Modal Analysis of Single Rectangular Cantilever Plate by Mathematically, FEA and Experimental

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056
Volume: 04 Issue: 08 | Aug -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 264
Modal Analysis of Single Rectangular Cantilever Plate by
Mathematically, FEA and Experimental
Ashish R. Sonawane1, Poonam S. Talmale2
1Research scholar, Late G. N. Sapkal College of Engineering, Nashik, Maharashtra, India
2Assistant Professor, Late G. N. Sapkal College of Engineering, Nashik, Maharashtra, India
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Abstract - Modal analysis is a major technique to
determine the vibration characteristics of engineering
structures and its component’s. It is process by which the
natural frequencies, mode shapes and damping factor of
structure can be determined with a relative ease. It should
be a major alternative to provide a helpful contribution in
understanding control of many vibration phenomenons
which encompasses in practice. In this work we compared
the natural frequency mathematically, FEA and
experimentally. The main objective of this paper is to
determine the natural frequency and mode shape of a single
rectangular cantilever beam condition and to compare the
results obtained by finite element analysis with
experimental results. The cantilever beam of rectangular
plate is designed and analyzed in ANSYS. A good correlation
between the mathematical, FEA and experimental result is
observed. The analysis result helps in depicting the failure
loads for different conditions. Aluminum single rectangular
cantilever plate is studied in this work. For mathematically
Euler’s Bernoulli’s beam theory is used. The results obtained
by both the methods are found to be satisfactory.
Key Words: Natural frequency, mode shapes, FEA.
1. INTRODUCTION
The aim of this paper is to provide efficient numerical
techniques for the prediction of the dynamic response of
single rectangular cantilever plate and to validate the
predictions via experimental tests. Vibration problems are
often occurred in mechanical structure. The structure
itself has a certain properties so it is necessary to
understand its characteristics. Especially in lightweight
structures, these additional contributions strongly affect
the modal properties of the overall structure and cannot
be neglected. At the same time, the precise knowledge of
the modal properties is the starting point for controllers
design. In this work a modal analysis by finite element
method is used. The main purpose of modal analysis is to
study the dynamic properties of structures like natural
frequency, damping and mode shapes. In the present
paper, we examine and compare different techniques for
modal analysis of simple rectangular cantilever beams.
The first technique is based on mathematical modeled by
the Euler–Bernoulli beam theory. Next, we test obtaining a
finite-dimensional version of the system finite-element
(FEA) method. The ABAQUS FEA software was used to
predict the natural frequencies, mode shapes. After
completing these two analysis, it is necessary to carry
experimental modal analysis to compare the results and to
analyze it for validation. An experimental set up is
prepared with the help of NI-Lab View software and data
acquisition hardware to obtain natural frequencies and
mode shapes. In this paper, we will be formulating the
equations of motion of a free cantilever beam. The natural
frequency of continuous beam system will be found out
different variables of beam using ANSYS 14.0. The results
compared with mathematically, FEA and experimentally
with each other. The modal analysis is used to understand
the dynamic properties of structure such as natural
frequency, damping ratio and mode shape. With modal
analysis, the modal parameters of the structures can be
extracted. The modal parameters, including natural
frequency, damping ratio, and mode shape, are the
fundamental elements that describe the movement and
response of a structures to free vibrations. After
completing these two analysis, it is necessary to carry
experimental modal analysis to compare the results and to
analyze it for validation. So knowing these modal
parameters helps to understand the structure’s response
to free vibration conditions as well as to perform design
validation.
2. LITERATURE SURVEY
Before starting with actual working, it’s always helpful to
study literature and work which is already carried out in
similar field. This study helps to decide the project outline
and flow. Some research papers, case study have
discussed.
Xiaocong He [1] Investigated the dynamic response of a
single rectangular plate numerically using ABAQUS
software & experimentally and compared both the results
with each other. The LMS CADA-X dynamic test software
and LMS DIFA Scads II 48 channels data acquisition
hardware is used in experimental measurements of
vibration of the beams. The comparable results are
obtained between the experimentally measured and
predicted dynamic response of the beams.
X.He, S.O. Oyadiji [2] In general, the transverse natural
frequencies of the single rectangular cantilevered beams
increase with an increase in the young’s modulus of

