Werner Hauger

Static and Kinetic Friction

Objectives: Bodies in contact exert a force on each other. In the case of ideally smooth surfaces, this force acts perpendicularly to the contact plane. If the surfaces are rough, however, there may also be a tangential force component. Students will learn that this tangential component is a reaction force if the bodies adhere, and an active force if the bodies slip. After studying this chapter, students should be able to apply the Coulomb theory of friction to determine the forces in systems with contact.

Center of Gravity, Center of Mass, Centroids

Objectives: In this chapter, definitions of the center of gravity and the center of mass are given. It is shown how to determine the centroids of bodies, areas and lines. Various examples demonstrate how to apply the definitions to practical problems.

Work and Potential Energy

Objectives: Students will become familiar with the concepts of work, conservative forces and potential energy. In addition, they will become acquainted with the principle of virtual work. After studying this chapter, students should be able to correctly apply this principle in order to determine equilibrium states in nonrigid systems as well as support reactions and internal forces and moments. Finally, it will be shown how to investigate the stability of equilibrium states of conservative systems with one degree of freedom.

Motion of a Point Mass

We will first learn how one describes the motion of a point mass by its position, velocity, and acceleration in different coordinate systems and how such quantities can be determined. Subsequently, we will concern ourselves with the equations of motion, which prescribe the relation between forces and motion. An important role will again be played by the free-body diagram with whose help we will be able to properly derive the equations of motion. Further, we will discuss important physical concepts such as momentum, angular momentum, and work-laws and their applications.

Dynamics of Systems of Point Masses

Up to now we have concerned ourselves with the study of single point masses. We now wish to extend the ideas developed in Chapter 1 to systems of interacting point masses. This includes the concepts of linearand angular momentum, the impulse law, and the work-energy theorem. The reader will learn how one studies such systems and how to systematically apply the laws of motion to them.

Numerische Methoden in der Mechanik

Die mathematische Formulierung mechanischer Probleme fuhrt auf Gleichungen, die fur konkrete Aufgabenstellungen gelost werden mussen. Diese Gleichungen konnen je nach Fragestellung von ganz unterschiedlichem Typ sein. Sie schliesen algebraische Beziehungen, Differentialgleichungen oder Variationsgleichungen ein. Beispiele dafur finden sich in den ersten drei Banden der Lehrbuchreihe und in den vorangegangenen Kapiteln dieses Buches. In der Elastostatik konnen wir die Gleichgewichtsbedingungen (algebraische Gleichungen) oder die Gleichung der Biegelinie eines Balkens (Differentialgleichung) nennen. In der Kinetik wird die Bewegung des Massenpunktes durch gewohnliche Differentialgleichungen beschrieben. Die Gleichungen fur die Scheibe in Kapitel 2 oder fur die Membran in Kapitel 3 stellen partielle Differentialgleichungen dar. Variationsgleichungen fur den Stab und den Balken sind in Abschnitt 2.7.3 angegeben.

Non-Inertial Reference Frames

According to Section 1.2.1 Newton’s law in the form m a = F is valid in a reference frame that is fixed in space. Such a reference frame is an inertial frame. The notion of an inertial frame will be discussed in more detail in Section 6.2.

Forces with a Common Point of Application

Objectives: In this chapter, systems of concentrated forces that have a common point of application are investigated. Such forces are called concurrent forces. Note that forces always act on a body; there are no forces without action on a body. In the case of a rigid body, the forces acting on it do not have to have the same point of application; it is sufficient that their lines of action intersect at a common point. Since in this case the force vectors are sliding vectors, they may be applied at any point along their lines of action without changing their effect on the body (principle of transmissibility). If all the forces acting on a body act in a plane, they are called coplanar forces.

General Systems of Forces, Equilibrium of a Rigid Body

Objectives: In this chapter general systems of forces are considered, i.e., forces whose lines of action do not intersect at a point. For the analysis, the notion moment has to be introduced. Students should learn how coplanar or spatial systems of forces can be reduced and under which conditions they are in equilibrium. They should also learn how to apply the method of sections to obtain a free-body diagram. A correct free-body diagram and an appropriate application of the equilibrium conditions are the key to the solution of a coplanar or a spatial problem.

Beams, Frames, Arches

Objectives: Beams are among the most important elements in structural engineering. In this chapter it is explained how the internal forces in a beam can be made accessible to calculation.