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Dynamics of hybrid shafts

H.B.H. Gubran

Mechanical Engineering Department, Faculty of Engineering, University of Aden, P.O. Box 5243, Yemen

Available online 10 March 2005

Abstract

In this paper the dynamic performance and cross-section deformation of shafts made of metals (steel and alumi-num), composites (CFRP and GFRP) and hybrids of metals and composites have been studied. A layered finite degen-erated shell element with transverse shear deformation and dynamic behavior is employed. Results obtained show thatimprovements in dynamic performance and reduction of cross-section deformation of hybrid shafts over metallic andcomposite shafts are possible. 2005 Elsevier Ltd. All rights reserved.

Keywords: Composite and metallic shafts; Hybrid shafts; Natural frequency; Cross-section deformation; Shell finite element

1. Introduction

In recent years, composite materials have been used in many advanced engineering structures such asmechanical, civil, aerospace, marine, etc. This is mainly due to excellent mechanical properties of suchmaterials, such as high stiffness-to-weight and high strength-to-weight ratios which can be tailored by vary-ing the fibre orientation and stacking sequence of different plies. Further the composites offer good environ-mental resistance. Many investigators (for example, Belingardi et al., 1990; Lee and Kim, 1999) havestudied the use of composite materials for driveshafts and rotating cylindrical shells. Improvement in thedynamic performance obtained by extending the design of uniform wall thickness shafts to tapered/variable

wall thickness shafts have been studied by Bauchau (1983); Kim et al. (1999) and Gubran and Gupta(2002b). In general, composite driveshafts are thin tubular shells for which studies of vibration behaviorand cross-section deformation are important. Formulations based on degenerated shell finite element havebeen used by several researchers (for example, Richardet et al., 2000; Gubran and Gupta, 2002a and Guoet al., 2002). In the present work, a layered nine-node isoparametric shell finite element is used to derive the

0093-6413/$ - see front matter 2005 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechrescom.2005.02.005

E-mail address: [email protected]

Mechanics Research Communications 32 (2005) 368–374

www.elsevier.com/locate/mechrescom

MECHANICS

RESEARCHCOMMUNICATIONS

mailto:[email protected]:[email protected]

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global mass and stiffness matrices. The eigenvalues are determined using the subspace iteration method.Later, displacements of different nodes along the circumferential direction of a certain cross-section at aparticular location along the shaft axial length are used to study the distortion of the cross-section at that

location.

2. Formulation

The cylindrical tube shown in Fig. 1 is assumed to be built up by a number of laminae perfectly bondedtogether. There are no relative displacements between adjacent layers. A degenerated nine nodded isopara-metric shell element (as shown in Fig. 1) is applied to the layers of the cylindrical tube. The formulationtakes into account transverse shear deformation and dynamic behavior. Referring to Fig. 1, the displace-ment field at a point in the element can be expressed in global coordinates as

½u v w

T

¼X

9

k ¼1 ½ N k

fd k

g ð1

Þ

where ½ N k is the generalized shape function matrix and {d k } = [uk vk wk a1k a2

k]T is the nodal displacementvector. The strain matrix [B ] relating the strain components in the local system to the element nodal vari-ables can be expressed as

feg ¼X9k ¼1

½ Bk fd k g ð2Þ

Nodes 1-12

section 1 (Y=0)

Nodes 25-36

section 3

Nodes 49-60

section 5

Nodes 73-84

section 7

Nodes 97-108

section 9 (Y=L)

Y

X

Z

t

Mid surface

1

2

k

N

h

h

h

1

h0

k

k-1

ζ

η

ξ

t / 2

Fig. 1. The laminated shaft and coordinate system.

H.B.H. Gubran / Mechanics Research Communications 32 (2005) 368–374 369

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As shown in Fig. 1, the natural coordinate f varies from 1 to +1, is determined at the middle point of eachlayer and strain–stress components and stiffness contributions are computed at the midsection of eachlayer. Consequently, the volume integral may be split into integrals over the area of the shell midsurface

and through the thickness t. Thus the stiffness can be written as

½ K e ¼

Z 11

Z 11

Z 11

½ BT

½ D½ B j J j dfdndg ð3Þ

where jJ j is the determinant of the Jacobian matrix for layer j . The matrix ½ D is the material transformedstiffness matrix which can be expressed as ½ D ¼ ½T 1½ D½T . Here [T ] and [D] are the transformation andmaterial stiffness with respect to the local coordinates matrices. Different elements of the [D] matrix canbe written as

½ D ¼

D1 D12 0 0 0

D12 D2 0 0 0

0 0 G 12 0 0

0 0 0 KG 12 0

0 0 0 0 KG 12

26666664

37777775

where D1 = E 1/D, D2 = E 2/D, D12 = E 2v12/D, D = 1 v12v21. K is shear correction factor. Element consis-tent mass matrix at layer j linking nodes i and j can be written as

½ M e ¼

Z 11

Z 11

Z 11

½ N iq½ N j dfdndg ð4Þ

The free vibration of an undamped system results into an eigenvalue problem. The generalized eigenvalueproblem can be expressed as

½½ K x2½ M f X g ¼ f0g ð5Þ

where [K ], [M ] and {X } are the global stiffness, mass and displacement matrices which are generatedthrough the assembly of elements local matrices; x is the undamped natural frequency.

