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Unformatted text preview: Lecture 4 Damped free vibrations So far we have been studying systems that have no friction. This is reasonable for a start. Many systems have dynamic properties that are dominated by their inertia, and friction can be neglected at the first pass. However, frictional processes are commonly present in en- gineering and in this lecture we begin the study of their effects on vibrations. The most common effect is that friction makes systems return to equilibrium in which forces and velocities are zero. This has broader consequences and as far as we know one day the universe is supposed to return to equilibrium too...but not before we make a lot of out of equilibrium rumble :-). Now, neglecting the fact that it is slowly killing the known universe, friction can be very useful. Remember how we said that often the job of a design engineer is to quench vibrations? This can be quite an important job when designing structures that are dominated by inertia and sit on shaky ground as is the case for tall buildings in California (not that we have any in Santa Barbara). In fact, engineers put large viscous dampers such as those shown in figure 1 in place where vibrational effects are bad. Such dampers dissipate the kinetic energy of seismic motion into heat, and are essential for per- sistence of structures (another design, driven by survival of occupants and economic issues uses plastic hinges that undergo plastic deforma- tions at large strains, but such structures can end up unusable after an earthquake). Viscous dampers consist of a closed cylinder containing a viscous fluid. The one that was passed around in class has air as the working fluid. But the usual choice is silicon oil due to its viscosity and easy flow properties. A piston rod is connected to a piston head with small holes in it. The piston can move in and out of the cylinder. As it does this, the oil is forced to flow through holes in the piston head causing friction. There are other types of friction whose effect on vibrations we are going to study, such as Coulomb friction, when two dry surfaces rub against each other. But lets start with viscous damping. 1 Figure 1: Dampers used for protection against seizmic effects in large build- ings Viscously damped free vibrations Viscous dampers, whose physical function was described above, are characterized by force-velocity relationship. Recall how the spring force was characterized by force-displacement relationship, and that rela- tionship could be linear or nonlinear. The same holds for force-velocity relationship of viscous dampers. Different types of such behavior are shown in figure 2, which shows magnitude of viscous friction force F against velocity. The force can be usually written as: F =- sgn ( x ) | x | , where sgn ( x ) means the sign of the velocity - direction of the viscous...
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- Winter '08