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Free vibration analysis of rotating Timoshenko beams with multiple delaminations
Composites: Part B 44 (2013) 733–739
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Free vibration analysis of rotating Timoshenko beams with multiple delaminations Yang Liu 1, Dong Wei Shu ⇑ School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
a r t i c l e
i n f o
Article history: Received 17 November 2011 Accepted 2 January 2012 Available online 12 January 2012 Keywords: A. Laminates B. Delamination B. Vibration C. Analytical modeling
a b s t r a c t Analytical solutions are developed to study the free vibrations of rotating Timoshenko beams with multiple delaminations. The Timoshenko beam theory and both the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration are adopted. Parametric studies are performed to study the influences of Timoshenko effect and rotating speed on delamination vibration. Results show that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and the rotating speed. Also, the comparison between ‘free mode’ assumption and ‘constrained assumption’ are conducted. Furthermore, the effect of delamination on mode shapes is also influenced by rotating speed and Timoshenko effect. For both Timoshenko effect and rotating speed, the influences on the second vibration mode shape are more significant. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The vibration characteristics of rotating composite beams are under frequent investigation due to its wide applications, such as helicopter blades, wind turbines and turbo-machinery. One common defect of laminate composite is delamination, which may significantly reduces the stiffness and strength of the structures [1], as well as the vibration characteristics. To study the influence of a through-width delamination on the free vibration of an isotropic beam, Wang et al. [2] presented an analytical model using four Euler–Bernoulli beams that are joined together. They assumed that the delaminated layers deform ‘freely’ without touching each other (‘free mode’ model) and will have different transverse deformations. While Mujumdar and Suryanarayan [3] assumed that the delaminated layers are in touch along their whole length all the time, but are allowed to slide over each other (‘constrained mode’ model). Thus, the delaminated layers are ‘constrained’ to have identical transverse deformations. This ‘constrained mode’ model was extended by Shu and Fan [4] on a bimaterial beam. However, the ‘constrained mode’ model failed to predict the opening in the mode shapes found in the experiments by Shen and Grady [5]. Analytical solutions for beams with multiple delaminations have been presented by many researchers. Shu [6] presented an analytical solution to study a sandwich beam with double delaminations. His study emphasized on the influence of the contact mode, ‘free’ and ‘constrained’, between the delaminated layers and the local deformations at the delamination fronts. Shu and
Della [7,8] and Della and Shu [9] used the ‘free mode’ and ‘constrained mode’ assumptions study a composite beam with various multiple delamination configurations. Their study emphasized on the influence of a second delamination on the first and second natural frequencies and the corresponding mode shapes of a delaminated beam. Della and Shu [10] also studied the free vibration of a delaminated bimaterial beam using Euler–Bernoulli beam theory. The free vibration characteristics of rotating beam have received considerable attention. Many previous studies have been based on Euler–Bernoulli beam theory and various approximate solution techniques [11,12]. Al-Ansary [13] studied the effects of rotary inertia on the extensional tensile force and on the eigenvalues of beams rotating uniformly about a transverse axis. Du et al. [14] presented a convergent power series expression to solve analytically for the exact natural frequencies and mode shapes of rotating Timoshenko beams. The effects of angular velocity, shear and rotary inertia are carefully studied. In the present work, analytical solutions are developed to study the free vibrations of delaminated rotating Timoshenko beams. The Timoshenko beam theory and both the ‘free mode’ and the ‘constrained mode’ assumption in delamination vibration are used. First, the vibration of a rotating Timoshenko beam with single delamination is formulated. Second, the present results are verified against previous published results. Third, a comprehensive study is conducted on how the effect of delaminations on natural frequencies is influenced by rotating speed and Timoshenko effect. Finally the first and second mode shapes are thoroughly investigated. 2. Formulation
⇑ Corresponding author. Tel.: +65 67904440. 1
E-mail addresses: [email protected] (Y. Liu), [email protected] (D.W. Shu). Tel.: +65 84327159.
1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2012.01.037
The analytical solution to the vibration of rotating Timoshenko beam, reported by Du et al. [14], is used here to solve the governing
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equations. Both ‘free mode’ and ‘constrained mode’ assumptions are adopted to investigate the delamination vibration problem. Fig. 1 shows a beam with length L and thickness H1. The beam is made wi(x, t) of two distinct layers, with Young’s modulus E2 and E3, and thickness H2 and H3. The beam is separated along the interface by a delamination with length a and located at a distance d from the center of the beam. The beam can then be subdivided into three span-wise regions, a delamination region and two integral regions. The delamination region is comprised of two segments (delaminated layers), beam 2 and beam 3, which are joined at their ends to the integral segments, beam 1 and beam 4. Each of the four beams is treated as rotating Timoshenko beams. 2.1. Governing equations and analytical solutions Let w0i and wi(x, t) and denote the slope of the deflection curve K when the shearing is neglected and the midplane deflection of beam i, respectively. The governing equations for the free vibration of a rotating Timoshenko beam with single delamination are: (i = 1–4).
ðEIi w0i Þ0 þ KAi Gi ðw0i wi Þ ¼ qi Ii w€i
ð1Þ
€i ¼ 0 ðKAi Gi ðw0i wi ÞÞ0 þ T i w0i Þ0 qi Ai w
ð2Þ
where qi is the mass density, Ai is the cross-sectional area of the beam, Gi shear modulus, Ii is the moment of area of the cross section and K is the shear coefficient. Ti denotes the centrifugal force exerted on the cross section of beam i. EIi (i = 1–4) is the bending stiffness of beam i. A closed form solution [14] is adopted to solve for the flexural natural frequencies and corresponding mode shapes. 2.2. Free mode model Eqs. (1) and (2) are applied to the four interconnected sub-beams, respectively (Fig. 1b). The appropriate boundary conditions that can be applied at the supports, x = x1 and x = x4, are W i ¼ 0; W 0i ¼ 0, if the end of the beam is clamped; W 00i ¼ 0; W 000 i ¼ 0, if free, where i = 1, 4.
