Vibration Characteristics of Carbon-fiber Reinforced Composite Drive Shafts Fabricated using Filament Winding Technology

Journal Pre-proofs Vibration Characteristics of Carbon-fiber Reinforced Composite Drive Shafts Fabricated using Filament Winding Technology Zeyu Sun, Jie Xiao, Xuduo Yu, Rogers Tusiime, Hongping Gao, Wei Min, Lei Tao, Liangliang Qi, Hui Zhang, Muhuo Yu PII: DOI: Reference:

S0263-8223(19)33732-8 https://doi.org/10.1016/j.compstruct.2019.111725 COST 111725

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

2 October 2019 21 November 2019 25 November 2019

Please cite this article as: Sun, Z., Xiao, J., Yu, X., Tusiime, R., Gao, H., Min, W., Tao, L., Qi, L., Zhang, H., Yu, M., Vibration Characteristics of Carbon-fiber Reinforced Composite Drive Shafts Fabricated using Filament Winding Technology, Composite Structures (2019), doi: https://doi.org/10.1016/j.compstruct.2019.111725

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Vibration Characteristics of Carbon-fiber Reinforced Composite Drive Shafts Fabricated using Filament Winding Technology Zeyu Sun1,2, Jie Xiao3, Xuduo Yu4, Rogers Tusiime1, Hongping Gao4, Wei Min1, Lei Tao1, Liangliang Qi1, Hui Zhang1,2* and Muhuo Yu1,2* 1 State Key Laboratory for Modification of Chemical Fibers and Polymer Materials, College of Materials Science and Engineering, Donghua University, Shanghai 201620, China; 2 Shanghai Key Laboratory of Lightweight Structural Composites, Donghua University, Shanghai 201620, China; 3 Research Center for Analysis and Measurement, Donghua University, Shanghai 201620, China; 4 Shanghai Fishman New Materials Technology Co., Shanghai 201620 China;

Abstract: The metal connector and tube of the composite drive shaft were bonded together generally. However, the glued shafts showed numerous risks due to the instability of chemical glues. To avoid potential risks, filament winding with integrated flanges was proposed to prepare carbon fiber-reinforced composite (CFRP) drive shafts in this study. The natural frequency and damping of CFRP shafts were investigated by FEA and experiment. Results revealed that the natural frequency of CFRP shafts embedded in metal flanges is slightly lower than that of the glued CFRP shafts. The metal flanges did not affect the changing trend of the natural frequency with various fiber ply angles or thicknesses, and the damping had little effect on the natural frequency. The natural frequency of CFRP tubes and drive shafts was predicted using FEA and agreed well with the experimental results. The results of this study will be of great significance in the design and application of CFRP drive shafts. Keywords: CFRP drive shaft; Metal flange; Filament winding; Natural frequency; Damping

1. Introduction Lightweighting is one of the effective ways to realize energy conservation and emission reduction in many fields[1-3]. Because of its light weight and high-strength performance, carbon fiber-reinforced plastic (CFRP) is a preferred alternative to traditional metals [4, 5]. Compared with metal drive shafts, CFRP drive shafts exhibit many advantages[6, 7] such as transmission efficiency, improved comfort, and emission reduction[8]. CFRP drive shafts have extensively been employed in the helicopter, automotive, marine industries[9, 10]. With the improvement in product performance, the application of high-speed shafts is gaining widespread acceptance[11]. In designing a high-speed shaft, vibration performance is an important characteristic that is considered for the service life, efficiency, reliability, and noise of the drive shaft. The vibration performance is determined by various parameters of the drive shaft system, such as natural frequency, modal mode, and damping[12-14]. Owing to the ultra-high specific strength and modulus of carbon-fiber composites, the natural frequency of CFRP drive shafts is more important than the torsional strength which is the main factor that influences the design of metal drive shafts[15]. Many researchers have studied the natural frequency of the composite tube, which is the uppermost part of the composite drive shaft[16]. Talib et al.[17] verified the influence of ply angles on the natural frequency of glass/carbon fiber-reinforced composite drive shafts. Badie et al.[18] studied the influence of fiber orientations and stacking sequences on the torsional stiffness, natural frequency, buckling strength, fatigue life, and failure modes of composite tubes using finite element analysis (FEA) and experiments. Montagnier et al.[19] determined the formulation for flexural vibrations, which was applied to composite drive shafts mounted on viscoelastic supports using a genetic algorithm. Aleksandr et al.[20] studied the multiple parameters in drive shaft design characterizing the composite material, such as fiber orientation angles, stacking sequence, and ply thicknesses.

