Modal Analysis of Rectangular Plate with Circular Cut out Subjected to Different Boundary Conditions
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This document presents a modal analysis of rectangular mild steel plates with circular cutouts subjected to different boundary conditions. A finite element analysis is performed to determine the natural frequencies and mode shapes. The cutout ratio is varied from 0 to 0.625 in steps of 0.25 under four boundary conditions: fully fixed-fixed-simply supported-simply supported (FFFS), fully fixed-fixed-simply supported (FFSS), fully simply supported-simply supported-simply supported (FSSS), and simply supported-simply supported-simply supported (SSSS). The results show that the natural frequencies decrease as the boundaries are freed up and also decrease with increasing cutout ratio. The boundary conditions have a
![International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 06 | June -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2075
Here we can observe that the natural frequencies are
negligible for SSSS, SSFF, SFFF as compared to SSSF
boundary condition.
CONCLUSION
The finite element formulation is used to study effect of cut
out on the free vibration of Mild steel plate. The numerical
results are presented and discussed in above. The broad
conclusions that can be made from the present study are
summarized as follows:
The fundamental natural frequencychangesonlymarginally
if a small cut out (either of the two cut out ratiosbeingsmall)
is made in the plate. However, for intermediate and large
size cut outs, the fundamental natural frequency increases
rapidly; the amount of increase depends on cut out ratios in
two directions.
For square plate with circular cut out, the natural frequency
increases with the increasing constraint at the boundary
irrespective of the size of the cut out. However, this increase
is not uniform; it is also dependent on the particularmodeof
vibration.
For square plate with circular cut out, an increase in cut out
size does not always result in an increase in the natural
frequency. Besides depending on the cut out size, the
increase or decrease also depends on the boundary
condition.
Hence depending upon the natural frequencyrequiredofthe
given component we can give the support to the system so
that there will not be any resonance during the operating
conditions.
Hence the required geometry of the componentcanbemade
use without causing resonance by making use of different
boundary conditions.
REFERENCES
[1]. Ramu I and Mohanty S. C. Modal Analysis of functionally
graded plates using finite element method, Procedia
Materials Science 6 ( 2014 ) 460 – 467,
[2]. Marco Alfano, Leonardo Pagnotta, Determining the
elastic constants of isotropic material s by modal vibration
testing of rectangular thin plate, Journal of Sound and
Vibration 293 (2006) 426–439, , Italy.
[3]. Kaushar H. Barad, D.S.Sharma, VishalVyas. Crack
detection in cantilever beam by frequency based method,
Procedia Engineering 51 ( 2013 ) 770 – 775,
[4]. A.Labib n, D. Kennedy, C. Feathersto , Free vibration
analysis of beams and frames with multiple cracks for
damage detection, Journal of Sound and Vibration, Cardiff
University, Cardiff CF24 3AA, UK ,.
[5]. P. K. Jenaa, D. N. Thatoib, J. Nandab, D. R. K. Parhic, Effect
of damage parameters on vibrationsignaturesofa cantilever
beam, International Conference on Modeling, Optimization
and Computing (ICMOC 2012), April 10 - 11, 2012,
[6]. JeslinThalapil, S.K.Maiti, Detectionoflongitudinal cracks
in long and short beams using changes in natural
frequencies, International Journal of Mechanical Sciences
83(2014)38–47.
[7]. O. Doerk, W. Fricke, C. Weissenborn, Comparison of
different calculation methods for structural
stresses at welded joints, International Journal of Fatigue 25
(2003) 359–369.
[8]. Kant T. and Swaminathan K., Analytical solutions for the
static analysis of laminated composite and sandwich plates
based on a higher order refined theory, Composite
Structures, vol.56 (2002): pp. 329-344.
[9]. Zhang Y.X. and KimK.S., Geometricallynonlinearanalysis
of laminated composite plates by two new displacement
based quadrilateral plate elements, Composite Structures,
vol.72 (2006): pp. 301-310.
