Study on the effect of curved fiber laying angle on the vibration free attenuation of composite laminates

Firstly, three groups of fiber curve laminates with different numbers of layers and different fiber curve angles were designed and prepared. A cantilever plate free vibration test was used to analyze the influence of the fiber curve angle on the vibration free attenuation characteristics of the laminates. The three groups of contrast test results show that the fiber curve angle has a significant effect on the loss factor of the laminate structure. When the angle change of the fiber curve was ±<45|60>, the loss factor of the laminates reached the maximum value, and the vibration damping performance of the laminates was the best. Secondly, modal tests were carried out on the laminates, the modeling and simulation of the variable stiffness laminate were completed, and the influence of the fiber curve angle on the natural frequency of the laminate was explored. The results showed that when the number of laminate layers was the same, the natural frequencies of the laminates first increased and then decreased with an increase in the fiber curve angle. The first three natural frequencies of the laminates reached the maximum value when the fiber curve angle was ±<45|60 > or ±<60|75>. Finally, it is concluded that when the fiber curve angle of the laminates varies between ±<45|60>, the damping ratio and the first three natural frequencies of the laminates reach the maximum value, and the vibration damping performance is the best.


Introduction
Composite laminates are widely used in the structural design of aircraft such as fuselages and wings. In the process of high-speed flight, the wings, walls and other structures of the aircraft inevitably produce vibrations owing to the disturbance of the air flow. Once the wing aerodynamic structure of an aircraft has resonance or instability, it causes damage to the aircraft, which results in immeasurable losses. Therefore, it is of great significance to explore the vibration characteristics of composite laminates for aircraft structure vibration reduction design.
Gurdal and Olmedo [1] first proposed that the variation in the fiber curve angle changes the stiffness of composite laminated plates. Variable stiffness laminates are also called variable angle laminates. Ganapathi et al [2] proved that different fiber curvatures and numbers of laminates have a significant influence on the vibration equation of laminates. Cai Bing et al [3] established a damping model for composite material laminates and studied the effects of the fiber volume fraction, thickness of each layer, number of layers, orientation of lay-up, and sequence of layers on the damping ratio and dynamic response of composite material laminates. Murakami et al [4] proposed the sawtooth theory of continuous in-plane interlaminar displacements, and used it to predict the in-plane displacements and in-plane stresses of symmetrically laminated plates. Abdalla et al [5] numerically analyzed the influence of the fiber angle curve variation on the natural frequency of laminates; however, there was no verified verification. Based on third-order shear deformation theory, Akhavan and Ribeiro [6] used the P-type finite element method to analyze the influence of using curved fibers instead of linear fibers on the vibration modes and natural frequencies of laminates. The results show that the use of curved fibers with variable stiffness laminates can significantly affect the natural frequency of the laminates by effectively changing their vibration modes, but this effect is not obvious for laminates with large thicknesses. Karimi et al [7] applied a modified shear deformation plate theory to investigate the free vibration of vertically laminated composite plates coupled to a sloshing liquid. Niu Xuejuan et al [8] explored the influence of the layering sequence on the natural frequency of composite laminates, and found that the layering of composite laminates has a significant influence on its natural frequency. Through test exploration, Liu Yuhua et al [9] found that the fundamental frequency of variable stiffness laminates first increased and then decreased with the beginning angle of the fiber curve, and increased with the ending angle of the fiber curve. Zhang Shanzhi et al [10] studied the natural frequency attenuation law of composite laminates under fatigue loads and obtained the natural frequency of laminates in each life stage. Zhang Ying et al [11] studied the modal node-line positions in different coupling regions of cantilever laminates and proposed a new modal confidence factor to characterize the vibration performance of bending coupling laminates. Xue Jianghong et al [12] adopted an accurate model of the interface contact effect to explore the influence of the delamination contact on the natural frequency of composite laminates containing layered composite materials. Chen Xinxin et al [13] simulated and analyzed the vibration frequency of composite laminates by considering different boundary conditions, thickness-to-span ratios and other factors. Liang Haofeng et al [14] conducted theoretical research on the longitudinal and transverse free vibration of composite laminates with stiffeners, and provided an optimization method for the stiffeners of composite laminates. Through numerical simulation, Yu Ruyu et al [15] found that the deformation coefficient of the laminate natural frequency increased with a decrease in the number of laminates, and the lower the number of layers, the greater the influence on the dispersion of the natural frequency of the laminate. Hu Han et al [16] conducted flutter research on composite laminates laid with fibers at various angles. The results showed that the flutter of composite laminates can be suppressed by a reasonable variable angle design. Liu Yanqing et al [17] defined the stiffness matrix of viscoelastic composites with frequency dependence based on the elasticviscoelastic correspondence principle and proposed a general method to solve the modal damping of composite structures using finite element software. Li Liang et al [18] analyzed the damping performance of eight layers of variable stiffness laminates using a free attenuation test and random test, and found that the damping ratio of eight layers of variable stiffness laminates was the largest when the fiber change angle was ±<45|60>, and the damping performance of variable stiffness laminates was significantly better than that of traditional straight fiber plied laminates.
At present, there are few comprehensive researches on the vibration characteristics of variable stiffness laminates in the research on variable stiffness laminates, and the researches are not in-depth enough [19], no specific guiding conclusions have been obtained for practical engineering applications. In this study, the laminate structure of aircraft wing siding is considered, and fiber curve laminates with 8 layers laminates, 12 layers laminates and 16 layers laminates, which are commonly used in engineering, are prepared by using automated fiber placement of composite materials. Using composite laminates with different fiber curve angles as the research object, the vibration reduction characteristics of variable stiffness laminates were verified through free vibration tests and finite element simulation, and the optimal range of variable angle was explored. The results of this study provide a reference for optimizing the vibration performance of fiber curve laminates.

Formula derivation
The free decay method is one of the most commonly used methods for modal frequency domain data acquisition [10]. Through a free vibration test, the logarithmic attenuation rate δ, energy loss factor β and damping ratio ξ of the structural system were obtained, and the stability of the system structure was evaluate [17]. Figure 1 shows the free decay curve of the structure. The specific waveform of the free decay curve was obtained from the vibration acceleration, and the modulus G' and loss factor β of the laminate were analyzed.
From the free decay curve [20], the relationship between the loss factor β and the viscoelastic modulus of Young's modulus E' can be derived [21].
The damping ratio in the above equation is: where C is the damping coefficient, C c is the critical damping coefficient, δ is the logarithmic attenuation rate.
Substituting equation (1) into equation (2), the mathematical equation of the free attenuation wave type of a viscoelastic material can be obtained [22]: where ω n represents the natural frequency of the undamped system, ω D the natural frequency of the damped system, K the real part of the complex tensile stiffness, M the structural mass of the vibration system. According to figure 1, the period from peak point x 1 to peak point x 3 is 2T, so: where E' refers to the real part of the complex tensile modulus, S e is the force bearing area, S I is the surface area of the free end, h is the thickness of the laminates, q is the structure shape coefficient. The loss factor β is obtained as follows: The loss factor obtained using equation (7), is the loss factor for the two attenuation periods. When the attenuation period is n, the loss factor of the viscoelastic material structure can be obtained as follows: In this study, the free decay method was used to conduct tests on composite laminates, the dynamic performance of the composite laminates was measured, and the G′(T) curve of the structural vibration shear modulus and β(T) curve of the structural vibration loss factor with time were obtained [24]. In the free vibration test, a data acquisition instrument was used to record the time-domain data. The specimen was towed to a specific angle by hand, the hand was released to allow the specimen to shake freely until it stopped, and the curve of the acceleration and force of the laminate with time were recorded.