Damping of drive shafts can trigger unstable vibrations at critical speed, which greatly affects the natural frequency[21]. Vibration damping is related to the material and structure[22]. Montagnier et al.[23] found that the damping of CFRP laminates is greater than that of metal materials. Sino et al. [6] studied the natural frequency and damping of composite drive shafts with different fiber volume content. They found that the natural frequency of rotation increased with the increase in fiber volume content and decrease in damping factor. The effects of material damping on the stability and self-excited vibration of composite shafts have also been researched[24, 25]. Drive shafts always end with metal connectors such as metal spline-forks and flanges (Fig. 1)[26, 27]. These indispensable metal connectors influence the vibration characteristics of drive shafts[28]. However, there are few systematic studies on the natural frequency and damping properties of CFRP drive shafts with flanges. Investigations of the vibration performance of composite drive shafts with mental flanges are essential and urgent. Metal connectors are often glued to composite tubes in regular CFRP drive shafts[29]. Hidden dangers such as fatigue and high hydrothermal characteristics exist during actual operations[18, 30]. To avoid the potential hazards of glued shafts, filament winding with integrated flanges is proposed to prepare CFRP drive shaft. After physical or chemical treatment, the flanges are directly embedded into the CFRP tubes during winding, which greatly improves the connective stability of the CFRP drive shaft. In this paper we introduced the following works. Firstly, natural frequency of CFRP tubes is investigated to verify the reliability of the simulation by FEA. Secondly, the vibration performance of CFRP tubes and CFRP drive shafts with different ply ways were comparatively analyzed to illustrate the effect of metal flanges on the natural frequency of the CFRP drive shafts by FEA. Finally, the natural frequency and damping of the CFRP drive shafts with various ply ways prepared by filament winding with integrated metal flanges were systematically investigated by FEA and the experimental pulse vibration excitation technique (PVET).

Composite tube Metal flange

Composite drive shaft

Fig. 1. The models of CFRP drive shaft

2. The natural frequency of the composite tube under the undamped and single degree-offreedom vibration Under undamped and single degree-of-freedom vibration, the natural frequency of shafts and tubes molded by the isotropic material is 1

𝐹0 = 2𝜋

48𝐸𝐼 𝑚𝐿3

(1)

where F0 is the natural frequency, L is the length of the shaft, E is the elasticity modulus of material, D is the external diameter of the tube, d is the inner diameter of the tube, and ρ is density of material.

A

B z'

x z y

x'

θ x

z y'

y

Fig. 2. Coordinate systems of composite laminate with unidirectional filament winding technology (A) and composite drive shaft reinforced by carbon fibers (B).

Material properties of the composite drive shaft were analyzed with classical lamination theory. The theory treats the linear elastic response of laminated composites under plane stress, and incorporates the Kirchhoff– Love assumption for bending and stretching of thin plates[31]. The stiffness matrix of the unidirectional filament layer in the x-y-z coordinate system (Fig. 2A) is [C]. During winding of the composite drive shaft, the fiber ply angle is represented by θ. The stiffness coefficient matrix of the x'-y'-z' coordinate (Fig. 2B) is [𝐶][32]. The relationship between[𝐶]and [C] is a complex function of the winding angle θ. According to engineering mechanics, the equivalent bending stiffness (EI) of the composite tube can be obtained by adding the bending stiffness of a single layer[6, 33]. 4

𝑁

𝜋[(𝐷𝑛) ― (𝑑𝑛)

EI = ∑𝑛 = 1

4

]

64

𝐸𝑛𝑥′

(2)

where N is the total number of layers, and 𝐷𝑛 and 𝑑𝑛 are the outer and inner radii of the nth group of the layers. 𝐸𝑛𝑥′ is the effective engineering modulus of the nth group of layers in the x' direction (Fig. 2B), which can be obtained with Eq. (3-6)[28, 34, 35]. 2 𝐴𝐴𝑛11𝐴𝐴𝑛22 ― (𝐴𝐴𝑛12)

𝐸𝑛𝑥′ =

𝐴𝑛11 =

𝐴𝐴𝑛22ℎ𝑛 𝐶213 (𝐶11 ― 𝐶 ) 33

𝐴𝑛12 = (𝐶12 ―

ℎ𝑛

𝑛 𝐶23𝐶13

𝐶33

) ℎ𝑛 𝑛

𝐶223

𝐴𝑛22 = (𝐶22 ― 𝐶 ) ℎ𝑛 33

𝑛

(3) (4) (5) (6)

where ℎ𝑛 is the thickness of the nth group of the layers, and 𝐶𝑖𝑗 is the angular conversion stiffness coefficient of the material spindle. From Eq. (4–6), we know that the effective engineering modulus 𝐸𝑛𝑥′ of the axial section for the composite drive shaft is also a complex function about winding angle θ. According to Eq. (2) and (3), the natural frequency of the composite shaft tube is 1

