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<p>A new model for Rolling Isolation Systems(RIS) is developed using Gauss's Principle
of Least Constraint (GPLC). Gauss Principle of Least Constraint is a versatile method
that can be used to model systems with constrained dynamics and is agnostic to the
type of constraints that need to be enforced. In GPLC models, constraints are enforced
in the acceleration level regardless of their classification and the equations of
constrained motion are then presented in KKT form. Since, enforcing constraints in
differentiated form may lead to numerical integration issues, a novel method, Direct
State Correction has been proposed and implemented. Direct State Correction, unlike
other constraint stabilization techniques, ensures that the constraints are exactly
satisfied at each time step by correcting the states of the system before computing
each state derivative. The widely used method of Baumgarte stabilization does not
stabilize constraint errors for the RIS system studied in this work and in addition
it appears to provide no additional stability or accuracy when implemented along with
Direct State Correction Method.</p><p>In order to optimize the RIS, a data set of
real recorded building motions from the Strong Motion Virtual Data Center is put together
with consideration of the type of buildings, earthquakes and locations the RIS is
expected to perform in. A total of 62 tri-axial floor motions were selected from 32
buildings and 5 representative earthquakes. It is discovered buildings undergo significant
vertical motion and the current isolation system is designed to only be effective
for horizontal motion, and therefore a vertical isolation system is added to the current
RIS. </p><p>The dynamics of the RIS are purely dependent on the dish shape and damping
present in the system, so with these design variables in mind, the new RIS with vertical
isolation is optimized under a probabilistic framework. The optimization uses Incremental
Dynamic Analysis (IDA) and the theorem of total probability in order to arrive at
an objective function that minimizes the overall rate of exceedance of a desired threshold
probability. The optimization is found to be sensitive to the initial starting guess
of the design variables, the input motion used to simulate the RIS and the thresholds
set for evaluating the performance of the RIS. The optimal shape of the dishes is
found to be a variation of a cubic profile and linearly increasing rolling resistance
damping. This new design for the RIS shows significant improvement over the current
system. Finally, an implementation of the proposed design of the RIS dish shape, damping
and vertical isolation system is presented.</p>
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