Theory

System dynamics is an important area of study in widespread engineering applications such as vibrations, electric circuits, and control systems. Dynamic performance characteristics of a system describe how the system responds to a varying input. The most useful mathematical model for representing system behavior is the ordinary linear differential equation with constant coefficients. Accordingly, the relationship between system input *x*(*t*) and system output *y*(*t*) may be written in the following form.

where *a* and *b* are constants dependent on system physical parameters.

The corresponding transfer function of the system is

where *s* is the Laplace operator.

Fig. 1 shows a block diagram of the system with input *x*(*t*) and output *y*(*t*).

**1. Test Inputs**

The following test inputs are normally used for testing the dynamics of a system. For characterizing system in time domain, the test inputs used are impulse, step, and ramp. A swept frequency sine wave is used to characterize system in frequency domain.

**1.1 Impulse Input**

The unit impulse function is defined as for

and is zero elsewhere.

The Laplace transform of unit impulse is given by

Unit impulse input

Unit impulse response

**1.2 Step Input**

The unit step function is defined as

The Laplace transform of unit impulse is given by

Unit step input

Unit step response

**1.3 Ramp Input**

The unit ramp function is defined as

The Laplace transform of unit ramp is given by

Unit ramp input

Unit ramp response

**1.4 Sine Wave Input**

The unit sine wave function *x*(*t*) is defined as

A swept-frequency sine wave input is used to characterize system in frequency domain and the frequency response is obtained. The frequency response consists of two plots: Gain versus frequency and phase versus frequency.

Frequency response , where .