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Latin American Journal of Solids and Structures

Print version ISSN 1679-7817On-line version ISSN 1679-7825

Lat. Am. j. solids struct. vol.12 no.9 Rio de Janeiro Sept. 2015

http://dx.doi.org/10.1590/1679-78251403 

Articles

Modeling of Cracked Beams by the Experimental Design Method

M. Seriera 

N. Benamaraa 

A. Meguenia 

K. Refassia 

aMechanics of Structures and Solids Laboratory. Faculty of Technolgy, University of Sidi-Bel-Abbes, Bp 89, cité Ben M'hidi sidi- Bel-Abbes 22000-Algeria


Abstract

The understanding of phenomena, no matter their nature is based on the experimental results found. In the most cases, this requires an important number of tests in order to put a reliable and useful observation served into solving the technical problems subsequently. This paper is based on independent and variables combination resulting from experimentation in a mathematical formulation. Indeed, mathematical modeling gives us the advantage to optimize and predict the right choices without passing each case by the experiment. In this work we plan to apply the experimental design method on the experimental results found by (Deokar, A, 2011), concerning the effect of the size and position of a crack on the measured frequency of a beam console, and validating the mathematical model to predict other frequencies

Keywords: Parameters; experimental design method; modeling; frequency; crack

INTRODUCTION

Several scientific works has been conducted on the experimental design method each in its field of application and its objective, (Bounazef M, Goupy J and Castro. L, 2009,1925,2004). Other authors (Aît Yala, 2009) have implemented experimental design using numerical results. Experimental design were used for the first time by (Fisher R, 1925), in the field of agriculture where the experimental parameters are numerous and significant which leads to mathematical modeling and therefore to optimize the model sought. In mechanical terms, for example, (Serier et al, 2013a) used experimental design method for optimizing the machining parameters in order to increase the life of the tool. These same authors (2013b) performed a work in the field of vibration where they showed the importance of the geometrical properties, the speed of rotation of a shaft (tree) on the opening and closing mechanism of crack in a rotating machine. It is clear that the phenomenon of cracking beams has negative consequences on the large industrial constructions, (Thomas, M, 1995). The work of (Deokar, A, 2011) has focused on the experimental investigation for the detection of a crack on a cantilever used a criteria based on natural frequencies. Our work consisted on the use of experimental design method for modeling and prediction of frequencies.

2 EXPERIMENTAL RESULTS FOR EACH FREQUENCY MODE

In the tables below, the experimental results of the variation of frequency ratio as a function of position and depth of modes 1, 2 and 3 are presented. These results came from the work of (Deokar, 2011).

Table 1 Frequency ratio of the various tests of the first mode and experience matrix. 

Physical values of parameters Coded values of parameters Experimental results
N Crack depth Crack location Frequency ratio first mode a0 X1 X2 I12 y
01 2 25 0,9887 1 -1 -1 1 0,9887
02 2 100 0,9956 1 -1 -0,14 0,14 0,9956
03 2 200 0,9996 1 -1 +1 -1 0,9996
04 6 25 0,91 1 -0,2 -1 0,2 0,91
05 6 100 0,9626 1 -0,2 -0,14 0,028 0,9626
06 6 200 0,9966 1 -0,2 +1 -0,2 0,9966
07 12 25 0,6465 1 +1 -1 -12 0,6465
08 12 100 0,8081 1 +1 -0,14 -1,68 0,8081
09 12 200 0,9766 1 +1 +1 12 0,9766

Table 2 Frequency ratio of the various tests of the second mode and experience matrix. 

Physical values of parameters Coded values of parameters Experimental results
N Crack depth Crack location Frequency ratio second mode a0 X1 X2 I12 y
01 2 25 0,9951 1 -1 -1 1 0,9951
02 2 100 0,9975 1 -1 -0,14 0,14 0,9975
03 2 200 0,995 1 -1 +1 -1 0,995
04 6 25 0,9641 1 -0,2 -1 0,2 0,9641
05 6 100 0,9663 1 -0,2 -0,14 0,028 0,9663
06 6 200 0,9316 1 -0,2 +1 -0,2 0,9316
07 12 25 0,8903 1 +1 -1 -12 0,8903
08 12 100 0,9039 1 +1 -0,14 -1,68 0,9039
09 12 200 0,7836 1 +1 +1 12 0,7836

Table 3 Frequency ratio of the various tests of the third mode and experience matrix. 

