MAS417 Mathematics for Mechatronics- Spring 2013, University of Agder, Grimstad, Norway

1. Mathematical Modeling and
Simulation of SAS System With
Magnetorheological (MR) Damper
MA417
Mathematics
for
Mechatronics
University of Agder-Spring 2013
Oreste Niyonsaba
Dimuthu Dharshana Arachchige
Subodha Tharangi Ireshika
Slide 1

2. Overview
•
•
•
•
•
•
•
Vibration isolation
MR dampers and SAS test rig
Mathematical modeling and stability
MR damper models
Vibration response analysis
Experimental comparison
Conclusion
Slide 2

3. Vibration Isolation
• In most mechanical systems the excess
energy that is created becomes vibration
• Vibration leads to
•
•
•
•
•
•
excessive wear of bearings
formation of cracks
loosening of fasteners
structural and mechanical failures
frequent and costly maintenance of machines
discomfort to humans
• A vibration isolation system is needed to
reduce vibrations
Slide 3

4. Isolation systems
Passive:
• No need of
external power
source
• Simple,
inexpensive and
reliable isolation
• Inherent
performance
limitations
Semi-active:
• Excellent
compromise
between passive
and active
systems
• Require low
power for signal
processing
• Improved
vibration
isolation
Active:
• Control forces
change with
excitation and
response
characteristics
• Need of external
energy source
• Can supply and
dissipate energy
Slide 4

5. Magneto-Rheological (MR Dampers)
MR Fluid
MR fluid is composed of oil and varying percentages of iron particles that
have been coated with an anti-coagulant material
Without Magnetic field
With Magnetic field
Slide 5

6. Modes of operation of MR fluid
a.Valve mode
b.Shear mode
c.Squeeze mode
Slide 6

7. MR Rotary damper and SAS test rig.
active MR fluid area
output axis
magnetic circuit(rotor)
magnetic circuit(stator)
coil
magnetic flux line
Viscosity is changed due to the generated magnetic field of
the coil, affecting to control the torque of the output axis
Semi Active Suspension (SAS)
system with MR rotary brake
Slide 7

8. Mathematical modeling of the
SAS system
Analysis of the upper beam
Analysis of the lower beam
Slide 8

9. Stability investigation
Current input to the MR damper
0A
0.4
2.5
0.3
2
1.5
Alpha2Dot(Rad/s)
Alpha1Dot(Rad/s)
0.2
0.1
0
-0.1
1
0.5
0
-0.5
-1
-0.2
-1.5
-0.3
-4
-3
-2
-1
Alpha1(Deg)
0
1
2
-2
20
25
30
35
40
45
Alpha2(Degree)
Equilibrium points
Slide 9
50

10. MR Damper models
a. The Bouc-Wen model
x
Torque (T) generated by the MR damper,
θ
γ=1, β=737,δ=843, n=1.9,
C1=0.0015, C2=17, α1=1,α2=17 [9]
Slide 10

11. Hysteresis behavior
Effect of the control current
Torque Vs Angular Displacement
Torque Vs Angular Velocity
100
100
i=0
i=1
i=2
i=3
80
60
60
40
20
20
Torque(Nm)
40
Torque(Nm)
i=0
i=1
i=2
i=3
80
0
-20
0
-20
-40
-40
-60
-60
-80
-80
-100
0
0.5
1
1.5
Angular Displacement(rad)
2
Current
2.5
-100
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Angular Velocity(rad/s)
0.6
Torque
Slide 11
0.8
1

12. Effect of MR Damper parameters on..
1. Displacement Hysteresis for 2A
Torque Vs Angular Displacement for different gamma values(i=2)
Torque Vs Angular Displacement for different delta values(i=2)
60
60
gamma=0.2
gamma=1
gamma=5
gamma=7
40
40
20
Torque(Nm)
Torque(Nm)
20
0
0
-20
-20
-40
-40
-60
-0.5
0
0.5
1
1.5
2
Angular Displacement(rad)
Torque Vs Angular Displacement for different n values(i=2)
-60
-0.5
2.5
0.5
1
1.5
Angular Displacement(rad)
2
2.5
80
n=1
n=1.9
n=5
n=8
60
40
beta=500
beta=600
beta=737
beta=900
60
40
20
Torque(Nm)
20
0
-20
0
-20
-40
-40
-60
-80
-0.5
0
Torque Vs Angular Displacement for different beta values(i=2)
80
Torque(Nm)
delta=600
delta=700
delta=843
delta=900
-60
0
0.5
1
1.5
Angular Displacement(rad)
2
2.5
-80
-0.5
0
0.5
1
1.5
Angular Displacement(rad)
2
2.5
Slide 12

