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School of Energy and Power Engineering，Jiangsu University of Science and Technology，Zhenjiang 212100，P.R.China

Abstract: In order to study the dynamic characteristics of multilayer fiber reinforced plastic（MFRP）shaft，the coupling model of three-dimensional equivalent bending stiffness theory and transfer matrix method is established，and the influence of thickness-radius ratio，length-radius ratio，layer angles，layer proportion，and stacked approaches on MFRP shaft dynamic characteristics is investigated. The result shows that the proposed coupling model has high accuracy in MFRP shaft dynamic performance prediction. The proportion of small-angle layers is the decisive factor of MFRP shaft natural frequency. With the increase of thickness-radius ratio and length-radius ratio，the natural frequency of MFRP shaft decreases. The natural frequency of MFRP shaft with the angle layers combination of±45° and ±90° is smaller compared with the metal shaft no matter in simple/free boundary condition or simple/simple supported boundary condition.

Key words：multilayer fiber reinforced plastic（MFRP）shaft；dynamic characteristics；natural frequency

0 Introduction

In recent years，multilayer fiber reinforced plas?tic（MFRP）shaft has been widely used in mechani?cal arm，drive shaft，electrical conduit，satellite truss structures，and unmanned aerial vehicle sys?tem（UAVs）［1］. Compared with the traditional met?al shaft，the MFRP shaft has the characteristics of light weight，high specific strength，and high specif?ic modulus［2-3］，which can improve its natural fre?quency and effectively reduce vibration and noise. In addition，stacking parameters of MFRP shaft can be changed，such as layer angles，number of layers，stacked approaches， discrete layer thickness，etc［4-5］，so the corresponding dynamic design can be carried out according to different requirements.

The stacking parameters of MFRP shaft have been deeply studied by domestic and foreign schol?ars. For layer angles，Badie et al.［6］investigated the effect of layer angles on the natural frequency of MFRP shaft. It shows that the natural frequency of MFRP shaft increases as the layer angles decrease and MFRP shaft containing ±45° angle layers has a high load-carrying capacity and torsion stiffness.Menshykova et al.［7］found that the MFRP shaft with 0° angle layers has a high ability to bear bend?ing load，while the MFRP shaft with 90° angle lay?ers has negligible improvement in bending resis?tance. Sun et al.［8］analyzed the natural frequency of carbon fiber reinforced plastic（CFRP）shaft embed?ded in metal flange by using finite element method，and further proposed that the effect of layer angles on the natural frequency of CFRP shaft is greater than layers thickness. In terms of the stacking se?quence，Gubran and Gupta［9］analyzed and opti?mized the dynamic stresses in composite shafts by using modified equivalent modulus beam theory，in?vestigated the effect of stacking sequence on the nat?ural frequency of composite shafts，and verified the accuracy of the model.

Also，several relevant dynamical modeling the?ories have been proposed. Gubran［10］proposed an equivalent modulus beam theory that took into ac?count the stacking sequence. Sino et al.［11］further proposed a simplified homogeneous beam theory uti?lizing stacking sequence and fiber orientation. Subse?quently，Sevkat and Saeed et al.［12-13］investigated the influence of composite stacking parameters on the shaft natural frequency by means of homoge?neous beam theory. To overcome the limitation that homogeneous beam theory can only solve for the equivalent bending stiffness of composite shafts with symmetrical laminated structure， Montagnier et al.［14］proposed a design for composite shafts based on a simply supported beam model，which can be used to study the dynamics of laminated structural shafts with non-symmetrical layers. Khoshravan et al.［15］designed the composite shaft based on classi?cal laminated theory and carried out modal simula?tions by using ANSYS. Ding et al.［16］analyzed the natural frequency of the composite shaft with the fi?nite element method（FEM），and verified the feasi?bility of FEM. Ren et al.［17］analyzed the effects of layer angles，rotation speed，and length-radius ratio on the natural frequency of CFRP transmission shaft based on the theory of composite thin-walled beam with the variational asymptotic method. Ding et al.［18］designed a lay-up scheme for CFRP drive shafts based on automotive drive shafts and verified the accuracy of the transfer matrix method.

