INSTITUTE OF TECHNOLOGY, NIRMA UNIVERSITY, AHMEDABAD 382 481, 08-10 DECEMBER, 2011 1 Experimental Investigation of Crack Detection in Cantilever Beam Using Natural Frequency as Basic Criterion A. A.V.Deokar, B. V.D.Wakchaure A & B., Department of Mechanical Engineering, Amrutvahini College of Engineering, Sangamner. Abstract-- Crack changes the dynamic behaviour of the structure and by examining this change, crack size and position can be identified. Non destructive testing (NDT) methods are used for detection of crack which are very costly and time consuming. Currently research has focused on using modal parameters like natural frequency, mode shape and damping. to detect crack in beams. In this paper a method for detection of open transverse crack in a slender Euler Bernoulli beam is presented. Experimental Modal Analysis (EMA) was performed on cracked beams and a healthy beam. The first three natural frequencies were considered as basic criterion for crack detection. To locate the crack, 3D graphs of the normalized frequency in terms of the crack depth and location are plotted. The intersection of these three contours gives crack location and crack depth. Out of several case studies conducted the results of one of the case study is presented to demonstrate the applicability and efficiency of the method suggested. Index Terms Crack, Euler Bernoulli, mode shape, natural frequency. M I. INTRODUCTION echanical structures in service life are subjected to combined or separate effects of the dynamic load, temperature, corrosive medium and other type of damages. The importance of an early detection of cracks appears to be crucial for both safety and economic reasons because fatigue cracks are potential source of catastrophic structural failure. Damage identification methods are mainly based upon the shifts in natural frequencies or changes in mode shapes. NDT methods are often employed for detection of cracks in machine and structural components. All of these NDT techniques require that the location of the damage is known a priori and that the portion of the structure being inspected is readily accessible. In order to detect a crack by this method, the whole component requires scanning. Their adoption becomes uneconomical for long beams and pipelines which are widely used in power plants, railway tracks, long pipelines etc. This makes the process tedious and time consuming. The drawbacks of traditional localized NDT methods have motivated development of global vibration based damage detection methods. It is well known that when a crack develops in a component it leads to changes in its vibration parameters, e.g. a reduction in the stiffness and increase in the damping and a reduction in the natural frequency. They may enable determination of location and size of a crack from the vibration data collected from a single or at most a few, points on the component. These changes are mode dependent. Hence it may be possible to estimate the location and size of the crack by measuring the changes in vibration parameters. The technique using changes in natural frequencies as the crack detection criterion has received considerable attention. The choice of using the natural frequency as a basis in the development of NDE (Non destructive evaluation) is most attractive. This is due to the fact that the natural frequencies of a beam can be measured from one single location on the beam, thus offering scope for the development of a fast and global NDE technique. Considerable efforts being made to make the method useful in practice. It results in a considerable saving in time, labour and cost for long beam like components, such as rails, pipelines, etc. Chaudhari and Maiti [1] proposed Frobenius method for solving an Euler-Bernoulli type differential equation. Solving inverse problem requires a lot of mathematical effort and it is time consuming. Lee [2] presented a method based on the Newton-Raphson iteration method. In this method proper selection of the initial guesses of the crack parameters is important. Rizos et al. [3] conducted experiments to detect crack depth and location from changes in the mode shapes of cantilever beams. A major disadvantage of using mode shape based technique is that obtaining accurate mode shapes involves arduous and meticulous measurement of displacement or acceleration over a large number of points on the structure before and after damage. The accuracy in measurement of mode shapes is highly dependent on the number and distribution of sensors employed. Owolabi et al. [4] used natural frequency as the basic criterion for crack detection in simply supported and fixed-fixed beams. The method suggested has been extended to cantilever beams to check the capability and efficiency. There is need to see if this approach can be used for fixed-free beams. There is a very limited data concerning experimental observations as in [1]- [3]. Most of the data available is theoretical. This paper validates the Owolabi et al. [4] approach to fixed-free beam based on experimental data.
