Control Design to Reduce the Effects of Torsional ...dml46/pages/Thesis/MS_Thesis_Pres.pdf · System Exhibiting Torsional Resonance 20 Bode Diagram (dB) (dB) 20 Bode Diagram a s Ts

Control Design to Reduce the Effects of Torsional Resonance in Coupledof Torsional Resonance in Coupled

Systems

By Daniel LofaroAdvisors:Dr. Tom ChmielewskiDr. Paul Kalata

1

Contents of Presentation

System Exhibiting Torsional Resonance (TR)ObjectiveObjectivePrimary methods of TR reduction (Industry)Designs to be Consideredg

– Resonance Equalization (RE) [1] – Linear Control– Sliding Mode Control (SMC) – Non-Linear Control

Control Design ConclusionControl Design ConclusionTR Test Rig DesignConclusion

2

System Exhibiting Torsional Resonance

System With Out TR System With TR

3


4


20Bode Diagram

(dB

)

(dB

)

20Bode Diagram

( )( )

a sT sθ

-60

-40

-20

0

Mag

nitu

de (d

B)

Mag

nitu

de (

Mag

nitu

de (

-60

-40

-20

0

Mag

nitu

de (d

B)

-80

-90

-45

0

ase

(deg

)

e (d

eg)

M

e (d

eg)

M -80

180

180.5

181

ase

(deg

)

101

102

103

-180

-135Pha

Frequency (rad/sec)

Pha

se

Pha

se

Desired System10

110

210

3179

179.5Pha

Frequency (rad/sec)

System With TR

5

Desired System( )( )

a sT sθ

System With TR( )( )

a sT sθ


B)

1

Anti-Resonant Frequency:

40

-20

0

20

gnitu

de (d

B)

Bode Diagram

Wr

Ja

delta dB

gnitu

de (d

B12

[1]car

L

KwJ

⎛ ⎞= ⎜ ⎟⎝ ⎠

-80

-60

-40

Mag

-45

0

g)

TR SystemOnly JaJL+Ja

JL+Ja

War

deg)

M

ag1

Resonance Frequency:

⎛ ⎞

101

102

103

-180

-135

-90

Phas

e (d

eg

Frequency (rad/sec)

Pha

se (d2

[1]cr

KwJ J

⎛ ⎞⎜ ⎟⎜ ⎟= ⎜ ⎟⎛ ⎞

6a L

a L

J JJ J

⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠

System With TR( )( )

a sT sθ


Decreases Gain MarginG i S ti B

)

Gain SeparationSwitching Effect

40

-20

0

20

gnitu

de (d

B)

Bode Diagram

Wr

Ja

delta dB

gnitu

de (d

B1040log r

ar

dB ωω⎛ ⎞

Δ = ⎜ ⎟⎝ ⎠[14]

-80

-60

-40

Mag

-45

0

g)


JL+Ja

War

deg)

M

agar⎝ ⎠[14]

101

102

103

-180

-135

-90

Phas

e (d

eg

Frequency (rad/sec)

Pha

se (d

7System With TR

( )( )

a sT sθ


Decreases Gain MarginG i S ti B

)

Gain SeparationSwitching Effect

40

-20

0

20

gnitu

de (d

B)

Bode Diagram

Wr

Ja

delta dB

gnitu

de (d

B– Pre-TR

( ) ( ) 1a La J J sθ + =

-80

-60

-40

Mag

-45

0

g)


JL+Ja

War

deg)

M

ag

– Post-TR

( ) 2( ) a LT s J J s=

+

( ) 1θ10

110

210

3-180

-135

-90

Phas

e (d

eg

Frequency (rad/sec)

Pha

se (d

8( )

2

( ) 1( )aa J

a

sT s J s

θ= System With TR

( )( )

a sT sθ


20Bode Diagram

B)

-40

-20

0

Mag

nitu

de (d

B)

nitu

de (d

B

-80

-60

M

0

TR Mag ChangedB

)

Mag

-135

-90

-45

Phas

e (d

eg)

TR Phase Changedeg

hase

(deg

910

110

210

3-180

Frequency (rad/sec)

Ph


Coupler

Motor / Actuator

Inertial Disc / Load

10

S t E hibiti T i lSystem Exhibiting Torsional Resonance

Differential Equations for a System Exhibitingfor a System Exhibiting Torsional Resonance Coupler

