dynamic design for l-type structure via frfs matching

7/29/2019 Dynamic design for L-type structure via FRFs matching

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Dynamic design for L-type structure via FRFs matching

C V ChandrashekaraProf. & Head

Department of Mechanical Engineering,

JSS Academy of Technical Education,

Noida 201301

S P Singh & T K KundraProfessor

Department of Mechanical Engineering,

Indian Institute of Technology Delhi,

New Delhi 110016

3rd Asian Conference on Mechanics of Functional Materials and Structures - 2012

5th to 8th Dec 2012

IITD, New Delhi, INDIA


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This paper address the issues related to dynamic design of structures

Many attempts have been made in the past, focusing to obtain a desiredfrequency. Earlier papers address the issues by adding mass or stiffness into

the system

This paper address the dynamic design procedure for simple and built-up

structures

Focus of the study is on obtaining a desired FRFs

Receptance based FRFs is considered

An earlier study was presented for a simple cantilever beam. It is extended

for an L-structure

An Optimization algorithm is developed and is used to obtain the desired

FRFs, while optimizing the different design variables

Main features of the present study


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Study has been conducted on L-structure mounted with Un-Constrainedlayer Damping (UCLD), Passive Constrained Layer Damping (ACLD) and

Active Constrained Layer Damping (ACLD)

Main features of the present study ..Contd.


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Presentation Flow

Structural dynamic Design

Problem formulation

Algorithm used for optimization

Numerical Case studies and result analysis

Experimental validation

Concluding remarks


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OBJECTIVE

DYNAMIC DESIGN

for L-type structure via FRFs matching to get Desired Dynamic

Characteristics

The design would include using anoptimal Minimum of

Structure Modification by the use of bothPassive and ControlElements

Desired Dynamic Characteristics includes,

Reduced Vibration Levels,

Shifting of Natural Frequencies,

Higher Dynamic Stability and

Desired FRFs


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STRUCTURAL DYNAMIC MODIFICATION (SDM)

SDM Techniques are computer based methods by whichdynamic behavior

of a structure is improved by predicting its modified behavior

brought about

by making suitable modifications like lumped masses, stiffness

and dampers or

variations in the configuration parameters of the structureitself


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STRUCTURAL DYNAMIC MODIFICATION (SDM)FE Model updating and structural modification based

Dynamic Design

Prototype Test

FE ModelAn Initial

Design

Update FE Model

Using Test Data

Is Correction

Acceptable?Validated FE

Model

Predict Dynamic

Characteristics

Perform

Structural

Dynamic

Modification

(SDM)

Are Predicted

DynamicCharacteristics

Acceptable?

Desired DynamicCharacteristics

A Dynamically

Sound Design

Yes

Yes

No

No

Experimental part NOT discussed here

Analytical Part discussed here


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650 700 750 800 850 900 950 1000 1050 1100

10-5

10-4

10-3

10-2

10-1

100

Frequency (Hz)

Receptance(m/N)

Initial FRF

Desired FRF

Achieved FRF

No

Generate thematching FRFs

Minimize the error &

Evaluate the optimum values

For Passive / Control Elements

FE Model

Evaluate Dynamic

Characteristics

Evaluate the error

(Difference)

Structure

Define the Target

Dynamic

Characteristics

Modify on the FE Model

Is the error

minimum?

Yes

END

Select the design variables

For modification

Flow Chart showing Structural Dynamic Modification using FRFs matching

FRFs Matching

FRFs Matching


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Problem Formulation


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L-Type structures can be realized in many practical applications

Drilling machine

Milling machines

Precision instruments

Attachments in satellites Civil structures

Solar Panel support


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100mm

100 mm

10 mm

1 mm

Finite Element Formulation for LType Structure

Element Stiffness and Mass Matrix for Vertical Column

Element Stiffness and Mass Matrix for Horizontal Column

Formulation

. .ve V V K R K R

. .

ve V V M R M R

. .he H H K R K R

. .he H H M R M R

Ref: (Kwon and Bang (1997) pp. 286)

(1)

(2)

(3)

(4)