The continuity conditions for deflection and slope at x = x2 are:
W 1 ¼ W 2 ¼ W 3 ; wi ¼ w2 ¼ w3
ð3Þ
The continuity condition for shear and bending moments at x = x2 are: 000 EI1 W 000 1 ¼ ðEI 2 þ EI 3 ÞW 2
EI1 þ
ðE2 H2 E3 H3 ÞH21 ðW 01 W 04 Þ ¼ EI2 W 002 þ EI3 W 003 4ðE2 H2 þ E3 H3 Þ
ð4Þ ð5Þ
similarly, we can derive the continuity conditions at x = x3. A non-trivial solution exists when the determinant of the coefficient matrix vanishes. 2.3. Constrained mode model The ‘constrained mode’ model is simplified by the assumption that the delaminated layers are constrained to have the same transverse deformations. The delaminated beam is analyzed as three beam segments I–III. The boundary conditions for the ‘constrained mode’ are identical to the boundary conditions of the ‘free mode’. The continuity conditions for deflection, slope, shear, and bending moments at x = x2 are:
W I ¼ W II ; w0I ¼ w0II
ð6Þ
000 EI1 W 000 I ¼ ðEI 2 þ EI 3 ÞW II
ð7Þ
EI1 þ
ðE2 H2 E3 H3 ÞH21 4ðE2 H2 þ E3 H3 Þ
ðW 0I W 0III Þ ¼ EI2 W 00II
ð8Þ
similarly, we can derive the continuity conditions at x = x3. A non-trivial solution exists when the determinant of the coefficient matrix vanishes. 2.4. Introduction on non-dimensional parameters To study the effect of rotating speed, as well as Timoshenko effect on delaminated beams, similar parameters are adopted to
Fig. 1. (a) A rotating Timoshenko beam is delaminated along the interface and (b) the delaminated beam is analyzed as four interconnected beams.
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Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739
represent the effect of rotating speed as well as Timoshenko effect as they are in the work of Du et al. [14]. To study the effect of shear deformation and rotary inertia, a relative slenderness ratio r = R/L is adopted, where R is the radius of gyration and L is the beam length. A higher value of r indicates a more prominent Timoshenko effect. pffiffiffiffiffiffiffiffiffiffiffiffiffi The effect of rotating speed is represented by g ¼ qA=EIXL2 , where X is the angular velocity. Also, the effect of root offset value is studied, we use a = r0/L, where r0 is the offset distance.
frequency of the delaminated beam k2 is normalized with respect to that of undelaminated beam k2d . As shown in Fig. 2, k2 =k2d decreases as r increases, indicating a higher influence of Timoshenko effect leads to lower natural frequencies. The results agree with Du’s work [14] (Fig. 5). The difference between natural frequencies of delamination length a/L = 0.2 and 0.8 becomes smaller as r increases, or as the thickness-wise location of delamination H2/H1 increases from 0.2 to 0.5. It shows that the effect of delamination on natural frequency becomes smaller with a bigger Timoshenko effect.
3. Results and discussion 3.1. Verification To validate the present study, comparisons with published results on static Euler–Bernoulli beam with a single delamination and undelaminated rotating Timoshenko beam are made. The first non-dimensional natural frequencies of a clamped–clamped beam with a single midplane central delamination having various lengths are compared with the analytical results of Wang et al. [2] and FEM results of Lee [15]. The first non-dimensional natural frequencies of a clamped–free rotating Timoshenko beams without delaminations are compared with the analytical results of Du et al. [14]. Tables 1 and 2 show good agreement between the present results and previous results. 3.2. Rotating Timoshenko beams with a single delamination 3.2.1. Static Timoshenko beam with a single delamination Fig. 2 shows the normalized frequency k2 =k2d of a static beam with a single central delamination versus r. This represents the influence of Timoshenko effect on natural frequency of beams with a single delamination of various lengths. The non-dimensional
Table 1 Non-dimensional fundamental frequency (k2) of a clamped–clamped isotropic static Euler–Bernoulli beam with a midplane delamination.
3.2.2. Rotating Euler–Bernoulli beam with a single delamination Fig. 3 shows the effect of rotating speed on the normalized ‘free mode’ natural frequency of beam with a single central delamination, neglecting the Timoshenko effect. As shown in Fig. 3, k2 =k2d increases as angular velocity g becomes higher, which agrees well with the work of Du et al. [14] (Table 7). The lower the rotating speed the higher the percentage the frequency decreases as delamination length a/L increases. It can be concluded that the effect of delamination is smaller when the beam is subjected to a bigger centrifugal force. When comparing the results between (a) and (b), it can be concluded that the effect of delamination length is more prominent when the thickness-wise location of delamination H2/H1 increases from 0.2 to 0.5. 3.2.3. Mode 1 and mode 2 ‘free mode’ frequencies Fig. 4 shows the results of the normalized frequency of modes 1 and 2. It incorporates the influence of rotating speed, Timoshenko effect and delamination. For both vibration modes, k2 =k2d increases as the rotating speed increase and decreases as r increases. Timoshenko effect exert a bigger influence on mode 2 natural frequency than on mode 1, while the effect of rotating speed has a smaller influence on mode 2 than on mode 1.
