modelling and simulation of sas system with mr damper dimuthu dharshana
DESCRIPTION
MAS417 Mathematics for Mechatronics- Spring 2013, University of Agder, Grimstad, NorwayTRANSCRIPT
Slide 1
Mathematical Modeling and Simulation of SAS System With Magnetorheological (MR) Damper
MA417 Mathematics
for Mechatronics
University of Agder-Spring 2013
Oreste NiyonsabaDimuthu Dharshana ArachchigeSubodha Tharangi Ireshika
Slide 2
• Vibration isolation• MR dampers and SAS test rig• Mathematical modeling and stability• MR damper models • Vibration response analysis• Experimental comparison• Conclusion
Overview
Slide 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
Isolation systems
Passive:• No need of
external power source
• Simple, inexpensive and reliable isolation
• Inherent performance limitations
Active:• Control forces
change with excitation and response characteristics
• Need of external energy source
• Can supply and dissipate energy
Semi-active:• Excellent
compromise between passive and active systems
• Require low power for signal processing
• Improved vibration isolation
Slide 4
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 fieldSlide 5
Modes of operation of MR fluid
a.Valve mode
b.Shear mode
c.Squeeze mode
Slide 6
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
Mathematical modeling of the SAS system
Analysis of the upper beam
Analysis of the lower beam
Slide 8
∑ 𝑇 𝑠𝑢𝑚= 𝐽 �̇�𝑻 𝒔𝒑𝒓𝒖𝒏𝒈𝒎𝒂𝒔𝒔=𝑻 𝑺𝒑𝒓𝒊𝒏𝒈−𝑻𝑮𝒓𝒂𝒗𝒊𝒕𝒚 𝑼𝒑𝒑𝒆𝒓 −𝑻𝑽𝒊𝒔𝒄𝒐𝒖𝒔−𝑻𝑴𝑹
𝐽 2
𝑑2𝛼2
𝑑𝑡 2 +𝑚2𝑔2𝑟2𝐶𝑜𝑠𝛼2−𝑟 2𝐾 𝑠 (𝑙𝑜𝑠−√(𝑟1𝐶𝑜𝑠𝛼1−𝑟2𝐶𝑜𝑠𝛼2)2+(𝑟2𝑆𝑖𝑛𝛼2−𝑟 1𝑆𝑖𝑛𝛼1)
2 )+𝑘2
𝑑𝛼2
𝑑𝑡=−𝐶 ( 𝑑𝛼1
𝑑𝑡−𝑑𝛼2
𝑑𝑡 )𝑻 𝒖𝒏𝒔𝒑𝒓𝒖𝒏𝒈𝒎𝒂𝒔𝒔=−𝑻𝑮𝒓𝒂𝒗𝒊𝒕𝒚 𝑳𝒐𝒘𝒆𝒓−𝑻 𝑺𝒑𝒓𝒊𝒏𝒈−𝑻𝑽𝒊𝒔𝒄𝒐𝒖𝒔+𝑻𝑻𝒊𝒓𝒆𝑬𝒍𝒂𝒔𝒕𝒊𝒄𝒊𝒕𝒚
+𝑻𝑻𝒊𝒓𝒆 ,𝑫𝒂𝒎𝒑𝒊𝒏𝒈−𝑻𝑴𝑹
𝐽 1
𝑑2𝛼1
𝑑𝑡2 +𝑚1𝑔1𝑟1𝐶𝑜𝑠 ( 𝛽−𝛼1 )+𝑟1 𝐾 𝑠 (𝑙𝑜𝑠−√(𝑟 1𝐶𝑜𝑠𝛼1−𝑟 2𝐶𝑜𝑠𝛼2)2+(𝑟 2𝑆𝑖𝑛𝛼2−𝑟1𝑆𝑖𝑛𝛼1)
2 )+𝑘1
𝑑𝛼1
𝑑𝑡−𝑘𝑔𝑅𝐶𝑜𝑠 (𝛽−𝛼1 ) ( 𝑙𝑜𝑔+𝑅𝑆𝑖𝑛 ( 𝛽−𝛼1 )+𝑟 −𝐷𝑥+𝑈𝑘𝑖𝑛)− 𝑓 𝑔( 𝑑 (𝐷𝑥−𝑢𝑘𝑖𝑛 )
𝑑𝑡−𝑑𝛼1
𝑑𝑡𝑅𝐶𝑜𝑠 (𝛽−𝛼1))=−𝐶 (𝑑𝛼1
𝑑𝑡−𝑑𝛼2
𝑑𝑡 )=−(𝑑𝛼1
𝑑𝑡−𝑑𝛼2
𝑑𝑡 )𝑀𝑀𝑅
