Vibrations: With Resonator and Stability Concepts

Vibrations: With Resonator and Stability Concepts

DOI: 10.4018/978-1-5225-3079-4.ch002
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Vibration concepts are reviewed. Single degree-of-freedom vibration (SDOF) are analyzed. Subsequently, the analysis is extended to two degrees-of-freedom (2DOF) systems and coupling in a 2DOF system. The analysis of parametric coupling is introduced. Two sections on energy flow and the modeling of damping follow. Normal modes and mode shapes for systems with multiple degrees-of-freedom (MDOF) will then be considered. By generalizing MDOF systems to continuous systems, we can analyze bending modes in plates. Experimental modal analysis is introduced to prepare the reader for later application of this technique to full-scale operational gates. The second section of this chapter reviews fundamental concepts of fluid-structure systems with resonance. The chapter concludes with a short discussion of stability concepts.
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Single Degree Of Freedom (Sdof) Damped Systems

A sufficiently compact body can be modeled as a concentrated point mass, m. When the concentrated mass is suspended on a linear spring with a spring constant, k, and exposed to viscous damping (proportional to and opposing the body velocity), c, the equation of motion for the natural vibration of the body with a single degree-of-freedom can be derived from 978-1-5225-3079-4.ch002.m01 yielding


After dividing by m and introducing the symbols 978-1-5225-3079-4.ch002.m03 to represent the square of the undamped natural frequency, and 978-1-5225-3079-4.ch002.m04 to represent the damping ratio, Equation 1 can be re-written as


If the system is driven by a sinusoidal force given by 978-1-5225-3079-4.ch002.m06, the equation of motion becomes


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