Tested details
Tested materials and preparation The raw material of the laminates was T300 carbon fiber produced by Jiangsu Hengshen Co., Ltd. and carbon fiber prepreg with EM118 resin. The specifications of the prepreg were EM118%-35%-A12-U-100gsm-1000, the volume fraction of resin content was approximately 35%, the density was about 1.51 g cm −3 , and the thickness of the monolayer was approximately 0.15 mm. The mechanical parameters of the monolayers are presented in table 1.
The fibers in the same layers of variable stiffness laminate change continuously within a certain angle range, showing a double 'S' shape. As shown in figure 2, below. Taking the curve layer angle of ±<0|15 > as an example, +<0|15>is the first layer, indicating that the included angle between the fiber tangent direction and the x-axis at any point changes continuously within 0°∼15°, −<0|15 > is the second layer, which means that the included angle between the fiber tangent direction and the x-axis at any point changes continuously within −15°∼0°, two layers of a cycle. Using trajectory planning software, the CAA was developed based on CATIA 3D software, and the fiber curve trajectory was generated by the translation method. Each track point has a corresponding coordinate value, and the angle between the tangent line of each point and the x-axis is within 0°∼15°. Post-processing software is used to output the track point into the machine NC code that can be recognized by an automatic fiber laying machine.
The process size of the fiber curve laminates was 350 mm×350 mm, and the machining size was 300 mm×300 mm. The prepreg was solidified using a vacuum bag pressing process. The pressure applied to the vacuum bag is 0.1Mpa. The prepreg was first heated to 80°C for insulation for 1 h, and then to 130°C for insulation for 2 h. The curing process curves are shown in figure 3.
In this test, 18 fiber curve laminates with six pieces of eight layers, six pieces of 12 layers, and six pieces of 16 layers were prepared by automatic fiber placement.  figure 4 shows the fiber curve laminates before and after curing.

Free vibration test
An LMS SCM05 data acquisition instrument was used in the test, and the sampling frequency was 1024Hz. The test platform is illustrated in figure 5. The data were calculated and analyzed using LMS TestLab software. The sensitivity of the force hammer was 0.23 mv N −1 , the bandwidth was 5000Hz, and the measuring range was 22KN. A lightweight PCB acceleration sensor with a sensitivity of 98.7Mv/g(10.06Mv/m/s 2 ) were selected. The equipment and machinery types used in the free vibration test are listed in table 2.
One end of the cantilever beam laminate is fixed with a threaded hole as shown in figure 5. The clamping device is made of a thick steel plate, which is bolted to the base of a large gantry type fiber laying machine. Free vibration tests were performed on fiber curve laminates, and the free delay curves were calculated using LMS data acquisition instrument. The free attenuation rate and damping ratio of laminates with different fiber curve angles were calculated for different thicknesses, and the influence of the fiber curve angle on the vibration attenuation of the laminates was compared and analyzed.
The specific operation and implementation steps of the free vibration test of the laminates were as follows:  (1)Before the free vibration test, the laminates were divided into nine excitation points, and central point 5 was selected as the pickup point. Figure 5 shows the tested excitation and pickup points.
(2)The laminate clamp wass fixed on the base of the vibration device, and the fiber curve laminate was then fixed on the fixture with screws. The laminate was in the cantilever plate state, as shown in figure 5.
(3)The PCB acceleration sensor for the free vibration test was fixed at point 5 of the laminate. To reduce the error in the tested data, a PCB acceleration sensor was pasted on the smooth side of the composite laminate.
(4)The vibration of the laminated plate was measured using point-by-point excitation and single point measurement. A force hammer was used to successively stimulate the nine measured points of the laminates, and each measured point was knocked five times to obtain the response signal for each measured point. Point 5 was the signal measurement collection point, and the response signal was transmitted to the data acquisition instrument.
