Composites: Part B 44 (2013) 733–739

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Composites: Part B

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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 2011Accepted 2 January 2012Available online 12 January 2012

Keywords:A. LaminatesB. DelaminationB. VibrationC. Analytical modeling

1359-8368/$ - see front matter � 2012 Elsevier Ltd. Adoi:10.1016/j.compositesb.2012.01.037

⇑ Corresponding author. Tel.: +65 67904440.E-mail addresses: liuy0059@ntu.edu.sg (Y. Liu), md

1 Tel.: +65 84327159.

a b s t r a c t

Analytical solutions are developed to study the free vibrations of rotating Timoshenko beams with multi-ple 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 influ-ences of Timoshenko effect and rotating speed on delamination vibration. Results show that the effect ofdelamination on both modes 1 and 2 natural frequencies is significantly influenced by Timoshenko effectand the rotating speed. Also, the comparison between ‘free mode’ assumption and ‘constrained assump-tion’ are conducted. Furthermore, the effect of delamination on mode shapes is also influenced by rotat-ing speed and Timoshenko effect. For both Timoshenko effect and rotating speed, the influences on thesecond vibration mode shape are more significant.

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1. Introduction Della [7,8] and Della and Shu [9] used the ‘free mode’ and ‘con-

The vibration characteristics of rotating composite beams areunder frequent investigation due to its wide applications, such ashelicopter blades, wind turbines and turbo-machinery. One com-mon defect of laminate composite is delamination, which may sig-nificantly reduces the stiffness and strength of the structures [1], aswell as the vibration characteristics.

To study the influence of a through-width delamination on thefree vibration of an isotropic beam, Wang et al. [2] presented ananalytical model using four Euler–Bernoulli beams that are joinedtogether. They assumed that the delaminated layers deform ‘freely’without touching each other (‘free mode’ model) and will have dif-ferent transverse deformations. While Mujumdar and Suryanara-yan [3] assumed that the delaminated layers are in touch alongtheir whole length all the time, but are allowed to slide over eachother (‘constrained mode’ model). Thus, the delaminated layers are‘constrained’ to have identical transverse deformations. This ‘con-strained mode’ model was extended by Shu and Fan [4] on a bima-terial beam. However, the ‘constrained mode’ model failed topredict the opening in the mode shapes found in the experimentsby Shen and Grady [5].

Analytical solutions for beams with multiple delaminationshave been presented by many researchers. Shu [6] presented ananalytical solution to study a sandwich beam with double delam-inations. His study emphasized on the influence of the contactmode, ‘free’ and ‘constrained’, between the delaminated layersand the local deformations at the delamination fronts. Shu and

ll rights reserved.

shu@ntu.edu.sg (D.W. Shu).

strained mode’ assumptions study a composite beam with variousmultiple delamination configurations. Their study emphasized onthe influence of a second delamination on the first and second nat-ural frequencies and the corresponding mode shapes of a delami-nated beam. Della and Shu [10] also studied the free vibration ofa delaminated bimaterial beam using Euler–Bernoulli beam theory.

The free vibration characteristics of rotating beam have re-ceived considerable attention. Many previous studies have beenbased on Euler–Bernoulli beam theory and various approximatesolution techniques [11,12]. Al-Ansary [13] studied the effects ofrotary inertia on the extensional tensile force and on the eigen-values of beams rotating uniformly about a transverse axis. Duet al. [14] presented a convergent power series expression to solveanalytically for the exact natural frequencies and mode shapes ofrotating Timoshenko beams. The effects of angular velocity, shearand rotary inertia are carefully studied.

In the present work, analytical solutions are developed to studythe free vibrations of delaminated rotating Timoshenko beams. TheTimoshenko beam theory and both the ‘free mode’ and the ‘con-strained mode’ assumption in delamination vibration are used.First, the vibration of a rotating Timoshenko beam with singledelamination is formulated. Second, the present results are verifiedagainst previous published results. Third, a comprehensive study isconducted on how the effect of delaminations on natural frequen-cies is influenced by rotating speed and Timoshenko effect. Finallythe first and second mode shapes are thoroughly investigated.