H2 /H1=0.2
1
Delamination length, a/L
Present constrained and free
Analytical FEM Wang et al. [2]
FEM Lee [15]
0.95
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
22.37 22.37 22.36 22.24 21.83 20.89 19.30 17.23 15.05 13.00
22.39 22.37 22.35 22.23 21.83 20.88 19.29 17.23 15.05 13.00
22.36 22.36 22.35 22.23 21.82 20.88 19.28 17.22 15.05 12.99
0.85
a/L=0.2
0.8
a/L=0.6
0.9
a/L=0.6
0.75
a/L=0.8
0.7 0.65 0.6
r 0
0.05
0.1
0.15
0.2
0.25
0.3
(a) H2 /H1=0.5
1 0.95
Table 2 Non-dimensional mode 1 frequency (k2) of a clamped–free rotating Timoshenko beam. r
g=4 Present
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
5.58 5.562 5.537 5.502 5.46 5.411 5.358 5.300 5.239 5.176
g=8 Du et al. [14] 5.58 5.564 5.539 5.505 5.463 5.415 5.363 5.307 5.249 5.191
Present 9.245 9.212 9.162 9.097 9.023 8.944 8.861 8.775 8.688 8.599
0.9 0.85
a/L=0.2 a/L=0.4 a/L=0.6 a/L=0.8
0.8 0.75
Du et al. [14] 9.246 9.215 9.167 9.106 9.036 8.963 8.889 8.815 8.744 8.677
0.7 0.65 0.6
r
0.55 0
0.05
0.1
0.15
0.2
0.25
0.3
(b) Fig. 2. Influence of rotary inertia and shear deformation with different delamination length on normalized mode 1 ‘free mode’ frequency k2 =k2d for a static homogeneous clamped–free beam with a single central delamination. (a) H2 = 0.2H1 and (b) H2 = 0.5H1.
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H 2/H1=0.2 4 3.5 3 2.5 2 1.5 1
a/L
0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) Fig. 5. Influence of rotary inertia and shear deformation on ‘free mode’ and ‘constrained mode’ frequencies for a homogeneous clamped–free beam with single delamination considering different rotating speed g. a/L = 0.4 and H2/H1 = 0.5.
H2 /H1=0.5 4 3.5 3 2.5 2 1.5 1
a/L
0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b) Fig. 3. Influence of rotating speed on normalized mode 1 ‘free mode’ frequency k2 =k2d for a rotating homogeneous clamped–free Euler–Bernoulli beam with single central delamination. (a) H2 = 0.2H1 and (b) H2 = 0.5H1.
(a)
(a) (b) Fig. 6. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for a homogeneous clamped–free beam with two double delaminations considering different r. d/L = 0. (a) a/L = 0.1 and (b) a/L = 0.4.
(b) Fig. 4. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for a homogeneous clamped–free beam with single delamination considering different r. (a) a/L = 0.1 and (b) a/L = 0.4.
For mode 1, the differences of natural frequencies between r = 0.01 and 0.1 increase monotonously as the rotating speed becomes higher. For mode 2, the differences of natural frequencies between r = 0.01 and 0.1 do not show a monotonous relation with rotating speed. Instead, the difference of natural frequencies decreases at first and then increase. The rotating speed corresponding to the minimum difference of nature frequencies is higher when delamination length a/L = 0.1 than that of a/L = 0.4. 3.2.4. Comparison between ‘free mode’ and ‘constrained mode’ Fig. 5 shows the influence of Timoshenko effect on the normalized frequencies k2 =k2d of beams with single central delamination,
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Fig. 7. Beams with two central delaminations of equal length.
(a)
(b) Fig. 8. Beams with two central enveloped delaminations. Fig. 9. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for a homogeneous clamped–free beam with two enveloping delaminations considering different r. d/L = 0, H4 = 0.33H1. (a) a1/L = 0.1, a2/L = 0.2 and (b) a1/L = 0.2, a2/L = 0.4.
where a/L = 0.4 and H2/H1 = 0.5. For both ‘free mode’ and ‘constrained mode’, k2 =k2d increases when the rotating speed g becomes higher. k2 =k2d decreases when the Timoshenko effect becomes more prominent. For both ‘free mode’ and ‘constrained mode’ a higher rotating speed leads to higher percentage decease of k2 =k2d . The natural frequency of ‘constrained mode’ is bigger than that of the ‘free mode’, which agrees with the established results of previous work [4]. The differences of natural frequency between the ‘constrained mode’ and the ‘free mode’ increase when rotating speed becomes higher. The differences of natural frequencies do not change monotonously as r increases. Generally, the difference between ‘constrained mode’ and ‘free mode’ increases at first then decreases as r becomes bigger. As the rotating speed becomes bigger, the value of r corresponding to the maximum difference between the ‘free mode’ and ‘constrained mode’ decreases. 3.3. Rotating Timoshenko beams with double delaminations of equal length Fig. 6 shows the case of beams with double delaminations of equal length (as shown in Fig. 7). Regarding the Timoshenko effect and the effect of rotating speed on mode 1 and mode 2 natural frequencies, similar results to Fig. 4 can be observed. Compared with beams with a single delamination, natural frequencies of beams with double delaminations is smaller when both cases have the same delamination length. However, the difference is less significant with a higher rotating speed or a more prominent Timoshenko effect. Regarding the differences of natural frequencies between r = 0.01 and 0.1, similar results to Fig. 4 can be observed for both mode 1 and mode 2.
Fig. 10. Influence of rotary inertia and shear deformation on ‘free mode’ and ‘constrained mode’ frequencies for a homogeneous clamped–free beam with two enveloping delaminations considering different rotating speed g. a1/L = 0.2, a2/ L = 0.4.
3.4. Rotating Timoshenko beams with two enveloped delaminations 3.4.1. Mode 1 and mode 2 ‘free mode’ frequencies The variations of the normalized frequency k2 =k2d with the rotating speed as well as Timoshenko effect for beams with two central enveloping delaminations (Fig. 8) are shown in Fig. 9. The thickness-wise location of two delaminations is equally distributed. Natural frequencies k2 =k2d of mode 1 and mode 2 are both included. The result shows the change of frequencies according to the rotating speed and they are compared among three different cases, each of which is under different influences of Timoshenko effect.