In summary，equivalent modulus beam theory，classical lamination theory，three-dimensional ana?lytical model，and finite element method are used to analyze the dynamics characteristics of CFRP shaft.For equivalent modulus beam theory，it does not re?flect the effect of stacking sequence. The classical lamination theory applies to the thin-walled condi?tion. Three-dimensional analytical model and finite element method can be used to solve the dynamics problem of the CFRP shaft. However，once the model is complicated，both of the two methods are time-consuming. In this paper，the coupling model which combines equivalent bending stiffness theory and transfer matrix method is proposed to analyze the dynamics characteristics of MFRP shaft. The equivalent bending stiffness theory is established based on three-dimensional analytical model. More?over，there are few studies on MFRP shaft consider?ing the effect of thickness-radius ratio，length-radius ratio，stacked approaches，and boundary conditions.Therefore，this paper studies the influence of layer angles，layer proportion and stacked approaches on the natural frequency of MFRP shafts under differ?ent thickness-radius ratios，length-radius ratios，and boundary conditions with the proposed coupling model.

1 Theoretical Equations

1.1 Equivalent bending stiffness theory based on three?dimensional analytical model

The stiffness matrix for the single-layer stressstrain relationship in a coordinate system can be ob?tained as［19］

whereQijis the coefficient of positive-axis compli?ance.σ1，σ2andσ3are the normal stresses，ε1，ε2andε3the strain components，τ23，τ13andτ12the shear stress components，andγ23，γ13andγ12the en?gineering shear strain components.

Based on the stress-strain theory of material mechanics，the positive?axis matrixQof a singlelayer can be obtained as

whereνijis the Poisson’s ratio in thei，jdirection，Gijthe shear modulus in thei，jdirection，andE1，E2，E3are the Young’s modulus.

To facilitate the calculation of the hollow shaft，the above positive?axis matrix is transformed to the off-axis and then the expression in column co?ordinates is obtained. At this point，the principal equation for the direction of fiber orientation at angleφto the axialzis given as

whereis the coefficient of off-axis compliance；r，θandzare the radial，hoop and axial coordinates.

In the above equations，11=m4Q11+2m2n2(Q12+ 2Q66)+n4Q22；33=Q33；44=m2Q44+n2Q55；12=m2n2(Q11+Q22-4Q66)+(m4+n4)Q12；36=mn(Q13-Q23)；13=m2Q13+n2Q23；16=mn[m2(Q11-Q12-2Q66)+n2(Q12-Q22+2Q66)］；45=mn(Q55-Q44)；55=n2Q44+m2Q55；22=n4Q11+2m2n2(Q12+2Q66)+m4Q22；23=n2Q13+m2Q23；Q22)+(n2-m2)2Q66；26=mn[n2(Q11-Q12-2Q66)+m2(Q12-Q22+2Q66)]（m=cosφ，n=sinφ）.

As a reference for the general production pro?cess of composite hollow shafts，alternating -φand +φangle layers are usually stacked and the thickness of the lay-ups is kept uniform. It is as?sumed that the bonding layer between the individual layers is negligibly thin and the thicknesses of the layer satisfy the thin-walled assumption. Treating the ±φangle layers as a minimum mechanical unit and then summing the corresponding intrinsic struc?ture matrix，Eq.（3）can be converted as

Notice that it is necessary to take into account the effects of stresses and strains in the cross-section when building a three-dimensional analytical model based on a thick-walled beam. The relationship be?tween the components of the cross-sectional internal strains and external strains can be expressed as

The stiffness matrix of the thick-walled shaft with thekth ±φangle layers can be obtained by combining Eq.（4）and Eq.（5），we have Eq.（6）be?low

where

The above equation eliminates some elements because the smallest mechanical unit contains a posi?tive and negative cross-laminations. Therefore，the above equations can be simplified as

Thick-walled shafts satisfy the equation for bending moment balance

whereMxis the bending moment in thex-axis，Athe axial cross-section area of MFRP shaft，dAa stress unit，ythe distance from the stress unit to the axis，andσzthe stress in thez-axis direction.

Converting a direct coordinate system into a column coordinate system

Substituting Eqs.（7，8）into Eq.（6）yields

whereεzis the strain in thez-axis direction andbthe curvature radius of the bending deformation layer.

Based on the classical layer-wise theory［20］，the integral of each layer of the moment balance equa?tion can be expressed as

where

Ultimately，the bending moment balance equa?tion can be obtained as

So the equivalent bending stiffness（EI）of the MFRP shaft can be obtained as

The lay-up scheme is expressed in matrix form：[Xi，Yj]t，whereXiandYjare the angles of the inner and outer layers，respectively；tis the stacked approaches，which means repeating internal layers whentis 2. Whentiss，it represents the way of symmetric stacking.

