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In the present paper, we consider nonlocal linear elastic Kirchhoff plates, where we restrict to thin, isotropic, and homogeneous plates under the action of static transverse forces, utilizing the differential equation form of the Eringen nonlocal continuum theory. For the case of simply supported plates of polygonal planform, we derive analogies between the solutions of the nonlocal Kirchhoff theory and its local counterpart. First, we extend the Marcus decomposition method for local Kirchhoff plates, where we show that, analogous to the local case considered by Marcus, the moment sum and the nonlocal deflection are both governed by Poisson boundary value problems, which correspond to auxiliary (local) membrane problems. In the present context, it eventually follows that there is a nonvanishing term responsible for a correction of the deflection due to the nonlocal effect, while the moments and shear forces of the local and the nonlocal plate coincide. The nonlocal correction of the deflection turns out to be governed by a membrane-type boundary value problem again. From this fact, it follows immediately that the correction for the deflection due to the nonlocal effect can be derived alternatively from both the local deflection and the local moment sum, which represents a substantial simplification of the nonlocal computations. For the sake of demonstration, we consider examples with closed form solutions. We first consider (as limiting case) infinite plates under the action of single forces, where the singular behavior of Green’s function of the nonlocal Kirchhoff theory can be clarified. Then we discuss the bending of nonlocal plate strips for comparison sake, as well as equilateral triangular plates. To our best knowledge, no results for bending of nonlocal triangular plates have been presented in the literature so far. The present paper has been strongly influenced by a former contribution on analogies between simply supported polygonal Kirchhoff plates rigid in shear and shear-deformable plate solutions. This paper was co-authored together with our teacher, the late Professor Franz Ziegler, to whom we dedicate the present paper. In light of this latter contribution, and extending an idea promoted by Professor Franz Ziegler and his co-workers in the framework of various other fields, we finally show in the Appendix that the deflections of simply supported nonlocal Kirchhoff plates can be considered as deflections of a local “background” Kirchhoff plate under the action of additional fictitious eigenstrain loadings, such as thermal moments.

The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt–pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure are determined. The contact pressure increases near the end point of the contact area; however, no concentrated contact force occurs.

A rational treatment of the relations of balance for mechanical systems with a time-variable mass and other non-classical supplies. - Systems with mass explicitly dependent on position. - Dynamics of the mass variable body. - Mechanics of multi-component media with exchange of mass and non-classical supplies. - Modeling of fluid-structure interaction: effects of added mass, damping and stiffness. - Dynamics and stability of engineering systems with moving continua.

An extension of Reissner’s geometrically exact theory for the plane deformation of beams allows a consistent constitutive modeling of various types of material behavior. In the present paper, such formulation is used to describe the coupled response of slender piezoelectric structures. Starting from the local equilibrium relations and the geometric description of the cross-sectional deformation, the constitutive equations in the structural mechanics framework are derived from quantities of the non-linear continuum theory in a step-by-step procedure.

The present contribution intends to promote an alternative form of Lagrange ’ s Equations, which rests upon the notion of momentum. We first present a short derivation of the proposed momentum based version of Lagrange’s Equations. From this derivation it becomes apparent that the derivatives of the kinetic energy with respect to the generalized coordinates must cancel out in the original kinetic energy based version of Lagrange’s Equations, and thus need not to be computed. The presented momentum based formulation of Lagrange’s Equations is valid for deformable bodies, modeled in the framework of the Ritz approximation technique, where rigid-body degrees-of-freedom may be present. After having stated this momentum based version of Lagrange’s Equations, we restrict to plane motions of rigid bodies, and demonstrate our proposed formulation for the case of a rotational degree of freedom, where we present an additional connection to the notion of momentum of the rigid body, particularly to angular momentum. Finally, we present the exemplary application to systems consisting of two rigid bodies, namely the pendulum with a point mass and movable support, and the Sarazin pendulum consisting of a rigid rotating disc and an attached point mass.

The present paper deals with a problem-oriented novel model reduction technique for elastic rotors with a high and/or rapidly changing axial speed. Numerical results are presented for various run-up simulations with different values of unbalance mass and internal damping ratio. An important result of this study is that the time-derivatives of the tilting angles and the flexible coordinates should not be considered as small when studying resonance and stability phenomena.

