Analysis of Active Vibration Control for a Cantilever Beam

Analysis of Active Vibration

Control for a Cantilever Beam

with the Use of the Lumped

Parameter Model

Jiří Tůma

Morivation

2 DYNAMICS OF MACHINES 2012, Prague, February 7 – 8, 2012

Henrik Åkesson: Analysis of Structural Dynamic Properties and Active Vibration Control Concerning Machine Tools and a Turbine Application. Blekinge Institute of Technology, Sweden

Outline

1. Lumped parameter model of a cantilever beam

2. Lagrange's equations of motion

3. Mass and stiffness matrices

4. Modal analysis

5. Frequency transfer function

6. Active vibration control

7. Controller type for undamped cantilever beam

8. Partial pole placement

9. Effect of a feedback on the beam response

10.Example

11.Conclusion

DYNAMICS OF MACHINES 2012, Prague, February 7 – 8, 2012 3

Lumped parameter model of a cantilever beam


1 2

ΔL

n n+1 n+2 N N-1

y1 y2 yn-1

yn+1 yn

yN yN-1

δn δN

δn+1

0

δ1 δ2 +x

+y

+z

Clamping

The bending stiffness Kδ relates the applied bending moment M to the resulting relative rotation Δδ of the elementary beam

A cantilever beam of the length L as a continuum is divided into discrete elements of the same length ΔL that are modelled using rigid-body dynamics

b

h

Cross-section Generalized coordinates … y1, y2, …, yN.

L

EIMK x

3

Small angles are assumed sin

Lagrange's equations of motion


Potential energy for horizontal position of the beam

Lagrange's equations of motion for a conservative system

N

n

nn

N

n

nnn

N

n

n

N

n

n yymgyyyL

KmgYKV

1

1

1

0

2

1121

1

0

2

2

12

22

1

1

0

2

1121

1

0

22

22

1 N

n

nnn

N

n

n

N

n

n yyyL

KmgYKV

N

n

nnxN

n

nnN

n

nx

n

t

yy

L

J

t

yym

tJ

t

YmT

1

2

1

21

2

1

1

22

d

d

2d

d

8d

d

2

1

d

d

2

1

Kinetic energy

Nny

V

y

T

y

T

t nnn

,...,2,1,0d

d

0GKyyM

Matrix form

Vertical position

Horizontal position

Mass and stiffness matrices


Parameters

2BA

ABA

ABA

ABA

AB

M

2

2

13

11

2

,13

11

4

L

hmB

L

hmA

121

2541

14641

1464

146

2

L

KK

5.0

1

1

1

1

1

2

1

mg

y

y

y

y

y

N

N

n Gy

Steffness matrix

Mass matrix Vectors

Modal analysis


Let a beam be parameterized by L = 0.5 m, b = 0.04 m, h = 0.005 m, N = 10.

Node 1 2 3 4 5 6 7 8 9 10

Frequency [Hz] 26.6 167 470 922 1522 2256 3093 3965 4755 5309

A) Deflection of the cantilever beam's own self-weight

B) The first 5 of 10 modal shapes of the cantilever beam

Frequency transfer functions


Excited vibration, undamped system

Nqrvv

HN

n n

qnnr

qr ,...,2,1,,1

22,

N

k nn

rsn

nn

rsnrs

jj

A

jj

AjH

1

*

pKyyM

pKyyCyM

Excited vibration, damped system, viscous damping

Frequency transfer function

elementthatactingforce

elementthontdisplaceme,

q

rfH qr

r

q

Hrs H =

r sensed element yr

q acting force pq

Matrix

Active vibration control

DYNAMICS OF MACHINES 2012, Prague, February 7 – 8, 2012 9

Localization of sensors and actuators

1 2 n n+1 N Actuators

Sensors

Controller Feedback

1 2 n n+1 N Actuator

Sensor

Controller Feedback

A co-located pair

Sensors:

-Displacement

-Velocity

-Acceleration

Actuator (source of force):