Fig.1 shows the structure and loading types of MFRP shaft，where Fig.1（b）is a schematic dia?gram of the laminated structure concerning the sec?tion of Fig.1（a）and Fig.1（c）shows different load?ing types of MFRP shaft. As can be seen in Fig.1，ois the axis of the MFRP shaft，direction 1 the di?rection of the fiber，direction 2 the direction perpen?dicular to the fiber，direction 3 the direction of the composite material layers thickness，φthe fiber ori?entation angle，Lthe length of MFRP shaft，Rthe radius of MFRP shaft，andhthe thickness of MFRP shaft.rkoandrkiare the outer and inner radius ofkth layer，respectively.Fig.2 is a schematic dia?gram of the part stacking structure of MFRP shaft.

Fig.1 Diagram of structure and loading types of MFRP shaft

Fig.2 Schematic diagram of the part of stacking structure of MFRP shaft

1.2 Natural frequency calculation equation of MFRP shaft

1.2.1 Empirical calculation equation of natu?ral frequency

The MFRP shaft is regarded as a beam structure with equal section，and the natural fre?quency of MFRP shaft in the free state can be calculated according to the empirical calculation equation in Ref.［21］，as shown in Eq.（14）.

whereEIis the equivalent bending stiffness of MFRP shaft andρthe density of composite mate?rial.

1.2.2 Natural frequency calculation equation based on the transfer matrix method

The MFRP shaft is divided into the disk and axis elements. Transfer matrix theory［22］is used to transfer the parameters from the beginning to the end，and the specific transfer mode is given as follows：Theith node represents theith axis segment so that the state variable at the right end of theith unit can be represented by the state at the right end of the（i+1）th unit.

Applying Newton’s second law to theith mass-disk，the state of theith mass-disk can be obtained

where R refers to the right end，L the left end，yithe radial displacement，α ithe deflection an?gle，M ithe bending moment of cross-section，S ithe shear force，mithe mass of mass-disk，Kithe supporting stiffness which acts on the mass-disk（Ki= 0 N·m2），ωthe natural frequency of the shaft，Ipthe polar moment of inertia of the shaft，andIdthe moment of inertia of the diameter of the shaft.

Convert Eq.（15）into matrix form，and then obtain the point transfer matrix of two ends，as shown in Eq.（16）.

Based on the bending and deformation theory of beam，the state between two ends can be de?scribed as

wherelis the length of the shaft section anduthe shear coefficient（u=0）.

Convert Eq.（17）into matrix form，and then obtain the field transfer matrix， as shown in Eq.（18）.

Substitute Eq.（18）into Eq.（16）to obtain the disk-axis transfer matrix，as shown in Eq.（19）.

Terminal state parameters can be expressed as

For the MFRP shaft with free/free support，its boundary conditions at both ends are

Substituting Eq.（21）into Eq.（20）yields

For the MFRP shaft with simple/free support?ed state，its boundary conditions at both ends are

Substituting Eq.（23）into Eq.（24）yields

Boundary conditions for simple supported cases at both ends are

Substituting Eq.（25）into Eq.（24）yields

By solving the non-zero solutions of Eqs.（22，24，26），the natural frequency of MFRP shaft can be obtained under different boundary conditions（free/free support，simple/free support，simple/simple support）.

2 Numerical Calculation and Mod?el Validation

To verify the correctness of the equivalent bending stiffness theory of the three-dimensional analytical model established in this paper，the cal?culated results are compared with the experimental results in Ref.［19］，and the comparison results are shown in Table 1. At the same time，in order to verify the correctness of the natural frequency of the MFRP shaft obtained by the transfer matrix theoretical model established in this paper，MFRP shaft with an outer diameter（D= 0.44 m）and length（L= 2.5 m）is taken as the research ob?ject. The calculation results obtained by transfer matrix theory are compared with classical empirical calculation. The comparison results are shown in Table 2.

Table 1 Comparison of calculation results with experimental results in Ref.[19]

Table 2 Comparison of calculation results with classical empirical calculation

As shown in Table 1，the theoretical calcula?tion results of equivalent bending stiffness by the three-dimensional analytical model in this paper are very close to the experimental results in Ref.［19］，which are controlled within 6%. Thus，it can be considered that the three-dimensional analytical model established in this paper has high accuracy.Moreover，as shown in Table 2，the calculation de?viation of MFRP shaft natural frequency between the empirical equation and transfer matrix method is less than 0.6%，so it can also be considered that the transfer matrix method has high accuracy in solving the natural frequency of MFRP shaft.

