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Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis
Alexandria Engineering Journal (2018) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis M. Elshamy, W.A. Crosby, M. Elhadary * Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt Received 5 August 2018; revised 22 September 2018; accepted 23 October 2018
KEYWORDS Breathing crack detection; Natural frequency; FEA
Abstract The investigation of a crack in a cantilever beam is very imperative in various mechanical applications such as turbo machinery impeller blades, propeller shafts and airplane wings. As a result, a full study is performed in this paper to investigate the crack consequences on a cantilever beam; this is carried out by monitoring the natural frequency of the beam, taking into consideration the probe mass used in data collection. The investigation is divided into two phases; the first phase is experimentally performed by measuring the natural frequency of a specimen with different thicknesses and widths while changing specimen material to illustrate the specimen dimensional and material effect on natural frequency reduction. This is carried out on different specimen configuration by changing crack location and depth. The second phase is to validate the acquired results from the first phase. This is achieved by using finite element analysis (FEA) techniques, this is performed by modeling the experimental phase conditions into ANSYS 16.2, one time considering the probe mass and other time without putting probe mass into consideration. Finally, the FEA results are compared with experimental phase results. Ó 2018 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction Several methodologies have been implemented to monitor the condition of mechanical structure, each of these techniques depends on the corresponding phenomenon in the device related to the properties of the material tested. Cracks are the most common defect in mechanical structure. A crack * Corresponding author. E-mail address: [email protected] (M. Elhadary). Peer review under responsibility of Faculty of Engineering, Alexandria University.
introduces flexibility thus resulting in an overall drop in system stiffness, leading to a total failure of the system. For determining system integrity and remaining useful life, an inspection plan should be established, to anticipate failure at an early stage. Cracks are initiated as a surface hair crack in the transverse direction. Along the service of the specimen, the surface cracks propagate along the depth. This is one of the most popular and common forms of crack spreading that perhaps arise out of fatigue loading condition. This type of crack in cantilever beams is obvious in various applications such as turbo machinery impeller blades. This crack is initiated from intensive cycles of fatigue due to highly dynamic stresses
https://doi.org/10.1016/j.aej.2018.10.002 1110-0168 Ó 2018 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
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Nomenclature List of symbols D crack depth [mm] T specimen thickness [mm] X crack location from fixation [mm] L specimen length from fixation to free end [mm] W specimen width [mm] wnH natural frequency of healthy specimen [Hz]
caused by rotor vibration and blades resonances. In aviation application, this type of crack is presented in aero- plane’s wings due to variation of lift forces along wing service. Despite, the wing’s material is composite; this method can be also applied on it. A vibration-based method is submitted as an efficient and quick approach of detecting depth and location of crack in the specimen. As a nondestructive test method, it trends the severity crack progression in mechanical structures. In order to identify structural defect by vibration-based method monitoring, this method is based on tracking changes in natural frequency of the specimen, resulted from the change in flexibility and it’s reciprocal (stiffness) of the structure due to the crack initiation and propagation. Identification of crack initiation is determined in many studies by tracking 1st mode natural frequency change of the specimen. However, for determining exact crack location and depth, vibration 2nd and 3rd modes should be detected and tracked to observe their changes. For every crack location and depth, there is only unique ratio of change for the first three modes of vibration. Pratibha M. et al. [1] evaluated the first three natural frequencies of a single cracked cantilever beam and a healthy beam using vibration measurements. Vikram K. et al. [2] Monitor the vibration behavior of beams both experimentally and using FEM software ANSYS. In the present work, vibration analysis is carried out. Nitesh A. M. et al. [3] describe finite elemental analysis of a cracked cantilever beam and analyze the relation between the modal natural frequencies with crack depth, modal natural frequency with crack location. Also, the relation among the crack depth, crack location and natural frequency has been analyzed. Only single crack at different depth and at different location are evaluated and the analysis revels relationship between crack depth and modal natural frequency. Kaushar H. B. et al. [4] Detect crack presence on the surface of beam-type structural element using natural frequency. First two natural frequencies of the cracked beam have been obtained experimentally and used for detection of crack location and size. Pankaj C. et al. [5] concluded that faults detection of a single cracked beam by theoretical and experimental analysis using vibration signatures. The results of theoretical analysis and experimental analysis are compared and are found to be good correlation between them. H. Nahvi et al. [6] find results that the finite element model of the cracked beam is constructed and used to determine its natural frequencies and mode shapes. Leonard F. et al. [7] proposed a study on spectrograms of the free-decay responses showed a time drift of the frequency and damping, the usual hypothesis of constant modal parameters is no longer appropriate, since
wn a b O
natural frequency of cracked specimen [Hz] crack depth (D) to specimen thickness (T) ratio [Dimensionless] crack location from fixation (X) to specimen length (L) ratio [Dimensionless] cracked to healthy natural frequency ratio [Dimensionless]
the latter are revealed to be a function of the amplitude. Orhan S. [8] studied free and forced vibration analysis of single and two-edged cracked beam and suggested that free vibration analysis provides suitable information for the detection cracks. Owolabi G. et al. [9] Detected crack damage experimentally on aluminum beam by measuring acceleration frequency response at seven different points on each beam model using dual channel frequency analyzer, the damage detection schemes used in their study depend on the measured changes in the first three natural frequencies and corresponding amplitudes of the measured acceleration frequency response functions. Leonard F. et al. [10] based their study on cracks that occurred in metal beams obtained under controlled fatiguecrack propagation. Where the beams are clamped in a heavy vise and struck in order to obtain a clean impulse model response, spectrograms of the free-decay responses showed a time drift of the frequency and damping. Rizos P.F. et al. [11] measured flexure vibrations of cantilever beam with rectangular cross-section having transverse surface crack extended uniformly along width, where crack location and depth can be estimated accurately by measuring amplitude at two positions of the structure. Robert Y. et al. [12] developed a method for crack location identification and quantification of damage, where this method adopts rotational mass less springs in the beam element as a mechanical model to represent flexibility introduced by crack. For a given natural frequency and a damage location, the characteristic equation can be solved to provide numerical value of the stiffness of the rotational spring.
Table 1
Specimen Configurations.
Configuration ID
L (mm)
W (mm)
T (mm)
1 2 3 4
400 400 400 400
48 48 48 32
6.5 7 7.5 6.5
Table 2
Material Properties.
