- Email: [email protected]
Vibration analysis of a cracked rotating tapered beam using the p-version finite element method
Finite Elements in Analysis and Design 47 (2011) 825–834
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel
Vibration analysis of a cracked rotating tapered beam using the p-version finite element method Yue Cheng a,b,n, Zhigang Yu b, Xun Wu b, Yuhua Yuan b a b
School of Aerospace Science and Engineering, Beijing Institute of Technology, Beijing 100081, China Qinghe Dalou Zi, Haidian District, Beijing 100085, China
a r t i c l e i n f o
abstract
Article history: Received 18 April 2010 Received in revised form 11 January 2011 Accepted 25 February 2011 Available online 17 March 2011
Vibration characteristics of cracked rotating tapered beam are investigated by the p-version finite element method. In the present formulation, the shape functions enriched with the shifted Legendre orthogonal polynomials are employed to represent the transverse displacement field within the rotating tapered beam element. The crack element stiffness matrix and the p-version finite element model of the structural system are obtained by using fracture mechanics and the Lagrange equation, respectively. The common integral terms appeared in the rotating tapered beam element matrices are derived in analytical forms so as to speed up the generation of these matrices during the implementation of the present method. Comparisons have been made for several configurations between results obtained by the present method with those available in literature, and the proposed method shows excellent accuracy and convergence. The effects of crack location, crack size, rotating speed and hub radius on vibration characteristics of a cracked rotating tapered beam are investigated. The mode shapes of the cracked rotating tapered beam are also obtained and analyzed with the spatial wavelet transform approach to detect the slight perturbation at the crack position. The present method is of general applicability and can be applied to investigate vibration of arbitrarily continuous non-uniform rotating beams. & 2011 Elsevier B.V. All rights reserved.
Keywords: Rotating tapered beam Crack Vibration characteristics Finite element method p-version
1. Introduction The design and analysis of engineering structures such as helicopter rotor blades, wind turbine blades and airplane propeller blades, have attracted considerable interest in the vibration behavior of rotating beams, which greatly advance elimination of undesirable resonance phenomena and determination of suitable vibration control strategies [1–3]. Compared with enormous research on vibration analysis of intact rotating beams, less attention has been focused on vibration characteristics of cracked rotating beams. However, cracks frequently occur in practical rotating structures for the sake of cyclic fatigue or manufacturing defects. Dynamic modeling and analysis of cracked rotating beams are very important to the mechanical design as well as vibration-based inspections, which have gained popularity recently [4]. Intact rotating beams are usually simplified to one-dimensional Euler–Bernoulli or Timoshenko beams subjected to centrifugal tension force. The governing differential equations of
n Corresponding author at: School of Aerospace Science and Engineering, Beijing Institute of Technology, Beijing 100081, China. Tel.: þ86 10 68019709. E-mail address: [email protected] (Y. Cheng).
0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2011.02.013
motion for rotating tapered beams contain variable coefficients introduced due to the variation of the centrifugal force and the geometry along the beam length. The closed-form solutions for such equations therefore are difficult to obtain. As a result, different approximate methods have been suggested to analyze the vibration problems of rotating beams. Hodegs and Rutkowski [5] presented a variable-order finite element method to capture the free vibration properties of rotating beams. Naguleswaran [6] studied lateral vibration of a rotating uniform Euler–Bernoulli beam based on the general solution of the mode shape equation as the superposition of four linearly independent functions. Rao and Gupta [7] applied the finite element to obtain natural frequencies and mode shapes of a rotating twisted and tapered beam. Gunda et al. [8] proposed a superelement, whose shape functions were composed by polynomials and trigonometric functions for dynamic analysis of rotating tapered beams. Gunda and Ganguli [9] then assumed the transverse displacement to vary as a fourth order function and obtained new shape functions that would satisfy the static part of the homogeneous governing differential equation. Later Gunda et al. [10,11] proposed a new finite element for free vibration analysis of high-speed rotating beams using shape functions that are a linear combination of the solution of the governing static differential equation of a stiff string and a cubic polynomial. Bazoune
826
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
et al. [12] investigated the modal characteristics of a rotating tapered Timoshenko beam using the finite element analysis. Bazoune [13] discussed the relationship between the in-plane and out-of-plane frequencies in terms of Southwell coefficient. Bazoune [14] studied free vibration of a rotating tapered beam with the finite element method in which the mass, elastic and centrifugal stiffness matrices were explicitly expressed in terms of the taper ratios. Yoo and Shin [15] investigated the modal characteristics of rotating cantilever beams in the flapwise and chordwise directions coupled with the stretching motion by the assumed mode method. Yoo et al. [16] studied dynamic characteristics of rotating pretwisted blades with a concentrated mass simultaneously. Chung and Yoo [17] conducted vibration and transient analysis of a rotating uniform cantilever beam by the finite element method in which the stretch, chordwise and flapwise deformations are included. Attarnejad and Shahba [18] used the basic displacement functions obtained by solving the governing static differential equation of flapwise motion of rotating tapered Euler–Bernoulli beams to establish the finite element formulations. Vinod et al. [19] presented an approximate spectral finite element with two different interpolating functions for free vibration and wave propagation analysis of uniform and tapered rotating beams. Banerjee et al. [20] utilized the dynamic stiffness method to study the free vibration of rotating tapered Euler–Bernoulli beams, whose height and/or breadth vary linearly along the beam length. Wang and Wereley [21] also applied the dynamic stiffness method based on the Frobenius method to obtain natural frequencies of a rotating tapered beam. Ozdemir and Kaya [22] studied the vibration characteristics of a rotating tapered cantilever Euler–Bernoulli beam with particular crosssection employing the differential transform method. It has been well recognized that crack on a beam will introduce a local flexibility and consequently alter its dynamic properties such as natural frequencies and mode shapes. Therefore, it is possible to estimate the crack location and size from measured vibration parameters of the cracked beam. The effect of crack has been widely investigated in the literature and various approaches for crack modeling have been developed. The state-ofthe-art reviews on crack modeling and identification techniques have been presented by many researchers such as Dimarogonas [23], Doebling et al. [24] and Chasalevris and Papadopoulos [25]. As for vibration of cracked rotating beams, Chen and Chen [26] investigated vibration and stability of cracked thick rotating blades based on a finite element model. Chang and Chen [27] applied the finite element method to analyze a cracked thick rotating blade and then proposed a spatial wavelet based approach to detect cracks on this blade. Kim and Kim [28] used the finite element approach to treat a rotating composite beam with a breathing crack. Kuang and Huang [29] and Huang [30] studied vibration localization and stability of the rotating uniform and tapered blades with a local crack defect using the Galerkin method, respectively. Masoud and Al-Said [31] derived a mathematical model using the Lagrange equation and the assumed mode method to describe the lateral vibration of a rotating cracked Timoshenko beam. The conventional finite element methods (CFEM) are widely used in the problems of rotating cracked beams. However, to accurately acquire the higher modes, the element size should be small enough to match the corresponding high frequencies. Thus a large number of degrees of freedom are required for accurately predicting the higher-order modes, which will greatly increase the computational cost. As a result, appropriate methods able to obtain accurate results with low computational effort are desirable. The p-version of the finite element method (p-FEM) appears to provide an efficient solution. In the p-FEM, any desired degree of accuracy can be obtained by simply increasing the number of
shape functions, while a coarse mesh is kept fixed. Another interesting feature of this p-FEM is that with respect to equivalent degrees of freedom, it offers more rapid convergence than the CFEM, which increases accuracy of results by refining the finite element mesh [32]. Thus, computational efforts can be remarkably saved. Different types of the p-FEM have been applied to treat engineering problems [33,34]. Nowadays, non-uniform rotating beams find extensive applications due to satisfying special functional requirements and achieving a better distribution of strength and weight [35]. Hence vibration analysis of the cracked non-uniform rotating beams is of practical significance. In this paper, the p-FEM is applied to analysis of the transverse vibration characteristics of a cracked rotating tapered beam. In the current formulation, the shape functions of the rotating tapered beam element consist of conventional Hermite cubics and the shifted Legendre orthogonal polynomials. The stiffness matrix of the crack element and the finite element equations of a cracked rotating tapered beam are formulated based on the fracture mechanics and the Lagrange equation, respectively. The convergence and accuracy of the present approach are examined by comparisons with the published results in literature. The effects of crack location, crack size, rotation speed and hub radius of a cracked rotating tapered beam on the vibration properties are investigated. Furthermore, the mode shapes are obtained and analyzed by the spatial wavelet transform approach to detect the crack location.
2. Formulation Consider a rotating tapered Euler–Bernoulli beam with an edge crack in transverse vibration, as shown in Fig. 1. The beam is attached to a rigid hub of radius R and rotates about a fixed axis at a constant angular speed O. The height of the tapered beam varies linearly along the beam length. In the present study, the cracked rotating tapered beam is discretized into three elements at the crack cross-section, see Fig. 1. The crack is represented by a torsional spring element with equivalent stiffness kc. It is also assumed that the presence of the crack does not change the mass distribution along the beam. The centrifugal tension forces exerting on any section of the rotating tapered elements on the left and right sides can be calculated as follows: Z l1 Z l2 F1 ðx1 Þ ¼ ArO2 ðR þ xÞdxþ ArO2 ðR þl1 þ xÞdx ð1Þ x1
F2 ðx2 Þ ¼
Z
l2
0
ArO2 ðR þ l1 þ xÞdx
ð2Þ
x2
y
Ω R b
a hc
h1
h2
lc
z
R
L l1=lc
x
L
kc
l2=L-lc
Fig. 1. Sketch of a cracked rotating tapered beam and finite element mesh.
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
where x1 and x2 stand for the local coordinates in the left and right elements, respectively. Then the expressions for the strain energy and the kinetic energy of the rotating tapered beam with a crack can be written as U ¼ U1 þ U2 þ U3 2 2 2 Z li 2 Z li X X EIðxÞ d2 w dw ¼ dx þ F ðxÞ dx i 2 2 dx dx i¼1 0 i¼1 0 þ
T¼
1 ½MðyR yL Þx ¼ l1 2
2 Z X i¼1
li
2
rAðxÞ @w
0
2
@t
2.1. p-version rotating tapered element For simplicity and convenience in mathematical formulation, the local non-dimensional element coordinate xi ¼2(x–xi)/li–1 is introduced. Then the geometrical parameters for the rotating tapered beam element are as follows: a þ 1 a1 þ hðxÞ ¼ hL x ð5Þ 2 2 1 X
qk x
k
U1 ¼
2 1X uT ½K1 u 2i¼1
ð12Þ
U2 ¼
2 1X uT ½K2 u 2i¼1
ð13Þ
T¼
2 1X u_ T ½Mu_ 2i¼1
u ¼ ½w1 , y1 ,w2 , y2 ,w5 ,w6 , ,wn T
w ¼ w1 j1 þ y1 j2 þ w2 j3 þ y2 j4 þ
n X
½K2 ¼
ð7Þ
½M ¼
wi ji
ð8Þ
i¼5
where w1, y1, w2 and y2 are the transverse displacements and slopes at the two end nodes of the rotating beam element, wi (i44) are generalized internal degrees of freedom, j1–j4 are the four standard Hermite cubics as follows: 1 j1 ¼ ð23x þ x3 Þ, 4 1 j3 ¼ ð2 þ 3xx3 Þ, 4
L j2 ¼ e ð1xx2 þ x3 Þ 8 L j4 ¼ e ð1x þ x2 þ x3 Þ 8
2 li
ð9Þ
polynomials. These shifted Legendre orthogonal polynomials have the following forms: ½ði1Þ=2 X
ð1Þk ð2i2k7Þ!!
k¼0
2k k!ði2k1Þ!
xi2k1 , i4 4
Z
T dN dN Fi ðxÞ dx, dx dx 1 1
rA0 li 2
Z
1
1
1 X
ð17Þ
! qk x
k
NT Ndx
ð18Þ
k¼0
To speed-up the generation of the above matrices, the common integral terms appearing in these matrices related to the higher-order shape functions are extracted as k Ii,j ðrÞ ¼
Z
1
1
xr fki fkj dx, Iik ðrÞ ¼
Z
1 1
xr fki dx, ði,j 4 4Þ
ð19Þ
where k is the order of the derivatives, i and j are the row and column number of the corresponding matrix element, respectively. By using the method of integration by parts repeatedly, the integrals in Eq. (19) can be obtained in the following explicit forms:
k Ii,j ðrÞ ¼
8 r > > c=2 >
> s¼0 > > :
2s1 s!