3 Calculation and Analysis

The MFRP shaft is composed of T700 carbon fiber and YPH-308 epoxy resin［19］，and the specific mechanical parameters are presented in Table 3.

Table 3 Mechanical parameters of MFRP

3.1 Influences on equivalent bending stiffness

The effect of different layer angles on the equiv?alent bending stiffness of MFRP shaft with different thickness-radius ratios was studied，as shown in Fig.3. Meanwhile，the control variable method was adopted to study the influence of factors（layer an?gles，layer proportion，and stacked approaches）on the equivalent bending stiffness of MFRP shaft（D=0.44 m，L=2.5 m）under different thicknessradius ratios（h/R），as shown in Fig.4.

Fig.3 Effect of different layer angles on the equivalent bending stiffness of MFRP shaft with different thick?ness-radius ratios

As shown in Fig.3，the equivalent bending stiffness of MFRP shaft for different layer angles within ±90° shows a normal distribution，and it can be seen that the smaller layer angles within ±45°，the higher the equivalent bending stiffness of MFRP shaft，while the layer angles beyond ±45° have less influence. In addition，the graph also shows that the equivalent bending stiffness of MFRP shaft comes higher with the increase of thickness-radius ratio，and the effect of thickness-radius ratios on MFRP shaft equivalent bending stiffness is greater when the layer angles are within ±45°，and the smaller the layer angle，the greater the difference in magni?tude.

Combined with Fig.4（a）and Fig.4（b），it can be seen that regardless of the stacked approaches，the proportion of small-angle layers is the main influ?encing factor on MFRP shaft equivalent bending stiffness when the thickness-radius ratio is certain.Specifically，the MFRP shaft equivalent bending stiffness comes higher with increasing the number of small-angle layers. Moreover，the position of the small-angle layers in the inner and outer of the MFRP shaft determines its equivalent bending stiffness，be?cause the MFRP shaft’s equivalent bending stiffness is higher when the small-angle layers are located on the outer side. In addition，the equivalent bending stiffness of the MFRP shaft increases integrally with the increase of thickness-radius ratio（h/R）.

Fig.4 Effect of factors on the equivalent bending stiffness of MFRP shaft with different thickness-radius ratios

3.2 Influences on the natural frequency

The effect of factors（layer angles，layer pro?portion，and stacked approaches）on the natural fre?quency of MFRP shaft with different thickness-radi?us ratios are investigated（L/R=10，free/free sup?port），as shown in Fig.5（a）. Meanwhile，the effect of factors （layer angles， layer proportion， and stacked approaches）on the MFRP shaft natural fre?quency with different length-radius ratios are investi?gated（h/R=1/2，free/free support），as shown in Fig.5（b），F(xiàn)ig.6（a）and Fig.6（b）.

Fig.5 Influence of factors on the natural frequency of MFRP shaft with different thickness-radius and length-radius ratios

Fig.6 Influence of factors on the MFRP shaft natural fre?quency with different length-radius ratios

As shown in Fig.5（a），it can be seen that the first-order and the second-order natural frequency of the MFRP shaft decrease with the increase of thick?ness-radius ratio for a certain proportion of layers laid at each angle，and the difference between the natural frequency of MFRP shaft obtained by each lay-up method is amplified. As shown in Fig.5（b），the difference in the first-order natural frequency of the MFRP shaft obtained by each stacked approach is small after the length-radius ratio reaches 11，while the difference on the second-order natural fre?quency of the MFRP shaft obtained by each stacked approach is small after the length-radius ratio reach?es 15. This means that the length-radius ratio has lit?tle effect on the natural frequency of the MFRP shaft after the length-radius ratio is greater than 15.Combining Fig.5（a）and Fig.5（b），it can be seen that the natural frequencies of the MFRP shafts with different layer angles and stacked approaches subsequently decrease with increasing thickness-ra?dius ratio and length-radius ratio.

Combining Fig.6（a）and Fig.6（b），it can also be seen that the proportion of small-angle layers is the decisive factor for the MFRP shaft natural fre?quency when the length-diameter ratio is certain.Specifically，the natural frequency of the MFRP shaft comes higher when increasing the number of small-angle layers. For either stacked approach，the position of the small-angle layers on the inner and outer of the MFRP shaft determines the natural fre?quency of the MFRP shaft，because the natural fre?quency of the MFRP shaft is higher when the smallangle layers are located on the outer side. The rea?son for this phenomenon is that the radius of the small-angle layers is different when their location varies. Their cross-sectional area is larger when the small-angle layers are located on the outer side of the MFRP shaft，which is in line with the abovementioned conclusion that the proportion of smallangle layers plays a decisive role in the natural fre?quency of the MFRP shaft.