Parameter
Value
Commercial name Modulus of elasticity Density Poisson ratio
ASTM A283 Grade C Carbon Steel 190 GPa 7900 kg/m3 0.32
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
Crack detection of cantilever beam
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Salawu O. et al. [13] reviewed four methods and made comparison of their effectiveness. One of the methods exploits the changes in Eigen parameters while the other three utilize system identification/model updating procedures. These three methods were originally developed to locate errors in mathematical models. For the comparison studies, response data were obtained from simulation analysis of a simply supported steel beam. Deokar A. et al. [14] performed experimental modal analysis on cracked beams and healthy beam. The first three natural frequencies were considered as basic criterion for crack detection, to locate crack a 3D graph normalize frequency in terms
Fig. 1
Table 3 Feature adjusted in Data Collector and Analyzer. Feature
Value
Maximum frequency (Hz) Number of lines Pre-trigger (G’s) Window type
100 1600 5 Rectangular
of the crack depth and location were plotted, where the intersection of these three contours gives crack location and crack
Experimental Configuration (All dimensions are in mms).
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
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depth. Baviskar P. et al. [15] observed changes in natural frequency due to crack propagation both theoretically using finite element analysis software and experimentally using fast Fourier transform analyzer, and they obtained that results by both methods were in a good agreement. Douka E. et al. [16] studied the nonlinear behavior of the system by using time frequency methods as an alternative to Fourier analysis methodology. A further investigation and study are performed to understand effect of specimen dimensions and specimen material on natural frequency reduction, furthermore study is done to illustrate effect of probe mass on results.
maximum frequency 100 ¼ ¼ 0:0625 Hz number of lines 1600 3. Experimental results 3.1. Bump test spectra See Figs. 2 and 3.
2. Experimental preparation 2.1. Specimen configuration The test specimens have rectangular cross-section with total length of 600 mm, while the length of fixed part of specimen is 200 mm. The cross-section dimensions of specimen configurations are shown in Table 1, while the material properties of specimen are listed in Table 2. 2.2. Experiment procedures (1) Specimen has been fixed between upper and lower halves of support, as shown in Fig. 1. (2) Vibration probe is placed at free end of specimen. (3) Vibration data is measured while the specimen is impacted using impact hummer. (4) Crack is initiated on the previous mentioned specimen at a certain distance from the fixation using wire cut method to ensure accurate crack depth, and then vibration data are collected as well. (5) Increase crack depth while repeating step 3. (6) Repeat the entire experiment at different crack location and depth.
Fig. 2 Bump test spectrum for a healthy specimen (ID 1, wn ¼ 21:5 Hz).
2.3. Data collector and analyzer 2.3.1. Theory of working Natural frequency is measured by applying an impact force on the specimen and then detects all excited natural frequencies resulting from this excitation using EMERSON CSI 2140 data collector and analyzer, in which many natural frequencies can appear in the collected data if impact force injects more energy to specimen. It is very important to select right hummer tip where the size of the hummer and stiffness of the tip greatly affect the results. Natural frequencies would be detected in low frequency zone if we perform bump test on a large mass specimen, then a heavy hummer with soft tip is advised to be utilized, as a heavy hummer generates a large force and a soft tip ensures that input energy is concentrated in lower frequency range.
Fig. 3 Bump test spectrum for defected specimen (ID 1,wn ¼ 15:7 Hz).
3.2. Bump test results: See Table 4. 4. Analysis using finite element method (FEM)
2.3.2. Device set up To ensure accurate results of data collection, vibration analyzer setup was as listed in Table 3. Therefore, minimum difference in frequencies that can be detected would be as follow;
4.1. Software ANSYS R16.2.
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
Crack detection of cantilever beam Table 4 D
0 1 2 2.5 3 3.5 4 4.5 5 D
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
5
Bump Test Results. wn for ID 1 b = 2.5%
b = 12.5%
b = 25%
b = 37.5%
b = 50%
21.5 21.31 20.94 20.5 N/M 18.5 N/M 15.69 N/M
21 20.88 20.62 N/M 19.88 19.31 18.62 16.69 N/M
20.5 20.44 20.25 N/M 19.81 19.12 18.62 17.5 N/M
21 20.87 20.75 N/M 20.56 N/M 19.875 19 17.5
20.56 20.5 20.44 N/M 20.25 N/M 19.81 19.12 N/M
wn for ID 3
wn for ID 4
b = 2.5%
b = 25%
b = 2.5%
b = 2.5%
20.56 N/M 20.5 N/M 20 N/M 19.25 18.75 17.69 N/M
20.75 N/M 20.56 N/M 20.44 N/M 19.81 19.31 N/M 18.25
26.19 26.12 25.94 25.75 25.56 25.19 24.94 24.44 23.75 22.31
20.25 N/M 20.12 N/M 19.81 N/M 18.81 17.88 16.69 15.44
wn for ID 2
Fig. 4
Rectangular crack cross-section.
4.2. Crack representation
extruded from specimen free end with a height equivalent to real probe mass.
Crack is represented as an open crack with a rectangular end with a constant width of 1 mm, as shown in Fig. 4.
4.4. Meshing
4.3. Probe representation
There are two basic types of element used in analysis: –
Probe weight is about 200 g and specimen’s weight ranges from 700 gm to 1600 gm, therefore probe must be represented in analysis. The vibration probe is magnetically mounted to the specimen, then probe is simulated in FEA as a cylinder
– The first type is used to mesh healthy specimen (with no crack) and specimen without probe, as shown in Fig. 5. – The second type of element is used during analysis of cracked specimen, the quality of mesh has been refined at
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
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Table 5 D
Fig. 5
Solid 186 Element Type.
Fig. 6
Solid 187 Element Type.
Results with considering probe in analysis. wn for ID 1 b = 2.5%
b = 12.5%
b = 25%
b = 37.5%
b = 50%
0 1 2 3 3.5 4 4.5
24.872 24.647 24.091 22.993 22.083 20.811 18.844
24.872 24.732 24.294 23.429 22.722 21.605 20.036
24.872 24.771 24.473 23.874 23.325 22.511 21.235
24.872 24.807 24.631 24.243 23.365 N/C 22.436
24.872 24.838 24.741 24.526 24.007 N/C 23.421
D
wn for ID 2
wn for ID 3
wn for ID 4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
b = 2.5%
b = 25%
b = 2.5%
b = 2.5%
27.221 N/C 26.988 N/C 26.444 N/C 25.383 24.55 23.424 21.768
27.221 N/C 27.117 N/C 26.824 N/C 26.254 25.777 25.061 24.045
29.585 29.496 29.349 29.124 28.809 28.373 27.786 26.998 25.957 24.591
22.406 N/C 22.228 N/C 21.746 N/C 20.766 19.942 18.758 17.051
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Crack detection of cantilever beam
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two locations; the crack location and the probe location, as shown in Fig. 6.
