## 352 Basic Principles and Methods of Analysis

where xg(t) is ground acceleration.

FIGURE 35.2 SDOF structure model.

FIGURE 35.3 SDOF structure model with passive damper element.

Consider now the addition of a generic passive or active control element into the SDOF model, as indicated in Figure 35.3. The response of the system is now influenced by this additional element. The symbol r in Figure 35.3 represents a generic integrodifferential operator, such that the force corresponding to the control device is written simply as Gx. This permits quite general response characteristics, including displacement-, velocity-, or acceleration-dependent contributions, as well as hereditary effects. The equation of motion for the extended SDOF model then becomes, in the case of an earthquake load, m€ + cx + kx + Gx = — (m + m )xg

with m representing the mass of the control element.

The specific form of Gx needs to be specified before Equation 35.3 can be analyzed, which is necessarily highly dependent on the device type. For passive energy dissipation systems, it can be represented by a x g m force-displacement relationship such as the one shown in Figure 35.4, representing a rate-independent elastic-perfectly plastic element. For an active control system, the form of Tx is governed by the control law chosen for a given application. Let us first note that, denoting the control force applied to the structure in Figure 35.1 by u(t), the resulting dynamical behavior of the structure is governed by Equation 35.3 with

Suppose that a feedback configuration is used in which the control force u(t) is designed to be a linear function of measured displacement x(t) and measured velocity x(t). The control force u(t) takes the form u(t)= g1x(t)+g2X (t) (35.5)

In view of Equation 35.4, we have

The control law is, of course, not necessarily linear in x(t) and X(t) as given by Equation 35.5. In fact, nonlinear control laws may be more desirable for civil engineering applications (Wu and Soong 1995). Thus, for both passive and active control cases, the resulting Equation 35.3 can be highly nonlinear.

Assume for illustrative purposes that the base structure has a viscous damping ratio Z = 0.05 and that a simple massless yielding device is added to serve as a passive element. The force-displacement relationship for this element, depicted in Figure 35.4, is defined in terms of an initial stiffness and a yield force fc.. Consider the case where the passively damped SDOF model is subjected to the 1940 El Centro S00E ground motion as shown in Figure 35.5. The initial stiffness of the elastoplastic passive device is specified as k = k, while the yield force fc is equal to 20% of the maximum applied ground force. That is, fy = 0.20 Max{ m|€g|}

The resulting relative displacement and total acceleration time histories are presented in Figure 35.6. There is significant reduction in response compared to that of the base structure without the control element, as shown in Figure 35.7. Force-displacement loops for the viscous and passive

FIGURE 35.4 Force-displacement model for elastic-perfectly plastic passive element.
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