FREE VIBRATION ANALYSIS OF BEAMS AND PLATES ARE EXPERIMENTALLY INVESTIGATED BY USING ACCELEROMETER, DAQ, AND LABVIEW.

1. Project Final Viva-Voice Examination
1
Department of Mechanical Engineering
The National Institute of Engineering, Mysuru

2. 2
•Avoid catastrophic failure and reduce maintenance cost.
Introduction
•Vibration analysis can be used as a trouble shooting tool.
•Vibration can be used to detect faults at an early stage.
•Compare to other studies like Fatigue and Fracture analysis Vibration analysis is
easy to carry out and required less time.
•Structural health monitoring. Because of safety purpose and reduce the damage
cost. Etc...
•To better understand any structural vibration problem, the resonant frequencies of
a structure need to be identified and quantified.

3. 3

4. 4
Fig:- Experimental Setup for Clamped-Clamped
Beam
Fig:- Results Obtained by Using DAQ

5. 5
1. Vibration analysis of cantilever beam with
diffrent end masses attached.
2. He used aluminium and iron as a specimen
material.
3.Compared the experimental results up to 3
modes with ANSYS results.
Fig:- Experimental Setup for Cantilever Beam
With End Mass Attached

6. 6
Fig:- Experimental Setup for Free-Free Beam
1. Study the mass effect of accelerometer on free
vibration analysis of beam.
2.Compare the experimental results with
analytical (Euler-Bernoulli) results.

7. 7
Fig:- Experimental Setup for Cantilever Beam
1. He studied the crack detection of cantilever beam
by vibration analysis method.
2.He compared the frequency results with and
without cracked cantilever beam.
3.He also has done FEM analysis by using MATLAB.

8. 8
Fig:- Experimental Setup for Free-Free
Boundary Condition of Composite Plate
1. Author studied the free vibration analysis of
woven fibre and epoxy matrix composite plates.
2. First order shear deformation theory is used to
calculate natural frequency analytically.

9. 9
Fig:- Experimental Setup for Free-Free Boundary
Condition of Epoxy Plate by Hanging a Rubber Band
Fig:- Experimental Setup for Free-Free Boundary
Condition of Epoxy Plate by Mounting on a Sponge
He studied the free vibration analysis of epoxy composite plate by free-free boundary
condition by two methods.

10. 10Fig:- Experimental Setup for Using Shaker System Fig:- Experimental Setup for Using TV Holographic System
1. He studied the forced vibration analysis of aluminium plate by diffrent type of
boundary conditions (CCCC, SSSS, CSCS,).
2. He said that it is not possible to acquire the data by single accelerometer at higher
frequencies on spectrum analysis. But higher frequencies identified by the virtue of the
TV Holographic method.
3. He compared the experimental results with analytical (Rayleigh-Ritz)results.

11. 11
Fig:- Experimental Setup for Clamped-Free BC’s of Plate Fig:- Experimental Setup for Clamped-Clamped BC’s of
Plate
1. She has taken stainless steel and aluminium plate to study the free vibration analysis.
2. Plate dimensions 100x50x10mm.
3. Compared the experimental results with ANSYS results.

12. 12
Free vibration analysis of square and rectangular plates with different end conditions by
using Rayleigh–Ritz method.
Fig:- Numerical Results of First Four Modes of Rectangular Plate Under S-S-F-F BC’s

13. 13
1. To develop experimental setup for free vibration testing of beams and plates
2. To prepare specimens of beam and plate with different materials for various aspect ratios
3. To determine properties of the materials experimentally
4. To acquire the free vibrations data using National Instruments Data Acquisition system
5. To obtain the natural frequencies and damping ratios by experimental method
6. To validate the experimental results comparing it with theoretical and numerical results
7. To study the effect of different parameters on free vibration characteristics
Objectives of the Project

14. 14
1
Literature Review
3
Acquiring Free Vibration Using NI-DAQ
2
Design and Fabrication of
Experimental Setup
4
Validation with Theoretical
and Numerical Results
5
Study of Effect of Parameters on Free
Vibration
Methodology

15. 15
Experimental Details

16. 16
Bottom View
Assembly View Left Side View
Front View
Fig:- Design of Experimental Setup
Detailed View of Designed Model