Results are represented as a relation between dimension-less parameters
5. FEA results
– relation between a (X-Axis) and O (Y-Axis) at constant b – Relation between b (X-Axis) and O (Y-Axis) at constant a
See Tables 5 and 6.
Table 6
6. Results comparison and validation
Results without Considering Probe in Analysis.
D
wn for ID 1 b = 2.5%
b = 12.5%
b = 25%
b = 37.5%
b = 50%
0 1 2 3 3.5 4 4.5
33.4 33.07 32.254 30.646 29.326 27.409 24.741
33.4 33.167 32.564 31.333 30.286 28.787 26.432
33.4 33.256 32.876 32.068 31.337 30.267 28.603
33.4 33 33.105 32.617 31.506 N/C 30.443
33.4 33 33.257 33.036 32.82 N/C 32.454
D
wn for ID 2
wn for ID 3
wn for ID 4
0 0.5 1 1.5 2 3 3.5 4 4.5
Table 7
b = 2.5%
b = 25%
b = 2.5%
b = 2.5%
35.964 N/C 35.631 N/C 34.834 33.331 32.165 30.53 28.233
35.946 N/C 35.82 N/C 35.446 34.677 34.059 33.156 31.816
38.529 38.401 38.193 37.87 37.427 35.975 34.893 33.47 31.515
33.314 N/C 32.978 N/C 32.17 30.555 29.236 27.388 24.745
ID 1 results.
Experimental X b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.9912 0.9943 0.9971 0.9948 0.9971
0.9740 0.9819 0.9878 0.9881 0.9942
0.9140 0.9467 0.9663 0.9790 0.9849
0.8050 0.8867 0.9083 0.9495 0.9635
0.7298 0.7948 0.8537 0.9048 0.9300
FE with considering Probe X b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.9909 0.9941 0.9959 0.9973 0.9986
0.9685 0.9765 0.9840 0.9903 0.9947
0.9244 0.9417 0.9599 0.9747 0.9861
0.8086 0.8686 0.9051 0.9394 0.9652
0.7584 0.8053 0.8538 0.9020 0.9417
FE without considering Probe X b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.9901 0.9930 0.9957 0.9973 0.9989
0.9657 0.9750 0.9843 0.9912 0.9957
0.9175 0.9381 0.9601 0.9766 0.9891
0.8206 0.8619 0.9062 0.9433 0.9717
0.7407 0.7914 0.8564 0.9115 0.9550
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First Mode (EXP) 0.15
(EXP) 0.31
(EXP) 0.46
(EXP) 0.61
(EXP) 0.69
(FEA Probe) 0.69
(FEA Probe) 0.15
(FEA Probe) 0.31
(FEA Probe) 0.46
(FEA Probe) 0.61
(FEA No Probe) 0.69
(FEA No Probe) 0.15
(FEA No Probe) 0.31
(FEA No Probe) 0.46
(FEA No Probe) 0.61
1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.0%
10.0%
20.0%
Fig. 7
30.0%
40.0%
50.0%
60.0%
First mode results comparison.
6.1. Results of specimen’s ID 1 6.1.1. First mode results For specimen ID 1, b ranges from 2.5% to 50% at each location crack depth varies till about 70% of thickness. Frequency reduction ratio (X) is calculated using finite element analysis with and without considering probe in analysis and is detected experimentally using vibration measurement.
Table 7 shows the outputs from the experimental and finite element analysis. These results are compared against each other, as shown in Fig. 7, to validate the experimental results. Fig. 8 shows second natural frequency at various crack location and depth for ID 1 without considering probe in the FEA, while Fig. 9 shows third natural frequency at various crack location and depth for ID 1 without considering probe in the FEA.
Second mode =0.154
1.02
=0.31
=0.45
=0.61
=0.69
1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.0%
Fig. 8
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
Second natural frequency at various crack location and depth for ID 1.
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Crack detection of cantilever beam
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Third mode =0.154
=0.31
=0.45
=0.61
=0.69
1.02 1 0.98 0.96 0.94 0.92 0.9 0.88 0.0%
Fig. 9
Table 8
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
Third natural frequencies at various crack location and depth for ID 1.
Error distribution at different crack location and depth for ID 1. Difference between FEA probe & EXP
b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.02% 0.01% 0.11% 0.65% 0.16%
0.55% 0.53% 0.39% 0.22% 0.06%
N/A 0.50% 0.67% 0.45% 0.12%
N/A 2.07% 0.36% 1.07% 0.18%
3.68% 1.34% 0.01% 0.30% 1.26%
Difference between FEA No probe & EXP b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.11% 0.13% 0.14% 0.66% 0.18%
0.86% 0.71% 0.35% 0.31% 0.16%
N/A 0.91% 0.65% 0.26% 0.42%
N/A 2.88% 0.23% 0.66% 0.84%
1.48% 0.43% 0.32% 0.30% 2.62%
ID1 EXP
ID2 EXP
ID1 FE2
ID2 FE2
ID1 FE1
ID2 FE1
1.02 1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 10 Effect of thickness on O at b = 25% (EXP Experimentally, FE1 Finite element with probe, FE2 Finite element without considering probe).
6.2. Error analysis
6.3. Effect of specimen thickness
At certain crack location and depth, difference in frequency reduction O between experimental and Finite Element Analysis results are listed in Table 8.
Figs. 10 and 11 show a comparison of O between three specimens’ configurations (ID 1, ID 2 and ID 3) at two different crack locations, in which the results for each specimen config-
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M. Elshamy et al. ID1 EXP
ID2 EXP
ID3 EXP
ID3 FE2
ID1 FE2
ID1 FE 1
ID2 FE1
ID3 FE1
ID2 FE2
1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 11 Effect of thickness on O at b = 2.5% (EXP Experimentally, FE1 Finite element with probe, FE2 Finite element without considering probe).
ID1 EXP
ID4 EXP
ID1 FE 1
ID4 FE1
ID1 FE2
ID4 FE2
1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0
Fig. 12 probe).
Table 9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Effect of width on O at b = 2.5% (EXP Experimentally, FE1 Finite element with probe, FE2 Finite element without considering
First mode X of ID 1 at different a for three different materials. Steel
Copper
Aluminum
D
a
wn
O
wn
O
wn
O
0 1 2 3 3.5 4 4.5
0.000 0.143 0.286 0.429 0.500 0.571 0.643
35.964 35.631 34.834 33.331 32.165 30.53 28.233
1.000 0.991 0.969 0.927 0.894 0.849 0.785
26.007 25.76 25.193 24.129 23.301 22.138 20.5
1.000 0.991 0.969 0.928 0.896 0.851 0.788
36.142 35.802 35.01 33.523 32.367 30.743 28.458
1.000 0.991 0.969 0.928 0.896 0.851 0.787
uration are compared experimentally and using finite element method, to illustrate the effect of changing specimen thickness on the results. 6.4. Effect of specimen width Fig. 12.