(5)The data acquisition instrument was connected to a computer and the tested data were transmitted. Through calculation and analysis of the free vibration test data collected by the data acquisition instrument in the computer, the vibration free decay curve of each laminate was obtained, and the vibration free attenuation rate and damping ratio of each laminate were obtained.

Modal test
The tested platform shown in figure 5, was used to conduct modal tests on the fiber curve laminate. The test procedure was similar to that of the free vibration test. The frequency response transfer function curve graph was collected using an LMS SCM05 data acquisition instrument, and then the impact testing module of LMS TestLab was used to calculate the first three order frequencies of the laminate.The test procedure was repeated, and the modal parameter information of each laminate was obtained. The influence of the fiber curve angle on the laminate frequency was compared and analyzed.

Finite element analysis
Owing to the continuous change in the fiber curve angle, it is necessary to complete the modeling of variable stiffness laminates by giving each element a specific angle. A continuous change in the element angle is used instead of the fiber curve angle to complete the modeling of the variable stiffness laminates. The establishment process is illustrated in figure 6: The key to script development is discretizing the mesh elements of the finite element model. First, a component model was created in Abaqus. The dimensions of the laminate were 300 mm × 300 mm. The model is then divided into meshes, and the attributes of the meshes are selected as quadrilaterals. Considering the calculation time cost and accuracy, the global size of the meshes was 10 mm, and the number of meshes was 900. Finally, the job is submitted and the inp file of the finite element model of the laminate is output. After the inp file was output, the script program of the unit angle assignment was written in MATLAB. The starting angle T 0 , ending angle T 1 , Length A and Width B of the finite element model, and other parameters were successively inputted in MATLAB. Then the node and unit information of the inp file are read and input into the script program successively for calculation. Number of each unit, and assign the corresponding curve angle for each mesh. Finally, the number of meshes and the corresponding curve angle were output. The mesh discretization and curve angle assignment of the laminate model were performed using MATLAB. The discrete field of the mesh cell was established in Abaqus, and the mesh information was imported into the cell ID and component of the field data. By endowing the fiber curve angle information to each element mesh, the information interaction between them was realized, and the finite element modeling of the composite fiber curve laminates was completed. Figure 7 shows the fiber curve angle for each mesh.

Results and discussion
This section present the result analysis of the influence law of the curve fiber angle change on the loss factor. Three groups of 8, 12 and 16 layers were selected as the control analysis group to quantitatively study the effect of the fiber curve angle on the vibration reduction performance of laminates based on different numbers of layers. Figures 8, 10, and 12 show the free decay curves of the laminates with six different fiber curve angles changing when the number of layers was 8, 12, and 16, respectively. According to the free decay curve, the free attenuation rate δ, loss factor β and damping ratio ξ of the laminates with different curve angles were calculated using equations (2) and (10) above. In the decay curve, two peak points separated by 40 cycles are used to calculate the loss factor, that is, n in equation (10) was set to 20. The calculation results of each parameter of vibration attenuation for the three groups of laminates with different numbers of layers are shown in tables 3, 4 and 5, respectively.
The greater the loss factor β of the laminates, the greater the vibration energy that can be absorbed. The faster the vibration energy dissipated, the better the vibration reduction performance. The loss factor values of the fiber curve laminates with 8 layers were analyzed. It can be seen from table 3 that when the fiber curve angle changes between ±<45|60>, the loss factor reaches a maximum value of 3.65%, and when the fiber curve angle changes between ±<0|15>, the loss factor gets a minimum value of 2.38%. When the fiber curve angle was changed between ±<45|60>, the loss factor increased by 53.4% compared to that when the fiber curve angle was changed between ±<0|15>.
