# Lab 3 - Second Order System

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I.     MASS-SPRING-DAMPER SYSTEM
A.     OBJECTIVE
The purpose for this experiment is to analyze a mass-spring-damper system and then
model it as a second order system using the results.

B.     PROCEDURE
In order to get a general overview for the system, the motor was first run from very low
frequencies up to the highest frequency.      Observations were made on the system to help
understand what will be seen later in the experiment. Then, a calibration was performed by
measuring the output voltages, mass displacements, and base displacements. To determine the
natural frequency, the frame attached to the spring was removed and the oscillations of the spring
were modeled on the oscilloscope to record the natural frequency. The next action was to record
some time response data of the system with a relatively high damping value. The mass was
manually pulled down as far as possible; it was then released in order to record the results from
the oscilloscope.    Using the same damping position, the frequency response was then
investigated. Five different frequencies were generated by the motor for the damping position
providing several measurements to take in order to characterize the system.         Once all the
measurements were recorded, the same procedure was used starting from the time response
process was performed but using different damping settings.

C.     EXPERIMENTAL RESULTS
The measurements taken from the initial calibration are displayed in Table I.1.

Table I.1. Calibration Results

Mass (kg) Displacement (cm) Output Voltage (V)
0             0                 0
1.0123          0.9                2
2.0246          1.7                4
3.0369          2.6                6

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A plot was created from the data, resulting in a linear curve fit that helps in evaluating the
spring constant of the system as illustrated in Figure I.1.

0.03

0.025
Displacement (m)

0.02

0.015

0.01

0.005

0
0     5       10      15      20          25       30   35     40      45      50
Force (N)

Figure I.1 Displacement versus Force Graph

From this graph, a linear fit trendline reveals the slope to be 0.0009 m/N. This is
equivalent to the value of 1/k. The value for k is found to be 1111.1 N/m. The theoretical
natural frequency from the measurements can be calculated using Equation I.1.

k
n                                                      (I.1)
m
The mass in the previous equation was the mass of two plates and the mass of the carrier.
After plugging in the data, the theoretical natural frequency was found to be 17.27 rad/s.
Because this measurement was not in the correct units for comparison, Equation I.2 was used to
determine the natural frequency in units of hertz.

n
f                                                      (I.2)
2
The value for the theoretical natural frequency was then calculated to be 2.75 Hz. Table
I.2 shows the results from the time response analysis of the mass-spring-damper system.

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Table I.2. Time Response of Second Order System

Damping tp(ms)          t d(ms)    x1(mV)    xn (mV) xss (mV)
Low           240        375        18        2.8      410
High          360        330        110       10       490
Medium        260        380        380       70       390

To acquire another comparison for the second order system, a collection of data
from the frequency response method was inputted into Table I.3.

Table I.3. Frequency Response of Second Order System

Damping Frequency (Hz) Vout (V) Vin (V) D t (ms) f (degrees)
1           0.680     0.680      0         0
2.5          1.92      0.680     155      139.5
Low             3           2.24      0.680     70        75.6
4           0.600     0.680     115      165.6
5           0.360     0.680     95        171
1           0.660     0.660     40        14.4
2.5          0.720     0.660     325      292.5
High            3           0.580     0.660     260      280.8
4           0.400     0.660     165      237.6
5           0.260     0.660     120       216
1           0.640     0.660     20        7.2
2.5          1.86      0.660     360       324
Medium            3            1.8      0.660     240      259.2
4           0.560     0.660     140      201.6
5           0.320     0.660     105       189

The equation for lag, represented by phi in Table I.3, is shown in Equation I.3.

f  Dt  f  360 o                                  (I.3)
Because some of the values are greater than 1800, the values were converted to the
corresponding angle between 1800 and -1800 to aid in the numerical modeling.

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D.     ANALYSIS AND DISCUSSION
Theoretical Natural Frequency
The measurement of the natural frequency was taken experimentally with no damping
and found to be 3.23 Hz. The percent error between the theoretical natural frequency and the
experimental values comes to 14.86%. This value shows the uncertainty in the calculation of the
natural frequency when using the graph and its slope to estimate the value for the spring
constant. The estimated value of natural frequency was 2.75 Hz as stated earlier.