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cLLccLLc

LccLcc

aa

cLLccLLc

LccLcc

aa

k

22

22

460260

61206120

0000

260460

61206120

0000

dLLddLLd

LddLddff

dLLddLLd

LddLdd

ff

m

22

22

42203130

22156013540

00200

31304220

13540221560

00002

3/

/

LEIc

LEAa

6/

420/

ALf

ALd

Where,

Mass and Stiffness Matrix

(5)

(6)


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Where,

cos sin 0 0 0 0

sin cos 0 0 0 0

0 0 1 0 0 0

0 0 0 cos sin 0

0 0 0 sin cos 0

0 0 0 0 0 1

v v

v v

V

v v

v v

R

cos sin 0 0 0 0

sin cos 0 0 0 0

0 0 1 0 0 0

0 0 0 cos sin 0

0 0 0 sin cos 00 0 0 0 0 1

h h

h h

H

h h

h h

R

/ 2

0

v

h

(7)

(8)


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0 1 0 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 0 0 1 0 00 0 0 0 0 1

VR

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

HR

/ 2

0

v

h

RV&RHfurther reduces to,With

(9)

(10)


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The equation of motion for an element can be written as,

e e e e e

M q K q F (11)

Mq Kq F

The global equations of motion are obtained by assembling the elemental equations

and applying appropriate boundary conditions as follows:

(12)


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Equation (12) can be solved to extract any FRF element, such as

jk ( ) (Receptance), and express it explicitly in a series forms as follows,

2 2 2

1

N

jkr r r r

jr kr

i

(13)

where,

= Mass-normalized eigenvectors of the system

r = Any mode of interest

r=Resonance frequency at rth mode

= Frequency of interest

N = Maximum mode of interest

j = Point of excitation

k = Point of response

r = System modal loss factor at rth mode

Using equation (13), over a selected frequency range one can determine the frequency

response functions (FRFs) of the system covering any number of modes. The same equation

is used in the optimization routine to evaluate the optimum values of selected design

variables of the system to generate FRFs to match the desired FRFs.


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Material characteristics and the dimensions

Details Value

Base-beam (mild steel)

Length (Vertical column) 0.100 m

Length (Horizontal column) 0.100 m

Width 0.01 m

Thickness 0.001 m

Mass density 7800 kg/m3

Young's modulus 2.1 x 1011 N/m2

Mode 1 2 3 4

Frequency

(Hz)

27.9 76.0 375.6 550.6

First five natural frequency-

L-type base structure


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Finite Element Formulation for L- type structure with UCLD patches

Finite element formulation for L-type structure mounted with UCLD patches is

developed.

The stiffness matrix of an element with UCLD patch is obtained by replacing

the stiffness terms a and c in equation (5) by the term av, and cv. Theseequivalent stiffness terms av, and cv are now given as follows,

22b b v v b n v bv nv

E bt E bt E D E t Da

L

(14)

where,

and2

v v bv b vn bv

v v b b

E t t t tD t

E t E t

3

cv

D bcL

(15)

c b vD D D 3

2

12

b bb b b n

E tD E t D

3

2

12

v vv v v bv n

E tD E t t D

where,


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the elemental mass matrix of an element with UCLD patch is obtained by

replacing the inertial terms d and f in equation (6) by the equivalent inertial

terms dv, andfv. These terms dv, andfv are now given as follows,

6

cv

r bLd

420

cv

r bLf

c b b v vr r t r t where,

(16) (17)

tv = Thickness of the UCLD patch in m

Ev = Complex modulus of UCLD patch in N/m2

v = Mass density per unit length of UCLD patch in kg/m3

The suffix v in the derivations represents the viscoelastic material, i.e., UCLD

patch. Equation (13) holds good to evaluate the FRFs for L-type structure with

UCLD patches.


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Material characteristics and the dimensions of Unconstrained

Layer Damping (UCLD)Viscoelastic material

Details Value

Width

Thickness

0.01 m

0.0015 m

Mass density 650 kg/m3

Complex modulus 2.5x109 (1 + 0.5 i) N/m2


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22

650 700 750 800 850 900 950 1000 1050 1100

10-5

10-4

10-3

10-2

10-1

100

Frequency (Hz)

Receptanc

e(m/N)

Initial FRF

Desired FRF

Achieved FRF

METHODOLOGY

Initial and Desired Frequency Response Function (FRF)

