This study was aimed at investigating the dynamic behaviors of the linear driven feeding stage by means of the analytical and finite element modeling approaches. To assess the dynamic characteristics of the stages with different linear guide arrangements, the finite element model of the stages was created, in which the linear components with rolling interface were accurately modeled based on the Hertzian theory. On the other hand, the analytically mathematical model was derived to determine how the linear guide arrangement affects the dynamic characteristics of the stage. Results of the modal analysis show that the vibration behaviors of the positioning stage are dominated by the rigidity of the linear components and the platform. In addition, comparisons of the results from the two approaches further indicate that the platform rigidity is an important factor determining the accuracy of the prediction of the vibration frequencies by the analytically mathematical model. As a conclusion of the study, the analytically mathematical model can approximate well to the finite element model when the linear stage is designed with appropriate structure rigidity.
In recent years, high precision positioning systems constructed of various linear rolling components have been widely used in the field of CNC machine tool, semiconductor manufacturing equipment, or inspection apparatus. In these linear feeding stages, there exist rolling interfaces between the rolling balls and the raceways of the linear components. Essentially, the bearing stiffness of the rolling elements is determined by the preloaded state of the rolling balls and has been shown to have significant influences on the dynamic characteristics of the stage [
Evaluation and optimization of a machining system with different configurations are complex processes in the design phase. Therefore, virtual prototyping technology is considered as a promising approach to avoid the traditionally time and costconsuming development process [
In finite element method, the machine structure with linear components can also be appropriately modeled with sufficient accuracy by introducing the modeling schemes of the rolling elements into the analysis model [
This study was aimed at modeling the dynamic behaviors of the linear driven positioning or feeding stages by means of the finite element analysis and analytical modeling approaches. For this purpose, different finite element models of the stages with different linear components were created for modal analysis. In order for the analysis to be realistic, the interface characteristics existing between the rolling elements and the ball grooves or raceways were accurately described based on Hertzian contact theory. Then, the analytical model was proposed to serve as an alternative tool to rapidly evaluate the influence of the design change or the arrangement of the linear components on the dynamic characteristics of a positioning stage. The applicability of analytical model was validated by the simulation results of finite element modeling of the stages designed with different structure properties.
A typical feed drive servomechanism for precision positioning, such as those found in machine tools, is usually constructed of machine base, planar platform or table, and linear driven mechanisms. The platform or table is supported by the carriages or slide blocks on linear guide pairs and driven with ball screw/nut mechanism, as shown in Figure
(a) A typical feed drive servomechanism for precision positioning; (b) a simplified small scaled stage.
For linear components, such as ball screw/nut and linear guides, the rolling interfaces between rolling balls and grooves or raceways primarily contribute to the structural dynamic characteristics of the feeding stage [
Schematic of contact geometry of a rolling guide with four rows of rolling balls, forming a twopoint contact mode within each ball groove. The contact mode can be regarded as a type of Hertzian contact mode. In the figure,
Schematic of contact geometry of a rolling guide
Modeling of linear rolling guide
In general, the contact force between a rolling ball and the raceway can be related to the local deformation at the contact point by the Hertzian expression [
In the above equations,
As revealed in (
The finite element model of the feeding stages with driven ball screw mechanism was created for modal analysis, as shown in Figure
(a) Finite element model of the positioning stage; (b) modeling of ball screw and ball bearings.
The stage model with driven mechanism was meshed with brick elements of 23242 elements and 31105 nodes. Based on the main specifications of the linear guide modules (ball diameter
In a similar way, the contact mode of the rolling balls between ball screw and ball nut was simplified as circular contact pairs with the introduction of spring elements at the rolling interface. The overall stiffness was estimated to be 152 N/
For finite element analysis, the material used for all components was carbon steel with an elastic modulus
In mathematical modeling, the stage system was considered as the mass spring system with multiple degrees of freedom. The planar platform was modeled as a rigid plate supported by various spring elements simulating the different linear components. The coordinate system for vibration motions of the stage is illustrated in Figure
Schematic of the dynamic structural model of the ball screw driven stage with linear rolling guides.
For such a spring mass system, we can define the vibration mode associated with the motion degrees of freedom. As shown in Figure
The linear guide modulus supporting the platform was simulated by linear springs with adequate stiffness that was determined by their preload condition based on the Hertzian contact theory, as discussed in previous section. The rigidity of the linear guide modulus is governed by the contact stiffness of the rolling interface between the rolling ball and raceway because the guide rail and carriage block are more rigid in structure. Therefore, by neglecting rolling balls and rail, each guide module was modeled by two spring elements, acting along the direction of the contact angle at each ball groove, respectively. The corresponding contact stiffness was, respectively, quantified with the stiffness
The governing equations of the five degrees of freedom model were derived in terms of the application of Lagrange’s approach to the potential energy of the platformspring system. Detailed derivations were listed in the appendix. The natural frequencies of the vibration of the stage platform including the yawing mode (
In (
Parameters of the ball screw driven stage.
Parameter  Value  Unit 

