Then, using th e Newton’s second law, the equation of motion of the motor shaft can be expressed by,
The mechanical configuration of the cart’s rack opinion system can be defined at the following relationship,
Substitute the equation 5 and 6 into equation 4. The new equation of armature inertial force is,
And
Then, substitute equation 2 and 7 into equation 8. By rearranging, the new equation can be expressed by,
From equation 9, it is expressed by second order differential equation in the cart position. Using the Laplace Transform technique and rearranging for open loop transfer function, the new equation of AMD can be expressed by,
5.0 Analytical Analysis
The block diagram of AFC and AMD is shown in Figure 3 below:
Fig 3 Block diagram of Active Force Controller and Active Mass Damper
The transfer function of AFC is:
AFC controller controller
Active Mass Damper AMD
G s
4 5
2 5
1 .8 3 6 9 5 .4 8 4 4
1 .1 2 4 3 e
H s
e s
e s
+ -
The transfer function of AMD is:
The final closed loop transfer function is:
6.0 Result and Discussion
This sub-chapter discusses the stability analysis for AMD without any controller involved and for the second analysis is AMD with AFC. In this stability analysis, four stability diagrams were used and they are:
i. Step response diagram analysis
ii. Bode diagram analysis
iii. Nyquist diagram analysis
iv. Root locus diagram analysis
a Step response diagram for AMD
b Bode diagram for AMD
c Nyquist diagram for AMD
d Root locus diagram for AMD
Fig 4 Stability analysis for AMD without any controller
Figure 4 a shows the step response diagram for AMD. According to this graph, the linear line is represented the behavior of AMD itself. It is means that, if the AMD in working condition, the vibration displacement is slightly
proportional to the amplitude and time. Based on the vibration condition, if vibration displacement proportional to amplitude and time, it has a big potential the building structural to collapse when the vibration displacement increase
based on time. Controller is needed to reduce this and finally avoid the phenomenon from happens. Figure 4 b shows bode diagram analysis of AMD. Based on this diagram, the value of peak gain is approximately 232 dB. The frequency
of this AMD is approximately
. The natural frequency of AMD system cannot be detected because the analysis in the experimental is only used a plant and not involved any disturbances to influent the system.
From Figure 4 c, the margin point 1 is infinity. It means the value cannot be read by the system and based on this value, the AMD system has a critical condition to control the vibration displacement when the seismic problem happen.
The closed loop for AMD system is under stabilizing. The phase margin of this system is approximatel y to 89.5 deg and the delay margin is 9.57 deg, respectively. The frequency of this system is 0.163 radsec. According to Figure 4 d,
it represents the root locus diagram for AMD system. For the first impression, the diagram shows the unstable condition because they have a linear line between point 1 and point 2. Theoretically, if the system is in a stable
condition, point 1 must be nearly to point 2.
From these four stability diagram, the AMD system is not in stable condition. Based on that, the controller is needed to reduce the unstable condition and finally to avoid the vibration displacement from slightly proportional to
the amplitude and time.
a Step response diagram for AMD with AFC
b Bode diagram for AMD with AFC
Step Response
Time sec
A m
p lit
u d
e
2 4
6 8
10 12
14 16
18 20
1000 2000
3000 4000
5000 6000
System: sys Peak amplitude = 5.41e+003
Overshoot : 0 At time sec 20
System: sys Settling Time sec: 14.4
System: sys Rise Time sec: 6.85
Bode Diagram
Frequency radsec 40
50 60
70 80
System: sys Peak gain dB: 74.7
At frequency radsec: 4.29e-013
M a
g n
itu d
e d
B
10
-2
10
-1
10 10
1
10
2
10
3
-90 -45
System: sys Phase Margin deg: 90
Delay Margin sec: 1.09e-005 At frequency radsec: 1.45e+005
Closed Loop Stable? Yes
P h
a s
e d
e g
c Nyquist diagram for AMD with AFC
d Root locus diagram for AMD with AFC
Fig 5 Stability analysis for AMD with AFC
Figure 5 a, shows the step response diagram for AMD with AFC. According to this figure, the overshoot of the system is approximately to 0 and this is for sure that the condition of the system is obeying the regulation of the
necessary to reduce and avoid the vibration displacement happen. Besides, the rise time of the system is 20 sec, peak
Nyquist Diagram
Real Axis
Im a
g in
a ry
A x
is
-1000 1000
2000 3000
4000 5000
6000 -2000
-1500 -1000
-500 500
1000 1500
2000
System: sys Peak gain dB: 74.7
Frequency radsec: 4.29e-013 System: sys
Phase Margin deg: 90 Delay Margin sec: 1.09e-005
At frequency radsec: 1.45e+005 Closed Loop Stable? Yes
Root Locus
Real Axis
Im a
g in
a ry
A x
is
-160 -140
-120 -100
-80 -60
-40 -20
20 -8
-6 -4
-2 2
4 6
8 0.999
1 1
1
0.88 0.97
0.988 0.995
0.997 0.999
1 1
1 20
40 60
80 100
120 140
160 System: sys
Gain: 0 Pole: -0.214
Damping: 1 Overshoot : 0
Frequency radsec: 0.214 0.88
0.97 0.988
0.995 0.997
System: sys Gain: 0
Pole: -46.8 Damping: 1
Overshoot : 0 Frequency radsec: 46.8
amplitude is and with rise time is 6.85 sec. Figure 5 b describes the natural frequency of the system. The
frequency of the system is approximately to where this frequency is referred as a natural
frequency of the system. Then, the peak gain of this system is 74.7 dB. Figure 5 c shows two points which is position in the circle. This means that the system is stable and the poles express the performance of the controller. In point 1,
the phase margin is 90 deg and frequency is . The peak gain is 74.7 dB and the frequency in point
2 is . For the root locus diagram, the overshoot is 0 and it shows that the system is in the stable
condition. Besides, it also has same values for the damping point and gain point. The differences are only at the value of the pole point and frequency. At point 1 has pole point -0.214 and point 2 has pole -46.8. For frequency, point 1 is
0.214 radsec while point 2 has 46.8 radsec.
7.0 Conclusion