L U


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23

,

Desired receptance

Achieved receptance

Lower frequency range

Upper frequency range

Number of frequency points on the FRFs

d

a

L

U

f

where

n

Notations used for the Optimization Algorithm Formulation

100

nff(x) =

n=U

n=L

d- a

d

[ ]

The multiplication term 100 indicates the percentage

The objective function percentage error is defined mathematically as,


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24

Objective function to minimize percentage error

min f(x)

subject to,

c1 (x) > 0,c2 (x) < ne,c3 (x) < cn

Length of each element,Number of elements and

Configuration number

x is the vector ofvariables, also called unknowns orparameters

f is the objective function, a function ofx to be minimized*

c is the vector ofconstraints that the unknowns must satisfy

This is a vector function of the variablesx

The number of components in c is the number of individual restrictions that we place on the

Variables

(* In this case, the area between the target FRFs and current FRFs over a specified frequency

range is sought to be minimized)


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25

Frequency ranges:

Pre selected by the designer to cover the desired modes of interest

Set of design variables:Location, length and thickness of patches, and displacement

and velocity gains- considered for modifications

Choice of target FRFs:

Depends on the dynamic designer and is application specific


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0 0 0 1 0 0 1 0 0 0 1 1

0 1 0 0 0 1 0 1 0 1 1 0

0 1 1 1 1 0 0 0 1 0 0 1

1 0 1 0 1 0 1 1 1 1 0 0

1 1 0 1 1 1 1 0 1 1 1 1

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m

)(n) (o)

Figure Shows Fifteen possible configurations of ULCD patches


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Example 1

l3

l2

ly2

ly1=0

lx1

l1 ly1=0

l3lx1

l1

FRFs

Position (m) Length (m)

ly1

ly2 lx1 l1 l2 l3

Desired 0 0.075 0.05 0.025 0.025 0.05

Achieved 0 0 0.025 0.025 0 0.075

DYNAMIC DESIGN FOR L-STRUCTURE

Configuration set

for the desired FRFsConfiguration achieved

for the achieved FRFs


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0 20 40 60 80 100 120-140

-130

-120

-110

-100

-90

-80

-70

-60

-50

Frequency (Hz)

Receptance-MAGindB

FRFs- L type structure with UCLD Patch

Base beam FRFs

Desired FRFs

Achieved FRFs

Example 1

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

40

Number of Iterations

%Error

betweenDesiredandAchievedF

RFs

Optimization Path with UCLD Patch Example 1

Mode

Frequency (Hz) System loss factor

Desired AchievedError

(%)Desired Achieved

Error

(%)

First 29 28 2 0.055865 0.045934 18

Second 79 77 3 0.061661 0.045145 27

DYNAMIC DESIGN FOR L-STRUCTURE WITH UCLD

Path involving 33 iterations followed by the

optimization algorithm is shown & Optimization

convergence occurs at 9th iteration onwards.

FRFs matching over a frequency range

0-120Hz covering first two natural frequencies


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Reduction in FRFs levels by specific percentage

Configurations considered

l2

ly2

ly1=0

l3

l1


ly1

ly2

lx1

l1 l2 l3

0 0.04 0.02 0.02 0.04 0.08

Thickness of the viscoelastic material

is considered as design variable

Desired to reduce FRF response level by 45%

keeping the initial value of viscoelastic materials

thickness as 0.01m, the algorithm achieved a reduction

of 43% at 0.002m at 23rd iterations.

Example 2


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0 10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4x 10

-5

X: 30.24

Y: 2.024e-005

X: 28.97

Y: 2.047e-005

X: 28.97

Y: 3.721e-005

X: 77.35Y: 3.81e-005

X: 77.35

Y: 2.096e-005

X: 79.26

Y: 2.126e-005

Frequency (Hz)

Response-(m/N)

Response v/s Frequency - L type structure with UCLD Patch

Initial Peak

Desired PeakAchieved Peak

Example 3

Mode Parameter Initial Desired Achieved

1

Frequency (Hz) 29 29 30

Response (m/N) 3.721 2.047 2.024

2

Frequency (Hz) 77 77 79

Response (m/N) 3.810 2.096 2.126


0 5 10 15 20 25-2

0

2

4

6

8

10

12x 10

-3


Thick

nessofVEM

-(m)

