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Research and Reviews on Experimental and Applied Mechanics
Volume 2 Issue 2
Experimental Investigations On Free Vibration Of Beams
Kemparaju H.R1, Prasanta Kumar Samal
2
1Associate Engineer, HCL Technology, Bengaluru, India.
2Assistant Professor, Department of Mechanical Engineering, NIE Mysuru, India.
E-mail: [email protected], [email protected]
ABSTRACT
In this article experimental investigation of the natural frequency, damping ratio and damping
constant of beams in different material i.e. steel, aluminium, copper with different boundary
conditions like Clamped-free, Clamped-clamped, clamped-simply support, Simply support-
simply support, and free-free(by using sponge, and by using rubber band) has been
investigated. Natural frequencies obtained using accelerometer, NI-DAQ (9234), and NI-
DAQ chassis (9137),and LABVIEW and MATLAB. Then the main objective of this paper is
to provide experimental data that can be used for checking the accuracy and reliability of
different theories and approaches like analytical, finite element method by using MATLAB
and ANSYS. The effects of different geometrical parameters including density are discussed
in above mentioned all boundary conditions in details with up to first 3 natural frequencies.
This study may provide valuable information for researchers and engineers in design
applications.
Keywords: Natural frequency, damping ratio, damping constant, and LABVIEW.
INTRODUCTION During last three decades the subject
Vibration Analysis has undergone
significant development. There has been
an urge and urgency of designing present
day modern and complex structures and
systems in their proper perspective, a task
which could not be conceived even three
decades back, this has been possible due to
the advent of the electronic digital
computer.
Daniel Ambrosini [1] . In this paper author
is carried out the experimental modal
analysis of the non-symmetrical thin
walled beam by using aluminium material.
The experiment is carried out in two
boundary conditions i.e. fixed-free and
fixed-fixed. Here accelerometer Kyowa
AS-GB and PCM�DAS16D=16 of 16 bit
data acquisition system is used to a modal
testing of beam. Then the experimental
results are compared with the analytical
method of vlasovs theory of thin walled
beam and numerical method by using
finite element software SAP2000N and
compared results. Here the author is shows
that the results are good agreement
between the all three methods. Mehmet
Avcar [2]. The free vibration analysis of
the square cross sectioned aluminium
beam is investigated by the analytical and
numerically under four boundary
conditions i.e. C-C, C-F, C-SS, SS-SS.
The Euler Bernoulli beam theory and
Newton Raphson methods are used to
analytical method. And finite element
method based software ANSYS is used to
find out numerical results for Free
Vibration Analysis of Beams. Then he
obtained a natural frequency and he
discussed first three modes of the
including boundary conditions, geometric
characteristics i.e. length, cross sectional
etc. Mr. P.Kumar, Dr. S.Bhaduri [3]. In
this paper the natural frequencies of two
different cantilever beams made of
Aluminum and Iron are measured
experimentally with and without the
presence of end masses. The finite element
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Research and Reviews on Experimental and Applied Mechanics
Volume 2 Issue 2
analysis is done using the developed codes
in MATLAB and also by ANSYS
software. In experimental setup consists of
FFT analyser with computer and its
accessories. For further analysis of the
effect of different masses on the natural
frequency of cantilever beam, free end of
cantilever is loaded with different masses
along with accelerometer and the result of
this experiment is observed on the
computer screen for the aluminium and
iron beams. Then the author is concluded
that numerical and experimental results are
much closed together. Yasha- vantha
kumar G.A [4]. The numerical study of the
free vibration analysis of the smart
composite beam by using the ANSYS is
explained in this paper. Here the author
composite beam made up of glass epoxy
and PZT patches are added in the surface
of the beam. Then the vibration analysis is
carried out under the clamped-free
condition of the beam. Kotambkar [5]. He
studied the mass loading effect of the
accelerometer on the natural frequency of
the beam under free-free boundary
condition. The analytical calculation is
taken b Euler-Bernoulli equation for
uniform cross section beam. For
experimental analysis accelerometer (B
and K make; weight 27(gram), FFT
analyser (DI22000) is used to obtained the
results. In most of the case mass of the
accelerometer is ignored,however when
lighter structures are investigated this
effect must be considered. Nikil T,
Chandrahas T,[6]. Here the author is
design and develop a test rig for
determining the vibration characteristics of
the beam with different boundary
conditions like C-F, C-C, and SS-SS. In
the test rig he provided a adjustable
eccentric weights are provided to vary the
force with approximately same frequency.