𝐹0 = 𝜋𝐿2

𝑁 2 2 3∑𝑛 = 1[(𝐷𝑛) + (𝑑𝑛) ]𝐸𝑛𝑥′

𝜌

(7)

3. The natural frequency and damping properties of the composite tube under the damped vibration Damping properties of a single-degree-of-freedom system is usually carried out by the attenuation ratio of the amplitude in the process curve of damping vibration. The ratio of two adjacent amplitude absolute values is called waveform attenuation coefficient. The greater the attenuation coefficient is, the faster the attenuation speed is, indicating the greater the material’s damping. Typical vibration attenuation curve with time is shown in Fig. 3.

Fig.3. Typical attenuation of vibration waveform.

Simply, attenuation coefficient is calculated by | 𝐴k

|𝐴k|

+ 1|

| 𝐴k

+ 2|

| 𝐴k

+ n|

φ = |𝐴 | = |𝐴 | = |𝐴 | = ⋯ = |𝐴 k +1 k + n + 1| k +2 k +3 (8) |𝐴k|

𝜑n = |𝐴

|𝐴k + 1| |𝐴k + 2| |𝐴k + n| |𝐴k| ∙ ∙ ⋯ ∙ = |𝐴k + n + 1| |𝐴k + n + 1| k + 2| |𝐴k + 3|

(9)

| ∙ |𝐴

k+1

where 𝜑 is the attenuation coefficient, 𝐴k is the amplitude of the kth peak and 𝐴k (k+n+1)th peak. In which n is calculated by

𝑛=2∗

𝑡𝑘 + 𝑛 + 1 ― 𝑡𝑘 + 1 𝑇

+1

is the amplitude of the

= 2𝐹0 ∗ (𝑡𝑘 + 𝑛 + 1 ― 𝑡𝑘 + 1)

(10) where F0 is the measure natural frequency under the undamped and single degree-of-freedom vibration. Further, the damping ratio 𝜉 is obtained by

ξ =

lgφ 1.862 + (lgφ)2

(11) Natural frequency of the composite tubes under the damped and free vibration is calculated by 𝐹𝑑 = 𝐹0 1 ― 𝜉2

(12)

where Fd is the natural frequency under the damped and free vibration. 4. Computational models and experimental setup The overall length of the CFRP tube was 1185 mm with inner diameter of 70 mm. The flange weighed 750 g. Twelve categories of CFRP drive shafts with different ply angles (Table 1) and ply thicknesses (Table 2) were prepared to study the vibration characteristics. The categories (a–f) (Table 1) are drive shafts with different ply angles while categories (g–l) (Table 2) are drive shafts with different ply thicknesses. Specimens x01 and x02 (x: a to l), represent the simulated and experimental CFRP tubes obtained by cutting away the metal flanges and transition area of the CFRP drive shafts, which aims to investigate the reliability of the simulation by FEA. Specimens x0 and x1 (x: a to l), represent the simulated CFRP tubes and CFRP drive shafts with metal flanges, respectively, which aims to investigate the influence of metal flanges on natural frequency. Specimens x2 (x: a to l) represent experimental drive shafts with metal flanges by PVET, which aims to investigate the influence of transition area of CFRP drive shafts by comparing with specimens x1 (x: a to l). Beside the damping properties are also studied in this part. Table. 1. CFRP tubes and CFRP drive shafts with various ply angles for FEA and PVET

Features

Ply angles

categories

a

specimens

Ply angles（° ）

Theoretical thicknesses（ mm）

a01 a02 a0

[±154/902]

3.00

With metal flanges

FEA

PV ET

√ √ √

b

c

d

e

f

a1 a2 b01 b02 b0 b1 b2 c0 c1 c2 d0 d1 d2 e01 e02 e0 e1 e2 f01 f02 f0 f1 f2 l3

√ √

√ √ √ √

[±254/902]

[±354/902] [±454/902]

√ √

3.00 √ √ 3.00

3.00

√ √ √

√ √

√ √ √

√ √

√ √ √

[±554/902]

3.00

√ √

√ √

√ √ √

[±654/902]

3.00

√ √

√ √ √

√ √

Table. 2. CFRP tubes and CFRP drive shafts with various ply thicknesses for FEA and PVET