•• • •

( ) ( ) ( )a a a c a c a c L c LT t J B B K K Bθ θ θ θ θ•• • •

= + + + − +

0 ( ) ( )L L L c L c L c a c aJ B B K K Bθ θ θ θ θ•• • •

= + + + − +Motor / Actuator

Inertial Disc / Load

11

S t E hibiti T i lSystem Exhibiting Torsional Resonance

Input:T B K B K− −⎡ ⎤

⎡ ⎤– Torque

Output– Actuator Angular

1

1 0 0 00 ( )

c c c c

a a a aa aa

a a

B K B KJ J J J

JT t

B K B K

θ θθ θθ θ

⎡ ⎤⎡ ⎤⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥= +⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥

&& &

&

&& &gPosition (θa) 0

00 0 1 0

c c c cL L

L L L LL L

B K B KJ J J J

θ θθ θ

⎢ ⎥ ⎢ ⎥− − ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

&

[ ]( ) 0 1 0 0

a

a

L

y t

θθθ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥

&

&

12

L

Lθ⎢ ⎥⎢ ⎥⎣ ⎦


Controllability Matrix Observability Matrix

1

00

aJB

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥

[ ]0 1 0 0C

CCA

Ob

=

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥

2 3

00

434.78 945.18 1.04 7 7.67 70 434 78 945 18 1 04 7

Co B AB A B A B

E EE

⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤= ⎣ ⎦

− −⎡ ⎤⎢ ⎥− −⎢ ⎥

2

3

0 1 0 01 0 0 0

2 17 2 39 4 2 17 2 39 4

ObCACA

ObE E

⎢ ⎥⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥0 434.78 945.18 1.04 7

0 658.76 7.24 6 5.35 70 0 658.76 7.24 6

( ) 4

ECo

E EE

Rank Co

⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦

=

2.17 2.39 4 2.17 2.39 42.39 4 8.82 4 2.38 4 8.82 4

( ) 4

E EE E E E

Rank Ob

⎢ ⎥− −⎢ ⎥− −⎣ ⎦

=

13 Values for all constants can be found in Table 1 and Table 2 of the thesis


Fully Observable:– Angular Position of Actuator as the Output– Angular Position of Load as the Output

Not Fully Observable:Angular Velocity of Actuator as the Output– Angular Velocity of Actuator as the Output

– Angular Velocity of Load as the Output

14


Input: Torque, Tin

A t t O t t θActuator Output: θa

Load Output: θL

1 ( )

( ) ( )

2

2 2

1( )( ) 1

L c ca a L

c a c L c a c La L a L

s J K sBs J J

T s ss s B J B J K J K JJ J J J

θ+ +

=⎛ ⎞

+ + + +⎜ ⎟⎝ ⎠

( )

( ) ( )2 2

1( )( ) 1

c ca LL


K sBJ Js


θ+

=⎛ ⎞

+ + + +⎜ ⎟⎝ ⎠

15 Assumptions: Kc>0, Bc>0, Ja>0, JL>0, Ba=0, BL=0


( )21( )( ) 1

L c ca a L

s J K sBs J J

T s sθ

+ +=

⎛ ⎞( ) ( )2 2( ) 1c a c L c a c L

a L a L


⎛ ⎞+ + + +⎜ ⎟

⎝ ⎠

1 ( )

( ) ( )2 2

1( )( ) 1

c ca LL

K sBJ Js

T s sB J B J K J K J

θ+

=⎛ ⎞⎜ ⎟

16( ) ( )2 2( )


s s B J B J K J K JJ J J J

+ + + +⎜ ⎟⎝ ⎠


Open Loop Poles:0 0 -1 845±j210 44 300

Pole/Zero Plot

– 0, 0, -1.845±j210.44Open Loop Zeros

– -0.76±j126.10100

200

300

Double Pole

-100

0

Imag

inar

y Ax

is( )

( ) ( )

2

2 2

1( )( ) 1

L c ca a L

L L

s J K sBs J J

T s ss s B J B J K J K J

θ+ +

=⎛ ⎞

+ + + +⎜ ⎟

System is Marginally Stable -2.5 -2 -1.5 -1 -0.5 0 0.5 1-300

-200

Real Axis

( ) ( )c a c L c a c La L a L

s s B J B J K J K JJ J J J

+ + + +⎜ ⎟⎝ ⎠



Relationship between θ and θθa and θL

2a L a c c

L c c

s J sB K sBK sB

θθ

⎛ ⎞+ + += ⎜ ⎟+⎝ ⎠



Bode Diagram20

M it d

-80

-60

-40

-20

0

Mag

nitu

de (d

B)

