Eigen value analysis of composite hollow shafts using modified EMBT formulation considering the shear deformation along the thickness direction

Pvni Udth , A.S.Sekhr , Velmurugn R

a Machine Design Section, Department of Mechanical Engineering, IIT Madras, Chennai, 600036, India

b Department of Aerospace Engineering, IIT Madras, Chennai, 600036, India

Keywords:Composite hollow shafts Bending natural frequency Modified EMBT formulation Thickness effect

ABSTRACT Composite hollow shafts are used in power transmission applications due to their high specific stiffness and high specific strength.The dynamic characteristics of these shafts are important for transmission applications.Dynamic modelling of these shafts is generally carried out using Equivalent Modulus Beam Theory (EMBT) and Layerwise Beam Theory (LBT) formulations.The EMBT formulation is modified by considering stacking sequence,shear normal coupling,bending twisting coupling and bending stretching coupling.It is observed that modified EMBT formulation is underestimating the shafts stiffness at lower length/mean diameter(l/dm)ratios.In the present work,a new formulation is developed by adding shear deformation along the thickness direction to the existing modified EMBT formulation.The variation of shear deformation along the thickness direction is found using different shear deformation theories,i.e.,first-order shear deformation theory(FSDBT),parabolic shear deformation theory(PSDBT),trigonometric shear deformation theory (TSDBT), and hyperbolic shear deformation theory (HSDBT).The analysis is performed at l/dm ratios of 5,10,15, 20, 25, 30, 35, and 40 for carbon/epoxy composites, E-glass/epoxy composites,and boron/epoxy composite shafts.The results show that new formulation has improved the bending natural frequency of the composite shafts for l/dm<15 in comparison with modified EMBT.The effect of new formulation is more significant for the second and third bending modes of natural frequencies.

1.Introduction

Today, composite materials are replacing many other engineering materials due to high specific stiffness, high specific strength, corrosion resistance, and tailoring of properties.Composite materials are widely used in various aerospace, naval,automobile, and construction industries.The composites as shafts are used in windmill transmission shafts, automobile propeller shafts, and helicopter propeller shafts.The first composite hollow shaft was manufactured by Zinberg and Symmonds [1], with ten layers of high-stiffness boron fiber/epoxy composite and calculated the critical speed of the shaft from experiments and compared with results using Equivalent Modulus Beam Theory (EMBT) formulation.The EMBT formulation is developed based on the Timoshenko beam theory, and the strain energy of the shaft is derived from bending and shear deformations.The stiffness properties of the shaft,i,e equivalent elastic modulus and in-plane shear moduli,are derived using classical laminate theory.The moduli used in the formulation are expressed as functions of invariants of the composite material.Hence,it is independent of the stacking sequence of the hollow shaft.Bauchau [2] designed and optimized the carbon/epoxy hollow shafts.Singh and Gupta [3] studied the effect of the ratio of thickness and radius(t/r),the ratio of length and radius(l/r)of the shaft with bending natural frequency (BNF), and the loss factor of the composite hollow shafts using EMBT formulation.Bert and Kim[4]have used Sanders's shell theory and Donnell's shallow shell theory to calculate the critical speed of the shafts.Singh and Gupta [5] developed layerwise beam theory (LBT) derived using shell theory,including the stacking sequence and couplings effects.Chen and Peng [6] also used Timoshenko beam theory with firstorder shear deformation theory (FSDBT) and finite element analysis for dynamic analysis of the shafts.Gurban and Gupta [7]developed the modified EMBT formulation by adding the stacking sequence effect, shear-normal coupling, bending-stretching coupling effect, and bending-twisting coupling effect to the EMBT formulation.The results obtained with modified EMBT theory match well with those of LBT formulation and are validated with the Zinberg and Symmonds[1]shaft.Recently,the authors[8]used the Modified EMBT formulation for dynamic analysis of carbon nanotube (CNT) reinforced composite hollow shafts.Ren et al.[9]developed a thin wall beam theory using the variational asymptotically method (VAM) and thin shell theory.They calculated the bending natural frequencies and unbalance response of composite hollow shafts with very low thickness.Arab et al.[10]developed a layerwise shaft theory (LST) derived from the shell finite element theory with a shear correction factor.They calculated the critical speeds of functionally graded hollow shafts.Cui et al.[11] developed an algorithm for modal analysis by considering the response of the shaft as the weighted modal response of different modes.The natural frequencies of the machine tool calculated using the algorithm have improved values compared to the finite element method.An analytical formulation for modal analysis of hollow shaft using the Kelvin-Voigt model (stress-strain relation) and Rayleigh-Ritz equation was developed by Cavalini et al.[12].The stiffness of the shaft was considered along the length direction;however, they neglected the stiffness along the through-thickness direction.Yongsheng et al.[9] also developed a mathematical solution for dynamic analysis of a thin-walled hollow shaft with a disk mounted at the centre using the Variational Asymptotically Method(VAM) and assumed that the variation of transverse shear stress along the thickness direction is linear.The present work includes the different shear deformation theories to understand transverse shear stress.

