Development and Experimental Verification of Multi-Parameter Test Bench for Linear Rolling Guide

Abstract

The linear rolling guide (LRG) is widely used in the computer numerical control machine tool industry and other industries. To accurately evaluate the performance of LRGs, a multi-parameter test bench was developed to measure motion accuracy, preload drag force (PDF), vibration, temperature rise, and fatigue life. The mechanical structure and measurement and control system of the test bench were designed based on established principles and methods. ANSYS 19.0 software was used for static analysis of the gantry, modal analysis of the upper bed, and simulation of the impact of loading block thickness on load distribution uniformity. At the same time, we used an impact hammer modal test to verify the correctness of the finite element analysis of the upper bed. The analysis results validated the structural design. To verify the test bench’s repeatability, comparative experiments were conducted with the Hilectro LGD35-type LRGs, focusing on motion accuracy, PDF, and fatigue life. The experimental results confirmed the test bench’s high repeatability and validated the derived equations for measuring motion accuracy.

1. Introduction

The linear rolling guide (LRG) is a specialized type of rolling bearing guide mechanism that was developed in the late 1970s and has improved progressively based on the fundamental principles of rolling bearings. The composition of this system includes a rail, rolling elements (balls or rollers), carriages, and other mechanical components, which serve the purposes of bearing loads and providing motion guidance for moving parts in mechanical transmissions [1,2]. A simple structure, low dynamic and static friction coefficients, high accuracy in positioning, and excellent accuracy retention are advantages of LRG. This emerging technology is widely applied in various pieces of high-accuracy mechanical equipment, such as machine tools, industrial robots, food machinery, medical devices, and integrated circuit packaging systems, and significantly influences the stability, accuracy, and reliability of the overall system [3].

Computer numerical control (CNC) machine tools represent advanced equipment in which the LRG serves as a critical part. Its load-bearing characteristics, accuracy retention, and reliability are key factors that determine the overall performance of the machine [4]. With the introduction of ‘Made in China 2025’, the Chinese government has begun to provide significant support for the research and development of CNC machine tools and related key components. Supported by national science and technology initiatives, leading Chinese industrial enterprises (e.g., Guangdong Kaite, Nanjing Craft, and Hanjiang Machine Tool) have made substantial advancements in the rigidity and comprehensive performance of their LRG products, gradually entering the mid-to-high-end market for rolling functional components. However, the overall quality of Chinese LRG products remains relatively limited in terms of product diversity, especially in accuracy retention, compared to well-established foreign brands, such as THK and NSK from Japan, as well as INA and Rexroth from Germany. The accuracy retention time of Chinese LRG is approximately two-thirds that of imported products, which makes it challenging for them to meet the high-accuracy and high-reliability requirements of advanced CNC machine tools. Therefore, the LRG used in Chinese high-end CNC machine tools still heavily relies on imported products. This is because the Chinese enterprises mainly focused on production and manufacturing in the LRG products, which has resulted in a lack of foundational technological theories and experimental studies related to LRGs [5].

In recent years, numerous researchers have made significant contributions to the development of LRG test benches. Xu et al. [6] designed a comprehensive performance testing bench for LRGs. This design improved the overall safety performance of the test bench. However, this test bench lacked a loading mechanism, which resulted in a limitation of its ability to measure the comprehensive performance parameters (e.g., motion accuracy and vibration) under realistic application conditions. Based on the principle of contact measurement, Wang et al. [7] designed a test bench for assessing the motion accuracy of LRGs. However, experimental verification demonstrated that the angle changes in different directions during the operation of LRG were minimal, leading to a waste of the sensor’s utilization. Additionally, the calculation formulas for the parallelism of top and side surfaces were complex and inconvenient to use. Xu [8] developed a bench for evaluating the lifespan of LRGs. However, the lifespan testing bench does not contain a corresponding monitoring system to prevent real-time observation of the experimental process and timely acquisition of fatigue failure data for the LRG. Furthermore, lifespan calibration for the LRG and a validation process for the functionality of the test bench are neglected in this study. Additionally, Li et al. [9] introduced a dynamic friction measurement system for LRGs. Li et al. [10] designed a resistance device for LRGs and conducted operational resistance tests. However, the design of the test bench is limited, as it is unable to measure multiple parameters of LRGs.