ðc2sÞ!!
ðc þ2j2s5Þ!!
0
,
c is even and c Z 0 else
ð20Þ
ji(i 44) are the double integrations of the Legendre orthogonal
ji ðxÞ ¼
ð15Þ
The elastic stiffness matrix [K1], the centrifugal stiffness matrix [K2] and the mass matrix [M] are as follows: ! !T Z 1 X 3 8EI d2 N d2 N k ½K1 ¼ 3 0 pk x dx, ð16Þ 2 2 li 1 k ¼ 0 dx dx
ð6Þ
where a ¼h2/h1 is the taper ratio, hL, A0 and I0 are the corresponding initial values of the element and pk and qk are the polynomial coefficients. Since Eqs. (6) and (7) can approximate arbitrarily continuous functions if the polynomial order is adequately high, the following formulations are able to be extended to analyze any non-uniform continuous rotating beams. The transverse displacement w within a rotating beam element can be expressed as
ð14Þ
where the dot (y) designates differentiation with respect to the time variable t and u is the vector of generalized nodal displacements
k¼0 3 X bh3 ðxÞ k ¼ I0 pk x IðxÞ ¼ 12 k¼0
ð11Þ
By substituting Eqs. (5)–(11) into Eqs. (3) and (4), the elastic strain energy, the centrifugal strain energy and the potential energy can be expressed as
ð4Þ
where U1, U2 and U3 are the elastic strain energy, the strain energy due to the centrifugal force and the strain energy due to the crack, w is the transverse displacement, E is the elastic modulus, I(x) is the area moment of inertia about the z axis, Fi(x) and li is the centrifugal force and the length for the ith rotating tapered beam element, M is the bending moment exerted on the crack section, yR and yL are the slopes of the right and left side of the crack, r is the mass density of the material of the beam, A(x) is the cross-sectional area and t is the time variable.
AðxÞ ¼ bhðxÞ ¼ A0
taking integer part. They are second derivative orthogonal with respect not only to themselves but also to the first four Hermite cubics [36]. Furthermore, they contribute the value of zero at each end of the element; hence, they do not affect the imposition of boundary conditions through nodal constraints alone. The shape functions then can be written in vector form N ¼ ½j1 , j2 , j3 , j4 , j5 , j6 , , jn
ð3Þ
dx
827
ð10Þ
where k!!¼ k(k–2)y2 or 1, 0!!¼(–1)!!¼ 1, and [ ] denotes
( Iik ðrÞ ¼
ð1Þ2k 2Ur! , ðr þ ikÞ!!d!!
d is even and d Z 0
0
else
ð21Þ
where c¼4þ rþ i–j–2k and d ¼r–i–kþ5. The values of the integrals can be obtained by simple algebraic manipulation and stored in a file that later will be used by the program that implements the rotating tapered beam high-order element. This process can greatly speed-up the generation of the element stiffness and mass matrices.
828
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
2.2. Crack modeling In this section, the crack is assumed to be always open due to the centrifugal tension forces and modeled as a torsional spring, whose equivalent stiffness will be derived by using the theory of fracture mechanics. The additional strain energy due to
Table 1 Comparison of non-dimensional natural frequencies of a clamped non-rotating pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tapered beam (O ¼ oL2 rA0 =EI0 ) by various approaches.
a
p-FEM
Frobenius Method [36]
NE¼ 1, DOF¼ 20
CFEM [37] NE¼ 9, DOF¼ 20
NE¼ 50, DOF ¼102
0.10
Om1 Om2 Om3
9.8846 27.0085 52.7081
9.8846 27.0084 52.7080
10.0526 27.6496 54.4770
9.8844 27.0067 52.7002
10.9209 29.9041 58.4341
10.9209 29.9041 58.4341
10.9983 30.2341 59.4421
10.9207 29.9022 58.4257
11.8417 32.4755 63.5118
11.8417 32.4755 63.5118
11.8819 32.6660 64.1524
11.8415 32.4734 63.5029
12.6886 34.8384 68.1728
12.6886 34.8384 68.1728
12.7116 34.9594 68.6201
12.6884 34.8362 68.1635
13.4832 37.0526 72.5368
13.4832 37.0527 72.5368
13.4974 37.1359 72.8753
13.4830 37.0504 72.5271
14.2381 39.1539 76.6753
14.2381 39.1539 76.6753
14.2474 39.2153 76.9489
14.2379 39.1515 76.6651
14.9616 41.1656 80.6348
14.9616 41.1656 80.6348
14.9679 41.2135 80.8681
14.9613 41.1631 80.6242
15.6593 43.1039 84.4478
15.6593 43.1039 84.4478
15.6639 43.1432 84.6555
15.6591 43.1013 84.4368
16.3356 44.9806 88.1382
16.3356 44.9806 88.1389
16.3390 45.0143 88.3291
16.3353 44.9779 88.1268
16.9935 46.8049 91.7238
16.9935 46.8049 N/A
16.9961 46.8348 91.9039
16.9932 46.8021 91.7120
17.6354 48.5836 95.2185
17.6354 48.5839 N/A
17.6376 48.6109 95.3919
17.6351 48.5807 95.2063
18.2634 50.3223 98.6336
18.2634 N/A N/A
18.2653 50.3479 98.8029
18.2631 50.3193 98.6210
18.8791 52.0255 101.9779
18.8791 N/A N/A
18.8806 52.0500 102.1454
18.8788 52.0225 101.9649
0.15
Om1 Om2 Om3 0.20
Om1 Om2 Om3 0.25
Om1 Om2 Om3 0.30
Om1 Om2 Om3 0.35
Om1 Om2 Om3 0.40
Om1 Om2 Om3 0.45
Om1 Om2 Om3 0.50
Om1 Om2 Om3
the crack can be expressed as [37] Z a 2 KI da Uc ¼ b 0 E
ð22Þ
in which KI is the stress intensity factor for the first mode crack and has the following form: pffiffiffiffiffiffi 6M pa KI ¼ f ðaÞ ð23Þ bh2c
Table 2 Comparison of non-dimensional natural frequencies of a rotating cantilever pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tapered beam for different non-dimensional rotation speeds l ¼ OL2 rA0 =EI0 . p-FEM NE¼1, DOF¼12
CFEM NE ¼5, DOF ¼12
Wang and Wereley [21]
Gunda and Ganguli [10]
Mode 1 1 2 3 4 5 6 7 8 9 10 11 12
3.9866 4.4368 5.0927 5.8788 6.7434 7.6551 8.5956 9.5540 10.5239 11.5015 12.4845 13.4711
3.9867 4.4368 5.0927 5.8788 6.7434 7.6552 8.5957 9.5542 10.5244 11.5023 12.4857 13.4729
3.9866 4.4368 5.0927 5.8788 6.7434 7.6551 8.5956 9.5540 10.5239 11.5015 12.4845 13.4711
3.9866 4.4368 5.0927 5.8788 6.7434 7.6551 8.5956 9.5540 10.5239 11.5015 12.4845 13.4711
Mode 2 1 2 3 4 5 6 7 8 9 10 11 12
18.4740 18.9366 19.6839 20.6852 21.9053 23.3093 24.8647 26.5437 28.3227 30.1827 32.1085 34.0877
18.4820 18.9442 19.6907 20.6912 21.9105 23.3138 24.8686 26.5472 28.3261 30.1864 32.1127 34.0929
18.4740 18.9366 19.6839 20.6852 21.9053 23.3093 24.8647 26.5437 28.3227 30.1827 32.1085 34.0877
18.4740 18.9366 19.6839 20.6852 21.9053 23.3093 24.8647 26.5437 28.3227 30.1827 32.1085 34.0877
Mode 3 1 2 3 4 5 6 7 8 9 10 11 12
47.4173 47.8716 48.6190 49.6456 50.9338 52.4633 54.2124 56.1595 58.2833 60.5639 62.9829 65.5237
47.5548 48.0068 48.7504 49.7722 51.0548 52.5782 54.3213 56.2626 58.3813 60.6576 63.0733 65.6121
47.4173 47.8716 48.6190 49.6456 50.9338 52.4633 54.2124 56.1595 58.2833 60.5639 62.9829 65.5237
47.4173 47.8717 48.6191 49.6456 50.9338 52.4633 54.2124 56.1595 58.2833 60.5639 62.9829 65.5237
l
0.55
Om1 Om2 Om3 0.60
Om1 Om2 Om3
Table 3 Comparison of non-dimensional natural frequencies of a uniform beam at high rotation speeds.
0.65
Om1 Om2 Om3 0.70
Om1 Om2 Om3
l ¼ 200
l ¼12
Mode Mode Mode Mode
1 2 3 4
Present
Gunda and Ganguli [10]
Present
Gunda and Ganguli [10]*
13.1702 37.6031 79.6145 140.534
13.1702 37.6031 79.6145 140.534
201.0781 492.6931 783.6959 1086.2823
201.0547 492.6843 783.7879 N/A
The symbol * denotes the values are measured from the corresponding figure in the reference.
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
Table 4 Comparison of non-dimensional natural frequencies of a tapered beam at high rotation speeds.
where the geometrical correction function is f ðaÞ ¼ 1:131:374
l ¼ 100
l ¼12 Present
Gunda et al. [11]*
14.0312 35.9064 72.8565 126.4005
14.0313 35.9064 72.8565 126.401
101.8196 218.4248 340.7309 470.4971
101.8233 218.4655 340.7654 471.6630
Uc ¼
Frequency of mode 2
Frequency of mode 1
1 M2 2 kc
ð24Þ
36.12
14.06 14.05 14.04 14.03
36.06 36.00 35.94 35.88
5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
180 Frequency of mode 4
81 Frequency of mode 3
2 3 a a a þ 5:749 4:464 hc hc hc
The additional strain energy can be expressed in an alternative form as
14.07
78
75
72
160
140
120 5
10
15 20 25 Degree of freedom
30
Fig. 2. Convergence of the natural frequencies for tapered rotating beam with l ¼ 12.
226 Frequency of mode 2
105.6
Frequency of mode 1
104.4
103.2
224 222 220
102.0 218 5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
540
360
Frequency of mode 4
1 2 3 4
Gunda et al. [11]
Frequency of mode 3
Mode Mode Mode Mode
Present
829
356 352 348 344 340
522 504 486 468
5
10
15 20 25 Degree of freedom
30
Fig. 3. Convergence of the natural frequencies for tapered rotating beam with l ¼ 100.
