# Research on Dynamic Modeling and Application of Kinetic Contact Interface in Machine Tool

Authors: Dan Xu; Zhixin Feng
Journal: Shock and Vibration
 (2016)

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## Abstract

A method is presented which is a kind of combining theoretic analysis and experiment to obtain the equivalent dynamic parameters of linear guideway through four steps in detail. From statics analysis, vibration model analysis, dynamic experiment, and parameter identification, the dynamic modeling of linear guideway is synthetically studied. Based on contact mechanics and elastic mechanics, the mathematic vibration model and the expressions of basic mode frequency are deduced. Then, equivalent stiffness and damping of guideway are obtained in virtue of single-freedom-degree mode fitting method. Moreover, the investigation above is applied in a certain gantry-type machining center; and through comparing with simulation model and experiment results, both availability and correctness are validated.


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## Body

## 1. Introduction

The dynamic analysis and simulation technology of numerical control machine tool are a very important research direction in modern advanced manufacturing technology and equipment technology. In a numerical control machine tool, the linear rolling guide is not only a significant functional component but also an important kinetic contact interface, meaning its characteristic has a direct effect on the machining precision and performance. Therefore, the modeling of contact interface, which takes the dynamic property of the NC machine tool into consideration, is a prerequisite research for establishing the dynamic overall model. Without this, it is impossible to get a practical conclusion.For the dynamic analysis and prediction of machine tool, the modeling of contact interface and accurate identifying of the parameters are the main difficulty. In recent years, the domestic and overseas research of contact interface is basic on three aspects: the mechanism of contact interface, modeling, and parameter identification. Some of the researchers, who are represented by Wen et al. [1, 2], put forward the scale-independent stiffness fractal modeling of the normal and tangential contact. At the same time, they reveal the nonlinear relationship between the contact stiffness and each interface parameter. The researchers, represented by Zhang et al. [3] and Mao et al. [4], build the fundamental property model of contact interface and multinode dynamic model; Dhupia et al. [5] applies the frequency-domain joint part model with weak nonlinear characteristic to the basic machine tool modeling and predicts the processing performance. When it comes to identifying the parameter of the joint part, there are several main ways such as frequency response function identification method [6], response coupling method [7], and contact interface parameter optimization method which is based on finite element modeling [8]. At present, the research of joint part is in the contending stage, and there are numerous problems of the mechanism and modeling to be solved.There are certain difficulties to apply the mechanism model to engineering, because although the fundamental performance model and parameter statistic obtained have a versatility, it should be based on a great amount of experiments.On the basis of the researches before [9–11] and starting from the material and structure aspects, this paper puts forward a combined “analysis-experiment” dynamic modeling method for linear guideway and studies the method of identifying the parameters of guideway contact interface. Furthermore, this paper builds a four-in-one joint part research method of “static stiffness model-vibration model-experimental parameter-parameter identification” and applies this method to the guideway contact part and overall dynamic modeling of the gantry machining center. The research results provide an effective and feasible solution channel and practical method to predict the machine performance in the pattern design phase.


## 2. Modeling of the Rolling Guideway Contact Part

### 2.1. Modeling Method

From the perspective of material parameters and structural distortion and based on the elastic mechanics and vibration mechanics theory, the guideway contact part dynamic modeling method is established from these four aspects, “static stiffness model, vibration model, experimental parameter, and parameter identification.” Figure1 clearly shows the relationship of each part of the method and the parameter that should be provided for next part. First of all, obtain the relationship between the stress and deformation through analyzing the static dynamics while the rolling linear guideway is under the general loading, solve the static stiffness via building the guideway static stiffness model, and provide the input parameter for the guideway vibration model. Secondly, employ the Lagrange method to analyze the vibration property of the linear rolling guideway and solve the basic modal vibration frequency of the vibration model through using analytic method, to provide the input requirement for the dynamic model parameter identification. At last, with the help of the hammer-hitting experiment, the multipoint frequency response function of the guideway is acquired. Combined with the guideway vibration basic modal frequency, the single degree modal fitting method is used to obtain the stiffness and damping value of the guideway equivalent model.Figure 1
Mind map of the guideway dynamic model research method.


