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Determination of Dynamic Performance Characteristics of Second Order System
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Theory

 

Systems

Systems are specified as zero-order, first-order, and second-order depending on the order (highest derivative term) of the differential equation. The following sections describe the system models, real life examples and equations of responses for the test inputs.

 

Second-Order System

The highest order term in the differential equation of the second-order systems is second derivative. The second-order systems follow the equation.

After rearranging the equation in the standard form, the equation for the second-order system is

Where   is called undammed natural frequency, is called damping ratio, and  

is the static sensitivity of the system.

                                                             

 

                                                                                                                          Fig .1      

 

The transfer function of the second-order system is

An example of a second-order measurement system is a mass-spring-damper assembly shown in above Fig. 1


 

 

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.   

                                                                                                       

                                                                            Unit impule input

 

The Laplace transform of unit impulse is given by

Unit impulse response 

 

1.2 Step Input

The unit step function u(t) 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 r(t) is defined as

                                                                    Unit ramp input

 

The Laplace transform of unit ramp is given by


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.

                                 

                                  where .

 

 


                   

 

                                                                                                                         

 

 

 

 

                                                              

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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