The finite element model of the seat rail is established with a springdamping element to simulate the ball in the rail joint part. The stiffness and damping parameters of the joint part are determined by the combination of finite element method and experiment. Firstly, the natural frequencies and modes of the guide rail are obtained by modal experiment. The stiffness of the springdamping element is optimized in the finite element software to make the natural frequencies and modes of the system consistent with the experimental ones. Secondly, the dynamic response curve of the key nodes is obtained through sweeping experiment, and the damping of the springdamping element is optimized in the finite element software to make the nodal response of the system output consistent with the experiment. Then, the gap of the joint part of the car seat rail is studied considering the factors of load and structure randomness. The influence factors of the gap are selected by Hammersley experimental design method. The results show that the gap is normally distributed, and therefore the confidence interval of the gap is obtained. Finally, the joint probability distribution of the gap is obtained under the condition that the load and the structure are all random, which provides the theoretical guidance for determining the reasonable gap of the joint.
With the progress of the times and the improvement of living standards, private cars have been rapidly entering thousands of households. The users’ requirements for private cars have also changed from a simple trip to a comfortable ride. Some of the advanced foreign car manufacturers in the early 1980s began to pay attention to body structure vibration and rough road bumps producing noise problems. In the 1990s, Volkswagen, GM, Ford, Toyota, and other wellknown car companies set up research centers to deal with the noise problems from vehicle structural vibration, noise, and rough road surface roughness [
The guide rail of car seat is the connection between the car seat and the car floor. Its role is to adjust the seat position back and forth and to protect the occupants’ safety. Seat rail quality determines the user experience. A good rail should not only be safe and reliable in the locked state, but also be smooth in the adjustment of seat position forth and back. However, when the vehicle is moving, the car seat rails will produce abnormal sound, which is usually solved by workers with their own experience when they encounter the problems. Their temporary solutions are inadequate to solve the seat rail abnormal sound completely. However, the increasing complaint and discontent of the users have caught seat manufacturers’ attention. The manufacturers have been putting great efforts to conduct experimental research to explore the causes of this problem. After long trials and exploration, engineers attribute the abnormal sound of rails to the uneven internal force of the balls. There are 4 slides between the inner rail and the outer rail, and 4 to 6 balls inside each slide. When the guide rail receives the random road excitation from the vehicle body, balls are subjected to uneven force. Some of them show the state of compression, and some of them show the state of separation. When the rail is subjected to the load of driving direction, noise is generated when the inner and outer rails are misaligned. Of course, the field engineer also gives the solution to this problem, which is to preload the rail before mounting. For example, if the nominal diameter of the steel ball is 6 mm, the actual spacing between the inner and outer rails is preloaded to 5.8 mm during installation so that the ball is actually in a preload state after the installation is complete. This solution is effective in the actual operation process. If the preload value is reasonable, the problem of abnormal sound in rails will be improved significantly. Therefore, the preload value for guidance is the key to solve the problem. If the preload is too little, noise will not be reduced effectively. On the other hand, if the preload is too large, serious slip will lock in the rail. Thus, the preload value, which is set to be the maximum value of the gap, may suppress the occurrence of the gap effectively, and it will be a reasonable preload value.
With the development of finite element method, it is an effective tool to solve the vibration and noise problem of vehicle seat by using finite element technology to simulate the dynamic characteristics of various parts of seat. To solve these problems, the dynamic finite element software can be used to analyze the dynamics of the steel ball. The gap between the steel ball and the inner and outer rails is analyzed by statistic method to observe the distribution of the gap and give the reasonable preloading value for guidance. However, many of the problems encountered will be the key to the successful solution. First of all, the interface between the steel ball and the inner and outer rail is actually a linear roller guide model. How can we establish the finite element model of the ball, so that the combination can be simulated exactly? It is also necessary to find a definite amount that can characterize the compacted and segregated state that occurs between the joint part. Secondly, this gap is affected by a number of uncertain factors, like random excitation of road, manufacturing error of balls, elastic modulus, and so on. However, deterministic finite element dynamic analysis will not show the influence factors other than external excitation, which cannot meet the actual demand of this issue. The stochastic finite element theory can treat the load, the strength of materials, and parts of the geometry as random variables to solve the problem. It is the numerical analysis theory based on the traditional finite element theory, and it is the combination of random field theory and finite element method [
In this paper, a finite element modeling method is studied for a vehicle seat rail. The springdamping modeling method for the joint part of linear roller is studied emphatically, and a complete method is proposed to optimize the stiffness and damping parameters of the joint using a combination of experiment and finite element analysis. Then, a series of dynamic analyses are carried out on the new dynamic model, and the distribution of the joint gap under the dynamic excitation is statistically analyzed. Finally, the distribution of the joint gap is investigated by using DOE and stochastic study considering the external excitation and the uncertainty of the structure itself.
As the thicknesses of the inner and outer guide rails are all 1.4 mm, the rails can be regarded as thinwalled parts. We adopt the method of extracting the middle layer. Using the CAE software, thinwalled parts are mathematically discretized by shell element, and a detailed finite element model is established. In this model, the outer rail is 450 mm long and the inner rail is 402 mm. The CQUAD4 unit with the basic dimension of 5 mm × 5 mm is used. There are 7913 nodes and 7406 units in total. The isotropic material is used to establish the material model. Its elastic modulus
Finite element model of seat rail
Schematic diagram of steel ball position
The handling of the internal steel ball is the key to this modeling. There are twenty guide balls in the rail, eight of which are Ф6, and twelve of which are Ф8. According to their position in the guide rail structure, the steel balls are divided into four groups, as shown in Figure
Figure
It is important to note that the CBUSH element needs to specify the local coordinate system, that is, the above six DOF directions are for the local coordinate system in which the CBUSH element is located. As the CBUSH element requires stiffness and damping parameters to represent the contact relationship, how to get the kinetic parameters of the junction becomes the key technology of the modeling method. The followup content will elaborate the method of combining the finite element method and the experiment to combine the dynamic parameter identification method.
The purpose of this modal experiment is to obtain the information of modal natural frequency, vibration mode, and damping ratio, which can provide reference for the revision of finite element calculation model by experimentally testing and analyzing a certain type of vehicle seat rails. It is necessary to determine the support mode, the excitation mode, the excitation point, and the response point in the modal experiment. The DH5927 dynamic testing and analysis equipment, threeway acceleration sensor, and impact hammer were used in modal experiment. The excitation adopts the hammering method. The excitation signal produced by the hammer has the advantage that the bandwidth of the signal can be controlled by different materials of the hammer. The higher the material stiffness, the wider the pulse signal spectrum will be. The adequate bandwidth can get more modes in one time. The hammering method is fixing the response point and moving the thumping point to compensate for the shortage of the number of acceleration sensors. For the excitation points and response points, the selected response points can reflect the basic outline of the specimen, while the nodes of main vibration modes should be avoided, and the test point which may have more local modes should be encrypted. In this modal experiment, four acceleration sensors are arranged. The transfer function is tested by moving the excitation point and fixing the response point.
According to vibration mechanics, the vibration of the structure is mainly due to the loworder natural frequency and its vibration mode. Due to the influence of the damping in the actual structure, the vibration modes of the higherorder frequencies will decay rapidly. In addition, when the sensitive frequency of the guide is less than 40 Hz, the first and second natural frequency and vibration mode can be found and used as the optimization target in stiffness optimization.
Table
Natural frequency and damping coefficient.
Order  Frequency (Hz)  Damping ratio (%)  Mode of vibration 

