PSI - Issue 18

Bilal L. Khan et al. / Procedia Structural Integrity 18 (2019) 108–118

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Bilal L. Khan et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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varies from 20% to 30% reduction. 5th floor and 6th floor, stiffness reduction varies from 40% to 50%. 7th and 8th Floor, stiffness reduction varies from 70 to 90%. Percentage change in damping from 1st floor to 4th floor varies from 10% to 15% reduction. 5th floor and 6th floor, damping reduction varies from 24% to 29%. 7th and 8th Floor damping reduction varies from 46 to 68%. Physical representation of base isolated structure is presented in figure 1. Notations k & c represent stiffness and damping of each storey respectively. u represent seismic force, x0 to x8 represent storey displacements, xg represents ground displacement due to earthquake, r represents relative displacement / inter-storey drift, k b & C b are stiffness and damping of passive base isolator and m b is the mass of base isolation layer.

Fig. 1. Simplified model of Base Isolated structure.

3. Methodology

3.1. Mathematical Modelling Technique

The state-space representation is a mathematical Technique for solving a system of ordinary differential equations. The model of a physical system represented by mass [M], stiffness [K] and damping [C] matrices along with input, output and state variables are transformed into the format of state-space equations. The state of the system in terms of a state vector is then determined using any mathematical technique. The dynamic equation of motion for a discrete mechanical system with n degrees of freedom can be written as [ ] ̈ + [ ] ̇ + [ ] = [ ] = [ ][ ] (1) The above equation can be reduced to first order differential equation and written in discrete time state space as +1 = + (2) = + (3) Where A is the system matrix, it defines the dynamics of the system (mass matrix. Stiffness matrix and damping matrix), B is the input matrix, C is output matrix and D is feed forward matrix, respectively. uk matrix contains excitation vectors usually acceleration time history, xk is the time state vector and yk is the output vector matrix at time instant k. The above state space representation does not contain any process and measurement errors that