According to the basic theoretical framework of composite mechanics, when an anisotropic material system undergoes deformation within the range of small deformation, the constitutive relationship between the internal stress field and strain field can be accurately described by the generalized Hooke’s law:

In the formula: from left to right are the stress component matrix [ σ ] , stiffness matrix [ C ] , and strain component matrix of the composite material [ ε ] , respectively. σ i ( i = 1 , 2 , 3 ) is the stress component in the i-direction; τ i j ( i , j = 1 , 2 , 3 ) is the shear stress in the i-j plane; ε i ( i = 1 , 2 , 3 ) is the normal strain in the i-direction; γ i j ( i , j = 1 , 2 , 3 ) is the shear strain in the i-j plane; C i j ; is the stiffness coefficient, From the analysis of material mechanical properties, it can be seen that in a three-dimensional fully elastic body, C i j = C j i ( i , j = 1 , 2 , 3 , 4 , 5 , 6 ). The stiffness matrix has the key characteristic of symmetry. For a fiber composite material system, its material coordinate system is generally defined as follows: the 1-axis corresponds to the principal fiber direction; the 2-axis denotes the in-plane transverse direction perpendicular to the fibers; and the 3-axis represents the thickness direction normal to the fiber orientation. With the establishment of this coordinate system, the original expression can be simplified:

The compliance matrix is used to represent the relationship between strain and stress, namely:

Thus the above equation can be simplified as:

where: S is the compliance matrix of the structure, and S i j denotes the stiffness coefficient.

Carbon Fiber Reinforced Polymer (CFRP), as a typical fiber-reinforced resin matrix composite, exhibits significant orthotropic properties. This material system has three mutually perpendicular planes of elastic symmetry, corresponding to the 2-3 plane, 3-1 plane, and 1-2 plane, respectively. According to this symmetry, the stress-strain relationship in its constitutive equation can be effectively simplified:

Therefore, having C 22 = C 33 , C 12 = C 13 , C 55 = C 66 , meanwhile, it is further derived based on the equality of strain potential energy density:

In this matrix, there are only C 11 , C 22 , C 12 , C 23 , C 44 , C 55 independent stiffness coefficients. In practical applications, the relationship between stress and strain is usually expressed in terms of engineering elastic constants for convenience in testing.

where: E 1 , E 2 , E 3 are the elastic moduli in different directions; E 1 , E 2 , E 3 are the shear moduli in different directions; ν 12 , ν 21 , ν 23 , ν 32 , ν 13 , ν 31 are the Poisson’s ratios in different directions. The relationships among them are as follows:

Composite components generally adopt a laminated structure composed of unidirectional materials stacked at various ply angles. According to the basic assumptions of Classical Laminated Plate Theory, the dimension in the thickness direction (i.e., direction 3) is much smaller than that in the in-plane directions (directions 1 and 2). Therefore, it can be approximated as σ 3 = 0 , τ 23 = τ 31 = 0 . At this time, the internal stress-strain relationship can be simplified as:

Based on the in-plane elastic properties of a single lamina in the principal material coordinate system, each component of its stiffness matrix can be characterized by engineering elastic constants. This matrix is adopted as a key parameter in the above theoretical calculation model:

However, the above stress-strain relationship is only applicable to the special case where the material coordinate system (1-2) of a single lamina coincides exactly with the global coordinate system (x-y) of the laminate. As illustrated in Figure 1, in practical engineering applications, there exists a specific ply angle θ between the principal fiber direction of each lamina and the principal axis of the laminate. Therefore, it is necessary to transform the off-axis stresses and strains in the global coordinate system obtained from the constitutive relation to the principal material directions of each individual lamina for analysis.

Figure 1. Angular difference between the two coordinate systems.

The relationship between the stiffness matrices in the two directions Q ¯ (off-axis direction) and Q (principal material direction) is given by:

where: T is the coordinate transformation matrix, whose specific elements can be expressed as:

Based on the deformation assumptions of laminate theory, the stress-strain relation of a single lamina in the global coordinate system can be expressed using the strain vector on the midplane and the curvature change vector of the midplane:

where: ε x 0 , ε y 0 , γ x y 0 represent the normal strains and shear strain on the midplane; k x , k y denote the bending curvatures of the midplane; k x y is the twisting curvature of the midplane; z k is the distance from the k-th lamina to the neutral surface. According to Equation (11), each stiffness coefficient of the off-axis stiffness matrix is given by:

Thus, it can be concluded that the global stiffness characteristics of composite laminated structures are mainly affected by the ply angles and stacking sequence of each individual lamina.