Sun et al.[13] fabricated carbon/epoxy hollow shafts and observed that the ply angle has more influence on bending natural frequency than the thickness of the shaft.The critical speed of the Carbon Fiber Reinforced Plastic (CFRP) drive shaft system used in heavy-duty machine tools was calculated by Yang et al.[14] using the Transfer Matrix Method (TMM) and Finite Element Method(FEM).They observed that the improvement in critical speed was 20% compared to the metal shaft system.Arslan et al.[15] fabricated carbon/epoxy and E-glass/epoxy composite hollow shafts for the driveshaft in a rear wheel drive of the four-wheel vehicle using filament winding technique and performed modal analysis using an impact hammer test.It is observed that the composite shaft could replace the two-piece steel shaft with 45% reduction in weight with the same torsional transmission capability.The shaft of the TURBO ZETA 85.14B truck was prepared using a single shaft made of Al/8 layered carbon epoxy composite and two-piece steel shafts [16].The hybrid shaft had higher critical speed and critical bucking torque of 18631 Nm by comparing the two shafts.The torsional analysis of drive shaft for the Ultralight Aircraft Propulsion system is performed by Poul et al.[17] using finite element method.The drive shaft made using ultrahigh-strength carbon fibers had higher torsional stiffness and elastic properties than Al alloy,and a hybrid design made of high-strength carbon composite and Al alloy.

Higher order theories are used to analyze functionally graded(FG) plate and sandwich members with FG plates.The strain distribution is considered using higher order shear deformation theory (HSDT) with sine shape function [18] and four variable HSDBT [19] for the hygro-thermo-mechanical bending analysis of advanced functionally graded (FG) ceramic metal plate.The quasi 3D -3D shear deformation theory was used for bending analysis[20]and vibration analysis of FG plate[21].Buckling analysis of the FG plate[22]and bi-directionally functionally graded plate[23]was performed by considering HSDT with parabolic shape function.The bending analysis of the FG plate [20], single-walled CNT added FG plate[24],was performed by considering the transverse shear.Free vibration analysis was also done using higher order quasi-3D formulation with cubic, sinusoidal and exponential shape functions.Later, the effect of visco-Pasternak foundations [25] and Pasternak foundation[26]on the bending behaviour of FG plates in hygrothermal environment was performed using integral HSDBT.

The modelling of displacement of sandwich(FG face sheets and FGM hardcore) is done using high-order shear and normal deformation theory.The calculated natural frequencies of the sandwich plate are close results with 3D elasticity[27].The buckling analysis of the same sandwich was performed under hygrothermal environment using HSDBT with sine shape function [28].The effect of visco-elastic foundation on the wave propagation behaviour of sandwich was studied using simple quasi-3D HSDT [29] and the natural frequencies of doubly curved FG sandwich [30].The bending,buckling,free vibration,and hygro-thermal analyses show that considering higher order theories is more appropriate for understanding the response of plate.