17. 17
Beam with C-F Boundary Condition
Design and Fabrication of Experimental Setup
Beam with C-C Boundary Condition
Beam with C-F Boundary Condition
Beam with C-C Boundary Condition

18. 18
Plate with C-C-C-C Boundary Condition
Design and Fabrication of Experimental Setup
Plate with C-C-C-C Boundary Condition
Plate with C-C-C-F Boundary Condition Plate with C-C-C-F Boundary Condition

19. 19
Design and Fabrication of Experimental Setup
Plate with C-C-F-F Boundary Condition
Plate with C-F-F-F Boundary Condition
Plate with C-C-F-F Boundary Condition
Plate with C-F-F-F Boundary Condition

20. • C-DAQ 9178 Chassis
• NI-9234 Module
• Triaxial Accelerometer (356A15)
• Impact Hammer (086c03)
• Cables
• LABVIEW Software
20
Data Acquisition System (DAQ)

21. • Constant voltage to be
supplied.
• Multiple mountings of
DAQ input and output
module system.
21
Fig:- C-DAQ 9178 Chassis
C-DAQ 9178 Chassis

22. • DAQ is a communication bridge between
sensors and computer.
• C series Sound and Vibration input
module.
• It supports both IEPE (integrated
electronic piezoelectric) and non IEPE sensors
like Accelerometer, Tachometers, and
Proximity Probes.
22
NI-9234 Module
Fig:- NI-9234 Modules

23. • Tri axial accelerometer.
• Sensitivity: 10.2 mV/(m/s²)
• Measurement Range: ±490 m/s²
• Frequency Range: (±5%) 2 to 5000 Hz
• Weight =10.3 gm
23
Accelerometer (356A15)
Fig:- Accelerometer (356A15)

24. • Sensitivity: 2.25 mV/N
• Measurement Range: ±2224 N
• Hammer Mass: 0.16 kg
• It is used to obtain a impact response of
the system
24
Impact Hammer ( Model 086c03)
Fig:- Impact Hammer

25. 25
Cables

26. • LABVIEW( Laboratory Virtual Instruments Engineering Workbench).
Advantages of LABVIEW.
• Graphical user interface.
• Drag-and-drop built-in functions.
• Multiple platforms.
• Flexibility and scalability.
26
LABVIEW

27. 27
Fig:- Block DiagramFig:- Front Panel
LABVIEW Programming

28. 28
Copper
Aluminium
Steel
Material Selection

29. 29
Fig:- Closed Image of Material TestingFig:- Material Testing is Conducted in UTM
Extensometer
Test Specimen
Material Testing

30. 30
Material Young’s modulus, E in
N/m2
Density in kg/m3
Steel 1.62X1011 7850
Copper 1.2X1011 8933
Aluminium 0.71X1011 2700
Material Testing Results

31. 31
For Beam
Dimension
LXbXt in mm
No. of
Aluminium
Specimens
No. of Steel
Specimens
No. of Copper
Specimens
Total No. of
Specimens
350X20X3 1 1 1 3
550X20X3 1 1 1 3
550X40X3 1 1 1 3
Total 09
Dimensions of Test Specimens

32. 32
For Plate
Dimension
LXbXt in
mm
Aspect
ratio= L/b
No. of
Aluminium
Specimens
No. of Steel
Specimens
No. of
Copper
Specimens
Total No. of
Specimens
500X500X3 1 1 1 0 2
400X200X3 2 1 1 1 3
300X200X3 1.5 1 1 1 3
200X200X3 1 1 1 1 3
Total 11
Dimensions of Test Specimens

33. 33
Name of
Experiments
Aluminium Copper Steel Number of
Experiments
BEAMS
Clamped-Free 3 3 3 9
Camped-Clamped 3 3 3 9
Free-Free 2 ----- ----- 2
Clamped-Simply
Support
1 ----- ----- 1
Pin-Pin 1 ----- ----- 1
PLATES
CCCC 2 1 1 4
CCCF 2 1 1 4
CCFF 2 1 1 4
CFFF 2 1 1 4
Free-Free 2 ----- ----- 2
Total 40
Number of Experiments Carried out