The purpose of this study is to present the design of the LRG’s multi-parameter test bench, which aims at measuring the comprehensive performance [11] and fatigue life [12] of the LRG. The comprehensive performance parameters include motion accuracy, preload drag force (PDF), vibration, temperature rise, and fatigue life; the multi-parameters mentioned in this paper are the five parameters. The mechanical structure and measurement and control system of the test bench are developed based on the experimental principles, with design validity confirmed through statics and modal analyses. Finally, a comparative analysis of the measurement results against relevant industry standards and theoretical calculations confirmed the repeatability of multi-parameter measurements conducted by the test bench. This work addresses the challenge of efficiently measuring multiple parameters of LRGs on a single test bench, offering practical value for verifying LRGs’ performance and confirming their certification.

2. Measurement Principles

2.1. Measurement Principle of Motion Accuracy

Figure 1 illustrates the installation position of three contact displacement sensors. Sensor 1 and Sensor 2 are positioned vertically against the reference surface a, mounted symmetrically on either side of the center of the carriage top surface. Sensor 3 contacts reference surface b at the rear end of the carriage. The direction of the X-axis corresponds to the motion direction of the LRG. The motion accuracy of the LRG includes horizontal and vertical parallelism as well as deflection angles in the X, Y, and Z directions, specifically the tilt, pitch, and yaw angles [13]. Horizontal parallelism is defined as the maximum variation in the distance from the side surface center of carriage to reference surface b during motion. Vertical parallelism is defined as the maximum distance variation from the top surface center of carriage to the reference surface, surface a, during movement. Experimental validation results indicate negligible changes in pitch and yaw angles; hence, only the tilt angle is considered in this study. The tilt angle is defined as the maximum deflection of the carriage relative to the LRG in the direction of the X-axis during operation.

The geometric relationship diagram used for calculating motion accuracy is presented in Figure 2. The data obtained from Sensor 1, Sensor 2, and Sensor 3 are denoted as deviations S₁, S₂, and S₃ from the initial zero point of the sensors, respectively. The distance between Sensor 1 and Sensor 2 is represented as L₁. Point A is located at the center of the top surface of the carriage, with a distance h₁ from the reference surface a. The distance from the mounting hole of Sensor 3 to the upper surface of the carriage is L₂, and to the side surface of the carriage is L₄. L₃ denotes the thickness of the carriage, and Point B is h₂ from the reference surface b. Additionally, the distances from the mounting holes of Sensor 1, Sensor 2, and Sensor 3 to their respective reference surfaces are denoted as H₁, H₂, and H₃. The tilt angle is represented by θ, while vertical and horizontal parallelism are indicated by ∆₁ and ∆₂, respectively.

Based on these geometric relationships, the equations for calculating motion accuracy are derived as follows:

$${\mathrm{s}}_1={\mathrm{H}}_1+{\mathrm{S}}_1$$

(1)

$${\mathrm{s}}_2={\mathrm{H}}_2+{\mathrm{S}}_2$$

(2)

$${\mathrm{s}}_3={\mathrm{H}}_3+{\mathrm{S}}_3$$

(3)

$$\mathsf{\theta }={\tan }^{-1}\left({\displaystyle \frac{{\mathrm{s}}_2-{\mathrm{s}}_1}{{\mathrm{L}}_1}}\right)$$

(4)

$${\mathrm{h}}_1={\displaystyle \frac{{\mathrm{s}}_2+{\mathrm{s}}_1}{2}}\cos \mathsf{\theta }$$

(5)

$${∆}_1={\mathrm{h}}_{1\max }-{\mathrm{h}}_{1\min }$$

(6)

$${\mathrm{h}}_2{=\mathrm{s}}_3\cos \mathsf{\theta }-{\left({{\mathrm{L}}_4}^2+{\left({\mathrm{L}}_2-{\displaystyle \frac{{\mathrm{L}}_3}{2}}\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}\sin {(\tan }^{-1}{\displaystyle \frac{{\mathrm{L}}_4}{\left({\mathrm{L}}_2-\frac{{\mathrm{L}}_3}{2}\right)}}-\mathsf{\theta })$$

(7)

$${\mathit{\Delta }}_2=h_{2\mathit{max}}-h_{2\mathit{min}}$$

(8)

Equations (1)–(3) illustrate the actual distances from each measurement point to the reference surface. Equation (4) is used to calculate the tilt angle. Additionally, Equation (5) indicates the actual distance from the center A of the carriage’s upper surface to reference surface a. Equation (6) defines the vertical parallelism to indicate the difference between the maximum and minimum values across all sampling points. Moreover, the actual distance from the center B of the carriage’s side surface to reference surface b is specified by Equation (7). Finally, Equation (8) describes the horizontal parallelism to capture the difference between the maximum and minimum values across all sampling points.