6.5. Effect of specimen material A comparison between 1st, 2nd and 3rd natural frequencies of three different materials (steel, copper and aluminum) was carried out to illustrate the effect of changing material on O. All of these materials have the same dimensional configuration (ID 2) with the same crack location (X = 10 mm) using finite
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
Crack detection of cantilever beam Table 10
11
Second mode X of ID 1 at different a for three different materials. Steel
Copper
Aluminum
D
a
wn
O
wn
O
wn
O
0 1 2 3 3.5 4 4.5
0.000 0.143 0.286 0.429 0.500 0.571 0.643
225.08 223.32 219.29 212.58 207.98 202.61 195.54
1.000 0.992 0.974 0.944 0.924 0.900 0.869
162.74 161.43 158.57 153.7 150.32 146.07 140.55
1.000 0.992 0.974 0.944 0.924 0.898 0.864
226.17 224.37 220.39 213.6 208.88 202.97 195.86
1.0000 0.9920 0.9744 0.9444 0.9236 0.8974 0.8660
Table 11
Third mode X of ID 1 at different a for three different materials. Steel
Copper
Aluminum
D
a
wn
O
wn
O
wn
O
0 1 2 3 3.5 4 4.5
0.000 0.143 0.286 0.429 0.500 0.571 0.643
242.53 241.61 239.15 234.38 230.78 226.28 220.72
1.0000 0.9962 0.9861 0.9664 0.9516 0.9330 0.9101
175.03 174.35 172.59 169.28 166.59 163.35 159.34
1.0000 0.9961 0.9861 0.9671 0.9518 0.9333 0.9104
243.37 242.43 239.98 235.23 231.62 227.11 221.53
1.000 0.996 0.986 0.967 0.952 0.933 0.910
1.05 1.00
Third mode steel Third mode copper
0.95
Third mode Aluminum 0.90
Second mode steel Second mode copper
0.85
Second mode Aluminum First mode steel
0.80
First mode copper First mode Aluminum
0.75 0
Fig. 13
0.1
0.2
0.3
0.4
0.5
0.6
0.7
First, second and third natural frequencies at different material types.
element method without taking into consideration probe mass (see Tables 9–11). 7. Discussion of results 1. The presence of crack initiation is determined by measuring first mode natural frequency of a healthy structure and comparing it with the defected one. 2. To determine exact crack location and depth, second and third mode natural frequencies should be detected and
tracked to observe their changes. For every crack location and depth, there is only unique ratio of change for the first three modes of vibration 3. Error analysis illustrated in Table 8 shows that, considering the probe in analysis lead to an error that ranges from (2.07% to 3.68%), while the error without considering probe in analysis ranges from (2.88% to 2.62%). 4. Error value is independent on crack location and depth. 5. Figs. 10 and 11 illustrate the effect of changing specimen thickness on O. It was observed that O at same b and a is almost equal, with maximum error of 3%.
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12 6. Fig. 12 illustrates the effect of changing specimen width on O. It was observed that O at same b and a is almost equal, with maximum error of 1.2%. 7. Fig. 13 illustrates effect of changing specimen material on O. It was observed that O at same b and a is almost equal, with maximum error of 0.5%.
8. Conclusion – Crack position and depth are the most important parameters in crack investigation. – Transverse breathing crack would introduce flexibility in cantilever beam, where system stiffness is affected and therefore the natural frequency is reduced. – Monitoring natural frequencies is essential to evaluate crack severity. – Natural frequency is dependent on probe mass, as probe mass increase, natural frequency measured decreased as well. While, natural frequency reduction ratio O is independent on probe mass. – Element type shows no effect on analysis, when comparing frequency reduction using element 186 to element 187, results would be approximately the same. – Element 187 is considered more accurate in meshing probe, due to curvature of probe. – From previous investigation and calculation, all results are represented as a dimensionless number, at same crack depth to specimen thickness also for same crack position from fixation to specimen length; frequency of cracked specimen to healthy one would be the same regardless specimen thickness, width and material. – Resolution setting of vibration analyzer is vital for accurate estimation of crack depth, till crack depth to specimen thickness ratio is about 0.2, after wards high resolution is not important due to great difference in frequency reduction.
9. Future work Further studies should be done to study the effect of – Changing cross sectional shapes (circular, pipe, I-Beam etc.) – Using different crack shapes and orientations. – Increasing number of cracks with same and different depths.
M. Elshamy et al. References [1] Pratibha M. Karandikar, Vaishnav Vivek, Dabhade Ganesh, Experimental investigation of crack detection of cantilever beam-FFT analyzer, IOSR J. Mech. Civil Eng. (IOSR-JMCE) 13 (6) (2016) 12–17. [2] Vikram S. Yendhe, V.L. Kadlag, R.S. Shelke, Vibration analysis of cracked cantilever beam for varying crack size and location, Int. Res. J. Eng. Technol. 3 (8) (2016) 1913–1919. [3] Nitesh A. Meshram, Vaibhav S. Pawar, Analysis of crack detection of a cantilever beam using finite element analysis, Int. J. Eng. Res. Technol. (IJERT) 4 (04) (2015) 713–718. [4] Kaushar H. Barad, D.S. Sharma, Vishal Vyas, Crack detection in cantilever beam by frequency-based method, Proc. Eng. 51 (2013) 770–775. [5] Pankaj Charan Jena, Dayal R. Parhi, Goutam Pohit, Faults detection of a single cracked beam by theoretical and experimental analysis using vibration signatures, IOSR J. Mech. Civil Eng. (IOSR-JMCE) 4 (3) (2012) 01–18. [6] H. Nahvi, M. Jabbari, Crack detection in beams using experimental modal data and finite element model, Int. J. Mech. Sci. 47 (10) (2005) 1477–1497. [7] F. Leonard, J. Lanteigne, S. Lalonde, Y. Turcotte, Free vibration behavior of cracked cantilever beam and crack detection, Mech. Syst. Sig. Process. 15 (3) (2001) 529–548. [8] Sadettin Orhan, Analysis of free and forced vibration of a cracked cantilever beam, NDT E Int. 40 (6) (2007) 443–450. [9] G.M. Owolabi, A.S.J. Swamidas, R. Seshadri, Crack detection in beams using changes in frequencies and amplitude of frequency response functions, J. Sound Vib. 265 (1) (2003) 1–22. [10] F leonard, J. Lanteigne, S. Lalonde, Free vibration behaviors of a cracked cantilever beam and crack detection, 15 (3) (2001) 529–548. [11] P.F. Rizos, N. Asparagathos, A.D. Dimarogonas, Identification of crack location and magnitude in a cantilever beam from the vibration modes, J. Sound Vib. 138 (3) (1990) 381–388. [12] Robert Y Liang, Fred K. Choy, Jealous Hu, Detection of cracks in beam structures using measurements of natural frequencies, J. Franklin Inst. 328 (4) (1991) 505–518. [13] O.S. Salawu, C. Williams, Structural damage detection using experimental modal analysis–a comparison of some methods, in: International Modal Analysis Conference, 1993, pp. 254–260. [14] A.V. Deokar, V.D. Wakchaure, Experimental Investigation of Crack Detection in Cantilever Beam Using Natural Frequency as Basic Criterion, Institute of Technology, Nimra University, 2011, pp. 382–481. [15] P.R. Baviskar, V.B. Tungikar, Analysis of crack in shaft of blower using finite element analysis and experimental technique, Int. J. Res. Rev. Appl. Sci. 8 (1) (2011) 30–35. [16] E. Douka, L.J. Hadjileontiadis, Time-frequency analysis of the free vibration response of a beam with a breathing crack, NDT&E Int. 38 (1) (2005) 3–10.
