Introduction to Dynamics

“Mechanics is the science of motion; we define as its task: to describe completely and in the simplest possible manner such motions as occur in nature.” With respect to engineering we should complete this statement by “as occur in nature and in technology.” This more than a hundred-year-old statement was made by Kirchhoff [34] and has lost neither its meaningfulness nor its assertion. Technical mechanics as a science must also be as simple as possible but conversely descriptively complete. If we consider motion as any kind of translation and rotation, even if only minimal as in the case of deformations, and also include the state of no motion (i.e., the state of rest), then motion describes mechanics as a whole. It comprises two fundamental aspects, that of geometry and kinematics describing positions and orientations, velocities and accelerations, and that of kinetics, describing the cause of motion. Regarding all possible interactions between material bodies or between material bodies and their environment, we consider those possibilities, which produce accelerations (or deformations) of these bodies. We call the driving magnitude of such interactions forces. Thus, the kinematics of bodies and their interaction with forces, their statics and kinetics, define mechanics.

In the presentation of the methods in Chapter 1, we have already assumed concrete conceptual models, that is rigid bodies with homogeneous, constant mass that are arbitrarily connected by constraints.

Discrete systems are composed of rigid bodies, the essential property of which is that the distance between two points in the interior of such bodies remains constant with time. Elastic bodies are continua, which can deform elastically.We assume that their masses are homogeneous and isotropic. Furthermore, we restrict ourselves to linear-elastic bodies and thus to small deformations. The vibrations of these bodies are determined by their mass and stiffness distributions, similar to masses and springs in the discrete case. Each vibrating elastic system is characterized by eigenfrequencies and mode shapes. For each eigenfrequency, there is a corresponding mode shape which the structure takes when it oscillates at this frequency. There are infinite many eigenfrequencies and modes, usually in a systematic order. This property makes linear-elastic vibration systems relatively easy to understand, but it does not apply to all continuous systems, such as rotating fluids.

The motion of any mechanical system composed of rigid bodies or discretized elastic bodies is always described by a set of nonlinear ordinary differential equations of second order of the form (Chapter 1)

The period need not be constant, but may depend on the amplitude (Fig. 4.7). In the following, we deal with vibrations with one or more constant periods and investigate the issue of vibration formation. In order to recognize the essentials, we restrict ourselves in many cases to oscillators with one degree of freedom x(t).