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Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739 2.2
a/L=0.1
2.1 2 1.9
a/L=0.2
1.8
a/L=0.5 1.7
a/L=0.8
1.6 1.5 1.4 0
0.2
0.4
0.6
0.8
1
α
Fig. 11. Effect of root offset value on normalized natural frequency of rotating Timoshenko beam with single delamination. g = 4, r = 0.05.
a/L=0.4
(a) a/L=0.1
(a)
4
a/L=0.4
(b) Fig. 12. Mode shapes of the first mode for clamped–free rotating Timoshenko beam with single delamination. (a) Consider different values of r with a constant rotating speed g = 4 and (b) consider different rotating speed g with a constant value of r = 0.05.
For mode 1 and mode 2, k2 =k2d increases as rotating speed g increases, decreases as r increases. Regarding the Timoshenko effect and the effect of rotating speed on mode 1 and 2 natural frequencies, similar results to Fig. 4 can be observed. For mode 1, the differences of natural frequencies between r = 0.01 and 0.1 increase monotonously as the rotating speed becomes higher. For mode 2, the differences of natural frequencies between r = 0.01 and 0.1 do not show a monotonous relation with rotating speed. Instead, the difference of natural frequencies decreases at first and then increase. The rotating speed corresponding to the minimum difference of nature frequencies is higher with delamination length a1/L = 0.1, a2/L = 0.2 than a1/L = 0.2, a2/L = 0.4.
(b) Fig. 13. Mode shapes of the second mode for clamped–free rotating Timoshenko beam with single delamination. (a) Consider different values of r with a constant rotating speed g = 4, when a/L = 0.1 and 0.4. (b) Consider different rotating speed g with a constant value of r = 0.05, when a/L = 0.1 and 0.4.
3.4.2. Comparison between ‘free mode’ and ‘constrained mode’ Fig. 10 shows the influence of Timoshenko effect on the normalized frequencies k2 =k2d with two central enveloping delaminations, results from both ‘free mode’ and ‘constrained mode’ are presented. The thickness-wise locations of two delaminations are equally distributed. For both ‘free mode’ and ‘constrained mode’, k2 =k2d increases when the rotating speed g becomes higher. k2 =k2d decreases
Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739
when the Timoshenko effect becomes more prominent. For both ‘free mode’ and ‘constrained mode’ k2 =k2d decreases faster with a higher rotating speed. The natural frequency of ‘constrained mode’ is bigger than the one with ‘free mode’, which agrees with the established results of previous work [7]. However, the difference between the ‘constrained mode’ and the ‘free mode’ does not change monotonously as r increases, which is similar to what we observe in the case of beams with single delamination.
3.5. Effect of root offset value Fig. 11 shows the effect of root offset value on normalized natural frequency when r = 0.05 and g = 4. As we can see it shows a linear relation between the frequency and root offset value for each delamination length under consideration.
3.6. Mode shape of delaminated rotating Timoshenko beam 3.6.1. Mode 1 Fig. 12 shows the ‘free mode’ first mode shape of delaminated beam under the influence of rotating speed and Timoshenko effect respectively. As shown in Fig. 12a, when the rotating speed is kept as g = 4, the change of mode shape due to delamination and Timoshenko effect is illustrated. Fig. 12b presents the case of the change of mode shape due the delamination and rotating effect, when the value of r is kept at 0.05. The amplitude is bigger in the middle when the delamination length is longer. Also, the amplitude is bigger when the rotating speed increases or with a bigger value of r. We can conclude that the bigger the effect of shear deformation and rotary inertia or the rotating speed, the bigger the amplitude of vibration.
3.6.2. Mode 2 As shown in Fig. 13, we have the similar results for mode 2 as those with mode 1. Furthermore, the effect of shear deformation and rotary inertia as well as rotating speed has a bigger effect on the second mode shape than the first one.
739
4. Conclusions Here the analytical solution to the vibration of rotating Timoshenko beams with multiple delaminations is developed. We study how Timoshenko effect and rotating speed influence the effect of delamination on natural frequencies and mode shapes. It can be concluded that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and rotating speed. The comparison between ‘free mode’ assumption and ‘constrained mode’ assumption is also investigated. Furthermore, Timoshenko effect and rotating speed are shown to influence the effect of delamination on the mode shapes. References [1] Tay TE. Characterization and analysis of delamination fracture in composites: an overview of developments from 1990 to 2001. Appl Mech Rev 2003;56(1):1–31. [2] Wang JTS, Liu YY, Gibby JA. Vibration of split beams. J Sound Vib 1982;84(4): 491–502. [3] Mujumdar PM, Suryanarayan S. Flexural vibrations of beams with delaminations. J Sound Vib 1982;125(3):441–61. [4] Shu D, Fan H. Free vibration of a bimaterial split beam. Composites: Part B 1996;27(1):79–84. [5] Shen M-HH, Grady JE. Free vibrations of delaminated beams. AIAA J 1992;30(5):1361–70. [6] Shu D. Vibration of sandwich beams with double delaminations. Compos Sci Technol 1995;54(1):101–9. [7] Shu D, Della CN. Vibrations of multiple delaminated beams. Compos Struct 2004;64(3–4):467–77. [8] Shu D, Della CN. Free vibration analysis of composite beams with two nonoverlapping delaminations. Int J Mech Sci 2004;46(4):509–26. [9] Della CN, Shu D. Free vibration analysis of composite beams with overlapping delaminations. Eur J Mech A 2005;24(3):491–503. [10] Della CN, Shu D. Free vibration of delaminated bimaterial beams. Compos Struct 2007;80:212–20. [11] Wright AD, Smith CE. Vibration modes of centrifugally stiffened beams. J Appl Mech 1982;49:197–202. [12] Lin WH, Yeh FH. Vibrations of nonuniform rotating beams. J Sound Vib 1987;119:379–84. [13] Al-Ansary MD. Flexural vibrations of rotating beams considering rotary inertia. Comput Struct 1998;69:321–8. [14] Du H, Lim MK, Liew KM. A power series solution for vibration of a rotating Timoshenko beam. J Sound Vib 1994;175:505–23. [15] Lee J. Free vibration analysis of delaminated composite beams. Comput Struct 2000;74(2):121–9.