Optimization Path- L type structure with UCLD Patch

Example 3


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ly1=0

l3lx2

l1

lx1 l2

configurationPosition (m) Length (m)

ly1 lx1 lx2 l1 l2 l3

Desired 0 0.02 0.08 0.04 0.04 0.02

Achieved 0 0.02 0.08 0.04 0.04 0.02

DYNAMIC DESIGN FOR L-STRUCTURE WITH PCLD

Configuration set

for the desired FRFs

Configuration achieved


Set of PCLD patches as design variable

Example 3


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0 50 100 150 200 250 300 350 400 450 500-180

-160

-140

-120

-100

-80

-60

-40

-20

Frequency (Hz)

R

eceptance-MAG

indB

FRFs- L type st ructure with PCLD patches Example 4

Base beam FRFs

Desired FRFs

Achieved FRFs

Mode

Frequency (Hz) System loss factor x 10

-9

Desired Achieved

Error

Desired Achieved

Error

(%) (%)

1 25 25 0 2.35 2.35 0

2 68 68 0 1.13 1.13 0

3 345 345 0 24.00 24.00 0

DYNAMIC DESIGN FOR L-STRUCTURE WITH PCLD

0 5 10 15 20 25 30 350

20

40

60

80

100

120

140

160


%ErrorbetweenDesiredandAchievedFRF

s

Optimization Path L type structure with PCLD patches Example 4

FRFs matching over a frequency

range 0-500Hz covering first three

natural frequencies are considered.

Converges starts at 16th and ends at

35th iterations


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Core layer thickness as design variable

ly1=0

l1

lx2

l2

lx1=0

l2

l1

ly2


ly1 ly2 lx1 lx2 l1 l2 l3 l4

0 0.04 0 0.04 0.02 0.04 0.02 0.04

Desired to reduce receptance response level by 35%,

Achieved reduction of receptance response level 34%

Initial value of viscoelastic materials thickness 0.001m,

Optimized thickness 0.0009m.

Example 4


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22 24 26 28 30 32

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 10-4

X: 27.53

Y: 0.0002925X: 27.06

Y: 0.0002895

X: 27.06Y: 0.0004454

Frequency (Hz)

Res

ponse-(m/N)

Response v/s Frequency - L type structure with PCLD Patch

Initial PeakDesired Peak

Achieved Peak

Example 5

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 67.5

8

8.5

9

9.5

10

10.5

11x 10

-4


Thickn

essofVEM

-(m)

Optimization Path- L type structure with PCLD Patch

Mode Parameter Initial Desired Achieved

1Frequency (Hz) 27.1 27.1 27.5

Peak (m/N) 0.00045 0.00029 0.00029

2Frequency (Hz) 72.4 72.4 73.8

Peak (m/N) 0.0013 0.00084 0.00085


Core layer thickness as design variable


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35

Hybrid Layer Damping (HLD) configurations

on a Cantilever beam

Other configurations available in the literature to be explained

u2

1 2

1

1 2u1

Constraining layer

Viscoelastic layer

Base beam

2

Piezoelectric Layer

PCLD patch

Passive Constrained layer

Controller

Amplifier

Base beam

Viscoelastic Layer

From Sensor

Fig. 1 Hybrid Layer Damping (HLD) configurations

Kinematics relationships


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36

Fig. 2 The Geometry and deformation of PCLD beam element

Kinematics relationships

us

ub

z

x

uc

Base Beam

Core Layer (VEM)

Constrained Layer

u2

1 2

1

1 2u1

Constraining layer

Viscoelastic layer

Base beam

2


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ly1=0

l3

lx2

l2

lx1=0

l4

l1

ly2



0 0.075 0 0.075 0.025 0.025 0.05 0.025

DYNAMIC DESIGN FOR L-STRUCTURE WITH ACLD

Configuration set

for the desired FRFs

Configuration achieved


Example 5

DYNAMIC DESIGN FOR L STRUCTURE WITH ACLD


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0 50 100 150 200 250 300 350 400 450 500-180

-160

-140

-120

-100

-80

-60

-40

Frequency (Hz)

Recept

ance-MAGindB

L type structure with ACLD

Base beam FRFs

Desired FRFs

Achieved FRFs

0 5 10 15 20 25 300

5

10

15

20

25


%ErrorbetweenD

esiredandAchievedFRFs

Optimization Path - L-type structure with ACLD

ModeFrequency

(Hz)gd gv

System loss factor

(x 10-9)

1 23 50 14 0.00539

2 63 25 10 1.41

3 314 17 8 2.13

DYNAMIC DESIGN FOR L-STRUCTURE WITH ACLDPath involving 29 iterations followed by the

optimization algorithm is shown & Optimization

convergence occurs at 11th iteration onwards.