adhesive, but do not appear to change significantly with an
increase in the poisons ratio. The investigations are
carried out using finite element method (FEM).
Yu Du, Lu Shi, [3] prepared the FE model for the specimen
used in this study was developed in the commercial FE
package ABAQUS. Validate with FEA and Experimentally
Ankit Gautam, Jai kumar Sharma, Pooja Gupta [4] the
theoretical and numerical modal analysis of beam are
performed. The numerical results are obtained using
ANSYS 14.5. The numerical and theoretical results are
found to have extremely good correlation.
Dr. Negahban [5] Beam studied in this paper are long, thin,
cantilever beam. Determine the equation vibration of
beams. Euler–Bernoulli beam theory is used.
Subhransu Mohan Satpathy, Praveen Dash [6] formulating
the equations of motion of a free cantilever beam. The
natural frequency of continuous beam system will be
found out at different variables of beam using ANSYS 14.0.
The results will be compared further using
experimentation by free vibration of a cantilever beam.
Using those results, able to compare the parameters in
Euler-Bernoulli.
3. MATHEMATICAL MODAL ANLYSIS
Euler–Bernoulli beam theory is used get natural frequency
of rectangular cross sectioned cantilever plate subjected to
natural vibration.
Consider an Euler-Bernoulli uniform cantilever beam
undergoing transverse vibration.
Fig -1: A cantilever beam
Nomenclature:
L = 1000mm Length of beam
b = 50mm Width of beam
h = 5mm Thickness of beam
E = 70 x 103 (N/mm2) Young’s modulus of Aluminum
v = 0.35 Poisons ratio of Aluminum
ρ = 2700 (Kg/m3) Density of Aluminum
Using Euler- Bernoulli Beam Theory,
(1)
Normal mode solution to the above equation,
( ) ( ) ( ) (2)
This make equation,
( ) ( )
(3)
(4)
Where √ (5)
(6)
( )√
(7)
Where (1.875, 4.694, 7.855,
10.996, 14.143, 17.286, 20.429, 23.571 etc)
Now, to find the natural frequencies for different modes,
1. First Natural Frequency
√
√
We know,
2
n
nf



2. Second Natural Frequency
√
√
We know

2
n
nf



3. Third Natural Frequency
√
√
13.24 rad/sec
We know
2
n
nf



4. Fourth Natural Frequency
√
√
28.094 rad/sec
We know
2
n
nf



5. Fifth Natural Frequency
√
√
We know
2
n
nf



Similarly, we can calculate natural frequencies for
remaining modes by this method. It is possible to compare
all natural frequencies with the software analysis as well
as the experimental modal analysis. The results of
theoretical natural frequencies obtained by using material
properties and dimensions of the beam.
Table -1: Theoretical natural frequencies of rectangular
cross sectioned cantilever plate
Mode
Shapes
Natural Frequencies
1 0.135
2 0.81
3 2.1
4 4.47
5 7.39
6 11.04
7 15.43
8 20.55
9 24.54
10 28.87
4. Finite Element Modeling
Modeling of single rectangular cantilever plate was done
with the help of CATIA software. Finite element model of
beam is constructed in ANSYS and then computational
modal analysis is performed to generate natural
frequencies and mode shapes. The material parameter is
taken from nomenclature of fig 1. Boundary conditions are
applied at end of beam. finite element analysis (FEA) of
specified objects are carried out to obtain necessary
parameters and the specimens are to be tested in the
software to analyze the dynamic response of the specimen
at cantilever beam condition and to get the natural
frequencies and mode shapes. Solving a practical problem
by FEA involves learning about the program, preparing a
mathematical model, discretizing it, doing the calculations
and checking the results.
Fig -2: First mode shape of rectangular cross sectioned
cantilever plate

Fig -3: Second mode shape of rectangular cross sectioned
cantilever plate
Fig -4: Third mode shape of rectangular cross sectioned
cantilever plate
Fig -5: Fourth mode shape of rectangular cross sectioned
cantilever plate
Fig -6: Fifth mode shape of rectangular cross sectioned
cantilever plate
Similarly, we can calculate natural frequencies for
remaining modes by this method.
Table -2: FEA Results
Mode
Shapes
Natural
Frequencies
1 0.29774
2 1.3296
3 1.8632
4 5.2034
5 8.2374
6 10.172
7 12.254
8 16.817
9 22.658
10 24.999
5. EXPERIMENTAL ANALYSIS
The basic experimental setup is shown in fig 7. The point of
impact and position of the accelerometer are chosen such a
way that the natural frequencies of the system can be easily
determined.
Fig -7: Experimental set up
The equipment’s which are used to perform the real
experiments are as follows.
1. Impact Hammer
2. Accelerometer
3. Data Acquisition Hardware (At least 2-Channel.)
4. A PC or a Laptop loaded with NI-Lab View
software for modal analysis.
5. Power supply for the PC and vibration analyzer,
connecting cables for the impact hammer and
accelerometer and adhesive/wax to fix the
accelerometer.
5.1 Experimental Procedure
1. A beam of a particular material (aluminum),
dimensions (L, w, d) and transducer (i.e.,
measuring device, e.g. strain gauge,
accelerometer, laser vibrato meter) was chosen.