2.2. Measurement Principle of PDF

To eliminate reverse backlash and enhance the dynamic rigidity and reliability of LRGs, various degrees of preload forces must be applied across most applications. The preload force is considered an internal force within the LRG, primarily determined by the interference of the rolling elements. The PDF is the force required to overcome the preload force, enabling the carriage to maintain a constant speed relative to the rail [14].

The measurement of the PDF is based on the fundamental principles of Newton’s First Law. As demonstrated in Figure 3, an S-type weighting sensor is employed to measure it. When the driving force F pulls the carriage from left to right, the S-type weighting sensor experiences a pulling force. In addition to its own weight G, the carriage is subject to the supporting force F_N from the rail, the external driving force F, and the frictional force F_f acting opposite direction of motion along the rail raceway. The frictional force F_f represents the PDF. The weight and supporting force are equal in magnitude and opposite in direction. When the carriage moves uniformly along the rail, the magnitude of the driving force matches that of the PDF. Therefore, measuring the driving force using the S-type weighting sensor allows for an indirect measurement of the PDF. The same principle applies when the driving force pushes the carriage to the left.

2.3. Test Principle of Fatigue Life

The rated lifespan of an LRG refers to the expected life of a single unit or batch of identical LRGs under consistent working conditions. It utilizes standard materials and adheres to manufacturing quality standards with a 90% reliability rate [15]. The criterion for determining the rated lifespan of the LRG is based on fatigue failure, which is identified by pitting or spalling on the raceway or rolling elements, commonly known as fatigue life. The basis for failure determination is that the spalling depth is ≥0.05 mm, the spalling area is ≥0.5 mm² for ball LRGs, and ≥1.0 mm² for roller LRGs [16]. The equation for the fatigue life calculation of the LRG is as follows [17]:

$$L_{\mathit{ar}}=L_f\times {\left({\displaystyle \frac{P_f}{P}}\right)}^n$$

(9)

where L_ar is the rated life. L_f is the test life. P_f is the test dynamic equivalent load. P is the equivalent dynamic load under normal working conditions. n is the exponent, n = 3 for ball LRGs, and n = 10/3 for roller LRGs.

3. Composition of the Test Bench

3.1. Mechanical Structure Design

3.1.1. Composition of Overall Structure

The overall structural composition(Pangong Industry, Ningbo, China) of the test bench is shown in Figure 4. The lower bed (1) acts as the foundation, with the dragging component (5) mounted on its upper surface. The upper bed (2) is connected to both the lower bed and the dragging component via an LRG and a lead screw. Two sets of measurement setups (4) are symmetrically installed on the sides of the upper bed. The loading assembly (3) spans across the lower bed and is securely connected to both sides, interfacing with the measurement setups of the test carriage through an S-type weighting sensor.

The lower and upper bed support and install other components. The main LRGs (7) are used to carry and guide the movement of the upper bed. The auxiliary LRG (14) enhances the lateral freedom of the installation platform (11). When the tested LRG is subjected to force, the slight movement of the auxiliary LRGs prevents bending deformation of the installation platform, ensuring that the loads are applied to both tested LRGs equally. During the comprehensive performance tests, loading is unnecessary. In this case, the four clamps (8) are locked to secure the lateral stability of the installation platform, thereby ensuring measurement accuracy.

The dragging and loading components facilitate the reciprocating motion of the tested LRGs under load conditions. In this setup, the carriage remains stationary while the rail moves, hence achieving relative reciprocation to the LRG. The baffle plate (13) and hydraulic cylinder (9) function as the force loading mechanism, employing Newton’s first law to subject the rails to action and reaction forces. The spoke-type force sensor (10) measures force, displaying values and providing feedback for the control system to regulate force on the LRGs. The ball screw (16) is selected as the dragging mechanism for LRG due to its high transmission efficiency, smooth operation, and substantial load capacity. The servo motor is directly connected to the ball screw using a coupling.

The test bench facilitates measurements of comprehensive performance parameters and fatigue life tests. As summarized in

Table 1. Test bench function table.

Test Type	Test Parameters	Yes/No Loading	Yes/No Dragging
Comprehensive performance test	Motion accuracy	No	Yes
	PDF
	Vibration
	Temperature rise
Fatigue life test	Pitting/spalling size	Yes	Yes
	Vibration
	Temperature rise

, comprehensive performance assessments require a no-load running-in procedure, allowing for the evaluation of running accuracy, PDF, vibration, and temperature rise of the LRG. For fatigue life testing, a loaded running-in procedure is necessary with simultaneous monitoring of vibration and temperature rise. Abnormalities in these metrics prompt an examination of the LRG’s raceway morphology, with pitting and spalling size to determine the end of its service life.