34. 34
Free Vibration Analysis of
Beams

35. 35
Fig:- Clamped-Clamped BeamFig:- Cantilever Beam
Experimental Setup for Beams
Fig:- Pin-Pin Beam Fig:- Clamped-Simply Supported Beam

36. 36
Fig:- Free-Free Beam by Using Sponge Fig:- Free-Free Beam by Using Rubber Band
Experimental Setup for Beams

37. 37
Experimental Results of Beams
Fig:- C-F Beam Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Al(550X40X3 mm) beam results

38. 38
Experimental Results of Beams
Fig:- C-C Beam Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Fig:- C-SS Beam Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Al(550X40X3 mm) beam results

39. 39
Experimental Results of Beams
Fig:- P-P Beam Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Fig:- F-F Beam Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Al(550X40X3 mm) beam results

40. Fig:- C-F Beam 1st Mode
Mode Shapes of the Beams obtained from ANSYS
Fig:- C-F Beam 2nd Mode
Fig:- C-F Beam 3rd Mode Fig:- C-F Beam 4th Mode
Al(550X40X3 mm) beam results

41. 41
Mode Shapes of the Beams obtained from ANSYS
Fig:- Finite Element Model of the Beam for Modal Analysis
Fig:- Mode Shapes of C-F Beam (Al=550X40X3 mm)

42. 42
Mode Shapes of the Beams obtained from ANSYS
Fig:- Mode Shapes of C-C Beam (Al=550X40X3 mm)
Fig:- Mode Shapes of C-SS Beam (Al=550X40X3 mm)

43. 43
Mode Shapes of the Beams obtained from ANSYS
Fig:- Mode Shapes of F-F Beam (Al=550X40X3 mm)
Fig:- Mode Shapes of P-P Beam (Al=550X40X3 mm)

44. 44
Fig:- Finite Element Programme in MATLAB
MATLAB Programme For Beams

45. 45
Theoretical Calculations For Beams
Mode shapes of the beam given by
Here the values of β changes with different boundary conditions remaining things will be same
1. Values β in Pin-Pin supported
2. Values β in Clamped-Free supported
3. Values β in Clamped-Clamped supported
4. Values β in Clamped-Pinned supported
5. Values β in Free-Free supported
L= Length of the beam in meter
E= Young's Modulus of the material in N/m2
I=Moment of inertial m2
M= Mass in kg
β= Constant

46. 46
Specimens Size BC’s Modes Theoretical ANSYS Experimental
550X40X3 (LXBXT)
in mm
C-F
1 8.2 7.87 8
2 52.34 49.3 45
3 144.16 138.1 133
C-C
1 52.34 49.3 47
2 144.2 138.1 142
3 282.7 271.9 281
C-P
1 35.95 34.5 32
2 116.8 111.8 107
3 242.98 233.5 233
P-P
1 23 22.77 23
2 92.23 91.16 117
3 207.5 205.3 196
F-F by using
rubber
1 51.9 49.8 48
2 144.13 137.4 138
3 282.5 269.6 258
F-F by using
Sponge
1 51.9 49.8 48
2 144.13 137.4 137
3 282.5 269.6 262
Comparison of Results
Table:- Natural Frequencies of Aluminium (550X40X3 mm) Beam in Hz for All Boundary Conditions

47. 47
Specimens
Size
BC’s Modes Theoretical MATLAB ANSYS Experimental
350X20X3
(LXBXT)in mm
C-F
1 20.13 20.13 20 18
2 129.3 112.9 125 110
3 356 311.4 350.2 345
C-C
1 129.3 112.9 127.8 103
2 356 311.4 350.2 345
3 698.2 660.4 689.7 612
550X20X3
(LXBXT)in mm
C-F
1 8.22 8.7 8.1 8
2 52.3 45.9 50.96 45
3 144.2 126.1 142.7 133
C-C
1 52.3 45.85 50.96 44
2 144.2 126.12 142.58 130
3 282.7 267.4 280.3 265
550X40X3
(LXBXT)in mm
C-F
1 8.2 8.7 7.87 7
2 52.3 45.85 49.3 44
3 144.2 126.1 138.1 130
C-C
1 52.3 45.85 49.3 45
2 144.2 126.1 138.1 133
3 282.7 267.4 271.9 281
Comparison of Results
Table:- Natural Frequencies of Aluminium Beams in Hz