13. Effect of MR Damper parameters on..
2. Velocity Hysteresis for 2A
Torque Vs Angular Velocity for different gamma values(i=2)
Torque Vs Angular Velocity for different beta values(i=2)
60
60
gamma=0.2
gamma=1
gamma=5
gamma=7
40
40
20
20
Torque(Nm)
Torque(Nm)
delta=600
delta=700
delta=843
delta=900
0
0
-20
-20
-40
-40
-60
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Angular Velocity(rad/s)
0.6
0.8
-60
-1
1
-0.8
-0.4
-0.2
0
0.2
0.4
Angular Velocity(rad/s)
0.6
0.8
1
Torque Vs Angular Velocity for different n values(i=2)
Torque Vs Angular Velocity for different beta values(i=2)
80
80
beta=500
beta=600
beta=737
beta=900
60
40
n=1
n=1.9
n=5
n=8
60
40
20
Torque(Nm)
20
Torque(Nm)
-0.6
0
0
-20
-20
-40
-40
-60
-60
-80
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-80
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Slide 13

14. Effect of MR damper parameters on the
vibration response
Vibration Response Vs Time for different gamma values(i=0.25A)
50
50
gamma=0.2
gamma=1
gamma=7
45
n=1
n=1.9
n=8
45
40
40
30
Vibration(Degrees)
35
35
30
35
30
25
25
25
20
20
20
15
0
0.2
0.4
0.6
0.8
1
1.2
Time(Time)
1.4
1.6
1.8
15
2
15
0
Vibration Response Vs Time for different beta values(i=0.25A)
0.2
0.4
0.6
0.8
1
1.2
Time(Time)
1.4
1.6
1.8
2
beta=500
beta=737
beta=900
0.4
0.6
0.8
1
1.2
Time(Time)
1.4
1.6
1.8
2
Vibration Response Vs Time for different C2 values(i=0.25A)
alpha2=12
alpha2=17
alpha2=20
45
c2=6
c2=10.5
c2=20
40
35
30
Vibration(Degrees)
40
Vibration(Degrees)
40
0.2
45
50
45
0
Vibration Response Vs Time for different alpha2 values(i=0.25A)
50
Vibration(Degrees)
delta=600
delta=843
delta=900
45
40
Vibration(Degrees)
Vibration(Degrees)
Vibration Response Vs Time for different delta values(i=0.25A)
Vibration Response Vs Time for different n values(i=0.25A)
50
35
30
25
20
30
25
25
20
35
15
0
0.2
0.4
0.6
0.8
1
1.2
Time(S)
1.4
1.6
1.8
2
15
20
15
0
0.2
0.4
0.6
0.8
1
1.2
Time(Time)
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8
1
1.2
Time(Time)
1.4
1.6
1.8
Slide 14
2

15. Comparison:
experiment and computer simulations
a. Bouc-Wen
Displacement Vs Time(i=.25)
Displacement Vs Time(i=1A)
50
45
Theoratical
Experimental
40
40
Displacement(Degrees)
Displacement(Degrees)
45
Theoratical
Experimental
35
30
25
30
25
20
20
15
35
0
5
10
15
Time(s)
20
25
15
0
5
10
15
20
Time(s)
Slide 15
25

16. Dhal model
T

z
K x (i)  K y (i) z
(
Kx
K a Kb i
Ky
K1 K 2 i
 z)
T : exerted torque of the MR brake
θ : angle
i : control current
z : dynamic hysteresis coefficient
Kx ,Ky, α: parameters which controls the
shape of the hysteric.
K1 5, K 2 1.5, K a 0.001, K b 0.001,
Slide 16
5

17. Hysteresis behavior
• Effect of the control current
Torque Vs Angular Displacement
Torque Vs Angular Velocity
8
6
6
4
4
2
2
Torque (Nm)
10
8
Torque (Nm)
10
0
-2
-4
-2
-4
i=0
i=1
i=2
i=3
-6
-8
-10
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Angular Displacement (rad)
1.6
1.8
-6
i=0
i=1
i=2
i=3
-8
2
-10
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Angular Velocity (rad/s)
0.6
0.8
Slide 17
1

18. Effect of MR damper parameters on..
• Displacement hysteresis for 2 A
Torque Vs Angular Displacement for different Ka values (i=2)
Torque Vs Angular Displacement for different K1 values(i=2)
20
15
15
10
10
Torque (Nm)
5
0
0
-5
-5
-10
K1=0
K1=5
K1=7
K1=10
-10
-15
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Angular Displacement (rad)
1.6
1.8
Ka=0.001
Ka=1
Ka=5
Ka=10
-15
-20
2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Angular Displacement (rad)
1.6
1.8
2
Torque Vs Angular Displacement for different Alpha values (i=2)
10
8
6
4
Torque (Nm)
Torque (Nm)
5
2
0
-2
-4
-6
Alpha=1
Alpha=5
Alpha=10
Alpha=15
-8
-10
-0.5
0
0.5
1
Angular Displacement (rad)
1.5
2
Slide 18