Table 2. Main design parameters of the test bench.

Design Parameters	Value
Overall dimension (m)	3.14 × 1.42 × 1.22
Weight (kg)	5113
Maximum test speed (m/s)	0.83
Maximum acceleration (m/s2)	8
Maximum stroke (m)	0.6
Maximum force (kN)	300
Motion accuracy (mm/m)	0.01/1000
Speed/force accuracy (%)	2%

presents the main design parameters of the test bench, which are designed to meet the requirements for an accelerated life test.

3.1.2. Measurement Setup Structural Design

Figure 5 illustrates the measurement setup. This component is installed on the installation platform via an adapter plate (8). The measuring frame (7) is fitted onto the loading block (3). An S-type weighting sensor (2) is used to measure the PDF, affixed to the connecting plate on the gantry side plate, ensuring the carriage of tested LRGs (1) remains stationary as the rail moves with the upper bed. Two acceleration vibration sensors (5) are positioned vertically and horizontally through magnetic attachment to capture vibration signals in both directions. A patch-type thermocouple (4) is affixed to the front end of the carriage due to the fact that the direction change and collision of rolling elements with the diverter generate considerable heat. Additionally, three contact displacement sensors (6) are installed in accordance with the measurement principle of motion accuracy.

3.2. Measurement and Control System Design

The measurement and control system of the test bench comprises an electronic control system and a measurement system. The electronic control system primarily manages the closed-loop control of the hydraulic cylinder’s output force and the ball screw’s rotational speed during the fatigue life test. The measurement system handles the acquisition, display, and storage of output signals from the contact displacement sensors, S-type weighting sensors, acceleration vibration sensors, and thermocouples for comprehensive performance parameter measurement. Additionally, signals from the acceleration vibration sensors(Lance, Qinhuangdao, China) and thermocouples(Dikewei, Yangzhou, China) can detect the fatigue failure of the tested LRG during the fatigue life test.

The electronic control system utilizes a programmable logic controller(PLC) as the central controller. The humanmachine interface (HMI) provides the target signals, while outputs from the spoke-type force sensors and encoder function as feedback signals. Using the proportional-integral-derivative (PID) function module of the PLC(Haitian Drive Co., Ltd., Ningbo, China), the trial-and-error method is applied to determine PID parameters. The PLC compares the target and feedback signals to generate error signals, which are transmitted to the servo motor drivers. Moreover, the driver subsequently controls the servo motor to achieve precise regulation of the target value. The electronic control principle is demonstrated in Figure 6.

The measurement system is implemented using LabView 2022 associated with relevant hardware for data acquisition. Figure 7 illustrates the hardware communication structure of the bench measurement system. The contact-type(KEYENCE, Osaka, Japan) displacement sensors convert internal changes in motion accuracy into electrical signals, which are transmitted to the industrial computer via RS232 serial communication through amplifiers and communication modules. The S-type weighting sensors(Yiceli, Wuxi, China) detect changes in external force, generating resistance signals which are converted into digital signals by signal amplifiers and displayed on measurement software via RS232 serial communication. Vibration signals from the acceleration sensors are captured by the industrial computer through a signal acquisition card. Additionally, the thermocouples generate the millivolt voltage signals from frictional heat between rolling elements and raceways of the tested LRG into a standard analog signal via temperature transmitters before entering the industrial computer. Each signal is processed, displayed, and stored by the acquisition program after entering the industrial computer.

4. Finite Element Analysis of the Test Bench

4.1. Static Analysis of the Gantry

During the lifespan test, the hydraulic cylinder exerts a vertical force on the tested LRG, while the gantry side plates absorb the reaction force and provide support. Under high force conditions, the gantry side plates, upper plate, and fixing screws will experience certain stress and strain. Therefore, a static analysis of the gantry is essential to verify the rationality of the structural design and the strength of the materials used.

For the convenience calculation, Figure 8a presents a simplified diagram of the structure, where the base is reduced to two fixed plates connected to the gantry side plates.

