Evgeniy Lebed

Nonlinear Theory of Vibration Protection Systems

This chapter contains information about the reasons for nonlinearity and the general properties of nonlinear vibration, and discusses the basics of the harmonic linearization method. Applications of this method for the analysis of free and forced vibrations of systems with one degree of freedom are presented. Different types of nonlinearities are considered. These include Duffing’s rigidity characteristic, a combination of nonlinear stiffness with viscous resistance, linear stiffness with dry friction. A nonlinear dynamic absorber is considered.

Structural Theory of Vibration Protection Systems

Modern theory of automatic control of dynamical systems contains in its arsenal an extremely valuable tool. We are talking about the structural representation of an arbitrary dynamical system. Such representation allows us to divert attention from the physical nature of a process (thermal, vibrational, diffusion, etc) to the physical nature of the elements (mechanical, pneumatic, etc). In the context of structural representation of a mechanical system, we can explore diverse aspects of dynamic processes (controllability, invariance, stability, etc.) [1–3]. The theory of vibration protection is a very attractive application area of structural theory for several reasons. First, many fundamental aspects and concepts of control theory in general and the theory of vibration protection coincide; these include input–output concepts, transfer function, etc. Second, a vibration protection system consists of pronounced blocks and can be represented in symbolic form by a functional block diagram. Successful attempts that consider the problems of vibration protection in terms of the structural theory have been performed by Kolovsky [4, 5], Eliseev [6], and Bozhko et al. [7]. Systematic exposition of the structural theory to systems with distributed parameters was presented by Butkovsky [8]. Structural representation of the system in conjunction with the vibration protection device is a common way of describing complex dynamical systems with lumped and distributed parameters. Structural theory allows us to easily introduce changes into a vibration protection system of the object and find a relationship between any coordinates of a system, while the differential equation of the system assumes a fixed input–output. The Simulink (MATLAB) package has a full set of blocks that allows us to implement just about any structural model.

Statistical Theory of the Vibration Protection Systems

Until now, we have assumed that the external exposure on the system can be represented as a known function of time. However, such representation of exposure on the system is not always possible. Many products operate on portable carriers (automotive and railway transport, ships, planes, rockets, guided missiles). In these cases, the products are subjected to excitations, which mostly have a random (stochastic) character [1]. In the case of stochastic excitation, the response of a system also has a random character; therefore, probability methods should be applied for analysis of such a system. Random factors may enter into dynamic analysis of a vibration protection system not only through exposures, but also through parameters of a system [2].

Vibration Suppression of Systems with Lumped Parameters

This chapter presents the theory of dynamic suppression of vibration of systems with lumped parameters. First, in the example of the simplest dynamic absorber, we consider the idea of suppressing of vibrations. Then we discuss the different types of absorbers have been considered by Babicky [1, Chap. 14, 2], Haxton and Barr [3], and Karamyshkin [4]. These include impact absorbers, gyroscopic vibration suppressors, and autoparametric vibration absorbers. Such devices can also be effectively used for reducing vibrations of systems with distributed parameters [5].

Active and Parametric Vibration Protection of Transient Vibrations

This chapter is devoted to the analysis of transient vibration of linear dynamical systems. The Laplace transform method and Heaviside expansion method are explained. These methods are applied to analysis of linear oscillators subjected to different types of force and kinematic excitation (shock, impulse, recurrent instantaneous pulses). Active vibration suppression through forces and kinematic methods, as well as parametric vibration protection, is discussed.

Shock and Spectral Theory

This chapter is devoted to the analysis of one degree of freedom systems subjected to shock excitation [1, 2, Chap. 9, 3], etc. Some important concepts are discussed. among which are types of shock excitation and different approaches to the shock problem. Fourier transformation of aperiodic functions and corresponding concepts are considered and are then applied to the shock phenomenon. The spectral shock theory method and the concepts of residual and primary shock spectrums are discussed [4]. The transient vibration caused by different force and kinematic shock excitation (Heaviside step excitation, step excitation of finite duration, impulse excitation) are considered. Dynamic and transmissibility coefficients are derived and discussed in detail.

Vibration Suppression of Structures with Distributed Parameters

This chapter is devoted to dynamic suppression of vibration of uniform homogeneous beams. The basic method for considering beams as a system with distributed parameters is the Krylov–Duncan method. Two types of absorbers are considered. They are the lumped m–k absorber and the distributed m–k vibration absorber of the beam subjected to harmonic excitation. Suppression of the transverse vibration of the cantilever beam subjected to force and kinematic excitation are considered. The absorber presents an extension rod, which is attached to the main beam.

Mechanical Two-Terminal and Multi-Terminal Networks of Mixed Systems

This chapter further develops the theory of the mechanical two-terminal network (M2TN) as applied to vibration protection of mixed systems. Such systems contain an arbitrary deformable structure and are fitted with vibration protection (VP) devices. Formulas are derived for determining the fundamental characteristics—input and output (transfer) impedance and mobility. The type of system and its peculiarities are not specified, nor is the structure of the VP devices or their parameters and location. Impedance and mobility are realized in constructing an optimal synthesized M2TN, with the number of its passive elements as optimality criterion.

Vibration Isolation of a System with One or More Degrees of Freedom

This chapter describes some general concepts, including design diagrams of vibration protection systems, the various means of vibrational excitation, and the complex amplitude method. We consider types of linear classic single-axis vibration isolators and special types of isolators (equal-frequency vibration isolator, isolator with dry friction, etc.) [1, 2].

Arbitrary Excitation of Dynamical Systems

This chapter deals with fundamental functions of linear dynamical systems including the transfer function, Green’s function, Duhamel’s integral, and standardizing function. We show their application to different problems dealing with dynamical systems.