The chapter discusses the dynamic properties of mechanical systems and the possibilities of their evaluation. The features of using the characteristic frequency equation, which allows determining the natural frequencies of the system, are shown. Methods of accounting for additional inertial type connections that can be used to improve the dynamic properties of technical systems are presented. The ideas about the possibility of including motion transformations devices in the system in order to change dynamic properties are presented.
TopCharacteristic Frequency Equation: Structure, Dynamic Stiffness, Features Of The Interaction Of System Elements
Interpretation of dynamic interactions of elements of mechanical systems, displayed by structural diagrams of dynamically equivalent automatic control systems, became the basis for the formation of non-traditional approaches, which initiated the introduction of a number of new issues related to the ideas of expanding ideas about a set of typical elements, ways of connecting them, and possibilities of structural transformations and identifying dynamic features (Khomenko et al., 2012; Belokobylsky et al., 2013; Khomenko et al., 2016). Structural mathematical modeling considers the connections and regulatory influences between elements, using tools from automatic control and mechanical circuits. This approach can expand the set of elements, introduce constraints, and transform models. One direction is to use dynamically equivalent automatic control systems as mathematical models, based on interactions between elements of mechanical and control systems. This approach is generalizable to systems with distributed parameters.
Figure 1a-f gives various interpretations in the representations of the initial physical model in the form of a mechanical oscillatory system with one degree of freedom. Such models are widely used in the tasks of machine dynamics and the protection of equipment and apparatus from vibrations, impacts, etc. The mechanical system in Figure 1a consists of the so-called typical mechanical elements in the form of a spring k, a damper bp and a mass-and-inertia unit m. A mass-and-inertia element can be considered as an object of control (in vibration protection tasks, an object of protection), which predetermines the possibility of using the analogy between automatic control systems.
The mathematical model of the system (Figure 1a) with a force perturbation in the form of a harmonic force Q has the form
.
(1)After the Laplace transform with respect to (1) with zero initial conditions, the differential equation (1) transforms to
,
(2) where
p = jω is a complex variable; the <–> symbol means a Laplace transform image (Eliseev et al., 2011).