Change in mode shape nodes of multiple cracked bar

Volume 35 Number 4 4 Vietnam Journal of Mechanics, VAST, Vol. 35, No. 4 (2013), pp. 299 – 311 CHANGE IN MODE SHAPE NODES OF MULTIPLE CRACKED BAR: II. THE NUMERICAL ANALYSIS N. T. Khiem∗, L. K. Toan, N. T. L. Khue Institute of Mechanics, VAST, 18 Hoang Quoc Viet, Hanoi, Vietnam ∗E-mail: ntkhiem@imech.ac.vn Abstract. In this paper it is numerically analyzed the change in position of mode shape nodes induced by multiple cracks in bar that has been theoretically investigated in pre- vious paper [1] of the authors. The focus is to analyze thoroughly dislocation of node located intermediately between two cracks in dependence upon the crack parameters without assumption on the smallness of crack. Keywords : Multiple cracked bar, crack detection, mode shape nodes, vibration method, modal analysis. 1. INTRODUCTION This paper is devoted to continue the study of change in node position induced by multiple cracks in bar by numerical examples. First, some theoretical aspects that were detailed in the previous paper are summarized to use for numerical computation. The emphasis is on the numerical investigating the case of double cracks and change in position of the node located between cracks. Two cases of boundary conditions: free-free and fixed-free ends are considered. 2. SUMMARIZED THEORETICAL ASPECTS For an uniform bar with Young’s modulus E, density ρ, cross section area A and length L that is assumed to be cracked at the locations e1, ..., en, free longitudinal vibration is described by the equation Φ′′(x) + λ2Φ(x) = 0, x ∈ (0, 1), λ = ωL/c0, c0 = √ E/ρ (1) with given boundary conditions at both ends x = 0, x = 1 and the compatibility conditions at the crack positions Φ(ej + 0) = Φ(ej − 0) + γjΦ ′(ej), γj = EA/LKj, j = 1, ..., n. (2) It was shown in previous paper [1] that general solution of Eqs. (1), (2) all over the bar length can be represented as 300 N. T. Khiem, L. K. Toan, N. T. L. Khue Φ(x) = Φ0(x) + n∑ j=1 µjK(x− ej, λ), (3) where Φ0(x) is general solution of Eq. (1) satisfying the left end condition and µj = γj[Φ ′ 0 (ej) + n∑ k=1 µkK ′(ej − ek, λ)], j = 1, ..., n; (4) K(x, λ) = { 0, x ≤ 0 cos λx, x  0 ; K ′(x, λ) = { 0, x ≤ 0 −λ sinλx, x  0 (5) Obviously, the solution Φ0(x) can be represented as Φ0(x) = CL0(x, λ), where C is a constant and L0(x, λ) called shape function is the solution of Eq. (1) satisfying the left boundary condition. For instance, in the case of general boundary condition α0Φ(0)+ β0Φ ′(0) = 0 the shape function is L0(λx) = α0 sinλx − β0λ cosλx. If the bar is fixed at the left end, x = 0, L0(x, λ) = sinλx and L0(x, λ) = cosλx for the free one. The constant C would be determined from the boundary condition at the right end x = 1 that can be expressed in the form α1Φ(1) + β1Φ ′(1) = 0. Substituting Eq. (3) with function Φ0(x) = CL0(x, λ) into the boundary condition at the right end x = 1 leads to C[α1L0(1, λ) + β1L ′ 0(1, λ)] + n∑ j=1 µj [α1K(1− ej , λ) + β1K ′(1− ej , λ)] = 0 from that one gets C = − [1/D0(λ)] n∑ j=1 µjK1(ej , λ); (6) D0(λ) = α1L0(1, λ) + β1L ′ 0(1, λ);K1(ej, λ) = α1K(1− ej, λ) + β1K ′(1− ej , λ) . Using Eq. (6) the Eqs. (3) and (4) can be rewritten as Φ(x) = n∑ j=1 µjα(x, ej, λ) , (7) α(x, e, λ) = [D0(λ)K(x− e, λ) – K1(e, λ)L0(x, λ)]/D0(λ) (8) and [Γ(γ)A(e, λ)−D0(λ)I]µ = 0, (9) where µ = (µ1, ..., µn) T ,γ = (γ1, ..., γn) T , e = (e1, ..., en) T ,Γ = diag{γ1, ..., γn}; A = [ajk = a(ej, ek, λ); j, k = 1, ..., n], (10) a(ej , ek, λ) = D0(λ)K ′(ej − ek, λ)− L ′ 0(ej, λ)K1(ek, λ); j, k = 1, ..., n. For existence of non-trivial vector µ as a solution of Eq. (10), it must be satisfied the condition D(λ, e,γ) ≡ det[Γ(γ)A(e, λ)−D0(λ)I] = 0. (11) Change in mode shape nodes of multiple cracked bar: II. The numerical analysis 301 This is a new form of the frequency equation in axial vibration for multiple cracked bar [2]. Obviously, the latter equation is determined by the determinant of order identical to the number of cracks. Solving Eq. (11) with respect to λ results in so-called eigenvalues λp, p = 1, 2, ... that relate to the natural frequencies ωp, p = 1, 2, ... by the relationship (λp/L) √ E/ρ = ωp. Each eigenvalue λp associates with a non-trivial solution of Eq. (13) that is so-called eigenvector µp of p-th mode. Thus the couple (λp, µp) enables to express the mode shape in the explicit form Φp(x) = Cp n∑ j=1 µpjα(x, ej, λp) . (12) The constant Cp is introduced in the latter equation because the eigenvector µp determined as a solution of Eq. (13) should contain an arbitrary constant that can be determined by choosing a normality condition, for example, max{Φp(x)} = 1. For a bar with a double crack the frequency Eq. (11) leads to D0(λ) + [γ1L ′ 0 (e1, λ)K1(e1, λ) + γ2L ′ 0 (e2, λ)K1(e2, λ)]+ +γ1γ2L ′ 0 (e1, λ)K1(e2, λ)K ′(e2 − e1, λ)] = 0. (13) The expression (12) for mode shape now can be written as φ(x, λ) = C[µ1α(x, e1, λ) + µ2α(x, e2, λ)] = C{L0(x, λ) + γ1L ′ 0 (e1, λ)K(x− e1)+ +γ2[L ′ 0 (e2, λ) + γ1L ′ 0 (e1, λ)K ′(e2 − e1, λ)]K(x− e2)}, that leads the mode shape node of the double cracked bar to be determined as root of the equation L0(x, λ) + γ1L ′ 0(e1, λ)K(x− e1) + γ2[L ′ 0(e2, λ) + γ1L ′ 0(e1, λ)K ′(e2 − e1, λ)]K(x− e2) = 0, or L0(x, λ) = 0, 0 ≺ x ≤ e1; (14) L1(x, λ) ≡ L0(x, λ) + γ1L ′ 0(e1, λ) cosλ(x− e1) = 0, e1 ≺ x ≤ e2; (15) L2(x, λ) ≡ L1(x, λ) + γ2[L ′ 0 (e2, λ) + γ1L ′ 0 (e1, λ)K ′(e2 − e1, λ)] cosλ(x− e2) = 0, e2 ≺ x ≺ 1. (16) In the case of free-free bar, when L0(x, λ) = cosλx, the frequency equation is sinλ− λ[γ1 sinλe1 sinλ(1− e1) + γ2 sinλe2 sinλ(1− e2)]+ +λ2γ1γ2 sinλe1 sinλ(e2 − e1) sinλ(1− e2) = 0. (17) The system of Eqs. (14), (15), (16) is reduced to cos λkx = 0, 0 ≺ x ≤ e1; (18) cosλkx− γ1λk sinλke1 cosλk(x− e1) = 0, e1 ≺ x ≤ e2; (19) cosλkx− γ1λk sinλke1 cos λ(x− e1)− γ2λk[sinλke2− −γ1λk sinλke1 sinλk(e2 − e1)] cosλ(x− e2) = 0, e2 ≺ x ≺ 1. (20) Analytical solution of Eqs. (18), (19) has been obtained in the previous paper, it remains Eq. (19) rewritten as tanλ0kθk = λkγ1 sin 2 λke1 1− λkγ1 sinλke1 cosλke1 , θk ∈ (∆1,∆2), (21) 302 N. T. Khiem, L. K. Toan, N. T. L. Khue where x = λ0k(x 0 kn − θk)/λk and ∆1 = x 0 kn − λke2/λ 0 k; ∆2 = x 0 kn − λke1/λ 0 k that has to be numerically solved. Owning solution θˆk of Eq. (21) the mode shape node determined can be calculated as xkn = λ 0 k(x 0 kn − θˆk)/λk, consequently, dislocation of the node from the generic one is δxkn ≡ xkn − x 0 kn = (δλk/λk)x 0 kn − (λ 0 k/λk)θˆk. (22) where δλk = λ 0 k − λk. If the bar is fixed at both the ends, L0(x, λ) = sinλx, the frequency and node are sought from the equations sinλ+ λ[γ1 cosλe1 cosλ(1− e1) + γ2 cosλe2 cos λ(1− e2)]− −λ2γ1γ2 cos λe1 sinλ(e2 − e1) cosλ(1− e2) = 0 (23) and one obtains equation similarly to the Eq. (21) as tanλ0kθk = λkγ1 cos 2 λke1 1 + λkγ1 sinλke1 cosλke1 , θk ∈ (∆1,∆2) (24) with ∆1 = x 0 kn − λke2/λ 0 k; ∆2 = x 0 kn − λke1/λ 0 k. For the bar with fixed-free ends, the characteristic equation is cos λ− λ[γ1 cosλe1 sinλ(1− e1) + γ2 cosλe2 sinλ(1− e2)]+ +λ2γ1γ2 cosλe1 sinλ(e2 − e1) sinλ(1− e2) = 0. (25) Though the characteristic Eq. (25) is different from the ones for the previous cases, the equations for the mode shape node in this case of boundary condition give solutions identical to those of the fixed-fixed bar. 3. NUMERICAL RESULTS To investigate effect of cracks on the shift of the intermediate node the dislocation of node is computed based on the numerical solution of Eqs. (21) and (24) in dependence on different combinations of position and depth of two cracks. The Eq. (21) represents the bar with free ends and Eq. (24) - the cantilevered bar. For simplicity, the crack located on the left (right) of a node is called left (right) crack and only the modes that have single and double