830
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
From Eqs. (22) and (24), the equivalent stiffness of the crack is obtained as Z a kc ¼ bh4 E=72pa2 s½f ðsÞ2 ds ð25Þ
conventional tapered beam elements to reach convergence. As can be observed, the present p-FEM can achieve high accuracy without numerical difficulty either for small or large taper ratios.
0
Now the strain energy due to the crack in Eq.(3) can be written
3.2. Un-cracked rotating tapered beams
as U3 ¼
1 1 1 ½MðyR yL Þ ¼ ½kc ðyR yL Þ2 ¼ yL yR ½K3 ½yL yR T x ¼ l1 x ¼ l1 2 2 2 ð26Þ
where the stiffness matrix of the crack element is " # kc kc ½K3 ¼ kc kc
ð27Þ
2.3. Finite element equation The Lagrange equation is applied to obtain the finite element equation of the cracked rotating tapered beam d @L @L ¼0 ð28Þ dt @u_ @u where the Lagrangian function L¼ T–U. In the free vibration, the periodic displacement vector can be expressed as uðtÞ ¼ ueiot , where o is the natural frequency. By substituting Eqs. (12)–(14) and (26) into Eq. (28), the FEM equation in matrix form for free vibration of the cracked rotating tapered beam are as follows: ð½Ko2 ½MÞu ¼ 0
ð29Þ
where [K] and [M] are the global stiffness and mass matrices, respectively. It is worth noting that the global stiffness matrix [K] has an additional degree of freedom of the slope to describe the effect of the crack and the global mass matrix [M] has no change compared with that of the un-cracked rotating tapered beam. The requirement of non-trivial solutions for Eq. (29) leads to the frequency equation detð½Ko2 ½MÞ ¼ 0
ð30Þ
where det denotes the determinant of matrix. Then the natural frequencies can be solved. The mode shape corresponding to each frequency may be obtained by back-substitution of the frequency into Eq. (29) in usual manner.
3. Validation In this section, accuracy and convergence studies of the present formulation are carried out. Vibration of beams with different crack and rotation statuses are considered. The results obtained by the proposed approach are compared with those available in literature. 3.1. Un-cracked non-rotating tapered beams The first three non-dimensional natural frequencies of a nonrotating tapered beam clamped at the two ends for different taper ratios a obtained by the p-FEM using sixteen enriched terms are shown in Table 1, where NE and DOF denote the number of elements and the degree of freedom of the system. The results are compared with the values given by Naguleswaran [38] and Chinchalkar [39]. Naguleswaran applied the Frobenius series method to determine the natural frequencies while failing to get effective values due to the ill-conditioned frequency equation for large values of taper ratio a. Chinchalkar employed 50
Here a rotating tapered beam of taper ratio a ¼0.5 is studied. The mass and transverse stiffness distributions along the length direction are given by m(x)¼ rA0(1–0.5x/L) and EI(x)¼ EI0(1–0.5x/L)3. The first three natural frequencies of the tapered beam for various nondimensional rotation speeds l obtained by the present approach are shown in Table 2. The results are compared with the results given by Wang and Wereley [21] and Gunda and Ganguli [10], respectively. Wang and Werelye applied the dynamic stiffness method based on the Frobenius power series solution to predict the natural frequencies and more than 80 terms were used. Gunda and Ganguli used the stiff-string basis function to establish the finite element. The comparison is very good. In addition, the first three natural frequencies, calculated using five conventional rotating tapered
Table 5 Comparison of natural frequencies by the p-FEM and the two-dimensional FEM. Crack
Method
Mode 1
Mode 2
Mode 3
Taper ratio a ¼0.2 Intact p-FEM 2d-FEM 0.5 0.288 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.6 0.292 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.95 0.399 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM
353.95 351.62 350.41 (0.99) 348.36(0.99) 340.28(0.96) 338.93(0.96) 346.36(0.98) 344.38(0.98) 326.91(0.92) 326.23(0.93) 307.55(0.87) 307.34(0.87) 279.14(0.79) 280.17(0.80)
1298.1 1271.5 1251.8(0.96) 1230.0(0.97) 1145.8(0.88) 1132.1(0.89) 1268.1(0.98) 1242.1(0.98) 1201.9(0.93) 1181.6(0.93) 1178.0(0.91) 1156.1(0.91) 1126.4(0.87) 1106.8(0.87)
3041.4 2925.3 3038.9(1.00) 2923.0(1.00) 3033.4(1.00) 2916.8(1.00) 2980.6(0.98) 2862.7(0.98) 2863.3(0.94) 2753.8(0.94) 2863.5(0.94) 2751.7(0.94) 2802.2(0.92) 2690.9(0.92)
Taper ratio a ¼0.4 Intact p-FEM 2d-FEM 0.6 0.306 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.7 0.304 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.8 0.3 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM
324.41 322.83 322.09(0.99) 320.63(0.99) 316.44(0.98) 315.35(0.98) 317.02(0.98) 315.93(0.98) 300.19(0.93) 300.14(0.93) 310.01(0.96) 309.17(0.96) 279.69(0.86) 280.68(0.87)
1442.02 1415.36 1374.9(0.95) 1349.3(0.95) 1245.0(0.86) 1229.1(0.87) 1384.2(0.96) 1357.1(0.96) 1278.9(0.89) 1259.0(0.89) 1436.8(1.00) 1404.3(0.99) 1426.5(0.99) 1394.0(0.98)
3630.2 3486.9 3534.5(0.97) 3373.2(0.97) 3384.6(0.93) 3224.2(0.93) 3588.0(0.99) 3424.2(0.98) 3516.7(0.97) 3358.2(0.96) 3476.5(0.96) 3318.4(0.95) 3218.7(0.89) 3070.2(0.88)
Taper ratio a ¼0.6 Intact p-FEM 2d-FEM 0.65 0.259 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.75 0.275 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.8 0.281 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM
308.15 306.85 308.11(1.00) 306.81(1.00) 307.96(1.00) 306.65(1.00) 305.95(0.99) 304.62(0.99) 298.72(0.97) 297.95(0.97) 302.55(0.98) 301.63(0.98) 286.35(0.93) 286.45(0.93)
1576.10 1538.13 1571.8(1.00) 1534.3(1.00) 1554.7(0.99) 1518.5(0.99) 1503.2(0.95) 1467.5(0.95) 1321.4(0.84) 1300.5(0.85) 1504.5(0.95) 1470.3(0.96) 1347.7(0.86) 1324.8(0.86)
4152.09 3935.17 4097.8(0.99) 3889.0(0.99) 3875.2(0.93) 3688.9(0.94) 4068.9(0.98) 3835.6(0.97) 3898.3(0.94) 3668.5(0.93) 4136.2(1.00) 3903.5(0.99) 4102.5(0.99) 3871.0(0.98)
b
a/hc
Natural frequency (Hz)
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
beam elements that have the same degrees of freedom as that present p-FEM are also listed in Table 2. To examine the accuracy of the p-FEM, a uniform rotating beam and a tapered rotating beam are considered at high and very high rotating speeds. The mass and transverse stiffness variations per unit length are given by m(x)¼ rA0(1–0.8x/L) and EI(x) ¼EI0(1–0.95x/L). Tables 3 and 4 show the lowest four nondimensional natural frequencies obtained by the present method and those obtained by Gunda and Ganguli [10] for the uniform
rotating beam and by Gunda et al. [11] for the tapered rotating beam, respectively. As observed, the present method works well both for high and very high rotating speeds. The convergence study of the present method for the above tapered beam at two different rotating speeds (l ¼12 and l ¼100) on the lowest four modes is conducted. Figs. 2 and 3 depict the convergence behavior of the proposed method, where the number of the enriched terms is increased from 2 to 26, and the total degrees of freedom correspondingly increase from 6 to 30. The
1.0
1.0
1
1
2
2
1
2
1
2
0.9 ωc /ω0
ωc /ω0
0.9
3
0.8
3 0.8
0.7
0.7 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
1.0
0.8
1.0
0.6
0.8
1.0
1.0
2
2 4
1
ωc /ω0
1
0.9
4
0.9
3
3
0.6
lc/L
lc/L
ωc /ω0
831
3
0.8
3
0.8 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
lc/L
lc/L
Fig. 4. Effect of crack location and size on the lowest four frequencies of a cracked rotating tapered beam with various crack sizes (1, 2 and 3 denote a/hc ¼0.1, 0.3 and 0.5, respectively, same as below).
1.00
1.00 1
1
2
2
0.96
1
2
ωc /ω0
1
2
ωc /ω0
0.96
3
0.92
3
0.92
0.88
0.88 0
5
10
15
20
0
5
λ
10
15
20
15
20
λ
1.00
1.00
1
1
2 2 3
0.96 4
ωc /ω0
3
ωc /ω0
0.96 4
3
3 0.92
0.92
0
5
10
λ Fig. 5. Effect of the rotation speed l ¼ OL2 crack sizes.
15
20
0
5
10
λ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rA0 =EI0 on the lowest four frequencies of a cracked rotating tapered beam with crack location lc/L¼ 0.45 and various
832
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
convergence of the proposed method is compared with the CFEM. It is found that with the same number of degrees of freedom, the present p-FEM converge more rapidly than the CFEM.
3.3. Cracked non-rotating tapered beams The beam to be considered is fixed at the large end. The geometrical and physical parameters are L ¼240 mm, b¼12 mm, h1 ¼20 mm, E¼210 GPa, material mass density r ¼7860 kg/m3 and Poisson ratio n ¼0.3. Vibration of this beam was previously studied by Chaudhari and Maiti [40]. They calculated the first three natural frequencies for different crack locations and sizes by 1.00
a two-dimensional finite element program. The first three natural frequencies are computed using the present method with 4 enriched terms in the shape functions, as shown in Table 5. The 2d-FEM results in Table 5 stand for the results presented by Chaudhari and Maiti. The notation b ¼1–(1–a)l1/L denotes the crack location. The numbers in the parentheses stand for the reduction ratios of the natural frequencies of the cracked beam over that of the un-cracked by the consistent method. From Table 3, the agreement for the two methods is good considering the difference in the modeling dimensions. In addition, an interesting phenomenon can be observed, that is, the reduction ratios of the first three natural frequencies obtained by the two methods are in excellent agreement and the maximal 1.00
1
1
2 ωc /ω0
1
ωc /ω0
2
0.96
2
0.92
3
2
1
3
0.96
0.92
0.88
0.88 0
5
10 R/L
15
20
0
1.00
5
1.00
10 R/L
15
20
4
ωc /ω0
20
2
0.98
3
0.96
4
3
ωc /ω0
0.98
15
1
1 2
10 R/L
3
0.94
0
3
0.96 0.94
5
10 R/L
15
20
0
5
Fig. 6. Effect of the hub radius on the lowest four frequencies of a cracked rotating tapered beam with crack location lc/L¼ 0.45 and various crack sizes.
1.0
1.0 1st mode shape
0.6
0
1
2
2nd mode shape
0
0.8
3
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
0.5
3
-0.5
0.0
0.2
0.4
x/L
0.6
0.8
1.0
x/L 1.0
0 0.5
1
2
4th mode shape
1.0 3rd mode shape
2
0.0
-1.0
1.0
1
3
0.0 -0.5
01 2 3
0.5 0.0 -0.5
0.0
0.2
0.4
0.6 x/L
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Fig. 7. Lowest four mode shapes of a rotating tapered beam with and without a crack located at lc/L¼0.45 (0 denotes the intact case here).