Contents lists available at SciVerse ScienceDirect
Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Free vibration analysis of rotating Timoshenko beams with multiple delaminations Yang Liu 1, Dong Wei Shu ⇑ School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore
a r t i c l e
i n f o
Article history: Received 17 November 2011 Accepted 2 January 2012 Available online 12 January 2012 Keywords: A. Laminates B. Delamination B. Vibration C. Analytical modeling
a b s t r a c t Analytical solutions are developed to study the free vibrations of rotating Timoshenko beams with multiple delaminations. The Timoshenko beam theory and both the ‘free mode’ and ‘constrained mode’ assumptions in delamination vibration are adopted. Parametric studies are performed to study the influences of Timoshenko effect and rotating speed on delamination vibration. Results show that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and the rotating speed. Also, the comparison between ‘free mode’ assumption and ‘constrained assumption’ are conducted. Furthermore, the effect of delamination on mode shapes is also influenced by rotating speed and Timoshenko effect. For both Timoshenko effect and rotating speed, the influences on the second vibration mode shape are more significant. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction The vibration characteristics of rotating composite beams are under frequent investigation due to its wide applications, such as helicopter blades, wind turbines and turbo-machinery. One common defect of laminate composite is delamination, which may significantly reduces the stiffness and strength of the structures [1], as well as the vibration characteristics. To study the influence of a through-width delamination on the free vibration of an isotropic beam, Wang et al. [2] presented an analytical model using four Euler–Bernoulli beams that are joined together. They assumed that the delaminated layers deform ‘freely’ without touching each other (‘free mode’ model) and will have different transverse deformations. While Mujumdar and Suryanarayan [3] assumed that the delaminated layers are in touch along their whole length all the time, but are allowed to slide over each other (‘constrained mode’ model). Thus, the delaminated layers are ‘constrained’ to have identical transverse deformations. This ‘constrained mode’ model was extended by Shu and Fan [4] on a bimaterial beam. However, the ‘constrained mode’ model failed to predict the opening in the mode shapes found in the experiments by Shen and Grady [5]. Analytical solutions for beams with multiple delaminations have been presented by many researchers. Shu [6] presented an analytical solution to study a sandwich beam with double delaminations. His study emphasized on the influence of the contact mode, ‘free’ and ‘constrained’, between the delaminated layers and the local deformations at the delamination fronts. Shu and
Della [7,8] and Della and Shu [9] used the ‘free mode’ and ‘constrained mode’ assumptions study a composite beam with various multiple delamination configurations. Their study emphasized on the influence of a second delamination on the first and second natural frequencies and the corresponding mode shapes of a delaminated beam. Della and Shu [10] also studied the free vibration of a delaminated bimaterial beam using Euler–Bernoulli beam theory. The free vibration characteristics of rotating beam have received considerable attention. Many previous studies have been based on Euler–Bernoulli beam theory and various approximate solution techniques [11,12]. Al-Ansary [13] studied the effects of rotary inertia on the extensional tensile force and on the eigenvalues of beams rotating uniformly about a transverse axis. Du et al. [14] presented a convergent power series expression to solve analytically for the exact natural frequencies and mode shapes of rotating Timoshenko beams. The effects of angular velocity, shear and rotary inertia are carefully studied. In the present work, analytical solutions are developed to study the free vibrations of delaminated rotating Timoshenko beams. The Timoshenko beam theory and both the ‘free mode’ and the ‘constrained mode’ assumption in delamination vibration are used. First, the vibration of a rotating Timoshenko beam with single delamination is formulated. Second, the present results are verified against previous published results. Third, a comprehensive study is conducted on how the effect of delaminations on natural frequencies is influenced by rotating speed and Timoshenko effect. Finally the first and second mode shapes are thoroughly investigated. 2. Formulation
⇑ Corresponding author. Tel.: +65 67904440. 1
E-mail addresses: [email protected] (Y. Liu), [email protected] (D.W. Shu). Tel.: +65 84327159.
1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2012.01.037
The analytical solution to the vibration of rotating Timoshenko beam, reported by Du et al. [14], is used here to solve the governing
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equations. Both ‘free mode’ and ‘constrained mode’ assumptions are adopted to investigate the delamination vibration problem. Fig. 1 shows a beam with length L and thickness H1. The beam is made wi(x, t) of two distinct layers, with Young’s modulus E2 and E3, and thickness H2 and H3. The beam is separated along the interface by a delamination with length a and located at a distance d from the center of the beam. The beam can then be subdivided into three span-wise regions, a delamination region and two integral regions. The delamination region is comprised of two segments (delaminated layers), beam 2 and beam 3, which are joined at their ends to the integral segments, beam 1 and beam 4. Each of the four beams is treated as rotating Timoshenko beams. 2.1. Governing equations and analytical solutions Let w0i and wi(x, t) and denote the slope of the deflection curve K when the shearing is neglected and the midplane deflection of beam i, respectively. The governing equations for the free vibration of a rotating Timoshenko beam with single delamination are: (i = 1–4).
ðEIi w0i Þ0 þ KAi Gi ðw0i wi Þ ¼ qi Ii w€i
ð1Þ
€i ¼ 0 ðKAi Gi ðw0i wi ÞÞ0 þ T i w0i Þ0 qi Ai w
ð2Þ
where qi is the mass density, Ai is the cross-sectional area of the beam, Gi shear modulus, Ii is the moment of area of the cross section and K is the shear coefficient. Ti denotes the centrifugal force exerted on the cross section of beam i. EIi (i = 1–4) is the bending stiffness of beam i. A closed form solution [14] is adopted to solve for the flexural natural frequencies and corresponding mode shapes. 2.2. Free mode model Eqs. (1) and (2) are applied to the four interconnected sub-beams, respectively (Fig. 1b). The appropriate boundary conditions that can be applied at the supports, x = x1 and x = x4, are W i ¼ 0; W 0i ¼ 0, if the end of the beam is clamped; W 00i ¼ 0; W 000 i ¼ 0, if free, where i = 1, 4.