As can be seen from figure 9, the loss factor of the fiber curve laminate presented a certain regularity with a continuous increase in the fiber curve angle. The loss factor of the laminate presented a trend of first increases and then decreases with an increase in the fiber curve angle. The loss factor was maximum when the fiber curve angle changed between ±<45|60>, and minimum when the fiber curve angle was between ±<0|15>. Compared with traditional linear laminates, variable stiffness laminates have longer in-plane fiber lengths and greater fiber friction, which play a greater role in the vibration energy dissipation of the material. The length of the fiber in the laminate was the longest at ±<45|60>, and its damping performance was the best. The above test results show that different fiber curve angles cause a difference in the loss factor values of laminates, and the vibration reduction ability of laminates can be improved by optimizing the fiber curve angles. The loss factor values of the fiber curve laminates with 12 layers were analyzed. It can be seen from table 4 that when the fiber curve angle changes between ±<45|60>, the loss factor reaches the maximum value 4.48%; When the fiber curve angle changes between ±<60|75>, the loss factor gets the minimum value, which is 1.74%. When the fiber curve angle was changed between ±<45|60>, the loss factor increased by 148% compared to that when the fiber curve angle was changed between ±<60|75>.
As shown in figure 11, the loss factor of the fiber curve laminates changed regularly with the fiber curve angle. With an increase in the fiber curve angle, the loss factor also exhibited a decreasing trend after the first increase. The loss factor was maximum when the fiber curve angle changed between ±<45|60>, and minimum when the fiber curve angle was ±<60|75>. According to the conclusion analysis in figure 8, the loss factor of the laminate is always optimal when the fiber curve angle changes between ±<45|60>. This shows that when the number of laminate layers increases, the size of the loss factor is still regular, and is only related to the fiber curve angle. And the closer the curve angle is to ±<45|60>, it can be found that the greater the loss factor, the better the damping characteristics.  The loss factor values of the fiber curve 16 layers laminates were analyzed. It can be observed from table 5 that when the fiber curve angle changes between ±<45|60>, the loss factor reaches a maximum value of 3.29%, and when the fiber curve angle changes between ±<75|90>, the loss factor gets a minimum value of 2.62%. When the fiber curve angle was changed between ±<45|60>, the loss factor increased by 25.6% compared to that when the fiber curve angle was changed between ±<75|90>. Figure 13 shows the trend of the loss factor of laminates with the change in the fiber curve angle when the number of laminates was 16. With an increase in the fiber curve angle, the loss factor of the laminates first decreases, then increases, and then decreases again. The loss factor was maximum when the fiber curve angle changed between ±<45|60 > and minimum when the fiber curve angle was ±<75|90>.
It can be found that the angle variation of fiber curve has a significant influence on the loss factors of laminates structure by analyzing the loss factors of fiber curve laminates under three groups of different layering quantities and combining the three groups of loss factor trend graphs of figures 9, 11 and 13. The three groups of control test data show that the loss factor and damping ratio of the fiber curve laminates reach a maximum value when the fiber curve angle changes between ±<45|60>, and the laminates have the best damping performance.  The change in the fiber curve angle has a significant effect on the damping performance. The continuous change in the fiber curve angle leads to different energy dissipations in different positions and directions, the vibration attenuation rate of the laminates changes, and the damping performance is different. In addition, compared with traditional linear laminates, variable stiffness laminates have longer in-plane fiber lengths and greater fiber friction, which play a greater role in the vibration energy dissipation of the material. When the fiber curve angle was ±<45|60>, the fiber length of the laminate was the longest; thus, its damping performance was the best.