3. Results and discussion

In this study, simply supported shafts of L/R and t/R ratios equal to 20 and 0.08 made of metals (steeland aluminum), composites (CFRP and GFRP) with materials properties as given in Table 1, are consid-ered. Shell finite element modeling (Gubran and Gupta, 2002a) with a grid of 6 · 4 elements, as shown inFig. 1 (i.e., 6 elements along the circumferential direction and 4 elements along the shaft axial length) is

used. Dynamic performance, deflection and cross-section deformation for different shafts are investigated.

Table 1Material properties used for hybrid shafts

Material property Steel Aluminum CFRP GFRP

E 1 (Gpa) 210 70 130 40.30E 2 (Gpa) 210 70 10 6.20G 12 (Gpa) 84 28 7 3.00v12 0.30 0.28 0.25 0.20q (kg/m3) 7830 2600 1500 1900

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3.1. Natural frequency analysis

The natural frequency for a shaft depends mainly on E 1/q ratio; this ratio is almost the same for shafts

made of steel or aluminum. This makes the dynamic performance of shafts made of steel or aluminum (asshown in Fig. 2) almost the same. However, E 1/q ratio for composites varies with the orientations of fibres,it is maximum at 0 and decreases as the fibre angle shifts towards 90. Accordingly, as shown in Fig. 2, thenatural frequencies of shafts made of composites (CFRP or GFRP) varies with fibre angle, having maxi-mum value at 0 and decreases as the fibre angle shifts towards 90. Comparing the dynamic performanceof shafts made of composites (CFRP or GFRP) and metallic (steel or aluminum) it is observed that, thenatural frequency of CFRP shaft with fibres oriented at 37 –38 is almost the same as that of steel and alu-minum shafts. However due to low E 1/q ratio, the natural frequencies of shafts made of GFRP with fibresoriented at different angles are lower than that made of metals. The performance of shafts made by hybrid-ization of metals and composites are studied. In general, the main objectives of hybridization of metals andcomposites are: (i) to have proper dynamic performance (for example, optimal placement of natural fre-quency with respect to shaft operating speed, (ii) cost effectiveness and (iii) easier mounting of metallic com-ponents (like gears, pulleys or mounting the shaft on bearings). The main emphasis here is to study dynamicperformance and cross-section deformation of hybrid shafts. The hybrid shafts considered are made byplacing metallic layers (i.e., steel or aluminum) at the outer and inner surfaces and composite layers (i.e.,CFRP or GFRP) at the middle surface. Referring to the results presented in Fig. 2, the natural frequencyof steel shaft has been increased by about 25% to 5% by hybridization with CFRP with fibres oriented atangles 0 –36, respectively. Similarly, the natural frequency of aluminum shaft has been increased by about40% to 7% by hybridization with CFRP with fibres oriented at angles in the same range as that of steel.Depending on E 1/q ratio, the natural frequencies of GFRP have been increased by about 20% and 16%at 0 fibre angle and 150% and 106% at 90 fibre angle by hybridization with steel and aluminum, respec-tively. The above results can be explained mainly by dependence of the natural frequencies of hybrid shaftson E 1/q ratios for metallic part, which is constant and composite part which varies with fibre orientation.

Fibre angle, Degrees

0 10 20 30 40 50 60 70 80 90

F r e q u

e n c y ,

H z

100

200

300

400

500

CFRP

GFRP

Steel Aluminum

(Hybrid)

Aluminum

Steel

CFRP

GFRP

Fig. 2. Natural frequencies of shafts made of different materials.


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This also gives an explanation for flat curves (almost constant natural frequencies) for hybrid shafts madeof metals and composites with fibre angles greater than 50 having lower E 1/q ratios.

12 24 36 48 60 72 84 96 108 I n - p l a n e d i s p l a c e m e n t , m m

0.0

0.5

1.0

1.5

2.0

2.5CFRP (0)4

CFRP (90)4

GFRP (0)4GFRP (90)4

Steel

AL

12 24 36 48 60 72 84 96 108

I n - p l a n e d i s p l a c e m e n t , m m

0.0

0.1

0.2

0.3

0.4

(St/CFRP,0)s

(St/GFRP,0)s

(AL/CFRP,0)s

(AL/GFRP,0)s

Node number

12 24 36 48 60 72 84 96 108

I n - p l a

n e d i s p l a c e m e n t , m m

0.0

0.1

0.2

0.3

0.4

0.5

0.6

(St/CFRP,90)s

(St/GFRP,90)s

(AL/CFRP,90)s

(AL/GFRP,90)s

(a)

(b)

(c)

Fig. 3. Deflection and cross-section deformation (a) shafts made of single material (b) hybrid shafts with 0 fibre angle and (c) hybridshafts with 90 fibre angle.