### 2.2. Static Stiffness Model

In the condition that the linear rolling guideway is under the load in vertical directionFV, the load in horizontal direction FH, and vertical moment Mγ (rotating around the y axis) (shown in Figure 2), the relation [12] between the stress and deformation is obtained via the mechanical analysis while the linear guideway is under general loading. The equation system is presented as(1)FV=f1δV,δH,γ,FH=f2δV,δH,γ,Mγ=f1δV,δH,γ.Figure 2
Bearing loading and guideway deformation graph.This nonlinear equation system can be solved by numerical computation method.Substitute a series of different external applied loadFV, FH, and Mγ into the algorithm above and then a series of relative displacement δV, δH, and γ under different loading of the guideway can be obtained. Using the computer calculating method, the vertical direction stiffness, horizontal direction stiffness, and the rotation angular rigidity can be, respectively, solved:(2)KV=∂FV∂δVFV,KH=∂FH∂δHFH,Kγ=∂Mγ∂γMγ.On the basis of the computing method above and MATLAB software, taking the Schneeberger MRB35 linear rolling guideway as an example, the parameters of the guideway measure are shown in Table1.Table 1
MRB35 rolling guideway model parameters.
Parameter Items Values The length of the sliderL6 103 mm The diameter of the rollerDa 4.5 mm The length of the rollerL 5.1 mm The mass of the sliderM 2.2 kg The moment of inertia aroundx-axis of the slider Jx 3.927626 × 10−3 kg m2 The moment of inertia aroundy-axis of the slider Jy 5.761594 × 10−3 kg m2 The moment of inertia aroundz-axis of the slider Jz 2.467506 × 10−3 kg m2 The number of the rollersZL 21 The horizontal distance between each rollere 32 mm The vertical distance between loaded point and upper rollersh1 23.5 mm The vertical distance between each rollerh0 10.7 mm Tight deformation amountΔ 0.00072 mm Contact angleβ 45°Figure3 shows the relation between the vertical bearing load and vertical direction deformation while the load is within the bearing capacity. In Figure 3, the line marked with red five-pointed stars represents computing data by theoretical model, and the line marked with blue points represents the experimental data provided by the guideway manufacturer. As is shown in the figure, the relation between the loading and deformation is moderate linear and it can be approximately treated as a linear relation. In the simplified model, it can be replaced by a linear model.Figure 3
The relationship of vertical stress and deformation of MRB35 guideway.


