 Open Access
 Total Downloads : 1856
 Authors : Girish Dalvi, Pankaj Kumar, Tejas Vispute, Anurag Sawant & Rohit Jagtap
 Paper ID : IJERTV4IS040847
 Volume & Issue : Volume 04, Issue 04 (April 2015)
 DOI : http://dx.doi.org/10.17577/IJERTV4IS040847
 Published (First Online): 22042015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Modal Analysis of Beam Type Structures
Pankaj Kumar, Tejas Vispute, Anurag Sawant, Rohit Jagtap
Students,
Department of Mechanical Engineering, Fr. C. Rodrigues Institute of Technology, Vashi, Navi Mumbai 400 703, India
Girish Dalvi
Asst. Professor, Department of Mechanical Engineering,
Fr. C. Rodrigues Institute of Technology, Vashi, Navi Mumbai 400 703, India
Abstract : The purpose of this project is to study Modal behaviour of Beam type structures. Beams under study include Cantilever, Simply Supported and Fixed beam. Mode shapes and natural frequencies of these three types of beams are obtained using Theoretical analysis, Simulation in ANSYS and Experiment using FFT analyser. Finally natural frequencies obtained from Simulation and Experiment are compared with Theoretical values of natural frequency. The mode shapes obtained from simulation and experiment are matching closely with analytical ones. Natural frequencies obtained by simulation are within 6% deviation when compared to theoretical results whereas for experimental natural frequencies the maximum deviation from theoretical values is 19.31%.
KeywordsModal Analysis, Beam type structure, FFT Analyzer, Natural Frequency, Mode Shapes

INTRODUCTION
Modal analysis is the study of the dynamic properties of structures under vibration excitation. The goal of modal analysis in structural mechanics is to determine the natural mode shapes and frequencies of an object or structure during free vibration.
The various research papers studied are based on evaluation of specific properties or characteristics of vibration of beams by various techniques. L.Rubios [4] work focuses on crack identification by means of modal parameters. P.urÃ¡nek et.al[6] work is on decaying rate of vibration in cantilever beam for which they used an aluminum frame as an accessory to increase decay rate. Farooq and B. Feenys[5] work is on new approach in theoretical modal analysis where they have used and evaluated the results experimentally for validity. H. Auweraer[2] has adopted a black box approach and evaluated them on industrial application. S. Mahalingam[1] has found changes occurring in modal parameters when support changes its position at an instance. A. Cusano et.al[3] used Bragg grating sensors instead of conventional accelerometer in experimental modal analysis and results were evaluated by experiment and simulation.
The literature survey shows that lot of efforts have been taken for determining the modal properties of beam type structures using numerous methods. Industry is focusing on reducing noise and vibration level for betterment of performance of various products. Beam type of structures are used in various application, hence it becomes an important structure to be studied for noise and vibration reduction. Mode shapes of beam type structures may provide more information to control vibration. The present study will
attempt to conduct experimental modal analysis of beam type structures namely Cantilever, Simply Supported and Fixed Beam. Thus, the scope involves:

Determination of Mode Shapes of Beam type structures analytically.

Simulation of Beam type structure in ANSYS.

Experimental Modal Analysis.


THEORETICAL ANALYSIS
Beams are slender members used for supporting transverse loading. It is a basic structural element that is capable of withstanding load primarily by resisting bending. Simply supported, cantilever and fixed beam are considered for analysis and description of them are given below.
Cantilever beam:
A beam which is supported on the fixed support and having the other end free is termed as a cantilever beam: Fixed support is obtained by building a beam into a brick wall, casting it into concrete or welding the end of the beam. Such a support provides both the translational and rotational constrain to the beam, therefore the reaction as well as the moments appears, as shown in the figure below.
Fig 1 Cantilever Beam
Simply supported beam:
The beams are said to be simply supported if their supports creates only the translational constraints. When both the supports of beams are roller supports or one support is roller and the other hinged, the beam is known as a simply supported beam.
Fig 2 Simply Supported Beam
Fixed beam:
A beam which is supported on the fixed support on both the ends is termed as a fixed beam. It provides both the translational and rotational constrain to the beam at both the ends.
Fig 3 Fixed Beam
Calculation of Natural Frequency
Natural frequencies for first five mode shapes of cantilever, simply supported beam and fixed beam are calculated in this section.
Using modified expression,
fn fn = natural frequency
C = constant
g = acceleration due to gravity E = youngs modulus
I = moment of inertia
w = weight per unit length l = length of beam
The value of constant (C) is different for different beam types which have been enlisted in Table 1.
Table 1: Values of c for different type of beams
Beam
1st
2nd
3rd
4th
5th
Cantilever
0.56
3.51
9.82
19.24
31.81
Simply Supported
1.57
6.28
14.14
25.13
39.27
Fixed
3.56
9.82
19.24
31.81
47.52
The dimensions of beam considered for all types of beam structures are shown in figure 4.
Length= 0.5 m
Width= 0.04 m
Depth= 0.005 m
Fig. 4 Dimensions of Beam
Natural Frequencies of three types of beam are calculated and listed in following table 2.
Table 2 Theoretical natural frequencies
Beam Type
Cantilever
Simply Supported
Fixed
1
16.68
46.78
104.10
2
104.30
187.12
286.75
3
292.52
420.92
562.36
4
573.66
748.02
929.52
5
948.21
1168.2
1384.99

