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EXPERIMENT 2
OPEN-LOOP DYNAMICS RESPONSE OF SECOND ORDER SYSTEMS
1. OBJECTIVES
1.1. To study the open loop time domain response of second order system with
various transfer function.
1.2. To determine the different of output from open loop response with RAMP, STEP
or IMPULSE input.
1.3. Software analysis using MATLAB/SIMULINK.
2. INTRODUCTION
If mechanical parts of a system or fluid in a system are subject to acceleration; the
dynamics behavior is described by second order equations. A current across an RLC
electrical circuit and a mass suspended from a spring are classical examples of second
order systems.
Consider the following first order system:
Figure 3.1: RLC Circuit
Figure 3.2: Spring –mass Damper System
The simple spring-mass-damper mechanical system shown here is described by
Newton's second law of motion. This system could represent for example an
automobile shock absorber. Therefore, we obtain:
M
d 2 y (t )
dy (t )
f
 Ky(t )  r (t )
2
dt
dt
Where K is the spring constant of the ideal spring and f is the friction constant.
This equation is a linear constant coefficient differential equation of second
order.
Alternatively, one may describe the electrical RLC circuit of the previous figure utilizing
Kirchhoffs current law. Then we obtain the following integrodifferential equation:
V (t )
dV (t ) 1 t
C
  V (t ).d (t )  r (t )
R
dt
L 0
The solution of the differential equation describing the process may be obtained by the
classical methods such as the use of integrating factor, the method of undetermined
coefficient or the Laplace transform method.
Then, we assume that a second order dynamic model can be put into the following
standard forms:
a.
an ordinary differential equation:
2
b.
d 2 y (t )
dy (t )
 2
 y (t )  Ku(t ) or by,
dt
dt
A transfer function:
G ( s) 
Y ( s)
K
 2
U ( s )  s  2s  1
Where
y(t)= Process output
u(t)= Process input
  Process time constant
K= Process(Steady State) gain
 = Damping Factor
G(s)=Process Transfer Function
The characteristic equation for this second order system is:

2
s  2s  1  0
The root of this equation may be real, equal or complex depending
on the numerical values of  and  . In the following, we shall
investigate the dynamics response of the second order system to unit
STEP input.
3. PROCEDURE
Consider the following second order system:
Figure 3.3: System Showing input and output
Note that   7 and
  0.5 for this system
3.1. Obtain the unit-STEP, RAMP and IMPULSE INPUT for this system using
simulink.
3.2. There are two parameters that completely define the dynamics of a second order
system, the time constant  and the damping factor  .
a) For above process, select 3 values of such that:
 1
i.
 1
ii.
iii. 0    1
 0
iv.
Use simulink to obtain unit STEP response only and superimpose the four
response. Obtain roots of the three different by the use off command
”roots” in MATLAB and manually plot the roots.
b) Make a small table that show the equivalence of the terminology:
i.
Overdamped, underdamped, undamped and critically
damped response.
ii.
Damping factor,   1 ,   1 , 0    1 and   0 .
iii.
Two real, two complex, two imaginary and two real
repeated root of the characteristics equation.
c) Investigate the significant of  by changing its value: e.g for 0    1 , try
  1,7 and 10. in each case obtain roots of the dominator of G(s).
3.3. For a typical of second order response, briefly define the following performance
criteria and where appropriate relate the by mathematical expression to the
parameters K,  ,  and M.
a. Rise time, t r
b. Time to first peak, t p
c. Settling time, t s
d. Overshoot
0.78
3.4. For G ( s ) 
, given a STEP input , using simulink find
2
49s  7 s  1
t p , t s , t r and percentage of overshoot.
4. APPLICATION
A unit step is given to the input of a second order system, the following is the plot of the
output response vs time:
Figure 3.4: Second Order Output Response.
From this plot determine:
a. Time to first peak, t p
b. Value at the first peak, M p
c. Settling time, t s
Then use mathematical relations below to calculate  n ,  and K. Write and analytical
expression in term of transfer function for this system. Use simulink to verify your
results.
tp 

d
d  n 1   2
   n
ts 
4

 
1
n
5. REPORT
Write your observations and calculations based on the experiment. The report should also
include answers and calculation to all questions.