T. Adachi: Energy Absorption of Axially-Impacted Column Controlled by Transverse Impact - C. Adam, T. Furtmuller: Seismic Performance of Tuned Mass Dampers - H. Bremer: Problems in Fast Moving Non-Holonomic Elastic Systems - F. Casciati, Z. Chen: Using GPS Sensors in Structural Mechanics - L. Faravelli, C. Fuggini, F. Ubertini: Hybrid Control Procedures in Mitigating Cable Vibrations - T. Furukawa: Thermal Stress Analysis in a Functionally Graded Material Considering Finite Thermal Wave Speed - U. Gabbert, S. Kari, N. Bohn, H. Berger: Numerical Homogenization and Optimization of Smart Composite Materials - R. Heuer, S. Mehdi Yousefi: Hybride Bell Tower Like Structures in Earthquake Environment - H. Irschik, M. Krommer, M. Gusenbauer: Tracking of Stresses: A Further Step Towards Ageless Structures - M. Ishihara, Y. Watanabe, N. Noda: Non-Linear Dynamic Deformation of a Piezothermoelastic Laminate - B. Jakoby, E. K. Reichel, F. Lucklum, B. Weiss, C. Riesch, F. Keplinger, R. Beigelbeck, W. Hilber: Determining Liquid Properties Using Mechanically Vibrating Sensors - R. Kawamura, H. Fujita, K. Heguri, Y. Tanigawa: Mathematical Analysis of Flexural Vibration for a Functionally Graded Material Plate and Vibration Suppression by Flexural Wave Control - M. Krommer, M. Zellhofer: Monitoring and Control of Multi-Storey Frame Structures by Strain-Type Actuators and Sensors - X. Ootao: Transient Piezothermoelastic Problem of a Functionally Graded Thermopiezoelectric Cylindrical Panel - M. Reichhartinger, M. Horn, A. Hofer: Control of an Electronic Throttle Valve for Drive-by-Wire Applications - T. Rittenschober, K. Schlacher: Output Regulation of Smart Structures, Theory and Practice - A. Saimoto: Analysis of Weld Induced Plasticity by BFM - Y. Shibuya: Evaluation of Internal Friction of Viscoelastic Composites with Meso-Scale Structures for Vibration Damping of Mechanical Structures - T. Tsuji, T. Kurimoto, T. Shibuya: An Identification Method of the Time Dependence of theImpact Force by Using Acoustic Response and FEM Analysis - S. Ueda: Infinite Row of Parallel Cracks in a Piezoelectric Material Strip Under Mechanical and Transient Thermal Loadings - K. Watanabe, N. Sumi: Elastodynamic Doppler Effects by a Moving Interface - C. Zehetner, J. Gerstmayr: Compensation of Flexible Vibrations in a Two-Link Robot by Piezoelectric Actuation - F. Ziegler: The Basis of Optimal Active (Static and Dynamic) Shape- and Stress-Controlby Means of Smart Materials

Basics of Continuum Mechanics (A. K. Belyaev).- A Treatise on the Equations of Balance and on the Jump Relations in Continuum Mechanics (H. Irschik).- The Rayleigh-Ritz Technique and the Lagrange Equationsin Continuum Mechanics: Formulations for Material and Non-Material Volumes (H. Irschik, H. J. Holl, F. Hammelmuller).- Basics of Analytical Mechanics (A. K. Belyaev).- Compensation of Deformations in Elastic Solids and Structures in the Presence of Rigid-Body Motions (H. Irschik,U. Pichler, M. Nader, Ch. Zehetner).- Computational Dynamics of an Elasto-Plastic Structural Element With Rigid-Body Degrees-of-Freedom (J. Gerstmayr, H. Irschik, M. Dibold).- High Frequency Dynamics of Engineering Structures (A. K. Belyaev).- Basic Differential Geometry for Mechanics and Control (K. Schlacher, K. Zehetleitner).- Port-Hamiltonian Systems: Network Modeling and Control of Nonlinear Physical Systems (A. J. van der Schaft).- Periodic Motion Induced by the Painleve Paradox (R. I. Leine, B. Brogliato, H. Nijmeijer).- Coordination of Rigid and Flexible Joint Robot Manipulators (A. Rodriguez-Angeles, H. Nijmeijer, H. A. van Essen).- Structural Control for Dynamic Hazard Mitigation (F. Casciati).- Some Applications of Differential Geometry in Control (K. Schlacher, S.Fuchshumer, J. Holl).- Some Applications of Differential Geometry in Mechanics (K. Schlacher, G.Grabmair, H. Ennsbrunner, R. Stadlmayr)