He used the accelerometer and NI 9234 to
acquire the vibration data. Then he
compared the results of transverse
vibration of aluminium and mild steel
beam and he also did the effect of the
geometric characteristics (length, c/s area)
on the frequency of the beam. M.N.
Hamdan [7]. Here the author showed that
the mass effect on the natural frequency of
the beam with different end conditions by
Galerkins methods. And compared the
results with the Euler-Bernoulli beam
equation results. H. Navi [8]. Study of this
paper shows that the effect of the crack
depth on the natural frequency of the beam
i.e. frequency decrease with the increase
crack depth. Then he compared the
frequency of the cantilever beam on with
crack without crack. And he also had done
the numerical results by using the
MATLAB.
In this paper, the effects of different
geometrical parameters including density
are discussed in above mentioned all
boundary conditions in details with up to
first 3 natural frequencies.
EXPERIMENTAL DETAILS
Material Testing and Selection
Aluminium, Steel, and Copper are selected
as component material.
Description of specimens
For BEAMS,
Table1: Dimensions of BEAMS
Dimensions in mm No of steel No of copper No of Aluminium Total
350*20*3 1 1 1 3
550*30*3 1 1 1 3
550*40*3 1 1 1 3
Tensile Testing
Tensile test were carried out using
Universal Testing Machine at NIE
Mysuru. Then calculating the young’s
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Research and Reviews on Experimental and Applied Mechanics
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modulus (E).after tensile testing young’s
modulus of the material is get as given in
below table
Fig. 1: Material tested in UTM
Fig. 2: Tested specimen
Table 2: Properties of Materials
sl.no Material Young’s modulus in N/m2 Density in kg/m3
1 Steel (ASTM-A36) 2X1011 7850
2 Copper 1.2X1011
8940
3 Aluminium 0.7X1011 2720
Experimental setup
Experimental modal analysis is a
technique used to study the dynamic
characteristics of a mechanical structure.
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Research and Reviews on Experimental and Applied Mechanics
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This technique can be used to describe a
structure in terms of its natural
characteristics, namely natural frequency,
damping ratio, and mode shapes. Two
widely used methods for performing
experimental modal analysis are impact
test and vibration shaker test.
In our experiments the tri axial
accelerometer (356A15) mounted at a test
specimens of Beams with suitable
position. The accelerometer connected to a
DAQ-9234 and the impact hammer is used
to give a initial disturbance. The impact
hammer also connected to a DAQ-9234.
And the DAQ-9234 is mounting on the
cDAQ-9178 chassis, the power supply is
given to the chassis. Then draw a block
diagrammed of the data acquisition
process on the LABVIEW block
diagrammed window. Here to obtain a
frequency response to FFT analyser is
used to convert time domain signal to
frequency signal. We can also save the
data into measurement files by using write
to measurement files tool. And later we
can plot it. The below figures shows the
experimental setups of beams with
different end conditions and corresponding
experimental results of natural frequency.
The below Figure3 shows the experimental
setup of clamped-free and clamped-
clamped beam and Figure4 shows the
experimental setup of free-free vibration in
two types c) mounting on a sponge, d) by
using on a rubber band. And Figure6
shows the simply supported at both ends
and clamped-simply supported.
Fig. 3: Photo Copy of Beam with Different Boundary Conditions a) C-F conditions, b) C-C
Fig. 4: Photo Copy of Beam with Free-Free Boundary Conditions c) by Using Sponge, d) by
Using Rubber Band
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Fig.5: Photo Copy of Beam with Different Boundary Conditions c) SS-SS, d) C-SS
Analytical and FEM methods
For beams, we shall assume that the
deflection is small. In addition, we shall
neglect the rotation of the beam elements
during oscillation. We shall consider the
deflections only due to the bending
moment [14] page (497-500).
1
Moreover, since we are neglecting the
rotational motion of the element, the total
moment about the y-axis must be zero.
2
Further, from the elementary beam theory,
we get (with the sign convention used).
