As described in the introduction, it is important, in determining the effect of damage, which is characterised by changes in natural frequencies and mode shape, to ensure frequency comparison and mode normalisation are performed using data corresponding to the same mode shape.
For example (Figure 4-4) for a βparallel crackβ (a crack arbitrarily located in the plate and parallel to one side), as the crack is moved parallel to one of the axes, and the length/depth are changed, the vibration characteristics (natural frequencies and mode shapes) will change, and the data obtained will contain many results obtained from different cases. A mode sequencing problem can occur when changing the location of the crack or increasing the severity of the crack, the natural frequencies of a higher-order mode swap over with those of a low-order mode. This plate problem is illustrated in Chapter 6 which shown in Figure 6-20 and Figure 6-21 including the characteristics including natural frequencies and mode shapes are shown for a plate having cracks located in different positions and having differing severity (length). The mode sequence is based on the intact plate. For Figure 4-4 (a) and (b), crack location x=0.01, the zerocontour line divides the plate into two parts in two directions. When moving the
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crack to location x=0.04 (Figure 4-4 (c) and (d)), zero-contour lines separate the plots into two parts in the opposite direction compared to x=0.01. However, at the same crack location x=0.04, when the crack length is increased from l=0.01 to l=0.08 (Figure 4-4 (e) and (f)), the zero-contour line changes to a third direction. If using the mode plot of the intact plate as a reference datum, for adjacent modes like mode 2 and mode 3, it is hard to recognise the sequence of modes from the natural frequency results or the contour plots of the mode shapes. However, in the damage detection problem, researchers need to use corresponding modes prior to and following the onset of damage to normalise data and find the crack location (section 7.3). It is important therefore to be able to sort the modes into the correct order. The development of a method to solve the sequencing problem is therefore essential for successful damage detection.
(a) (b) (c) (d)
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(e) (f)
Figure 4-4 Mode shape for an isotropic simply supported cracked plate with different crack location and different crack severity, crack.
(a) crack location x=0.01, crack length l=0.01, mode number r=2; (b) crack location x=0.01, crack length l=0.01, mode number r=3; (c) crack location x=0.04, crack length l=0.01, mode number r=2; (d) crack location x=0.04, crack length l=0.01, mode number r=3; (e) crack location x=0.04, crack length l=0.08, mode number r=2; (f) crack location x=0.04, crack length l=0.08, mode number r=3.
4.2.3.1 Automatic mode shape sign methodology
To ensure the correct mode is assigned to a particular natural frequency, a damaged plate model is first established. The W-W algorithm is then used to obtain the related natural frequencies, by default in ascending order. Then the obtained natural frequencies are substituted into the complete stiffness matrix and the modal displacements are found by solving the stiffness equations with a random force vector (Hopper and Williams 1977) on the right-hand side. A set of displacements is obtained with the default order. It is notable that, for a single cracked plate for example, when slightly moving the crack or increasing its severity (e.g. its depth or length), the mode shape will be similar for the cross-reference. The difference in the vertical displacements is then used as a reference variable to sort modes. Before sequencing the mode, a reference standard needs to be confirmed. In this example using a crack location as the standard, means that the next mode sequence of the longer crack length is based on the previous model with a smaller crack length at the same crack location.
As an illustration, for the crack starting point of Figure 4-4, the mode shape of the intact plate is used as the first reference standard. For case 1: π₯ = 0.01, π = 0.01 π = 1, the vertical displacements of example 1 are subtracted from those of the intact plate from the same default order of modes and add the absolute value of the differences. Extracting the minimum value from the above two data sets in terms of π1, repeat the same procedure
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for the next default order π = 2 with the intact plate mode π = 1 and the obtained minimum value π2. Comparing the minimum values from the two calculation, if π1 < π2,
the order of natural frequency and mode shape keeps the same sequence as default. Otherwise, the order is changed.
The actual value of displacement now settles down for case 1 and becomes the standard for case 2: π₯ = 0.01, π = 0.01, π = 2. Iteration 1 is stopped when the mode number meets the requirement. For the next iteration 2, case 3: π₯ = 0.01, π = 0.02 πππ π = 1, case 1 will be chosen as the first reference standard and the same procedure repeated as for the previous iteration until the crack length reaches its final value. Hence, a group of data where π₯ = 0.01, π = (0.01 π‘π ππππ’ππππ π£πππ’π), π = (1 π‘π ππππ’ππππ πππππ ) are obtained and are in the right order. Using the vertical displacement of π₯ = 0.01, π = 0.01 and π = (1 π‘π ππππ’ππππ πππππ ) as the first reference for case 4: π₯ = 0.02, π = 0.01 and π = (1 π‘π ππππ’ππππ πππππ ), the same calculation procedure is repeated as for iteration 1 and iteration 2. The iterations will stop when all parameters meet the desired values.
Table 4-1 Normalised nodal displacements for two hypothetical crack cases.
Crack cases Node Mode 1 Mode 2
Case 1 Node 1 1 0.7 Node 2 0.8 0.9 Node 3 0.6 1 Case 2 Node 1 0.65 1 Node 2 0.85 0.85 Node 3 1 0.65
This iteration procedure has been programmed into MATLAB and is attached in the Appendix B. Table 4-1 will use two hypothetical crack cases with assumed normalised displacement data to show the basic procedure described above which is called the mode shape sign method.