48. 48
Specimens
Size
BC’s Modes Theoretical MATLAB ANSYS Experimental
350X20X3
(LXBXT)in mm
C-F
1 19.6 19.5 19.7 17
2 125.1 109.2 123.6 110
3 344.3 301.2 346.1 317
C-C
1 125.1 109.3 123.6 110
2 344.3 301.2 346.1 317
3 675.2 638.7 679.7 558
550X20X3
(LXBXT)in mm
C-F
1 7.95 8.7 7.68 7
2 50.6 44.2 48.1 44
3 139.4 121.9 134.8 126
C-C
1 50.63 44.16 48.1 44
2 139.4 121.9 134.8 128
3 273.5 258.6 264.6 255
550X40X3
(LXBXT)in mm
C-F
1 7.95 8.7 7.8 7
2 50.6 44.2 48.7 44
3 139.4 121.9 136.4 128
C-C
1 50.62 44.2 49.7 42
2 139.4 121.9 136.8 120
3 273.5 258.6 268.3 235
Comparison of Results
Table:- Natural Frequencies of Steel Beams in Hz

49. 49
Specimens
Size
BC’s Modes Theoretical MATLAB ANSYS Experimental
350X20X3
(LXBXT)in mm
C-F
1 14.5 14.24 14.5 13
2 92.4 80.68 90.6 84
3 254.5 222.5 253.7 244
C-C
1 92.4 80.68 90.6 84
2 254.5 222.5 253.7 244
3 499 472 498.8 468
550X20X3
(LXBXT)in mm
C-F
1 5.87 5.96 5.85 6
2 37.4 32.69 36.65 41
3 103 90.12 102.6 119
C-C
1 37.4 32.69 36.65 36
2 103 90.12 102.6 107
3 202 191.13 201.5 211
550X40X3
(LXBXT)in mm
C-F
1 5.9 5.9 5.7 6
2 37.4 32.69 35.46 36
3 103 90.12 99.3 107
C-C
1 37.4 32.69 36.2 36
2 103 90.12 99.8 108
3 201.1 191.14 195.6 210
Comparison of Results
Table:- Natural Frequencies of Copper Beams in Hz

50. 50
Effect of Parameters on Free
Vibration of Beams

51. 51Fig:- Density Varying by Using Aluminium, Steel and Copper
Effect of Density

52. 52
Aluminum Steel Copper
Density (kg/𝒎 𝟑
) 2700 7850 8933
Mode 1 (Hz) 18 17 13
Mode 2 (Hz) 110 110 84
Mode 3 (Hz) 345 317 244
Effect of Density
Table:- Effect of Density on Free Vibration of Beams (350X20X3 mm)
The values of natural frequency of the beam decreases as its density increases

53. 53
Fig:- Different Length of Beams
Effect of Length

54. 54
Materials Aluminum Steel Copper
Length
350 mm 550 mm
350 mm
550 mm 350 mm 550 mm
Mode 1 (Hz)
18 7 17 7 13 6
Mode 2 (Hz)
110 110 110 44 84 41
Mode 3 (Hz)
345 317 317 128 244 119
Effect of Length
Table:- Effect of Length on Free Vibration of Beams (350X20X3 mm & 550X20X3 mm)
The values of natural frequency of the beam decreases as its length increases

55. 55
Fig:- Different C/s Area of Beams
Effect of C/s Area

56. 56
Effect of C/s Area
Materials Aluminum Steel Copper
Width
40 mm 20 mm
40 mm
20mm 40 mm 20 mm
Mode 1 (Hz)
7 8 7 7 6 6
Mode 2 (Hz)
44 45 44 44 36 41
Mode 3 (Hz)
130 133 126 128 107 119
Table:- Effect of C/s Area on Free Vibration of Beams (Al=550X20X3 mm & 550X40X3 mm)
The values of natural frequency of the beam decreases as its cross section area increases

57. 57
Fig:- Clamped-Pin Beam
Fig:- Experimental Setup for Clamped-Free Beam Fig:- Fixed-Fixed Beam
Fig:- Pin-Pin Beam
Effect of Boundary Conditions