The 3D model of the gantry was imported into ANSYS Workbench 19.0. The material properties were assigned as follows: the gantry was set to grade 45 steel, the base was set to gray cast iron, and the bolts were grade 12.9, which are set as alloy steel materials. The mesh generation tool was utilized to create the mesh, with refinement achieved by adjusting the transition and center of the span angle [18]. The completed mesh consisted of 28,949 elements and 53,030 nodes, as depicted in Figure 8b. A bolt preload force of 281,000 N was applied, and the maximum supporting force of the gantry was specified as 300 kN, based on the maximum load being 300 kN. Subsequently, a static analysis was conducted, and the results are presented in Figure 9.

According to the analysis results shown in Figure 9, the maximum strain occurs at the contact surface between the left gantry side plate and the baffle plate. This position takes a relatively large supporting force, and the side plate has a relatively thin wall thickness. However, the maximum strain of 0.45 mm indicates a small deformation, which has a minimal impact on the loading of the tested LRG and can be neglected. The maximum stress is found at the first row of bolts on the right side plate of the gantry. This is attributed to the bolts being the connection point between the right side plate of the gantry and the base, as well as their proximity to the support force location, resulting in significant axial tensile forces.

4.2. Modal Analysis of the Upper Bed

4.2.1. The Upper Bed Modal Finite Element Simulation

During the operation of the upper bed, the interaction with the main and auxiliary LRGs reduces its overall stiffness. Additionally, design and manufacturing errors contribute to vibrations. Therefore, external excitations can lead to local deformations, compromising the reliability of the test bench and test accuracy. Modal analysis can identify the inherent frequency and deformation-prone areas of the upper bed to validate the design’s rationality.

The simplified upper bed model is imported into ANSYS Workbench 19, where the material properties were specified. The installation platform was designated as ductile cast iron, while the LRG and motion plate were defined as grade 45 steel. The connecting surfaces between the various components of the upper bed were primarily linked by bolts; hence, they were defined as bonded contacts. A spring element is introduced at the LRG’s sliding joint to account for its impact on dynamic performance [19]. The normal stiffness is set to 305,000 N/mm for the mounting platform and auxiliary LRG, and 410,000 N/mm for the moving plate and main LRG. The model is meshed with 59,731 elements and 117,459 nodes, as depicted in Figure 10b.

Constraints were imposed on the upper bed, and the first six vibration modes were calculated. The results, depicted in Figure 11, reveal natural frequencies of 267.9, 318.4, 369.9, 421.9, 433.5, and 460.6 Hz, respectively.

In modal analysis, it is commonly accepted that the highest vibration energy is most pronounced in the lower-order modes [20]. As observed in Figure 11, the upper bed experiences significant vibrational deformation at the first-order vibrational mode. The calculated operating excitation frequency of the upper bed at maximum rotational speed is as follows [21]:

f = n/60

(10)

The motor’s rated speed n is 2000 r/ min, and substituting it into Equation (10) yields an excitation frequency of 33.3 Hz. The inherent frequency of the upper bed at the first-order vibrational mode is 267.9 Hz. Consequently, the operating vibrational frequency of the upper bed at maximum speed is approximately 13% of the first-order vibrational mode. Furthermore, since the inherent vibrational frequency and the operational excitation frequency do not fall within the same frequency range, the design of the upper bed meets the safety requirements.

4.2.2. The Upper Bed Hammer Impact Modal Test

To validate the simplification of the upper bed model and the finite element analysis results, a hammer impact modal test was designed. Modal analysis primarily involves measuring the system’s excitation and response signals, performing frequency response analysis on the measured signals, and identifying relevant dynamic parameters, such as inherent frequencies [22].

The results of the finite element analysis indicate that the amplitude positions of the first two modal shapes appear at the center of the side of the moving plate. In the hammer impact modal test, low-order modes are easily excited, so the accelerometers are placed at the center of the side of the moving plate, where the amplitude is the largest, to verify the low-order modes of the upper bed. Figure 12 shows the distribution of excitation and vibration points.

A hammer is used to strike each of the 20 excitation points twice, with each strike consisting of three impacts. The striking force and time intervals are kept uniform. The resulting force and vibration signals are collected and imported into a computer via a data acquisition card. Using a LabVIEW program on the computer, the force and vibration signals are captured and stored. Taking excitation point 1 as an example, the hammer strike force curve is shown in Figure 13a, and the acceleration sensor’s time-domain response curve is shown in Figure 13b. The saved files are imported into MATLAB 2022a for frequency response function (FRF) analysis to obtain the modal test results. The calculated FRF curve is shown in Figure 13c.

