Study on Preload of Bolted Connections in Pitch Bearing Based on Vibration Modal Analysis

. In wind turbine systems, bolted connections are frequently subjected to gravitational and centrifugal loads transmitted by the blades during operation. Tis can lead to the attenuation of bolt preloading, resulting in bolt loosening or uneven loading, which in turn afects the service life of the generator unit. Terefore, the study of bolt preloading variations is crucial. However, there are numerous factors infuencing bolt preloading, and the existing techniques struggle to precisely assess bolt preloading. Tis paper proposed a bolt preloading evaluation technique based on the hammer modal method. Focusing on the 42CrMo4 bolted connection of a pitch bearing, a test platform for bolt preloading assessment is constructed. Hammer modal tests are conducted under various preloading forces. By combining fnite element modal analysis, the correspondence between preloading changes and the bending frequencies and modes of the bolted connection is obtained. Te research illustrated that with changes in bolt preloading, variations occur in coherence, phase, and natural frequencies of frequency response functions. Te fundamental correlation between bolt preloading and the second-order bending frequency can be utilized to assess changes in preloading. Furthermore, the applicability of this method has been validated, ofering a reference for evaluating bolt preloading.


Introduction
Te pitch bearing in a wind turbine, as one of the essential components, serves to connect the hub and blades, working in conjunction with the pitch drive system to adjust the angle of the blades for wind power generation.During the operation of a wind turbine, the bolted connection of the pitch bearing is subjected to gravitational loads transmitted by the blades, centrifugal loads, and torque from the drive gear.Tese factors can signifcantly reduce the bolt preload, leading to phenomena such as bolt loosening or uneven loading, exacerbating fatigue wear of the pitch bearing.Over prolonged operation, bolt connection failures can occur, afecting the entire turbine's overall safety, reliability, and lifespan [1][2][3][4][5].Terefore, the assessment of bolt preload is of paramount importance.
Currently, hammer impact tests, torque tests, and ultrasonic tests are the main methods used for evaluating preload.
Traditional hammer impact tests rely on the subjective judgment of testing personnel based on the characteristic sound emitted by the vibration device, but they do not provide quantitative monitoring.Torque tests control bolt preload by using a torque wrench, but this method is mainly applicable to later-stage equipment maintenance and is challenging to accurately evaluate axial forces due to the changing friction characteristics of the contact surfaces with tightening.Ultrasonic testing for bolt preload primarily involves measuring the sound propagation delay of the bolt head [6][7][8][9], the ratio of longitudinal and transverse wave velocities [10][11][12][13], combined vibration modes and longitudinal ultrasonic waves [14], or the shift in resonance frequency [15][16][17][18].Due to the numerous factors infuencing bolt preload, the above methods have certain errors.Terefore, to address this issue, the most advanced technique currently is to directly measure preload by embedding strain gauges and fber optic grating sensors into the bolts.However, this method has limitations in bolt connection devices and is cost-intensive [19][20][21].Literature [22][23][24] has shown through simulation and experimental methods that the magnitude of preload can alter the natural frequency of the connected components, with greater preload resulting in increased natural frequency, providing a reference for testing residual bolt preload.Zhang [25] established stifness and fnite element models for bolted CFRP (carbon fber reinforced polymer) lap joints and quantitatively calculated bolt relaxation position and residual preload using artifcial neural network algorithms.Te results indicated a positive correlation between the bolt's third-order bending mode and bolt preload.Hou [26], starting from the microlevel perspective of the bolt contact surface, studied the infuence of contact stifness and damping on the dynamic characteristics of bolted connections.Trough mutual verifcation using hammer vibration tests and modal simulations, the modal damping ratio under diferent preloads was obtained, indicating that modal damping increases with increasing bolt preload.
Tis paper presented a method for assessing bolt preloading based on vibration modal analysis.A testing platform for evaluating bolt preloading was established and hammer modal tests on 42CrMo4 bolted connections of pitch bearings under various preloading conditions were conducted.By integrating fnite element modal analysis and harmonic response, the changes in bending frequencies and modes of bolted connections under diferent preloading forces were analyzed.Te evaluation of preloading variations was conducted based on the bending frequencies.Furthermore, the method was validated through hammer modal tests on bolted connections with varying dimensions and sizes.