2. Formulation

The analytical solution to the vibration of rotating Timoshenkobeam, reported by Du et al. [14], is used here to solve the governing

734 Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739

equations. Both ‘free mode’ and ‘constrained mode’ assumptionsare adopted to investigate the delamination vibration problem.Fig. 1 shows a beam with length L and thickness H1. The beam ismade wi(x, t) of two distinct layers, with Young’s modulus E2 andE3, and thickness H2 and H3. The beam is separated along the inter-face by a delamination with length a and located at a distance dfrom the center of the beam. The beam can then be subdivided intothree span-wise regions, a delamination region and two integralregions. The delamination region is comprised of two segments(delaminated layers), beam 2 and beam 3, which are joined at theirends to the integral segments, beam 1 and beam 4. Each of the fourbeams 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 Kwhen the shearing is neglected and the midplane deflection ofbeam i, respectively. The governing equations for the free vibrationof a rotating Timoshenko beam with single delamination are:(i = 1–4).

ðEIiw0iÞ0 þ KAiGiðw0i � wiÞ ¼ qiIi

€wi ð1Þ

ðKAiGiðw0i � wiÞÞ0 þ Tiw0iÞ

0 � qiAi €wi ¼ 0 ð2Þ

where qi is the mass density, Ai is the cross-sectional area of thebeam, Gi shear modulus, Ii is the moment of area of the cross sec-tion and K is the shear coefficient. Ti denotes the centrifugal forceexerted on the cross section of beam i. EIi (i = 1–4) is the bendingstiffness of beam i. A closed form solution [14] is adopted to solvefor the flexural natural frequencies and corresponding modeshapes.

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 canbe applied at the supports, x = x1 and x = x4, are Wi ¼ 0;W 0

i ¼ 0, if theend of the beam is clamped; W 00

i ¼ 0; W 000i ¼ 0, if free, where i = 1, 4.

Fig. 1. (a) A rotating Timoshenko beam is delaminated along the interface a

The continuity conditions for deflection and slope at x = x2 are:

W1 ¼W2 ¼W3; wi ¼ w2 ¼ w3 ð3Þ

The continuity condition for shear and bending moments atx = x2 are:

EI1W 0001 ¼ ðEI2 þ EI3ÞW 000

2 ð4Þ

EI1 þðE2H2E3H3ÞH2

1

4ðE2H2 þ E3H3ÞðW 0

1 �W 04Þ ¼ EI2W 00

2 þ EI3W 003 ð5Þ

similarly, we can derive the continuity conditions at x = x3.A non-trivial solution exists when the determinant of the coef-

ficient matrix vanishes.

2.3. Constrained mode model

The ‘constrained mode’ model is simplified by the assumptionthat the delaminated layers are constrained to have the sametransverse deformations. The delaminated beam is analyzed asthree beam segments I–III.

The boundary conditions for the ‘constrained mode’ are identi-cal to the boundary conditions of the ‘free mode’. The continuityconditions for deflection, slope, shear, and bending moments atx = x2 are:

WI ¼WII;w0I ¼ w0II ð6Þ

EI1W 000I ¼ ðEI2 þ EI3ÞW 000

II ð7Þ

EI1 þðE2H2E3H3ÞH2

1

4ðE2H2 þ E3H3ÞðW 0

I �W 0IIIÞ ¼ EI2W 00

II 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 Timoshenkoeffect on delaminated beams, similar parameters are adopted to

nd (b) the delaminated beam is analyzed as four interconnected beams.

Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739 735

represent the effect of rotating speed as well as Timoshenko effectas they are in the work of Du et al. [14].

To study the effect of shear deformation and rotary inertia, arelative slenderness ratio r = R/L is adopted, where R is the radiusof gyration and L is the beam length. A higher value of r indicatesa more prominent Timoshenko effect.

The effect of rotating speed is represented by g ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiqA=EI

pXL2,

where X is the angular velocity. Also, the effect of root offset valueis studied, we use a = r0/L, where r0 is the offset distance.

3. Results and discussion

3.1. Verification

To validate the present study, comparisons with published re-sults on static Euler–Bernoulli beam with a single delaminationand undelaminated rotating Timoshenko beam are made. The firstnon-dimensional natural frequencies of a clamped–clamped beamwith a single midplane central delamination having variouslengths are compared with the analytical results of Wang et al.[2] and FEM results of Lee [15]. The first non-dimensional naturalfrequencies of a clamped–free rotating Timoshenko beams withoutdelaminations are compared with the analytical results of Du et al.[14]. Tables 1 and 2 show good agreement between the present re-sults and previous results.