3
Where I is the second moment of beam
cross-section about its neutral axis. Using
(2) in (1) in conjunction with (3), the final
equation of motion we obtain is
4
Assuming a normal mode vibration in the
form x (z, t) = X(z) cos t, we can rewrite
the foregoing equation as
5
√
6
The general solution of (5) can be
rewritten as
X = cosh z+ sinh z+ cos z+
sin z 7
The values of can be obtained when the
boundary conditions of the beam are
prescribed. Once the is known, the
natural frequencies can be computed from
(6). Let us consider the following common
types of boundary conditions:
Both ends Simply-Supported
In this situation, the deflections and the
bending moments at the support cross-
sections must be equal to zero. Thus, at
such ends,
x = 0, M = 0, for all t.
This implies X=0, and
at the
simply supported ends. Therefore, the
boundary conditions can be written as
X=0,
Substituting these conditions in (1), we
get, for the nontrivial solutions, sin L = 0
or
(i=1, 2, 3…….) 8
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First three roots of this equation are
Hence, using (1) in (2), we find the natural
frequencies come out as
√
One end fixed and the other end free
(cantilever)
When one end of the beam is clamped,
both x and
should be zero for all t. This
yields, at the clamped end,
X=0,
For the free end of the beam, the bending
moment and the shear force are zero which
Yield
=0 for all t.
Therefore, the boundary conditions can be
written as,
X=0,
Using these conditions in (2) and
considering the nontrivial solutions, the
frequency equation we get has the form
Cos L cosh L = -1 9
First three roots of this equation are
For i > 3, the roots of 4 can be
approximated as
(
) 10
Both ends fixed
Both ends fixed proceeding in a manner
similar to that in previous conditions, the
frequency equation we obtain is
Cos L cosh L = 1 11
The first three roots and the asymptotic
solutions of (5) are
(
) i > 3 12
One end fixed and other end simply-
supported Here, the frequency equation is given by
tan L = tan L 13
The first three roots and the asymptotic
solutions of (7) are
(
) i > 3 14
Both ends free
The first three roots for Free-Free
boundary conditions.
(
) i =1, 2, 3 15
FEM method to use calculates natural
frequencies of beams.
The stiffness matrices of beam.
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[ ]
[
]
16
The mass matrices of beam.
[ ]
[
]
17
The nodal displacement vector,
q(t) = [ ]T
18
The equilibrium equations of motion of
entire beam is given by,
[M]= ̈ [ ] 19
To solve this above equation by applying a
boundary conditions of the beam we get
the solution.
ANSYS results
The below figures shows the ANSYS
results of aluminium beam (550*40*3 in
mm) in different boundary conditions.
Fig.6: Cantilever Beam Analysis Results
Fig. 7: Analysis of C-C Boundary Condition Results
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Fig. 8: Analysis of C-SS Boundary Condition Results
Fig. 9: Analysis of F-F Boundary Condition Results
Fig.10: Analysis of SS-SS Boundary Condition Results
Experimental Results The below figures shows the experimental
results of Time domain signals and
Frequency domain signals of aluminium
beam (550*40*3 in mm) with different
boundary conditions. Here due to the space
constrained not shown the other specimen
results. We discussed detailed in
aluminium beam (550*40*3 mm) similar
steps followed for others.
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Fig. 11: Cantilever Beam Experimental Results a) Time Domain Signal, b) Frequency
Domain Signal
Fig.12: C-C Beam Experimental Results c) Time Domain Signal, d) Frequency Domain
Signal
Results and Discussion
Free vibration analysis of beams
Below table shows the free vibration
analysis results of beams by varying
boundary conditions and geometry
conditions with analytical, FEM results by
using MATLAB, ANSYS and
experimental results are given in below
table’s.
Aluminium beam results B=20mm, L=350mm, t=3mm.
Table 3: Frequency Results of Aluminium C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 20.3 20.13 20 18
2 129.3 112.87 125 110
3 356 311.35 350.18 345
Table 4: Frequency Results of Aluminium C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 129.3 112.87 127.8 103
2 356 311.35 350.18 345
3 698.2 660.38 689.71 612
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Steel beam results B=20mm, L=350mm, t=3mm.
Table 5: Frequency Results of Steel C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 19.6 19.5 19.73 17
2 125.1 109.23 123.63 110
3 344.3 301.2 346 317
Table 6: Frequency Results of Steel C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 125.1 109.23 123.63 110
2 344.3 301.2 346 317
3 675 638.74 679.7 558
Copper beam results B=20mm, L=350mm, t=3mm.