Modal Experiment Principle
To obtain a feasible method for efectively monitoring the preloading force of bolts, the concept of smart washers as an elastic suspension cantilever, as proposed by Okugawa and Egawa [27], was considered.It was found that when the clamping force on the smart washer was sufciently high, its natural frequency was the same as when fully tightened.As the clamping force decreased, the natural frequency of the smart washer decreased, allowing the clamping status to be monitored by detecting changes in the natural frequency.Terefore, the bolted connection could be regarded as a cantilever with an elastic suspension consisting of translational and rotational springs.Te reduction in preloading force resulted in a decrease in the stifness of the bolted connection, which was refected in changes in the natural frequency.By conducting modal testing to measure the natural frequencies and mode shapes of the bolted connection structure, the preloading force could be efectively assessed.
Modal parameters, including frequency, mode shape, and damping ratio, were intrinsic characteristics of the structure and could be obtained through modal analysis [28].Modal testing involves analyzing the continuous physical signals obtained from the test and expressing the dynamic properties of any structure using an Nth-order diferential equation [26].
Equation ( 3) was the dynamic matrix of the structure, expressing the frequency response characteristics.Te frequency response function matrix was its inverse matrix, which is as follows: Terefore, X(s) is expressed as follows: Let s � jω; equation (5) gave the relationship between the excitation response and the excitation input as follows: Te frequency response function matrix is as follows: It could be seen from the above formula that when there was only excitation in coordinate j, the frequency response of coordinate i was inversely proportional to the excitation force.
Similarly, the impedance matrix could be obtained from equation (3) as follows: Te M and k matrices were weighted orthogonal to obtain the following equation: Terefore, the vibration mode matrix ϕ � [ϕ 1 , ϕ 2 , • • • ϕ n ]; the damping matrix was orthogonal to the vibration mode matrix, and the following equation could be obtained: Shock and Vibration We brought the above formula into equation (8) to get the following equation: Te transfer function of the jointly obtained structure above is as follows: In the formula, ω 2 r � k r /m r , ξ r � c r /2m r ω r , K is the stifness matrix, k r is the r-th order modal stifness matrix, N is the total order of structural mode, H ij (ω) is the transfer function of structure, m r is the modal mass, Φ ri , Φ rj is the i and j r-order modal shape of i and j point, ω r is the r-th modal frequency, and ξ r is the damping ratio.
Based on Okugawa's study [22] on the testing method for the clamping force of smart washers, a frequency equation was established.By using equation (17), the correlation between the natural frequency of the bolt and the pretightening force could be determined.Terefore, the variation in the pretightening force of the bolt could be evaluated by testing the changes in the natural frequency of the bolted connection structure.
In the formula, l is the bolt length, E is Young's modulus, I is the moment of inertia of cross section, A is the crosssectional area of the bolt, ρ is the density, ω is the frequency, k T is the bolt transverse spring coefcient, and k R is the bolt circumferential spring exchange coefcient.

Experimental Method and Setup of Bolted Connections
3.1.Test Structure.Te experimental structure, as shown in Figure 1, consisted of a bolt, nut, and 42CrMo4 cube with a side length of 35 mm.A throughhole with a diameter of 11 mm was drilled at the center of the specimen, and it was connected using an M10 × 55 bolt.In this experiment, square steel was used as the connecting component to reduce the infuence of frictional contact between multiple connected components on the natural frequency and mode shapes of the specimen, thus making the pretightening force testing method more intuitive and accurate.Referring to [18], points 1 to 6 on the specimen represented the locations for impact testing, and the red square on the opposite side of point 6 indicated the installation position of the accelerometer sensor.

Experimental Device.
Te experimental setup, as illustrated in Figure 2, consisted of four components: the excitation system, response system, dynamic signal acquisition and analysis system, and computer analysis software.Te excitation system primarily referred to the force hammer, while the response system comprised accelerometers, test bolts, and 42CrMo4 square steel components.Te dynamic signal acquisition and analysis system used the INV 3018 vibration and noise analyzer developed by COINV, and the computer analysis software used was DASP V11.Te basic parameters of the experimental setup are presented in Table 1.
During the setup of the experimental testing system, accelerometers were bonded to the designated positions on the bearing steel using adhesive to prevent any positional changes during the hammering process.Te accelerometers were connected to the INV 3018 vibration and noise analyzer, and the force hammer was also connected to the Shock and Vibration acquisition instrument.Finally, the acquisition instrument was connected to the computer for data collection and analysis.

