Improving Solutions of Nonlinear Differential Equations Using Control

Vibrations and dynamic chaos are undesired phenomenon in structures as they cause the 4D. They are: disturbance, discomfort, damage and destruction of the system or the structure. For these reasons, money, time and effort are spent to eliminate or control vibrations,noise and chaos or to minimize them. The main object of this thesis is the investigation of the behavior two systems. They are simple and spring pendulums. A tuned absorber (passive control), in the transverse direction and/or the longitudinal one is connected to both systems to reduce the oscillations. Negative velocity feedback or its quadratic or cubic value is applied to the systems (active control). Also active control is applied to the systems via negative acceleration feedback or negative angular displacement or its quadratic or cubic value. Multiple scale is applied to determine approximate closed form solutions for the differential equations describing the systems. Both frequency response equations and the phase plane technique are applied to study systems stability. Optimum working conditions of both systems are extracted when applying both passive and active control, to be used in the design of such systems.