As can be concluded from the foregoing analysis, the key to obtaining each order damping ratio of CFRP structural components lies in accurately extracting the stress and strain distribution data of the structure under each mode shape, followed by solving relevant parameters through numerical computation. On this basis, this study adopts the Ansys finite element analysis platform and applies the Lanczos mode ex-traction algorithm to calculate multiple natural frequencies and corresponding mode shapes of the Al/CFRP transmission shaft. Afterwards, the stress and strain field data of each finite element are exported via the APDL post-processing module. Finally, a dedicated calculation program is developed in the Matlab programming environment to complete the numerical solution of damping parameters. The detailed calculation procedure is illustrated in Figure 2.

Partial APDL command streams are listed as follows:

/POST1

*CREATE,READCODE,MAC

*DO,J,1,20

ALLSEL,ALL

SET,1,J

ETABLE,EJ,SENE

SSUM

*GET,EJ_ALL,SSUM,,ITEM,EJ

*CFOPEN,EJ_ALL%J%,TXT

*VWRITE,EJ_ALL

(F20.6)

*CFCLOS

*ENDDO

*DO,J,1,20

ALLSEL,ALL

SET,1,J

*DO,I,1,4 ESEL,S,MAT,,I

ETABLE,EJI%I%,SENE

SSUM

*GET,EJI_ALL%I%,SSUM,,ITEM,EJI%I%

*ENDDO

*CFOPEN,EJI_ALL%J%,TXT

*DO,I,1,4

*VWRITE,EJI_ALL%I%

(F20.6)

*ENDDO

*CFCLOS

*ENDDO

*END

READCODE

Figure 2. Flow chart of the damping ratio calculation program.

As shown in Figure 3, a shaft for engineering machinery has load positions at both ends, while the middle section bears torque. The modeling of the carbon fiber/aluminum alloy shaft is illustrated in Figure 4. To meet the vibration damping requirements and assembly requirements of the shaft, the dimensions of the shaft must remain unchanged.

Figure 3. Flow chart of the damping ratio calculation program.

Figure 4. Modeling diagram of Al/CFRP shaft.

The carbon fiber layers are made of T300/YPH-308 unidirectional prepreg tape. The volume fractions of carbon fiber and epoxy resin are 68% and 32%, respectively. Its basic mechanical properties are: E_x = 135 GPa, E_y = 10 GPa, G_xy = 5.5 GPa, ν_xy = 0.28, ρ = 1600 kg/m³. The specific material properties of the aluminum alloy shaft core are: E = 71 GPa, ν = 0.33, ρ = 2770 kg/m³.

The adhesive layer is made of epoxy adhesive DP-460, whose basic mechanical parameters are as follows: E = 1.65 GPa, v = 0.28, ρ = 1140 kg/m³, ƞ = 0.05%.

The ACP module in ANSYS Workbench can be used for composite material modeling and analysis. Figure 5 shows the interface for setting CFRP ply parameters. The specific operation procedure is as follows: First, select the composite material with defined material properties in the Fabric module and specify the lamina thickness of each ply. Then, set the fiber orientation angle of each ply through the Stackup module. Define the ply coordinate system direction by combining Rosettes and Oriented selection sets. Uniformly define the overall ply thickness in Modeling Groups. Convert ply data into solid geometry via the Solid Models module. Finally, couple and transfer data between ACP Pre and Mechanical Model according to the requirements of the analysis type. As shown in Figure 6, the mesh type adopted is Hex Dominant. The mesh size is 1 mm for carbon fiber layers and adhesive layers, and 3 mm for the metal shaft. The total number of elements is 203,938.

Figure 5. CFRP ply design interface.

Figure 6. Finite element mesh generation.

According to the actual operating state of the transmission shaft, the boundary conditions shown in Figure 7 are applied to it with simply supported constraints. Remote displacement constraints are configured as follows: all translational displacements along the X, Y and Z directions and all rotational displacements about the X and Z axes at end A are set to zero, while the translational displacement along the Y axis is left free. At end B, all translational displacements along the Y and Z directions and all rotational displacements about the X and Z axes are set to zero, and the translational displacement along the X direction (rotation about the Y axis) is set free.

Figure 7. Schematic diagram of constraint settings.

To eliminate the interference of mesh discretization errors on the calculated modal frequencies and damping loss factors, a mesh independence verification is carried out, and the corresponding schemes and results are listed in Table 1.

Table 1. Mesh sensitivity simulation results.