Boundary Conditions.
Te diferent constraints in experimental conditions could lead to signifcant diferences in the modal test results.In this experiment, the unconstrained condition was adopted because it was easier to simulate compared to the boundary-constrained condition.Te rubber ropes were selected as the suspension system on the test platform, as shown in Figure 3, to simulate the free boundary condition.However, due to the presence of rubber ropes, it was not an absolute unconstrained state, and thus there were certain diferences between the experimental and numerical simulation results.
Te experimental setup utilized the single-reference point test method, which involved fxing an accelerometer at a specifc location and moving the force hammer excitation point, as shown in Figure 4. Te test points on the specimen (1∼6) were sequentially struck, and three impacts were applied at the excitation point.During the hammering process, it was crucial to ensure that the direction of the force  Trough numerical simulations in the following section, it was determined that the 1st bending mode frequency of the model was around 20 kHz.Terefore, an aluminum force hammer tip was chosen for this experiment.

Modal Testing and Simulation
4.1.Experimental Modal Measurement.Te application of preloading force in the experimental model of the bolt was achieved by converting torque to preload force based on equation (18).Te tightening torque coefcient K varied due to diferences in the thread surface fnish and lubrication conditions.A surface-plated galvanized bolt of grade 12.9 was selected for the experiment, and no lubrication was applied during tightening.Referring to Table 2, the tightening torque coefcient K was chosen as 0.22.Torque was applied using an electronic torque wrench with a digital display, and the relationship between torque and preload force was obtained by utilizing equation (18), as shown in Table 3.
In the formula, T represents torque, F 0 represents bolt preload, K represents tightening torque coefcient, and d represents nominal diameter.
Te data acquisition was conducted with a sampling frequency of 102.4 kHz.Te triggering method involved initiating data collection when the impact force of the hammer exceeded 5 N. To capture the complete vibration signal, each sampling cycle was set to include 32,768 data points, with a sampling lag of 256 points.Tree trigger events were set for each measurement point.With the aforementioned parameter settings, three hammer strikes were applied to a specifc measurement point on the specimen, resulting in a complete set of time-domain vibration acceleration response signals, as shown in Figure 5. Te time it took for the acceleration response signal to attenuate to zero after a single excitation was approximately 4 ms, and the signal exhibited a symmetric distribution on both sides, indicating the reliability of the test data.Otherwise, retesting would be necessary after reapplying the hammer strikes.

Modal Simulation of Bolted Connections.
Te natural frequencies and mode shapes of the bolted connection were determined using fnite element analysis.Experimental investigations were conducted to study the infuence of varying bolt preloads on the natural frequencies of the specimen and to further validate a modal testing method for evaluating bolt preloads.Because mesh refnement in fnite element analysis not only afected the computation time but also impacted the accuracy of the solution, grid-independent analysis was   Shock and Vibration conducted.A convergence target value was set, and multiple automatic computations were performed to ascertain a meshindependent solution using Ansys Workbench.
According to Figure 1, a fnite element model consisting of bolts, nuts, and 42CrMo4 square steel was created in Workbench.An allowable change of 0.1% was set.Ansys Workbench automatically refned and solved the fnite element model's mesh and compared the results with the previous computations until the allowable change dropped below the set threshold.Te computation results are presented in Table 4 and Figure 6.Using the initial model with a unit size of 6 mm, after the frst computation of the frstorder bending mode frequency, a convergence insert was implemented.Tree additional computations were performed.Te frequency obtained in the fourth computation was 21992 Hz, with only a 0.038% diference from the third computation result, meeting the convergence requirement.
As shown in Figure 6, it could be observed that the infuence of mesh size variation on the 1st mode frequency gradually diminished, approaching convergence.Te smallest diference was observed in the fourth computation.However, with the reduction of mesh size, the mesh density and computation time increased substantially, signifcantly afecting computational efciency.Considering convergence analysis comprehensively, to ensure a balance between computational accuracy and efciency, a cubic mesh size of 2 mm was adopted during mesh partitioning.Additionally, to ensure precision in bolt connections, the mesh size for both bolts and nuts was defned as 1 mm, as illustrated in Figure 7. Te number of nodes was 85495.Material properties for the 12.9-grade high-strength bolts, nuts, and 42CrMo4 steel were defned according to Table 5.
Te fnite element model consisted of three contact pairs.Te contact between the bolt head and the 42CrMo4 square steel, as well as the contact between the nut and the 42CrMo4 square steel, were defned as bonded contacts.Te contact between the bolt and the nut was modeled using the MPC method.Te bonded contacts represented the fully loaded state with a bolt preload of 60 N•m.To represent the specimen in a free boundary condition, no additional constraints were added for modal analysis.Te modal results were then inputted into the harmonic response analysis module, with an excitation frequency range of 0 to 40 kHz and a modal damping ratio defned as 0.03.A 5 N excitation load was applied at the six measurement points shown in Figure 1 for numerical simulation and solution.7, and compared with the results from the fnite element analysis.Te frequency response functions, phase, and coherence all indicated that the experimental and simulation results exhibited similar trends.However, due to the inability to achieve complete free constraints in the experiment, there was a slight discrepancy between the results.