58. 58
Boundary
Conditions
C-C C-P P-P C-F
Mode 1 (Hz)
45 32 23 8
Mode 1 (Hz)
133 107 117 45
Mode 1 (Hz)
281 233 196 133
Effect of Boundary Conditions
Table:- Effect of Boundary Conditions on Free Vibration of Beams (Al=550X40X3 mm)
The values of natural frequency of the beam is more at C-C then C-P and P-P at last C-F boundary
conditions respectively

59. 59
Experimental Results of Damping Ratios for Beams
Table:- Experimental Results of Damping Ratios for Beams
Damping Ratio
Critical Damping
Coefficient
Damping Coefficient
BC’s C-C C-F C-C C-F C-C C-F
AL 1 0.01997 0.00705 11.68 2.04 0.2332 0.0144
AL 2 0.034 0.0148 7.841 1.25 0.2666 0.18495
AL 3 0.01979 0.0102 16.038 2.8512 0.317 0.0291
STL 1 0.0107 0.00294 36.267 5.605 0.38806 0.0165
STL 2 0.264 0.0147 22.8 3.63 0.6024 0.05321
STL3 0.0244 0.01929 43.5 7.25 1.0618 0.1399
COP 1 0.01554 0.01324 31.5 4.877 0.4896 0.0646
COP 2 0.0405 0.2354 21.2 3.5376 0.8591 0.0833
COP 3 0.0413 0.0413 42.5 7.25 0.1399 0.087
Where 1= 350X20X3 mm
2= 550X20X3 mm
3= 550X40X3 mm

60. 60
Boundary
Conditions
Damping Ratio
Critical Damping
Coefficient
Damping
Coefficient
C-C 0.05272 16.038 0.8455
C-P 0.0372 11.405 0.424
P-P 0.0294 8.197 0.241
C-F 0.0218 2.85 0.0621
Effect of Boundary Conditions on Damping Ratios
Table:- Effect of Boundary Conditions on Damping Ratios (Al=550X40X3 mm)
The values of damping ratio of beam is more at C-C then C-P and P-P at last C-F boundary
conditions respectively

61. 61
1.Experimental results are closer to the theoretical and numerical results
2. Natural frequency of the beam decreases with increasing its density
3. Natural frequency of the beam decreases with increasing its length
4. Natural frequency of the beam decreases with increasing its c/s area
5. Natural frequencies are highest for C-C boundary condition and lowest for C-F condition. The values of
natural frequencies decrease with sequence C-C, C-P, P-P and C-F boundary conditions
6. The values of damping ratio decrease with sequence C-C, C-P, P-P and C-F boundary condition
Conclusions from Beam Analysis

62. 62
Free Vibration Analysis of
Plates

63. 63
Fig:- Plate CCCC BC’s Condition
Experimental Setup for Plates
Fig:- Plate CCCF BC’s Condition
Fig:- Plate CFCF BC’s Condition Fig:- Plate CFFF BC’s Condition

64. 64
Fig:- Plate Free-Free BC’s by Using Rubber Band
Experimental Setup for Plates
Fig:- Plate Free-Free BC’s by Using Sponge

65. 65
Experimental Results for Plates
Fig:- CCCC Plate Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Fig:- CCCF Plate Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot

66. 66
Experimental Results for Plates
Fig:- CCFF Plate Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot
Fig:- CFFF Plate Experimental Results, a) Time Domain Plot, b) Frequency Domain Plot

67. 67
Mode Shapes of Plates obtained from ANSYS
Fig:- CCCC Plate 1st Mode Fig:- CCCC Plate 2nd Mode
Fig:- CCCC Plate 3rd Mode Fig:- CCCC Plate 4th Mode
(Al=550X40X3 mm)

68. 68
Mode Shapes of Plates obtained from ANSYS
Fig:- Finite Element Model of the Plate with FFFF for Modal Analysis
Fig:- Mode Shapes of Plate (Al=400X200X3 mm) with CCCC Condition

69. 69
Mode Shapes of Plates obtained from ANSYS
Fig:- Mode Shapes of Plate (Al=400X200X3 mm) with CCCF Condition
Fig:- Mode Shapes of Plate (Al=400X200X3 mm) with CCFF Condition

70. 70
Mode Shapes of Plates obtained from ANSYS
Fig:- Mode Shapes of Plate (Al=400X200X3 mm) with FFFF Condition
Fig:- Mode Shapes of Plate (Al=400X200X3 mm) with CFFF Condition