FRFs matching over a frequency range

0-500Hz covering first three natural frequencies


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ly1=0

l1

lx2

l2

lx1=0

l2

l1

ly2

10 15 20 25 30 35

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 10

-5

X: 23.87

Y: 4.563e-005

X: 23.87

Y: 2.282e-005

Frequency (Hz)

Response-(m/N)

Response v/s Frequency - L type structure with ACLD Patch

X: 23.4

Y: 2.461e-005

Initial Peak

Desired Peak

Achieved Peak



0 0.04 0 0.04 0.02 0.04 0.02 0.04

Feedback gains as design variablesExample 6

For first modal 35%

reduction desired

Achieved

reduction 34%



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0 5 10 15 20 25 30 35 40-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4

Number of iterations

Valueofgd

Optimization path for gd

0 5 10 15 20 25 30 35 400

20

40

60

80

100

120

140

160

Number of iterations

Valueofgv

Optimization path for gv

Optimization path for displacement gain Optimization path for velocity gain

Mode 1 2 3

FRFs

Frequency

(Hz)

Response

(m/N)

Frequency

(Hz)

Response

(m/N)

Frequency

(Hz)

Response

(m/N)

Initial 23.87 4.56E-05 66.05 0.00011 324.5 0.00053

Desired 23.87 2.28E-05 66.05 5.7E-05 324.5 0.00026

Achieved 23.4 2.46E-05 66.21 5.8E-05 324.5 0.00027



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Conclusions

A technique has been presented and applied for modification

of structures, so as to obtain the desired dynamic

characteristics.

The optimization strategy includes the varied parameters, of

actively controlled structures, including the position of patches

( a digital variable) , no of patches as well as control gains.

It is proved that the strategy works very well for getting the

desired system behavior since

FRF encompasses the resonance frequencies and damping

effects. The output gives the final size, thickness and number of

patches and where to place them and should be useful for

designers of smart structures.


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Thank You for Your Patience


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FRFs matching approach can successfully be used in dynamic design to obtain desired

FRFs for a built-up structure other than beam structure mounted with different kinds of

damping treatment. Design variables can be position, length and thickness of the

damping treatment considered, as well as the control parameters for the active elements.

The use pattern formation which was explained for cantilever beam structure can be

extended for L-structure and used for quick results of getting desired fundamentalfrequencies.

In the case of L-structure mounted with PCLD patches, reduction in receptance levels

by specific percentage for a given configuration can be extended involving the thickness

of both core layer and constraining layer as design variables.

STRUCTURAL DYNAMIC MODIFICATION (SDM) i FRF MATCHING


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1. Formulation of Finite Element Model

2. Developing the Computer Code

3. Evaluate initial Frequency Response Functions Matrix

4. Choose a Desired FRFs (D_FRFs)

5. Select Desired Design Variable

For Structural Dynamic Modification

6. Evaluate the Difference in area formed by the (I_FRF) & (D_FRF)

over a selected range of frequency and termed it as error

METHODOLOGY

STRUCTURAL DYNAMIC MODIFICATION (SDM) via FRFs MATCHING

Methodology

STRUCTURAL DYNAMIC MODIFICATION (SDM) ia FRFs MATCHING


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8. Minimize the error

leads to an optimum value for the Selected Design Variable

9. Repeat the step (7) to (9) for next selected design variable

Passive Controlled Structures,

m, k and cAs Design Variables

Actively controlled Structures,

Actuator Size(Area) / thickness/ location etc.,Displacement/ Velocity Gain As Design Variables

10. Suggest the best suited Structural Dynamic Modification Configuration

METHODOLOGY (Cont..)

STRUCTURAL DYNAMIC MODIFICATION (SDM) via FRFs MATCHING

dynamic design for l-type structure via frfs matching

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