Features

categories

g

h

i Ply thicknesses j

k

l

specimens g0 g1 g2 h0 h1 h2 i0 i1 i2 j0 j1 j2 k0 k1 k2 l1 l2 l3

Ply angles（° ）

Theoretical thicknesses（ mm）

With metal flanges

[±256/902]

4.20

√ √

[±255/902] [±254/902] [±253/902] [±252/902] [±25/902]

3.60

3.00

2.40

1.80

1.20

√ √ √ √ √ √ √ √ √ √

FEA

PV ET

√ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √ √

4.1 Materials The reinforcement material was carbon fiber (T700SC-12000-50C, Toray Industries). The matrix material was epoxy resin (Bisphenol A epoxy, Wuxi Phoenix Resin Company, China), and the curing agent was 1cyanoethylated imidazole (Sinopharm Chemical Reagent Co., Ltd, China). 4.2 Preparation of CFRP drive shafts by one- step winding The CFRP drive shafts with metal flanges (Fig. 4) were prepared by one-step wet winding in a four-axis winding machine (CRJ-12, LONGTEC, China). A 45-steel cylindrical mandrel with 1600 mm length and 70 mm diameter was the mold core used to wind the CFRP tubes and drive shafts. For the CFRP drive shaft, the thicknesses of each layer was 0.3 mm. To ensure the tension of the fiber in the radial direction, the outermost fiber layer of the tube was wound at 90°.

CFRP tube x02

CFRP tube x2

Fig. 4. CFRP drive shaft molded by filament winding.

The so-called filament winding with integrated flanges means that the flanges were directly embedded into the CFRP tube during fiber winding (Fig. 5). To realize continuous winding, the carbon fibers were wound in the transition region 70 mm from geodetic line to 90°.

Fig. 5. The winding process of CFRP drive shaft.

4.3 Finite element model Finite element models were generated and analyzed using ANSYS (Workbench 17.0) commercial software (Fig. 6). A three-dimensional model of x01and x0 composite tubes were also developed and meshed with linear quadrilateral elements. The element number of x01and x0 composite tubes respectively were 9064 and 2674. Three-dimensional models of the metal flange were developed and meshed with tetrahedron elements. The element number of the flange was 3921. A coordinate system was defined to align the material axis of the layup, and the finite element model was established. Only gravity was applied to the drive shaft as load and no constraints were applied to the model.

CFRP tube x01

x1

x0

Fig. 6. The models of FEA.

According to the laminated plate theory, the real laying angle of the transition zone is highly complex in the FEA process. Thus, the layering model of the transition zone in the CFRP drive shaft molded by filament winding technology was simplified. The angle of the CFRP tube in the CFRP drive shaft was set to a uniform fiber angle (Fig. 7). In fact, this simplified layering model can also be used as a model of the CFRP shaft joined by glue.

Fig. 7. The fiber angle model of the transition zone in the CFRP drive shaft.

The material parameters are listed in Table. 3. The free constraint was set to obtain natural frequencies and vibration nephograms. Table. 3. Material parameters for FEA Material

Tensile modulus E /GPa

Shear modulus G/GPa

Poisson's ratio μ

Density/kg·m-3

Structural steel

200.0 128.7 (E1) 7.8 (E2) 7.8 (E3)

76.0 5.0 (G12) 3.1 (G23) 5.0 (G13)

0.30 0.27 (μ12) 0.42 (μ23) 0.27 (μ13)

7.8 × 103

CFRP

1.6 × 103

4.4. PVET to study the vibration characteristics The vibration characteristics of CFRP drive shafts were obtained by the free-free modal test. The modal testing system schematic is shown in Fig.8 PVET was adopted to test the natural frequency of the CFRP drive shaft; it has also been widely used to describe the dynamic characteristics of vibrating systems[36-38].

Fig.8. Modal testing system schematic.

Both ends of all specimens were put on the foam to simulate the free-free boundary conditions. Different points along the axial direction were evenly assigned on the specimens. An impulse hammer (Type 9722A500; sensitivity at 100 Hz, 12.09 mV/N; KISTLER) was used to impact the assigned points to induce and excite vibrations of the specimens. The vibration responses were detected using an acceleration transducer (Type 8640A5; sensitivity at 159 Hz, 100.64 mV/m/s2; KISTLER). Then, the excitation and response signals were collected and processed by the 640UEZ-analyst dynamic signal analyzers connected to a computer monitor. The fast Fourier transform method was used to determine the frequency responses and modal parameters, including natural frequencies and damping. The center of the drive shaft was hit three times by the hammer to reduce experimental errors. 5. Results and Discussion