From the reference, it is observed that the composite hollow shafts are good solution for drive shafts in automobiles with sufficient propulsion with,sufficient dynamic and stiffness properties and lesser weight.In all the theories discussed above, the shear deformation along the thickness direction is considered by assuming a specific shear correction factor.The improvement in modelling FG plates and sandwiches with FG plates was observed while considering the higher order deformation theories(different shape functions).However, the actual variation of deformation depends on the length and mean diameter of the hollow shafts,i.e.,the l/dmratio.Further, the consideration of shear correction factor may not be suitable for all the l/dmratios.Therefore,it is sensible to modify the shear correction factor with shear deformations to understand the potential of the shaft.The different shear deformations are considered,such as linear and higher order,such as parabolic,trigonometric and hyperbolic variations.The study aims to develop a new formulation by considering higher order shear deformation theories and the effect of the l/dmratio and thickness of the composite shafts on the natural frequencies.The higher order theories predicted higher BNF values than existing theories i,e EMBT and Modified EMBT.The formulation is applied to perform the modal analysis of tube shaft with three different composite materials (E-glass/epoxy, carbon/epoxy, and boron/epoxy composites), and the effect of shaft material and transverse shear on BNFs are highlighted.

2.Shear deformation theories

The shaft used in the present study is considered a simply supported beam with a circular cross-section and finite thickness,as shown in Fig.1.For the present formulation, the masses of the discs M1and M2are considered zero.The thickness of the shaft consists of four layers of composite, and the fiber orientation θ is shown in Fig.1.The displacements along the X,Y and Z axes are u,v and w.The bending and shear deformations of the shaft are considered along the XY and XZ planes, and shear deformation is considered along the thickness direction.The variation of transverse shear deformations along the thickness direction is considered using first order shear deformation theory(FSDBT), parabolic shear deformation theory (PSDBT), trigonometric shear deformation theory (TSDBT), and hyperbolic shear deformation theory(HSDBT).

Fig.1.Coordinate system of the rotor and cross-section of the hollow shaft.

The displacements of the neutral plane along X,Y and Z axes are u0, v0, and w0, respectively.

The bending slopes along XY and XZ planes are φxand φy,respectively.The axial displacement due to elongation in length due to bending is zero.The total axial displacement is due to bending and shear deformations.The equations of displacement in the XY plane, including higher order shear deformation, are [31].

And displacements in the XZ plane are

Here, f(z) is the shape function that varies with shear deformation theory.The shape functions for FSDBT, PSDBT, TSDBT, and HSDBT are presented below [31].

Here z is a variable along the thickness,and h is the thickness of the shaft.

3.Modified-EMBT formulation with shear deformation along the thickness direction

Modified EMBT formulation [7] assumed that the shear deformation along the thickness direction of the shaft as linear i,e First order shear deformation theory.The shear deformation along the thickness depends upon the length/thickness ratio, and the linear variation may not give close results with the increase in the thickness.S?ims?ek [31] has explained the different shear deformation theories for a functionally graded beam.Hence, the present formulation is developed by combining higher order shear deformations [31] and Modified EMBT formulation [7].This formulation is an upgradation Modified EMBT with higher order shear deformation effect.This formulation is used for the dynamic analysis of shafts.Below are the strains induced in the shaft due to bending and shear deformations.

The stress-strain relations of the composite along the fiber direction are given below[32].

Three variable refined plate theory for analysis of composite plates, and the theory is developed by considering the transverse shear stress along with in-plane stresses and classical laminate theory.The constitutive equations of generally orthotropic lamina with XY as in-plane and Z-direction along the thickness direction[33].

The axial and transverse shear strains induced in the shafts are εxxand γxy, γxz, respectively.The strain energy stored in the shaft considering the strains along the length direction is given below

The stress-strain relations by considering the strains along the x-direction and other strains (εyy, γyz) are negligible.The stressstrain relations of the lamina by considering these strains is given below

Substituting the stress-strain relations in Eq.(8).