Table 7: Frequency Results of Copper C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 14.5 14 14.5 13
2 92.4 80.7 90.6 84
3 254.5 222.5 253.7 244
Table 8: Frequency Results of Copper C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 92.4 80.68 90.6 84
2 254.5 222.5 253.7 244
3 499 472 498.8 468
Aluminium beam B=20mm, L=550mm, t=3mm.
Table 9: Frequency Results of Aluminium C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 8.22 8.72 8.1 7
2 52 45.85 50.9 44
3 144 126 142.68 130
Table 10: Frequency Results of Aluminium C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 52.34 45.85 50.9 44
2 144 126 142 130
3 282.7 267 280 265
Steel beam B=20mm, L=550mm, t=3mm.
Table 11: Frequency Results of Steel C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 7.95 8.72 7.7 7
2 50.6 44 48 44
3 139.4 121.7 134.8 128
Table 12: Frequency Results of Steel C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 50.6 44 48 44
2 139.4 121.9 134.8 128
3 273.5 258.6 264.6 255
Copper B=20mm, L=550mm, t=3mm.
Table 13: Frequency Results of Copper C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 5.8 5.9 5.8 6
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2 37.4 32.6 36.7 36
3 103 90 102.6 107
Table 14: Frequency Results of Copper C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 37.4 32.7 36.7 36
2 103 90 102.6 107
3 201 191 201 211
Aluminium B=40mm, L=550mm, t=3mm.
Table 15: Frequency Results of Aluminium C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 8.2 8.7 7.9 8
2 52 45.8 49 45
3 144 126 138 133
Table 16: Frequency Results of Aluminium C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 52 45 49.5 45
2 144 126 138 133
3 282.7 267.4 271.9 281
Steel B=40mm, L=550mm, t=3mm.
Table 17: Frequency Results of Steel C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 7.9 8.7 7.8 7
2 50.6 44 48.7 44
3 139.4 121.9 136 126
Table 18: Frequency Results of Steel C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 50.6 44 49.7 42
2 139 121.9 136.8 120
3 273.5 258.6 268.3 235
Copper B=40mm, L=550mm, t=3mm.
Table 19: Frequency Results of Copper C-F Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 5.9 5.9 5.7 6
2 37 32.7 35 41
3 103 90 99 119
Table 20: Frequency Results of Copper C-C Beam in Hz Modes Analytical MATLAB ANSYS Experimental
1 37 32.7 36 36
2 103 90 99.8 108
3 201 191 195 210
Comparison of beam results
Comparison of free vibration results of
beam with by varying Density, Length,
Cross section area, and different boundary
conditions as given below,
1. Above all table results shows that there
are good agreement between the all four
analytical, FEM methods by using
MATLAB, ANSYS, and experimental
results.
2. Comparison of Density v/s Frequency of
the BEAMS.
The ANSYS and experimental results are
taken to compare the results.
Beam dimension B=20mm, L=350mm,
t=3mm, C-F, BC’S.
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Aluminium (2700 kg/m3)
Steel (6585 kg/m3)
Copper 8933 kg/m3)
Table 21: Density v/s Frequency of the Cantilever Beam in Hz Mode
Number
ANSYS
(AL) Exp (AL)
ANSYS
(STL) Exp (STL) ANSYS (COP)
Exp (COP)
1 20 18 19.7 17 14.5 13
2 125 110 123.6 110 90.6 84
3 350 345 346 317 253.7 244
From above Table21 we can observe that
density increases natural frequency of the
system decreases.
Comparison of Length v/s Frequency of
the Cantilever beams. Beam, B=20mm,
t=3mm,
L1 = 350mm; and L2 = 550mm;
Table 22: Length v/s Frequency of the Cantilever Beam in Hz Mode umber AL(L1) AL(L2) STL(L1) STL(L2) COP(L1) COP(L2)
1 18 7 17 7 13 6
2 110 44 110 44 84 36
3 345 130 317 128 244 107
From above Table22 we can observe that
length increases natural frequency of the
system decreases.
4. Comparison of c/s Area v/s Frequency
of the beam. Beam L=550mm, t=3mm,
B1 = 40mm; and B2 = 20mm.