3.2. Rotating Timoshenko beams with a single delamination

3.2.1. Static Timoshenko beam with a single delaminationFig. 2 shows the normalized frequency k2=k2

d of a static beamwith a single central delamination versus r. This represents theinfluence of Timoshenko effect on natural frequency of beams witha single delamination of various lengths. The non-dimensional

Table 1Non-dimensional fundamental frequency (k2) of a clamped–clamped isotropic staticEuler–Bernoulli beam with a midplane delamination.

Delaminationlength, a/L

Present constrainedand free

Analytical FEM Wanget al. [2]

FEM Lee[15]

0.00 22.37 22.39 22.360.10 22.37 22.37 22.360.20 22.36 22.35 22.350.30 22.24 22.23 22.230.40 21.83 21.83 21.820.50 20.89 20.88 20.880.60 19.30 19.29 19.280.70 17.23 17.23 17.220.80 15.05 15.05 15.050.90 13.00 13.00 12.99

Table 2Non-dimensional mode 1 frequency (k2) of a clamped–free rotating Timoshenkobeam.

r g = 4 g = 8

Present Du et al. [14] Present Du et al. [14]

0.01 5.58 5.58 9.245 9.2460.02 5.562 5.564 9.212 9.2150.03 5.537 5.539 9.162 9.1670.04 5.502 5.505 9.097 9.1060.05 5.46 5.463 9.023 9.0360.06 5.411 5.415 8.944 8.9630.07 5.358 5.363 8.861 8.8890.08 5.300 5.307 8.775 8.8150.09 5.239 5.249 8.688 8.7440.10 5.176 5.191 8.599 8.677

frequency of the delaminated beam k2 is normalized with respectto that of undelaminated beam k2

d .As shown in Fig. 2, k2=k2

d decreases as r increases, indicating ahigher influence of Timoshenko effect leads to lower natural fre-quencies. The results agree with Du’s work [14] (Fig. 5). The differ-ence between natural frequencies of delamination length a/L = 0.2and 0.8 becomes smaller as r increases, or as the thickness-wiselocation of delamination H2/H1 increases from 0.2 to 0.5. It showsthat the effect of delamination on natural frequency becomessmaller with a bigger Timoshenko effect.

3.2.2. Rotating Euler–Bernoulli beam with a single delaminationFig. 3 shows the effect of rotating speed on the normalized ‘free

mode’ natural frequency of beam with a single central delamina-tion, neglecting the Timoshenko effect.

As shown in Fig. 3, k2=k2d increases as angular velocity g be-

comes higher, which agrees well with the work of Du et al. [14](Table 7). The lower the rotating speed the higher the percentagethe frequency decreases as delamination length a/L increases. Itcan be concluded that the effect of delamination is smaller whenthe beam is subjected to a bigger centrifugal force.

When comparing the results between (a) and (b), it can be con-cluded that the effect of delamination length is more prominentwhen the thickness-wise location of delamination H2/H1 increasesfrom 0.2 to 0.5.

3.2.3. Mode 1 and mode 2 ‘free mode’ frequenciesFig. 4 shows the results of the normalized frequency of modes 1

and 2. It incorporates the influence of rotating speed, Timoshenkoeffect and delamination. For both vibration modes, k2=k2

d increasesas the rotating speed increase and decreases as r increases. Timo-shenko effect exert a bigger influence on mode 2 natural frequencythan on mode 1, while the effect of rotating speed has a smallerinfluence on mode 2 than on mode 1.

(a)

(b)

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.05 0.1 0.15 0.2 0.25 0.3r

H2/H1=0.2

a/L=0.2

a/L=0.6

a/L=0.6

a/L=0.8

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 0.05 0.1 0.15 0.2 0.25 0.3

r

H2 /H1=0.5

a/L=0.2a/L=0.4a/L=0.6a/L=0.8

Fig. 2. Influence of rotary inertia and shear deformation with different delamina-tion length on normalized mode 1 ‘free mode’ frequency k2=k2

d for a statichomogeneous clamped–free beam with a single central delamination. (a) H2 = 0.2H1

and (b) H2 = 0.5H1.