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
10-6 ×
10-5 1.0 × Wavelet coefficients of 2nd mode shape
Wavelet coefficients of 1st mode shape
1.0 0.5 0 -0.5 -1.0
0.5 0 -0.5 -1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
x/L
0.6
0.8
1.0
0.6
0.8
1.0
x/L
× 10-5
10-5 1.6 × Wavelet coefficients of 4th mode shape
Wavelet coefficients of 3rd mode shape
1.6
833
0.8 0 -0.8 -1.6
0.8 0 -0.8 -1.6
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
x/L
0.4 x/L
Fig. 8. Wavelet coefficients for the lowest four mode shape of a cracked rotating tapered beam with crack location lc/L ¼0.45 and size a/hc ¼ 0.1.
difference between them does not exceed 1%. According to this phenomenon, the reduction ratio of natural frequencies can serve as a robust damage indicator in crack identification.
4. Vibration analysis In this section, the present formulation is used to analyze the vibration characteristics of a cracked rotating tapered beam. The basic geometrical and material parameter values are the same as those used in section 3.3. Unless specially stated, the taper ratio, hub radius and rotation speed of the rotating beam are a ¼ 0.5, R/L¼0 and l ¼1, respectively. It is assumed that a crack of depth a/hc ¼0.3 occurs at the position lc/L¼0.45. In the following study we use 4 terms to enrich the shape functions of the rotating tapered beam element. Reduction ratios of the lowest four natural frequency of the cracked rotating tapered beam over those of the intact case are computed for different crack locations, crack sizes, rotation speeds and hub radii. The effects of the non-dimensional crack location lc/L and crack size a/hc on the lowest four natural frequencies are shown in Fig. 4. It can be seen that when the crack size is increased, the lowest four natural frequencies of the cracked rotating tapered beam decrease except that the crack is exactly located at the vibration nodes. For a given crack size, the fundamental frequency decreases monotonically when the crack location varies from the free end to the fixed root, and the frequencies of higher modes reduce rapidly as the crack goes away from the vibration nodes. The effects of non-dimensional rotation speed l and hub radius R/L on the natural frequencies of the cracked rotating tapered beam are shown in Figs. 5 and 6, respectively. It is observed that as the rotation speed and the hub radius increase, the lowest four natural frequencies tend to increase. This can be explained by the fact that the rotating centrifugal forces are proportional to O2(Rþx), the increase of the rotation speed as well as hub radius can improve the stiffening effect, which will compensate for the stiffness loss due to the presence of the crack.
The lowest four vibration mode shapes of the cracked rotating tapered beam are shown in Fig. 7. With the increase of crack size, the mode shapes of cracked rotating tapered beam change significantly. A local perturbation on the mode shapes gradually becomes distinct at the crack location. However, when the crack size is very small, the perturbation on the mode shapes cannot be identified. In order to examine this slight perturbation, the spatial wavelet transform approach is used. The lowest four mode shapes for crack size a/hc ¼0.1 are spatially decomposed with the Daubechies Db4 wavelet. The wavelet coefficients of the lowest four modes along the beam length are plotted in Fig. 8. As can be seen, the crack location can be easily detected from the local perturbation based on the wavelet coefficient distributions.
5. Conclusions The p-FEM for vibration analysis of cracked rotating tapered beams is formulated. The shifted Legendre orthogonal polynomials linearly combined with the Hermite cubics are applied to approximate the displacement field within a rotating tapered beam element and the common integrals in the elastic stiffness, centrifugal stiffness and mass matrices of the rotating tapered beam element are extracted and derived in analytical form, which will speed-up the generation of these matrices. The fracture mechanics and the Lagrange equation are used to construct the equivalent stiffness matrix of the crack element and the p-version finite element model of the structural system, respectively. Comparisons of the results for different configurations of beams calculated by the proposed method with those published in literature are made. It shows that the proposed method has favorable accuracy and can yield higher accuracy than the CFEM with equal degrees of freedom. Vibration analysis of cracked rotating tapered beams has been carried out. Effects of the crack location and size, rotation speed and hub radius on vibration characteristics of a cracked rotating tapered beam are investigated. The mode shapes of the cracked rotating tapered beam are
834
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
obtained and analyzed by the spatial wavelet transform based approach to detect the crack location. The present work provides helpful information for structural design and diagnosis and can be extended to study vibration of arbitrarily non-uniform cracked rotating beams. References [1] K. Wei, G. Meng, S. Zhou, J. Liu, Vibration control of variable speed/ acceleration rotating beams using smart materials, Journal of Sound and Vibration 298 (2006) 1150–1158. [2] S.-M. Lin, PD control of a rotating smart beam with an elastic root, Journal of Sound and Vibration 312 (2008) 109–124. [3] J.B. Yang, L.J. Jiang, D.C. Chen, Dynamic modelling and control of a rotating Euler–Bernoulli beam, Journal of Sound and Vibration 274 (2004) 863–875. [4] H. Sohn, C.R. Farrar, F.M. Hemez, D.D. Shunk, D.W. Stinemates, B.R. Nadler, A review of structural health monitoring literature: 1996–2001, Los Alamos National Laboratory Report, LA-13976-MS, 2003. [5] D.H. Hodges, M.J. Rutkowski, Free vibration analysis of rotating beams by a variable-order finite-element method, American Institute of Aeronautics and Astronautics Journal 19 (1981) 1459–1466. [6] S. Naguleswaran, Lateral vibration of a centrifugally tensioned Euler– Bernoulli beam, Journal of Sound and Vibration 176 (1994) 613–624. [7] S.S. Rao, R.S. Gupta, Finite element vibration analysis of rotating Timoshenko beams, Journal of Sound and Vibration 242 (2001) 103–124. [8] J.B. Gunda, A.P. Singh, P.S. Chhabra, R. Ganguli, Free vibration analysis of rotating tapered blades using Fourier-p superelement, Structural Engineering and Mechanics 27 (2007) 243–257. [9] J.B. Gunda, R. Ganguli, New rational interpolation functions for finite element analysis of rotating beams, International Journal of Mechanical Sciences 50 (2008) 578–588. [10] J.B. Gunda, R. Ganguli, Stiff String basis functions for vibration analysis of high speed rotating beams, Journal of Applied Mechanics 75 (2008) 245021–245025. [11] J.B. Gunda, R.K. Gupta, R. Ganguli, Hybrid stiff-string-polynomial basis functions for vibration analysis of high speed rotating beams, Computers and Structures 87 (2009) 254–265. [12] A. Bazoune, Y.A. Khulief, N.G. Stephen, Further results for modal characteristics of rotating tapered timoshenko beams, Journal of Sound and Vibration 219 (1999) 157–174. [13] A. Bazoune, Relationship between softening and stiffening effects in terms of Southwell coefficients, Journal of Sound Vibration 287 (2005) 1027–1030. [14] A. Bazoune, Effect of tapering on natural frequencies of rotating beams, Shock and Vibration 14 (2007) 169–179. [15] H.H. Yoo, S.H. Shin, Vibration analysis of rotating cantilever beams, Journal of Sound Vibration 212 (1998) 807–828. [16] H.H. Yoo, J.Y. Kwak, J. Chung, Vibration analysis of rotating pre-twisted blades with a concentrated mass, Journal of Sound and Vibration 240 (2001) 891–908. [17] J. Chung, H.H. Yoo, Dynamic analysis of a rotating cantilever beam by using the finite element method, Journal of Sound and Vibration 249 (2002) 147–164. [18] R. Attarnejad, A. Shahba, Basic displacement functions for centrifugally stiffened tapered beams, Communications in Numerical Methods in Engineering (2010). doi:10.1002/cnm.1365.
[19] K.G. Vinod, S. Gopalakrishnan, R. Ganguli, Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements, International Journal of Solids and Structures 44 (2007) 5875–5893. [20] J.R. Banerjee, H. Su, D.R. Jackson, Free vibration of rotating tapered beams using the dynamic stiffness method, Journal of Sound and Vibration 298 (2006) 1034–1054. [21] G. Wang, N.M. Wereley, Free vibration analysis of rotating blades with uniform tapers, American Institute of Aeronautics and Astronautics Journal 42 (2004) 2429–2437. [22] O. Ozdemir, M.O. Kaya, Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method, Journal of Sound and Vibration 289 (2006) 413–420. [23] A.D. Dimarogonas, Vibration of cracked structures: a state-of-the-art review, Engineering Fracture Mechanics 55 (1996) 831–857. [24] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage identification methods, The Shock and Vibration Digest 30 (1998) 91–105. [25] A.C. Chasalevris, C.A. Papadopoulos, Identification of multiple cracks in beams under bending, Mechanical Systems and Signal Processing 20 (2006) 1631–1673. [26] L.-W. Chen, C.-L. Chen, Vibration and stability of cracked thick rotating blades, Computers and Structures 28 (1988) 67–74. [27] C.-C. Chang, L.-W. Chen, Damage detection of cracked thick rotating blades by a spatial wavelet based approach, Applied Acoustics 65 (2004) 1095–1111. [28] S.-S. Kim, J.-H. Kim, Rotating composite beam with a breathing crack, Composite Structures 60 (2003) 83–90. [29] J.H. Kuang, B.W. Huang, The effect of blade crack on mode localization in rotating bladed disks, Journal of Sound and Vibration 227 (1999) 85–103. [30] B.W. Huang, Effect of number of blades and distribution of cracks on vibration localization in a cracked pre-twisted blade system, International Journal of Mechanical Sciences 48 (2006) 1–10. [31] A.A. Masoud, S. Al-Said, A new algorithm for crack localization in a rotating Timoshenko beam, Journal of Vibration and Control 15 (2009) 1541–1561. [32] B. Szabo, I. Babuska, Finite Element Analysis, Wiley, New York, 1991. [33] A. Houmat, Three-dimensional free vibration analysis of plates using the h–p version of the finite element method, Journal of Sound and Vibration 290 (2006) 690–704. [34] P. Ribeiro, Forced periodic vibrations of laminated composite plates by a p-version, first order shear deformation, finite element, Composites Science and Technology 66 (2006) 1844–1856. [35] H.H. Yoo, J.E. Cho, J. Chung, Modal analysis and shape optimization of rotating cantilever beams, Journal of Sound and Vibration 290 (2006) 223–241. [36] E.T. Whittaker, G.N. Watson, A Course Of Modern Analysis, 4th edition, Cambridge Mathematical Library, 1996. [37] W.M. Ostachowicz, M. Krawczuk, Analysis of the effect of crack on the natural frequecy of a cantilever beam, Journal of Sound and Vibration 150 (1991) 191–201. [38] S. Naguleswaran, A direct solution for the transverse vibration of eulerbernoulli wedge and cone beams, Journal of Sound and Vibration 172 (1994) 289–304. [39] S. Chinchalkar, Determination of crack location in beams using natural frequencies, Journal of Sound and Vibration 247 (2001) 417–429. [40] T.D. Chaudhari, S.K. Maiti, Modelling of transverse vibration of beam of linearly variable depth with edge crack, Engineering Fracture Mechanics 63 (1999) 425–445.
