik

Die Form der Bewegungsgleichungen in der Rotordynamik unterscheidet sich von anderen strukturdynamischen Problemen durch die Unsymmetrie in den Systemmatrizen und durch spezielle, meist lokale Nichtlinearitaten. Die Nichtlinearitaten bei rotordynamischen Problemen sind hauptsachlich mit dem Verhalten der hydrodynamischen Gleitlager und damit nur mit wenigen Freiheitsgrade verbunden.

On the extension of global vibration modes with Ritz-vectors needed for local effects

In Ritz-vector based model reduction techniques, the problem-oriented combination of different kind of Ritz-vectors may significantly influence the quality of the reduction base. It is common to combine global vibration modes with Ritz-vectors, which are necessary to characterize local effects. Even if the global vibration modes and the local Ritz-vectors may be separately orthogonal with respect to the mass and stiffness matrix, the combined reduced system is usually not decoupled. By using common decoupling strategies the separation of the two mode groups is lost. In this contribution, we will present a transformation procedure in order to obtain a combined and decoupled mode base, which is still separable into global vibration modes and Ritz-vectors due to local effects. Due to the clear separation of the two kinds of modes it is possible to give a frequency limit for the relevance of the inertia effects of the second mode group. In case the inertia effects of the second mode group may be neglected, the dimension of the differential equation of motion can be reduced once more again. At our theoretical considerations, an example is presented for the sake of illustration.

Analysis of Variable Mass Rotordynamic Systems with Semi-analytic Time-Integration

The analysed rotordynamic system is modeled as a non-linear variable mass system and represents a part of a production line where an axially moving material is coiled on a rotating drum. The suitable and accurate simulation of the vibrations in a coiling process is important to predict the vibrations during standard operation and for special non-steady operation conditions. Variable parameters are present and bending vibrations of the rotor with the coiling drum and the transversal oscillations of the elastic strip are coupled. The longitudinal and transversal motion of the axially moving strip and the bending deflection of the coiling drum are considered by Rayleigh-Ritz approximations which involve the application of the extended equation of Lagrange for open systems. Simulations are performed for a non-linear rotordynamic system for different operation conditions. The results computed with a semi-analytic time-integrations algorithm are shown.

LEAN PANEL BENDER – Einige mechanische Aspekte der Modellierung in Echtzeit für Produktion in Losgröße 1

ZusammenfassungNachfolgend werden einige wissenschaftliche Aspekte der Echtzeit-Materialerkennungsstrategie MAC diskutiert, die in der neuen Maschinenfamilie LEAN PANEL BENDER der Firma Salvagnini Maschinenbau realisiert ist und die eine hochpräzise und hocheffiziente Herstellung von komplex geformten Blechprodukten sowohl bei Losgröße 1 als auch in der Serie erlaubt.AbstractIn this paper, we discuss some scientific aspects of the real-time material detection strategy MAC, which is realized in the novel LEAN PANEL BENDER generation of the company Salvagnini Maschinenbau GmbH and which allows the production of complex metal devices in a highly efficient and accurate manner for both, single-slot production and production in series.

Berechnung von Schwingungen beim Aufwickeln eines Stahlbandes in einem Steckelwalzwerk

ZusammenfassungDie Verringerung von Schwingungen im Steckelwalzwerk während des Bearbeitungsprozesses sind für die Qualität des erzeugten Flachproduktes wichtig. Das abgeleitete mechanische Modell lässt den Einfluss von verschiedenen geometrischen und Betriebsparameter studieren. Es ist dafür notwendig, ein vereinfachtes und reduziertes mechanisches Modell für das nichtlineare dynamische und elastisch gekoppelte Mehrkörpersystems abzuleiten. Mit einer geeigneten Zeitintegrationsmethode werden die Lösungen für das dynamische System für den langen Zeitbereich des Wickelvorganges berechnet, wobei ein geeigneter Zeitschritt verwendet wurde, der eine konvergierte Lösung sicherstellt. Aus den Berechnungsergebnissen lassen sich die Einflüsse der Parameter und Betriebszustände auf die Schwingungen ermitteln und verbesserte Einstellungen für den Produktionsprozess finden. Weiters ist ein reduziertes Modell mit möglichst genauen Parametern notwendig, um einen Reglerentwurf durchzuführen.AbstractThe reduction of vibrations in a Steckel-Mill during the production process is important in order to produce high quality flat products. With the derived mechanical model, it is possible to study the influence of geometric and process parameters. It is necessary to get a reduced mechanical model for the non-linear dynamic and elastically coupled multi-body system. A suitable time-integration method computes the solution for the dynamic system for a rapid long-term coiling process with a time-step, which guarantees a converged solution. The analysis of the computation results shows the influence of the parameters and operation conditions to the vibration amplitudes and serves in finding improved parameters for the production process. Furthermore a reduced mechanical model with adjusted parameters allows the combination with a controller.