Physical values of parameters Coded values of parameters Experimental results
N Crack depth Crack location Frequency ratio third mode a0 X1 X2 I12 y
01 2 25 0,9986 1 -1 -1 1 0,9986
02 2 100 0,9946 1 -1 -0,14 0,14 0,9946
03 2 200 0,9927 1 -1 +1 -1 0,9927
04 6 25 0,9888 1 -0,2 -1 0,2 0,9888
05 6 100 0,9569 1 -0,2 -0,14 0,028 0,9569
06 6 200 0,9441 1 -0,2 +1 -0,2 0,9441
07 12 25 0,9574 1 +1 -1 -12 0,9574
08 12 100 0,8279 1 +1 -0,14 -1,68 0,8279
09 12 200 0,7995 1 +1 +1 1 0,7995

3 CALCULATIONS OF THE EFFECTS OF FACTORS

Each factor xi is affected (acted on) the behavior of the beam and it's defined by the effect of ai . It is possible that the factors interact with each other, this is the case in our work, and therefore we are left with three factors, instead of two, including the average of these latter. In other words any response yi depends on the action of all factors together xi. Analytically, and the relationship between the response factor can exist only when a certain proportionality exists between them. This leads us to write:

The Solve of system of equations is based on the least squares method, and the solution is noted a.

This solution is given by the following formula derived from the theory of matrix calculation.

Therefore

Table 4 Coefficients that make up the mathematical model 

Coefficients Mode 1 Mode 2 Mode 3
a0 0,92 0,93 0,93
a1 -0,090 -0,070 -0,070
a2 0,076 -0,026 -0,036
I12 0,081 -0,028 -0,037

The models are thus written in the following forms

    Mode 1. :y 1 = 0.92 - 0.090(X 1) + 0.076(X 2)+0.081(X 1 X 2)

    Mode 2. :y 2 = 0.93 - 0.70(X 1) - 0.026(X 2) - 0.028(X 1 X 2)

    Mode 3. :y 3 =0.93 - 0.70(X 1) - 0.036(X 2) - 0.037(X 1 X 2)

4 ANALYSES WITH ONE VARIABLE FACTOR

4.1 Effect of Each Factor for the Three Modes

The figures below represent the effect of the two most principals factors (the crack depth and position of the crack) for each mode of vibration. We note that the ratio of the frequency is proportional to the crack depth; the slope is negative regardless of the active mode. This confirms that the propagation of a crack causes a decrease in the ratio of the frequency. In the case of the effect of crack position on the frequency, we observed a change of the upward frequency in the first mode. In other words, the more crack is remote from the recess the more important is the frequency, however in the case of other modes (2 and 3) an opposite change is noticed with slopes more or less important. This is due to the values of the amplitudes of modes 2 and 3.

Figure 1 Effect of the depth of the crack on the frequency for the three modes. 

Figure 2 Effect of the position of the crack on the frequency for the three modes. 

5 INTERACTION ANALYSE

The analyze of the interaction between the depth and crack location was done by the help of a representation of iso-courbes

Table 5 The results of combined effects (mode1). 

Mode 1 Crack depth (2mm) Crack depth(12mm)
Crack location (25mm) 0,98 0,701
Crack location (200mm) 0,99 0,94

Thefigure 3arepresents the effects of the two factors versus the ratio of frequency in mode 1. We observed that there is just one near the abscissa of embedding and at low depths frequency ratio reached the value of 0.98. This ratio decreases to great depths. For abscissa near the edge of the beam, the depth of the crack has not effect. In the second and third mode analysis of iso-curves (fig.3b,3c) shows that the ratio reaches its maximum when the two factors are the lowest values (tableaux 6and7)

Figure 3 Effect of the interaction of two factors on the frequencies for the three modes. 

Table 6 The results of combined effects (mode 2). 

Mode 2 Crack depth (2mm) Crack depth(12mm)
Crack location (25mm) 0,99 0,78
Crack location (200mm) 0,98 0,90

Table 7 The results of combined effects (mode 3). 

Mode 3 Crack depth (2mm) Crack depth(12mm)
Crack location (25mm) 1,00 0,93
Crack location (200mm) 0,99 0,79

6 ANALYZE WITH THREE FACTORS

The mathematic models already established, they allowed us to calculate the predicted frequencies and the residues for each mode (table 8,9and10).