Contents lists available at ScienceDirect
Finite Elements in Analysis and Design journal homepage: www.elsevier.com/locate/finel
Vibration analysis of a cracked rotating tapered beam using the p-version finite element method Yue Cheng a,b,n, Zhigang Yu b, Xun Wu b, Yuhua Yuan b a b
School of Aerospace Science and Engineering, Beijing Institute of Technology, Beijing 100081, China Qinghe Dalou Zi, Haidian District, Beijing 100085, China
a r t i c l e i n f o
abstract
Article history: Received 18 April 2010 Received in revised form 11 January 2011 Accepted 25 February 2011 Available online 17 March 2011
Vibration characteristics of cracked rotating tapered beam are investigated by the p-version finite element method. In the present formulation, the shape functions enriched with the shifted Legendre orthogonal polynomials are employed to represent the transverse displacement field within the rotating tapered beam element. The crack element stiffness matrix and the p-version finite element model of the structural system are obtained by using fracture mechanics and the Lagrange equation, respectively. The common integral terms appeared in the rotating tapered beam element matrices are derived in analytical forms so as to speed up the generation of these matrices during the implementation of the present method. Comparisons have been made for several configurations between results obtained by the present method with those available in literature, and the proposed method shows excellent accuracy and convergence. The effects of crack location, crack size, rotating speed and hub radius on vibration characteristics of a cracked rotating tapered beam are investigated. The mode shapes of the cracked rotating tapered beam are also obtained and analyzed with the spatial wavelet transform approach to detect the slight perturbation at the crack position. The present method is of general applicability and can be applied to investigate vibration of arbitrarily continuous non-uniform rotating beams. & 2011 Elsevier B.V. All rights reserved.
Keywords: Rotating tapered beam Crack Vibration characteristics Finite element method p-version
1. Introduction The design and analysis of engineering structures such as helicopter rotor blades, wind turbine blades and airplane propeller blades, have attracted considerable interest in the vibration behavior of rotating beams, which greatly advance elimination of undesirable resonance phenomena and determination of suitable vibration control strategies [1–3]. Compared with enormous research on vibration analysis of intact rotating beams, less attention has been focused on vibration characteristics of cracked rotating beams. However, cracks frequently occur in practical rotating structures for the sake of cyclic fatigue or manufacturing defects. Dynamic modeling and analysis of cracked rotating beams are very important to the mechanical design as well as vibration-based inspections, which have gained popularity recently [4]. Intact rotating beams are usually simplified to one-dimensional Euler–Bernoulli or Timoshenko beams subjected to centrifugal tension force. The governing differential equations of
n Corresponding author at: School of Aerospace Science and Engineering, Beijing Institute of Technology, Beijing 100081, China. Tel.: þ86 10 68019709. E-mail address: [email protected] (Y. Cheng).
0168-874X/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.finel.2011.02.013
motion for rotating tapered beams contain variable coefficients introduced due to the variation of the centrifugal force and the geometry along the beam length. The closed-form solutions for such equations therefore are difficult to obtain. As a result, different approximate methods have been suggested to analyze the vibration problems of rotating beams. Hodegs and Rutkowski [5] presented a variable-order finite element method to capture the free vibration properties of rotating beams. Naguleswaran [6] studied lateral vibration of a rotating uniform Euler–Bernoulli beam based on the general solution of the mode shape equation as the superposition of four linearly independent functions. Rao and Gupta [7] applied the finite element to obtain natural frequencies and mode shapes of a rotating twisted and tapered beam. Gunda et al. [8] proposed a superelement, whose shape functions were composed by polynomials and trigonometric functions for dynamic analysis of rotating tapered beams. Gunda and Ganguli [9] then assumed the transverse displacement to vary as a fourth order function and obtained new shape functions that would satisfy the static part of the homogeneous governing differential equation. Later Gunda et al. [10,11] proposed a new finite element for free vibration analysis of high-speed rotating beams using shape functions that are a linear combination of the solution of the governing static differential equation of a stiff string and a cubic polynomial. Bazoune
826
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
et al. [12] investigated the modal characteristics of a rotating tapered Timoshenko beam using the finite element analysis. Bazoune [13] discussed the relationship between the in-plane and out-of-plane frequencies in terms of Southwell coefficient. Bazoune [14] studied free vibration of a rotating tapered beam with the finite element method in which the mass, elastic and centrifugal stiffness matrices were explicitly expressed in terms of the taper ratios. Yoo and Shin [15] investigated the modal characteristics of rotating cantilever beams in the flapwise and chordwise directions coupled with the stretching motion by the assumed mode method. Yoo et al. [16] studied dynamic characteristics of rotating pretwisted blades with a concentrated mass simultaneously. Chung and Yoo [17] conducted vibration and transient analysis of a rotating uniform cantilever beam by the finite element method in which the stretch, chordwise and flapwise deformations are included. Attarnejad and Shahba [18] used the basic displacement functions obtained by solving the governing static differential equation of flapwise motion of rotating tapered Euler–Bernoulli beams to establish the finite element formulations. Vinod et al. [19] presented an approximate spectral finite element with two different interpolating functions for free vibration and wave propagation analysis of uniform and tapered rotating beams. Banerjee et al. [20] utilized the dynamic stiffness method to study the free vibration of rotating tapered Euler–Bernoulli beams, whose height and/or breadth vary linearly along the beam length. Wang and Wereley [21] also applied the dynamic stiffness method based on the Frobenius method to obtain natural frequencies of a rotating tapered beam. Ozdemir and Kaya [22] studied the vibration characteristics of a rotating tapered cantilever Euler–Bernoulli beam with particular crosssection employing the differential transform method. It has been well recognized that crack on a beam will introduce a local flexibility and consequently alter its dynamic properties such as natural frequencies and mode shapes. Therefore, it is possible to estimate the crack location and size from measured vibration parameters of the cracked beam. The effect of crack has been widely investigated in the literature and various approaches for crack modeling have been developed. The state-ofthe-art reviews on crack modeling and identification techniques have been presented by many researchers such as Dimarogonas [23], Doebling et al. [24] and Chasalevris and Papadopoulos [25]. As for vibration of cracked rotating beams, Chen and Chen [26] investigated vibration and stability of cracked thick rotating blades based on a finite element model. Chang and Chen [27] applied the finite element method to analyze a cracked thick rotating blade and then proposed a spatial wavelet based approach to detect cracks on this blade. Kim and Kim [28] used the finite element approach to treat a rotating composite beam with a breathing crack. Kuang and Huang [29] and Huang [30] studied vibration localization and stability of the rotating uniform and tapered blades with a local crack defect using the Galerkin method, respectively. Masoud and Al-Said [31] derived a mathematical model using the Lagrange equation and the assumed mode method to describe the lateral vibration of a rotating cracked Timoshenko beam. The conventional finite element methods (CFEM) are widely used in the problems of rotating cracked beams. However, to accurately acquire the higher modes, the element size should be small enough to match the corresponding high frequencies. Thus a large number of degrees of freedom are required for accurately predicting the higher-order modes, which will greatly increase the computational cost. As a result, appropriate methods able to obtain accurate results with low computational effort are desirable. The p-version of the finite element method (p-FEM) appears to provide an efficient solution. In the p-FEM, any desired degree of accuracy can be obtained by simply increasing the number of
shape functions, while a coarse mesh is kept fixed. Another interesting feature of this p-FEM is that with respect to equivalent degrees of freedom, it offers more rapid convergence than the CFEM, which increases accuracy of results by refining the finite element mesh [32]. Thus, computational efforts can be remarkably saved. Different types of the p-FEM have been applied to treat engineering problems [33,34]. Nowadays, non-uniform rotating beams find extensive applications due to satisfying special functional requirements and achieving a better distribution of strength and weight [35]. Hence vibration analysis of the cracked non-uniform rotating beams is of practical significance. In this paper, the p-FEM is applied to analysis of the transverse vibration characteristics of a cracked rotating tapered beam. In the current formulation, the shape functions of the rotating tapered beam element consist of conventional Hermite cubics and the shifted Legendre orthogonal polynomials. The stiffness matrix of the crack element and the finite element equations of a cracked rotating tapered beam are formulated based on the fracture mechanics and the Lagrange equation, respectively. The convergence and accuracy of the present approach are examined by comparisons with the published results in literature. The effects of crack location, crack size, rotation speed and hub radius of a cracked rotating tapered beam on the vibration properties are investigated. Furthermore, the mode shapes are obtained and analyzed by the spatial wavelet transform approach to detect the crack location.
2. Formulation Consider a rotating tapered Euler–Bernoulli beam with an edge crack in transverse vibration, as shown in Fig. 1. The beam is attached to a rigid hub of radius R and rotates about a fixed axis at a constant angular speed O. The height of the tapered beam varies linearly along the beam length. In the present study, the cracked rotating tapered beam is discretized into three elements at the crack cross-section, see Fig. 1. The crack is represented by a torsional spring element with equivalent stiffness kc. It is also assumed that the presence of the crack does not change the mass distribution along the beam. The centrifugal tension forces exerting on any section of the rotating tapered elements on the left and right sides can be calculated as follows: Z l1 Z l2 F1 ðx1 Þ ¼ ArO2 ðR þ xÞdxþ ArO2 ðR þl1 þ xÞdx ð1Þ x1
F2 ðx2 Þ ¼
Z
l2
0
ArO2 ðR þ l1 þ xÞdx
ð2Þ
x2
y
Ω R b
a hc
h1
h2
lc
z
R
L l1=lc
x
L
kc
l2=L-lc
Fig. 1. Sketch of a cracked rotating tapered beam and finite element mesh.
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
where x1 and x2 stand for the local coordinates in the left and right elements, respectively. Then the expressions for the strain energy and the kinetic energy of the rotating tapered beam with a crack can be written as U ¼ U1 þ U2 þ U3 2 2 2 Z li 2 Z li X X EIðxÞ d2 w dw ¼ dx þ F ðxÞ dx i 2 2 dx dx i¼1 0 i¼1 0 þ
T¼
1 ½MðyR yL Þx ¼ l1 2
2 Z X i¼1
li
2
rAðxÞ @w
0
2
@t
2.1. p-version rotating tapered element For simplicity and convenience in mathematical formulation, the local non-dimensional element coordinate xi ¼2(x–xi)/li–1 is introduced. Then the geometrical parameters for the rotating tapered beam element are as follows: a þ 1 a1 þ hðxÞ ¼ hL x ð5Þ 2 2 1 X
qk x
k
U1 ¼
2 1X uT ½K1 u 2i¼1
ð12Þ
U2 ¼
2 1X uT ½K2 u 2i¼1
ð13Þ
T¼
2 1X u_ T ½Mu_ 2i¼1
u ¼ ½w1 , y1 ,w2 , y2 ,w5 ,w6 , ,wn T
w ¼ w1 j1 þ y1 j2 þ w2 j3 þ y2 j4 þ
n X
½K2 ¼
ð7Þ
½M ¼
wi ji
ð8Þ
i¼5
where w1, y1, w2 and y2 are the transverse displacements and slopes at the two end nodes of the rotating beam element, wi (i44) are generalized internal degrees of freedom, j1–j4 are the four standard Hermite cubics as follows: 1 j1 ¼ ð23x þ x3 Þ, 4 1 j3 ¼ ð2 þ 3xx3 Þ, 4
L j2 ¼ e ð1xx2 þ x3 Þ 8 L j4 ¼ e ð1x þ x2 þ x3 Þ 8
2 li
ð9Þ
polynomials. These shifted Legendre orthogonal polynomials have the following forms: ½ði1Þ=2 X
ð1Þk ð2i2k7Þ!!
k¼0
2k k!ði2k1Þ!
xi2k1 , i4 4
Z
T dN dN Fi ðxÞ dx, dx dx 1 1
rA0 li 2
Z
1
1
1 X
ð17Þ
! qk x
k
NT Ndx
ð18Þ
k¼0
To speed-up the generation of the above matrices, the common integral terms appearing in these matrices related to the higher-order shape functions are extracted as k Ii,j ðrÞ ¼
Z
1
1
xr fki fkj dx, Iik ðrÞ ¼
Z
1 1
xr fki dx, ði,j 4 4Þ
ð19Þ
where k is the order of the derivatives, i and j are the row and column number of the corresponding matrix element, respectively. By using the method of integration by parts repeatedly, the integrals in Eq. (19) can be obtained in the following explicit forms:
k Ii,j ðrÞ ¼
8 r > > c=2 >
> s¼0 > > :
2s1 s!
