
## 4 Response Prediction

The basic idea behind the proposed methodology is to predict the structural response of single degree of freedom system i.e. single storey building subject to various earthquake forces. Two cases viz without damping and with damping have been considered for the analysis. 

## Case(I) **: Without Damping**

Let M be the mass of the generalized one storey structure, K the stiffness of the structure and x be the displacement relative to the ground then the equation of motion may be written as: 
Mx&& + Kx = −Ma&& )1(
&&

&&

a Ground **acceleration.**
x **Displacement,** x Response **acceleration,**
=
= =
where, Equation (1) may be written as, 
)2( 2 x&& +ω x = −a&&
Where ω 2**=K/M, is the natural frequency parameter of the undamped structure.** 
The solution of equation (2) [Ref. 2] is given by 

$\eqref{eq:walpha}$
= − ∫ −
t

```
          0
           (a )sin[ t( )]d )3(
        1
)t(x τ ω τ τ
        ω
         
           &&

```

From this solution the response of the structure viz. acceleration is obtained for no damping. 