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H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis M. Elshamy, W.A. Crosby, M. Elhadary * Department of Mechanical Engineering, Faculty of Engineering, Alexandria University, Alexandria, Egypt Received 5 August 2018; revised 22 September 2018; accepted 23 October 2018
KEYWORDS Breathing crack detection; Natural frequency; FEA
Abstract The investigation of a crack in a cantilever beam is very imperative in various mechanical applications such as turbo machinery impeller blades, propeller shafts and airplane wings. As a result, a full study is performed in this paper to investigate the crack consequences on a cantilever beam; this is carried out by monitoring the natural frequency of the beam, taking into consideration the probe mass used in data collection. The investigation is divided into two phases; the first phase is experimentally performed by measuring the natural frequency of a specimen with different thicknesses and widths while changing specimen material to illustrate the specimen dimensional and material effect on natural frequency reduction. This is carried out on different specimen configuration by changing crack location and depth. The second phase is to validate the acquired results from the first phase. This is achieved by using finite element analysis (FEA) techniques, this is performed by modeling the experimental phase conditions into ANSYS 16.2, one time considering the probe mass and other time without putting probe mass into consideration. Finally, the FEA results are compared with experimental phase results. Ó 2018 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).
1. Introduction Several methodologies have been implemented to monitor the condition of mechanical structure, each of these techniques depends on the corresponding phenomenon in the device related to the properties of the material tested. Cracks are the most common defect in mechanical structure. A crack * Corresponding author. E-mail address: [email protected] (M. Elhadary). Peer review under responsibility of Faculty of Engineering, Alexandria University.
introduces flexibility thus resulting in an overall drop in system stiffness, leading to a total failure of the system. For determining system integrity and remaining useful life, an inspection plan should be established, to anticipate failure at an early stage. Cracks are initiated as a surface hair crack in the transverse direction. Along the service of the specimen, the surface cracks propagate along the depth. This is one of the most popular and common forms of crack spreading that perhaps arise out of fatigue loading condition. This type of crack in cantilever beams is obvious in various applications such as turbo machinery impeller blades. This crack is initiated from intensive cycles of fatigue due to highly dynamic stresses
https://doi.org/10.1016/j.aej.2018.10.002 1110-0168 Ó 2018 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
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Nomenclature List of symbols D crack depth [mm] T specimen thickness [mm] X crack location from fixation [mm] L specimen length from fixation to free end [mm] W specimen width [mm] wnH natural frequency of healthy specimen [Hz]
caused by rotor vibration and blades resonances. In aviation application, this type of crack is presented in aero- plane’s wings due to variation of lift forces along wing service. Despite, the wing’s material is composite; this method can be also applied on it. A vibration-based method is submitted as an efficient and quick approach of detecting depth and location of crack in the specimen. As a nondestructive test method, it trends the severity crack progression in mechanical structures. In order to identify structural defect by vibration-based method monitoring, this method is based on tracking changes in natural frequency of the specimen, resulted from the change in flexibility and it’s reciprocal (stiffness) of the structure due to the crack initiation and propagation. Identification of crack initiation is determined in many studies by tracking 1st mode natural frequency change of the specimen. However, for determining exact crack location and depth, vibration 2nd and 3rd modes should be detected and tracked to observe their changes. For every crack location and depth, there is only unique ratio of change for the first three modes of vibration. Pratibha M. et al. [1] evaluated the first three natural frequencies of a single cracked cantilever beam and a healthy beam using vibration measurements. Vikram K. et al. [2] Monitor the vibration behavior of beams both experimentally and using FEM software ANSYS. In the present work, vibration analysis is carried out. Nitesh A. M. et al. [3] describe finite elemental analysis of a cracked cantilever beam and analyze the relation between the modal natural frequencies with crack depth, modal natural frequency with crack location. Also, the relation among the crack depth, crack location and natural frequency has been analyzed. Only single crack at different depth and at different location are evaluated and the analysis revels relationship between crack depth and modal natural frequency. Kaushar H. B. et al. [4] Detect crack presence on the surface of beam-type structural element using natural frequency. First two natural frequencies of the cracked beam have been obtained experimentally and used for detection of crack location and size. Pankaj C. et al. [5] concluded that faults detection of a single cracked beam by theoretical and experimental analysis using vibration signatures. The results of theoretical analysis and experimental analysis are compared and are found to be good correlation between them. H. Nahvi et al. [6] find results that the finite element model of the cracked beam is constructed and used to determine its natural frequencies and mode shapes. Leonard F. et al. [7] proposed a study on spectrograms of the free-decay responses showed a time drift of the frequency and damping, the usual hypothesis of constant modal parameters is no longer appropriate, since
wn a b O
natural frequency of cracked specimen [Hz] crack depth (D) to specimen thickness (T) ratio [Dimensionless] crack location from fixation (X) to specimen length (L) ratio [Dimensionless] cracked to healthy natural frequency ratio [Dimensionless]
the latter are revealed to be a function of the amplitude. Orhan S. [8] studied free and forced vibration analysis of single and two-edged cracked beam and suggested that free vibration analysis provides suitable information for the detection cracks. Owolabi G. et al. [9] Detected crack damage experimentally on aluminum beam by measuring acceleration frequency response at seven different points on each beam model using dual channel frequency analyzer, the damage detection schemes used in their study depend on the measured changes in the first three natural frequencies and corresponding amplitudes of the measured acceleration frequency response functions. Leonard F. et al. [10] based their study on cracks that occurred in metal beams obtained under controlled fatiguecrack propagation. Where the beams are clamped in a heavy vise and struck in order to obtain a clean impulse model response, spectrograms of the free-decay responses showed a time drift of the frequency and damping. Rizos P.F. et al. [11] measured flexure vibrations of cantilever beam with rectangular cross-section having transverse surface crack extended uniformly along width, where crack location and depth can be estimated accurately by measuring amplitude at two positions of the structure. Robert Y. et al. [12] developed a method for crack location identification and quantification of damage, where this method adopts rotational mass less springs in the beam element as a mechanical model to represent flexibility introduced by crack. For a given natural frequency and a damage location, the characteristic equation can be solved to provide numerical value of the stiffness of the rotational spring.