19. Effect of MR damper parameters on..
• Velocity hysteresis for 2A
Torque Vs Angular Velocity for different Ka values (i=2)
Torque Vs Angular Velocity for different K1 values (i=2)
20
15
15
10
10
Torque (Nm)
5
0
0
-5
-5
-10
K1=0
K1=5
K1=7
K1=10
-10
-15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Angular Velocity (rad/s)
0.6
0.8
Ka=0.001
Ka=1
Ka=5
Ka=10
-15
-20
-1
1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Angular Velocity (rad/s)
0.6
0.8
1
Torque Vs Angular Velocity for different Alpha values (i=0)
6
4
2
Torque (Nm)
Torque (Nm)
5
0
-2
Alpha=1
Alpha=5
Alpha=10
Alpha=15
-4
-6
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Angular Velocity (rad/s)
0.6
0.8
1
Slide 19

20. Effect of MR damper parameters on the
vibration response
Vibration Response Vs Time for different K1 values (i=1)
50
K1=0
K1=5
K1=7
Vibration Response (Degrees))
45
40
35
30
25
20
Ka=0.001
Ka=0
Ka=10
15
Time (s)
20
25
30
Alpha=0
Alpha=5
Alpha=7
45
Vibration Response (Degrees))
Vibration Response (Degrees))
10
50
40
35
30
40
35
30
25
25
20
5
Vibration Response Vs Time for different Alpha values (i=1)
Vibration Response Vs Time for different Ka values(i=1)
50
45
0
20
0
5
10
15
Time (s)
20
25
30
0
5
10
15
Time (s)
20
25
30
Slide 20

21. Experimental task for hysteresis
measurement
Torque from the MR damper,
M MR
d 1
dt
d 2
dt
d2 2
J2
dt 2
M MR
d 1
dt
d 2
dt
d2 2
J2
dt 2
d 2
k2
dt
M spring
r2 k s los
RG2 cos
k2
d 2
dt
(r2 sin
2
RG2 cos
2
r1 sin
1
2
)2
r1 cos
1
r2 cos
2
Slide 21

22. Hysteresis behavior of the MR damper
• Displacement Hysteresis
Torque Vs Displacement (i=0.25)
Torque Vs Displacement (i=1)
Torque Vs Displacement (i=1.5)
10
10
0
0
0
-10
-10
-10
-20
Torque (Nm)
20
Torque (Nm)
20
10
Torque (Nm)
20
-20
-20
-30
-30
-30
-40
-40
-40
-50
-50
-50
-60
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Displacement (rad)
0.05
0.1
0.15
-60
-0.4
0.2
-0.3
-0.2
-0.1
0
Displacement (rad)
0.1
0.2
-60
-0.3
0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Displacement (rad)
0.05
0.1
0.15
0.2
2
2.5
• Velocity Hysteresis
Torque Vs Velocity (i=1)
Torque Vs Velocity (i=0.25)
Torque Vs Velocity (i=1.5)
10
10
0
0
0
-10
-10
-10
-20
Torque (Nm)
20
Torque (Nm)
20
10
Torque (Nm)
20
-20
-20
-30
-30
-30
-40
-40
-40
-50
-50
-50
-60
-2
-1.5
-1
-0.5
0
0.5
Velocity (rad/s)
1
1.5
2
-60
-2
-1.5
-1
-0.5
0
0.5
Velocity (rad/s)
1
1.5
2
2.5
-60
-2
-1.5
-1
-0.5
0
0.5
Velocity (rad/s)
1
1.5
Slide 22

23. Comparison: experiment and computer
simulations
b. Dhal
Displacement Vs Time(i=0.25)
Displacement Vs Time(i=1)
50
50
Theoritical
Experiment
45
40
Displacement (Degrees)
Displacement (Degrees)
45
35
30
25
20
15
Theoritical
Experiment
40
35
30
25
20
0
5
10
15
Time (s)
20
25
15
0
5
10
15
20
Time (s)
Slide 23
25

24. Conclusion
• Easy to analyze MR damper with SAS test rig which
supports Matlab Simulink environment.
• Both theoretical and experimental models, magnitude of
torque in hysteresis behavior lies in a common range.
• If model parameters are diligently tuned, a similar vibration
response can be obtained for both theoretical and
experimental models.
• Bouc-Wen model stands taller as far as the more realistic,
accurate results are concerned.
• Semi-active dampers provide remarkable improvements
over passive suspensions.
Slide 24

25. Thank You...!
Slide 25

26. Reference Slides
26