Natural Frequency and
From Figure 8, it could be observed that the model exhibited two natural frequencies near 20 kHz and 35 kHz, confrming the conclusion of a symmetric model with repeated roots.Te extracted natural frequencies and mode shapes from both the experimental and simulation results are shown in Table 5, with the simulation mode shapes referenced to the bolt's central axis as the cross-section.
Based on Table 6, the mode shapes from the experiment and simulation were consistent with each other.Te frst and second modes were symmetric bending modes, with deformation primarily occurring at the bolt head and the bolt's center position.Te third and fourth bending modes showed deformation primarily at the bolt head, with no deformation observed in the connected 42CrMo4 square steel in the frst four bending modes.However, the third and fourth bending modes were close to the frequency testing range of the acceleration sensor, resulting in larger errors.Terefore, the  Shock and Vibration relationship between natural frequencies and bolt preload will be investigated in next section based on the frst-and second-mode frequencies to determine a method for evaluating preload.

Evaluation of Bolt Preload
Variations.For analysis, experimental data were extracted for bolt torques of 10 N•m, 30 N•m, and 60 N•m.Te frequency response functions are shown in Figure 9.It could be seen that as the preload force increased, the coherence, phase, and amplitude of the frequency response function varied around the peak near kHz, indicating an increase in the natural frequency with increasing preload force.Since the model exhibited repeated roots around 20 kHz, there was an overlap between the frst bending mode frequency at 60 N•m and the second bending mode frequency at 30 N•m within a certain frequency range.Tis could lead to ambiguity in evaluating the magnitude of the preload force.Terefore, in this model, the second bending mode frequency was used to assess the bolt preload force.
and 60 N•m were applied to the bolts, and two modal tests were conducted for each torque level.Five bolts were used for repeated tests, resulting in the corresponding relationship between the second bending mode frequency and the tightening torque as shown in Figure 10.From Figure 10, it could be observed that the bolt frequency was positively correlated with the tightening torque, and the frequency variation range was more pronounced for preloads less than 30 N•m, leading to more accurate preload evaluation.As the tightening torque increased, the frequency variation tended to stabilize, introducing some errors in preload assessment.Terefore, it was feasible to evaluate the trend of preload changes by measuring the variation in the second bending mode frequency of the bolted joint structure.Te experiment also demonstrated that diferent bolts of the same size and model had minimal impact on the structure's natural frequencies.