Slide 9
Stability investigation
20 25 30 35 40 45 50-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Alpha2(Degree)
Alp
ha
2D
ot(
Ra
d/s
)
-4 -3 -2 -1 0 1 2-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
Alpha1(Deg)
Alp
ha
1D
ot(
Ra
d/s
)
Current input to the MR damper 0A
Upper beam initial excitation():
Lower beam default position():
Equilibrium points : :
MR Damper models
γ=1, β=737,δ=843, n=1.9, C1=0.0015, C2=17, α1=1,α2=17 [9]
a. The Bouc-Wen model
Torque (T) generated by the MR damper,
α,C: Damping Coefficents depends on current iγ,β,δ,n : Parameters control the shape of the hysteresis z: hysteretic displacement
Slide 10
Bouc-Wen
x
θ
Hysteresis behavior
0 0.5 1 1.5 2 2.5-100
-80
-60
-40
-20
0
20
40
60
80
100Torque Vs Angular Displacement
Angular Displacement(rad)
Tor
que(
Nm
)
i=0
i=1i=2
i=3
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-100
-80
-60
-40
-20
0
20
40
60
80
100Torque Vs Angular Velocity
Angular Velocity(rad/s)
Tor
que(
Nm
)
i=0
i=1i=2
i=3
Effect of the control current
Current Torque
Slide 11
Bouc-Wen
Effect of MR Damper parameters on..
1. Displacement Hysteresis for 2A
-0.5 0 0.5 1 1.5 2 2.5-60
-40
-20
0
20
40
60Torque Vs Angular Displacement for different gamma values(i=2)
Angular Displacement(rad)
Tor
que(
Nm
)
gamma=0.2
gamma=1gamma=5
gamma=7
-0.5 0 0.5 1 1.5 2 2.5-80
-60
-40
-20
0
20
40
60
80Torque Vs Angular Displacement for different beta values(i=2)
Angular Displacement(rad)
Tor
que(
Nm
)
beta=500
beta=600beta=737
beta=900
-0.5 0 0.5 1 1.5 2 2.5-60
-40
-20
0
20
40
60Torque Vs Angular Displacement for different delta values(i=2)
Angular Displacement(rad)
Tor
que(
Nm
)
delta=600
delta=700delta=843
delta=900
-0.5 0 0.5 1 1.5 2 2.5-80
-60
-40
-20
0
20
40
60
80Torque Vs Angular Displacement for different n values(i=2)
Angular Displacement(rad)
Tor
que(
Nm
)
n=1
n=1.9n=5
n=8
Slide 12
Bouc-Wen
Effect of MR Damper parameters on..