Time Response
Time response data collected in laboratory for the highly damped and nearly undamped
scenarios was used in log decrement analysis to find the damped natural frequencies and the
damping ratio for the highly and nearly undamped cases. The log decrement formula, Equation
I.4, is used to obtain the damping ratio since

1  x0                       2
ln          n t d                                        (I.4)
n  xn


                    1 2

where n is the theoretical natural frequency,  is the damping ratio, xo is the amplitude of the

first peak measured, xn is the amplitude of the nth consecutive peak, and td is the damped period.
The log decrement formula was simplified to directly solve for the damping ratio in Equation I.5.

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 2        
  1 
  t      
                                       (I.5)
 n d       
The damped natural frequency is a function of the damping ratio. Equation I.6 was used
to calculate the damped natural frequency, where d is the damped natural frequency.

d  n 1  2                                           (I.6)
The time for the highly damped and nearly undamped step response scenarios to reach
their respective first peak overshoot was calculated using Equation I.7.


tp                                                  (I.7)
d
Table I.4 shows the breakdown for the three calculations as described above.

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Table I.4 Step Response Analysis

Damping Ratio, ζ  n (rad/s) Peak Time, tp,(ms)
Highly Damped              0.4870                  19.04   165
Nearly Undamped             0.6399                 16.755   188

To provide an understanding of these results, they can be compared to the time response
results as shown in Table I.2. The nearly undamped case proves to have more accurate values
when compared to the calculated value from the frequency response; whereas, the highly damped
case does not reveal any similarity. This shows that unless the data is highly accurate at the
initial source, then the final calculations do not prove to be a good representation of the system.
Because so many calculations are used in the process of evaluating, the error in the values tends
to increase significantly.

Frequency Response
Frequency response data collected in laboratory for the highly damped and nearly
undamped cases can be used to calculate the damping ratio and the damped natural frequency.
The damping ratio is found using Equation I.8, where Mr is the magnitude of the resonant peak
and is found using Equation I.9,

1
1 1             2
Mr
                                                        (I.8)
2
A max
Mr                                                       (I.9)
A max-low -frequency
where Amax is the amplitude at resonance. Amax-low-frequency is an amplitude at a very low
frequency. The natural frequency can be calculated for the highly damped and nearly undamped
cases using Equation I.10.

r
n                                                   (I.10)
1 2

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The values for both damping cases from the three previous equations are displayed in
Table I.5. These values are used to derive the peak time and settling time for the highly damped
case.

Table I.5 Frequency Response Analysis

Highly Damped   1.091              0.548         18.779        15.708
Nearly Undamped 3.294              0.1536        19.076        18.850

The peak time is calculated in the same manner it was calculated for the time response
case. The damped natural frequency, d, is calculated via Equation I.6. The subject peak time is
calculated by plugging in the damped natural for the frequency response case into Equation I.7.
The peak time for the frequency response scenario is found to be 167 ms. The settling time for
the highly damped case is found by Equation I.11.

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ts                                                  (I.11)
 n
Substituting 18.779 rad/s as the natural frequency and 0.548 as the damping ratio of the
highly damped case, the resulting settling time is 388 ms. The peak time from the frequency
response does not resemble the measured value any better than from the step response. The
settling time from the highly damped case is a better approximation of the measured value than
the peak time; however, neither one is close to the actual measure. It may be reasonable to
assume that the measurements taken from the step response could be misrepresented since the
calculated peak times from each both types of responses are similar to each other.

Numerical Modeling
An effective way to visualize the analysis described in the previous subsections is to use
a MATLAB command called “bode”.               This command produces a plot using the natural
frequency and damping ratio established from the time response data. Figures I.2 through I.5
show the experimental frequency response gain and phase data from each damping scenario.

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Bode Diagrams

From: U(1)
10

0

Phase (deg); Magnitude (dB)                                       -10

-20

-30

0

-50
To: Y(1)

-100

-150

-200
0                         1           2
10                     10           10

Figure I.2. Highly Damped Case with Natural Frequency and Damping Ratio of Time Response
Case.

Bode Diagrams

From: U(1)
0

-10
Phase (deg); Magnitude (dB)

-20

-30

-40

0

-50
To: Y(1)

-100

-150

-200
-1                0         1
10               10          10

Figure I.3. Highly Damped Case with Gain and Phase Shift.