SIMULATION
Simulation of modal analysis is done on FEA software
ANSYS for three different types of beam structures which are cantilever, simply supported and fixed beam.
Cantilever Beam
The following parameters have been used in simulation.
Youngs Modulus = 2.1 Ã— 1011 N/mm2 Poissons ratio = 0.3
Density = 7886 kg/m3
The Grid size has been gradually increased from 20 to 85 to reach a point where Natural frequencies obtained in simulation matches very closely with that of Analytical results. As the results best match at a mesh size of 85 all beam elements are further given a mesh size of 85 for analysis.
The natural frequencies of cantilever beam are found with the help ANSYS software and shown in the following table 3
Table 3 Natural Frequency of Cantilever beam
Set
Natural Frequency(Hz)
1
16.75
2
105.02
3
294.05
4
575.09
5
952.08
Mesh model of cantilever beam is shown in fig 5 and first five mode shapes obtained using Ansys are shown in figure 6, 7, 8, 9 and 10.
Fig 5 Meshed Model of Cantilever Beam
Fig 6 First mode shape
Fig 7 Second mode shape
Fig 8 Third mode shape
Fig 9 Fourth mode shape
Fig 10 Fifth mode shape
Similarly mode shapes and natural frequencies are found out for Simply Supported and Fixed beam. Natural frequencies obtained by simulation for different beam structures are given in Table 4.
Table 4 Natural Frequencies of Beam Type Structure
Beam Type
Natural Frequencies
1
2
3
4
5
Cantilever
16.75
105.02
294.05
575.09
952.08
Simply Supported
47.05
188.18
423.22
752.43
1175
Fixed
105.56
293.97
575.20
952.20
1422

EXPERIMENTAL SETUP
In this chapter, the various types of beams studied in the project are realized. Natural frequencies and mode shapes of different beams are obtained using FFT analyzer.

Cantilever Beam
A Cantilever beam can be made by restricting all degrees of freedom of beams one end only. This arrangement can be realized using the same setup of fixed beam by eliminating its second support as shown in figure 11.
Fig 11 Cantilever Beam Setup

Simply Supported Beam
Simply supported beam can be made if the supports create only translational constraint at one end and only vertical reaction at other end. The set up prepared is shown in figure12.
Fig 12 Combined Setup for Simply Supported & Fixed Beam

Fixed Beam
Fixed beam can be made by restricting all degrees of freedom of beam at ends as shown in figure 5.1. Fixed beam arrangement consists of two identical Isections, two plates and mild steel strip of dimensions 70Ã—4Ã—0.5 cm. Two I sections are welded to base, which is in the form of C section, at 50 cm. Ends of strip are sandwiched between upper flange of Isection and plate and then bolted tightly. The setup is shown in figure 13.
Fig 13 Fixed Beam Setup