Fig. 5. Influence of rotary inertia and shear deformation on ‘free mode’ and‘constrained mode’ frequencies for a homogeneous clamped–free beam with singledelamination considering different rotating speed g. a/L = 0.4 and H2/H1 = 0.5.

(a)

(b)

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9a/L

H2/H1=0.2

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9a/L

H2/H1=0.5

Fig. 3. Influence of rotating speed on normalized mode 1 ‘free mode’ frequencyk2=k2

d for a rotating homogeneous clamped–free Euler–Bernoulli beam with singlecentral delamination. (a) H2 = 0.2H1 and (b) H2 = 0.5H1.

(a)

(b)Fig. 4. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for ahomogeneous clamped–free beam with single delamination considering different r.(a) a/L = 0.1 and (b) a/L = 0.4.

(a)

(b)Fig. 6. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for ahomogeneous clamped–free beam with two double delaminations consideringdifferent r. d/L = 0. (a) a/L = 0.1 and (b) a/L = 0.4.

736 Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739

For mode 1, the differences of natural frequencies betweenr = 0.01 and 0.1 increase monotonously as the rotating speedbecomes higher. For mode 2, the differences of natural frequenciesbetween r = 0.01 and 0.1 do not show a monotonous relation withrotating speed. Instead, the difference of natural frequencies de-creases at first and then increase. The rotating speed correspondingto the minimum difference of nature frequencies is higher whendelamination 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 normal-

ized frequencies k2=k2d of beams with single central delamination,

Fig. 7. Beams with two central delaminations of equal length.

Fig. 8. Beams with two central enveloped delaminations.

(a)

(b)Fig. 9. Mode 1 and mode 2 ‘free mode’ frequencies versus the rotating speed for ahomogeneous clamped–free beam with two enveloping delaminations consideringdifferent 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.

Fig. 10. Influence of rotary inertia and shear deformation on ‘free mode’ and‘constrained mode’ frequencies for a homogeneous clamped–free beam with twoenveloping delaminations considering different rotating speed g. a1/L = 0.2, a2/L = 0.4.

Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739 737

where a/L = 0.4 and H2/H1 = 0.5. For both ‘free mode’ and ‘con-strained mode’, k2=k2

d increases when the rotating speed g becomeshigher. k2=k2

d decreases when the Timoshenko effect becomes moreprominent.

For both ‘free mode’ and ‘constrained mode’ a higher rotatingspeed leads to higher percentage decease of k2=k2

d . The natural fre-quency 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 ‘constrainedmode’ and the ‘free mode’ increase when rotating speed becomeshigher. The differences of natural frequencies do not change monot-onously as r increases. Generally, the difference between ‘con-strained mode’ and ‘free mode’ increases at first then decreases asr becomes bigger. As the rotating speed becomes bigger, the valueof r corresponding to the maximum difference between the ‘freemode’ and ‘constrained mode’ decreases.

3.3. Rotating Timoshenko beams with double delaminations of equallength

Fig. 6 shows the case of beams with double delaminations ofequal length (as shown in Fig. 7). Regarding the Timoshenko effectand the effect of rotating speed on mode 1 and mode 2 naturalfrequencies, similar results to Fig. 4 can be observed.

Compared with beams with a single delamination, naturalfrequencies of beams with double delaminations is smaller whenboth cases have the same delamination length. However, the dif-ference is less significant with a higher rotating speed or a moreprominent Timoshenko effect. Regarding the differences of naturalfrequencies between r = 0.01 and 0.1, similar results to Fig. 4 can beobserved for both mode 1 and mode 2.

3.4. Rotating Timoshenko beams with two enveloped delaminations

3.4.1. Mode 1 and mode 2 ‘free mode’ frequenciesThe variations of the normalized frequency k2=k2

d with the rotat-ing speed as well as Timoshenko effect for beams with two centralenveloping delaminations (Fig. 8) are shown in Fig. 9. The thick-ness-wise location of two delaminations is equally distributed.Natural frequencies k2=k2

d 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.