2 INTERNATIONAL CONFERENCE ON CURRENT TRENDS IN TECHNOLOGY, NUiCONE 2011 II. LITERATURE REVIEW Different researchers have discussed damage detection of vibrating structures in various ways. Chaudhari et al. [1] proposed modelling of transverse vibration of a beam of linearly variable depth and constant thickness in the presence of an open edge crack normal to its axis using the concept of a rotational spring to represent the crack section and the Frobenius method to enable possible detection of location of the crack based on the measurement of natural frequencies. Lee [2] presented a method to detect a crack in a beam. The crack was not modeled as a massless rotational spring, and the forward problem was solved for the natural frequencies using the boundary element method. The inverse problem was solved iteratively for the crack location and the crack size by the Newton-Raphson method. The present crack identification procedure was applied to the simulation cases which use the experimentally measured natural frequencies as inputs, and the detected crack parameters are in good agreements with the actual ones. The present method enables one to detect a crack in a beam without the help of the massless rotational spring model. Rizos et al. [3] investigated the flexural vibrations of a cantilever beam with a rectangular cross-section having a transverse surface crack which was modelled as a massless rotational spring. They also assumed that the crack was fully open and has uniform depth. As an experimental study, they forced the beam by a harmonic exciter to vibrate at one of the natural modes of vibration and measured the amplitudes at two positions. Owolabi et al. [4] reported an ongoing research on the experimental investigations of the effects of cracks and damages on the integrity of structures, with a view to detect, quantify, and determine their extents and locations. Two sets of aluminum beams were used for this experimental study. Each set consisted of seven beams, the first set had fixed ends, and the second set was simply supported. Cracks were initiated at seven different locations from one end to the other end (along the length of the beam) for each set, with crack depth ratios ranging from 0.1d to 0.7d (d was the beam depth) in steps of 0.1, at each crack location. Measurements of the acceleration frequency responses at seven different points on each beam model were taken using a dual channel frequency analyzer. The damage detection schemes used in this study depended on the measured changes in the first three natural frequencies and the corresponding amplitudes of the measured acceleration frequency response functions. A substantial amount of work has been conducted on natural frequency and mode shape based damage detection methods in the past. Some of the approaches used the methods which are iterative and requires an initial guess. As a result the error in the solution is remarkably influenced by the initial guess. Most of the researchers studied the effect of a single crack on the dynamics of structures. A lot of studies using natural frequency as a damage detection tool are being carried out in the vibration based damage detection field. Recently, a new vibration based damage detection technique that utilizes a shift in natural frequencies has been the focus in this paper. Results obtained from these studies seem more promising. However, in actual practice structural members such as beams or shafts are highly susceptible to transverse crosssectional cracks due to fatigue. Slender components are encountered in many practical applications, e.g. railway tracks, crane girders, brackets; long intermittently supported pipelines, etc. A new vibration based damage detection technique that utilizes a shift in natural frequencies has been focused in this paper. The objective of this paper is to analyze experimentally the vibration characteristics of the cracked cantilever beam. The data of experimental modal analysis is used to detect crack location and calculate crack depth using changes in natural frequencies. III. EXPERIMENTAL SET-UP AND PROCEDURE 1) Experimental Model Description Mild steel beams were used for this experimental investigation. The set consisted of 49 beam models with the fixed-free ends. Each beam model was of cross-sectional area 20mmX20mm with a length of 300 mm from fixed end. It had the following material properties: Young s modulus, E= 2.06X10 5 MPa, density, ρ=7850kg/m 3, the Poisson ratio, µ=0.35. Fig.1. Experimental set-up. 2) Experimental Procedure The fixed free beam model was clamped at one end, between two thick rectangular steel plates, supported over a short and stiff steel I-section girder. The beam was excited with an impact hammer. The first three natural frequencies of the uncracked beam were measured. Then, cracks were generated to the desired depth using a wire cut EDM (around 0.35mm thick); the crack always remained open during dynamic testing Total 49 beam models were tested with cracks at different locations starting from a location near to fixed end. The crack depth varied from 1.5mm to 14mm at