The continuity conditions for deflection and slope at x = x2 are:
W 1 ¼ W 2 ¼ W 3 ; wi ¼ w2 ¼ w3
ð3Þ
The continuity condition for shear and bending moments at x = x2 are: 000 EI1 W 000 1 ¼ ðEI 2 þ EI 3 ÞW 2
EI1 þ
ðE2 H2 E3 H3 ÞH21 ðW 01 W 04 Þ ¼ EI2 W 002 þ EI3 W 003 4ðE2 H2 þ E3 H3 Þ
ð4Þ ð5Þ
similarly, we can derive the continuity conditions at x = x3. A non-trivial solution exists when the determinant of the coefficient matrix vanishes. 2.3. Constrained mode model The ‘constrained mode’ model is simplified by the assumption that the delaminated layers are constrained to have the same transverse deformations. The delaminated beam is analyzed as three beam segments I–III. The boundary conditions for the ‘constrained mode’ are identical to the boundary conditions of the ‘free mode’. The continuity conditions for deflection, slope, shear, and bending moments at x = x2 are:
W I ¼ W II ; w0I ¼ w0II
ð6Þ
000 EI1 W 000 I ¼ ðEI 2 þ EI 3 ÞW II
ð7Þ
EI1 þ
ðE2 H2 E3 H3 ÞH21 4ðE2 H2 þ E3 H3 Þ
ðW 0I W 0III Þ ¼ EI2 W 00II
ð8Þ
similarly, we can derive the continuity conditions at x = x3. A non-trivial solution exists when the determinant of the coefficient matrix vanishes. 2.4. Introduction on non-dimensional parameters To study the effect of rotating speed, as well as Timoshenko effect on delaminated beams, similar parameters are adopted to
Fig. 1. (a) A rotating Timoshenko beam is delaminated along the interface and (b) the delaminated beam is analyzed as four interconnected beams.
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represent the effect of rotating speed as well as Timoshenko effect as they are in the work of Du et al. [14]. To study the effect of shear deformation and rotary inertia, a relative slenderness ratio r = R/L is adopted, where R is the radius of gyration and L is the beam length. A higher value of r indicates a more prominent Timoshenko effect. pffiffiffiffiffiffiffiffiffiffiffiffiffi The effect of rotating speed is represented by g ¼ qA=EIXL2 , where X is the angular velocity. Also, the effect of root offset value is studied, we use a = r0/L, where r0 is the offset distance.
frequency of the delaminated beam k2 is normalized with respect to that of undelaminated beam k2d . As shown in Fig. 2, k2 =k2d decreases as r increases, indicating a higher influence of Timoshenko effect leads to lower natural frequencies. The results agree with Du’s work [14] (Fig. 5). The difference between natural frequencies of delamination length a/L = 0.2 and 0.8 becomes smaller as r increases, or as the thickness-wise location of delamination H2/H1 increases from 0.2 to 0.5. It shows that the effect of delamination on natural frequency becomes smaller with a bigger Timoshenko effect.
3. Results and discussion 3.1. Verification To validate the present study, comparisons with published results on static Euler–Bernoulli beam with a single delamination and undelaminated rotating Timoshenko beam are made. The first non-dimensional natural frequencies of a clamped–clamped beam with a single midplane central delamination having various lengths are compared with the analytical results of Wang et al. [2] and FEM results of Lee [15]. The first non-dimensional natural frequencies of a clamped–free rotating Timoshenko beams without delaminations are compared with the analytical results of Du et al. [14]. Tables 1 and 2 show good agreement between the present results and previous results. 3.2. Rotating Timoshenko beams with a single delamination 3.2.1. Static Timoshenko beam with a single delamination Fig. 2 shows the normalized frequency k2 =k2d of a static beam with a single central delamination versus r. This represents the influence of Timoshenko effect on natural frequency of beams with a single delamination of various lengths. The non-dimensional
Table 1 Non-dimensional fundamental frequency (k2) of a clamped–clamped isotropic static Euler–Bernoulli beam with a midplane delamination.
3.2.2. Rotating Euler–Bernoulli beam with a single delamination Fig. 3 shows the effect of rotating speed on the normalized ‘free mode’ natural frequency of beam with a single central delamination, neglecting the Timoshenko effect. As shown in Fig. 3, k2 =k2d increases as angular velocity g becomes higher, which agrees well with the work of Du et al. [14] (Table 7). The lower the rotating speed the higher the percentage the frequency decreases as delamination length a/L increases. It can be concluded that the effect of delamination is smaller when the beam is subjected to a bigger centrifugal force. When comparing the results between (a) and (b), it can be concluded that the effect of delamination length is more prominent when the thickness-wise location of delamination H2/H1 increases from 0.2 to 0.5. 3.2.3. Mode 1 and mode 2 ‘free mode’ frequencies Fig. 4 shows the results of the normalized frequency of modes 1 and 2. It incorporates the influence of rotating speed, Timoshenko effect and delamination. For both vibration modes, k2 =k2d increases as the rotating speed increase and decreases as r increases. Timoshenko effect exert a bigger influence on mode 2 natural frequency than on mode 1, while the effect of rotating speed has a smaller influence on mode 2 than on mode 1.