Table 1
Specimen Configurations.
Configuration ID
L (mm)
W (mm)
T (mm)
1 2 3 4
400 400 400 400
48 48 48 32
6.5 7 7.5 6.5
Table 2
Material Properties.
Parameter
Value
Commercial name Modulus of elasticity Density Poisson ratio
ASTM A283 Grade C Carbon Steel 190 GPa 7900 kg/m3 0.32
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Crack detection of cantilever beam
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Salawu O. et al. [13] reviewed four methods and made comparison of their effectiveness. One of the methods exploits the changes in Eigen parameters while the other three utilize system identification/model updating procedures. These three methods were originally developed to locate errors in mathematical models. For the comparison studies, response data were obtained from simulation analysis of a simply supported steel beam. Deokar A. et al. [14] performed experimental modal analysis on cracked beams and healthy beam. The first three natural frequencies were considered as basic criterion for crack detection, to locate crack a 3D graph normalize frequency in terms
Fig. 1
Table 3 Feature adjusted in Data Collector and Analyzer. Feature
Value
Maximum frequency (Hz) Number of lines Pre-trigger (G’s) Window type
100 1600 5 Rectangular
of the crack depth and location were plotted, where the intersection of these three contours gives crack location and crack
Experimental Configuration (All dimensions are in mms).
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depth. Baviskar P. et al. [15] observed changes in natural frequency due to crack propagation both theoretically using finite element analysis software and experimentally using fast Fourier transform analyzer, and they obtained that results by both methods were in a good agreement. Douka E. et al. [16] studied the nonlinear behavior of the system by using time frequency methods as an alternative to Fourier analysis methodology. A further investigation and study are performed to understand effect of specimen dimensions and specimen material on natural frequency reduction, furthermore study is done to illustrate effect of probe mass on results.
maximum frequency 100 ¼ ¼ 0:0625 Hz number of lines 1600 3. Experimental results 3.1. Bump test spectra See Figs. 2 and 3.
2. Experimental preparation 2.1. Specimen configuration The test specimens have rectangular cross-section with total length of 600 mm, while the length of fixed part of specimen is 200 mm. The cross-section dimensions of specimen configurations are shown in Table 1, while the material properties of specimen are listed in Table 2. 2.2. Experiment procedures (1) Specimen has been fixed between upper and lower halves of support, as shown in Fig. 1. (2) Vibration probe is placed at free end of specimen. (3) Vibration data is measured while the specimen is impacted using impact hummer. (4) Crack is initiated on the previous mentioned specimen at a certain distance from the fixation using wire cut method to ensure accurate crack depth, and then vibration data are collected as well. (5) Increase crack depth while repeating step 3. (6) Repeat the entire experiment at different crack location and depth.
Fig. 2 Bump test spectrum for a healthy specimen (ID 1, wn ¼ 21:5 Hz).
2.3. Data collector and analyzer 2.3.1. Theory of working Natural frequency is measured by applying an impact force on the specimen and then detects all excited natural frequencies resulting from this excitation using EMERSON CSI 2140 data collector and analyzer, in which many natural frequencies can appear in the collected data if impact force injects more energy to specimen. It is very important to select right hummer tip where the size of the hummer and stiffness of the tip greatly affect the results. Natural frequencies would be detected in low frequency zone if we perform bump test on a large mass specimen, then a heavy hummer with soft tip is advised to be utilized, as a heavy hummer generates a large force and a soft tip ensures that input energy is concentrated in lower frequency range.
Fig. 3 Bump test spectrum for defected specimen (ID 1,wn ¼ 15:7 Hz).
3.2. Bump test results: See Table 4. 4. Analysis using finite element method (FEM)
2.3.2. Device set up To ensure accurate results of data collection, vibration analyzer setup was as listed in Table 3. Therefore, minimum difference in frequencies that can be detected would be as follow;
4.1. Software ANSYS R16.2.
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Crack detection of cantilever beam Table 4 D
0 1 2 2.5 3 3.5 4 4.5 5 D
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
5
Bump Test Results. wn for ID 1 b = 2.5%
b = 12.5%
b = 25%
b = 37.5%
b = 50%
21.5 21.31 20.94 20.5 N/M 18.5 N/M 15.69 N/M
21 20.88 20.62 N/M 19.88 19.31 18.62 16.69 N/M
20.5 20.44 20.25 N/M 19.81 19.12 18.62 17.5 N/M
21 20.87 20.75 N/M 20.56 N/M 19.875 19 17.5
20.56 20.5 20.44 N/M 20.25 N/M 19.81 19.12 N/M
wn for ID 3
wn for ID 4
b = 2.5%
b = 25%
b = 2.5%
b = 2.5%
20.56 N/M 20.5 N/M 20 N/M 19.25 18.75 17.69 N/M
20.75 N/M 20.56 N/M 20.44 N/M 19.81 19.31 N/M 18.25
26.19 26.12 25.94 25.75 25.56 25.19 24.94 24.44 23.75 22.31
20.25 N/M 20.12 N/M 19.81 N/M 18.81 17.88 16.69 15.44
wn for ID 2
Fig. 4
Rectangular crack cross-section.
4.2. Crack representation
extruded from specimen free end with a height equivalent to real probe mass.
Crack is represented as an open crack with a rectangular end with a constant width of 1 mm, as shown in Fig. 4.
4.4. Meshing
4.3. Probe representation
There are two basic types of element used in analysis: –
Probe weight is about 200 g and specimen’s weight ranges from 700 gm to 1600 gm, therefore probe must be represented in analysis. The vibration probe is magnetically mounted to the specimen, then probe is simulated in FEA as a cylinder
– The first type is used to mesh healthy specimen (with no crack) and specimen without probe, as shown in Fig. 5. – The second type of element is used during analysis of cracked specimen, the quality of mesh has been refined at
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Table 5 D
Fig. 5
Solid 186 Element Type.