5.1 Natural frequency of CFRP tubes with different ply angles and reliability verification of simulation The ply angle is an important parameter in the design of composites [39, 40]. General speaking, the ply angle is defined relative to the axial direction. The bending stiffness of composite tube is the main factor affecting its natural frequency. The ply angle affects the load distribution of the tube, which results in the difference of the natural frequency. In this part, the influence of ply angle on the natural frequency of CFRP tubes obtained by cutting away the metal flanges and transition area of the CFRP drive shafts is studied by FEA and PVET, which verified the reliability of simulation by FEA furtherly. As shown in Table 2, four categories of parameters (a, b, e and f) were set to study the effect of ply angles on the natural frequency of CFRP tubes and verify the reliability of simulation furtherly. The natural frequency and vibration nephogram of CFRP tubes with different ply angles are shown in Fig. 9. The natural frequency of CFRP tubes decreased with the increase in ply angles. Specifically, the natural frequency was reduced by 62.71% when the ply angle increased from 15° to 65°. The vibration modes of CFRP tubes were the bending modes. The results indicate that the ply angles can significantly change the natural frequency. Specimen a01: 539.53Hz

Specimen b01: 486.84 Hz

Specimen e01: 236.41 Hz

Specimen f01: 201.19 Hz

Fig.9. The natural frequency and bending patterns of CFRP tubes with different ply angles by FEA.

The natural frequency results of CFRP tubes with different ply angles by PVET are shown in Fig. 10. With the increase in ply angles, the natural frequency of CFRP tubes decreased drastically. Specifically, the natural frequency was reduced by 63.30% when the ply angle increased from 15° to 65°. It can be seen that ply angle has great influence on the natural frequency of CFRP tubes.

Accelerated speed (m/s2)

0.06

a02

0.05

b02 e02

0.04

545 Hz 480 Hz

f02

0.03 235 Hz 200 Hz

0.02 0.01 0.00 0

200 400 Frequency (Hz)

600

Fig.10. Diagram of the nature frequency of CFRP tubes with different ply angles by PVET.

The relationship between the natural frequency of CFRP tubes and ply angles is shown in Fig. 11. The ply angle plays an important role in improving the natural frequency of CFRP drive shafts. The high coincidence of the natural frequency by FEA and PVET verifies the reliability of the simulation, which provides a basis for the natural frequency analysis of the drive shaft with metal flanges.

550 x01

500

x02

Frequency(Hz)

450 400 350 300 250 200 150 10

20

30

40

θ (°)

50

60

70

Fig. 11. Graph of changes in natural frequency with ply angles. x01: CFRP tubes by FEA and x02: CFRP tubes by PVET.

5.2 Vibration characteristics of CFRP drive shafts with different ply angles As shown in Table 2, six categories of parameters (a–f) were set to study the effect of ply angles on the natural frequency of CFRP drive shafts. The natural frequency and vibration nephogram of CFRP tubes with different ply angles are shown in Fig. 12. The natural frequency of CFRP tubes decreased with the increase in ply angles. Specifically, the natural frequency was reduced by 63.07% when the ply angle increased from 15° to 65°. The vibration modes of CFRP tubes were the bending modes. The results indicate that the ply angles can significantly change the natural frequency. Specimen a0: 432.69 Hz

Specimen c0: 305.76 Hz

Specimen e0: 187.75 Hz

Specimen b0: 388.49 Hz

Specimen d0: 239.84 Hz

Specimen f0: 159.79 Hz

Fig.12. The natural frequency and bending patterns of CFRP tubes with different ply angles by FEA.

The natural frequency and vibration nephogram of the CFRP drive shafts by FEA are shown in Fig.13. The natural frequency was much lower and the downward trend gradually slowed down with the increase in ply

angle. In particular, the natural frequency of the CFRP drive shafts was reduced by 63.52% when the ply angle increased from 15° to 65°. The reduction ratios of the natural frequencies of the CFRP tubes and CFRP drive shafts were close at approximately 63% (63.07% and 63.53%) when the ply angle increased from 15° to 65°. Specimen a1: 244.86 Hz

Specimen b1: 218.42 Hz

Specimen c1: 171.35 Hz

Specimen d1: 134.23 Hz

Specimen e1: 104.94 Hz

Specimen f1: 89.32 Hz

Fig. 13. The natural frequency and bending pattern of CFRP drive shaftss with different ply angles by FEA.

The natural frequency results of CFRP drive shafts with different ply angles by PVET are shown in Fig. 14. With the increase in ply angles, the natural frequency of CFRP drive shafts decreased.

Accelerated speed (g)

0.008

a2

b2

c2

d2

e2

f2 206 Hz

0.006 0.004

235 Hz

155 Hz 118 Hz 100 Hz 87 Hz

0.002 0.000 50

100

150

200

250

Frequency (Hz) Fig. 14. Diagram of the nature frequency of CFRP drive shafts with different ply angles by PVET.