### 2.3. Dynamic Model

Ignoring the mass of the roller, linear rolling guideway vibration model only takes the normal direction stiffness of the roller (the normal direction means the direction that is vertical to the contact interface) and simplifies the roller between the two contact interfaces as a spring that is vertical to the contact interface, whose stiffness isKn. It is shown in Figure 4.Figure 4
Simplified linear rolling guideway model.The solution for the stiffnessKn [9, 10] is as follows.The simplified spring normal stiffness of the single sphere roller is(3)Kn=dFdα=32khα1/2.The simplified spring normal stiffness of the single cylinder roller is(4)Kn=dQdδ=109klL8/9δ1/9orKn=KV4Zsin⁡β,where Kn means the spring normal stiffness which is simplified from the single roller; kh means the roller contact rigidity of the spherical guideway; α means the elastic deformation of the roll ball; kl means the roller contact stiffness of the roller guideway; L means the length of the roller guideway; δ means the elastic deformation of the roller guideway; KV means the vertical stiffness of the linear guideway; Z means the number of the single row roller; β means the contact angle of guideway and groove.Using the Lagrange equation, the dynamic equation of the linear guideway can be detected. Lagrange equation uses the variation of the generalized coordinates to present the virtual displacement of the particle in the particle system. This kind of system is more useful than Newton’s laws of motion while solving some problems (e.g., the small oscillation theory and rigid body dynamics).The Lagrange equation of linear rolling guideway is given in the following:(5)ddt∂T∂qj˙-∂T∂qj=Qjj=1,2,…,k,where T means the kinetic energy of the system; qj means the generalized coordinate of the system; qj˙ means the generalized velocity of the system; Qj means the generalized force which corresponds to the generalized coordinate of the system; k means the number of the generalized coordinates of the system, meaning the free degree of the system.If all the force applied on the particle is potential force, usingV to represent the potential function of the system; the generalized force corresponding to the generalized coordinate qj is(6)Qj=-∂V∂qj.DefiningL=T-V, L is named the Lagrange function or dynamic potential. Then the Lagrange function in potential field can be written as(7)ddt∂L∂qj˙-∂L∂qj=0j=1,2,…,k.The Lagrange function of the linear rolling guideway system shown in Figure4 can be established directly.The overall dynamic energyEk can be expressed as follows.(8)Ek=12Mu˙2+12Mv˙2+12Jxθ˙2+12Jyψ˙2+12Jzϕ˙2.The potential functionEp can be expressed as(9)Ep=Ep1=12Kn∑i=1ZLδ1i2+δ2i2+δ3i2+δ4i2,Ep1=12Kn∑i=1ZLu+h1ϕ+l1iψcos⁡β+v-eϕ-l1iθsin⁡β2+-u-h1ϕ-l2iψcos⁡β+v+eϕ-l2iθsin⁡β2+u+h3ϕ+l3iψcos⁡β+-v+eϕ+l3iθsin⁡β2+-u-h3ϕ-l4iψcos⁡β+-v-eϕ+l4iθsin⁡β2,ZL1=ZL2=ZL3=ZL4=ZL,l1i=l2i=l3i=l4i=ZL-2i+12ZL-1lLi=1,2,…,ZL,h3=h1+h0,where u means the displacement in X direction; v means the displacement in Y direction; θ means the angular displacement around x-axis (pitching motion); ψ means the angular displacement around y-axis (yawing motion); ϕ means angular displacement around z-axis (rolling motion); M means the mass of slider; Jx means the moment of inertia around x-axis of the slider; Jy means the moment of inertia around x-axis of the slider; Jz means the moment of inertia around y-axis of the slider; e means the moment of inertia around z-axis of the slider; h1 means Y direction distance from the center of roller 1 to the origin of coordinate; h0 means Y direction distance from the center of roller 1 to the center of roller 3; β means the angle between the normal direction of the contact interface and the horizontal X direction; Ep1 means the elastic potential energy of the normal direction spring; ZLj (j=1,2,3,4) means the number of the rollers in theith raceway; lji (j=1,2,3,4) means the z-axis coordinate of theith roller injth raceway as is shown in Figure 5; the angle between Y′ axis and y-axis is (π/2-β); lL means the length of single raceway roller.Figure 5
The position of the roller inY′Z plane.Establishing the linear guideway system dynamic function,(10)Mu¨+4ZLKncos2βu+2ZLKnh1+h3cos2βϕ=0,(11)Mv¨+4ZLKnsin2βv=0,(12)Jxθ¨+Kn∑i=1ZLl1i2+l2i2+l3i2+l4i2sin2βθ=0,(13)Jyψ¨+Kn∑i=1ZLl1i2+l2i2+l3i2+l4i2cos2βψ=0,(14)Jzϕ¨+2ZLKnh1+h3cos2βu+2ZLKnh12+h32cos2β+4ZLKne2sin2β+4ZLKnh0esin⁡βcos⁡βϕ=0.Detected from (11) the vibration frequency of the direction that vertical to y-axis of the guideway can be obtained:(15)fV=1πZLKnsin2βM.Detected from (12), the vibration frequency of the pitching motion around x-axis of the guideway can be obtained:(16)fXP=12πKn∑i=1ZLl1i2+l2i2+l3i2+l4i2sin2βJx.Detected from (4), the vibration frequency of the yawing motion around y-axis of the guideway can be obtained:(17)fYY=12πKn∑i=1ZLl1i2+l2i2+l3i2+l4i2cos2βJy.The displacement alongX direction u and the angular displacement around z-axis couple together so (10) and (14) are simultaneously solved to get the vibration frequency.(18)Mu¨+4ZLKncos2βu+2ZLKnh1+h3cos2βϕ=0,Jzϕ¨+2ZLKnh1+h3cos2βu+2ZLKnh12+h32cos2β+4ZLKne2sin2β+4ZLKnh0esin⁡βcos⁡βϕ=0.Suppose u=Uejωt and ϕ=Φejωt, while ω represents the frequency and j=-1.Then the simultaneous equation above can be written as follows.(19)c1-Mω2c2c2c3-Jzω2UΦ=0.In this expression,(20)c1=4ZLKncos2β,c2=2ZLKnh1+h3cos2β,c3=2ZLKnh12+h32cos2β+4ZLKne2sin2β+4ZLKnh0esin⁡βcos⁡β.Solve (19):(21)MJzω4-c3M+c1Jzω2+c1c3-c22=0.Therefore,(22)ω1,22=c3M+c1Jz∓c3M+c1Jz2-4MJzc1c3-c222MJz.The displacement alongX direction u and the angular displacement around z-axis couple together so we suppose that the frequency obtained is low order rolling frequency fRL and high order rolling frequency fRH:(23)fRL=ω12π,fRH=ω22π.The solution of the vibration model above is the vibration frequency of the linear rolling guideway.The parameters of the experimental guideway Schneeberger MRB35-V2 are displayed in Table1. The result of the vertical moving frequency, which is computed by the analytical method from the vibration model, is fV = 3529 Hz.