ðc2sÞ!!
ðc þ2j2s5Þ!!
0
,
c is even and c Z 0 else
ð20Þ
ji(i 44) are the double integrations of the Legendre orthogonal
ji ðxÞ ¼
ð15Þ
The elastic stiffness matrix [K1], the centrifugal stiffness matrix [K2] and the mass matrix [M] are as follows: ! !T Z 1 X 3 8EI d2 N d2 N k ½K1 ¼ 3 0 pk x dx, ð16Þ 2 2 li 1 k ¼ 0 dx dx
ð6Þ
where a ¼h2/h1 is the taper ratio, hL, A0 and I0 are the corresponding initial values of the element and pk and qk are the polynomial coefficients. Since Eqs. (6) and (7) can approximate arbitrarily continuous functions if the polynomial order is adequately high, the following formulations are able to be extended to analyze any non-uniform continuous rotating beams. The transverse displacement w within a rotating beam element can be expressed as
ð14Þ
where the dot (y) designates differentiation with respect to the time variable t and u is the vector of generalized nodal displacements
k¼0 3 X bh3 ðxÞ k ¼ I0 pk x IðxÞ ¼ 12 k¼0
ð11Þ
By substituting Eqs. (5)–(11) into Eqs. (3) and (4), the elastic strain energy, the centrifugal strain energy and the potential energy can be expressed as
ð4Þ
where U1, U2 and U3 are the elastic strain energy, the strain energy due to the centrifugal force and the strain energy due to the crack, w is the transverse displacement, E is the elastic modulus, I(x) is the area moment of inertia about the z axis, Fi(x) and li is the centrifugal force and the length for the ith rotating tapered beam element, M is the bending moment exerted on the crack section, yR and yL are the slopes of the right and left side of the crack, r is the mass density of the material of the beam, A(x) is the cross-sectional area and t is the time variable.
AðxÞ ¼ bhðxÞ ¼ A0
taking integer part. They are second derivative orthogonal with respect not only to themselves but also to the first four Hermite cubics [36]. Furthermore, they contribute the value of zero at each end of the element; hence, they do not affect the imposition of boundary conditions through nodal constraints alone. The shape functions then can be written in vector form N ¼ ½j1 , j2 , j3 , j4 , j5 , j6 , , jn
ð3Þ
dx
827
ð10Þ
where k!!¼ k(k–2)y2 or 1, 0!!¼(–1)!!¼ 1, and [ ] denotes
( Iik ðrÞ ¼
ð1Þ2k 2Ur! , ðr þ ikÞ!!d!!
d is even and d Z 0
0
else
ð21Þ
where c¼4þ rþ i–j–2k and d ¼r–i–kþ5. The values of the integrals can be obtained by simple algebraic manipulation and stored in a file that later will be used by the program that implements the rotating tapered beam high-order element. This process can greatly speed-up the generation of the element stiffness and mass matrices.
828
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
2.2. Crack modeling In this section, the crack is assumed to be always open due to the centrifugal tension forces and modeled as a torsional spring, whose equivalent stiffness will be derived by using the theory of fracture mechanics. The additional strain energy due to
Table 1 Comparison of non-dimensional natural frequencies of a clamped non-rotating pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tapered beam (O ¼ oL2 rA0 =EI0 ) by various approaches.
a
p-FEM
Frobenius Method [36]
NE¼ 1, DOF¼ 20
CFEM [37] NE¼ 9, DOF¼ 20
NE¼ 50, DOF ¼102
0.10
Om1 Om2 Om3
9.8846 27.0085 52.7081
9.8846 27.0084 52.7080
10.0526 27.6496 54.4770
9.8844 27.0067 52.7002
10.9209 29.9041 58.4341
10.9209 29.9041 58.4341
10.9983 30.2341 59.4421
10.9207 29.9022 58.4257
11.8417 32.4755 63.5118
11.8417 32.4755 63.5118
11.8819 32.6660 64.1524
11.8415 32.4734 63.5029
12.6886 34.8384 68.1728
12.6886 34.8384 68.1728
12.7116 34.9594 68.6201
12.6884 34.8362 68.1635
13.4832 37.0526 72.5368
13.4832 37.0527 72.5368
13.4974 37.1359 72.8753
13.4830 37.0504 72.5271
14.2381 39.1539 76.6753
14.2381 39.1539 76.6753
14.2474 39.2153 76.9489
14.2379 39.1515 76.6651
14.9616 41.1656 80.6348
14.9616 41.1656 80.6348
14.9679 41.2135 80.8681
14.9613 41.1631 80.6242
15.6593 43.1039 84.4478
15.6593 43.1039 84.4478
15.6639 43.1432 84.6555
15.6591 43.1013 84.4368
16.3356 44.9806 88.1382
16.3356 44.9806 88.1389
16.3390 45.0143 88.3291
16.3353 44.9779 88.1268
16.9935 46.8049 91.7238
16.9935 46.8049 N/A
16.9961 46.8348 91.9039
16.9932 46.8021 91.7120
17.6354 48.5836 95.2185
17.6354 48.5839 N/A
17.6376 48.6109 95.3919
17.6351 48.5807 95.2063
18.2634 50.3223 98.6336
18.2634 N/A N/A
18.2653 50.3479 98.8029
18.2631 50.3193 98.6210
18.8791 52.0255 101.9779
18.8791 N/A N/A
18.8806 52.0500 102.1454
18.8788 52.0225 101.9649
0.15
Om1 Om2 Om3 0.20
Om1 Om2 Om3 0.25
Om1 Om2 Om3 0.30
Om1 Om2 Om3 0.35
Om1 Om2 Om3 0.40
Om1 Om2 Om3 0.45
Om1 Om2 Om3 0.50
Om1 Om2 Om3
the crack can be expressed as [37] Z a 2 KI da Uc ¼ b 0 E
ð22Þ
in which KI is the stress intensity factor for the first mode crack and has the following form: pffiffiffiffiffiffi 6M pa KI ¼ f ðaÞ ð23Þ bh2c
Table 2 Comparison of non-dimensional natural frequencies of a rotating cantilever pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tapered beam for different non-dimensional rotation speeds l ¼ OL2 rA0 =EI0 . p-FEM NE¼1, DOF¼12
CFEM NE ¼5, DOF ¼12
Wang and Wereley [21]
Gunda and Ganguli [10]
Mode 1 1 2 3 4 5 6 7 8 9 10 11 12
3.9866 4.4368 5.0927 5.8788 6.7434 7.6551 8.5956 9.5540 10.5239 11.5015 12.4845 13.4711
3.9867 4.4368 5.0927 5.8788 6.7434 7.6552 8.5957 9.5542 10.5244 11.5023 12.4857 13.4729
3.9866 4.4368 5.0927 5.8788 6.7434 7.6551 8.5956 9.5540 10.5239 11.5015 12.4845 13.4711
3.9866 4.4368 5.0927 5.8788 6.7434 7.6551 8.5956 9.5540 10.5239 11.5015 12.4845 13.4711
Mode 2 1 2 3 4 5 6 7 8 9 10 11 12
18.4740 18.9366 19.6839 20.6852 21.9053 23.3093 24.8647 26.5437 28.3227 30.1827 32.1085 34.0877
18.4820 18.9442 19.6907 20.6912 21.9105 23.3138 24.8686 26.5472 28.3261 30.1864 32.1127 34.0929
18.4740 18.9366 19.6839 20.6852 21.9053 23.3093 24.8647 26.5437 28.3227 30.1827 32.1085 34.0877
18.4740 18.9366 19.6839 20.6852 21.9053 23.3093 24.8647 26.5437 28.3227 30.1827 32.1085 34.0877
Mode 3 1 2 3 4 5 6 7 8 9 10 11 12
47.4173 47.8716 48.6190 49.6456 50.9338 52.4633 54.2124 56.1595 58.2833 60.5639 62.9829 65.5237
47.5548 48.0068 48.7504 49.7722 51.0548 52.5782 54.3213 56.2626 58.3813 60.6576 63.0733 65.6121
47.4173 47.8716 48.6190 49.6456 50.9338 52.4633 54.2124 56.1595 58.2833 60.5639 62.9829 65.5237
47.4173 47.8717 48.6191 49.6456 50.9338 52.4633 54.2124 56.1595 58.2833 60.5639 62.9829 65.5237
l
0.55
Om1 Om2 Om3 0.60
Om1 Om2 Om3
Table 3 Comparison of non-dimensional natural frequencies of a uniform beam at high rotation speeds.
0.65
Om1 Om2 Om3 0.70
Om1 Om2 Om3
l ¼ 200
l ¼12
Mode Mode Mode Mode
1 2 3 4
Present
Gunda and Ganguli [10]
Present
Gunda and Ganguli [10]*
13.1702 37.6031 79.6145 140.534
13.1702 37.6031 79.6145 140.534
201.0781 492.6931 783.6959 1086.2823
201.0547 492.6843 783.7879 N/A
The symbol * denotes the values are measured from the corresponding figure in the reference.
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
Table 4 Comparison of non-dimensional natural frequencies of a tapered beam at high rotation speeds.
where the geometrical correction function is f ðaÞ ¼ 1:131:374
l ¼ 100
l ¼12 Present
Gunda et al. [11]*
14.0312 35.9064 72.8565 126.4005
14.0313 35.9064 72.8565 126.401
101.8196 218.4248 340.7309 470.4971
101.8233 218.4655 340.7654 471.6630
Uc ¼
Frequency of mode 2
Frequency of mode 1
1 M2 2 kc
ð24Þ
36.12
14.06 14.05 14.04 14.03
36.06 36.00 35.94 35.88
5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
180 Frequency of mode 4
81 Frequency of mode 3
2 3 a a a þ 5:749 4:464 hc hc hc
The additional strain energy can be expressed in an alternative form as
14.07
78
75
72
160
140
120 5
10
15 20 25 Degree of freedom
30
Fig. 2. Convergence of the natural frequencies for tapered rotating beam with l ¼ 12.
226 Frequency of mode 2
105.6
Frequency of mode 1
104.4
103.2
224 222 220
102.0 218 5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
5
10
15 20 25 Degree of freedom
30
540
360
Frequency of mode 4
1 2 3 4
Gunda et al. [11]
Frequency of mode 3
Mode Mode Mode Mode
Present
829
356 352 348 344 340
522 504 486 468
5
10
15 20 25 Degree of freedom
30
Fig. 3. Convergence of the natural frequencies for tapered rotating beam with l ¼ 100.