The relationship between the natural frequency of CFRP drive shafts and ply angles is shown in Fig. 15. The result shows that the flanges greatly affected the natural frequency of the CFRP drive shafts. The ply angle plays an important role in improving the natural frequency and even the critical speed. Furthermore, the natural frequency of the CFRP tubes is lower than that of the CFRP shafts with metal flanges at different ply angles. The reduction ratios of approximately 44% changed with the increase of the fiber ply angles. The metal flanges had no influence on the relationship between the natural frequency and ply angles. The deviation in natural frequency in FEA and PVET first increased and then decreased with the increase in the ply angles, which is attributed to the transition area in the filament winding method (Fig.5). This deviation was also attributed to the omission of the damping factor during simulating by FEA. The deviation reached the maximum at ply angle of

45°. This is mainly because the natural frequency of the CFRP drive shaft with small angle fibers was very high, and the decrease in natural frequency caused by the transition zone was relatively small. The fiber angle in the transition zone did not change much when the fibers were wound at a large angle, which caused the natural frequency to decrease slightly. This result indicates that the natural frequency of the CFRP shafts molded by filament winding technology is lower than that of the glued CFRP shafts. The magnitude of reduction changed with the increase in fiber ply angle, and reached the maximum at ply angle of 45°. 187.83 Hz 43.4% 170.07 Hz 43.8% 134.41 Hz 43.96%

450

Frequency(Hz)

400 350 300

4.19%

250

82.81 Hz 44.11%

6.02% x2 x1 x0 Polynomial fit of x2 Polynomial fit of x1 Polynomial fit of x0

200 150 100 50 10

105.61 Hz 44.03%

2.67% 20

10.54%

70.47 Hz 44.10%

13.71% 4.94%

30

40

50

60

70

θ (°) Fig. 15. Graph of changes in natural frequency with ply angles. x0: CFRP tubes by FEA, x1: CFRP drive shafts by FEA, and x2: CFRP drive shafts using filament winding technology by PVET.

When the structure is subjected to impact acceleration, the amplitude generated gradually decreased over time until it disappears; this can be attributed to the damping property of the material and structure [41].The relationships between the amplitude of the CFRP drive shafts and different ply angles and time are shown in Fig. 16. In the diagram, the amplitude gradually decreased with vibration time, which can be used to characterize vibration damping of the shaft[41]. From 0.50 s to 0.10 s, it can be seen that the larger the winding angle of the CFRP drive shafts, the shorter the vibration period of the shafts, and the greater the damping.

0.072 0.000 -0.072 -0.144 a2 b2

a2 d2

2

Accelerated speed (m/s )

0.144

b2 e2

c2 f2

c2 d2 e2

0.4

0.6

0.8

1.0

1.2

Time (s)

1.4

f2 0.5

0.6

0.7

Time(s)

0.8

0.9

1.0

Fig. 16. Vibration damping of CFRP drive shafts with different ply angles by PVET.

The natural frequencies and calculated damping ratio of drive shafts is displayed in Table.4 by the equations (9-12) above. With the increasing of ply angles, natural frequency calculated by theoretical formulas considering the damping factor changed little compared with the natural frequency of simulation. It indicates that damping has little effect on the natural frequency. Besides, measured natural frequency by PVET shows to be a little lower than the calculated natural frequency by theoretical formulas, which indicates that the transition area does affect the natural frequency of the drive shafts. Table.4. Damping properties of CFRP drive shafts with different ply angles by PVET, and natural frequency by PVET, FEA and calculation.

Specimens

Attenuation coefficient

Damping ratio

a2

1.01019

0.00323

Measured natural frequency (Hz) 235

b2

1.01074

0.00340

c2

1.02125

0.00669

d2

1.03569

e2

1.04112

f2

1.04936

Simulated natural frequency (Hz)

Calculated natural frequency (Hz)

244.86

244.85867

206

218.42

218.41874

155

171.35

171.34617

0.01116

118

134.23

134.22164

0.01282

100

104.94

104.93138

0.01533

87

89.32

89.30950

Results in Fig.17. shows that damping ratios increased with the increase in ply angles and the slope of the polynomial curve reached the maximum at ply angle of 45°. y = Intercept + B1*x^1 + B2*x^2 + B3*x^3

0.016

Equation

0.014

Residual Sum of Squares

Damping ratio

Weight

0.012 0.010

No Weighting 1.54021E-6 0.96997

Adj. R-Square

B

Intercept B1 B2 B3

Value Standard Error 0.00913 0.00591 -7.56209E-4 5.32208E-4 2.72528E-5 1.42392E-5 -2.18572E-7 1.16416E-7

0.008 0.006 Calculated damping ratios Polynomial fit of calculated damping ratio

0.004 0.002 10

20

30

40

50

60

70

θ (°)

Fig.17. Graph of changes in damping ratios with ply angles of CFRP drive shafts using filament winding technology by PVET.