The mesh sensitivity simulation results indicate that after refining the mesh size of carbon fiber layers and adhesive layers to 1 mm and the mesh size of the metal shaft to 3 mm, the first-order natural frequency and damping loss factor tend to stabilize with a relative error less than 1%. Considering both convergence accuracy and computational cost comprehensively, this mesh size is adopted as the uniform standard mesh in this paper.

Response Surface Methodology, also known as regression design, explores the influence laws of various experimental factors and their interactions on the target response by designing multi-factor and multi-level experimental combinations. On the basis of experimental data, an accurate mathematical regression model is established, which can effectively predict and optimize the investigated indicators [16].

In this paper, the Box-Behnken Design (BBD) principle of response surface methodology is adopted for factor design of ply angle, stacking sequence, adhesive layer thickness and laying ratio, to reveal the influence laws of different ply structures and bonding interface parameters on the damping loss factor and first-order natural frequency of the shaft, and to fit its prediction model.

It can be concluded from literature results that the shaft exhibits optimal damping performance when the ply angle is approximately 30˚, the layup sequence is symmetric, the adhesive layer thickness is 0.2 mm, and the laying proportion is 80%. The number of carbon fiber plies and the thickness of each single ply are set to be identical across different schemes of ply parameters. The levels and coding of each parameter are listed in Tables 2-5.

Table 2. Ply angle and coding.

Table 3. Stacking sequence and coding.

Table 4. Adhesive layer thickness and coding.

Table 5. Adhesive lay-up ratio and coding.

Since the layup sequence is a categorical variable, it cannot be input as a continuous value. Therefore, a discrete design variable z₂ is introduced, where z₂ = 1, 2, 3 correspond to three lamination stacking configurations [θ]8, [θ/-θ]4, [0˚/θ/-θ/90˚]S, respectively.

Figure 8. Box-Behnken experimental design plan.

According to the level ranges of ply parameters and adhesive layer parameters in Tables 2-5, a four-factor, three-level experimental design was carried out, yielding 29 test points as shown in Figure 8.

According to the experimental scheme shown in Figure 8, simulation analysis was carried out using ANSYS Workbench. The simulation results of 29 groups of damping loss factors obtained are listed in Table 6 and input into the design scheme of Figure 8.

Table 6. Adhesive lay-up ratio and coding.

The experimental data of damping loss factor in Table 6 were subjected to multiple regression analysis using Design-Expert software, and the prediction model of the first-order damping loss factor is obtained as follows:

where: z 1 , z 2 , z 3 and z 4 represent the ply angle, stacking sequence, adhesive layer thickness and adhesive lay-up ratio, respectively.

Analysis of Variance (ANOVA) is a statistical technique that examines the statistical significance of the regression model and evaluates the model reliability by comparing the ratio of the regression mean square to the residual mean square (F-statistic). By decomposing the total variation of data into components derived from different factors, this method can effectively verify the fitting effect of the model. The analysis results can identify the significant main factors, secondary factors and interaction effects between various factors on the dependent variable [17]. Table 7 shows the ANOVA of the prediction model for damping loss factor.

Table 7. ANOVA of the prediction model for damping loss factor.

As can be seen from Table 7, the P-value of the model is much less than 0.05, indicating that the model is extremely significant and has strong prediction ability. Among them, the significance levels of ply angle, stacking sequence and adhesive lay-up ratio are P < 0.01, so these three factors all have significant effects on damping performance. In the parameter interactions, the interaction between stacking sequence and adhesive lay-up ratio has a significance level of P = 0.003 < 0.05, indicating that this interaction term has a significant influence on damping performance. All quadratic terms in the model have significance levels of P < 0.05, suggesting that there is a clear optimal region for this process. In contrast, the significance level of adhesive layer thickness is P > 0.05, so its influence on damping performance is not significant. However, since its quadratic term is significant, it cannot be completely neglected.

Considering the interactive effects of different parameter combinations on the damping of the shaft, response surface plots were fitted for the above damping loss factor prediction model using Design-Expert software, as shown in Figures 9-14.

Figure 9. Box-Behnken experimental design plan.

Figure 10. Interactive effect of ply angle and adhesive layer thickness.

Figure 11. Interactive effect of ply angle and adhesive lay-up ratio.

Figure 12. Interactive effect of stacking sequence and adhesive lay-up ratio.

Figure 13. Interactive effect of stacking sequence and adhesive lay-up ratio.

Figure 14. Interactive effect of adhesive layer thickness and adhesive lay-up ratio.

