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}).
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 |
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 |
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 a_{i} . 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 y_{i} 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
Coefficients | Mode 1 | Mode 2 | Mode 3 |
---|---|---|---|
a_{0} | 0,92 | 0,93 | 0,93 |
a_{1} | -0,090 | -0,070 | -0,070 |
a_{2} | 0,076 | -0,026 | -0,036 |
I_{12} | 0,081 | -0,028 | -0,037 |
The models are thus written in the following forms
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.
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
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 6and^{7})
Mode 2 | Crack depth (2mm) | Crack depth(12mm) |
---|---|---|
Crack location (25mm) | 0,99 | 0,78 |
Crack location (200mm) | 0,98 | 0,90 |
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,^{9}and^{10}).
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 |
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 |
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.
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:
H_{i} = << a_{i} ≠ 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: t_{i} x s_{i} = 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
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 |
Mode 2:
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}
s2 = (0.00057)
The variance calculated for each effect is then:
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]
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 |
Mode 3:
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} = (0.00076)
The variance calculated for each effect is then:
s ^{2} = 0.000084
s_{i} = 0.0092
Therefore: t_{i} x s_{i} = 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
[a_{i}- t (α,ν) s_{i} ; a_{i} + t (α,ν) s_{i}]
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 |
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