node are considered. The positive displacement of a node implies that it moves to the right and negative displacement - to the left. First scenario is examined for cracks of equal depth and varying positions in both sides of generic node with purpose to highlight influence of crack position on the node dislocation. In the second scenario the contours of zero displacement of nodes are plotted in the plane of position and depth of left crack with different fixed parameters of right crack. This task is accomplished to reveal influence of cracks depth on the direction by which a node move due to the cracks. Numerical results are shown in Figs. 1-12 that allow for making the following discussion. For the bar with free ends, single node at 0.5 moves toward the crack that is more close to it if the cracks have the same depth. It would move toward farther crack only if depth of the crack exceeds that of other one. Absolute displacement of the single node increases monotonically with reduction of distance from node to closer crack. The node Change in mode shape nodes of multiple cracked bar: II. The numerical analysis 303 0 .5 0 .5 5 0 .6 0 .6 5 0 .7 0 .75 0 .8 0 .8 5 0 .9 0 .9 5 1 0 0 ,0 5 0 ,1 0 ,1 5 0 ,2 0 ,2 5 0 ,3 0 ,3 5 0 ,4 0 ,4 5 0 e1 e2 1 ,6 1 ,4 1 ,2 1 ,0 0 ,8 0 ,6 0 ,4 0 ,2 0 ,1 0 -0 ,1 -0 ,2 -0 ,4 -0 ,6 -0 ,8 -1 ,0 -1 ,2 -1 ,4 -1 ,6 Fig. 1. The free-free bar: contours for dislocation of single node (first mode) versus positions of cracks of equal depth a/h = 30% 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 + 20% 30% 30% 20% 10% 10% 5% 5% 10% 40% 30% 20% a1/h e1 40% 40% 5% Fig. 2. The free-free bar: contours for zero displacement of single node (first mode) versus position and depth of the left crack in different depth from 5% – 40% of the right crack at positions 0.55 (solid); 0.75(dash dot) and 0.9 (dot) does not change its position for the cracks of equal depth located at the same distance from both sides of the node, see Figs. 1, 2. 304 N. T. Khiem, L. K. Toan, N. T. L. Khue 0.3 0 .4 0 .5 0.6 0.7 0 .8 0 .9 1 0 0.05 0.1 0.15 0.2 0.25 3 2,5 2 1,5 1 0,5 -1 -2 -3 -4 -5 -6 2 ,5 2 1.5 1 0,5 0 -7 -8 -9 -10 -7 -8 -9 -10 3 e2 e1 0 Fig. 3. The free-free bar: contours for dislocation of first node (second mode) versus positions of cracks of equal depth a/h = 30% 0 0.05 0.1 0.15 0.2 0.25 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 30% 20% 20% 10% 30% 40% 40% 30% 20% 10% 5% 40% 10% 5% 5% a1/h e1 Fig. 4. The free-free bar: contours for zero displacement of first node (second mode) versus position and depth of the left crack in different depth 5% - 40% of the right crack at positions 0.4 (solid); 0.7 (dash dot) and 0.95 (dot) Change in mode shape nodes of multiple cracked bar: II. The numerical analysis 305 It is observed more complicated behavior of nodes of second mode that has two nodes, the first is at 0.25 and the second one at 0.75, and one crack-unaffected point (ICP) at the middle of bar (0.5), see Tab. 1 [1]. Namely, it is apparent from Fig. 3 that in the case when the right crack is positioned in the segments from the node (0.25) to the ICP (0.5) and from second node (0.75) to the right end of bar the first node behaviors almost similarly to that is discussed above for the single node. The difference is only that zero displacement contour shifts to the left end of bar, perhaps, because the left end is a node and the middle is a ICP of the bar (no symmetry). For the right crack located in the segment from the ICP to the second node behavior of the first node is symmetrical about the axes e2 = 0.5 and e2 = 0.75 to its behavior in mentioned above two segments. This allows making a conclusion that for a given displacement of the first node it can be found 0.75 0.8 0.85 0.9 0.95 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 10 10 9 8 7 9 8 7 6 5 4 3 2 1 -0.5 -1.0 -2.0 -2.5 0 -0.5 -1.0 -1.5 -1.5 -2.0 -2.5 -3 -3 0 0 e2 e1 Fig. 5. The free-free bar: contours for dislocation of second node versus positions of cracks of equal depth a/h = 30% one position preceding the node for the left crack and three positions belonging to three bar segment (0.25, 0.5); (0.5, 0.75) and (0.75, 1) respectively for the right crack. Otherwise speaking, the right crack at three positions symmetrical about the ICP, 0.5, and second node, 0.75, causes the same displacement of the first node. Moreover, the first node moves toward the left crack located from point 0.1 to the node (0.25) for arbitrary positions of the right crack. It may move to the right only for the position of left crack less than 0.1. The contour of zero displacement for the first node exists and is almost unchanged regarding to the crack depth. The negative and positive displacement domains for the first node versus position and depth of the left crack in different magnitudes of the right crack are shown in Fig. 4 that is similar to Fig. 2 for the single node. A little difference observed 306 N. T. Khiem, L. K. Toan, N. T. L. Khue between the Figures shows again the fact that a node can move toward the closer crack of less depth. The contours for dislocation of second node for equal crack depth plotted in the plane (e1, e2) are shown in Fig. 5. It is easily to verify that Fig. 5 can be exactly obtained from Fig. 3 for the first node by the replacement of variables e1 = 1 − e ′ 2 ; e2 = 1 − e ′ 1 . Therefore, a discussion similar to that has been made above for the first node can be simply conducted with taking account of the change in crack positions. However, zero displacement contours for the second node shown in Fig. 6 are considerably different from those for the first node. Namely, every contour for zero displacement is undetermined at the ICP, 0.5, and has two minimums at the two nodes (0.25 and 0.75). The uncertainty indicates that the left crack appeared at the ICP makes no effect on the second node. On the other hand, the minimum points of the contours say about positions the left crack at which makes largest effect on the second node. This exhibits the importance of positions for generic node and ICP in the representation of node shift caused by multiple cracks. 0 0,1 0,2 0,25 0,3 0,4 0,5 0,6 0,7 0,75 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 20% 10% 5% 5% 20% 5% 20% 10% 5% 40% 30% 10% 10% 20% 40% 30% 30% 40% 30% 20% 10% 5% 30% 40% 5% 10% 20% 30%a1/h e1 Fig. 6. The free-free bar: contours of zero displacement of second node (second mode) versus position and depth of the left crack in different depth from 5% – 40% of the right crack at positions 0.8 (solid); 0.85 (dash dot) and 0.9 (dot) In the case of cantilevered bar, single node exists at the position 2/3 for second mode that has also a single ICP at the position 1/3. The third mode has two generic nodes 0.4 (the first) and 0.8 (the second) and two ICPs at 0.2 and 0.6 respectively. The contours for various dislocation of the single node given in Fig. 7 show that all the contours have the same axis of symmetry e1 = 1/3 crossing the single ICP, the middle of the segment from left end to the node (0, 2/3). The symmetry points out that it can be found one position Change in mode shape nodes of multiple cracked bar: II. The numerical analysis 307 0,7 0,75 0,8 0,85 0,9 0,95 1 0 0,1 0,2 0,3 1/3 0,4 0,5 0,6 2/3 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0 - 4.0 - 2.5 - 3.0 - 3.5 - 4.0 - 0.5 - 1.0 - 1.5 - 2.0 - 0.5 e2 e1 - 3.5 - 3.0 - 2.5 - 2.0 - 1.5 - 1.0 - 0.5 Fig. 7. The cantilever bar: contours for dislocation of single node (second mode), 2/3, versus positions of cracks of equal depth a/h = 30% 0 0,1 0,2 0,3 1/3 0,4 0,5 0,6 2/3 0,05 0,1 0,15 0,2 0,25 0,3 0,35 0,4 0,45 0,5 40% 30% 5% 5% 10% 20% 5% 10% 20% 30% 40% 30% 20% 10% 5% 5% 10% 20% 30% 40% 20% 10% 