DEIM for the Efficient Computation of Contact Interface Stresses

The computational effort for the simulation of reduced order models containing contact stresses is determined by these nonlinear terms. Recent publications suggest the utilization of interpolation methods to overcome this bottleneck. The applicability of the Discrete Empirical Interpolation Method (DEIM) for the efficient computation of contact stresses is demonstrated. The modeling of a mechanical structure containing an interface using zero thickness elements is outlined first. This is followed by a reduction method using joint interface modes as extension to the well known Craig Bampton approach. The basic idea of interpolation methods and a summary of the applied DEIM algorithm is given. Finally the numerical example of a bolted cantilever is investigated for two loadcases and the results are discussed for different trial function bases. It is clearly shown that DEIM can be used to significantly improve the computational efficiency for this type of problems while keeping accuracy at an acceptable level.

Rechnergestützte nichtlineare Modellierung von Abkantautomaten

ZusammenfassungDieser Beitrag beschreibt computergestützte und messtechnische Untersuchungen, welche als eine wissenschaftliche Grundlage für die Analyse und Entwicklung einer neuen Generation von Abkantautomaten zur Produktion von kompliziert geformten Profilen aus ebenen Blechteilen dienen. Es erfolgt einerseits eine Modellbildung unter Verwendung der Mehrkörperdynamik, um das Verhalten von starren Maschinenteilen zu modellieren, andererseits werden elastische Komponenten über eine Diskretisierung mittels nichtlinearer Finite Elemente abgebildet. Für die Beschreibung der Blechumformung werden Formulierungen der Plastizitätstheorie und der Kontaktmechanik herangezogen. Besonderer Wert wird auf die Verbindung dieser fortschrittlichen computergestützten Methoden mit entsprechend konzipierten Messungen an gebauten Prototypen, sowie mit dem Expertenwissen gelegt. Methoden der Schwingungslehre und Festigkeitslehre werden interaktiv in den Untersuchungen und als Grundlage für den Konstruktionsprozess angewendet. Im Rahmen von umfangreichen Messerien wurde eine ausgezeichnete Übereinstimmung zwischen den Rechenergebnissen und den Versuchsergebnissen erzielt und daraus Verbesserungsvorschläge hinsichtlich des Entwurfs und des Betriebs abgeleitet. Das Studium von Detailproblemen im Rahmen von Modellverfeinerungen resultiert in neuen Erkenntnissen für den Umformprozess.AbstractIn this paper investigations with computational and experimental methods are documented, which serve as the scientific basis for the analysis and development of a new design concept for a machine known as a panel bender, which forms sophisticated profiles out of plane sheets. The derivation of the mechanical model makes use of the multi-body dynamics for the modeling of the rigid machine parts whereas for the elastic components a discretisation with non-linear finite elements was involved. For the description of the deformation of the sheet formulations of the theory of plasticity and contact mechanics have been applied. Special emphasis is laid on the combination of the advanced computational methods with suitably planned measurements on prototypes, where also the expert knowledge was involved. Methods of dynamics, vibration analysis and strength of materials are incorporated interactively in the process of analysis and design. During these investigations the machines are modeled as multi-body systems where friction is also taken into account at the contact surfaces. Nonlinear Finite Elements and Matlab/Simulink simulations of the machine are also performed to establish the most useful design parameters. Extensive measurements have been carried out, which show good agreement between the computational and experimental results and some improvements for the operation have been derived. Based on these findings suggestions for an improvement for the operation and design of the panel bender have been derived. The analysis of serious and promising further engineering problems and further model refinements are currently under investigation and give an increasing insight.