Total stage mass, 
36.866  kg 
Moments of inertia, 
0.45085, 0.5115, 0.95064  Kg m^{2} 
Geometry dimensions, 
140  mm 

83.5  mm 

27.962  mm 

20, 700, 350, 10.5  mm 
Spring constant (linear guide) 
139.2 
N/m 
Spring constant (ball screw) 
0.879 
N/m 
Spring constant (ball screw) 
0.0359 
Nm/rad 
The first five vibration mode shapes of the positioning stage predicted by modal analysis are shown in Figure
The first five vibration mode shapes of the positioning stage.
The fundamental vibration frequencies of linear stage predicted from finite element modal analysis are listed in Table
Modal frequencies of the feeding stage predicted by finite element approach.
Stage model  Vibration mode  

Yawing mode  Pitching mode  Lower rolling mode  Vertical mode  Higher rolling mode  
Guide span (200 mm)  323  459  477  604  677 
Guide span (280 mm)  322  444  563  528  784 


Difference (%)  0.4  3.3  18.0  12.6  11.4 
The natural frequencies associated with the fundamental vibration modes of the stage were calculated by substituting the parameters in Table
Modal frequencies of the feeding stage predicted by analytical approach.
Stage model  Vibration mode  

Yawing mode  Pitching mode  Lower rolling mode  Vertical mode  Higher rolling mode  
Guide span (200 mm)  343  468  517  657  754 
Guide span (280 mm)  341  468  608  657  885 


Difference (%)  0.7  0.0  17.5  0.0  17.3 
The natural frequencies predicted by finite element simulation and analytical calculation are further compared in Figure
Comparisons of the frequencies of the stage predicted by finite element simulation and analytical calculation.
Comparisons of the results give the implication that the inconsistence in predicting the vibration frequency between the analytically mathematical model and the finite element model is determined by the relative rigidity between the platform and the linear component.
Based on the above analysis, we could understand that structure rigidity of stage platform plays an important role in deciding whether the analytical model can approach the finite element model. To realize the applicability of proposed analytical mode, in this section, we created another three stage models by thickening the planar platform from original model by a ratio of 20, 30, and 50%, respectively. On the other hand, since the modal stiffness is one measure of the ability of the structure to resist the deformation under a specific resonance mode vibration of the structure, which also contributes to a part of the overall structure, we further defined the rigidity ratio as the ratio of the lowest modal stiffness of the platform to the bearing stiffness of the linear components of the stage.
With the rigidity ratio, we can effectively determine the structure component that dominates the overall structural behavior of the stages. The rigidity ratio of the four stage models was calculated as 0.41, 1.03, 2.40, and 4.02, respectively. The natural frequencies of these different stages were further estimated by analytical calculation and finite element simulation, respectively. According to the results in Table
Natural frequencies of the stage with different rigidity ratios predicted by analytical calculation and finite element simulation.
Rigidity ratio of stage  Method  Yawing mode  Pitching mode  Lower rolling mode  Vertical mode  Higher rolling mode 