Fig. 6
Solid 187 Element Type.
Results with considering probe in analysis. wn for ID 1 b = 2.5%
b = 12.5%
b = 25%
b = 37.5%
b = 50%
0 1 2 3 3.5 4 4.5
24.872 24.647 24.091 22.993 22.083 20.811 18.844
24.872 24.732 24.294 23.429 22.722 21.605 20.036
24.872 24.771 24.473 23.874 23.325 22.511 21.235
24.872 24.807 24.631 24.243 23.365 N/C 22.436
24.872 24.838 24.741 24.526 24.007 N/C 23.421
D
wn for ID 2
wn for ID 3
wn for ID 4
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
b = 2.5%
b = 25%
b = 2.5%
b = 2.5%
27.221 N/C 26.988 N/C 26.444 N/C 25.383 24.55 23.424 21.768
27.221 N/C 27.117 N/C 26.824 N/C 26.254 25.777 25.061 24.045
29.585 29.496 29.349 29.124 28.809 28.373 27.786 26.998 25.957 24.591
22.406 N/C 22.228 N/C 21.746 N/C 20.766 19.942 18.758 17.051
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Crack detection of cantilever beam
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two locations; the crack location and the probe location, as shown in Fig. 6.
Results are represented as a relation between dimension-less parameters
5. FEA results
– relation between a (X-Axis) and O (Y-Axis) at constant b – Relation between b (X-Axis) and O (Y-Axis) at constant a
See Tables 5 and 6.
Table 6
6. Results comparison and validation
Results without Considering Probe in Analysis.
D
wn for ID 1 b = 2.5%
b = 12.5%
b = 25%
b = 37.5%
b = 50%
0 1 2 3 3.5 4 4.5
33.4 33.07 32.254 30.646 29.326 27.409 24.741
33.4 33.167 32.564 31.333 30.286 28.787 26.432
33.4 33.256 32.876 32.068 31.337 30.267 28.603
33.4 33 33.105 32.617 31.506 N/C 30.443
33.4 33 33.257 33.036 32.82 N/C 32.454
D
wn for ID 2
wn for ID 3
wn for ID 4
0 0.5 1 1.5 2 3 3.5 4 4.5
Table 7
b = 2.5%
b = 25%
b = 2.5%
b = 2.5%
35.964 N/C 35.631 N/C 34.834 33.331 32.165 30.53 28.233
35.946 N/C 35.82 N/C 35.446 34.677 34.059 33.156 31.816
38.529 38.401 38.193 37.87 37.427 35.975 34.893 33.47 31.515
33.314 N/C 32.978 N/C 32.17 30.555 29.236 27.388 24.745
ID 1 results.
Experimental X b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.9912 0.9943 0.9971 0.9948 0.9971
0.9740 0.9819 0.9878 0.9881 0.9942
0.9140 0.9467 0.9663 0.9790 0.9849
0.8050 0.8867 0.9083 0.9495 0.9635
0.7298 0.7948 0.8537 0.9048 0.9300
FE with considering Probe X b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.9909 0.9941 0.9959 0.9973 0.9986
0.9685 0.9765 0.9840 0.9903 0.9947
0.9244 0.9417 0.9599 0.9747 0.9861
0.8086 0.8686 0.9051 0.9394 0.9652
0.7584 0.8053 0.8538 0.9020 0.9417
FE without considering Probe X b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.9901 0.9930 0.9957 0.9973 0.9989
0.9657 0.9750 0.9843 0.9912 0.9957
0.9175 0.9381 0.9601 0.9766 0.9891
0.8206 0.8619 0.9062 0.9433 0.9717
0.7407 0.7914 0.8564 0.9115 0.9550
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First Mode (EXP) 0.15
(EXP) 0.31
(EXP) 0.46
(EXP) 0.61
(EXP) 0.69
(FEA Probe) 0.69
(FEA Probe) 0.15
(FEA Probe) 0.31
(FEA Probe) 0.46
(FEA Probe) 0.61
(FEA No Probe) 0.69
(FEA No Probe) 0.15
(FEA No Probe) 0.31
(FEA No Probe) 0.46
(FEA No Probe) 0.61
1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0.0%
10.0%
20.0%
Fig. 7
30.0%
40.0%
50.0%
60.0%
First mode results comparison.
6.1. Results of specimen’s ID 1 6.1.1. First mode results For specimen ID 1, b ranges from 2.5% to 50% at each location crack depth varies till about 70% of thickness. Frequency reduction ratio (X) is calculated using finite element analysis with and without considering probe in analysis and is detected experimentally using vibration measurement.
Table 7 shows the outputs from the experimental and finite element analysis. These results are compared against each other, as shown in Fig. 7, to validate the experimental results. Fig. 8 shows second natural frequency at various crack location and depth for ID 1 without considering probe in the FEA, while Fig. 9 shows third natural frequency at various crack location and depth for ID 1 without considering probe in the FEA.
Second mode =0.154
1.02
=0.31
=0.45
=0.61
=0.69
1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0.82 0.0%
Fig. 8
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
Second natural frequency at various crack location and depth for ID 1.
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Crack detection of cantilever beam
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Third mode =0.154
=0.31
=0.45
=0.61
=0.69
1.02 1 0.98 0.96 0.94 0.92 0.9 0.88 0.0%
Fig. 9
Table 8
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
Third natural frequencies at various crack location and depth for ID 1.
Error distribution at different crack location and depth for ID 1. Difference between FEA probe & EXP
b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.02% 0.01% 0.11% 0.65% 0.16%
0.55% 0.53% 0.39% 0.22% 0.06%
N/A 0.50% 0.67% 0.45% 0.12%
N/A 2.07% 0.36% 1.07% 0.18%
3.68% 1.34% 0.01% 0.30% 1.26%
Difference between FEA No probe & EXP b
a = 0.15
a = 0.3
a = 0.45
a = 0.6
a = 0.7
2.5% 12.5% 25% 37.5% 50%
0.11% 0.13% 0.14% 0.66% 0.18%
0.86% 0.71% 0.35% 0.31% 0.16%
N/A 0.91% 0.65% 0.26% 0.42%
N/A 2.88% 0.23% 0.66% 0.84%
1.48% 0.43% 0.32% 0.30% 2.62%
ID1 EXP
ID2 EXP
ID1 FE2
ID2 FE2
ID1 FE1
ID2 FE1
1.02 1 0.98 0.96 0.94 0.92 0.9 0.88 0.86 0.84 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 10 Effect of thickness on O at b = 25% (EXP Experimentally, FE1 Finite element with probe, FE2 Finite element without considering probe).