Load

Actuator

war

wrMagnitude

(dB)

270

360

eg)

-120

-100

LoadActuator

Load

ActuatorPhase

101

102

103

0

90

180

Phas

e (d

e

F ( d/ )

Load

(deg)

19

Frequency (rad/sec)

Assumptions: Kc>0, Bc>0, Ja>0, JL>0, Ba=0, BL=0


Torque Input Angular Velocity Output on Actuator Shaft with( )θ on Actuator Shaft with and with out TR

( )( )

a sT sθ



Angular Velocity Output on Actuator Shafton Actuator Shaft

– System With TR (Yellow)– System Without TR (Purple)

( )θ ( )( )

a sT sθ

21


Root Locus of Closed Loop System Containing p y gTR with negative Feedback


Objective

Primary:– Design a linear controller to reduce the effects of

TR, on a mechanically coupled system, using V.Rizzo’s Resonance Equalization [1] methodV.Rizzo s Resonance Equalization [1] method

Extension:– Design a non-linear controller to reduce the g

effects of TR using Sliding Mode Control (SMC)– Implement controllers on physical system

23

Torsional Resonance (TR) Overview: TR Reduction

Primary Methods of TR Reduction– Reducing Bandwidth– Increase Coupler’s Spring Constant

N t h Filt– Notch Filters– Addition of Damping

24

Torsional Resonance (TR) Overview: TR Reduction - Reducing Bandwidth

Reducing BandwidthPPro

– Low Cost– Effective

agni

tude

(dB

)Con

– Reduction in BandwidthSl R e

(deg

)

Ma

– Slower Response

Theta_abutter

Pha

se

25Theta _L

2 Theta _a1

TR System

T

Theta_L

Low Pass FilterGain

KpU1

Torsional Resonance (TR) Overview: TR Reduction – Increase Coupler’s Spring Constant

20Bode Diagram

Stiffer and higher quality parts such as anti-backlash

h th i

-60

-40

-20

0

20

Mag

nitu

de (d

B)

K =55

gears push the spring constant upPro

– Increase TR Frequencies nitu

de (d

B)

-100

-80

-45

0

deg)

K

c=55

Kc=110

Kc=110Kc=55q

– Increase Useable Bandwidth

ConIncreases Cost de

g)

Mag

n

101

102

103

-180

-135

-90

Phas

e (d

Frequency (rad/sec)

– Increases Cost– Low Return Cost

Doubling KcLess Than Doubled wr

Pha

se (d

26

Torsional Resonance (TR) Overview: TR Reduction – Notch Filter

P

Notch Filter Designed to Remove wr

Pro– Reduces the effects

of TR -50

0

50

Mag

nitu

de (d

B)

Bode Diagram

LoadActuatorActuator (No Filter)

War

Wr

Actuator (No Filter)

Actuator

nitu

de (d

B)

– Low Cost

ConIneffective if TR

-150

-100

M

0

180

360

e (d

eg)

ar

Load

deg)

M

agn

– Ineffective if TR changes

101

102

103

-540

-360

-180Ph

ase

Frequency (rad/sec)

Theta_abutter

Pha

se (d

27Theta _L

2 Theta _a1

TR System

T

Theta_L

Notch FilterGain

KpU1

Torsional Resonance (TR) Overview: TR Reduction – Addition of Damping

Addition of DampingP

Bode Diagram

Pro– Low Cost– Effective -40

-20

0

20

Mag

nitu

de (d

B)

System: Gtf6Frequency (rad/sec): 203Magnitude (dB): -28.5

System With Out DampingSystem With Damping

System without Damping

gnitu

de (d

B)

Con– Does Not Reduce ΔdB

-80

-60System: Gtf6Frequency (rad/sec): 130Magnitude (dB): -47.3

-45

0

System with Damping

eg)

Mag

101

102

103

-180

-135

-90

Phas

e (d

eg)

Pha

se (d

28

Frequency (rad/sec)

D i t b C id dDesigns to be Considered:

Designs to be Considered

– Resonance Equalization (RE) [1] – Linear Control

– Sliding Mode Control (SMC)- Non-Linear Control

29

D i t b C id dDesigns to be Considered: Resonance Equalization

W. J. Bigley and V. Rizzo presented a t h i f th d ib it “ li i titechnique for, as they describe it, “eliminating the destabilizing effect of mechanical resonance in feedback control systems ”resonance in feedback control systems. [1][16]Linear ControlLinear ControlMethod: Resonance Equalization (RE)

30


Current at Maximum Velocity is at a Minimum

nitu

de (d

B)

[1]

M

agn

Phas

e (d

eg)

31 Velocity Current

P

D i t b C id d( )a sθ

Designs to be Considered: Resonance Equalization

( )( )

a

inV s

DC Motor Design DC Motor Bode Plot

20

30Bode Diagram

[15]

10

0

10

gnitu

de (d

B)

-30

-20

-10M

ag

3210

110

210

310

4-40

Frequency (rad/sec)


V.Rizzo’s Resonance Equalized Rate Loop [1]

DC Motor

33 Resonance Equalization


Open Loop With O t RE

Open Loop With REWith Out RE With RE

nitu

de (d

B)

ude

(dB

)

(deg

) M

agn

eg)

Mag

nitu

Phas

e (

Phas

e (d

e

34( )( )

a

in

sV sθ ( )

( )a

in

sV sθ


Open Loop With RE

ude

(dB

)

de (d

B)

eg)

Mag

nitu

g)

Mag

nitu

Phas

e (d

e

Phas

e (d

eg

35( )( )

a

in

sV sθ ( )

( )a

in

sV sθ


36 Unity Gain Closed Loop With Kp=50


Closed LoopWith O t RE

Closed LoopWith RE– With Out RE – With RE

tude

(dB

)

ude

(dB

)

deg)

M

agni

t

eg)

Mag

nitu

Phas

e (d

Phas

e (d

e



Closed LoopWith RE– With RE

tude

(dB

)

ude

(dB

)

deg)

M

agni

t

eg)

Mag

nitu

Phas

e (d

Phas

e (d

e



R ti JΔ dB

O L Δ dB RETR Mag Ch

TR Phase ChRatio

(JL:Ja)JL

oz-in-sec2Open Loop

(dB)Δ dB RE

(dB)Change

(dB)Change

(deg)100 0.23 40.1 0 12.53 82.1

10 0.023 20.8 0.4 2.5 18.22 0.0046 9.5 0.5 0.4 4.6

1.435 0.0033 7.7 0.3 0.2 3.50.5 0.00115 3.5 0.5 0.1 1.60.1 0.00023 0.8 0 0 0.3

0 01 0 000023 0 1 0 0 0 1

39

0.01 0.000023 0.1 0 0 0.1

Open Loop With REJa=0.0023 oz-in-sec2

Kc=55 oz/in

D i t b C id dDesigns to be Considered: Sliding Mode Control

Sliding Mode ControlNon Linear Control– Non-Linear Control

– Deal with one or more unknown bounded parametersp

– Tracks multiple states– Use with systems with

un-modeled dynamicsy– Once within sliding

boundary stays in the sliding boundary

40


41 Effective Inertia Unknown: Ja≤J≤(Ja+JL)


SMC Input– Derivations in Chapter IV of Thesis– States Used:

Angular Position of the ActuatorAngular Position of the ActuatorAngular Velocity of the Actuator

– Requires Full State Feedback

( ) ( )1 1ˆ ˆ ˆ ˆ1 ( )du b x f x b F u sign x xλ β η β λ− −⎡ ⎤ ⎡ ⎤= − − − − + − +⎣ ⎦ ⎣ ⎦& &&& % % %

42


Input to Track: θa=sin(20t)

θaθ

aθ&

43


Resonance Equalization Sliding Mode Control( )θ Input: Torque ( ) Output: θT

40Bode Plot

( )( )

a

in

sV sθ a

a a

Input: Torque ( ) Output: θ

Track: θ and θ

T&

dB)

dB)

101 102 103-20

0

20

Mag

(dB

)

Mag

nitu

de (d

Mag

nitu

de (d

0

200

400

Pha

se (d

eg)

ase

(deg

)

M

ase

(deg

)

M

44101 102 103

Frequency (rad/sec)

Pha

Pha


Δ dB TR Mag TR Phase Ratio

(JL:Ja)JL

oz-in-sec2Open Loop

(dB)Δ dB(dB)

gChange

(dB)Change

(deg)