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Bode Diagrams

From: U(1)
10

0

-10
Phase (deg); Magnitude (dB)

-20

-30

-40

0

-50
To: Y(1)

-100

-150

-200
0                1         2
10              10          10

Figure I.4. Nearly Undamped Case with Natural Frequency and Damping Ratio
of Time Response Case.

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Bode Diagrams

From: U(1)
0

Phase (deg); Magnitude (dB)               -20

-40

-60

-80

0

-50
To: Y(1)

-100

-150
-1            0                     1                2
10             10                    10               10

Figure I.5. Nearly Undamped Case with Gain and Phase Shift

The graphs in the previous figures relate qualitatively with what is expected had they
been taken directly from an oscilloscope or some other such device. The phase, as plotted with
respect to frequency, is theoretically expected to start at 00 and extend downward to -1800, which
is exactly what Figure I.3 shows. The plotted gain behaves as expected from a high frequency
response, although the resonance value may not fall on the exact points. As for the nearly
undamped case, the phase is expected to start at zero and extend downward to -900. This is
clearly what happens in the graph. The general shape of the gain also shows resemblances to
what is expected, although the resonance is not accurately portrayed.
SIMULINK is an additional program that allows the data from the step response to be
modeled using the natural frequency and damping ratio determined from the frequency response.
Figures I.6 through I.9 show the block diagrams created in SIMULINK which are used in
plotting the desired graph.

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High Damped

1.4

1.2

1

0.8
Displacement(mm)

0.6

0.4

0.2

0
02

39

77

52

76

76

63

38

86

01

14

99

47
5

3
0

0

5

5
52

02
0.

0.
00

00

09

56

60

65

72

78

84

94

13

24

41
0.

1.
0.

0.

0.

0.

0.

0.

0.

0.

0.

0.

1.

1.

1.
-0.2
Time(s)

Figure I.6 High Damped SIMULINK Graph

time
Clock
To Workspace

362.522
x
Step                                         s2 +18.545s+362.522
To Workspace1
Transfer Fcn

Figure I.7 SIMULINK Block Diagram for High Damped Case

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Nearly Undamped

3.5

3

2.5
Displacement(mm)

2

1.5

1

0.5

0
59

99

32

73

95

24

32

94

91

35

31

24

81

16

68

21

63

08

93

6
0

73
2

77
0.
24

33

42

52

61

83

95

07

20

35

51

67

84

03

23

47

74

03

36
0.

3.
0.

0.

0.

0.

0.

0.

0.

1.

1.

1.

1.

1.

1.

2.

2.

2.

2.

3.

3.
Time(s)

Figure I.8 Nearly Undamped SIMULINK Graph

time
Clock
To Workspace

280.73
x
Step                                                      s2 +21.443s+280.73
To Workspace1
Transfer Fcn

Figure I.9 SIMULINK Block Diagram for Nearly Undamped Case

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Both of these graphs display almost exactly what would be expected theoretically. The
values for peak time, maximum overshoot, settling time, and similar components may not
numerically agree with experimental measurements; however, when comparisons are made with
what is typically expected, they clearly follow experimental trends.

Quality of the System
The quality of the system at each damping value can be calculated using the theoretical
natural frequency. Equation I.12 shows a general definition for the quality factor.

n
Q                                                (I.12)
D
The bandwidth is represented by D and can be determined from the magnitude plot of
frequency response for each of the damping values. It is the difference in frequency at a point
that is a factor of 0.7071 to the maximum amplitude. Estimated from Figures I.3 and I.5, the
bandwidth and quality are shown in Table I.5.

Table I.5 Quality of the System

Bandwidth, D  (Hz) Quality
Highly Damped         0.2            16.15
Nearly Undamped        0.4            8.075

The relationship between the quality of the system and the sharpness of the resonant peak
shows that as the damping values decrease, the peak becomes sharper and the quality decreases.
This means that at lower damping values, the resonance frequency is more difficult to distinguish
because the plot of the magnitude is more flat than a higher damping value. The sharper the
peak gets, the lower the bandwidth becomes, making the quality increase, since the bandwidth is
in the denominator of the equation.
The methods used in the experiment were fairly complex.              After completing the
calculations, the time response seemed easier to implement because of its linear relationship.
However, the frequency response produced more reliable results since the natural frequency as
approximated by it was only a few percentage points different from the experimental value.

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MATLAB and SIMULINK provided a clear picture as to how the system physically responds,
although the difficulty in implementing them may have been more trouble than it was worth.

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