Single Degree of Freedom, Damped, Forced Systems.

a. Vibrations of foundation-soil systems can adequately be represented by simple mass-spring-dashpot systems.  The model for this simple system consists of a concentrated mass, m, supported by a linear elastic spring with a spring constant, k, and a viscous damping unit (dashpot) having a damping constant, c.  The system is excited by an external force, e.g., Q = Qo sin (ωt), in which Qo is the amplitude of the exciting force, ω = 2πfo is the angular frequency (radians per second) with fo the exciting frequency (cycles per second), and t is time in seconds.

Response curves for the single-degree-of-freedom system with viscous damping.
Figure 17-2.  Response curves for the single-degree-of-freedom system with viscous damping.


b. If the model is oriented as shown in the insert in figure 17-2(a), motions will occur in the vertical or z direction only, and the system has one degree of freedom (one coordinate direction (z) is needed to describe the motion).  The magnitude of dynamic vertical motion, Az, depends upon the magnitude of the external excitation, Q, the nature of Qo, the frequency, fo, and the system parameters m, c, and k.  These parameters are customarily combined to describe the "natural frequency" as follows:

c. Figure 17-2(a) shows the dynamic response of the system when the amplitude of the exciting force, Qo, is constant.  The abscissa of the diagram is the dimensionless ratio of exciting frequency, fo, divided by the natural frequency, fn, in equation (17-1).  The ordinate is the dynamic magnification factor, Mz, which is the ratio of Az to the static displacement, Az  = (Qo/k). Different response curves correspond to different values of D.

d. Figure 17-2(b) is the dynamic response of the system when the exciting force is generated by a rotating mass, which develops:

e. The ordinate Mz.  (fig 17-2(b)) relates the dynamic displacement, Az, to me e/m.  The peak value of the response curve is a function of the damping ratio and is given by the following expression:

For small values of D, this expression becomes 1/2D. These peak values occur at frequency ratios of f




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