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3.2. Deflection and cross-section deformation

Shafts considered earlier for natural frequency analysis are subjected to point static load of 1 kN at mid-span. Referring to Fig. 1, with shell finite element modeling, the shaft is divided to 12 sections along theaxial length. Each section contains 12 nodes along the circumferential direction. Variation in the displace-ments of different nodes located at the same section along the shaft length, represents the distortion of thecross-section and hence deformation of the cross-section. Gubran and Gupta (2002a) have studied theoret-ically and experimentally, the deformation of ±45 CFRP shaft. They have observed the dependence of deflection and cross-section deformation on the longitudinal and circumferential modulii of the shaft. This

study analyzes the deflection and cross-section deformation of shafts made of CFRP and GFRP. Further,the possibility of reducing both of shaft deflection and cross-section deformation by hybridization of com-posites (i.e., CFRP or GFRP) and metals (i.e., steel or aluminum) has been attempted. Results presented in(Figs. 3a–c) show the deflection and cross-section deformation for shafts made of different materials. It isobserved that (Fig. 3a) shafts made of GFRP with 0 and 90 fibre orientation have maximum cross-sectiondeformation and deflection, respectively. This is expected due to low circumferential and longitudinalmodulii compared to that of CFRP and metals. Deflection and cross-section deformation of shafts madeof metals are minimum. A drastic reduction in shaft deflection and cross-section deformation are obtainedfor shafts made by hybridization of GFRP or CFRP with metals. These are clearly observed in Fig. 3b forfibres oriented at 0 and (Fig. 3c) for fibres oriented at 90. Taking the case of GFRP with fibres oriented at0 hybrid with steel and aluminum, it can be observed, (Fig. 3b) and pictorial views shown in Fig. 4a–c, a

significant reduction in the shaft cross-section deformation is obtained. The identical deflection and cross-section deformation for hybrid shafts made from CFRP or GFRP with fibres oriented at 90 and steel, (Fig.3c), can be explained by the fact that, for fibres oriented at 90 the longitudinal modulus is very low forboth of CFRP and GFRP compared to that of steel. This makes shafts deflections much dependent onthe contribution of the steel part.

4. Conclusions

The dynamic performance, deflection and cross-section deformation of shafts made of metals (i.e., steeland aluminum), composites (i.e., CFRP and GFRP) and hybrids of metals and composites have been stud-

0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

Un-deformed

-0.06

-0.03

0.00

0.03

0.06

0.00

0.25

0.50

0.75

1.00

-0.06-0.03

0.000.03

(a) (b) (c)

Deformed

-0.06

-0.03

0.00

0.03

0.06

-0.06

-0.03

0.00

0.03

0.06

-0.06-0.03

0.000.03

-0.06-0.03

0.000.03

Fig. 4. Cross-section deformation (a) GFRP with 0 fibre angle, (b) GFRP with 0 fibre angle hybrid with aluminum and (c) GFRPwith 0 fibre angle hybrid with steel.


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ied. Possibilities of improvement in both of dynamic performance, deflection and cross-section deformationby hybridization of metals and composites have been investigated. Followings are the main conclusions:

1. Depending on E 1/q ratio for metals and fibre angle for composites, the natural frequencies of hybridshafts can be optimally placed.2. Deflection and cross-section deformation are reduced by employing hybrid shafts made of metals and

composites.3. Assembling of metallic components (such as, bearings, gears, pulleys, etc.) can be made easier for hybrid

shafts.

References

Bauchau, O.A., 1983. Optimal design of high speed rotating graphite/epoxy shafts. Journal of Composite Materials 17, 170–181.

Belingardi, G., Calderale, P.M., Rosetto, M., 1990. Design of composites material drive shaft for vehicular applications. InternationalJournal of Vehicle Design 11 (6), 553–563.

Gubran, H.B.H., Gupta, K., 2002a. Cross-section deformation of tubular composite shafts subjected to static loading conditions.Mechanics Research Communications 29, 367–374.

Gubran, H.B.H., Gupta, K., 2002b. Composite shaft optimization using simulated annealing, part I: natural frequency. InternationalJournal of Rotating Machinery 8 (4), 275–283.

Guo, M., Harik, I.E., Ren, W.X., 2002. Free vibration analysis of stiffened laminated plates using layered finite element method.Structural Engineering and Mechanics 14 (3), 245–262.

Kim, W., Argento, A., Scott, R.A., 1999. Free vibration of a rotating tapered composite Timoshenko shaft. Journal of Sound andVibration 226 (1), 125–147.

Lee, Y.S., Kim, Y.W., 1999. Nonlinear free vibration analysis of rotating hybrid cylindrical shells. Computers and Structures 70, 161– 168.

Richardet, J., Chatelet, E., Lornage, D., 2000. A three dimensional modeling of the dynamic behavior of composite rotors. Proceedingsof ISROMAC-8 (The 8th International Symposium on Transport Phenomena and Dynamics of Rotating Machinery), March 2000,Honolulu, Hawaii, USA, pp. 988–994.


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