Then, a modal test of the fiber curve laminates with different angles was carried out, and the corresponding frequency response function was obtained after the modal test of the fiber curve laminates with 8, 12, and 16 layers. The first three frequencies of each variable stiffness laminate were calculated according to the frequency response function. Figures 14, 15, and 16 show the frequency response function curves of laminates with different fiber curve angles when the number of layers was 8, 12, and 16, respectively. The calculation results of the first three order frequencies of the three groups of laminates with different numbers of layers are listed in tables 6, 7, and 8, respectively. Table 6 lists the first three order frequencies of the laminates with 8 layers. It can be seen from table 6 that when the fiber curve angle changes between ±<60|75>, the first-order natural frequency reaches the maximum,  which is 65.45 Hz; When the fiber curve angle changes between ±<60|75>, the second-order natural frequency reaches the maximum, which is 106.95 Hz; When the fiber curve angle changes between ±<45|60>, the thirdorder natural frequency reaches the maximum, which is 166.73 Hz. The higher the natural frequency of the fiber curve laminate, the better the structural stability of the fiber curve laminate. The fiber curve laminate has a better ability to bear vibration loads, better impact resistance, and more stable structural performance under various actual working conditions, and it is not easy to cause structural failure, such as structural resonance. Table 7 shows the first three order frequencies of the laminates with 12 layers. It can be seen from table 7 that when the fiber curve angle changes between ±<60|75>, the first-order natural frequency reaches the maximum, which is 54.65 Hz; When the fiber curve angle changes between ±<60|75>, the second-order natural frequency reaches the maximum, which is 96.57 Hz; When the fiber curve angle changes between ±<45|60>, the thirdorder natural frequency reaches the maximum, which is 168.59 Hz.
According to the modal data of laminates in table 8, it can be seen that the first three order natural frequencies of composite laminates with 16 layers reach the maximum value when the fiber curve angle is ±<60|75>.  The simulation results were analyzed to determine the influence of different angle changes in the fiber curve on the natural frequency of the laminates. Taking 8, 12, and 16 layers as examples, the influence of the curve change of the fiber angle on the natural frequency of the laminates is explored. According to the modeling method of the fiber curve laminates above, the fiber curve angles are ±<0|15>, ±<15|30>, ±<30|45>, ±<45|60>, ±<60|75 > and ±<75|90>, respectively.  The vibration simulation results of the laminates with different fiber curve angles are shown in figure 17. It can be seen from the figure 17 that the distribution of stress and strain at each position is different when laminates with different fiber curve angles are vibrated, resulting in different vibration characteristics. When the fiber curve angle is ±<45|60 > and ±<60|75>, the force distribution of the laminates is relatively uniform, indicating that the vibration characteristics of the laminates under these curve angles are relatively stable. When the fiber curve angle was ±<0|15 > and ±<30|45>, the stress distribution of the laminates was more concentrated and mostly concentrated at the edge of the laminates. The natural frequency of the laminates is affected by the stiffness of the laminates and presents different results.
Based on the modal frequency analysis of Abaqus, the influence of the number of layers and fiber curve angle on the first three order natural frequencies of the laminates was compared and analyzed. The first three order frequencies of the fiber curve laminates with 8, 12, and 16 layers are shown in tables 9, 10, and 11, respectively.
According to the above three groups of simulation results, it can be seen that when the angle of the fiber curve is different, the first three orders of natural frequencies f 1 , f 2 , f 3 of the laminates are different values, which indicates that the angle of the fiber curve has obvious influence on the natural frequency of the laminates. When the number of plies was the same, the natural frequency of the laminate first increased and then decreased with an increase in the fiber curve angle. The first three natural frequencies of the laminates are maximized when the fiber curve angle is ±<45|60 > or ±<60|75>. The results are in good agreement with those of the tested results. According to the numerical simulation results in the above three groups of tables, the influence of the variation range of the starting angle T 0 and ending angle T 1 in the fiber curve angle ±<T 0 |T 1 > on the vibration characteristics of laminates was simulated and calculated. The distribution cloud diagrams of the first three orders of natural frequencies of laminates with different fiber curve angles are shown in figures 18, 19, and 20, respectively, where the abscissa represents the value of the starting angle T 0 and the ordinate represents the value of the ending angle T 1 .  