830
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
From Eqs. (22) and (24), the equivalent stiffness of the crack is obtained as Z a kc ¼ bh4 E=72pa2 s½f ðsÞ2 ds ð25Þ
conventional tapered beam elements to reach convergence. As can be observed, the present p-FEM can achieve high accuracy without numerical difficulty either for small or large taper ratios.
0
Now the strain energy due to the crack in Eq.(3) can be written
3.2. Un-cracked rotating tapered beams
as U3 ¼
1 1 1 ½MðyR yL Þ ¼ ½kc ðyR yL Þ2 ¼ yL yR ½K3 ½yL yR T x ¼ l1 x ¼ l1 2 2 2 ð26Þ
where the stiffness matrix of the crack element is " # kc kc ½K3 ¼ kc kc
ð27Þ
2.3. Finite element equation The Lagrange equation is applied to obtain the finite element equation of the cracked rotating tapered beam d @L @L ¼0 ð28Þ dt @u_ @u where the Lagrangian function L¼ T–U. In the free vibration, the periodic displacement vector can be expressed as uðtÞ ¼ ueiot , where o is the natural frequency. By substituting Eqs. (12)–(14) and (26) into Eq. (28), the FEM equation in matrix form for free vibration of the cracked rotating tapered beam are as follows: ð½Ko2 ½MÞu ¼ 0
ð29Þ
where [K] and [M] are the global stiffness and mass matrices, respectively. It is worth noting that the global stiffness matrix [K] has an additional degree of freedom of the slope to describe the effect of the crack and the global mass matrix [M] has no change compared with that of the un-cracked rotating tapered beam. The requirement of non-trivial solutions for Eq. (29) leads to the frequency equation detð½Ko2 ½MÞ ¼ 0
ð30Þ
where det denotes the determinant of matrix. Then the natural frequencies can be solved. The mode shape corresponding to each frequency may be obtained by back-substitution of the frequency into Eq. (29) in usual manner.
3. Validation In this section, accuracy and convergence studies of the present formulation are carried out. Vibration of beams with different crack and rotation statuses are considered. The results obtained by the proposed approach are compared with those available in literature. 3.1. Un-cracked non-rotating tapered beams The first three non-dimensional natural frequencies of a nonrotating tapered beam clamped at the two ends for different taper ratios a obtained by the p-FEM using sixteen enriched terms are shown in Table 1, where NE and DOF denote the number of elements and the degree of freedom of the system. The results are compared with the values given by Naguleswaran [38] and Chinchalkar [39]. Naguleswaran applied the Frobenius series method to determine the natural frequencies while failing to get effective values due to the ill-conditioned frequency equation for large values of taper ratio a. Chinchalkar employed 50
Here a rotating tapered beam of taper ratio a ¼0.5 is studied. The mass and transverse stiffness distributions along the length direction are given by m(x)¼ rA0(1–0.5x/L) and EI(x)¼ EI0(1–0.5x/L)3. The first three natural frequencies of the tapered beam for various nondimensional rotation speeds l obtained by the present approach are shown in Table 2. The results are compared with the results given by Wang and Wereley [21] and Gunda and Ganguli [10], respectively. Wang and Werelye applied the dynamic stiffness method based on the Frobenius power series solution to predict the natural frequencies and more than 80 terms were used. Gunda and Ganguli used the stiff-string basis function to establish the finite element. The comparison is very good. In addition, the first three natural frequencies, calculated using five conventional rotating tapered
Table 5 Comparison of natural frequencies by the p-FEM and the two-dimensional FEM. Crack
Method
Mode 1
Mode 2
Mode 3
Taper ratio a ¼0.2 Intact p-FEM 2d-FEM 0.5 0.288 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.6 0.292 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.95 0.399 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM
353.95 351.62 350.41 (0.99) 348.36(0.99) 340.28(0.96) 338.93(0.96) 346.36(0.98) 344.38(0.98) 326.91(0.92) 326.23(0.93) 307.55(0.87) 307.34(0.87) 279.14(0.79) 280.17(0.80)
1298.1 1271.5 1251.8(0.96) 1230.0(0.97) 1145.8(0.88) 1132.1(0.89) 1268.1(0.98) 1242.1(0.98) 1201.9(0.93) 1181.6(0.93) 1178.0(0.91) 1156.1(0.91) 1126.4(0.87) 1106.8(0.87)
3041.4 2925.3 3038.9(1.00) 2923.0(1.00) 3033.4(1.00) 2916.8(1.00) 2980.6(0.98) 2862.7(0.98) 2863.3(0.94) 2753.8(0.94) 2863.5(0.94) 2751.7(0.94) 2802.2(0.92) 2690.9(0.92)
Taper ratio a ¼0.4 Intact p-FEM 2d-FEM 0.6 0.306 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.7 0.304 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.8 0.3 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM
324.41 322.83 322.09(0.99) 320.63(0.99) 316.44(0.98) 315.35(0.98) 317.02(0.98) 315.93(0.98) 300.19(0.93) 300.14(0.93) 310.01(0.96) 309.17(0.96) 279.69(0.86) 280.68(0.87)
1442.02 1415.36 1374.9(0.95) 1349.3(0.95) 1245.0(0.86) 1229.1(0.87) 1384.2(0.96) 1357.1(0.96) 1278.9(0.89) 1259.0(0.89) 1436.8(1.00) 1404.3(0.99) 1426.5(0.99) 1394.0(0.98)
3630.2 3486.9 3534.5(0.97) 3373.2(0.97) 3384.6(0.93) 3224.2(0.93) 3588.0(0.99) 3424.2(0.98) 3516.7(0.97) 3358.2(0.96) 3476.5(0.96) 3318.4(0.95) 3218.7(0.89) 3070.2(0.88)
Taper ratio a ¼0.6 Intact p-FEM 2d-FEM 0.65 0.259 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.75 0.275 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM 0.8 0.281 p-FEM 2d-FEM 0.5 p-FEM 2d-FEM
308.15 306.85 308.11(1.00) 306.81(1.00) 307.96(1.00) 306.65(1.00) 305.95(0.99) 304.62(0.99) 298.72(0.97) 297.95(0.97) 302.55(0.98) 301.63(0.98) 286.35(0.93) 286.45(0.93)
1576.10 1538.13 1571.8(1.00) 1534.3(1.00) 1554.7(0.99) 1518.5(0.99) 1503.2(0.95) 1467.5(0.95) 1321.4(0.84) 1300.5(0.85) 1504.5(0.95) 1470.3(0.96) 1347.7(0.86) 1324.8(0.86)
4152.09 3935.17 4097.8(0.99) 3889.0(0.99) 3875.2(0.93) 3688.9(0.94) 4068.9(0.98) 3835.6(0.97) 3898.3(0.94) 3668.5(0.93) 4136.2(1.00) 3903.5(0.99) 4102.5(0.99) 3871.0(0.98)
b
a/hc
Natural frequency (Hz)
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
beam elements that have the same degrees of freedom as that present p-FEM are also listed in Table 2. To examine the accuracy of the p-FEM, a uniform rotating beam and a tapered rotating beam are considered at high and very high rotating speeds. The mass and transverse stiffness variations per unit length are given by m(x)¼ rA0(1–0.8x/L) and EI(x) ¼EI0(1–0.95x/L). Tables 3 and 4 show the lowest four nondimensional natural frequencies obtained by the present method and those obtained by Gunda and Ganguli [10] for the uniform
rotating beam and by Gunda et al. [11] for the tapered rotating beam, respectively. As observed, the present method works well both for high and very high rotating speeds. The convergence study of the present method for the above tapered beam at two different rotating speeds (l ¼12 and l ¼100) on the lowest four modes is conducted. Figs. 2 and 3 depict the convergence behavior of the proposed method, where the number of the enriched terms is increased from 2 to 26, and the total degrees of freedom correspondingly increase from 6 to 30. The
1.0
1.0
1
1
2
2
1
2
1
2
0.9 ωc /ω0
ωc /ω0
0.9
3
0.8
3 0.8
0.7
0.7 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
1.0
0.8
1.0
0.6
0.8
1.0
1.0
2
2 4
1
ωc /ω0
1
0.9
4
0.9
3
3
0.6
lc/L
lc/L
ωc /ω0
831
3
0.8
3
0.8 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
lc/L
lc/L
Fig. 4. Effect of crack location and size on the lowest four frequencies of a cracked rotating tapered beam with various crack sizes (1, 2 and 3 denote a/hc ¼0.1, 0.3 and 0.5, respectively, same as below).
1.00
1.00 1
1
2
2
0.96
1
2
ωc /ω0
1
2
ωc /ω0
0.96
3
0.92
3
0.92
0.88
0.88 0
5
10
15
20
0
5
λ
10
15
20
15
20
λ
1.00
1.00
1
1
2 2 3
0.96 4
ωc /ω0
3
ωc /ω0
0.96 4
3
3 0.92
0.92
0
5
10
λ Fig. 5. Effect of the rotation speed l ¼ OL2 crack sizes.
15
20
0
5
10
λ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rA0 =EI0 on the lowest four frequencies of a cracked rotating tapered beam with crack location lc/L¼ 0.45 and various
832
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
convergence of the proposed method is compared with the CFEM. It is found that with the same number of degrees of freedom, the present p-FEM converge more rapidly than the CFEM.
3.3. Cracked non-rotating tapered beams The beam to be considered is fixed at the large end. The geometrical and physical parameters are L ¼240 mm, b¼12 mm, h1 ¼20 mm, E¼210 GPa, material mass density r ¼7860 kg/m3 and Poisson ratio n ¼0.3. Vibration of this beam was previously studied by Chaudhari and Maiti [40]. They calculated the first three natural frequencies for different crack locations and sizes by 1.00
a two-dimensional finite element program. The first three natural frequencies are computed using the present method with 4 enriched terms in the shape functions, as shown in Table 5. The 2d-FEM results in Table 5 stand for the results presented by Chaudhari and Maiti. The notation b ¼1–(1–a)l1/L denotes the crack location. The numbers in the parentheses stand for the reduction ratios of the natural frequencies of the cracked beam over that of the un-cracked by the consistent method. From Table 3, the agreement for the two methods is good considering the difference in the modeling dimensions. In addition, an interesting phenomenon can be observed, that is, the reduction ratios of the first three natural frequencies obtained by the two methods are in excellent agreement and the maximal 1.00
1
1
2 ωc /ω0
1
ωc /ω0
2
0.96
2
0.92
3
2
1
3
0.96
0.92
0.88
0.88 0
5
10 R/L
15
20
0
1.00
5
1.00
10 R/L
15
20
4
ωc /ω0
20
2
0.98
3
0.96
4
3
ωc /ω0
0.98
15
1
1 2
10 R/L
3
0.94
0
3
0.96 0.94
5
10 R/L
15
20
0
5
Fig. 6. Effect of the hub radius on the lowest four frequencies of a cracked rotating tapered beam with crack location lc/L¼ 0.45 and various crack sizes.
1.0
1.0 1st mode shape
0.6
0
1
2
2nd mode shape
0
0.8
3
0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
0.5
3
-0.5
0.0
0.2
0.4
x/L
0.6
0.8
1.0
x/L 1.0
0 0.5
1
2
4th mode shape
1.0 3rd mode shape
2
0.0
-1.0
1.0
1
3
0.0 -0.5
01 2 3
0.5 0.0 -0.5
0.0
0.2
0.4
0.6 x/L
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
x/L
Fig. 7. Lowest four mode shapes of a rotating tapered beam with and without a crack located at lc/L¼0.45 (0 denotes the intact case here).