2. Velocity Hysteresis for 2A
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-60
-40
-20
0
20
40
60Torque Vs Angular Velocity for different gamma values(i=2)
Angular Velocity(rad/s)
Tor
que(
Nm
)
gamma=0.2
gamma=1gamma=5
gamma=7
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-80
-60
-40
-20
0
20
40
60
80Torque Vs Angular Velocity for different beta values(i=2)
Angular Velocity(rad/s)
Tor
que(
Nm
)
beta=500
beta=600beta=737
beta=900
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-60
-40
-20
0
20
40
60Torque Vs Angular Velocity for different beta values(i=2)
Angular Velocity(rad/s)
Tor
que(
Nm
)
delta=600
delta=700delta=843
delta=900
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-80
-60
-40
-20
0
20
40
60
80Torque Vs Angular Velocity for different n values(i=2)
Angular Velocity(rad/s)
Tor
que(
Nm
)
n=1
n=1.9n=5
n=8
Slide 13
Bouc-Wen
Effect of MR damper parameters on the vibration response
Slide 14
Bouc-Wen
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 215
20
25
30
35
40
45
50
Time(Time)
Vib
ratio
n(D
egre
es)
Vibration Response Vs Time for different gamma values(i=0.25A)
gamma=0.2
gamma=1gamma=7
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 215
20
25
30
35
40
45
50
Time(Time)
Vib
ratio
n(D
egre
es)
Vibration Response Vs Time for different delta values(i=0.25A)
delta=600
delta=843delta=900
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 215
20
25
30
35
40
45
50
Time(Time)
Vib
ratio
n(D
egre
es)
Vibration Response Vs Time for different n values(i=0.25A)
n=1
n=1.9n=8
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 215
20
25
30
35
40
45
50
Time(Time)
Vib
ratio
n(D
egre
es)
Vibration Response Vs Time for different alpha2 values(i=0.25A)
alpha2=12
alpha2=17alpha2=20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 215
20
25
30
35
40
45
Time(Time)
Vib
ratio
n(D
egre
es)
Vibration Response Vs Time for different C2 values(i=0.25A)
c2=6
c2=10.5c2=20
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 215
20
25
30
35
40
45
50
Time(S)
Vib
ratio
n(D
egre
es)
Vibration Response Vs Time for different beta values(i=0.25A)
beta=500
beta=737beta=900
Comparison: experiment and computer simulations
0 5 10 15 20 2515
20
25
30
35
40
45
50
Time(s)
Dis
plac
emen
t(D
egre
es)
Displacement Vs Time(i=.25)
Theoratical
Experimental
Bouc-Wen
Slide 15
0 5 10 15 20 2515
20
25
30
35
40
45
Time(s)
Dis
plac
emen
t(D
egre
es)
Displacement Vs Time(i=1A)
Theoratical
Experimental
a. Bouc-Wen
Dhal model
ziKiKT yx )( )(
) ( zz
T : exerted torque of the MR brakeθ : anglei : control current z : dynamic hysteresis coefficientKx ,Ky, α: parameters which controls the shape of the hysteric.iKKK
iKKK
y
bax
21
5 ,001.0 ,001.0 ,5.1 ,5 21 ba KKKK
Slide 16
Dhal
Hysteresis behavior
• Effect of the control current
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-10
-8
-6
-4
-2
0
2
4
6
8
10
Angular Displacement (rad)
Torq
ue (N
m)
Torque Vs Angular Displacement
i=0
i=1i=2
i=3
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-10
-8
-6
-4
-2
0
2
4
6
8
10
Angular Velocity (rad/s)
Torq
ue (N
m)
Torque Vs Angular Velocity
i=0
i=1i=2
i=3
Slide 17
Dhal
Effect of MR damper parameters on..
• Displacement hysteresis for 2 A
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-15
-10
-5
0
5
10
15
Angular Displacement (rad)
Tor
que
(Nm
)
Torque Vs Angular Displacement for different K1 values(i=2)
K1=0
K1=5K1=7
K1=10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-20
-15
-10
-5
0
5
10
15
20
Angular Displacement (rad)
Tor
que
(Nm
)
Torque Vs Angular Displacement for different Ka values (i=2)
Ka=0.001
Ka=1Ka=5
Ka=10
-0.5 0 0.5 1 1.5 2-10
-8
-6
-4
-2
0
2
4
6
8
10
Angular Displacement (rad)
Tor
que
(Nm
)
Torque Vs Angular Displacement for different Alpha values (i=2)
Alpha=1
Alpha=5Alpha=10
Alpha=15
Slide 18
Dhal
Effect of MR damper parameters on..