Table 8 Residues for the mode 1. 

y1 = 0.92 - 0.090(X1) + 0.076(X2) + 0.081(X1X2) Ypre Yobse Residus = Yobse - Ypre
y 11 = 0.92 - 0.090(-1) + 0.076(-1) + 0.081(1) 1,02 0,9887 -0,0263
y 12 = 0.92 - 0.090(-1) + 0.076(-0.14) + 0.081(0.14) 1,01 0,9956 -0,0151
y 13 = 0.92 - 0.090(-1) + 0.076(1) + 0.081(-1) 1,01 0,9996 -0,0054
y 14 = 0.92 - 0.090(-0.2) + 0.076(1) + 0.081(-0.2) 0,88 0,91 0,0318
y 15 = 0.92 - 0.090(-0.2) + 0.076(-0.14) + 0.081(0.028) 0,93 0,9626 0,032972
y 16 = 0.92 - 0.090(-0.2) + 0.076(1) + 0.081(-0.2) 1,00 0,9966 -0,0012
y 17 = 0.92 - 0.090(1) + 0.076(-1) + 0.081(-1) 0,67 0,6465 -0,0265
y 18 = 0.92 - 0.090(1) + 0.076(-0.14) + 0.081(-0.14) 0,80 0,8081 0,0081
y 19 = 0.92 - 0.090(1) + 0.076(1) + 0.081(1) 0,99 0,9766 -0,0104

Table 9 Residues for the mode 2. 

y2 = 0.93 - 0.070(X1) + 0.026(X2) + 0.028(X1X2) Ypre Yobse Residus = Yobse - Ypre
y 21 = 0.93 - 0.070(-1) + 0.026(-1) + 0.028(1) 1,00 0,9951 -0,0029
y 22 = 0.93 - 0.070(-1) + 0.026(-0.14) + 0.028(0.14) 1,00 0,9975 -0,00222
y 23 = 0.93 - 0.070(-1) + 0.026(1) + 0.028(-1) 1,00 0,995 -0,007
y 24 = 0.93 - 0.070(-0.2) + 0.026(1) + 0.028(-0.2) 0,96 0,9641 -0,0003
y 25 = 0.93 - 0.070(-0.2) + 0.026(-0.14) + 0.028(0.028) 0,95 0,9663 0,019444
y 26 = 0.93 - 0.070(-0.2) + 0.026(1) + 0.028(-0.2) 0,92 0,9316 0,008
y 27 = 0.93 - 0.070(1) + 0.026(-1) + 0.028(-1) 0,91 0,8903 -0,0237
y 28 = 0.93 - 0.070(1) + 0.026(-0.14) + 0.028(-0.14) 0,87 0,9039 0,03634
y 29 = 0.93 - 0.070(1) + 0.026(1) + 0.028(1) 0,81 0,7836 -0,0224

Table 10 Residues for the mode 3. 

y3 = 0.93 - 0.070(X1) + 0.036(X2) + 0.037(X1X2) Ypre Yobse Residus = Yobse - Ypre
y 31 = 0.93 - 0.070(-1) + 0.036(-1) + 0.037(1) 1,00 0,9986 -0,0004
y 32 = 0.93 - 0.070(-1) + 0.036(-0.14) + 0.037(0.14) 1,00 0,9946 -0,00526
y 33 = 0.93 - 0.070(-1) + 0.036(1) + 0.037(-1) 1,00 0,9927 -0,0083
y 34 = 0.93 - 0.070(-0.2) + 0.036(1) + 0.037(-0.2) 0,97 0,9888 0,0162
y 35 = 0.93 - 0.070(-0.2) + 0.036(-0.14) + 0.037(0.028) 0,95 0,9569 0,008896
y 36 = 0.93 - 0.070(-0.2) + 0.036(1) + 0.037(-0.2) 0,92 0,9441 0,0287
y 37 = 0.93 - 0.070(1) + 0.036(-1) + 0.037(-1) 0,93 0,9574 0,0244
y 38 = 0.93 - 0.070(1) + 0.036(-0.14) + 0.037(-0.14) 0,87 0,8279 -0,04232
y 39 = 0.93 - 0.070(1) + 0.036(1) + 0.037(1) 0,79 0,7995 0,0125

With the help of the statistics calculations, we were able to define the effects the most significant of the factors and their gaps of confidence (trust), all with calculating the residues ei. The residues are the difference between the experimental value and predicted value by the mathematic model (figure 4) and they are linked by the linear regression.

Figure 4 Distribution of the experimental points from the mathematical model for each mode. 