H2 /H1=0.2
1
Delamination length, a/L
Present constrained and free
Analytical FEM Wang et al. [2]
FEM Lee [15]
0.95
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90
22.37 22.37 22.36 22.24 21.83 20.89 19.30 17.23 15.05 13.00
22.39 22.37 22.35 22.23 21.83 20.88 19.29 17.23 15.05 13.00
22.36 22.36 22.35 22.23 21.82 20.88 19.28 17.22 15.05 12.99
0.85
a/L=0.2
0.8
a/L=0.6
0.9
a/L=0.6
0.75
a/L=0.8
0.7 0.65 0.6
r 0
0.05
0.1
0.15
0.2
0.25
0.3
(a) H2 /H1=0.5
1 0.95
Table 2 Non-dimensional mode 1 frequency (k2) of a clamped–free rotating Timoshenko beam. r
g=4 Present
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10
5.58 5.562 5.537 5.502 5.46 5.411 5.358 5.300 5.239 5.176
g=8 Du et al. [14] 5.58 5.564 5.539 5.505 5.463 5.415 5.363 5.307 5.249 5.191
Present 9.245 9.212 9.162 9.097 9.023 8.944 8.861 8.775 8.688 8.599
0.9 0.85
a/L=0.2 a/L=0.4 a/L=0.6 a/L=0.8
0.8 0.75
Du et al. [14] 9.246 9.215 9.167 9.106 9.036 8.963 8.889 8.815 8.744 8.677
0.7 0.65 0.6
r
0.55 0
0.05
0.1
0.15
0.2
0.25
0.3
(b) Fig. 2. Influence of rotary inertia and shear deformation with different delamination length on normalized mode 1 ‘free mode’ frequency k2 =k2d for a static homogeneous clamped–free beam with a single central delamination. (a) H2 = 0.2H1 and (b) H2 = 0.5H1.
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H 2/H1=0.2 4 3.5 3 2.5 2 1.5 1
a/L
0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(a) Fig. 5. Influence of rotary inertia and shear deformation on ‘free mode’ and ‘constrained mode’ frequencies for a homogeneous clamped–free beam with single delamination considering different rotating speed g. a/L = 0.4 and H2/H1 = 0.5.
H2 /H1=0.5 4 3.5 3 2.5 2 1.5 1
a/L
0.5 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b) Fig. 3. Influence of rotating speed on normalized mode 1 ‘free mode’ frequency k2 =k2d for a rotating homogeneous clamped–free Euler–Bernoulli beam with single central delamination. (a) H2 = 0.2H1 and (b) H2 = 0.5H1.
(a)
(a) (b) Fig. 6. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for a homogeneous clamped–free beam with two double delaminations considering different r. d/L = 0. (a) a/L = 0.1 and (b) a/L = 0.4.
(b) Fig. 4. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for a homogeneous clamped–free beam with single delamination considering different r. (a) a/L = 0.1 and (b) a/L = 0.4.
For mode 1, the differences of natural frequencies between r = 0.01 and 0.1 increase monotonously as the rotating speed becomes higher. For mode 2, the differences of natural frequencies between r = 0.01 and 0.1 do not show a monotonous relation with rotating speed. Instead, the difference of natural frequencies decreases at first and then increase. The rotating speed corresponding to the minimum difference of nature frequencies is higher when delamination length a/L = 0.1 than that of a/L = 0.4. 3.2.4. Comparison between ‘free mode’ and ‘constrained mode’ Fig. 5 shows the influence of Timoshenko effect on the normalized frequencies k2 =k2d of beams with single central delamination,
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Fig. 7. Beams with two central delaminations of equal length.
(a)
(b) Fig. 8. Beams with two central enveloped delaminations. Fig. 9. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for a homogeneous clamped–free beam with two enveloping delaminations considering different r. d/L = 0, H4 = 0.33H1. (a) a1/L = 0.1, a2/L = 0.2 and (b) a1/L = 0.2, a2/L = 0.4.
where a/L = 0.4 and H2/H1 = 0.5. For both ‘free mode’ and ‘constrained mode’, k2 =k2d increases when the rotating speed g becomes higher. k2 =k2d decreases when the Timoshenko effect becomes more prominent. For both ‘free mode’ and ‘constrained mode’ a higher rotating speed leads to higher percentage decease of k2 =k2d . The natural frequency of ‘constrained mode’ is bigger than that of the ‘free mode’, which agrees with the established results of previous work [4]. The differences of natural frequency between the ‘constrained mode’ and the ‘free mode’ increase when rotating speed becomes higher. The differences of natural frequencies do not change monotonously as r increases. Generally, the difference between ‘constrained mode’ and ‘free mode’ increases at first then decreases as r becomes bigger. As the rotating speed becomes bigger, the value of r corresponding to the maximum difference between the ‘free mode’ and ‘constrained mode’ decreases. 3.3. Rotating Timoshenko beams with double delaminations of equal length Fig. 6 shows the case of beams with double delaminations of equal length (as shown in Fig. 7). Regarding the Timoshenko effect and the effect of rotating speed on mode 1 and mode 2 natural frequencies, similar results to Fig. 4 can be observed. Compared with beams with a single delamination, natural frequencies of beams with double delaminations is smaller when both cases have the same delamination length. However, the difference is less significant with a higher rotating speed or a more prominent Timoshenko effect. Regarding the differences of natural frequencies between r = 0.01 and 0.1, similar results to Fig. 4 can be observed for both mode 1 and mode 2.
Fig. 10. Influence of rotary inertia and shear deformation on ‘free mode’ and ‘constrained mode’ frequencies for a homogeneous clamped–free beam with two enveloping delaminations considering different rotating speed g. a1/L = 0.2, a2/ L = 0.4.
3.4. Rotating Timoshenko beams with two enveloped delaminations 3.4.1. Mode 1 and mode 2 ‘free mode’ frequencies The variations of the normalized frequency k2 =k2d with the rotating speed as well as Timoshenko effect for beams with two central enveloping delaminations (Fig. 8) are shown in Fig. 9. The thickness-wise location of two delaminations is equally distributed. Natural frequencies k2 =k2d of mode 1 and mode 2 are both included. The result shows the change of frequencies according to the rotating speed and they are compared among three different cases, each of which is under different influences of Timoshenko effect.