The change trend of the first natural frequency of the fiber curve laminates with the starting angle T 0 and ending angle T 1 is shown in figure 18. It can be observed that the first natural frequency of the fiber curve laminates changes uniformly with the starting angle T 0 and ending angle T 1 . With the simultaneous increase in  T 0 and T 1 at the same time, the first-order natural frequency of the laminates shows an increasing trend. When the angle values of T 0 and T 1 are close to 90°at the same time, the first-order natural frequency of the laminates tends to the maximum. Figure 19 shows the distribution cloud of the second-order natural frequency of the fiber curve laminates with the change in the starting angle T 0 and ending angle T 1 . It can be observed from the figure that when T 0 is   between 40°and 70°and T 1 is between 50°and 80°, the second-order natural frequency region of the fiber curve laminates is maximized. The second-order frequency maximization region is generally located in the upper part of the cloud image, that is, when T 1 is greater than T 0 , higher vibration characteristics are obtained. Overall, when the fiber curve angle was ±<45|60 > and ±<60|75>, the second-order natural frequency of the laminates reached the optimal value. The third-order natural frequency of the fiber curve laminates varies with the starting angle T 0 and ending angle T 1 , as shown in figure 20. Its trend law is similar to that of the second-order natural frequency; however, the angle range of its curve is relatively narrow. When T 0 is between 40°and 50°and T 1 is between 60°and 75°, the third-order natural frequency of the fiber curve laminates achieves the maximum value. By comparing the above three groups of simulation results, it was found that the first three natural frequencies of the laminates reached the maximum value when the fiber curve angle was ±<45|60 > or ±<60|75>. The second-order and third-order frequencies of the fiber curve laminate were improved to different degrees compared with other fiber curve angles, and the overall vibration damping performance of the fiber curve laminate was improved.
The results of the laminate mode analysis show that when the fiber curve angle of the laminate changes between ±<45|60>, the vibration damping performance of the laminate is optimal. Compared with the results of the loss factor and damping ratio of the laminate in the previous free vibration test, there is little difference in the value interval of the fiber curve angle, which is consistent, confirming the correctness of the conclusion.

Conclusion
This study mainly examines the law of influence of the change in the fiber curve angle on the free attenuation of vibration of composite laminates, explores the correlation between the curve angle and the natural frequency, and studies the influence of changes in the fiber curve angle on the free attenuation rate, loss factor, damping ratio and natural frequency of fiber curve laminates. The main conclusions are as follows: (1)The tested verification and result analysis show that the vibration characteristics of the laminates are obviously different under the influence of different fiber curve angles when the number of layers is the same.
(2)When the fiber curve angle changes between ±<45|60>, the loss factor of the variable stiffness laminate reaches the maximum value, which is 3.65% for 8 layers, 4.48% for 12 layers and 3.29% for 16 layers.
(3)The loss factor was used to evaluate the damping effect of the laminate. The results show that when the fiber curve angle of the laminate changes between ±<45|60>, the loss factor and damping ratio of the laminate achieved the maximum value, that is, the damping performance of the laminate under the curve angle is good. (4)Modal tests of fiber curve laminates with different angles were carried out, and it can be seen that the first three natural frequencies of the laminates all reached the maximum value when the fiber curve angle was ±<45|60 > or ±<60|75>.
(5)By analyzing the simulation results of the first three natural frequencies of the laminates, it was found that different fiber curve angles had obvious effects on the natural frequencies of the laminates. When the number of plies is the same, with an increases in the fiber curve angle, the natural frequency of the laminate shows a trend of first increases and then decreases. The first three natural frequencies of the laminates all reached the maximum value when the fiber curve angle was ±<45|60 > or ±<60|75>.
(6)When the starting angle T 0 and the ending angle T 1 were both close to 90°at the same time, the first-order natural frequency of the fiber curve laminate tended to be maximized. When the fiber curve angle changes between ±<45|60>, the second order natural frequency of the fiber curve laminate is optimal. When the fiber curve angle changes between ±<60|75>, the third order natural frequency of the fiber curve laminates is optimal.