40% 30% 20% 10% 5% 30%a1/h e1 Fig. 8. The cantilever bar: contours for zero displacement of single node (second mode) versus position and depth of the left crack in different depth from 5 - 40% of the right crack at positions 0.7(dash dot); 0.8 (solid) and 0.9 (dot) 308 N. T. Khiem, L. K. Toan, N. T. L. Khue of the right crack and two symmetrical positions of the left crack for a given displacement of the node. Furthermore, the node cannot moves to the left if position of the right crack is less than 0.83 for wherever occurred the left crack of the same depth. The positive displacement of single node increases monotonically as the right crack approaching to the node. The negative displacement domain of single node consists of two separated small parts at the right corners enclosed by contour of zero displacement. The symmetry is observed more apparently also in Fig. 8 where there are shown various contours for zero displacement of the node plotted versus the left crack parameters for the right crack with depth from 5% to 40% at three positions 0.7, 0.8 and 0.9. It can be noted from the latter figure that effect of the left crack on the dislocation of the single node becomes more significant when the crack is more close to the left end and the node. 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,05 0,1 0,15 0,2 0,25 0,3 0,35 35 4.5 4 3.5 3 2.5 2 1 0 0 - 3 - 4 - 5 - 6 1 1.5 1.5 2 2.5 4 3.5 4.5 5 xe-3 0 e1 e2 - 1 - 1 - 2 - 1 - 2 - 1 - 2 - 3 - 4 - 5 - 6 - 7 - 8 - 6 - 7 - 8 - 5 - 4 - 3 - 8 - 6 - 7 - 5 - 4 - 3 - 2 - 8 - 7 0 0 0 0 Fig. 9. The cantilever bar: contours for dislocation of first node 0.4 (third mode) versus positions of cracks with equal depth a/h = 30% Behavior of double node for the third mode of cantilever bar is presented in Figs. 9-12. The contours for dislocation of the first node, 0.4, given in Figs. 9, 10 show that all the contours have one axis of symmetry, e1 = 0.2 crossing the first ICP. So that one can find two positions for the left crack and one position for the right crack that make the same shift of the first node. The node cannot move to the right if the left crack appeared at positions less than 0.075 and more than 0.325 for the right crack appeared wherever in the segment (0.4, 1.0). This implies also that the first node may move to the right if the left crack appears in the segment (0.075, 0.325). The behavior of the first node for the right crack appeared in the segments (0.4, 0.6) and (0.8, 1.0) is fully similar and dislocation of the node for the right crack located between 0.6 and 0.8 is symmetrical to that mentioned Change in mode shape nodes of multiple cracked bar: II. The numerical analysis 309 0 0.05 0.1 0.15 0.2 0 .25 0.3 0.35 0.4 0 0.05 0.1 0 .15 0.2 0 .25 0.3 0 .35 0.4 0 .45 0.5 40% 30% 20% 10% 5% e1 a1/h Fig. 10. The cantilever bar: contours for zero displacement of first node 0.4 (third mode) versus position and depth of the left crack in different depth from 5% – 40% of for the right crack at positions 0.5 (solid); 0.7 (dash dot) and 0.9 (dot) 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 10 0 10 9.0 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 0 -1.5-1.0 xe-3 0 -0.5 -1.0 -1.5 -2.0 -2.5 -0.5 -1.0 -1.5 -2.0 -2.5 0 0 e1 e2 -2.5-2.0-0.5 Fig. 11. The cantilever bar: contours for dislocation of second node 0.8 (third mode) versus positions of cracks of equal depth a/h = 30% 310 N. T. Khiem, L. K. Toan, N. T. L. Khue previously. Consequently, for a given position of the left crack and displacement of the first node it can be found three positions for the right crack. The node is really unmoved if both the cracks coincide with the ICPs, 0.2 and 0.6. The typical differences observed in the figures are intersection of zero displacement contour and existence of closed contours of positive displacement and effect of the left crack becomes more significant as it approaching to the fixed end of bar. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 40% 30% 20% 10%10% 30% 40% 30% 20% 10% 5% 5% 10% 20% 5% 40% 5% 20% 20% 10% 40% 20% 5% 40% 30% 30% 20% 10% 5% 5% 40% 30% 10% 5% 30% 10% 20% 30% 30% 20% 10% 5% a1/h e1 Fig. 12. The Cantilever bar: contours for zero displacement of second node 0.8 (third mode) versus position and depth of the left crack in different depth from 5% - 40% of the right crack at positions 0.85 (solid); 0.9 (dash dot) and 0.95 (dot) The contours for dislocation of second node (0.8) shown in Figs. 11, 12 remain totally symmetrical about the axis e1 = 0.4 and two the symmetrical portions in turn have also local axes of symmetry e1 = 0.2, e1 = 0.6 crossing the ICPs of this mode. These symmetries lead to the fact that there are four positions of the left crack associated with one value of node displacement and a position of the right crack. Moreover, the negative displacement domain exists only for the right crack located at the positions exceeding 0.935 and consists of three separated parts connected each to other by the ICPs 0.2 and 0.6. In this case, the most affected positions of the left crack for the second are fixed end and two generic nodes of the bar, see Fig. 12. Change in mode shape nodes of multiple cracked bar: II. The numerical analysis 311 4. CONCLUSIONS The obtained numerical results show that the intermediate single node is generally arranged to move toward the crack that is closer to it. It may move toward farther crack if depth of the crack is larger than that of the rest. In the case of multiple nodes, the effect of a crack on dislocation of a node becomes more significant if the crack approaches to the other node and the effect is eliminated when the crack occurred at an inactive crack point. Multiple cracks at different positions may produce the same change in a node position, but the different positions of multiple cracks are related each to other by symmetry about certain axes. There exists always an arrangement of two cracks that can maintain an intermediate node unmovable. ACKNOWLEDGEMENT This work has been completed with the financial supports from NAFOSTED of Vietnam under Grant No. 107.04.12.09, for which the authors are sincerely thankful. REFERENCES [1] N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: I. The theoretical study, Vietnam Journal of Mechanics, 35(3), (2013), pp. 175-188. [2] Ruotolo R., Surace C., Natural frequencies of a bar with multiple cracks, Journal of Sound and Vibration, 272, (2004), pp. 301–316. Received October 16, 2012 VIETNAM ACADEMY OF SCIENCE AND TECHNOLOGY VIETNAM JOURNAL OF MECHANICS VOLUME 35, N. 4, 2013 CONTENTS Pages 1. Nguyen Manh Cuong, Tran Ich Thinh, Ta Thi Hien, Dinh Gia Ninh, Free vibration of thick composite plates on non-homogeneous elastic foundations by dynamic stiffness method. 257 2. Vu Lam Dong, Pham Duc Chinh, Construction of bounds on the effective shear modulus of isotropic multicomponent materials. 275 3. Dao Van Dung, Nguyen Thi Nga, Nonlinear buckling and post-buckling of eccentrically stiffened functionally graded cylindrical shells surrounded by an elastic medium based on the first order shear deformation theory. 285 4. N. T. Khiem, L. K. Toan, N. T. L. Khue, Change in mode shape nodes of multiple cracked bar: II. The numerical analysis. 299 5. Tran Van Lien, Trinh Anh Hao, Determination of mode shapes of a multiple cracked beam element and its application for free vibration analysis of a multi- span continuous beam. 313 6. Phan Anh Tuan, Pham Thi Thanh Huong, Vu Duy Quang, A method of skin frictional resistant reduction by creating small bubbles at bottom of ships. 325 7. Nguyen Thoi Trung, Bui Xuan Thang, Ho Huu Vinh, Lam Phat Thuan, Ngo Thanh Phong, An effective algorithm for reliability-based optimization of stiffened Mindlin plate. 335