6.2. Error analysis
6.3. Effect of specimen thickness
At certain crack location and depth, difference in frequency reduction O between experimental and Finite Element Analysis results are listed in Table 8.
Figs. 10 and 11 show a comparison of O between three specimens’ configurations (ID 1, ID 2 and ID 3) at two different crack locations, in which the results for each specimen config-
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
10
M. Elshamy et al. ID1 EXP
ID2 EXP
ID3 EXP
ID3 FE2
ID1 FE2
ID1 FE 1
ID2 FE1
ID3 FE1
ID2 FE2
1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Fig. 11 Effect of thickness on O at b = 2.5% (EXP Experimentally, FE1 Finite element with probe, FE2 Finite element without considering probe).
ID1 EXP
ID4 EXP
ID1 FE 1
ID4 FE1
ID1 FE2
ID4 FE2
1.05 1 0.95 0.9 0.85 0.8 0.75 0.7 0
Fig. 12 probe).
Table 9
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Effect of width on O at b = 2.5% (EXP Experimentally, FE1 Finite element with probe, FE2 Finite element without considering
First mode X of ID 1 at different a for three different materials. Steel
Copper
Aluminum
D
a
wn
O
wn
O
wn
O
0 1 2 3 3.5 4 4.5
0.000 0.143 0.286 0.429 0.500 0.571 0.643
35.964 35.631 34.834 33.331 32.165 30.53 28.233
1.000 0.991 0.969 0.927 0.894 0.849 0.785
26.007 25.76 25.193 24.129 23.301 22.138 20.5
1.000 0.991 0.969 0.928 0.896 0.851 0.788
36.142 35.802 35.01 33.523 32.367 30.743 28.458
1.000 0.991 0.969 0.928 0.896 0.851 0.787
uration are compared experimentally and using finite element method, to illustrate the effect of changing specimen thickness on the results. 6.4. Effect of specimen width Fig. 12.
6.5. Effect of specimen material A comparison between 1st, 2nd and 3rd natural frequencies of three different materials (steel, copper and aluminum) was carried out to illustrate the effect of changing material on O. All of these materials have the same dimensional configuration (ID 2) with the same crack location (X = 10 mm) using finite
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
Crack detection of cantilever beam Table 10
11
Second mode X of ID 1 at different a for three different materials. Steel
Copper
Aluminum
D
a
wn
O
wn
O
wn
O
0 1 2 3 3.5 4 4.5
0.000 0.143 0.286 0.429 0.500 0.571 0.643
225.08 223.32 219.29 212.58 207.98 202.61 195.54
1.000 0.992 0.974 0.944 0.924 0.900 0.869
162.74 161.43 158.57 153.7 150.32 146.07 140.55
1.000 0.992 0.974 0.944 0.924 0.898 0.864
226.17 224.37 220.39 213.6 208.88 202.97 195.86
1.0000 0.9920 0.9744 0.9444 0.9236 0.8974 0.8660
Table 11
Third mode X of ID 1 at different a for three different materials. Steel
Copper
Aluminum
D
a
wn
O
wn
O
wn
O
0 1 2 3 3.5 4 4.5
0.000 0.143 0.286 0.429 0.500 0.571 0.643
242.53 241.61 239.15 234.38 230.78 226.28 220.72
1.0000 0.9962 0.9861 0.9664 0.9516 0.9330 0.9101
175.03 174.35 172.59 169.28 166.59 163.35 159.34
1.0000 0.9961 0.9861 0.9671 0.9518 0.9333 0.9104
243.37 242.43 239.98 235.23 231.62 227.11 221.53
1.000 0.996 0.986 0.967 0.952 0.933 0.910
1.05 1.00
Third mode steel Third mode copper
0.95
Third mode Aluminum 0.90
Second mode steel Second mode copper
0.85
Second mode Aluminum First mode steel
0.80
First mode copper First mode Aluminum
0.75 0
Fig. 13
0.1
0.2
0.3
0.4
0.5
0.6
0.7
First, second and third natural frequencies at different material types.
element method without taking into consideration probe mass (see Tables 9–11). 7. Discussion of results 1. The presence of crack initiation is determined by measuring first mode natural frequency of a healthy structure and comparing it with the defected one. 2. To determine exact crack location and depth, second and third mode natural frequencies should be detected and
tracked to observe their changes. For every crack location and depth, there is only unique ratio of change for the first three modes of vibration 3. Error analysis illustrated in Table 8 shows that, considering the probe in analysis lead to an error that ranges from (2.07% to 3.68%), while the error without considering probe in analysis ranges from (2.88% to 2.62%). 4. Error value is independent on crack location and depth. 5. Figs. 10 and 11 illustrate the effect of changing specimen thickness on O. It was observed that O at same b and a is almost equal, with maximum error of 3%.
Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002
12 6. Fig. 12 illustrates the effect of changing specimen width on O. It was observed that O at same b and a is almost equal, with maximum error of 1.2%. 7. Fig. 13 illustrates effect of changing specimen material on O. It was observed that O at same b and a is almost equal, with maximum error of 0.5%.
8. Conclusion – Crack position and depth are the most important parameters in crack investigation. – Transverse breathing crack would introduce flexibility in cantilever beam, where system stiffness is affected and therefore the natural frequency is reduced. – Monitoring natural frequencies is essential to evaluate crack severity. – Natural frequency is dependent on probe mass, as probe mass increase, natural frequency measured decreased as well. While, natural frequency reduction ratio O is independent on probe mass. – Element type shows no effect on analysis, when comparing frequency reduction using element 186 to element 187, results would be approximately the same. – Element 187 is considered more accurate in meshing probe, due to curvature of probe. – From previous investigation and calculation, all results are represented as a dimensionless number, at same crack depth to specimen thickness also for same crack position from fixation to specimen length; frequency of cracked specimen to healthy one would be the same regardless specimen thickness, width and material. – Resolution setting of vibration analyzer is vital for accurate estimation of crack depth, till crack depth to specimen thickness ratio is about 0.2, after wards high resolution is not important due to great difference in frequency reduction.
9. Future work Further studies should be done to study the effect of – Changing cross sectional shapes (circular, pipe, I-Beam etc.) – Using different crack shapes and orientations. – Increasing number of cracks with same and different depths.
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Please cite this article in press as: M. Elshamy et al., Crack detection of cantilever beam by natural frequency tracking using experimental and finite element analysis, Alexandria Eng. J. (2018), https://doi.org/10.1016/j.aej.2018.10.002