1  7.22  3.74  Torsion of outer rail 
2  12.65  2.57  Swing of Internal rail 
The stable diagram of the mode experiment for a certain point of force hammer.
The mode shape of first order
The mode shape of second order
The purpose of this experiment is to obtain the timeacceleration curve of the excitation point and the timeacceleration response curve of the characteristic node by sweeping a certain type car seat rail, which provides the basis for the optimization of contact damping in finite element calculation. The DH5927 dynamic test and analysis system, threeway acceleration sensor, and HEV20 exciter were used in modal experiment. There are 5 experimental output points, of which three timeacceleration curves will be used to fit the results of the finite element analysis output, and the remaining two points are used to validate the fitted parameters. The selection and installation of the sensor have a significant impact on the measurement results. The sensor should be installed in the condition of sufficient rigidity and without increasing the structural quality and then measure the true direction of the vibration signal. The excitation source of the sweep experiment needs to meet the following conditions: the amplitude level should satisfy certain conditions and the component being excited should have antiinterference ability when the exciter has tiny nonlinear behavior. In engineering fields, the sinusoidal signal is often chosen as the excited signal. The excited point selects the bolt installation hole at the bottom of the guide rail, which is consistent with the excited point in the practical working condition of the guide rail. The rails are suspended using a flexible rope during the entire excitation to simulate the unrestrained state.
The frequency sweep experiment uses the exciter as a frequency generator. As the exciter and the track are composed of openloop system, it can be seen that the collected excitation and response signals have a significant signal amplification at the resonance, as shown in Figure
Excitation/response signal collected by frequency sweep experiment. The abscissa is for the sweep time (s), and the vertical axis is for the excitation amplitude (g).
Based on the above two experiments, the stiffness and damping of the springdamper element embedded in the joint part are optimized using OptiStruct and HyperStudy of FEM software. The equivalent contact stiffness and damping obtained from optimization characterize the contact properties in the joint part and thus allow us to establish a more accurate dynamic model of vehicle seat guide.
The least squares method is a kind of technique of mathematical optimization, minimizing the sum of squares of errors to find the best function of the data match [
The specific approach to data fitting is as follows: for the given data
In the geometric sense, it means searching for the curve
The function
Solve formula
The general method of polynomial fitting can be summarized as follows:
Use the known data to draw a rough graphic of functions—Scatter plot—to determine the number of times of fitting polynomials.
Calculate
Write a formal equation and find
Write the fitting polynomial:
Under the OptiStruct in HyperWorks, the Lanczos method is used to extract the real eigenvalues of free modal for guide rail model by mass matrix normalization. In the initial analysis, a set of initial stiffness values are given for the spring, in which the rail calculation mode is in the same order of magnitude as the experimental mode natural frequency. The modal results are found to be not sensitive to the four sets of rotation angles, freedom, and stiffness, so only the stiffness of four axial displacements of these springs is considered. When the 12 rigidities are specified as 500 N/m, the first and second mode natural frequency of the guide rail system is given in Table
In the OptiStruct module, the stiffness of the spring element is taken as the design variable and the least squares method is used to optimize the stiffness.
where
The natural frequency of the rail model after optimization.
Order  Experimental values (Hz)  Optimization value (Hz)  Vibration mode 