According to the above figures, it can be concluded that the damping loss factor first increases and then decreases with the increase of ply angle, shows a similar trend with the increase of adhesive layer thickness, and first increases and then decreases with the decrease of adhesive lay-up ratio with a relatively gentle initial trend. The damping loss factor is optimal when the ply angle is 21˚ - 33˚, the stacking sequence is symmetric, the adhesive layer thickness is 0.19 mm - 0.23 mm, and the adhesive lay-up ratio is 80% - 100%, corresponding to better damping performance of the shaft.

Based on the experimental data of the first-order natural frequency in Table 6, multiple regression analysis was carried out using Design-Expert software, and the prediction model of the first-order natural frequency is obtained as follows:

where: z 1 , z 2 , z 3 and z 4 represent the ply angle, stacking sequence, adhesive layer thickness and adhesive lay-up ratio, respectively.

Table 8 shows the analysis of variance for the prediction model of the first-order natural frequency of the shaft.

As can be seen from Table 8, the P-value of the model is much less than 0.05, indicating that the model is extremely significant and has strong prediction ability. Among them, the significance levels of ply angle and stacking sequence are P < 0.01, so they have significant effects on the first-order natural frequency. In the parameter interactions, the significance level of the interaction between ply angle and stacking sequence is P = 0.0002 < 0.05, indicating that this interaction term has a significant influence on the first-order natural frequency.

Table 8. ANOVA of the prediction model for damping loss factor.

The response surfaces were fitted based on the above first-order natural frequency prediction model, as shown in Figures 15-17.

Figure 15. Interactive effect of ply angle and stacking sequence.

Figure 16. Interactive effect of ply angle and adhesive layer thickness.

Figure 17. Interactive effect of ply angle and adhesive lay-up ratio.

As can be seen from the above figures, the first-order modal frequency decreases linearly with the increase of ply angle, while adhesive layer thickness and adhesive lay-up ratio have little effect on the first-order modal frequency.

According to the optimization results presented above, aluminum alloy/carbon fiber (Al/CFRP) shaft specimens were fabricated. The main manufacturing process of the Al/CFRP shaft specimen includes preparation and surface treatment of the aluminum alloy core, preparation and coating of the adhesive layer, cutting and laying of carbon fiber prepreg, high-temperature curing, and post-processing. The finished specimen is shown in Figure 19.

(a) Aluminum alloy shaft.

(b) Al/CFRP shaft.

Figure 19. Physical photograph of the specimen.

According to the actual construction of the experimental platform for modal analysis of the specimen in this paper, the setup is shown in Figure 20. The impact hammer method is adopted for pulse excitation. The fixing method of the specimen is shown in Figure 21. In the measurement system, the model of the force sensor is 2302-10, which is mounted on the impact hammer. The model of the triaxial accelerometer is 356A25. Seven measuring points are arranged on the shaft surface, as shown in Figure 22. The seven measuring points are divided into four groups for testing: the first three groups measure two points each, and the last group measures the middle single point.

Figure 20. Modal test platform.

Figure 21. Fixing method of the shaft.

Figure 22. Distribution of measuring points.

Finite element analysis and modal test analysis were carried out respectively for the Al/CFRP shaft and the aluminum alloy shaft. The first four natural frequencies and damping loss factors obtained are shown in Table 10 and Table 11.

Table 10. Simulation and experimental data of the first 4 natural frequencies.

Table 11. Simulation and experimental data of the first 4 damping loss factors.

It can be seen from the tables that some differences between simulation and experimental data are inevitable, which are caused by boundary conditions, material impurities, curing temperature of carbon fiber, and other factors. For the aluminum alloy shaft, the deviations of the first four natural frequencies between finite element simulation and experimental data are 3.45%, 4.12%, 5.67%, 6.78% in sequence, and the deviation of the first four damping loss factors is 10%. For the Al/CFRP shaft, the deviations of the first four natural frequencies between finite element simulation and experimental data are 2.8%, 2.5%, 3.6%, 4.8% in sequence, and the deviations of the first four damping loss factors are 4.17%, 7.32%, 2.94%, 17.39% in sequence. All deviations are within a reasonable range, indicating that the simulation model has high accuracy.

By comparing the experimental data of the aluminum alloy shaft and the Al/CFRP shaft, the first four natural frequencies of the Al/CFRP shaft are increased by 7.69%, 7.1%, 3.38%, and 2.6% in sequence compared with the aluminum alloy shaft, and the first four damping loss factors are increased by 283.3%, 111.1%, 83.3%, and 5.56% in sequence.