Normalized Equations of Motion

Since the behavior of seismic isolated structures can be significantly affected by the nonlinear behavior of both spring and damping elements, it is convenient to recast the equations of motion in terms of parameters that relate to the strength of both the isolation system and the superstructure. These are parameters with which designers are often familiar.

For example, an elastomeric isolation system may be designed with an aggregate lead core strength of , where W is the seismic weight of the superstructure. For a friction pendulum bearing, this is equivalent to a design friction coefficient of 0.07. Similarly, the superstructure may be designed for a base shear of . This corresponds to a pushover curve for the superstructure that reaches a plateau at 25% of the structure’s seismic weight. Such behavioral parameters are particularly convenient, because they are meaningful to designers even in cases where details of the supported structure are either unknown or approximate. It is useful, however, to non-dimensionalize the equations of motion used in the studies reported herein so that they can be applied to a broad class of structures rather than only specific ones such as the case just mentioned. Normalization of the equation of motion for SDOF oscillators with bilinear springs subjected to both harmonic pulse and near-field earthquake excitations was represented in Makris and Black [2004a, 2004b]. While this work showed self-similarity of the response for a fixed yield displacement, the normalization

1 1 1 1

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below is for a 2-DOF structure, and is for the purpose of defining the nonlinear system of equations in terms of convenient non-dimensional design parameters.

To develop a system of equations of motion that are in terms of non-dimensional design parameters, we first express the structural displacements in terms of element deformations, rather than global displacements relative to an absolute frame of reference.

This transformation was used by Kelly [1996] to approximate dynamic characteristics of linear isolated structures. The extension here is to nonlinear behavior in the isolation system and superstructure. Let where the square equilibrium transformation matrix B is defined in Equation (6.1). Recalling that B is both square and full rank, the global displacements can then be expressed as . Substituting this transformation into Equation (6.3) yields

(6.12)

Premultiplying this by gives

(6.13)

To obtain the transformed mass matrix on the right side of Equation (6.13), it is necessary to solve for a modified influence vector r from . Doing so yields , and; hence Equation (6.13) becomes

(6.14)

where

(6.15)

The forces and are defined above in Equations (6.5) and (6.8), respectively. The linear portions of this system follow the results of Kelly [1996].

The system of equations in (6.14) is not in its most convenient form for parametric nonlinear analysis, since the explicit appearance of a mass matrix indicates that information about the supported structure is necessary. Additionally, in most common applications viscous damping associated with the supported structural elements may be considered small,

= T

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and therefore approximated as being linear. Therefore, the elements of are simply . To normalize this system of equations, each DOF is considered separately.

6.3.1.2.1 First Degree of Freedom

In developing a normalized system of equations of motion, consider the first row of Equation (6.14):

(6.16)

Diving by , this becomes

(6.17)

Following the notation of Kelly [1996], the superstructure mass ratio is defined as . Recognizing the viscous damping ratio of the isolated structure, where the supported structure is completely rigid, as ,

(6.18)

This formulation is practical, since the designer often wishes to establish the elastic isolated frequency of the structure, , before proceeding to a more detailed analysis.

Here, we define a frictional displacement of the isolation system as , or the ideal elastic displacement of the isolation system when subjected to the characteristic strength qy1. This parameter has been introduced to solve the differential equation of Coulomb-damped oscillators [Jacobsen and Ayre, 1958]. Note that this frictional displacement is not the same as the traditional yield displacement, which we define for the superstructure as

. The yield displacement of the superstructure is defined as , noting that is the elastic stiffness of the superstructure. If normalized displacements

and (for ) are substituted into Equation (6.18), the equation of motion becomes

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(6.19)

Here, we define a normalized isolator spring force , or the spring force as a fraction of the weight above the isolation system. This normalized spring force has the advantage of (a) being non-dimensional; and (b) approximating the pseudo-acceleration demand as a percentage of gravity. Therefore, a normalized yield strength follows the same definition, that is, . The ratio of isolator spring force to yield force is defined as . Here, we also define a normalized ground acceleration

. Applying these definitions, and substituting yields

(6.20)

This equation of motion has units of [sec^-2], so it is desirable to multiply by . Doing so leads to the non-dimensional nonlinear equation of motion below:

(6.21)

6.3.1.2.2 Second Degree of Freedom We now consider the second row of (6.14):

(6.22)

Dividing by , and recognizing that the viscous damping ratio of the superstructure can be expressed as , this becomes

(6.23)

Recalling the normalized displacement and substituting it into Equation (6.23), the equation of motion becomes

(6.24)

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It is noted that, by its above definition, the normalized displacement is also the superstructural ductility demand. It is also convenient to introduce a normalized superstructure spring force , or the spring force as a fraction of the weight of the superstructure (analogous to the spectral acceleration in units of g). Therefore, a normalized yield strength follows the same definition, that is, . The ratio of spring force to yield force is defined as . Applying these definitions, and

substituting yields

(6.25)

Premultiplying by leads to the non-dimensional equation of motion below:

(6.26)

6.3.1.2.3 Results of Normalization

Combining the developed normalized equations of motion in a convenient matrix form gives the following non-dimensional system of nonlinear differential equations

(6.27)

Substituting definitions for each matrix and vector above, we express compactly as

(6.28)

This non-dimensional form may be implemented in a numerical integration algorithm to determine the nonlinear response to ground acceleration given parameters that are generally of practical interest to the designer and have important physical interpretations. These equations form the basis for the analyses presented in subsequent 2-DOF parametric studies.

y2

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