Substituting the strain expressions and the modified strain energy equation becomes

And the coefficients in the above equation are [31]

The off-axis properties of each play are [33]

The cross section of the tube shaft is expresses as the integration of concentric circular layers, and the same is expressed below

The stiffness matrix elements of each layer are expressed as a function of the properties of the lamina.The equations are given below[32]

Here,k is the number of layers,NL is the number of layers in the composites, ρkis the density, and dkinnerand dkouterare inner and outer diameters of the kth composite layer.

The kinetic energy due to variation in the displacement with time along the X, Y and Z directions is given below.

Here Vx,Vy, and Vzare the velocities along X,Y, and Z directions,ρ(z)is density variation of the composite layer along the thickness.Where the coefficients are [31]

The total energy of the shafts is expressed by using Lagrange expression.

The solution series simply supported boundary conditions with time variation used are [34]

The simply supported boundary conditions are displacements,and slopes are zero at x = 0, and x = l.

Here Wj，Vj，Aj，and Bjare the solution coefficients and wj（x），vj（x），aj（x） and bj（x） are assumed as trigonometric functions,which satisfy the boundary conditions [34].

Solution equations are obtained by applying the stationary condition with respect to solution coefficients, i.e.differentiating the Lagrangian with solution coefficients and equating the derivative with zero.

The total number of equations is 4 N with Ω2(Ω is the eigenvalue variable), and Wj, Vj, Aj, and Bjare unknowns.The total equations yield into eigenvalue problem, and the natural frequency of the shafts is the square root of eigenvalue.

Here {X} is {W1,W2,W3, …,WN; V1,V2,V3, …,VN; A1,A2,A3, …,AN;B1,B2,B3, …,BN;}.

[K] and [M] are symmetric matrices.The non-zero elements of matrices are given below

The matrices are calculated for the first three modes,i.e.at j=1,2, and 3.The present formulation is used for obtained natural frequencies, and the damping matrix [D] is considered null matrix.The eigenvalues are calculated using MATLAB14a for each set of[K]matrix and [M] matrix.The first two eigenvalues represent the normal modes along Y and Z axes and are close to zero, third and fourth eigenvalues are bending modes along XY plane and XZ plane.The bending natural frequencies of the composite shaft calculated along XY plane(mode 1,mode 3 and mode 5)and XZ plane(mode 2,mode 4 and mode 6) are same for all the modes, respectively.Hence, the results are presented for the first three bending modes(mode 1,mode 2 and mode 3),representing modes along XY and XZ planes.

4.Composite hollow shaft modelling and validation

The composite hollow shaft having four carbon/epoxy layers is considered in the present study.The properties of every layer were considered to understand the stacking sequence effect and calculate the stiffness and mass matrix of the shaft.The shaft length,mean diameter, and thickness of each layer are 1000, 100, and 1 mm, respectively.The modal analyses are performed for three materials: carbon/epoxy composite, E-glass/epoxy composite and boron/epoxy composite.The physical and elastic properties of the shaft materials are presented in Table 1.

The boron/epoxy composite shaft used by Zinberg and Symmonds [1] is considered to validate the present formulation.The properties of the shaft:length 2.47 m,mean radius 0.0635 m,wallthickness 0.001321 m, longitudinal modulus 210 GPa, transverse modulus 24.1 GPa, shear modulus 6.9 GPa, Poisson's ratio 0.36,density 1970 kg/m3,and stacking sequence[90°,45°,45°,[0]6,90°].The l/dmratio of the shaft used in Ref.[1] is 38.89.The Bending Natural Frequency (BNF) of boron/epoxy composite shaft is calculated using the present formulation,and the values are presented in Table 2.The values of the present formulation with FSDBT, PSDBT,TSDBT, and HSDBT are close to those obtained by Zinberg and Symmonds [1].

Table 1Properties of the composite materials.

Table 2The critical speed of the shafts using present formulation results with Zinberg and Symmonds [1] shaft.