1  7.22  7.220026  Torsion of outer rail 
2  12.65  12.65000  Swing of Internal rail 
Contact stiffness parameter after optimization (N/m).
Number of Spring group 




1  205.5  155.2  158.3 
2  223.3  151.9  154.7 
3  201.3  152.1  152.9 
4  205.0  155.7  155.9 
After comparison, the optimized vibration mode of the guide rail system coincides with the corresponding vibration mode of the experimental mode, so we believe that the optimized spring stiffness is the equivalent contact stiffness of the joint to be obtained.
The timedomain signal of the excitation and response of the sweep test is obtained in the previous section, as shown in Figure
The output of the finite element calculation and experiment.
The red curve is the timeacceleration response signal getting from software before optimization, and the blue curve is the experimental response signal after sparse processing in Figure
It can be seen that the peaks of these two curves are close to each other. The finite element calculation curves are well consonant with the test data. On the other hand, the previous optimization of the binding site of the stiffness value is consistent with the actual rail.
Then, we further fit the FE output to the experimental output curve to obtain a set of optimized damping coefficients, which is exactly the joint contact damping we are looking for. In HyperStudy curve fitting, we must explicitly design variables, objective function, design goals, and other objects.
The data sheet of design variable (Ns/m).
Design variable  Notes  Lower limit  Initial value  Upper limit 

