Dynamics
Within the chapter on dynamics the transient behavior of the acting loads on the structure will be introduced additionally into the analysis. The procedure for the analysis of dynamic problems depends essentially on the character of the time course of the
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Dynamics
Abstract Within the chapter on dynamics the transient behavior of the acting loads on the structure will be introduced additionally into the analysis. The procedure for the analysis of dynamic problems depends essentially on the character of the time course of the loads. At deterministic loads the vector of the external loads is a given function of the time. The major amount of problems in engineering, plant and vehicle construction can be analyzed under this assumption. In contrast to that, the coincidence is relevant in the case of stochastic loads. Such cases will not be regarded here. For deterministic loads a distinction is drawn between • periodic and non-periodic, • slow and fast changing load-time functions (relatively related to the dynamic eigenbehaviour of the structure). In the following chapter linear dynamic processes will be considered, which can be traced back to an external stimulation. The field of self-excited oscillation will not be covered.
13.1 Principles of Linear Dynamics The point of origin is an elastic continuum with mass which is, in contrast to previous problems, stressed with time-dependent loads. The mass with density ρ extends itself over the volume Ω (Fig. 13.1). For dealing with dynamic problems in the context of the finite element method various model assumptions can be discussed [1–7]: 1. the distribution of the mass and 2. the treatment of the time dependency of all involved parameters. Within the framework of the FE method the continuum will be discretized. A first model assumes that the distribution of the masses is not influenced by the discretization. The masses are also distributed continuously in the discretized condition. Figure 13.2a shows the continuously distributed mass for a bar. Another model assumes that A. Öchsner and M. Merkel, One-Dimensional Finite Elements, DOI: 10.1007/978-3-642-31797-2_13, © Springer-Verlag Berlin Heidelberg 2013
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13 Dynamics
Fig. 13.1 Elastic continuum with mass under time-dependent load
(a)
(b) m
Fig. 13.2 Models of dynamic systems with a continuously and b discrete distributed masses
the originally continuously distributed mass can be concentrated on discrete points (see Fig. 13.2b). The total mass n mi (13.1) m= i
of the system remains. The connection between the points with mass will be applied with elements without mass, which may represent further physical properties, for example stiffness. Regarding the time dependency of the state variables, both the loads and the deformations as response of the system to the external loads are time changeable. Depending on the character of the external load, different problem areas are distinguished in dynamics (see Fig. 13.3) and pursue different strategies for the solution: • Modal analysis Here the vibration behavior is considered without external loads. Eigenfrequency and eigenmodes are determined. • Forced vibrations An external periodic force excites the component to resonate in the excitation frequency. • Transient analysis The external stimulating force F(t) is
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