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Abstract Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. The oscillations may be periodic, such as the motion of a pendulum or random, such as the movement of a tire on a gravel road. Mechanical vibrations span amplitudes from meters (civil engineering) to nanometers (precision engineering). In many cases, vibration is undesirable and causes detrimental effects on various systems as follows: Failure: vibrations may cause structure failure such as, excessive strain during transient events when a building responses to an earthquake, and flutter phenomenon in bridges and plane wings when subjected to wind excitation. Also failure may occur due to fatigue in mechanical parts in machines. Discomfort: vibrations reduce comfort such as noise and vibration in helicopters, car suspensions and wind-induced sway of buildings. Reduce accuracy during production of precision devices: many precision industrial processes cannot take place if machinery is being affected by vibration, for example, the production of semiconductor wafers. Detrimental effects of vibrations increase more and more and should be eliminated especially when mechanical resonance occurs. Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the excitation frequency matches than it does at other frequencies. It may cause great amplitude motions and even catastrophic failure in improperly constructed structures including bridges, buildings, and airplanes in a phenomenon known as resonance disaster. So vibration reduction is of vital importance especially at resonance cases. The basic concepts used to reduce vibrations are stiffening, damping, and isolation. 2 Stiffening shifts the resonance frequency of the structure beyond the frequency band of excitation. Damping reduces the resonance peaks by dissipating the vibration energy. Isolation depends on preventing the propagation of disturbances to sensitive parts of the systems. Vibration control systems can be broadly classified as passive, active, and semi-active. Passive control generally consists of spring and damper elements and no computer control is associated with this type of control. The characteristics of passive vibration absorbers are fixed by the mass, spring, and damper elements. Passive vibration absorbers do not automatically change or optimize their spring or damper characteristics based upon a changing environment. Therefore, they are almost effective over a narrow range of disturbance inputs. The passive vibration absorbers are designed based on a nominal mass load and disturbance environment expected to be most encountered over the design life of the structure. Passive vibration absorbers represent the majority of vehicular vibration absorbers in use today due to their low cost of manufacturing and maintenance when compared to other vibration absorbers. Also passive vibration vibration absorbers may take the form of basic structural changes or the addition of passive elements such as masses, springs, and fluid dampers or addition of smart materials like piezoelectric materials (PZT). Authors in [1] studied extensively the passive vibration absorbers. The response of a harmonically excited system consisting of a nonlinear shaker emulates the machine coupled to a passive vibration absorber is investigated in chapter 1 in this thesis. The model Active vibration controller depends on application of a dynamic force in an opposite fashion to the forces imposed by external vibration so that the amplitude of the resultant force and the system response to external vibrations 3 decrease. Active vibration controllers are typically composed of a spring element and some type of force actuator. The primary damping or energy dissipation is provided by the active force actuator which replaces the passive damper used in passive vibration absorbers. Theoretically, both the spring and the damper in passive vibration absorber could be replaced by a force actuator; however, this is not typical due to practical system constraints. At a minimum, a spring is usually necessary to provide support, even in the most advanced system designs. Although they can be manufactured with a high degree of reliability, the rare possibility of failure in operation must be taken into consideration. A passive system may be required as a backup to prevent such failure from making a complete disaster. The mathematical control law is developed and implemented in software on a real-time computer processor, which drives the force actuator through the interface circuits and thus determines the actuation force. Also sensors which detect or measure the vibration and convert it to signals are needed. The linear electromagnetic motor can be used as a dynamic force actuator as in case of Bose Corporation automotive suspension system. Active control methods are more costly than passive ones, but some vibration problems are so intensive that active control methods alone can cure them. Active vibration controllers are studied extensively in [3]. A semi-active vibration controller is typically composed of a spring type element and a damper that is continuously adjustable such as Magneto-Rheological damper or shortly MR damper. The damper characteristics are continuously variable and can be controlled by a computer algorithm. Since the adjustable damper is not capable of supplying energy to the system, the system performance is limited compared to the capability of the active vibration absorber. However, the semi-active vibration controller is more stable than active vibration controllers. Semi-active vibration controllers provide an alternative to active systems when the performance improvement of the semi-active system over the passive system is 4 adequate. In addition, a semi-active vibration absorbers may be less costly and potentially more reliable than an active vibration absorbers. Semi-active vibration controllers are studied extensively in [4]. Unfortunately, active and semi-active vibration controllers may suffer from time/loop delays. Time delays are inevitable in any active control system as a result of measuring system states, processing the control algorithms, control interfaces, transport delay, and actuation delay. Presence of time delays imposes strict limitations on the control system. With delays in measurement, the ays in actuation. Thus time delay reduces the compensation efficiency to the effect of disturbances. So controller design and operation become complicated. Time delays can affect the stability of the system. Thus control system with time delays became a subject systems with loop delays. Many active vibration control techniques are studied in this thesis to reduce the vibrations of a nonlinear beam in chapters 2 to 4. This beam is subjected to dynamic instability due to occurrence of flutter phenomenon. The mathematical models under study in this thesis are represented by a second order ordinary differential equation, and the vibration controller is represented also by another second order ordinary differential equation. So the closed loop system consists of a coupled system of differential equations and the main interest of the thesis is reducing vibrations of coupled systems. Flutter is a dangerous phenomenon which occurs when elastic structures are subjected to aerodynamic forces. This aerodynamic forces are exerted on the elastic structure by the fluid flow due to relative motion between the structure ed 5 by the fluid flow can cause a positive feedback to the structure. This positive feedback increases oscillations which may lead to instability and cause flutter. example the wing of a plane has two basic degrees of freedom or natural modes of vibration: pitch and plunge (bending). where the pitch mode is rotational and the bending mode is a vertical up and down motion at the wing tip. This includes aircraft, buildings, telegraph wires, and bridges. The mathematical -Bernoulli beam with nonlinear curvature. This model was given by Warminski in [6]. The beam is subjected to external harmonic excitation close to its first natural frequency, also the beam is subjected to fluid flow which is modeled by a nonlinear damping causes a positive feedback which is proportional to velocity of motion, so oscillation system draws energy from the fluid flow and can increase vibrations even in the case of free vibrations. However, it vanishes when motion stops so vibrations resulted from fluid flow are called a self-excited vibrations. Selfexcited oscillations are studied extensively in [7, 8]. The beam model under study model. Interaction between forced vibrations and self-excited vibrations may cause flutter. So our main purpose here is to reduce vibrations of the bending mode in order to prevent occurrence of flutter. Chapter 2 presents a quantitative analysis on the nonlinear behavior of the studied beam coupled to a positive position feedback controller PPF. In chapter 3, we use the delayed positive position feedback controller to reduce the bending mode vibrations of the beam under study. In chapter 4, an improved saturation controller and a velocity feedback controller are used together to eliminate the vibrations of the studied beam. |