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Unformatted text preview: S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19- Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free vibrations of systems. (The concept of free vibrations is important; this means that although an outside agent may have participated in causing an initial displacement or velocityor both of the system, the outside agent plays no further role, and the subsequent motion depends only upon the inherent properties of the system. This is in contrast to forced motion in which the system is continually driven by an external force.) We shall consider only undamped systems for which the total energy is conserved and for which the frequencies of oscillation are real. This forms the basis of the approach to more complex studies for forced motion of damped systems. We saw in Lecture 13, that the free vibration of a mass-spring system could be described as an oscillatory interchange between the kinetic and potential energy, and that we could determine the natural frequency of oscillation by equating the maximum value of these two quantities. (The natural frequency is the frequency at which the system will oscillate unaffected by outside forces. When we consider the oscillation of a pendulum, the gravitational force is considered to be an inherent part of the system.) The general behavior of a mass-spring system can be extended to elastic structures and systems experiencing gravitational forces, such as a pendulum. These systems can be combined to produce complex results, even for one-degree of freedom systems. We begin our discussion with the solution of a simple mass-spring system, recognizing that this is a model for more complex systems as well. 1 In the figure, a) depicts the simple mass spring system: a mass M, sliding on a frictionless plane, restrained by a spring of spring constant k such that a force F ( x ) = kx opposes the displacement x . (In a particular problem, the linear dependence of the force on x may be an approximation for small x.) In order to get a solution, the initial displacement and initial velocity must be specified. Common formulations are: x (0) = 0, and dx (0) = V (The mass responds to an initial impulse.); or x (0) = X and dx (0) = (The mass is given dt dt an initial displacement.). The general formulation is some combination of these initial conditions. From Newtons law, we obtain the governing differential equation d 2 x m = kx (1) dt 2 with x (0) = X , and dx (0) = V . dt The solution is of the general form, x ( t ) = Re ( Ae it ), where, at this point in the analysis, both A and are unknown . That is, we assume a solution in which both A and are unknown, and later when the solution is found and boundary conditions are considered, we will end up taking the real part of the expression....
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- Fall '09