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Determination of Dynamic Performance Characteristics of Higher Order Systems
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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  .

 

 

                             

 

                               

 

 

 

 

 

 

 

 


                                                                                                                                                               

                                                                                                                     

 

 

 

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