[10]. Akavci S.S., Yerli H.R. and Dogan A., The first order
shear deformation theory for symmetrically laminated
composite plates on elastic foundation, The Arabian Journal
for Science and Engineering,
[11]. Krishna Murty A.V. and Vellaichamy S., On higherorder
shear deformation theory of laminated composite panels,
Composite Structures, vol. 8 (1987): pp. 247-270.
[12]. Kant T. and Swaminathan K., Analytical solutions for
free vibration analysis of laminatedcompositeandsandwich
plates based on a higher order refined theory, Composite
Structures, vol.53, (2001): pp. 73-85.](https://image.slidesharecdn.com/irjet-v4i6406-180219083025/85/Modal-Analysis-of-Rectangular-Plate-with-Circular-Cut-out-Subjected-to-Different-Boundary-Conditions-4-320.jpg)
Modal Analysis of Rectangular Plate with Circular Cut out Subjected to Different Boundary Conditions
- 1. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 06 | June -2017 www.irjet.net p-ISSN: 2395-0072 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2072 Modal Analysis of Rectangular Plate with Circular Cut out Subjected to Different Boundary Conditions. Praveen Mirji1, Venkatesh Deshpande2 1,2 Asst.Professor,Dept of Mechanical Engg,GIT,Belgaum,India ---------------------------------------------------------------------***--------------------------------------------------------------------- Abstract - The aim of this paper is to perform a modal analysis to determine the natural frequencies and mode shapes for the mild steel specimen. This research presents an experimental of the Model analysis of Square Mild steel plate with circular cut out. The Various sizes of square mild steel plate with circular cut outs are consideredinordertoexamine the effect of cut out size and boundary condition on the fundamental natural frequency of the square plate. The Natural frequencies of the finite element (FE) were analyzed by the Optistuct software, which was found by analysis. Key Words: Circular Cut out; Model Analysis; Boundary Condition; Square plate; FEM. 1. INTRODUCTION The paper aims to calculate the natural frequencies and accompanying mode shapes of the plate which are the key parameters when considering noise and acoustic parameters. To determine these parameters modal analysis will be used. Vibration problems are often occurred in mechanical structure. It is important to prevent such problems because it can cause structures damage by fatigue. The structure itself has a certain properties so it is necessary to understand its characteristics. 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 and mode shapes. This can also be used for some purposes such as, troubleshooting i.e. direct insight into the root cause of vibration problems, find structural flexibility properties quickly to monitor incremental structural changes, design optimization-designaccordingtonoiseandvibrationtargets, enhance performance and reduce component and overall vibration & fast, test based evaluation of redesign for dynamics etc. So it is important to determine dynamic properties of a square plate of mild steel material. First the modal analysis of a square plate is done by finite element analysis software method. Modal analysis is a method of determining the natural acoustic characteristics or dynamic response of materials. The analysis involves imposing an excitation into the structure and finding the frequencies at which the structure resonates (when the excitation and vibration response match). A typical modal analysis will return multiple frequencies each with an accompanying displacement field which the structure experiencing at that frequency. Each frequency is known as mode and displacement field is known as mode shape. In this work the free vibration responses of a square mild steel plate with different circular cut outs at the centre and subjected to different boundary conditions is tabulated and analyzed. 2. PROBLEM MODELING Square plate with circular cut out Various sizes of centrally located circular cut outs are considered in order to examine the effect of cut out size on the natural frequency of the square plate. The plate has thickness of 5mm. The material properties of the plate are given in Table 1. The cut out ratio (i.e. d/a) along x-axis and y-axis is varied from 0.0 to 0.6 in steps of 0.25. Plate geometries for plates with cut out ratio 0.0, 0.125, 0.375, 0.625 are shown in Fig 1 0 800 300 Figure 1: Geometry of the plate 100 500300 800 1 2 3 4