2 0.0046 9.5 0.5 0.4 4.6

1.435 0.0033 7.7 0.3 0.2 3.5

0.5 0.00115 3.5 0.5 0.1 1.6

RE

2 0.0046 9.5 0 24.652 130.84

1.435 0.0033 7.7 0 10.2 23.77

0 5 0 00115 3 5 0 0 012 1 8145

SMC

45

0.5 0.00115 3.5 0 0.012 1.8145Ja=0.0023 oz-in-sec2

Kc=55 oz/in Open Loop With RE

C l iConclusion:Designs to be Considered

Resonance Equalization– Highly Effective in Reducing the Effect of TR– Effective in Reducing ΔdB– Works for wide range of JL:Ja– Can be used with conventional control methods

Required Measurements– Required MeasurementsMotor Current or TorqueAngular Velocity

Sliding ModeSliding Mode– Accurate Model Not Needed– No Extra Control Needed to Track Desired System– Required Measurements

46

qAngular PositionAngular Velocity

TR T t Ri D iTR Test Rig Design:Overview

-Real Time Control

142 H S li R tMotor Shaft Angel (Rad)

-142 Hz Sampling Rate

-Matlab/Simulink Interface

Measures θ and θMotor Vin = + is Clockwise, Angel out = + is clockwise

Voltage_In

Load Shaft Angel (Rad)

10

151693.08

d) we (d

B)

-Measures θa and θL

0

5

Ang

el M

agna

tude

(rad

wr

warMag

nitu

de

47101

-5

Frequency (Hz)

M

Frequency (Hz)

Acknowledgements and Thanks

Superus Thanks– Mother, Father, Andy, and Jenny

Advising:– Dr. Tom Chmielewski– Dr. Paul Kalata

Funding and Support:Funding and Support:– Dr. Moshe Kam– Mr. Robert Shaffer– Dr. Edward GarGiulo– Dade Behring Inc.– Dr. Paul Oh

Special ThanksJessica Finkowski Bella Sorkin Rachel Back Kevin Lynch Nate

48

– Jessica Finkowski, Bella Sorkin, Rachel Back, Kevin Lynch, Nate Fried, Trey Davis , Jenn Voss, Daniel Luig, Brian Kravitz, Jeremy Wakeman, IEEE Drexel Student Branch

Work Cited

[1] W.J. Bigley; V Rizzo. Resonance Equalization in Feedback Control Systems: An ASME Publication, 78-WA/DSC-24, 1978.[2] Nise, Norman S. Control Systems Engineering Fourth Edition: John Wiley and Sons INC, 2004[3] Chen, Chi-Tsong. Linear System Theory and Design Third Edition: Oxford University Press, New York Oxford, 1999.[ ] , g y y g y , ,[4] Ametek: Technical and Industrial Products. Pittman Elcom ST N2314 Series Brushless DC Servo Motor. Datasheet, www.ametektip.com.[5] Kwatny, Harry G; Blankenship, Gilmer L. Nonlinear Control and Analytical Mechanics - A Computational Approach: Birkhauser, Birkhauser Boston, 2000.[6] Li, Weiping; Slotine, Jean-Jacques E. Applied Nonlinear Control Third Edition: Prentice Hall, 2001. [7] Korondi, Peter; Hashimoto, Hideki; Utkin, Vadim. Direct Torsion Control of Felexible Shaft in an Observer-Based Discrete-Time Sliding Mode: IEEE Transaction on Industrial Electronics Vol. 45, No. 2, April 1998.g , , p[8] 80/20 Inc. 80/20 1”x 1”x L Rods. http://www.8020.net/, Columbia City, IN. 2008-02-23.[9] US Digital. E6S-2500-375-HIM3 Quadrature Optical Encoder 2500CPR. Vancouver, Washington. 2008-01-20.[10] US Digital. E5S-1024-315-IG Quadrature Optical Encoder 1024CPR. Vancouver, Washington. 2008-01-20.[11] US Digital. AD4-B-S RS232 Quadrature Encoder Reader. Vancouver, Washington. 2008-01-20.[12] xPC. xPC Target 3.3 Real-Time Hardware in the Loop Solutions. The MathWorks Inc, Novi, MI. http://www.mathworks.com/products/xpctarget/, 2007-11-12.[13] MicroMo Electronics Faulharber Group MVP 2001A01+ MOD2527 Single Axis Intelligent Drive with Integrated PWM[13] MicroMo Electronics, Faulharber Group. MVP 2001A01+ MOD2527 Single Axis Intelligent Drive with Integrated PWM Amplifier RS232C Communication. Clearwater, FL. http://www.micromo.com/, 2007-09-21.[14] Chmielewski, Tom. Private Conversation. 2008[15] http://robotics.ee.uwa.edu.au/courses/embedded/tutorials/tutorials/tutorial_6/images/DC_motor_model.gif[16] Rizzo, Jincent J; Bigley, William J Jr; US Patent Number 4295081. Lockheed electronics Co. Inc. Plainfield NJ. App No: 05/866,394. Filed January 6, 1978.

