Vibration Dynamics and Control

Over the last decades there has been much work concerned with the vibration control of different dynamical systems. The objective in writing this textbook was to help students wishing to get deeper knowledge on structural dynamics and vibration control, while providing an overview of the potential of smart materials based sensor and actuator technologies for active vibration control. The textbook is aimed at first towards graduate and postgraduate students following Master and PhD programmes related to structural dynamics, mechatronics, control engineering, automotive engineering noise and vibrations. The only prerequisite for reading this book is some background in structural dynamics and in automatic control. The contents of the textbook consist of five parts: Vibration dynamics (Part 1), Passive and semi-active vibration control (Part 2), Active and hybrid vibration control (Part 3), Applications (Part 4), and Supplementary mathematics, List of Matlab codes and Answers and hints for the exercises (Part 5). The Part 1 of the textbook is called “Vibration dynamics”. It consists of Chapter 2 and Chapter 3. In Chapter 2 we present three approaches, which are usually used for developing of mathematical models of vibration dynamics of mechanical systems. These approaches are: free-body diagram approach, energy method and Lagrange formalism. Chapter 3 is devoted to elements of vibration dynamics analysis. The focus is set primarily on simple and widely recognizable vibrating mechanical systems. Attention is paid to analysis of vibration dynamics under harmonic excitations, transmissibility and vibration isolation. Some specific properties and phenomena that occur in nonlinear vibrating systems (like parametric resonance) are discussed. Vibration dynamics of a mechanical system can be affected by changing the initial state, or/and by changing the system’s structural/design parameters, or/and by varying the external force/torque excitations acting on the system. This type of problems is considered in the Part 2 of the textbook which is called “Passive and semi-active vibration control”. The Part 2 consists of Chapters 4 - 7. In Chapter 4 we consider so-called passive vibration control (PVC) problems. To this class belong the control problems dealing with determination of the initial state of the system or/and its structural (design) parameters which together with given external force/torque excitations guarantee prescribed (desirable) properties of vibration dynamics of the considered system. Dynamics of undamped as well as damped vibrating systems with tuned mass dampers are studied and analyzed. In many cases mechanical systems are inherently stable to begin with, and external control is applied to improve the performance. But, unfortunately, the introducing the active control, let say for vibration control by using feedback control, can often make the system unstable. Thus analysis of stability of the vibrating system after a control strategy is designed and applied to the system is an important step in engineering practice. In Chapter 5 the elements of the theory of stability which can be used for designing of active vibration control strategies which make the closed loop vibrating system stable, are presented. It is desirable to design active vibration control which uses real-time measurement data as a response of a system in question. In this case the control is mathematically represented as a function of parameters of the system response, e.g. as a function of positions and/or velocities. Such a control is called feedback control. Chapter 6 presents several details of physical and mathematical representations of feedback control. Some important properties of a system to be controlled such as controllability and observability are defined and discussed. In Chapter 7 we consider semi-active vibration control problems. The semi-active vibration control method is defined here as an approach which gives possibility to change damping or/and stiffness properties of functional components of a system, e.g., damping or/and stiffness coefficients of shock absorbers, and in this way to control vibration dynamics of the system in question. Some well-know semi-active control strategies based on skyhook approach are presented. A brief overview of magnetorheological (MR) fluids technology including modelling of MR dampers is presented and their applications for semi-active control are discussed. The Part 3 of the textbook is called “Hybrid and active vibration control”. It consists of Chapters 8 - 9. Chapter 8 presents several approaches and methods for designing of optimal control laws and algorithms for vibration attenuation and vibration suppression. Focus is put on LQR optimization technique, the calculus of variations approaches, the methods which are used first integrals of a vibrating system to be controlled, and the method for optimal vibration control based on Pontryagin maximum principle. The term hybrid control generally refers to a combined passive and active control system. Since a portion of the control objective is accomplished by the passive system, less active control effort, implying less power, is required. A side benefit of hybrid and semi-active control systems is that, in the case of a power failure, the passive components of the control still offer some degree of control, unlike a fully active control system. Chapter 9 presents the elements of theory of hybrid control techniques. A mathematical statement of the optimal control problem which is suitable for modelling of controlled motion and optimization of semi-passively actuated mechanical systems is proposed. A methodology and numerical algorithms for solving the control and optimization problems for semi-passively actuated mechanical systems are described. Special emphasis is put on the study of controlled mechanical systems having different degrees and types of actuation (underactuated and overactuated systems, external powered drives, unpowered spring-damper like drives, etc.). The solutions of energy-optimal control problems are presented for different kinds of semi-passively actuated multibody systems (closed-loop chain semi-passively actuated robot, multibody system modelled the human locomotor apparatus with above-knee prosthesis). The methodology and numerical algorithms described and implemented for particular control problems are also suitable for design of energy efficient active vibration control algorithm for nonlinear vibration mechanical systems. The Part 4 of the textbook is called “Applications”. Here a brief overview is given of the research on vibration dynamics and control performed at the Mechanical systems group of the division of Dynamics, department of Applied Mechanics, Chalmers University of Technology. The focus is on current doctoral projects which are related to vibration dynamics and control problems having applications in high speed train industry, automotive engineering, home appliances design and smart material based power harvesting from vibrations. During the last decades interest in research and development of smart actuators, sensors and power generators that use giant magnetostrictive materials has been continually growing. Both academia and industry are actively looking for broad utilization of this technology for different applications (active vibration and noise control, structural health monitoring, self-powered electronic equipments and systems, MEMS, robotics, biomedical engineering, etc.). Recent developments in miniaturized sensors, digital processors, self-powered electronics and wireless communication systems have many desirable applications. The realization of these applications however, is limited by the lack of a similarly sized power sources. Powering the above mentioned systems can be a significant engineering problem, as traditional solutions such as batteries are not always appropriate. The one issue that still needs to be resolved is a method to generate sufficient energy to power the electronics. The Chapter 10 deals with application of smart materials, namely giant magnetostrictive materials, for power harvesting from vibration. Mathematical modelling and design of magnetostrictive electric generators (MEG) are considered. A mathematical model, physical prototype of MEG and test rig have been developed for simulation and experimental study of conversion of mechanical energy of vibration into electrical energy using Terfenol-D as an active material. Simulation and experimental results have confirmed functionality of the designed MEG. The textbook ends with the Part 5 which comprises Supplementary mathematics, List of Matlab codes, and Answers and hints for the exercises. List of references consists of only those books and scientific papers which were used during preparation of the textbook or which were recommended for additional information on a studied topic