5.Results and discussion

Modal analyses of the composite shafts(using Table 1 data)are carried out to find the influence of thickness, l/dmratio, and shaft material on critical speeds.The thickness of the shaft has varied from 3 to 10 mm,while the fiber orientation from 0°to 90°.The l/dmvalues of 5,10,15, 20, 25, 30, 35 and 40 are considered in this study.Further, l/dmvalues are obtained by changing the length of the shaft and dmof the shaft is kept constant.The shaft's first three Bending Natural Frequencies (BNFs) are calculated.Three shaft materials are considered in the present study.

5.1.Effect of thickness and fiber orientation on BNF using FSDBT

Modal analysis are performed on composite shafts with different fiber orientations and thicknesses using FSDBT.The first three BNFs are calculated at l/dm= 10 (l - 1000 mm) for carbon/epoxy,E-glass/epoxy and boron/epoxy shafts.The variation of BNFs of carbon/epoxy composite shafts is presented in Fig.2 with increasing the fiber orientation from 0°to 90°and with increasing the thickness from 3 mm to 10 mm.The same trends are observed for E-glass/epoxy and boron/epoxy shafts.The BNFs of all three modes are decreased with the increase in the fiber orientation (0°to 90°),and the percentage decrease in BNF for the three modes at 3 mm shaft thickness are 69.7%, 60.4%, and 51%, respectively.The BNFs are increased with thickness; however, the improvement is less, depending on the fiber orientation.The effect of increased thickness is more effective from 0°to 40°,the effect is very less for the first bending mode.For the second bending mode, the maximum improvement is at close to 45°, while the effect is minimum at 0°and 90°fiber orientations.For the third mode,the effect is increased with fiber orientation up to 70°and later, it is constant.The increase in the first three BNFs with the thickness increased from 3 mm to 10 mm are 20.5,29.3 and 27 Hz at 0°fiber orientation, respectively, and 9, 29.2 and 50.5 Hz at 90°fiber orientation, respectively.For all the bending modes, the effect of thickness variation from 3 mm to 10 mm on BNF improvement is less.Since FSDBT is first order deformation theory,it shows only the linear variation of shear deformation.Hence, higher order deformations theories are necessary for understanding the thickness effect on shear deformation.The following sections discuss the effect of thickness using PSDBT, TSDBT, and HSDBT.

Fig.2.Variation of BNFs of carbon/epoxy composite shafts at l/dm = 10 with orientation and thickness varied from 3 to 10 mm.

5.2.Natural frequencies using different theories at l/dm = 5

The natural frequencies of the composite hollow shafts at l/dm=5(l-500 mm)are calculated using FSDBT,PSDBT,TSDBT,and HSDBT.The fiber orientation is 0°, and the shaft's thickness varies from 3 mm to 10 mm.The BNFs determined for carbon/epoxy shafts using all the theories are compared with FSDBT in the following subsections.

Fig.3.BNFs of carbon/epoxy shafts at different l/dm ratios using different theories at thickness 3 mm: (a) Mode-1; (b) Mode-2; (c) Mode-3.

Fig.4.Percentage increase in BNFs with the increase in thickness from 3 to 10 mm at different l/dm ratios using different theories: (a) Mode-1; (b) Mode-2; (c) Mode-3.

Fig.5.BNF values E-glass/epoxy composite shafts at different l/dm ratios using different theories at thickness 3 mm: (a) Mode-1; (b) Mode-2; (c) Mode-3.

Fig.6.BNF values boron/epoxy composite shafts at different l/dm ratios using different theories at thickness 3 mm: (a) Mode-1; (b) Mode-2; (c) Mode-3.

5.2.1.First order shear deformation theory (FSDBT)

Modal analyses are performed for composite shafts using FSDBT.The BNFs for the first three modes at 3 mm thickness are 1202.58,2803.5 and 4354.54 Hz,respectively.With the increase of thickness from 3 mm to 10 mm, the BNF for the first three bending modes increased by 29.34, 23.11, and 17.09 Hz (2.44%, 0.82%, 0.39%),respectively.The influence of thickness is not significant at higher bending modes.