G  Structural damping  0.1  0.2  0.3 
C11 

0.2  2  20 
C12 

0.2  2  20 
C13 

0.2  2  20 
C21 

0.2  2  20 
C22 

0.2  2  20 
C23 

0.2  2  20 
C31 

0.2  2  20 
C32 

0.2  2  20 
C33 

0.2  2  20 
C41 

0.2  2  20 
C42 

0.2  2  20 
C43 

0.2  2  20 
Four sets of spring have been defined in the model as shown in Figure
where
According to the objective function, the target value OBJ before optimization is 274.64. After HyperStudy optimization, the target OBJ reduces to 37.81. The number of iterations is 17 as shown in Figure
The convergent curve of the objective function.
The optimized output and experimental output of the finite element software.
The blue curve is the experimental response signal after sparse processing in Figure
After optimization is completed, the optimal solution of a set of contact damping is shown in Table
Results of damping optimization (Ns/m).
Number of spring group 




1  14.2  3.4  4.7 
2  17.3  13.7  1.8 
3  2.7  9.5  1.3 
4  6.1  19.1  5.3 
So far, the contact stiffness and contact damping of the joint of the rail model are all optimized, and the finite element model is reestablished based on the optimized contact damping and contact stiffness. The relative displacement between the two ends of the spring element can be output by the appropriate excitation signal to the guide rail model, which is between the steel ball and the inner/outer rail.
Under uncertain external loads which depend on timedomain or frequency domain, the springdamper element embedded in the joint part will behave irregularly, resulting in a gap. In order to statistically analyze the confidence interval of the gap, the transient analysis can be carried out on the dynamic model of the guide rail, and the gap of the guide joint is studied using statistical method. The frequency response analysis is used to verify the results. Finally, the interval confidence intervals of the joints are calculated under dynamic excitation. Seat mass is converted to forces which are applied to both ends of the rail and the excitation location, as shown in Figure
Seat position
Excitation location
Gap calculation formula:
Figure
As the amplitude of actual road conditions is random, the results of the road spectrum will reflect the actual rail gap situation more accurately. Therefore, the road spectrum of the rail bodyconnection point is collected when the car is driving on the road, and the data is used as the excitation signal of the rails, as shown in Figure
The
The
The relative displacementtime curve of the two ends of the spring under the excitation of 60 km/h.
It is shown that the maximum value of the gap is 0.23 mm in the timehistory curve with the time span of 100 seconds. In order to further determine the confidence interval of the seat guide gap, the relative displacement between the two ends of 20 springs in Figure
The distribution of relative displacement
The distribution of gap interval
Figure
Since Figure
According to the normal distribution table, 90% of the grasp of the gap does not exceed 0.116 mm; 95% of the grasp of the gap does not exceed 0.137 mm; 99% of the grasp of the gap does not exceed 0.180 mm, as shown in Table
Interval confidence table.
Confidence  Gap value 

90%  0.116 mm 
95%  0.137 mm 
99%  0.180 mm 
The conversion from the time domain to the frequency domain signal requires a Fourier transform. The ideal Fourier transform is defined as follows:
Since the idealized Fourier transform is not windowed, the time is considered to be from negative infinity to positive infinity, which leads to an increase in the magnitude of the frequency domain by
Figure
Frequency domain signal after Fourier transform
Low frequency domain signal
According to Figure
According to formula (
The relative displacementfrequency curve of the two ends of the spring.
It can be seen that the maximum relative displacement of the two ends of the spring is
Substituting formula (
In order to study the influence of eight factors on the gap of the joint, experiments need to be designed to screen out the factors that have an important impact to focus on inspection. The firstround selection is determined by using the Hammersley design method, which is uniform and robust in multidimensional problems. The values of each factor are interval uncertain values, which belong to an infinite number of theoretical levels. The optimization problem is defined in HyperStudy, including defining eight factors as design variables and assigning the initial value and the upper and lower limits, as shown in Table
Valve table of experimental design variable.
Factor  Notes  Lower limit  Initial value  Upper limit 


