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
Position (m) Length (m)
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
Reduction in FRFs levels by specific percentage
0 5 10 15 20 25-2
0
2
4
6
8
10
12x 10
-3
Number of Iterations
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
for the achieved FRFs
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
Number of Iterations
%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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Reduction in FRFs levels by specific percentage
Core layer thickness as design variable
ly1=0
l1
lx2
l2
lx1=0
l2
l1
ly2
Position (m) Length (m)
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
Number of Iterations
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
Reduction in FRFs levels by specific percentage
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
Position (m) Length (m)
ly1 ly2 lx1 lx2 l1 l2 l3 l4
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
for the achieved FRFs
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
Number of Iterations
%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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Reduction in FRFs levels by specific percentage
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
Position (m) Length (m)
ly1 ly2 lx1 lx2 l1 l2 l3 l4
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%
Reduction in FRFs levels by specific percentage
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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
Reduction in FRFs levels by specific percentage
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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
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