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

0 0.2 0.4 0.6 0.8 1

a/L=0.2

a/L=0.5

a/L=0.8

α

Fig. 11. Effect of root offset value on normalized natural frequency of rotatingTimoshenko beam with single delamination. g = 4, r = 0.05.

(a)

(b)Fig. 12. Mode shapes of the first mode for clamped–free rotating Timoshenko beamwith single delamination. (a) Consider different values of r with a constant rotatingspeed g = 4 and (b) consider different rotating speed g with a constant value ofr = 0.05.

4

a/L=0.1

a/L=0.4

a/L=0.1

a/L=0.4

(a)

(b)Fig. 13. Mode shapes of the second mode for clamped–free rotating Timoshenkobeam with single delamination. (a) Consider different values of r with a constantrotating speed g = 4, when a/L = 0.1 and 0.4. (b) Consider different rotating speed gwith a constant value of r = 0.05, when a/L = 0.1 and 0.4.

738 Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739

For mode 1 and mode 2, k2=k2d increases as rotating speed g

increases, decreases as r increases. Regarding the Timoshenkoeffect and the effect of rotating speed on mode 1 and 2 naturalfrequencies, similar results to Fig. 4 can be observed.

For mode 1, the differences of natural frequencies betweenr = 0.01 and 0.1 increase monotonously as the rotating speed be-comes higher. For mode 2, the differences of natural frequenciesbetween r = 0.01 and 0.1 do not show a monotonous relation withrotating speed. Instead, the difference of natural frequencies de-creases at first and then increase. The rotating speed correspondingto the minimum difference of nature frequencies is higher withdelamination length a1/L = 0.1, a2/L = 0.2 than a1/L = 0.2, a2/L = 0.4.

3.4.2. Comparison between ‘free mode’ and ‘constrained mode’Fig. 10 shows the influence of Timoshenko effect on the normal-

ized 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 equallydistributed. For both ‘free mode’ and ‘constrained mode’, k2=k2

d in-creases when the rotating speed g becomes higher. k2=k2

d decreases

Y. Liu, D.W. Shu / Composites: Part B 44 (2013) 733–739 739

when the Timoshenko effect becomes more prominent. For both‘free mode’ and ‘constrained mode’ k2=k2

d decreases faster with ahigher rotating speed. The natural frequency of ‘constrained mode’is bigger than the one with ‘free mode’, which agrees with theestablished results of previous work [7]. However, the differencebetween the ‘constrained mode’ and the ‘free mode’ does notchange monotonously as r increases, which is similar to what weobserve 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 nat-ural frequency when r = 0.05 and g = 4. As we can see it shows alinear relation between the frequency and root offset value for eachdelamination length under consideration.

3.6. Mode shape of delaminated rotating Timoshenko beam

3.6.1. Mode 1Fig. 12 shows the ‘free mode’ first mode shape of delaminated

beam under the influence of rotating speed and Timoshenko effectrespectively. As shown in Fig. 12a, when the rotating speed is keptas g = 4, the change of mode shape due to delamination and Timo-shenko effect is illustrated. Fig. 12b presents the case of the changeof mode shape due the delamination and rotating effect, when thevalue of r is kept at 0.05. The amplitude is bigger in the middlewhen the delamination length is longer. Also, the amplitude is big-ger when the rotating speed increases or with a bigger value of r.We can conclude that the bigger the effect of shear deformationand rotary inertia or the rotating speed, the bigger the amplitudeof vibration.

3.6.2. Mode 2As shown in Fig. 13, we have the similar results for mode 2 as

those with mode 1. Furthermore, the effect of shear deformationand rotary inertia as well as rotating speed has a bigger effect onthe second mode shape than the first one.

4. Conclusions

Here the analytical solution to the vibration of rotating Timo-shenko beams with multiple delaminations is developed. Westudy how Timoshenko effect and rotating speed influence the ef-fect of delamination on natural frequencies and mode shapes. Itcan be concluded that the effect of delamination on both modes1 and 2 natural frequencies is significantly influenced by Timo-shenko effect and rotating speed. The comparison between ‘freemode’ assumption and ‘constrained mode’ assumption is alsoinvestigated. Furthermore, Timoshenko effect and rotating speedare shown to influence the effect of delamination on the modeshapes.

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