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a/L=0.1
2.1 2 1.9
a/L=0.2
1.8
a/L=0.5 1.7
a/L=0.8
1.6 1.5 1.4 0
0.2
0.4
0.6
0.8
1
α
Fig. 11. Effect of root offset value on normalized natural frequency of rotating Timoshenko beam with single delamination. g = 4, r = 0.05.
a/L=0.4
(a) a/L=0.1
(a)
4
a/L=0.4
(b) Fig. 12. Mode shapes of the first mode for clamped–free rotating Timoshenko beam with single delamination. (a) Consider different values of r with a constant rotating speed g = 4 and (b) consider different rotating speed g with a constant value of r = 0.05.
For mode 1 and mode 2, k2 =k2d increases as rotating speed g increases, decreases as r increases. Regarding the Timoshenko effect and the effect of rotating speed on mode 1 and 2 natural frequencies, similar results to Fig. 4 can be observed. For mode 1, the differences of natural frequencies between r = 0.01 and 0.1 increase monotonously as the rotating speed becomes higher. For mode 2, the differences of natural frequencies between r = 0.01 and 0.1 do not show a monotonous relation with rotating speed. Instead, the difference of natural frequencies decreases at first and then increase. The rotating speed corresponding to the minimum difference of nature frequencies is higher with delamination length a1/L = 0.1, a2/L = 0.2 than a1/L = 0.2, a2/L = 0.4.
(b) Fig. 13. Mode shapes of the second mode for clamped–free rotating Timoshenko beam with single delamination. (a) Consider different values of r with a constant rotating speed g = 4, when a/L = 0.1 and 0.4. (b) Consider different rotating speed g with a constant value of r = 0.05, when a/L = 0.1 and 0.4.
3.4.2. Comparison between ‘free mode’ and ‘constrained mode’ Fig. 10 shows the influence of Timoshenko effect on the normalized frequencies k2 =k2d with two central enveloping delaminations, results from both ‘free mode’ and ‘constrained mode’ are presented. The thickness-wise locations of two delaminations are equally distributed. For both ‘free mode’ and ‘constrained mode’, k2 =k2d increases when the rotating speed g becomes higher. k2 =k2d decreases
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when the Timoshenko effect becomes more prominent. For both ‘free mode’ and ‘constrained mode’ k2 =k2d decreases faster with a higher rotating speed. The natural frequency of ‘constrained mode’ is bigger than the one with ‘free mode’, which agrees with the established results of previous work [7]. However, the difference between the ‘constrained mode’ and the ‘free mode’ does not change monotonously as r increases, which is similar to what we observe in the case of beams with single delamination.
3.5. Effect of root offset value Fig. 11 shows the effect of root offset value on normalized natural frequency when r = 0.05 and g = 4. As we can see it shows a linear relation between the frequency and root offset value for each delamination length under consideration.
3.6. Mode shape of delaminated rotating Timoshenko beam 3.6.1. Mode 1 Fig. 12 shows the ‘free mode’ first mode shape of delaminated beam under the influence of rotating speed and Timoshenko effect respectively. As shown in Fig. 12a, when the rotating speed is kept as g = 4, the change of mode shape due to delamination and Timoshenko effect is illustrated. Fig. 12b presents the case of the change of mode shape due the delamination and rotating effect, when the value of r is kept at 0.05. The amplitude is bigger in the middle when the delamination length is longer. Also, the amplitude is bigger when the rotating speed increases or with a bigger value of r. We can conclude that the bigger the effect of shear deformation and rotary inertia or the rotating speed, the bigger the amplitude of vibration.
3.6.2. Mode 2 As shown in Fig. 13, we have the similar results for mode 2 as those with mode 1. Furthermore, the effect of shear deformation and rotary inertia as well as rotating speed has a bigger effect on the second mode shape than the first one.
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4. Conclusions Here the analytical solution to the vibration of rotating Timoshenko beams with multiple delaminations is developed. We study how Timoshenko effect and rotating speed influence the effect of delamination on natural frequencies and mode shapes. It can be concluded that the effect of delamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effect and rotating speed. The comparison between ‘free mode’ assumption and ‘constrained mode’ assumption is also investigated. Furthermore, Timoshenko effect and rotating speed are shown to influence the effect of delamination on the mode shapes. References [1] Tay TE. Characterization and analysis of delamination fracture in composites: an overview of developments from 1990 to 2001. Appl Mech Rev 2003;56(1):1–31. [2] Wang JTS, Liu YY, Gibby JA. Vibration of split beams. J Sound Vib 1982;84(4): 491–502. [3] Mujumdar PM, Suryanarayan S. Flexural vibrations of beams with delaminations. J Sound Vib 1982;125(3):441–61. [4] Shu D, Fan H. Free vibration of a bimaterial split beam. Composites: Part B 1996;27(1):79–84. [5] Shen M-HH, Grady JE. Free vibrations of delaminated beams. AIAA J 1992;30(5):1361–70. [6] Shu D. Vibration of sandwich beams with double delaminations. Compos Sci Technol 1995;54(1):101–9. [7] Shu D, Della CN. Vibrations of multiple delaminated beams. Compos Struct 2004;64(3–4):467–77. [8] Shu D, Della CN. Free vibration analysis of composite beams with two nonoverlapping delaminations. Int J Mech Sci 2004;46(4):509–26. [9] Della CN, Shu D. Free vibration analysis of composite beams with overlapping delaminations. Eur J Mech A 2005;24(3):491–503. [10] Della CN, Shu D. Free vibration of delaminated bimaterial beams. Compos Struct 2007;80:212–20. [11] Wright AD, Smith CE. Vibration modes of centrifugally stiffened beams. J Appl Mech 1982;49:197–202. [12] Lin WH, Yeh FH. Vibrations of nonuniform rotating beams. J Sound Vib 1987;119:379–84. [13] Al-Ansary MD. Flexural vibrations of rotating beams considering rotary inertia. Comput Struct 1998;69:321–8. [14] Du H, Lim MK, Liew KM. A power series solution for vibration of a rotating Timoshenko beam. J Sound Vib 1994;175:505–23. [15] Lee J. Free vibration analysis of delaminated composite beams. Comput Struct 2000;74(2):121–9.