- 2. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 06 | June -2017 www.irjet.net p-ISSN: 2395-0072 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2073 1) Plate geometry with cut out ratio 0. 2) Plate geometry with cut out ratio 0.125. 3) Plate geometry with cut out ratio 0.375. 4) Plate geometry with cut out ratio 0.625. Four types of boundary condition namely FFFS, FFSS, FSSS and SSSS (where F and S stand for a free, a simply supported) are considered in order to examine the effect of boundary condition on thefundamental natural frequencyof a square mild steel plate with a centrally located circular cut out. The FFFS, FFSS, FSSS, SSSS stand for four types of boundary conditions. Table 1: Material properties of the Plate. Material Constant Young’s modulus Density Poissons ratio Values 2E5N/mm2 7800Kg/m3 0.3 The meshed model of square plate with cutoutratioof0.375 is as shown in figure2 Figure 2: Square plate with cut out ratio of 0.375 Cut out ratio is varied from 0.0 to 0.625 in steps of 0.25.Plate geometries for plates with cut out ratio 0.0, 0.125, 0.375, 0.625, are shown in Figure 1. The Results of Natural frequencies for various d/a ratios with different boundary conditionsaresummarizedinTable 2. Table2: Natural frequencies for various d/a ratios with different boundary conditions Condit ions Modes d/a ratio 0 0.125 0.375 0.625 1 1.64 1.65 3.96E-05 2.75E-05 FFFS 2 4.28 4.27 6.69E-05 5.78E-05 3 10.55 10.6 7.68E-05 8.89E-05 4 17.06 16.9 3.27E-04 3.26E-04 1 3.11 8.65E-05 3.71E-05 7.37E-05 FFSS 2 14.43 9.18E-05 5.46E-05 8.11E-05 3 25.79 9.48E-05 7.51E-05 8.28E-05 4 41.58 4.11E-05 3.29E-04 3.11E-04 1 8.84 8.88E-05 3.71E-05 1.854 FSSS 2 23.89 9.21E-05 5.46E-05 4.281 3 36.38 9.54E-05 7.51E-05 12.18 4 44.05 4.10E-04 3.29E-04 14.37 1 18.68 8.52E-05 3.71E-05 7.38E-05 2 37.76 9.48E-05 5.46E-05 8.11E-05 SSSS 3 37.76 9.94E-05 7.51E-05 8.28E-05 4 45.89 3.93E-04 3.30E-04 3.11E-04 3. RESULT ANALYSIS: Below table shows the first 4 Natural frequencies for the steel plate with cut outs and for various boundary conditions. Case1: Plate with No hole. Table3: Natural frequencies for Plate with No hole condition SSSS SSSF SSFF SFFF F1 18.68 8.84 3.11 1.64 F2 37.76 23.89 14.43 4.28 F3 37.76 36.38 25.79 10.55 F4 45.89 44.05 41.58 17.06
- 3. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 06 | June -2017 www.irjet.net p-ISSN: 2395-0072 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2074 Figure3 Graph variation of frequencies for plate with no hole condition. We can observe that all the frequencies are decreasingaswe go on freeing up the corners of the plate. Hence we can conclude that the natural frequency will decrease as we go on freeing up the boundaries. Case2: Plate with cut out (d/a) ratio of 0.125. Table4: Natural frequencies for Plate with cut out (d/a) ratio of 0.125 SSSS SSSF SSFF SFFF F1 8.52E-05 8.88E-05 8.65E-05 1.65 F2 9.48E-05 9.21E-05 9.18E-05 4.27 F3 9.94E-05 9.54E-05 9.48E-05 10.6 F4 3.93E-04 4.10E-04 4.11E-05 16.9 Figure4: Graph variation of frequencies for plate with cut out (d/a) ratio of 0.125 We can observe that all the frequencies are very less for SSSS, SSSF, SSFF as compared to SFFF boundary condition. Case3: Plate with cut out (d/a) ratio of 0.375 Table5: Natural frequencies for Plate with cut out (d/a) ratio of 0.375 SSSS SSSF SSFF SFFF F1 3.71E-05 3.71E-05 3.71E-05 3.96E-05 F2 5.46E-05 5.46E-05 5.46E-05 6.69E-05 F3 7.51E-05 7.51E-05 7.51E-05 7.68E-05 F4 3.30E-04 3.29E-04 3.29E-04 3.27E-04 Figure5: Graph variation of frequencies for plate with cut out (d/a) ratio of 0.375 We can observe that all the frequencies are very negligible for all the conditions. Case 4: Plate with cut out (d/a) ratio of 0.625 Table6: Natural frequencies for Plate with cut out (d/a) ratio of 0.625 SSSS SSSF SSFF SFFF F1 7.38E-05 1.854 7.37E-05 2.75E-05 F2 8.11E-05 4.281 8.11E-05 5.78E-05 F3 8.28E-05 12.18 8.28E-05 8.89E-05 F4 3.11E-04 14.37 3.11E-04 3.26E-04 Figure6 : Graph variation of frequencies for plate with cut out (d/a) ratio of 0.625
