Second Order Step Response

How 2nd order systems (those with 2 energy storing elements) respond to step inputs.

Standard form

$$\frac{1}{{\omega }_n^2}\frac{d^2}{d{t}^2}y(t)+\frac{2\zeta }{{\omega }_n}\frac{d}{dt}y(t)+y(t)=u(t)$$

• ${\omega }_n$ is the undamped frequency of the system response
  • Indicates the speed of the response
• $\zeta$ is the damping factor
  • Indicates the shape of the response

Forced Response

• Forces response is the response to a non-zero input, namely
  • Step
  • Sinusoidal
• Initial conditions are zero, it $y(0)=0$, $\frac{d}{dt}y(0)=0$
• The response is the solution to a non-homogeneous second order differential equation

Damped Response

There are 4 different cases for system response:

The response of a system to the same input with varying damping factors is shown in the graph below, from the data book. The equations are also given in the data book.

Undamped

The system is not damped at all and is just a normal sinusoidal wave.

$$y(t)=H(1-\cos {\omega }_{n}t)$$

Underdamping

The amplitude of the sinusoidal output decreases slowly over time to a final "steady state" value.

$$y(t)=H[1-\frac{e^{-\zeta {\omega }_{n}t}}{\sqrt{1-{\zeta }^2}}\sin (\sqrt{1-{\zeta }^2}{\omega }_{n}t+\varphi )]$$ $$\tan \varphi =\frac{\sqrt{1-{\zeta }^2}}{\zeta }$$

Critical Damping

This gives the fastest response, where the output rises to its final steady state value.

$$y(t)=H(1-({\omega }_{n}t)e^{-{\omega }_{n}t})$$

Overdamping

The output rises slowly to its steady state value $$y(t)=H[1-\frac{e^{-\zeta {\omega }_{n}t}}{\sqrt{{\zeta }^2-1}}\sinh (\sqrt{{\zeta }^2-1}{\omega }_{n}t+\varphi )]$$ $$\tan \varphi =\frac{\sqrt{{\zeta }^2-1}}{\zeta }$$