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
10-6 ×
10-5 1.0 × Wavelet coefficients of 2nd mode shape
Wavelet coefficients of 1st mode shape
1.0 0.5 0 -0.5 -1.0
0.5 0 -0.5 -1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
x/L
0.6
0.8
1.0
0.6
0.8
1.0
x/L
× 10-5
10-5 1.6 × Wavelet coefficients of 4th mode shape
Wavelet coefficients of 3rd mode shape
1.6
833
0.8 0 -0.8 -1.6
0.8 0 -0.8 -1.6
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
x/L
0.4 x/L
Fig. 8. Wavelet coefficients for the lowest four mode shape of a cracked rotating tapered beam with crack location lc/L ¼0.45 and size a/hc ¼ 0.1.
difference between them does not exceed 1%. According to this phenomenon, the reduction ratio of natural frequencies can serve as a robust damage indicator in crack identification.
4. Vibration analysis In this section, the present formulation is used to analyze the vibration characteristics of a cracked rotating tapered beam. The basic geometrical and material parameter values are the same as those used in section 3.3. Unless specially stated, the taper ratio, hub radius and rotation speed of the rotating beam are a ¼ 0.5, R/L¼0 and l ¼1, respectively. It is assumed that a crack of depth a/hc ¼0.3 occurs at the position lc/L¼0.45. In the following study we use 4 terms to enrich the shape functions of the rotating tapered beam element. Reduction ratios of the lowest four natural frequency of the cracked rotating tapered beam over those of the intact case are computed for different crack locations, crack sizes, rotation speeds and hub radii. The effects of the non-dimensional crack location lc/L and crack size a/hc on the lowest four natural frequencies are shown in Fig. 4. It can be seen that when the crack size is increased, the lowest four natural frequencies of the cracked rotating tapered beam decrease except that the crack is exactly located at the vibration nodes. For a given crack size, the fundamental frequency decreases monotonically when the crack location varies from the free end to the fixed root, and the frequencies of higher modes reduce rapidly as the crack goes away from the vibration nodes. The effects of non-dimensional rotation speed l and hub radius R/L on the natural frequencies of the cracked rotating tapered beam are shown in Figs. 5 and 6, respectively. It is observed that as the rotation speed and the hub radius increase, the lowest four natural frequencies tend to increase. This can be explained by the fact that the rotating centrifugal forces are proportional to O2(Rþx), the increase of the rotation speed as well as hub radius can improve the stiffening effect, which will compensate for the stiffness loss due to the presence of the crack.
The lowest four vibration mode shapes of the cracked rotating tapered beam are shown in Fig. 7. With the increase of crack size, the mode shapes of cracked rotating tapered beam change significantly. A local perturbation on the mode shapes gradually becomes distinct at the crack location. However, when the crack size is very small, the perturbation on the mode shapes cannot be identified. In order to examine this slight perturbation, the spatial wavelet transform approach is used. The lowest four mode shapes for crack size a/hc ¼0.1 are spatially decomposed with the Daubechies Db4 wavelet. The wavelet coefficients of the lowest four modes along the beam length are plotted in Fig. 8. As can be seen, the crack location can be easily detected from the local perturbation based on the wavelet coefficient distributions.
5. Conclusions The p-FEM for vibration analysis of cracked rotating tapered beams is formulated. The shifted Legendre orthogonal polynomials linearly combined with the Hermite cubics are applied to approximate the displacement field within a rotating tapered beam element and the common integrals in the elastic stiffness, centrifugal stiffness and mass matrices of the rotating tapered beam element are extracted and derived in analytical form, which will speed-up the generation of these matrices. The fracture mechanics and the Lagrange equation are used to construct the equivalent stiffness matrix of the crack element and the p-version finite element model of the structural system, respectively. Comparisons of the results for different configurations of beams calculated by the proposed method with those published in literature are made. It shows that the proposed method has favorable accuracy and can yield higher accuracy than the CFEM with equal degrees of freedom. Vibration analysis of cracked rotating tapered beams has been carried out. Effects of the crack location and size, rotation speed and hub radius on vibration characteristics of a cracked rotating tapered beam are investigated. The mode shapes of the cracked rotating tapered beam are
834
Y. Cheng et al. / Finite Elements in Analysis and Design 47 (2011) 825–834
obtained and analyzed by the spatial wavelet transform based approach to detect the crack location. The present work provides helpful information for structural design and diagnosis and can be extended to study vibration of arbitrarily non-uniform cracked rotating beams. References [1] K. Wei, G. Meng, S. Zhou, J. Liu, Vibration control of variable speed/ acceleration rotating beams using smart materials, Journal of Sound and Vibration 298 (2006) 1150–1158. [2] S.-M. Lin, PD control of a rotating smart beam with an elastic root, Journal of Sound and Vibration 312 (2008) 109–124. [3] J.B. Yang, L.J. Jiang, D.C. Chen, Dynamic modelling and control of a rotating Euler–Bernoulli beam, Journal of Sound and Vibration 274 (2004) 863–875. [4] H. Sohn, C.R. Farrar, F.M. Hemez, D.D. Shunk, D.W. Stinemates, B.R. Nadler, A review of structural health monitoring literature: 1996–2001, Los Alamos National Laboratory Report, LA-13976-MS, 2003. [5] D.H. Hodges, M.J. Rutkowski, Free vibration analysis of rotating beams by a variable-order finite-element method, American Institute of Aeronautics and Astronautics Journal 19 (1981) 1459–1466. [6] S. Naguleswaran, Lateral vibration of a centrifugally tensioned Euler– Bernoulli beam, Journal of Sound and Vibration 176 (1994) 613–624. [7] S.S. Rao, R.S. Gupta, Finite element vibration analysis of rotating Timoshenko beams, Journal of Sound and Vibration 242 (2001) 103–124. [8] J.B. Gunda, A.P. Singh, P.S. Chhabra, R. Ganguli, Free vibration analysis of rotating tapered blades using Fourier-p superelement, Structural Engineering and Mechanics 27 (2007) 243–257. [9] J.B. Gunda, R. Ganguli, New rational interpolation functions for finite element analysis of rotating beams, International Journal of Mechanical Sciences 50 (2008) 578–588. [10] J.B. Gunda, R. Ganguli, Stiff String basis functions for vibration analysis of high speed rotating beams, Journal of Applied Mechanics 75 (2008) 245021–245025. [11] J.B. Gunda, R.K. Gupta, R. Ganguli, Hybrid stiff-string-polynomial basis functions for vibration analysis of high speed rotating beams, Computers and Structures 87 (2009) 254–265. [12] A. Bazoune, Y.A. Khulief, N.G. Stephen, Further results for modal characteristics of rotating tapered timoshenko beams, Journal of Sound and Vibration 219 (1999) 157–174. [13] A. Bazoune, Relationship between softening and stiffening effects in terms of Southwell coefficients, Journal of Sound Vibration 287 (2005) 1027–1030. [14] A. Bazoune, Effect of tapering on natural frequencies of rotating beams, Shock and Vibration 14 (2007) 169–179. [15] H.H. Yoo, S.H. Shin, Vibration analysis of rotating cantilever beams, Journal of Sound Vibration 212 (1998) 807–828. [16] H.H. Yoo, J.Y. Kwak, J. Chung, Vibration analysis of rotating pre-twisted blades with a concentrated mass, Journal of Sound and Vibration 240 (2001) 891–908. [17] J. Chung, H.H. Yoo, Dynamic analysis of a rotating cantilever beam by using the finite element method, Journal of Sound and Vibration 249 (2002) 147–164. [18] R. Attarnejad, A. Shahba, Basic displacement functions for centrifugally stiffened tapered beams, Communications in Numerical Methods in Engineering (2010). doi:10.1002/cnm.1365.
[19] K.G. Vinod, S. Gopalakrishnan, R. Ganguli, Free vibration and wave propagation analysis of uniform and tapered rotating beams using spectrally formulated finite elements, International Journal of Solids and Structures 44 (2007) 5875–5893. [20] J.R. Banerjee, H. Su, D.R. Jackson, Free vibration of rotating tapered beams using the dynamic stiffness method, Journal of Sound and Vibration 298 (2006) 1034–1054. [21] G. Wang, N.M. Wereley, Free vibration analysis of rotating blades with uniform tapers, American Institute of Aeronautics and Astronautics Journal 42 (2004) 2429–2437. [22] O. Ozdemir, M.O. Kaya, Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method, Journal of Sound and Vibration 289 (2006) 413–420. [23] A.D. Dimarogonas, Vibration of cracked structures: a state-of-the-art review, Engineering Fracture Mechanics 55 (1996) 831–857. [24] S.W. Doebling, C.R. Farrar, M.B. Prime, A summary review of vibration-based damage identification methods, The Shock and Vibration Digest 30 (1998) 91–105. [25] A.C. Chasalevris, C.A. Papadopoulos, Identification of multiple cracks in beams under bending, Mechanical Systems and Signal Processing 20 (2006) 1631–1673. [26] L.-W. Chen, C.-L. Chen, Vibration and stability of cracked thick rotating blades, Computers and Structures 28 (1988) 67–74. [27] C.-C. Chang, L.-W. Chen, Damage detection of cracked thick rotating blades by a spatial wavelet based approach, Applied Acoustics 65 (2004) 1095–1111. [28] S.-S. Kim, J.-H. Kim, Rotating composite beam with a breathing crack, Composite Structures 60 (2003) 83–90. [29] J.H. Kuang, B.W. Huang, The effect of blade crack on mode localization in rotating bladed disks, Journal of Sound and Vibration 227 (1999) 85–103. [30] B.W. Huang, Effect of number of blades and distribution of cracks on vibration localization in a cracked pre-twisted blade system, International Journal of Mechanical Sciences 48 (2006) 1–10. [31] A.A. Masoud, S. Al-Said, A new algorithm for crack localization in a rotating Timoshenko beam, Journal of Vibration and Control 15 (2009) 1541–1561. [32] B. Szabo, I. Babuska, Finite Element Analysis, Wiley, New York, 1991. [33] A. Houmat, Three-dimensional free vibration analysis of plates using the h–p version of the finite element method, Journal of Sound and Vibration 290 (2006) 690–704. [34] P. Ribeiro, Forced periodic vibrations of laminated composite plates by a p-version, first order shear deformation, finite element, Composites Science and Technology 66 (2006) 1844–1856. [35] H.H. Yoo, J.E. Cho, J. Chung, Modal analysis and shape optimization of rotating cantilever beams, Journal of Sound and Vibration 290 (2006) 223–241. [36] E.T. Whittaker, G.N. Watson, A Course Of Modern Analysis, 4th edition, Cambridge Mathematical Library, 1996. [37] W.M. Ostachowicz, M. Krawczuk, Analysis of the effect of crack on the natural frequecy of a cantilever beam, Journal of Sound and Vibration 150 (1991) 191–201. [38] S. Naguleswaran, A direct solution for the transverse vibration of eulerbernoulli wedge and cone beams, Journal of Sound and Vibration 172 (1994) 289–304. [39] S. Chinchalkar, Determination of crack location in beams using natural frequencies, Journal of Sound and Vibration 247 (2001) 417–429. [40] T.D. Chaudhari, S.K. Maiti, Modelling of transverse vibration of beam of linearly variable depth with edge crack, Engineering Fracture Mechanics 63 (1999) 425–445.