5.2.2.Parabolic shear deformation theory (PSDBT)

The BNFs are calculated using PSDBT,and the values of the first three modes of shaft at 3 mm thickness are 1810.6, 5721.2 and 10269.09 Hz, respectively.The BNFs calculated using PSDBT show higher values than FSDBT,and the percentages increase in BNFs for the first three modes are 50%,104% and 135%, respectively.This is due to the variation of shear deformation from linear to cubic and the increase in the strain energy of the shaft.The modal analysis is performed by varying the thickness from 3 to 10 mm.The increases in the BNFs for the first three modes are found to be 107.86,247.29,and 368.13 Hz (5.96%, 4.32% and 3.59%), respectively, indicating a significant influence of thickness.Thus it shows that when considering the shaft thickness, PSDBT is more effective than FSDBT.

5.2.3.Trigonometric shear deformation theory (TSDBT) & higher order shear deformation theory (HSDBT)

It can be noted that the variation of shear deformation for TSDBT and HSDBT are very close.For both cases, the increased values in BNFs of shafts for the first three modes compared to FSDBT are 63%,149%and 210%,respectively.The BNFs of the shaft using TSDBT and HSDBT are a little higher than PSDBT.This is due to the present theories having more shear deformations along the thickness direction compared to PSDBT and enhancement of strain energy.The increase in the thickness of the shafts from 3 to 10 mm has improved the BNF.The percentage improvement in BNFs for the first three modes are 6.61%, 4.87% and 3.37%, respectively.The thickness effect on BNF is high in the first mode, and the effect decreases with further modes.

5.3.Effect of l/dm on BNF of carbon/epoxy composite shafts using higher order deformations theories

The bending natural frequencies are calculated using FSDBT,PSDBT, TSDBT and HSDBT are presented in Fig.3.The thickness of the shafts is 3 mm;dmis 100 mm and the fibers are oriented along the length direction.The percentage increase in mode-1 BNFs calculated using HSDBT compared to FSDBT at l/dmratios 5,10,15,20,25,30,35,and 40 are 64%,21%,9.8%,5.7%,3.7%,2.6%,1.9%,and 1.4%, respectively.The difference between BNFs obtained using higher order theories, and FSDBT is more for l/dm= 5, decreasing with an increase in l/dm.This is due to the shear deformations along the thickness direction,which are more effective at lower values of l/dmratio, and the effect is lower at higher l/dmratio.The percentage increase in the BNFs with higher order theories is increased with mode number.The increase in mode number leads to the fluctuations of slope along the length direction, and the strain energy stored is increased due to shear deformation and gives the increase in BNF with the increase in mode numbers.Fig.3 shows that the increase in BNFs using higher order theories is considerable up to l/dm<15 and at l/dm<15,BNFs of higher order theories coincide with FSDBT.It is more appropriate to use HSDBT for the shafts with l/dm< 15 and FSDBT for those with l/dm> 15.

5.4.Effect of thickness and l/dm on BNF of carbon/epoxy composite shafts using higher order theories

The modal analyses of hollow shafts are performed using FSDBT,and higher order shear deformation theories (PSDBT, TSDBT and HSDBT)with shaft thickness varied from 3 mm to 10 mm and at l/dmratios 5,10, 20, 25, 30, 35 and 40.The percentage increase in BNFs of the first three modes with the increase in thickness from 3 to 10 mm are calculated using all the theories and presented in Fig.4.The improvement in BNF with thickness is observed for all the theories and ranking order of maximum increase is given by HSDBT, TSDBT, PSDBT and FSDBT and the maximum percentage increase is less than 8% for all the three modes.The percentage increase in the first BNF is less for FSDBT compared to higher order theories for the shafts with l/dm< 15 ratios and the value is increased with an increase in l/dmratio and become close to higher order theories at l/dm>15 ratios.All the higher order theories have more BNF at a lower l/dmratio, and the improvement is very less.The influence of thickness on BNF is more at a lower l/dmratio,and the same is observed using higher order theories compared to FSDBT.The FSDBT shows the linear variation of percentage increase in BNF with l/dmwith the increase in mode number.The higher order theories show that the percentage increase in BNFs decreased with an increase in mode number for l/dm< 15; later, it is almost constant.Hence,the dynamic analysis of hollow shafts with higher l/dmratios can be performed with FSDBT and lower l/dmratios with HSDBT.