2. One end of the beam was clamped as the
cantilever beam support.
3. An accelerometer (with magnetic base) was
placed at the free end of the cantilever beam, to
observe the free vibration response.
4. An initial deflection was given to the cantilever
beam and allowed to oscillate on its own. To get
the higher frequency it is recommended to give
initial displacement at an arbitrary position apart
from the free end of the beam (e.g. at the mid
span).
5. This could be done by bending the beam from its
fixed equilibrium position by application of a
small static force at the free end of the beam and
suddenly releasing it, so that the beam oscillates
on its own without any external force applied
during the oscillation.
6. The free oscillation could also be started by giving
a small initial tap at the free end of the beam.
7. The data obtained from the chosen transducer
was recorded in the form of graph.
8. The procedure was repeated for 5 to 10 times to
check the repeatability of the experimentation.
9. The whole set of data was recorded in a data base.
By using experimental procedure we find natural
frequencies and mode shapes of single rectangular plate
which is helpful to validate with mathematical and FEA
modal analysis.
Table -3: Experimental natural frequencies
Mode
Shapes
Natural
Frequencies
1 0.233
2 1.142
3 1.737
4 4.799
5 8.32
6 9.36
7 11.67
8 15.013
9 23.617
10 23.944
6. RESULT AND DISCUSSION
In this work, theoretical modal analysis of single
rectangular cantilever plate has been carried out in
various conditions and natural frequencies are obtained.
The theoretical results are validating with ANSYS 14.0
software and experimental results. From this analysis it is
observed that the natural frequencies obtained are almost
close to each other except the 20 % of the error.
Table -1: Comparative table of natural frequencies at each
node
Mode Theoretical FEA Experimental
1 0.135 0.29774 0.233
2 0.81 1.3296 1.142
3 2.1 1.8632 1.737
4 4.47 5.2034 4.799
5 7.39 8.2374 8.32
6 11.04 10.172 9.36
7 15.43 12.254 11.67
8 20.55 16.817 15.013
9 24.54 22.658 23.617
10 28.87 24.999 23.944
Chart -1: Comparison of theoretical, FEA and
experimental results.
3. CONCLUSIONS
In this paper the theoretical, FEA and experimental modal
analysis of beam are performed. In this report, we
compared the Euler-Bernoulli models by ANSYS and
experimentally. The FEA and experimental results are
found to have extremely good correlation. The equation of
motion and the boundary conditions were obtained and
the natural frequencies were also obtained for different
modes. The results obtained from Finite element modeling
and experimentation has found good agreement. It can be
found out that Euler-Bernoulli equation is valid for long
and slender beams where we neglect shear deformation
effects.
REFERENCES
[1] Xiaocong He Numerical and Experimental
Investigations of Dynamic Response of Bonded Beams
with a Single Lap Joint, International Journal of
Adhesion and Adhesives 37 (2012) 79-85.
0
10
20
30
40
1 2 3 4 5 6 7 8 9 10
NaturalFrequenciesinHertz
Number of Modes
Natural Frequency in Hertz Theoretical
Natural Frequency in Hertz FEA
Natural Frequency in Hertz Experimental

[2] X.He, S.O. Oyadiji Influence of adhesive characteristics
on the transverse free vibration of single lap-jointed
cantilevered beams, Journal of material processing
technology 119 (2001) 366-373.
[3] Subhransu Mohan Satpathy, Praveen Dash Dynamic
analysis of cantilever beam and its experimental
validation” National Institute of Technology Rourkela
[4] Yu Du, Lu Shi, Effect of vibration fatigue on modal
properties of single lap adhesive joints, International
Journal of adhesion and adhesives 53 (2014) 72-79.
[5] Ankit Gautam, Jai kumar Sharma, Pooja Gupta Modal
analysis of beam through analytically and FEM
International conference
[6] Free vibration of cantilever beam, Virtual Labs for
Mechanical Vibration, Mechanical Engineering.
[7] Hassan Jalali a,1, Hamid Ahmadian a, John E.
Mottershead b, Identification of nonlinear bolted lap-
joint parameters by force-state mapping, International
Journal of Solids and Structures 44 (2007) 8087–
8105.
[8] Y.B. Patil, R.B. Barjibhe, Modal analysis of adhesively
bonded Joints of different Materials, International
Journal of Modern Engineering Research, Vol.3,
Issue.2, March-April.( 2013) pp-633-636.

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