Impulse Response Function

Describes when a sudden force is applied to a mass spring system

Transfer Function

Given:

$$a{y}^{\prime \prime }+b{y}^{\prime }+cy=g(t);y(0)=y^{\prime }(0)=0$$

The transfer function is:

$$H(s)=\frac{Y(s)}{G(s)}=\frac{1}{a{s}^2+bs+c}$$

where $Y$ and $G$ are the Laplace Transform of the respective function

The impulse response function is:

$$h(t)={\mathcal{L}}^{-1}\{H(s)\}$$

Solving Using

1. Find the solution to the Homogeneous Linear Equation (with initial conditions)
2. Determine the impulse response function (above)
3. $y(t)=y_h(t)+(h\ast g)$
   • Convolution of $h$ and $g$
   • Just $h(t)$ when $g(t)$ is the Dirac Delta Function and there are no initial conditions