49

QuestionsQuestions

2008-06-06

50

TR Test Rig Design

Real-Time System

Matlab/Simulink Interface

142Hz Sampling Rate

51


52


53


54

TR Test Rig Design:Real-Time System xPC/Simulink

System Hardware in the Loop BlockT=0.007sec (142Hz) Fixed Step Size

Voltage_In

Motor Shaft Angel (Rad)

Motor Vin = + is Clockwise, Angel out = + is clockwise


55

TR Test Rig Design:Real-Time System xPC/Simulink

Inside System Hardware in the Loop Block

In1 Out1

E bl Di bl ith C i

In OutVoltage_In

1

Motor Input ConversionEnnable Disable with Conversion

Motor Shaft Angel (Rad)1

To Serial Com 1

In1 Out1

Tick Per T to current angel (rad)

In1 Out1

Manual SwitchGet Distance travled In This Period

Bin Dist_Per_T

FIFO bin read 1

FIFORead BINARY

F 1

56Constant

0

TR T t Ri D iTR Test Rig Design:Verification

Real-Time Test Setup

be sure to set trun for the sample period before you compile

this file will go to a target pc using xPC and output on com 1 the MVP pwm out values Scope (xPC) 5

File ScopeId: 6

File ScopeId: 5Scope (xPC) 3

Target ScopeId: 4

Out11

Scope (xPC) 4

Record in mRad /sec

1

Frequency (rad/sec)

Square Wave

Motor_Angel

Terminator Scope (xPC) 1

Target ScopeId: 3

Scope (xPC)

Target ScopeId: 1


Voltage_In


Load Shaft Velos (Rad/sec)

Kp

1

Frequency Ouput

Sin wave

Motor_Angel

Load Velos

Load VelosLoad Velos

57Zero-Order

Hold

Scope (xPC) 2

Target ScopeId: 2

Product 1Manual Switch 2

Constant 2

0


3.122e6

Theta _Out_11

2nd Order Motor Model

s +1017 s+1.413e52Vin1

1

Theta _Out_22

Voltage In


2

Motor Vin + is Clock ise Angel o t + is clock ise

Voltage_In


Vin22

58



Two Pole Motor Model (Pink)(Pink)Physical System Matlab/Hand Calc (Red)Physical SystemPhysical System Matlab/Computer Calc (Green)Physical SystemPhysical System Labview/Hand Calc (Black)

( )a sθ

59

( )( )

a

inV s


Theta _Out_11

Extra Pole

50

s+70

2nd Order Motor Model

3.122e6

s +1017 s+1.413e52Vin1

1

Theta _Out_22Motor Shaft Angel (Rad)

Voltage_In


Vin22

60Motor Vin = + is Clockwise, Angel out = + is clockwise


Three Pole Motor Model (Pink)(Pink)Physical System Matlab/Hand Calc (Red)Physical SystemPhysical System Matlab/Computer Calc (Green)Physical SystemPhysical System Labview/Hand Calc (Black)

( )a sθ

61

( )( )

a

inV s

TR T t Ri D iTR Test Rig Design:TR Measurement

151693.08

10

w

5

Ang

el M

agna

tude

(rad

) wr

0

A

war

62101

-5

Frequency (Hz)

C l iConclusion:TR Test Rig Design

Measurements are taken properly and t laccurately

Runs in Real-Time properly

Motor driver needs to be replaced in order for system to function properly

63

Documents

Control Design to Reduce the Effects of Torsional ...dml46/pages/Thesis/MS_Thesis_Pres.pdf · System Exhibiting Torsional Resonance 20 Bode Diagram (dB) (dB) 20 Bode Diagram a s Ts