7 REALIZATION OF THE TEST OF EFFECTS SIGNIFICANCE

The test used is the Student test << t>>. An effect is said significant (that is to say that the interaction or the variable associated with it has an influence on the response), if, for a given risk significantly different from 0. So we test the hypothesis

Hypothesis:

Hi = << ai ≠ 0>>

For this, we calculate

Student table is then used to v= n - p degrees of freedom (n is the number of experiments and p is the number of effects including the constant). The risk of a first species is selected (usually 1% or 5%) and is read in the table of the Student t value, using the part of the table related to a bilateral test.

Mode 1:

The variance calculated for each effect is then:

Therefore: ti x si = 0.037

  • The effect of crack depth : |-0.090| > 0.037 → significatif

  • The effect of crack location : |+0.076| > 0.037 → significatif

  • The effect of general average : |+0.920| > 0.037 → significatif

  • The effect of the interaction : |-0.810| < 0.037 → significatif

Confidence interval of model effects of the first mode

Table 11 Confidence interval of model effects of the first mode. 

ai - (t (α,ν)*si) ai ai + (t (α,ν)*si)
-0,127 Crack depth -0,09 -0,053
0,039 Crack location 0,076 0,113
0,044 Interaction 0,081 0,118

Figure 5 Confidence interval of model effects of the first mode. 

Mode 2:

s 2 =Σ

s 2 =Σ(-0.000008)2 + (-0.000004)2 + (0.00000009)2 + (0.0003)2 + (-0.00006)2 + (-0.0005)2 + (-0.0013)2 + (-0.0005)2

s 2 =(0.0028)

s2 = (0.00057)

The variance calculated for each effect is then:

s2 =(0.00057)

s2 = 0.000060

si = 0.0077

Therefore: ti x si = 0.030

Then:

  • The effect of crack depth: |+0.070| > 0.032 → significatif

  • The effect of crack location: |-0.026| < 0.032 → non significatif

  • The effect of general average: |+0.930| > 0.032 → significatif

  • The effect of the interaction: |-0.028| < 0.032 → non significatif

Confidence interval of model effects of the first mode

[ai- t (α, ν) si ; ai + t (α, ν) si]

Table 12 Confidence interval of model effects of the second mode. 

ai - (t (α,ν)*si) ai ai + (t (α,ν)*si)
-0,098 Crack depth -0,07 -0,038
-0,05824 Crack location -0,026 0,006
-0,06024 Interaction -0,028 0,004

Figure 6 confidence interval of model effects of the second mode. 

Mode 3:

s2 =Σ

s 2 =Σ(-0.0004)2 + (-0.00002)2 + (0.000068)2 + (0.0003)2 + (-0.00026)2 + (-0.00007)2 + (-0.00082)2 + (-0.00059)2 + (-0.0017)2 + (-0.00015)2

s 2 =(0.0038)

s 2 = (0.00076)

The variance calculated for each effect is then:

s 2 =(0.00076)

s 2 = 0.000084

si = 0.0092

Therefore: ti x si = 0.0370

Then:

  • The effect of crack depth: |-0.070| > 0.037 → significatif

  • The effect of crack location: |-0.036| < 0.037 → non significatif

  • The effect of general average: |+0.930| > 0,037 → significatif

  • The effect of the interaction: |-0.037| > 0.032 → significatif

Confidence interval of model effects of the first mode

[ai- t (α,ν) si ; ai + t (α,ν) si]

Table 13 Confidence interval of model effects of the third mode. 

ai - (t (α,ν)*si) ai ai + (t (α,ν)*si)
-0,107 Crack depth -0,07 -0,033
-0,073 Crack location -0,036 0,001
-0,074 Crack depth -0,037 0

Figure 7 Confidence interval of model effects of the third mode. 

8 CONCLUSIONS

Using the method of experimental design allows to extract the maximum information at a reasonable cost and minimum time. In our case, we have:

    1. ) Modeled the vibration behavior of a cracked beam console

    2. ) Plotted the iso-curves for different modes of vibration

These curves allow the choice according to the needs of the operating state (condition) of the beam.

It further notes that:

The depth of the crack remains significantly regardless of the current mode vibration, while this is not the case for the location of the crack which is significant only in the first mode.

The interaction between the two factors of an unexplained way varies from one mode to another; in the first mode of interaction is significant, it is only slightly in the third mode when it no longer is in the second mode

References

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Fisher,R. (1925). Statistical Methods for Research Workers. Oliver and Boyd;. PMid: 17246289 [ Links ]

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Received: June 11, 2014; Accepted: February 17, 2015

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