0.41  Analytical analysis  343.2  467.9  517.3  657.3  754.3 
Finite element 
323.3  459.3  477.2  604.1  677.3  
Difference 




 


1.03  Analytical analysis  299.1  403.0  452.7  572.2  659.6 
Finite element 
287.8  401.7  420.1  536.9  634.4  
Difference 




 


2.40  Analytical analysis  245.8  325.8  346.1  469.6  584.8 
Finite element 
240.3  328.1  334.3  452.8  559.2  
Difference 




 


4.02  Analytical analysis  213.6  279.6  286.3  407.7  531.1 
Finite element 
210.0  282.2  277.7  395.5  507.5  
Difference 





To get clear insight into the effect of the rigidity ratio, the natural frequencies predicted by the two methods for different stages are depicted in Figures
Comparison of the natural frequencies of the stages between the finite element simulation (solid line) and the analytical calculation (dotted line). The four stage models (1 to 4) are rated with different structure rigidity ratios as 0.40, 1.03, 2.40, and 4.02, respectively (guide span = 200 mm).
Comparison of the natural frequencies of the stages between the finite element simulation (solid line) and the analytical calculation (dotted line). The four stage models (1 to 4) are rated with different structure rigidity ratios as 0.40, 1.03, 2.40, and 4.02, respectively (guide span = 280 mm).
As observed from analysis results, the analytical model predicted results deviate more from the finite element solutions for stage with the low rigidity ratio, but the differences between them decrease with the increasing in rigidity ratio of the stage. As was defined, a stage with high rigidity ratio means that the platform of the stage is more rigid than the supporting components. If a stage is designed with enough rigidity ratios, the linear components will give more influence on the vibration characteristics of the stage than the structure rigidity of platform. In other words, the influence of the platform structure on the overall structure stiffness of the stage gradually lessens with the increasing rigidity ratio. Such phenomenon has been verified in the analytical models of the sage with a rigidity ratio higher than 2.4, which shows the predicted vibration frequencies approaching those obtained from finite element models.
Basically, in derivation of the analytically mathematical model, the machine structure was assumed as rigid bodies mounted on elastic supports of different types. This makes the analytical model unable to stand for a real machine structure accurately, hence resulting in a greater deviation from the finite element model in predicting the vibration characteristics. However, current analysis results obviously suggest that the analytically mathematical model derived for a linear stage designed with higher rigidity ratio in structure configuration can approximate well with the finite element model. This also gives an implication that the analytical approximation could be an effective tool in modeling the dynamic behavior of a feeding system when it was constructed with adequate rigidity of the structure modulus and linear components.
In this study, the dynamic behaviors of the feeding stages constructed with different arrangements of linear guides were analyzed by means of the finite element analysis and analytical modeling approaches. According to the finite element modal analysis, it is found that the dynamic behaviors of the stage platform are mainly governed by the fundamental vibration modes of the carriage blocks of linear guides and partly accompanied by the elastic deformation of the platform. As realized from this analysis, vibration characteristics are mainly determined by the bearing stiffness of the linear guide modulus and the rigidity of the platform. The dominance of the two properties can be quantified by the rigidity ratio which is defined as the ratio of the modal stiffness of platform to the bearing stiffness of linear guides. A feeding stage with high rigidity ratio means that the linear guide system dominates the dynamic behavior. On the contrary, for a stage with low rigidity ratio, the flexibility of the platform affects the dynamic behavior more than the supporting guide system.
Moreover, comparisons of the results predicted by the two approaches show that the rigidity ratio of the stage platform is an important factor affecting the accuracy in modeling the vibration behavior based on the analytically mathematical model. Conclusions of this study suggest that the analytically mathematical model approximates well to the finite element models in investigating the dynamic behavior of a stage which is designed with appropriate structure rigidity.
Using the notation as described in Section
The potential energy
Applying Lagrange’s approach to (
In (
It is noticed that (
The natural frequency of the vertical vibration of the stage is
Since in (
Finally, the two solutions
The authors declare that there is no conflict of interests regarding the publication of this paper.