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Note that the amplitude of a motion depends on the given initial conditions and that all the motions are in the same manner except they are displaced relative to each other along time t. The relative magnitude in radians between x, x, and x is called the phase angle.

3.2.2.2 Damped Vibration with Initial Conditions

In the previous discussion, we assumed an ideal vibrating system free from internal and external damping. Damping maybe defined as a force that resists motion at all times. Therefore, a free undamped vibration continues in motion indefinitely without its amplitude diminishing or its frequency changing. Real systems, however, do not possess perfectly elastic springs nor are they surrounded by a frictionless medium. Various damping agents — such as the frictional forces of structural joints and bearing supports, the resistance of surrounding air, and the internal friction between molecules of the structural materials — always exist.

It is difficult, if not impossible, to derive a mathematical formula for damping resistance that represents the actual behavior of a physical system. A simple yet realistic damping model for mathematical analysis is that the damping force is proportional to velocity. This model can represent structural damping of which the force is produced by the viscous friction and is therefore called viscous damping. Figure 3.4 shows a vibration model consisting of an ideal spring and dashpot in parallel. The dashpot exerts a damping force, cx, proportional to the relative velocity, in which c is a proportionality and is called the coefficient of viscous damping. The governing differential equation is

of which the standard solution is x = C1eait + C2ea2 f (3.6)

where C1 and C2 are integration constants, and a1 and a2 may be expressed as

After substituting Equations 3.7 and 3.8 for the corresponding terms in Equation 3.6, possible solutions can be obtained for three cases of c2/4M2 = KM, c2/4M2 > KM, and c2/4M2 < KM, corresponding to critical damping, overdamping, and underdamping, respectively. When c2/4M2 = KM, the value of c is called critical damping and takes the form

The ratio of c/ccr is called viscous damping factor or simply damping factor, p, and may be expressed as c c c cp p = ccr = 2Mp = 2vtKM) = 2K (3:10)

In most structural and mechanical systems, the assumption of underdamping is justified, that is, p < 1. For this case, the motion equation is

The displacement response may be obtained from x = e-ppt (A cos a/1 - p2 pt + B sin \/1 — p2 pt) (3.12a)

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