INSTITUTE OF TECHNOLOGY, NIRMA UNIVERSITY, AHMEDABAD 382 481, 08-10 DECEMBER, 2011 3 each crack position. Each model was excited by an impact hammer. This served as the input to the system. It is to be noted that the model was excited at a point, which was a few millimeters away from the center of the model. This was done to avoid exciting the beam at a nodal point (of a mode), since the beam would not respond for that mode at that point. The dynamic responses of the beam model were measured by using light accelerometer placed on the model as indicated in Fig. 1. The response measurements were acquired, one at a time, using the FFT analyzer. TABLE I FUNDAMENTAL NATURAL FREQUENCY RATIO (ω c/ω) AS A FUNCTION OF CRACK LOCATION (X) AND CRACK DEPTH (a) IV. RESULTS AND DISCUSSIONS 1) Results The FRFs obtained were curve-fitted using the B&K PULSE 14.1.1 software package. The experimental data from the curve-fitted results were tabulated, and plotted (in a three dimensional plot) in the form of frequency ratio (ω c /ω) (ratio of the natural frequency of the cracked beam to that of the uncracked beam) versus the crack depth (a) for various crack location (X). Tables I III show the variation of the frequency ratio as a function of the crack depth and crack location for beams with fixed-free ends. TABLE II SECOND NATURAL FREQUENCY RATIO (ω c/ω) AS A FUNCTION OF CRACK LOCATION (X) AND CRACK DEPTH (a) 2) Changes in Natural Frequencies Fig. 2 to 4 shows the plots of the first three frequency ratios as a function of crack depths for some of the crack positions. Fig.5 to Fig.7 shows the frequency ratio variation of three modes in terms of crack position for various crack depths respectively. From Fig.2 it is observed that, for the cases considered, the fundamental natural frequency was least affected when the crack was located at 265mm from fixed end. The crack was mostly affected when the crack was located at 25mm from the fixed end. Hence for a cantilever beam, it could be inferred that the fundamental frequency decreases significantly as the crack location moves towards the fixed end of the beam. This could be explained by the fact that the decrease in frequencies is greatest for a crack located where the bending moment is greatest. It appears therefore that the change in frequencies is a function of crack location. From Fig.3 it is observed that the second natural frequency was mostly affected for a crack located at the center for all crack depths of a beam due to the fact that at that location the bending moment is having large value. The second natural frequency was least affected when the crack was located at 265mm from fixed end. From Fig.4 it is observed that the third natural frequency of beam changed rapidly for a crack located at 200 mm. The third natural frequency was almost unaffected for a crack located at the center of a cantilever beam; the reason for this zero influence was that the nodal point for the third mode was located at the center of beam. TABLE III THIRD NATURAL FREQUENCY RATIO (ω c/ω) AS A FUNCTION OF CRACK LOCATION (X) AND CRACK DEPTH (a) From Fig.5 it is observed that, for the cases considered, the fundamental natural frequency was least affected when the crack depth was 4.5mm. The crack was mostly affected when the crack depth was 14mm. Hence for a cantilever beam, it could be inferred that the fundamental frequency decreases significantly as the crack depth increase to 70% of beam depth. This could be explained by the fact that the decrease in frequencies is greatest for a more crack depth because as more material gets removed the stiffness of the beam decrease and hence the natural frequency. It appears therefore that the change in frequencies is a function of crack depth also.
4 INTERNATIONAL CONFERENCE ON CURRENT TRENDS IN TECHNOLOGY, NUiCONE 2011 combination of different crack depths and crack locations (for a particular mode) could be plotted in a curve with crack location and crack depth as its axes. Fig.2. Fundamental natural frequency ratio in terms of crack depth for various crack positions (x=25,100,150,200,265mm) Fig.4. Third natural frequency ratio in terms of crack depth for various crack positions (x=25, 100,150,200,265mm). Fig.3. Second natural frequency ratio in terms of crack depth for various crack positions (x=25, 100,150,200,265mm) From Fig.6 it is observed that the second natural frequency was mostly affected for a crack depth of 14mm at the crack location 175mm. The second natural frequency was least affected when the crack depth was 2mm. From Fig.7 it is observed that the third natural frequency of beam changed rapidly for a crack depth of 14mm.Third natural frequency was remained unaffected when crack depth was 4.5mm. Third natural frequency was remained unchanged at crack locations 40mm, 200mm, and 265mm due to the presence of node point at that position. Fig.8 to Fig.10 show the three dimensional plots of Normalized Frequency versus Crack Location and Crack Depth for first, second and third mode respectively for crack location of 100mm and crack depth of 7.5mm. To get these three dimensional plots program is written in MATLAB. In Fig.8 to Fig.10, the contour line are not present due to the presence of node points. 3) Crack Identification Technique Using Changes In Natural Frequencies As stated earlier, both the crack location and the crack depth influence the changes in the natural frequencies of a cracked beam. Consequently, a particular frequency could correspond to different crack locations and crack depths. This can be observed from the three-dimensional plots of the first three natural frequencies of cantilever beams as shown in Fig.8 to Fig.10. On this basis, a contour line, which has the same normalized frequency change resulting from a Fig.5. First Mode Frequency Ratio in Terms of Crack Position for Various Crack Depths (a= 4.5, 7.5, 10, 14mm) Fig.6. Second Mode Frequency Ratio in Terms of Crack Position for Various Crack Depths (a= 4.5, 7.5, 10, 14mm) To identify the presence of crack in the beam, an essential step is to measure a sufficient number of natural frequencies of the beam, and then use the technique explained in this section to estimate the crack location, and depth. Measuring the first three natural frequencies will be sufficient to determine the crack location, and the crack depth for a beam with a single crack.