Table 23: C/s area v/s Frequency of the Cantilever Beam in Hz Mode umber AL (B1) AL (B2) STL (B1) STL (B2) COP (B1) COP (B2)
1 7 8 7 7 6 6
2 44 45 44 44 36 41
3 130 133 128 126 107 119
From above Table23 we can observe that
c/s area increases natural frequency of the
system decreases slightly. But in steel it is
same for first 2 frequency and 3 frequency
is reverse to the others
5. Free-Free vibration analysis of beam.
Dimension of Aluminium beam
L=550mm, B=40mm, t=3mm.
Free vibration analysis of beam under
Free-Free boundary conditions by using
rubber band and mounting on a sponge.
Then the results are compared with the
analytical and ANSYS results are shown
in below Table24. Both experimental
results are near to the analytical and
ANSYS results.
Table 24: Frequency of Free-Free BC’S beam results in Hz Mode Number Analytical ANSYS by using Rubber by using Sponge
1 51.9 49.8 48 48
2 144 137.4 138 137
3 282.5 269.6 258 262
6. Both the ends Simply Supported BEAM
Dimension of Aluminium beam
L=550mm,
B=40mm, t=3mm.
Free vibration analysis of beam under both
the ends simply supported boundary
conditions. Then the results are compared
with the analytical and ANSYS results are
shown in below Table25.Experimental
results are near to the analytical and
ANSYS results.
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Table 25: Frequency of both the ends simply supported beam results in Hz Mode Number Analytical ANSYS Experimental
1 23 22.8 23
2 92 91.2 117
3 207.5 205.3 196
7. Clamped and Simply Supported BEAM
Dimension of Aluminium beam
L=550mm, B=40mm, t=3mm.
Free vibration analysis of beam under
Free-Free boundary conditions by using
rubber band and mounting on a sponge.
Then the results are compared with the
analytical and ANSYS results are shown
in below Table26. Both experimental
results are near to the analytical and
ANSYS results.
Table 26: Frequency of Clamped-Simply supported beam results in Hz Mode Number Analytical ANSYS Experimental
1 35.9 34.5 32
2 116.8 111.8 107
3 242.98 233.5 233
8. Experimental results of aluminium
beam L=550mm, B=40mm, t=3mm, in all
Boundary conditions as shown in below
table
From below Table27 we can observe that
natural frequency is higher at free-free
boundary conditions and next clamped-
clamped boundary conditions next
clamped-simply support then simply
support-simply supported and clamped-
free conditions results respectively.
Table 27: Frequency of Aluminium Beam (L=550mm,B=40mm, t=3mm)in all Boundary
Conditions in Hz Mode umber C-F C-C C-SS SS-SS F-F by Rubber F-F by sponge
1 8 45 32 23 48 48
2 45 133 107 117 138 137
3 133 281 233 196 258 262
Experimental results of Damping factor (
), and Damping constant(C) for Beams
Logarithmic decrement method is one of
the most basic method used to measure the
damping factor of an under-damped
system.
Fig.13: Logarithmic Decrement Method of Calculating Damping Factor
In such a system, the vibration amplitude
decays with time and the natural log of
amplitudes of any two successive peak is
called the logarithmic decrement or log
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decrement. Logarithmic decrement ( ) can
be calculated by the equation
20
Where d is the log decrement ( ), is the
maximum amplitude of first cycle and
X is the maximum amplitude of nth cycle.
Damping ratio, ( ) can be calculated using
log decrement as follows
√
21
Then the critical damping constant (Cc)
can be obtained
Cc = 2mωn in
22
Thus the damping constant(C) is given by,
C = Cc in
23
The below Table28 shows the dimension
of specimens.
Table 28: Dimension wise Beam Naming Specimen Name Dimensions (L*B*h) in mm
1 350*20*3
2 550*20*3
3 550*40*3
9. Experimental results of Damping factor
( ), and Damping constant(C) for beams
the below Table29 shows the Damping
factor ( ), Critical damping factor (Cc),
damping constant(C) for aluminium, steel,
and copper with C-F and C-C. In that table
we can observe that the values of damping
factor ( ) and critical damping factor (Cc)
in
, and damping constant (C) in
clamped-clamped boundary conditions is
more than the clamped-free boundary
conditions.