5.5.Effect of thickness and l/dm on BNF of composite shafts using higher order theories

Modal analyses are performed on shafts made of E-glass/epoxy composite and boron/epoxy composite using all the theories using the properties present in Table 1 and varied thickness from 3 to 10 mm.The BNFs of the first three modes of E-glass/epoxy composite shafts and boron/epoxy composite shafts at l/dmratios are presented in Figs.5 and 6, respectively.From Figs.3, 5 and 6, it is observed that the first three BNFs of the shafts made of carbon/epoxy, E-glass/epoxy and boron/epoxy composites are improved with higher order theories for l/dm< 15 and shafts with l/dm> 15 are coinciding with the FSDBT.The increasing order of BNF of shafts is carbon/epoxy composite, boron/epoxy composite and E-glass/epoxy composite.The improvement in the BNF for the second and third modes is close to the two times of the first mode and three times of the first mode, respectively.It is observed that for the shafts with l/dm>15,the shaft is becoming long,the stiffness of the shaft becomes less, and the deformations along the thickness direction are negligible compared to that in the length direction.

The percentage improvement in BNFs of E-glass/epoxy composite shafts and boron/epoxy composite shafts with increasing thickness from 3 to 10 mm and l/dmratios 5 to 40 are presented in Figs.7 and 8,respectively.From Figs.4,7 and 8,it is observed that the maximum percentage is close to 8%for all three modes and the three composites.For carbon/epoxy composite and E-glass/epoxy composite,all the higher order theories are very close to each other.The effect of thickness is seen in higher order theories at l/dm<15 for all the composites, and later the difference between higher order theories and FSDBT is becoming less.The difference between higher order theories (HSDBT) and FSDBT is increased with the mode number for all the materials.For all the shafts, HSDBT gives the highest BNF and the highest percentage improvement in BNFs with increasing thickness.

Fig.8.Percentage increase in BNF of boron/epoxy shafts with the increase in thickness from 3 to 10 mm at different l/dm ratios using different theories:(a)Mode-1;(b)Mode-2; (c) Mode-3.

6.Conclusions

A new formulation for the dynamics analysis of composite hollow shafts with shear deformation along the thickness direction is developed.The existing theories for eigenvalue analysis of composite tube shafts considered the shear correction factor and didn't consider the shear deformations.The addition of shear deformation along thickness direction has improved the stiffness of the shafts and hence the natural frequency.The shear deformation variations along the thickness direction are taken using FSDBT,TSDBT, TSDBT and HSDBT.Modal analysis of carbon/epoxy composite,E-glass/epoxy composite,and boron/epoxy composite shafts are performed using the present formulation at l/dmratios 5,10,15,20,25,30,35,and 40.The bending natural frequencies of the shafts increase with thickness from 3 to 10 mm.The increase in BNF of the shaft due increase in thickness is less with the increase in fiber orientation from 0 to 90.The higher order theories give higher BNF values for shafts with l/dm< 15 than FSDBT.The BNFs of the shaft with l/dm>15 are almost the same for all theories.The shear deformation along the thickness direction influences the natural frequencies at lower l/dmratios.The present formulation with higher order shear deformation theories is more appropriate for dynamic analysis of the shafts with l/dm< 15.The variation of bending natural frequencies by varying the thickness from 3 to 10 mm is marginal.The present formulation clearly explained the importance of transverse shear deformation on the stiffness and eigenvalues of the tube shaft.The methodology used in the present work is simple, and the results are comparable with the experimental results.The present formulation's results can guide the shafts' selection for torque transmission applications.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.