-Electrodynamic shaker

-Piezoactuator

sHN 1, sR +

-

yN f1

Controller Beam

Reduced arrangement of the AVC system to a pair of a noncolocated sensor and actuator

Controller type for undamped cantilever beam


Cantilever beam transfer function

sH qr , sR +

-

yr fq

Controller Beam SP

N

n

N

nkk

kqnrn

N

n

n

N

n

N

nkk

kqnrn

N

n n

qnrn

N

n n

qnrn

qr

qr

SPr

svvsRs

svvsR

s

vvsR

s

vvsR

sHsR

sHsRsH

1 1

22

1

22

1 1

22

122

122

,

,

,

11

~

Nqrs

vvsH

N

n n

qnnr

qr ,...,2,1,,1

22,

N

n

N

nkk

kqnrn

N

n

n

nDN

n

n

nD

N

n

N

nkk

kqnrn

svv

sT

sRsTsvvsR

1 1

22

1

12

,

1

12

,

1 1

22

Closed-loop transfer function

Structurally stable system

Poles lie on the imaginary axis (stability margin)

Polynomial of s2

Controller at the margin

of stability

D-type controller is

inappropriate Polynomial of s2

Suitable type of controller

sTKsR D 0

Displacement sensor combined with velocity sensor

Partial pole placement


Pole placement

sH qr , sR +

-

yr fq

Controller Beam SP

Closed-loop transfer function

sHsTK

sHsTK

sHsR

sHsRsH

qrD

qrD

qr

qr

SPr

,0

,0

,

,

,11

~

kkk

kkk

k

k

j

j

j

j

*

01

01

*

,0

,0

kqrkkD

kqrkkD

HjTK

HjTK

.1

1

**

,0

*

,

,0,

Dkkqrkqr

Dkkqrkqr

THKH

THKH

Denominators of the transfer function System of two linear equations

Solution for K0 and TD

kkqrkqrkkqrkqr

kqrkqr

D

kkqrkqrkkqrkqr

kkqrkkqr

HHHH

HHT

HHHH

HHK

,

*

,

**

,,

*

,,

,

*

,

**

,,

,

**

,

0

Re

Im

*

k

k kj

kj

Poles

Effect of a feedback on the beam response


Equation of motion

It is assumed that the actuator acts at q-th element and the vibration is sensed on r-th element. The transfer function relating the sensor to the control variable is as follows

,

,2

yTF

pbyKCM

Tsu

uss

.2 pybFKbTCM TT ss

Mottershead, J.E., Tehrani, M.G., Ram, Y.M.: An Introduction to the Receptance Method in Active Vibration Control. In Proceedings of the IMAC-XXVII, February 9-12, 2009 Orlando, Florida USA.

yTFT

N

rD

su

y

y

y

sTKu

r

1

0 00

elementth

Actuator configuration

elementth

0

1

0

q

b

After rearangement

For r = N and q = 1

00

0

00

0 0

D

T

T

T

K

CbTC

KbFK

Example


The response is calculated using Newmark’s method

Equation of motion

.2 pybFKbTCM TT ss

K0 = 66.8, TD = 20.7 s

KMC

Rayleigh Damping α = 0.159, β = 0.0000411

100

101

102

103

104

10-2

10-1

100

101

frequency [Hz]

ksi [-

]

Rayleigh

experiment

Conclusion

14

The lumped-parameters model of the cantilever beam was designed using the method

based on the modal analysis. It was proved that the beam vibration can be actively damped

by a force which is proportional to the velocity and displacement.

The lumped-parameters model of the cantilever beam was designed using the method

based on the modal analysis.

It was proved that the beam vibration can be actively damped by a force which is

proportional to the velocity and displacement.

The response of a mechanical system with active vibration control, designed with the use

of pole placement, can be calculated from the equations of motion in the matrix form.

DYNAMICS OF MACHINES 2012, Prague, February 7 – 8, 2012

Than you

for your attention

http://homel.vsb.cz/~tum52


Documents

Analysis of Active Vibration Control for a Cantilever Beam