71. 71
Specimens Size BC’s Modes ANSYS Experimental
500X500X3 (LXBXT)in
mm
CCCC
1 85.2 83
2 173.8 160
3 173.8 161
CCCF
1 56.4 53
2 94.2 91
3 180 172
CCFF
1 52.4 49
2 61.98 59
3 102.2 100
CFFF
1 8.2 7
2 49.9 49
3 63.8 62
Comparison of Results
Table:- Natural Frequencies of Aluminium Square Plates in Hz

72. 72
Specimens Size BC’s Modes ANSYS Experimental
400X200X3 (LXBXT)in
mm
CCCC
1 270.9 280
2 375.3 377
3 557.6 529
CCCF 200mm
Side Free
1 242.7 243
2 289.8 284
3 398.3 371
CCFF Both
200mm Side
Free
1 240.9 246
2 255.8 253
3 311.6 314
CFFF 400mm
Side Fixed
1 37.14 41
2 124.4 125
3 232.5 234
Comparison of Results
Table:- Natural Frequencies of Steel (400X200X3) Plates in Hz

73. 73
Specimens Size BC’s Modes ANSYS Experimental
400X200X3 (LXBXT)in
mm
CCCC
1 202.4 187
2 280.2 277
3 426 402
CCCF 200mm
Side Free
1 180.8 188
2 215.7 216
3 296.3 291
CCFF Both
200mm Side
Free
1 177.1 185
2 187.6 192
3 228.4 234
CFFF 400mm
Side Fixed
1 27.7 21
2 44.7 43
3 91 97
Comparison of Results
Table:- Natural Frequencies of Copper (400X200X3) Plates in Hz

74. 74
Specimens Size BC’s Modes ANSYS Experimental
400X200X3 (LXBXT)in
mm
CCCC
1 280 270
2 387 408
3 576 620
CCCF 200mm
Side Free
1 250.5 262
2 298.8 294
3 410.6 406
CCFF Both
200mm Side
Free
1 245.3 209
2 260 259
3 317.2 320
CCFF Both
400mm Side
Free
1 76.9 75
2 123.7 117
3 211.9 210
Comparison of Results
Table:- Natural Frequencies of Aluminium (400X200X3) Plates in Hz

75. 75
Specimens Size BC’s Modes ANSYS Experimental
400X200X3 (LXBXT)in
mm
CFFF 400mm
Side Fixed
1 38.3 42
2 126.7 131
3 239.7 218
CFFF 200mm
Side Fixed
1 11.9 12
2 50.6 50
3 74.5 81
FFFF by
Using Rubber
Band
1 74.23 75
2 80.5 82
3 180.7 174
FFFF by
Using Sponge
1 74.23 70
2 80.5 82
3 180.7 174
Comparison of Results
Table:- Natural Frequencies of Aluminium (400X200X3) Plates in Hz

76. 76
Free Vibration Analysis of
Square Plates

77. 77
Mode Degeneracy
Free vibration analysis of square plate

78. 78
Aluminum
Square plate
ANSYS Experimental
Mode 1 (Hz) 85.23 84
Mode 2 (Hz)
173.79 160
Mode 3 (Hz)
173.79 161
Fig:- Square Plate (500X500X3 mm)
CCCC BC’s Condition Experimental Results
Table:- Square plate (500X500X3 mm) CCCC BC’s Condition
Results
Mode Degeneracy

79. 79
Effect of Parameters on Free
Vibration of Plates

80. 80
Aluminum Steel Copper
Density(kg/𝒎 𝟑
) 2700 7850 8933
Mode 1 (Hz)
270 258 187
Mode 2 (Hz)
408 377 277
Mode 3 (Hz)
620 529 402
Table:- Rectangular (400X200X3 mm) Plate CCCC BC’s Condition Results
Effect of Density
The natural frequency of the plate decreases with increase its density

81. 81
Boundary
Conditions
CCCC CCCF CCFF CFFF
Mode 1 (Hz)
270 262 209 42
Mode 1 (Hz)
408 294 259 131
Mode 1 (Hz)
620 406 320 218
Table:- Aluminium (400X200X3 mm) Plate CCCC BC’s Condition Results
Effect of Boundary Conditions
The values of natural frequency of the plate is more at CCCC then CCCF and CCFF at last CFFF
boundary conditions respectively