The results show that the first two natural frequencies of the upper bed are 271.4 Hz and 312.3 Hz, which are in good agreement with the finite element results. Further analysis of the signals at the other excitation points reveals that the first inherent frequency of the system is within the range of 250–290 Hz, and the second inherent frequency is within the range of 290–330 Hz, with an error rate within 2%. These errors may be caused by system errors in the acquisition system, etc., and are considered within the normal range. The results confirm the rationality of the simplification of the upper bed model and the effectiveness of the finite element analysis.

4.3. Comparative Analysis of the Load Uniformity of the Tested LRG

In practice, LRGs typically experience relatively uniform forces [23]. To accurately simulate loading conditions, it is essential to consider the impact of loading block thickness on the uniformity of the forces applied. Empirical observations suggest that a thicker loading block results in a more uniform distribution of force across the upper surface of the tested LRGs. However, excessive thickness can increase the spatial requirements of the testing platform, and an overly heavy loading block may adversely affect both the motion accuracy and the accuracy of the PDF measurements. Therefore, selecting an appropriate loading block thickness is critical.

Three different loading block thicknesses were established for this study, corresponding to the thicknesses of the tested carriage at 1, 1.5, and 2 times, specifically, 47 mm, 70.5 mm, and 94 mm, respectively, as illustrated in Figure 14.

Simulation analysis of the three models was conducted using ANSYS Workbench 19.0. The contact area between the loading front and the loading block was defined to establish the load application surface, with a loading force set at 5 tons, which corresponds to the maximum load of the 35 series LRG.

As can be seen from Figure 15, the uniformity of stress on the carriage’s upper surface significantly improves when the loading block thickness is increased to 70.5 mm compared to 47 mm. Although the area of lower stress regions at the edges slightly decreases when the loading block thickness is further increased to 94 mm, the overall improvement in stress uniformity is minimal. A similar conclusion can be drawn from the strain distribution cloud diagrams shown in Figure 16 for the different loading block thicknesses. Considering the impact of loading block thickness on the uniformity of the stress on the carriage surface, as well as the need to conserve space and material, a loading block thickness of 70.5 mm, which is 1.5 times the carriage thickness, is selected.

5. Experimental Verification

This experimental verification contains the assessment of the motion accuracy, PDF, and accuracy of the fatigue life testing of the multi-parameter LRG. The bench consists of a main body, hydraulic system, and electric control cabinet, as illustrated in Figure 17. Prior to experimentation, both software and hardware are rigorously tested to ensure the proper functioning of keys, display screens, test software, and sensors. Additionally, the overall structure of the test bench underwent vibration and durability testing to verify its stability under various environmental and operational conditions. Repeatability tests were conducted under identical experimental conditions to validate the repeatability of the testing platform [24].

5.1. Verification of Motion Accuracy

This study evaluates manual measurements of the motion accuracy for LRGs classified as accuracy levels P3 and P4. Both LRGs were of the same model, specifically the LGD35 type. The test length of the LRG is 1500 mm, the operating speed is 0.33 m/s, and the experimental acceleration is set to 1 m/s². Three contact displacement sensors were installed appropriately for these measurements. Figure 18 presents the measurement data from the three sensors for both accuracy grades. Due to the arbitrary initial positions of the displacement sensors being set to random values, the measured data exhibited notable discrepancies in displacement value.

Figure 18a illustrates the displacement data for the P3 accuracy LRG. Sensors 1 and 2 are used to measure vertical parallelism, while Sensor 3 measures horizontal parallelism. The data from Sensors 1 and 2 exhibit a similar trend, with minor variations in parallelism at both ends and greater variations in the middle section. This indicates that the vertical parallelism of the middle section of the LRG is inferior to that of the ends. The data from Sensor 3 show considerable overall volatility, suggesting a relatively uniform trend of horizontal parallelism across the entire guide. Figure 18b displays the operational data for the P4 accuracy LRG. Similar to the previous data, Sensors 1 and 2 display comparable trends with small variations in parallelism at both ends and larger variations in the middle. This also indicates that the vertical parallelism in the middle section of this LRG is poorer compared to that at the end. The data from Sensor 3 once again display significant variability, reflecting a relatively uniform trend of horizontal parallelism throughout the entire guide.

By substituting the sensor data into Equations (1)–(8), the vertical parallelism, horizontal parallelism, and tilt angle for both the P3 and P4 accuracy LRGs can be calculated. These calculated values were then compared with the manually measured vertical and horizontal parallelism values, as shown in

Table 3. Comparison table of motion accuracy measurement results.