In this Lecture, we are concerned with the range of applicability of the balance relations in continuum mechanics. These relations have been already introduced in Lecture 1 of the present book as an important foundation of the dynamics and control of structures and machines, Belyaev (2004). In the present Lecture, we start with the general form of the relations of balance, which we afterwards specialise to the equations of balance of mass, momentum and energy (the first law of thermodynamics) and to the balance relation of entropy (the second law of thermodynamics), the latter in the form of the Clausius-Duhem inequality. We also discuss the frequently used equations of balance of kinetic energy (the law of power), and we point out the consequences of inserting the latter into the first and second law of thermodynamics, leading to the balance relations for the internal energy and to the Clausius-Planck inequality, respectively. We lay special emphasis on reviewing the continuity conditions that must be satisfied in order that the global and local forms of the relations of balance do hold. When a surface of discontinuity, a so-called singular surface, is present, across which some entity shows different values when approaching from the two sides of the surface, jump relations are needed in order to connect the local forms of the relations of balance at the two sides. We summarise and extend a recent formulation, which allows to connect the local equations of balance of mass, momentum, energy, kinetic energy and internal energy in a consistent manner, Irschik (2003). In the latter reference, it has been shown that surface growth terms must be introduced for the sake of consistency, and relations between these surface growth terms have been derived. In the present Lecture, these results are extended with respect to the second law of thermodynamics, and with respect to the resolution of a contradictory result in the literature on the jump relations of energy and of internal energy.

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).

The present Lecture is concerned with vibrations of linear elastic solids and structures. Some part of the boundary of the structure is suffering a prescribed large rigid-body motion, while an imposed external traction is acting at the remaining part of the boundary, together with given body forces in the interior. Due to this combined loading, vibrations take place. The latter are assumed to remain small, such that the linear theory of elasticity can be applied. As an illustrative example for the type of problems in hand, we mention the flexible wing of an aircraft in flight. In this example, the rigid-body motion is defined through the motion of the comparatively stiff fuselage to which a part of the boundary of the wing is attached. The goal of the present paper is to derive a time-dependent distribution of actuating stresses produced by additional eigenstrains, such that the deformations produced by the imposed forces and the rigid-body motion are exactly compensated. This is called a shape control problem, or a deformation compensation problem. We show that the distribution of the actuating stresses for shape control must be equal to a quasi-static stress distribution that is in temporal equilibrium with the imposed forces and the inertia forces due to the rigid-body motion. Our solution thus explicitly reflects the non-uniqueness of the inverse problem under consideration. The present Lecture extends previous results by Irschik and Pichler (2001, 2004) for problems without rigid-body degrees of freedom. As a computational example, we present results for a rectangular domain in a state of plane strain under the action of a translatory support motion.

In the present Lecture, we study the motion of a single elasto-plastic body that represents a moving element of a structure or machine, where we present equations that are applicable to three-dimensional motions and bodies of arbitrary shape. We assume the displacements and strains of the body to be small with respect to a floating reference configuration, and we present the corresponding small-strain elasto-plastic-constitutive relations. We then point out the necessity of refined computational procedures for obtaining the plastic parts of strain in the case of a reversed loading, a problem often to be encountered in practice. In a Rayleigh-Ritz procedure, the flexible coordinates, which are coupled to the rigid-body degrees of freedom via the equations of motion, must be brought into connection with the plastic parts of strain. Often, the influence of the plastic parts of strain upon the motion of the body can not be neglected. We sketch an advantageous iterative numerical procedure for computing the plastic parts of strain, and we eventually discuss their influence upon the equations of motion in more detail.