82. 82
Boundary
Conditions
CCFF CFFF
Fixed Edge Length 400 mm 200 mm 400 mm 200 mm
Mode 1 (Hz) 209 75 42 12
Mode 2 (Hz) 259 117 131 50
Mode 3 (Hz) 320 210 218 81
Table:- Aluminium (400X200X3 mm) Plate Results
Effect of Fixed Edge Length
The natural frequency of rectangular plate has higher value when longer edge is fixed compared to the
smaller edge fixed conditions

83. 83
Specimens Size BC’s Damping Factor
Critical
Damping
Coefficient
Damping
Coefficient
Aluminum 400X200X3
(LXBXT)in mm
CCCC 0.415 523.13 217
CCCF 0.3676 507.63 186.6
CCFF 0.3046 408.8 124.5
CFFF 0.2124 81.4 17.3
Steel
400X200X3 (LXBXT)in mm
CCCC 0.4466 1323.3 590.97
CCCF 0.2616 1148.42 300.46
CCFF 0.2096 1162.6 243.3
CFFF 0.1844 193.8 35.73
Copper 400X200X3
(LXBXT)in mm
CCCC 0.381 1198.7 456.7
CCCF 0.262 1205.1 315.7
CCFF 0.193 1182.9 228.7
CFFF 0.1653 134.6 22.24
Experimental Results of Damping Ratio for Plates
Table:- Experimental Results of Damping Ratio for Plates

84. 84
Boundary Conditions Damping Ratio
Critical Damping
Coefficient
Damping Coefficient
CCCC 0.415 523.13 217
CCCF 0.3676 507.63 186.6
CCFF 0.3046 408.8 124.5
CFFF 0.2124 81.4 17.3
Table:- Aluminium (400X200X3 mm) Plate Results
Effect of Boundary Conditions on Damping Ratios
The values of damping ratios of the plate is more at CCCC then CCCF and CCFF at last CFFF
boundary conditions respectively

85. 85
Conclusions From Plate Analysis
1. The values of natural frequencies are decreasing with increase in density of plates
2. The effect of boundary conditions on plate vibration was found that the natural frequency is descending in
the order of CCCC, CCCF, CCFF and CFFF
3. The value of natural frequencies of a rectangular plate with longer edge fixed is more as compared to that of
the smaller edge fixed under CFFF and CCFF boundary conditions
4. The FFFF boundary conditions of the plate is obtained by mounting the specimen on sponge and by rubber
bands. Both mountings give the same values of natural frequencies
5. The effect of boundary conditions on damping was found that the damping ratio is descending in the order
of CCCC, CCCF, CCFF and CFFF

86. • Signal leakage error
• In analytical or numerical method of free vibration damping is neglected,
but in actual practice damping occurred
• Weight of the accelerometer
• Electrical noise
• geometrical imperfection
• Environmental effect. Etc....
86
Causes for Error

87. 87
Conclusions
1. The values of natural frequencies of the beams and plates obtained from all the analyses were compared and
found to be in a good agreement within a deviation of about 5 percent and 7 percent respectively. This proves
that the developed experimental setup is good enough to study the free vibration characteristics of beams and
plates.
2. The Free-Free boundary conditions of beams and plates obtained by mounting the specimen on sponge and
by rubber bands. Both mountings give the same values of natural frequencies.
3. It is observed that the values of natural frequencies are decreases with increase in density and cross
sectional area.
4. It is observed that the values of natural frequencies are Boundary phenomena. Natural frequencies are
higher at all the edges having the fixed boundary conditions and lower at one end fixed and other ends are free
boundary conditions.

88. 88
Future Work
1. To vary the plate and beam thickness to measure the mode shapes
2. To Study the vibration behaviour of the tapered section
3. To study the impact response of the beam and plate
4. To study the vibration analysis by using composite beam and plates
5. To study the damping factor for different length, different aspect ratio and different material of the beam
and plates
6. To develop a analytical and FEM solution for free vibration analysis of plates

89. 89
Special Thanks to
Samal Sir
TEQUIP-2
Our Beloved Faculty
And My Dear Friends

90. 90
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