Accuracy Level	Manual Measurement (mm)		Test Bench Measurement (mm)		Repeatability	Yes/No Conform to the Standard
P3	Vertical parallelism	0.012	Vertical parallelism	0.01362	86.5%	Yes
	Horizontal parallelism	0.01	Horizontal parallelism	0.01165	83.8%	Yes
	Tilt angle	—	Tilt angle	4.68″	—	Yes
P4	Vertical parallelism	0.021	Vertical parallelism	0.01834	87.3%	Yes
	Horizontal parallelism	0.018	Horizontal parallelism	0.01588	88.2%	Yes
	Tilt angle	—	Tilt angle	12.24″	—	Yes

. The measurement repeatability for the vertical and horizontal parallelism of the P3-grade LRG was found to be 85.8% and 87.9%, respectively, while the P4-grade LRG exhibited measurement repeatability of 85.7% for vertical parallelism and 85.8% for horizontal parallelism. These repeatabilities are notably high, and all measurement results conform to the acceptance criteria specified by industry standards, thereby validating both the motion accuracy measurement methods and the derived equations.

5.2. Verification of PDF

This study measures the manual PDFs of three LGD35BH LRGs at light, medium, and heavy preload levels. The test bench is utilized to analyze PDFs under non-lubricated conditions. The testing length of the guide was set at 1500 mm, with an operational speed of 0.33 m/s. To enhance the length of the uniform speed measurement phase, the experimental acceleration was set to 6 m/s². An S-type weighting sensor is accurately installed. To verify the repeatability of the test bench in measuring the PDF, sensor data from several running-in processes were collected. Figure 19 presents the drag force data for the three preload levels.

As shown in Figure 19, during the tensile phase of the sensor, the data present negative values, while during the compression phase, the data show positive values. Notably, the values recorded during the tensile phase are generally slightly lower than those during the compression phase. This phenomenon is attributed to the grinding marks created during the machining process of the LRG, resulting in a minor difference in the coefficient of friction between the two directions, which ultimately affects the magnitude of the PDF in each direction. The minor fluctuations in the tensile and compressive measurement data depicted in the figure are due to the accuracy of the sensor. However, these variations are minimal and can be considered negligible.

The data from the sensors were taken exclusively from the tensile phase, and the average values were calculated to represent the PDF of the LRG.

Table 4. Comparison table of PDF measurement results.

Model	Standard Requirements (N)	Manual Measurement (N)		Test Bench Measurement (N)		Repeatability
LGD35	9.4 ± 1.5	Light preload level	10	Light preload level	9.8	98%
	13.9 ± 1.8	Medium preload level	15	Medium preload level	14.9	99.3%
	21.3 ± 1.9	Heavy preload level	20	Heavy preload level	19.6	98%

presents the measured PDF values obtained from the test bench for the three preload levels that conform to the standard requirements. Furthermore, when compared to manual measurements, the repeatability for light, medium, and heavy preload was found to be 98%, 99.3%, and 98%, respectively. This confirms that the test bench can accurately measure the PDF values of LRGs under different preload levels.

5.3. Verification of Fatigue Life

To verify the accuracy of the multi-parameter test bench in simulating the operating conditions of the LRG and measuring its fatigue life, a comparative analysis was performed between the actual experimental lifespan and the theoretical lifespan.

First, the theoretical calculation for the fatigue life of the LRG was conducted, using the basic equation for the rated lifespan of the LRG as provided in the following [25]:

$$\mathrm{L}={\left({\displaystyle \frac{f_h{f}_t{f}_c{f}_a}{f_{\omega }}}·{\displaystyle \frac{C}{F}}\right)}^3\times 50$$

(11)

where $L$ represents the rated life. $C$ represents the rated dynamic load. $F$ represents the test equivalent load. $f_h$ represents the hardness coefficient. When the raceway hardness is not less than HRC58, $f_h$ is taken as 1. $f_t$ represents the temperature coefficient. In this study, the test temperature is 25 °C ≤ 100 °C, and $f_t$ is taken as 1. $f_c$ represents the lubrication coefficient. There is one carriage on the tested LRG, and $f_c$ is taken as 1. $f_a$ indicates the accuracy coefficient. The tested LRG is classified as P3 accuracy and $f_a$ is taken as 1. Finally, $f_{\omega }$ represents the impact load coefficient. Under the working conditions of negligible vibrations with a medium speed range of 0.25 m/s ≤ v ≤ 1 m/s, $f_{\omega }$ is taken as 1.75.