INSTITUTE OF TECHNOLOGY, NIRMA UNIVERSITY, AHMEDABAD 382 481, 08-10 DECEMBER, 2011 5 frequencies; (2) normalization of the measured frequencies; (3) plotting of contour lines from different modes on the same axes; and (4) location of the point(s) of intersection of the different contour lines. The point(s) of intersection, common to all the three modes, indicate(s) the crack location, and crack depth. This intersection will be unique due to the fact that any normalized crack frequency can be represented by a governing equation that is dependent on crack depth (a), crack location (X). Therefore a minimum of three curves is required to identify the two unknown parameters of crack location and crack depth. Fig.7. Third Mode Frequency Ratio in Terms of Crack Position for Various Crack Depths (a= 4.5, 7.5, 10, 14mm) Fig.10. Three-dimensional plot with contour lines of normalized natural frequency versus crack location and crack depth for third mode for crack location of 100mm and crack depth of 7.5mm Fig.8. Three-dimensional plot with contour lines of normalized natural frequency versus crack location and crack depth for first mode for crack location of 100mm and crack depth of 7.5mm Fig.9. Three-dimensional plot with contour lines of normalized natural frequency versus crack location and crack depth for second mode for crack location of 100mm and crack depth of 7.5mm For a beam with a single crack with unknown parameters, the following steps are required to predict the crack location, and depth, namely, (1) measurements of the first three natural From Tables I III, it is observed that for a crack depth of 7.5mm located at a distance of 100mm from fixed end of the beam, the normalized frequencies are 0.9398 for the first mode, 0.9663 for the second mode and 0.9334 for the third mode. The contour lines with the values of 0.9398, 0.9663 and 0.9334 were retrieved from the first three modes with the help of MINITAB software as shown in Fig.11 to Fig.13 and plotted on the same axes as shown in Fig.14. From the Fig.14 it could be observed that there are two intersection points in the contour lines of the first and the second modes. Consequently the contour of the third mode is used to identify the crack location (X=100mm) and the crack depth (a=7.5mm), uniquely. The three contour lines gave just one common point of intersection, which indicates the crack location and the crack depth. Since the frequencies depend on the crack depth and location, these values can be uniquely determined by the solution of a function having solutions one order higher (in this case, three) than the number of unknowns (in this case, two, namely crack depth and location) to be determined. This is the reason for the requirement of three modes. If there were more parameters that influence the response (besides the crack depth and location), then one will require more modes to identify the unknown crack depth and crack location.
6 INTERNATIONAL CONFERENCE ON CURRENT TRENDS IN TECHNOLOGY, NUiCONE 2011 location, crack depth and mode number. A simple method for predicting the location and depth of the crack based on changes in the natural frequencies of the beam is also presented, and discussed. This procedure becomes feasible due to the fact that under robust test and measurement conditions, the measured parameters of frequencies are unique values, which will remain the same (within a tolerance level), wherever similar beams are tested and responses measured. The experimental identification of crack location and crack depth is very close to the actual crack size and location on the corresponding test specimen. Fig.11. Frequency contour plot of mode-1 for normalized frequency 0.9398 Fig.14. Crack identification technique by using frequency contours of the first three modes of beam (mode 1, normalized frequency (0.9398); mode 2, normalized frequency (0.9663); and 3: mode 3, normalized frequency (0.9334). Fig.12. Frequency contour plot of mode-2 for normalized frequency 0.9663 VI. REFERENCES [1] T.D.Chaudhari, S.K. Maiti, Modelling of transverse vibration of beam of linearly variable depth with edge crack, Engineering Fracture Mechanics vol. 63, pp. 425-445, 1999. [2] J. Lee, Identification of a crack in a beam by the boundary element method, Journal of Mechanical Science and Technology, vol. 24 (3), pp. 801-804, 2010. [3] Rizos R.F., N.Aspragathos, A.D.Dimarogonas, (1990), Identification of crack location and magnitude in a cantilever beam from the vibration modes, Journal of Sound and Vibration 138(3) 381 388. [4] G.M. Owolabi, A.S.J. Swamidas, R. Seshadri, Crack detection in beams using changes in frequencies and amplitudes of frequency response functions, Journal of Sound and Vibration, vol. 265 (1), pp. 1 22, 2003. [5] Y. Narkis, Identification of crack location in vibrating simply supported beams, Journal of Sound and Vibration, vol. 172(4), pp. 549 558, 1994. [6] A.D.Dimarogonas, Vibration of cracked structures: a state of the art review, Engineering Fracture Mechanics, vol. 55, pp. 831-857, 1996. Fig.13. Frequency contour plot of mode-3 for normalized frequency 0.9334 V. CONCLUSIONS Detailed experimental investigations of the effects of crack on the first three modes of vibrating cantilever beams have been presented in this paper. From the results it is evident that the vibration behavior of the beams is very sensitive to the crack