3.3 Influences on different boundary condi?tions

The first-order natural frequency of the MFRP shaft（h/R=1/2，D=0.44 m，L=2.5 m）with dif?ferent layer angles，layer proportion，and stacked approaches was investigated for simple/free support and simple/simple support， respectively， which compared with the first-order natural frequency of the same size metal shaft（See Fig.7）. The mode shapes of the shaft under different boundary condi?tions are shown in Fig.8.

Fig.7 Comparison of the natural frequency of MFRP shaft and metal shaft under different boundary conditions

Fig.8 Mode shapes of the shaft under different bound?ary conditions

As shown in Fig.7，for all boundary condi?tions，it can be seen that the proportion of small-an?gle layers is the decisive factor for the natural fre?quency of MFRP shaft when the length-diameter ra?tio is certain. Specifically，the natural frequency of MFRP shaft comes higher with the increase of small-angle layers. For either stacked approach，the position of the small-angle layers on the inner and outer of the MFRP shaft determines the magnitude of the natural frequency of the MFRP shaft，the nat?ural frequency of the MFRP shaft is greater when the small-angle layers are located on the outer side.The natural frequency of the MFRP shaft is greater in the simple/free boundary condition than in the simple/simple supported boundary condition for ei?ther stacked approach. Under the same boundary conditions，the natural frequency of the MFRP shaft with the angle layers ombination of ±45° and±90° is less than the metal shaft，while the remain?ing combinations of angle layers are greater than the metal shaft. The above phenomena suggest that a certain proportion of small-angle layers（small-angle layers within ±45°）needs to be met to obtain a MFRP shaft design that is superior to the natural frequency of the metal shaft section.

As shown in Fig.8，the mode shape of MFRP shaft is the same as metal shaft under the same boundary conditions. Because mode shape reflects the inherent properties of the system，and it is the normalized deformation under its natural frequency.Thus the mode shape will be determined once the boundary condition is given. However，in forced vi?bration calculation，the displacement response var?ies due to the existence of damping in MFRP shaft.

In summary，the dynamics of MFRP shaft are mainly influenced by its stacking parameters. The variation of MFRP shaft natural frequency under dif?ferent boundary conditions is the same as metal shaft.

4 Conclusions

Based on the coupling model of three-dimen?sional equivalent bending stiffness theory and trans?fer matrix method，this paper studies the influence of layer angles，layer proportion，and stacked ap?proaches on the natural frequency of MFRP shaft under different thickness-radius ratios，length-radius ratios，and different boundary conditions. Several conclusions are as follows：

（1）The coupling model of three-dimensional equivalent bending stiffness theory and transfer ma?trix method is applicable to the dynamics characteris?tics of MFRP shaft. And its prediction accuracy is high.

（2）The proportion of small-angle layers is a decisive factor in the natural frequency of the MFRP shaft.

（3）For either stacked approach，the natural frequency of the MFRP shaft decreases with increas?ing thickness-radius ratio and length-radius ratio for different layer angles and layer proportion.

（4）The natural frequency of MFRP shaft with the angle layers combination of ±45° and ±90° is less than the metal shaft no matter in simple/free boundary condition or simple/simple supported boundary condition，while the remaining combina?tions of angle layers are greater than the metal shaft.

Authors Mr. QIAN Haiyureceived the B.S. degree in New Energy Science and Engineering from Yangzhou Uni?versity in 2019. Since 2019，he has been studying for a mas?ter’s degree at Jiangsu University of Science and Technol?ogy.His research is focused on rotor dynamics.

Dr. ZHU Junchaoreceived the B.S. degree in Physics from Jiangsu University of Science and Technology in 2010 and the Ph.D. degree in Transportation Engineering from Wuhan University of Science and Technology in 2019，respectively.He joined in Jiangsu University of Science and Technology in February 2020. His research is focused on rotor dynamics and bearing lubrication.

Author contributionsMr. QIAN Haiyu complied with the models and wrote the manuscript. Dr.ZHU Junchao contributed to the background of the study and designed the study. Prof.WEN Huabing revised and modified the manuscript. Mr.HE Congshuai translated and proofread the text. All authors commented on the manuscript draft and approved the submission.

Competing interestsThe authors declare no competing interests.