Active and Passive Vibration Control of Structures

The Chapter ‘Mechanical Systems: Equations of Motion and Stability’ corresponds to the material presented in five lectures given at the CISM Course no. 418. The first parts deal with the form of the equations of motion of mechanical systems, in particular the linearized equations and the influence and importance of the different terms (inertia terms, damping, gyroscopic terms, restoring terms and circulatory terms as well as with their physical origin). This is done both for discrete systems, and the corresponding material is part of the recent book Hagedorn & Hochlenert, Technische Schwingungslehre, Verlag Harri Deutsch, Frankfurt, 2012, as well as for continuous systems, the material being adapted from Hagedorn & DasGupta, Vibrations and Waves in Continuous Mechanical Systems, Wiley, Chichester, 2007. Almost all the material is presented in typical elementary vibration courses, but here certain aspects will be highlighted, which are not always stressed in basic vibration courses. The third part deals with Liapounov stability, the material is from the author’s earlier book Hagedorn, Non-Linear Oscillations, 2nd edition, Oxford Science Publications, 1988. The material of these five lectures is used in the other lectures of the course.

The author prepared most of the material in 2012 and 2013, while staying at the University of Canterbury in Christchurch, New Zealand. The author thanks the Department of Mechanical Engineering of the UC for providing the infrastructure and assistance.

The chapter ’Variational principles in mechanics and control’ summarizes the material presented in six lectures in the CISM course no. 418. The first part considers the derivation of equations of motion for discrete and continuous systems. Founding on the basics of the calculus of variations the relation of the principle of virtual work to Lagrange’s equations and Hamilton’s principle is discussed. The concepts are useful for an efficient modeling of control systems. In the second part variational methods are used to introduce basics of optimal control and control system design.

This chapter compares three different ways of mitigating the dynamic response of buildings: Dynamic Vibration Absorber (DVA), Active Mass Damper (AMD) and Hybrid Mass Damper (HMD). The methodology is illustrated with a shear frame example subjected to a random seismic input. Two different ways of implementing the HMD are considered, one called passive starting from a mistuned DVA, and one called active starting from a tuned DVA and using a control system with two feedback loops. It is shown that a well designed HMD may produce performances comparable to that of an AMD while significantly reducing the actuator force and stroke requirements. Besides, the active implementation is immune to control system breakdown, because the HMD is reduced to a properly tuned DVA with optimum performances for a passive system.

Hamilton invented state space models of nonlinear dynamic systems with his generalized momenta work in the 1800s, but, at that time, the lack of computational tools prevented broad acceptance of the first order form of dynamic equations. With the rapid development of computers in the 1960s, State Space models evoked a formal control theory for minimizing a scalar function of control and state, propelled by the calculus of variations and Pontryagin’s maximal principle.

High damping is desirable to attain low vibration and noise levels whereas low damping is desirable for increased sensitivity in sensors and certain precision instrumentation.

Damping is most obvious at resonance where the stiffness and inertia forces become equal. As a result, damping is a key factor in predicting vibration response of structures.

As we will see in the following sections, there are numerous paths to damping and in a complex structure several means of damping may take place simultaneously at different locations throughout the structure. Accordingly, in determining the response of a vibrating structure, the total effect of all types of damping that may be distributed throughout a structure must be taken into account.

Measurements of damping normally indicate the total damping a system experiences. It is difficult to isolate a component or a subsystem or a material within a system and measure its damping. In describing the various damping mechanisms, we will examine each through its effect on a single-degree-of-freedom (sdof) oscillator.

In this section, we will review the response of a simple oscillator and examine the role of damping on it and review the basic methods of measurement criteria for damping properties of structures. However, we will not consider here the role of damping in dynamic behaviors such as chaos, stability, etc.

Dissipation of vibratory energy takes place in both fluid and solid media, initiated by a number of possible macro activities. Accordingly, we will consider damping methods to reflect the media in which dissipation takes place when addressing damping methods in the next section. Models of fundamental dissipation mechanisms that describe energy transfer from ordered energy to disordered or thermalized energy are briefly summarized in the last section.

Mechatronic components are getting more and more common in mechanical systems. As an example Active Magnetic Bearings (AMB) are often used in Rotating Machinery. Besides their function of an oil-, contact- and frictionless levitation of the rotor, they are best suited to be used as an exciter and measurement instrument to extract more information from the system under observation. In this paper it is shown, how Active Magnetic Bearings can be used for identification, diagnosis and optimization purposes.