Section 5.1.3 of the international standard “ISO 14728-1 Rolling bearings—Linear rolling bearings—Dynamic load ratings and rated life” [26] states the calculation equation for the basic dynamic load rating C of the linear rolling LRG as follows:

$$\begin{gathered} C=b_m·f_c·i^{0.7}·{d_w}^{2.1}·{l_t}^{{\displaystyle \frac{1}{30}}}·{Z_t}^{{\displaystyle \frac{2}{3}}}·\cos \alpha \\ {f}_c=24.5·\epsilon ·{{\displaystyle \frac{2\times r_g}{2\times r_g-d_w}}}^{0.41} \end{gathered}$$

(12)

where $C$ represents the rated dynamic load, $Z_t$ demotes the number of steel balls within the effective contact length, $i$ is the number of rows of rolling elements, $\alpha$ is the contact angle, $d_w$ is the ball diameter, $r_g$ is the raceway radius, and $l_t$ is the effective length of the raceway during the calculation process. The specific model of the LRG used for the fatigue life verification test in this study is LGD35BH1PA3L1500IYW, which is a 35-series ball rectangular guide with an accuracy grade of P3 and light preload. The accuracy grade is P3, and the preload level is light. By obtaining the corresponding geometric parameters for this model and substituting them into Equation (12), the rated dynamic load of the LRG is calculated to be 49,470.4 N.

Substituting the above parameters into Equation (11), the theoretical calculation equation for the fatigue life of the LRG of LGD35BH can be derived as follows:

$$\mathrm{L}={\left({\displaystyle \frac{28268.8}{F}}\right)}^3\times 50$$

(13)

The actual fatigue life test was conducted using three randomly selected LRGs from a sample batch, installed on the multi-parameter test bench. The test maintained constant loading force and operating speed, with forces set at 15,000 N, 20,000 N, and 25,000 N, and each less than half the rated dynamic load. The test bench used a vertical loading method, hence he equivalent load during the experiment was equivalent to the applied loading force. The operation speed and acceleration for operational stability were set at 0.5 m/s and 1 m/s², respectively. Pitting and spalling were taken every 1 km after detecting abnormal vibration or temperature rises, continuing until spalling met fatigue failure criteria. Figure 20 illustrates the fatigue failure of the carriage and rail.

The fatigue lives of three LRGs were determined under three distinct loading force conditions. Theoretical fatigue lifetimes for the same three LRGs were also calculated. The measurement repeatability of the test bench in assessing the fatigue lives of the LRG was subsequently evaluated. As shown in

Table 5. Comparison table of fatigue life measurement results.

Model	Loading Force (N)	Theoretical Life (km)	Test Life (km)	Repeatability
LGD35	15,000	334.46	302.41	90.4%
	20,000	141.19	150.19	93.5%
	25,000	72.19	79.66	89.7%

, the comparison of the tested and theoretical fatigue life results indicates that the test bench exhibits a high level of repeatability in measuring the fatigue life of the LRG.

Figure 21 illustrates a relationship between the theoretical life of the LRG and the test equivalent load, as derived from Equation (13). The curve displays an overall descending trend, indicating that as the experimental equivalent load increases, the theoretical lifespan decreases. The hexagram markers represent the experimentally measured fatigue life, which aligns closely with theoretical predictions. This concordance confirms the test bench’s repeatability in simulating the operating conditions of the LRG and in measuring fatigue life.

6. Conclusions

This study introduces the experimental principles for evaluating the motion accuracy, PDF, and fatigue life of LRGs. Based on these principles, a multi-parameter test bench for LRGs was designed, including its mechanical structure and measurement-control system. A static analysis of the gantry and a modal analysis of the upper bed were carried out, and the simulation results were validated using an impact hammer test. Furthermore, the influence of the loading block thickness on the force uniformity of the tested LRGs was analyzed, thereby verifying the rationality of the test bench’s mechanical structure design. Finally, comparative verification experiments were conducted, where the measured values of motion accuracy and PDF were compared with manually measured values, and the actual fatigue life was compared with the theoretical life. The comparison results demonstrate that the test bench achieves high repeatability in measuring motion accuracy, PDF, and fatigue life of LRGs, and verify the correctness of the derived formula for motion accuracy. The development of this test bench addresses the problem of efficiently conducting multi-parameter testing of comprehensive performance and fatigue life on the same test bench and provides a targeted solution for measuring the comprehensive performance and verifying the fatigue life of LRGs.