5.3. Vibration characteristics of CFRP drive shafts with different ply thicknesses CFRP drive shafts were processed by filament winding method. Drive shafts with different thicknesses were monitored by changing the number of plies. Here, six categories of drive shafts (categories g-l in Table 2) with different ply thicknesses were prepared to study the influence of natural frequency and vibration characteristics. As shown in Table 2, the numbers of plies are 14 (category g), 12 (category h), 10 (category i), 8 (category j), 6 (category k), and 4 (category l), respectively. The vibration mode of the CFRP tubes with different thicknesses calculated by FEA method is shown in Fig. 18. The natural frequency decreased gradually with the decrease in ply thicknesses. In particular, the natural frequency decreased by 20.53% as ply thicknesses decreased from 4.20 mm to 1.20 mm. Vibration models showed it to be bending vibration, which also verifies the importance of the bending strength of the drive shaft. Specimen g0: 403.28 Hz

Specimen i0: 388.49 Hz

Specimen h0: 396.71 Hz

Specimen j0: 377.45 Hz

Specimen k0: 359.57 Hz

Specimen l0: 320.48 Hz

Fig. 18. The natural frequency and bending pattern of CFRP tubes with different tube thicknesses by FEA.

The vibration mode of the CFRP drive shafts in Fig. 19 shows that the natural frequency decreased with the decrease in ply thicknesses. As the ply thicknesses decreased from 4.20 mm to 1.20 mm, the natural frequency of the CFRP drive shaft was reduced by 33.79%. Compared with ply angles, ply thicknesses have a greater influence on the decrease in the natural frequency of CFRP drive shaft (33.79% compared with 20.53%). Specimen h1: 229.78 Hz

Specimen g1: 239.80 Hz

Specimen i1: 218.42 Hz

Specimen j1: 205.02 Hz

Specimen k1: 187.46 Hz

Specimen l1: 158.76 Hz

Fig. 19. The natural frequency and bending pattern of CFRP drive shafts with different tube thicknesses by FEA.

The natural frequency of the CFRP drive shafts with different ply thicknesses by PVET are shown in Fig. 20. With the decrease in ply thicknesses, the natural frequency of CFRP drive shafts decreased. 148 Hz

g2 h2

177 Hz

2

Accelerated speed (m/s )

0.025 0.020

i2 j2 k2

0.015

l2

191 Hz

0.010

221 Hz 213 Hz 206 Hz

0.005 0.000 120

150

180

210

240

Frequency (Hz) Fig. 20. The diagrams of the nature frequency of CFRP drive shafts with different ply thicknesses by PVET.

The relationship between natural frequency and ply thicknesses of the CFRP drive shafts is shown in Fig. 21. The fitted plot of the natural frequency under different ply thicknesses corresponds to the parabolic relation of the CFRP tubes and CFRP drive shafts. With the increase in ply thicknesses, the rate of increase of the natural frequency of all the shafts flattened out. Furthermore, the natural frequency of the CFRP tubes was lower than that of the CFRP shafts at different ply thicknesses. The reduction ratios increased with the increase in ply thicknesses from 40.53 % to 47.86 %, while the metal flanges had no influence on the relationship between natural frequency and ply thicknesses. The deviation between natural frequency in the FEA and PVET results was approximately 7%. This deviation was attributed to the omission of the damping factor during simulating by FEA. Furthermore, winding transition area in the CFRP drive shaft molded by filament winding technology also caused the deviation. When the ply thicknesses changed, the fiber angle of each layer in the transition zone

remained unchanged; the deviation in natural frequency between FEA and PVET results basically did not change. This also indicates that the natural frequency of the CFRP shaft molded by filament winding technology was lower than that of the CFRP shaft joined by glue at different ply thicknesses. The magnitude of reduction did not change with the increase in fiber ply thicknesses. 600

x2

x1

550

x0

500

polynomial Fit of x2 polynomial Fit of x1

Frequency(Hz)

450

161.72Hz 50.46%

polynomial Fit of x0

400 350

170.07Hz 43.7%

166.93Hz 42.08%

300 163.48Hz

172.11Hz 47.86%

172.43Hz 45.68%

40.53%

250 200 150

6.03%

7.88%

8.51%

1.2

1.8

2.4

7.27%

5.91%

7.34%

3.0

3.6

4.2

Thickness (mm) Fig.21. Graph of changes in natural frequency with ply thicknesses. x0: CFRP tubes by FEA, x1: CFRP drive shafts by FEA, and x2: CFRP drive shafts using filament winding technology by PVET.