Table 29: Experimental results of beam for ( ),Cc, C for CC and CF boundary conditions
Specimen for CC Cc
for
CC
C
for
CC for CF
Cc
for
CF C
for CF
AL 1 0.01997 11.68 0.2332 0.00705 2.0412 0.0144
AL 2 0.034 7.841 0.2666 0.0148 1.2474 0.18495
AL 3 0.01979 16.038 0.317 0.0102 2.8512 0.0291
STL 1 0.0107 36.267 0.38806 0.00294 5.605 0.0165
STL 2 0.264 22.801 0.6024 0.0147 3.63 0.05321
STL 3 0.0244 43.52 1.0618 0.01929 7.25 0.1399
COP 1 0.01554 31.51 0.4896 0.01324 4.877 0.0646
COP 2 0.0405 21.23 0.8591 0.02354 3.5376 0.0833
COP 3 0.0413 42.45 0.1399 0.0413 7.25 0.087
Error Causes of Results
It can see from the above table the
experimental results are not exactly equal
to the numerical results due to some of the
error occurred during the experimental
process. And some of assumption taken in
analytical method to simplify the
calculations. The causes of error are listed
in below
1. Signal Leakage error.
2. In calculation vibration damping is
neglected, but in actual practice damping
occurred.
3. Weight of the accelerometer.
4. Electrical noise.
5. Geometrical imperfection.
6. Environmental effect. Etc...
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Leakage error
1. Difference between the time period of
captured signal and original signal is
called leakage error.
2. To avoid this leakage error to match the
time period or multiply with the integer
after capturing the signal.
3. Windowing techniques are used to
reduce the leakage error by multiplying the
weights.
Ex:- middle of signal multiply with higher
value and end of the signal multiply with
smaller value then joining this type of two
signal we get smooth joining curve so
leakage is less.
Strategies for Choosing Windows
Fig.14: Types windows available in LABVIEW
Each window has its own characteristics,
and different windows are used for
different applications. To choose a spectral
window, you must guess the signal
frequency content. If the signal contains
strong interfering frequency components
distant from the frequency of interest,
choose a window with a high side lobe
roll-off rate. If there are strong interfering
signals near the frequency of interest,
choose a window with a low maximum
side lobe level.
Selecting a window function is not a
simple task. In fact, there is no universal
approach for doing so. However, Table
below can help you in your initial choice.
Always compare the performance of
different window functions to find the best
one for the application.
Table 30: Initial Window Choice Based on Signal Content Window Signal Content
Hanning 1. Sine wave or combination of sine wave.
2. Narrow band random signal (vibration data)
3. Unknown data.
Flat top Sine wave (amplitude accuracy is important).
Uniform Broadband random (white noise).
uniform, Hanning Closely spaced sine waves.
Force Excitation signals (hammer blow).
Exponential Response signal.
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Electrical noise.
1. Random electron motion (micro volts) it
creates heat on surface during measure.
2. Local magnetic fields arcing (in mille
volts).
3. Earth loop faults.
Geometrical imperfection
1. Chemical combination of material is not
linear.
2. Dimension of specimen is not accurate.
Environmental effect.
1. Temperature variation.
2. Noise in wind flow.
3. Dust particle etc.
CONCLUSIONS
The following are the important
conclusions of this study:
4.1. Free vibration analysis of BEAMS
1. Experimental results are closer to the
numerical results.
2. C-F Beam 2nd frequency is first
frequency of the C-C beam modal
analysis.
3. Density increases frequency decreases.
4. For Free-Free BCs placed on Sponge
and hanging by rubber band both are gives
same results.
5. All boundary conditions of beam BCs is
experimentally satisfied.
6. C/s area increases frequency is slightly
decreases.
7. We can observe that natural frequency
is higher at free-free boundary conditions
and next clamped-clamped boundary
conditions next clamped-simply support
then simply support-simply supported and
clamped-free conditions results
respectively.
8. We can observe that the values of
damping factor ( ) and critical damping
factor (Cc) in
, and damping constant
(C) in clamped-clamped boundary
conditions is more than the clamped-free
boundary conditions.
ACKNOWLEDGEMENTS This work was performed using facilities
at NIE, Mysuru, funded by the NIE,
TEQUIP-2. The authors are grateful to the
NIE, Mysuru for their support and
encouragement.
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