EXPERIMENTAL ANALYSIS
Experimental analysis is performed on three types of beam using FFT analyzer. Modes shapes and Natural Frequencies of Cantilever Beam are shown in this section. Figure 14 shows five peaks corresponding to five natural frequencies. Figure 15 to 19 represents first five mode shapes obtained experimentally.
Fig 14 Peaks obtained for Cantilever Beam
(m/sÂ²/Newton) 25.0000
EWaterfall_H1_2,1(f)
22.0000
20.0000
18.0000
16.0000
14.0000
12.0000
10.0000
8.0000
6.0000
4.0000
2.0000
0
1.5000
0
1.00
2.00
3.00 4.00 5.00
Time (seconds)
6.00
7.00
8.00
9.00
Freq = 25.00 Hz
Fig 15 First mode shape for Cantilever Beam
(m/sÂ²/Newton) 88.0000
75.0000
60.0000
45.0000
30.0000
15.0000
EWaterfall_H1_2,1(f)
0
15.0000
30.0000
45.0000
60.0000
75.0000
90.0000
105.0000
120.0000
135.0000
140.0000
0
1.00
2.00
3.00
4.00 5.00
Time (seconds)
6.00
7.00
8.00
9.00
Freq = 160.00 Hz
Fig 16 Second mode shape for Cantilever Beam
(m/sÂ²/Newton) 240.0000
200.0000
EWaterfall_H1_2,1(f)
160.0000
120.0000
80.0000
40.0000
0
40.0000
80.0000
120.0000
160.0000
200.0000
240.0000
280.0000
320.0000
0
1.00
2.00
3.00
4.00 5.00
Time (seconds)
6.00
7.00
8.00
9.00
Freq = 443.00 Hz
Fig 17 Third mode shape for Cantilever Beam
Table 6 % Deviation of Simulation and Theoretical values
Beam Type
Natural Frequencies
1
2
3
4
5
Cantilever
0.42
0.69
0.52
0.25
0.41
Simply Supported
5.59
5.64
5.71
5.68
5.78
Fixed
1.39
2.51
2.28
2.43
2.67
Beam Type
Natural Frequencies
1
2
3
4
5
Cantilever
2.62
4.72
5.34
9.35
4.75
Simply Supported
15.37
9.05
18.40
4.89
4.18
Fixed
19.31
17.00
2.90
1.99
1.29
Table 7 % Deviation of Experimental and Theoretical values
34.0000
24.0000
16.0000
12.0000
4.0000
8.0000
16.0000
20.0000
0
500.0000
400.0000
300.0000
200.0000
100.0000
(m/sÂ²/Newton) 650.0000
600.0000
(m/sÂ²/Newton)
28.0000
EWaterfall_H1_2,1(f)
20.0000
8.0000
4.0000
12.0000
24.0000
27.5000
0
1.00
2.00
3.0
4.00
Time (seconds)
5.00
6.00
7.00
8.00
9.00
Freq = 897.00 Hz
Fig 18 Fourth mode shape for Cantilever Beam
EWaterfall_H1_2,1(f)
0
0
1.00
2.00
3.00
4.00 5.00
Time (seconds)
6.00
7.00
8.00
9.00
Freq = 1446.00 Hz
100.0000
200.0000
300.0000
400.0000
475.0000
Fig 19 Fifth mode shape for Cantilever Beam
It is observed that maximum percentage deviation is 5.78 for theoretical analysis and simulation and that for theoretical and experimental results it is from 19.31. For experimental analysis larger deviation are observed.
VII. CONCLUSION
Based on theoretical, analytical & experimental results it is hereby concluded that:

Results obtained by simulation are matching closely with theoretical values. The maximum percentage deviation is 5.78%.

Results obtained by experimental analysis deviate more from theoretical analysis compared to simulation. The maximum percentage deviation is 19.13%.

In experimental analysis of Simply Supported Beam and Fixed Beam some extra peaks are observed along with peaks corresponding to natural frequencies.

Modes shapes obtained from simulation and experiment are
Similarly mode shapes and natural frequencies are found out for Simply Supported and Fixed beam. Natural frequencies obtained experimentally for different beam structures are given in Table 5.
Beam Type
Natural Frequencies
1
2
3
4
5
Cantilever
16
99.3
276
520
903
Simply Supported
57
217
366
758
1195
Fixed
84
238
546
911
1367
Table 5 Natural Frequencies of beams by Experiment


RESULTS AND DISCUSSION
This chapter compares results obtained by simulation and experimental analysis with theoretical values. Percentage deviation of experimental and simulation values from theoretical value is calculated and listed in following two tables. Table 6 gives percentage deviation of simulation values from theoretical and table 7 gives percentage deviation of experimental values from theoretical values.
in agreement with the theoretical ones.
ACKNOWLEDGEMENT
We would like to take this opportunity to thank all those who have whole heartedly lent their support and contributed in this project.
We are sincerely thankful to our HOD, Dr. S. M. Khot and our principal, Dr. Rollin Fernandes for giving us an opportunity to work on this project.
We are sincerely thankful to our guide, Prof. Girish Dalvi for his valuable, inspiring and timely guidance and assistance throughout the course of the project.
We would like to sincerely thank Mr. K. P. Rajesh and Mr. Moreshwar Kor for helping us in preparation of experimental setup of the beam structures.
Lastly, we would like to thank all those people who knowingly and unknowingly contributed to our project.
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