Verification of Applicability of Bolt Preload Evaluation Method
To validate the applicability of the modal method for evaluating preload force, verifcation was conducted on bolts of diferent sizes and models, as shown in Table 7.Based on the test results from the previous section, it was observed that bolts of the same model but diferent sizes exhibited little dispersion in the second bending mode frequency under the same preload force.Terefore, in the experiment of this section, each group of bolts was subjected to three repeated tests, and the average values of the three test results were calculated to obtain the corresponding second bending mode frequencies.
Te A∼D experiments primarily investigated the applicability of the testing method for diferent bolt lengths, and the results are shown in Figure 11.From the graph, it could be observed that the bolt length had some infuence on the natural frequency but still followed the positive correlation between preload force and the second bending mode frequency.Terefore, this method could be used to evaluate the trend of preload force changes in bolts.It should be noted that for the D bolt, which had a longer length extending beyond the length of the connected component, the second mode frequency exhibited a larger variation range before reaching 10 kN•m.
Te bolt connections in wind turbine pitch bearings mainly used grade 12.9 high-strength bolts.In the previous section, the modal preload testing using grade 12.9 highstrength bolts provided consistent results.However, in   Shock and Vibration engineering applications, grade 8.8 and grade 4.8 bolts are more commonly used.Terefore, it was necessary to validate the applicability of this method for diferent strength grades.By conducting modal preload tests on B, E, and F bolts, the relationship between preload force and natural frequency was obtained, as shown in Figure 12.Te results indicated that, for the three diferent strength grades of bolts, the second bending mode frequency exhibited similar variations with changes in preload force.Hence, this method was suitable for grade 8.8 and grade 4.8 bolts.
When structural components were bolted together in engineering applications, it was often necessary to place washers between the nut and the connected component.To study the applicability of washers in this method, a washer was added to the G bolt connection, as shown in Figure 13(a), while the B bolt connection was tested without a washer, as shown in Figure 13(b).Te infuence of the presence or absence of washers on the second bending mode frequency was investigated through modal preload testing using the hammering method.Te test results are shown in Figure 14.When washers were used in the bolt connection, the test results aligned with the fndings of the previous section.However, the G bolt connection with washers exhibited a higher second bending mode frequency under full load compared to the washerless B bolt connection.At a preload force of 5 kN•m, the G bolt connection with washers showed a lower second bending mode frequency than the washerless B bolt connection, indicating a wider range of second bending mode frequencies for the G bolt connection with washers.

Conclusion
Tis study proposed a vibration modal testing method for evaluating bolt preloading based on modal testing principles.A testing platform for assessing bolt preloading was established, and hammer modal tests were conducted on 42CrMo4 bolted connections of pitch bearings under different preloading conditions.Trough numerical simulations and experimental methods, the impact of changes in bolt preloading on natural frequencies and modes was investigated.Te study revealed that the frst four bending modes of the bolted connection model corresponded to each other.Additionally, by studying frequency response functions under diferent tightening torques, it was determined that changes in preloading led to alterations in the peak values of coherence, phase, and amplitude in frequency response functions.Te fundamental relationship of a positive correlation between bolt preloading and the secondorder bending frequency was established.Te experiments demonstrated the feasibility of evaluating bolt preloading using the second-order bending frequency of bolted connections and validated the applicability of the hammer modal testing method to bolted connections of diferent sizes and types.Te r-th order modal stifness matrix N: Te total order of structural mode H ij (ω): Te transfer function of the structure m r : Modal mass Φ ri : Te i r-order modal shape of i point Φ rj : Te j r-order modal shape of j point ω r : Te r-th modal frequency ξ r : Damping ratio l: Bolt length E: Young's modulus I: Moment of inertia of the cross-section A: Te cross-sectional area of the bolt ρ: Density ω: Frequency k T : Bolt transverse spring coefcient k R : Bolt circumferential spring exchange coefcient T: Torque F 0 : Bolt preload K: Tightening torque coefcient d: Nominal diameter.
Mode Shape Analysis.Te extracted force hammer input signals and acceleration output signals from measurement points 1 to 6 at a bolt preload of 60 N•m were used to calculate the transfer functions.Te frequency response functions, coherence, phase, and amplitude were determined, as shown by the

Figure 9 :Figure 10 :
Figure 9: Comparison diagram of the frequency response function.

Figure 11 :Figure 12 :
Figure 11: Efect of diferent screw lengths on second-order bending natural frequency.

Figure 13 :
Figure 13: Comparison diagram of bolt connection with or without gasket.(a) Bolted connection with gasket.(b) Bolted connection without gasket.

Figure 14 :
Figure 14: Efect of the gasket on the second-order natural frequency.
the hammer struck was perpendicular to the specimen surface.

Table 3 :
Corresponding relationship of torque preload.

Table 6 :
Natural frequency and vibration mode.

Table 7 :
Specifcations of bolts of diferent models.