Case 1 and Case 2 in Table 4-1 represent a crack occurring in the same plate structure with different locations or lengths, based on the normalised displacement results obtained by the mode shape calculation (section 5.3) at different nodes. The sequence of modes defined in case 1 is used as the reference, the recorded sequence of modes for case 2 needs to be determined by the mentioned method. Using (πππ π2, ππππ1) with
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(πππ π1, ππππ1βππππ2) to determine the first mode sequence of case 2. The summation of the difference in nodal displacements for those cases are:
1. Compare (πππ π2, ππππ1) with (πππ π1, ππππ1)
|0.65 β 1| + |0.85 β 0.8| + |1 β 0.6| = 0.8 2. Compare (πππ π2, ππππ1) with (πππ π1, ππππ2)
|0.65 β 7| + |0.85 β 0.9| + |1 β 1| = 0.1
The summation for these two comparisons, comparing the obtained summation: 0.1 < 0.8. Hence, the assumed (πππ π2, ππππ1) should be recorded as (πππ π2, ππππ2). Again, for the assumed (πππ π2, ππππ2), a similar procedure is followed:
3. Compare (πππ π2, ππππ2) with (πππ π1, ππππ1)
|1 β 1| + |0.85 β 0.8| + |0.65 β 0.6| = 0.1 4. Compare (πππ π2, ππππ2) with (πππ π1, ππππ2)
|1 β 7| + |0.85 β 0.9| + |0.65 β 1| = 0.7
According to the above procedure, (πππ π2, ππππ2) matches (πππ π 1, ππππ1), which should be corrected to (πππ π2, ππππ1). Through the above iterations, all modes will comply with the right order, and corresponding vibration characteristics will be recorded correctly. It should be noted that a few modes still need to have their sequence adjusted manually. For example, when a crack runs from (π₯, π¦) = (πΌπ, 0) π‘π (πΌπ, π½π) where Ξ± is a location parameter in the range 0.1 β€ πΌ β€ 0.9 and Ξ² is a length parameter in the range 0.2 β€ π½ β€ 0.8. For the cracked plated cases shown in Figure 4-5, Ξ² = 0.4 and πΌ is a location parameter in the range 0.1 β€ πΌ β€ 0.5. The contour plots show a singularity when cracks occur at the middle of a square plate compared with other cases due to the way the crack is simulated, the geometry conditions and the crack location.
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(m, n) = (1, 2): = 0.2 = 0.3 = 0.4 = 0.5
Figure 4-5 Variation of mode shapes with crack location πΌ, for π½ = 0.4 (Luo et al. 2019), crack.
4.3 Final Remarks
In this chapter, the W-W algorithm provides accurate solutions of the transcendental eigenvalue problem; an automatic mode shape sign method sorts modes to the right order; the bandwidth method significantly improves the computation behaviour in the iteration process. Besides those methodologies, other potential methods also could increase the speed of data processing, for example mentioned in section 4.2.1, parallel computation in MATLAB. In this study, MATLAB R2017A (MathWorks 2019) is used to program the hybrid cracked plate model. In the forward problem, a group of single cracked plate cases need to be analysed, and each case is independent. A parfor-loop in MATLAB processes a series of statements in the iteration calculation in parallel. A parfor-loop can provide significantly better behaviour than a for-loop because several MATLAB workers can calculate simultaneously on the same iteration. MATLAB works independently to compute iterations with no particular order, and if the number of workers is equal to the number of loop iterations, each worker can deal with one loop iteration simultaneously. For eight workers, the maximum performance could be 8 times faster than its analogous for-loop in MATLAB. For further efficiency improvements, a reasonable eigenvalue accuracy should be considered in the study which is generally no less than 0.0001.
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Free Vibration Modelling of Plates
Chapter 5--- Free Vibration Modelling of Plates
5.1 Introduction
The presence of cracks or delaminations in a plate element will change the nature of its structural mechanics and vibration characteristics as well as those of the structure in which it is assembled (Teughels et al. 2002; Escobar et al. 2005; Fang et al. 2005; Caddemi and Greco 2006; Danai et al. 2012 and Suliman 2018). A database of the effects of damage on natural frequencies and mode shapes needs to be prepared in order to solve the inverse problem of the damage detection based on the change of vibration characteristics. To this end, the forward problem of calculating the natural frequencies and mode shapes of an isotropic simply supported square plate with pre-existing cracks or delaminations is studied in this chapter.
As introduced in Chapter 2, a crack can be modelled as a rotational spring and is assumed to be always open. Any loss of mass at the crack is ignored. Hence, the problem is simplified to a linear problem. In this study, the crack will be arbitrarily located in the plate and have random severity; with the shear and axial stiffness remaining intact and the effects of structural damping ignored. The crack will be utilized in a modified hybrid damaged model (VFM). For the theoretical derivation of the stiffness matrix, the FEM part is based on Przemieniecki (1985), while the ESM part is from Wittrick and Williams (1974). The W-W algorithm (Wittrick and Williams 1971) is applied to calculate the natural frequencies numerically using a MATLAB code (Appendix C).
An isotropic simply supported plate example is used to study the effect of changing the location and severity of the crack on vibration parameters like the natural frequencies and mode shapes. All the displacements at the plate boundaries are assumed to be zero. Only the first six out-of-plane natural frequencies are needed in the study. These are decoupled from the in-plane natural frequencies which are much higher. Thus, it is possible to consider only the out-of-plane behaviour and ignore the dynamic effects.
Results obtained using different techniques (VICONOPT, FEM (MATLAB, ABAQUS) and VFM) are compared with previous studies by Stahl and Keer (1972), Liew et al. (1994), and Huang and Leissa (2009). As a result, an insight is given to the inverse problem of damage identification.
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