2

Accelerated speed(m/s )

The relationships between the amplitude of the CFRP drive shafts and different ply thicknesses and time are shown in Fig. 22. In the diagram, the amplitude gradually decreased with vibration time, which can be used to characterize vibration damping of the shaft. From 0.50 s to 0.10 s, it can be seen that the larger the winding thickness of the CFRP drive shaft, the shorter the vibration period of the shaft, and the higher the damping.

0.34 0.17 0.00

g2 j2

h2 k2

i2 l2

-0.17 -0.34

g2

h2 i2 j2 k2 l2 0.4

0.6

0.8

1.0

Time (s)

1.2

1.4

0.5

0.6

0.7

0.8

0.9

Time(s)

Fig. 22. Vibration damping mode of CFRP drive shafts with different tubes thicknesses.

1.0

The natural frequencies and calculated damping ratio of drive shafts is displayed in Table.5 by the equations (9-12) above. Natural frequency calculated by theoretical formulas has little change compared with the natural frequency of simulation, which confirmed that damping has little effect on the natural frequency furtherly. Similarly, measured natural frequency by PVET shows to be a little lower than the calculated natural frequency by theoretical formulas, which also indicates that the transition area does affect the natural frequency of the drive shafts. Besides, damping ratios increased slightly with the increase in ply thicknesses as shown in Fig.23. Table.5. Damping properties of CFRP drive shafts with different ply angles by PVET. Specimens g2 h2 i2 j2 k2 l2

Attenuation coefficient 1.01284 1.01247 1.01101 1.01074 1.00985 1.00956

Damping ratio 0.00406 0.00394 0.00349 0.00340 0.00312 0.00303

Measured natural frequency (Hz) 221 213 206 206 177 148

Simulated natural frequency (Hz) 239.80 229.78 218.42 205.02 187.46 158.76

Calculated natural frequency (Hz) 239.79802 229.77822 218.41867 205.01882 187.45909 158.75927

0.0042

0.0040

Equation Weight Residual Sum of Squares Pearson's r Adj. R-Square

0.97977 0.94994 Intercept Slope

?$OP:A=1

Damping ratio

y = a + b*x No Weighting 3.53333E-8

Value Standard Error 0.00252 1.08137E-4 3.66667E-4 3.74448E-5

0.0038

0.0036

0.0034

0.0032 Calculated damping ratios Linear fit of calculated damping ratios

0.0030 1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Ply thickness(mm) Fig.23. Graph of changes in damping ratios with ply angles of CFRP drive shafts using filament winding technology by PVET.

6. Conclusions The vibration performance of CFRP drive shafts prepared by filament winding with integrated metal flanges under various ply ways is developed and validated in this paper. The larger the ply angle or thickness, the greater the damping of the CFRP drive shaft, and the damping had little effect on the natural frequency. In addition, ply angles played much more importance on the natural frequency than that of ply thicknesses. The natural frequency of the CFRP shafts molded by filament winding technology is slightly lower than that of the glued CFRP shafts. The metal flanges as embedded parts had a great influence on the natural frequency of CFRP drive shafts, but did not affect the relationship between the natural frequency and various fiber ply angles or thicknesses. The natural frequency of CFRP tubes and drive shafts by FEA agreed well with the experimental results, which will be of great significance in the design of CFRP drive shafts. Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study. Acknowledgements This work was supported by Fundamental Research Funds for the Central Universities (No. 2232018A302), Science and Technology Committee of Shanghai Municipality (No. 18DZ1101003, No.16DZ112140).

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Declaration of interests

Manuscript title: Vibration Characteristics of Carbon-fiber Reinforced Composite Drive Shafts Fabricated using Filament Winding Technology 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. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Author statement Manuscript title:Vibration Characteristics of Carbon-fiber Reinforced Composite Drive Shafts Fabricated using Filament Winding Technology Zeyu Sun: Conceptualization, Methodology, Software. Jie Xiao: Writing- Original draft preparation. Xuduo Yu: Data curation. Rogers Tusiime: Writing- Reviewing. Hongping Gao: Software, Validation. Wei Min: Visualization. Lei Tao: Software. Liangliang Qi: Investigation. Hui Zhang: Writing- Reviewing and Editing. Muhuo Yu: Supervision.