• Velocity hysteresis for 2A
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-6
-4
-2
0
2
4
6
Angular Velocity (rad/s)
Tor
que
(Nm
)
Torque Vs Angular Velocity for different Alpha values (i=0)
Alpha=1
Alpha=5Alpha=10
Alpha=15
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-20
-15
-10
-5
0
5
10
15
20
Angular Velocity (rad/s)
Tor
que
(Nm
)
Torque Vs Angular Velocity for different Ka values (i=2)
Ka=0.001
Ka=1Ka=5
Ka=10
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-15
-10
-5
0
5
10
15
Angular Velocity (rad/s)
Tor
que
(Nm
)
Torque Vs Angular Velocity for different K1 values (i=2)
K1=0
K1=5K1=7
K1=10
Slide 19
Dhal
Effect of MR damper parameters on the vibration response
0 5 10 15 20 25 3020
25
30
35
40
45
50
Time (s)
Vib
ratio
n R
espo
nse
(Deg
rees
))
Vibration Response Vs Time for different K1 values (i=1)
K1=0
K1=5K1=7
0 5 10 15 20 25 3020
25
30
35
40
45
50
Time (s)
Vib
ratio
n R
espo
nse
(Deg
rees
))
Vibration Response Vs Time for different Ka values(i=1)
Ka=0.001
Ka=0Ka=10
0 5 10 15 20 25 3020
25
30
35
40
45
50
Time (s)
Vib
ratio
n R
espo
nse
(Deg
rees
))
Vibration Response Vs Time for different Alpha values (i=1)
Alpha=0
Alpha=5Alpha=7
Slide 20
Dhal
Experimental task for hysteresis measurement
22
22
22112
1122222
2
221
222
222
2
221
cos
coscos)sinsin(
cos
RGdt
dk
rrrrlkrdt
dJ
dt
d
dt
dM
RGdt
dkM
dt
dJ
dt
d
dt
dM
ossMR
springMR
Torque from the MR damper,
Slide 21
Dhal
Hysteresis behavior of the MR damper
• Displacement Hysteresis
• Velocity Hysteresis
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-60
-50
-40
-30
-20
-10
0
10
20
Displacement (rad)
Tor
que
(Nm
)
Torque Vs Displacement (i=0.25)
-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3-60
-50
-40
-30
-20
-10
0
10
20
Displacement (rad)
Tor
que
(Nm
)
Torque Vs Displacement (i=1)
-0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2-60
-50
-40
-30
-20
-10
0
10
20
Displacement (rad)
Tor
que
(Nm
)
Torque Vs Displacement (i=1.5)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2-60
-50
-40
-30
-20
-10
0
10
20
Velocity (rad/s)
Tor
que
(Nm
)
Torque Vs Velocity (i=0.25)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-60
-50
-40
-30
-20
-10
0
10
20
Velocity (rad/s)
Tor
que
(Nm
)
Torque Vs Velocity (i=1)
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-60
-50
-40
-30
-20
-10
0
10
20
Velocity (rad/s)
Tor
que
(Nm
)
Torque Vs Velocity (i=1.5)
Slide 22
Comparison: experiment and computer simulations
0 5 10 15 20 2515
20
25
30
35
40
45
50
Time (s)
Dis
plac
emen
t (D
egre
es)
Displacement Vs Time(i=0.25)
Theoritical
Experiment
0 5 10 15 20 2515
20
25
30
35
40
45
50
Time (s)D
ispl
acem
ent
(Deg
rees
)
Displacement Vs Time(i=1)
Theoritical
Experiment
Slide 23
Dhalb. Dhal
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
Thank You...!
Slide 25
Reference Slides
26