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

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