- 4. International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 06 | June -2017 www.irjet.net p-ISSN: 2395-0072 © 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 2075 Here we can observe that the natural frequencies are negligible for SSSS, SSFF, SFFF as compared to SSSF boundary condition. CONCLUSION The finite element formulation is used to study effect of cut out on the free vibration of Mild steel plate. The numerical results are presented and discussed in above. The broad conclusions that can be made from the present study are summarized as follows: The fundamental natural frequencychangesonlymarginally if a small cut out (either of the two cut out ratiosbeingsmall) is made in the plate. However, for intermediate and large size cut outs, the fundamental natural frequency increases rapidly; the amount of increase depends on cut out ratios in two directions. For square plate with circular cut out, the natural frequency increases with the increasing constraint at the boundary irrespective of the size of the cut out. However, this increase is not uniform; it is also dependent on the particularmodeof vibration. For square plate with circular cut out, an increase in cut out size does not always result in an increase in the natural frequency. Besides depending on the cut out size, the increase or decrease also depends on the boundary condition. Hence depending upon the natural frequencyrequiredofthe given component we can give the support to the system so that there will not be any resonance during the operating conditions. Hence the required geometry of the componentcanbemade use without causing resonance by making use of different boundary conditions. REFERENCES [1]. Ramu I and Mohanty S. C. Modal Analysis of functionally graded plates using finite element method, Procedia Materials Science 6 ( 2014 ) 460 – 467, [2]. Marco Alfano, Leonardo Pagnotta, Determining the elastic constants of isotropic material s by modal vibration testing of rectangular thin plate, Journal of Sound and Vibration 293 (2006) 426–439, , Italy. [3]. Kaushar H. Barad, D.S.Sharma, VishalVyas. Crack detection in cantilever beam by frequency based method, Procedia Engineering 51 ( 2013 ) 770 – 775, [4]. A.Labib n, D. Kennedy, C. Feathersto , Free vibration analysis of beams and frames with multiple cracks for damage detection, Journal of Sound and Vibration, Cardiff University, Cardiff CF24 3AA, UK ,. [5]. P. K. Jenaa, D. N. Thatoib, J. Nandab, D. R. K. Parhic, Effect of damage parameters on vibrationsignaturesofa cantilever beam, International Conference on Modeling, Optimization and Computing (ICMOC 2012), April 10 - 11, 2012, [6]. JeslinThalapil, S.K.Maiti, Detectionoflongitudinal cracks in long and short beams using changes in natural frequencies, International Journal of Mechanical Sciences 83(2014)38–47. [7]. O. Doerk, W. Fricke, C. Weissenborn, Comparison of different calculation methods for structural stresses at welded joints, International Journal of Fatigue 25 (2003) 359–369. [8]. Kant T. and Swaminathan K., Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures, vol.56 (2002): pp. 329-344. [9]. Zhang Y.X. and KimK.S., Geometricallynonlinearanalysis of laminated composite plates by two new displacement based quadrilateral plate elements, Composite Structures, vol.72 (2006): pp. 301-310. [10]. Akavci S.S., Yerli H.R. and Dogan A., The first order shear deformation theory for symmetrically laminated composite plates on elastic foundation, The Arabian Journal for Science and Engineering, [11]. Krishna Murty A.V. and Vellaichamy S., On higherorder shear deformation theory of laminated composite panels, Composite Structures, vol. 8 (1987): pp. 247-270. [12]. Kant T. and Swaminathan K., Analytical solutions for free vibration analysis of laminatedcompositeandsandwich plates based on a higher order refined theory, Composite Structures, vol.53, (2001): pp. 73-85.