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elements are displayed in Figure 35.8. In this case, the size of these loops indicates that a significant portion of the energy is dissipated in the control device. This tends to reduce the forces and displacements in the primary structural elements, which of course is the purpose of adding the control device.

### 35.2.2 Multi-Degree-of-Freedom Structural Systems

In light of the preceding arguments, it becomes imperative to accurately characterize the behavior of any control device by constructing a suitable model under time-dependent loading. Multiaxial representations may be required. Once that model is established for a device, it must be properly incorporated into a mathematical idealization of the overall structure. Seldom is it sufficient to employ an SDOF idealization for an actual structure. Thus, in the present subsection, the formulation for dynamic analysis is extended to a MDOF representation.

The finite element method (FEM) (e.g., Zienkiewicz and Taylor 1989) currently provides the most suitable basis for this formulation. From a purely physical viewpoint, each individual structural member is represented mathematically by one or more finite elements having the same mass, stiffness, and damping characteristics as the original member. Beams and columns are represented by one-dimensional elements, while shear walls and floor slabs are idealized by employing two-dimensional finite elements. For more complicated or critical structural components, complete three-dimensional models can be developed, and incorporated into the overall structural model in a straightforward manner via substructuring techniques.

The FEM actually was developed largely by civil engineers in the 1960s from this physical perspective. However, during the ensuing decades the method has also been given a rigorous mathematical foundation, thus permitting the calculation of error estimates and the utilization of adaptive solution strategies (e.g., Szabo and Babuska 1991). Additionally, FEM formulations can now be derived from variational principles or Galerkin weighted residual procedures. Details of these formulations is beyond our scope. However, it should be noted that numerous general-purpose finite element software packages currently exist to solve the structural dynamics problem, including ABAQUS, ADINA, ANSYS, and NASTRAN. While none of these programs specifically addresses the special formulations needed to characterize structural protective systems, most permit generic user-defined elements. Alternatively, one can utilize packages geared exclusively toward civil engineering structures, such as SAP 2000, ETABS, DRAIN, and IDARC, which in some cases can already accommodate typical passive elements.

Via any of the above-mentioned methods and programs, the displacement response of the structure is ultimately represented by a discrete set of variables, which can be considered the components of a generalized relative displacement vector x(t) of dimension N. Then, in analogy with Equation 35.3, the N equations of motion for the discretized structural system, subjected to uniform base excitation and time varying forces, can be written as

where M, C, and K represent the mass, damping, and stiffness matrices, respectively, while T symbolizes a matrix of operators that model the protective system present in the structure. Meanwhile, the vector xg contains the rigid body contribution of the seismic ground displacement to each degree-of-freedom. The matrix M represents the mass of the protective system.

There are several approaches that can be taken to solve Equation 35.8. The preferred approach, in terms of accuracy and efficiency, depends upon the form of the various terms in that equation. Let us first suppose that the protective device can be modeled as direct linear functions of the acceleration, velocity, and displacement vectors. That is,

Then, Equation 35.8 can be rewritten as

in which

Equation 35.10 is now in the form of the classical matrix structural dynamic analysis problem. In the simplest case, which we will now assume, all of the matrix coefficients associated with the primary structure and the passive elements are constant. As a result, Equation 35.10 represents a set of N linear second-order ordinary differential equations with constant coefficients. These equations are, in general, coupled. Thus, depending upon N, the solution of Equation 35.10 throughout the time range of interest could become computationally demanding. This required effort can be reduced considerably if the equation can be uncoupled via a transformation; that is, if M, C, and K can be diagonalized. Unfortunately, this is not possible for arbitrary matrices M, C, and K. However, with certain restrictions on the damping matrix C, the transformation to modal coordinates accomplishes the objective via the modal superposition method (see, e.g., Clough and Penzien 1975).

As mentioned earlier, it is more common having Tx in Equation 35.9 nonlinear in x for a variety of passive and active control elements. Consequently, it is important to develop alternative numerical approaches and design methodologies applicable to more generic passively or actively damped structural systems governed by Equation 35.8. Direct time-domain numerical integration algorithms are most useful in that regard. The Newmark beta algorithm, for example, is one of these algorithms and is used extensively in structural dynamics.

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