Internal rail thickness (mm)  1.30  1.40  1.50 

Outer rail thickness (mm)  1.30  1.40  1.50 

Elastic modulus of guide material (Mpa)  200000  210000  220000 
nu  Poisson’s ratio of guide material  0.27  0.3  0.33 

Guide material density ( 




Ridge radius of inner rail (mm)  −1  0  1 

Ridge radius of outer rail (mm)  −1  0  1 

Manufacturing error of ball diameter (mm)  −1  0  1 
As can be seen from Figure
In Table
Gap responses of different confidences.
Response  Confidence  Definitions 


90% 


95% 


99% 


100% 

After the design variables and response definitions are complete, the main effect is calculated using the DOE method of the Hammersley sampling design, and several factors that have the greatest effect on the response are selected as shown in Figure
The sampling design and the main effect of the value of Hammersley.
Variable name  Main effect 


−0.08 

−0.101 

0.007 
nu  0.001 

0.005 

−0.007 

−0.006 

−0.02 
The schematic diagram of the main effect of the design of the Hammersley sampling.
It can be determined that the inner rail thickness (
The internal rail thickness (
Random distribution table of design variables.
Factor  Random distribution  Parameter 
Parameter 


Normal distribution ( 
1.40  0.0015 

Normal distribution ( 
1.40  0.0015 

Normal distribution ( 
0  0.1 
According to the distribution of the parameters in Table
Under a transient load, the clearance corresponding to the outside confidence is defined as shown in the definition in Table
Random distribution of
Random distribution of
Random distribution of
Random distribution of
Figure
Confidences of junction gap (mm).
Load confidence  Structural confidence External  

90%  95%  99%  100%  
90%  0.1510  0.1542  0.1621  0.1638 
95%  0.1776  0.1814  0.1903  0.1931 
99% 

0.2095  0.2210  0.2314 
100%  0.2010  0.2260  0.2350  0.2454 
It can be seen that with the increase of the load confidence and the structural confidence, the joint gap tends to increase, and the effect of the load uncertainty is greater than the structural uncertainty.
Based on the above conclusions, consider the gap value of 0.1968 mm as the recommended installation preload value, using 90% of the structure confidence external and 99% of the load confidence. The specific basis is as follows.
The impact of load uncertainty is higher than the degree of structural uncertainty, which can be key consideration.
In reality, structural uncertainty cannot be measured directly. It is difficult to accurately control.
In engineering fields, seat manufacturers preload the rail to the value of 0.2 mm when they solve the problem of abnormal noise, which also shows that the research work done in this paper accurately predicts the rail preload value, providing theoretical basis to solve this problem.
With passengers’ increasing requirements of comfort, noise, vibration, and harshness (NVH) have become one of the major concerns of car manufacturers. In this paper, the study of the gap between the car seat rail joints is a componentlevel NVH problem. By studying the finite element dynamic model of the car seat rail joint, the problem of the gap distribution of the joint is analyzed according to a series of dynamic responses under the uncertainty of the external load. Furthermore, in the case that the structural parameters of the guide rail itself are uncertain, the randomness of the gap was also studied in this paper. The significance of this paper is predicting the distribution of gap in the joint of car seat guide rail by finite element simulation and providing the guiding value of installation preloading, which provides not only a solution but also the theoretical basis for field engineer to solve the problem of track abnormal sound.
The research results completed in this paper are listed as follows:
There are no conflicts of interest regarding the publication of this paper.
Thanks are due to the support of the CivilMilitary Integration Project of Shanghai, no. 201643.