## 2.1. Modeling Method

From the perspective of material parameters and structural distortion and based on the elastic mechanics and vibration mechanics theory, the guideway contact part dynamic modeling method is established from these four aspects, “static stiffness model, vibration model, experimental parameter, and parameter identification.” Figure1 clearly shows the relationship of each part of the method and the parameter that should be provided for next part. First of all, obtain the relationship between the stress and deformation through analyzing the static dynamics while the rolling linear guideway is under the general loading, solve the static stiffness via building the guideway static stiffness model, and provide the input parameter for the guideway vibration model. Secondly, employ the Lagrange method to analyze the vibration property of the linear rolling guideway and solve the basic modal vibration frequency of the vibration model through using analytic method, to provide the input requirement for the dynamic model parameter identification. At last, with the help of the hammer-hitting experiment, the multipoint frequency response function of the guideway is acquired. Combined with the guideway vibration basic modal frequency, the single degree modal fitting method is used to obtain the stiffness and damping value of the guideway equivalent model.Figure 1
Mind map of the guideway dynamic model research method.


## 2.2. Static Stiffness Model

In the condition that the linear rolling guideway is under the load in vertical directionFV, the load in horizontal direction FH, and vertical moment Mγ (rotating around the y axis) (shown in Figure 2), the relation [12] between the stress and deformation is obtained via the mechanical analysis while the linear guideway is under general loading. The equation system is presented as(1)FV=f1δV,δH,γ,FH=f2δV,δH,γ,Mγ=f1δV,δH,γ.Figure 2
Bearing loading and guideway deformation graph.This nonlinear equation system can be solved by numerical computation method.Substitute a series of different external applied loadFV, FH, and Mγ into the algorithm above and then a series of relative displacement δV, δH, and γ under different loading of the guideway can be obtained. Using the computer calculating method, the vertical direction stiffness, horizontal direction stiffness, and the rotation angular rigidity can be, respectively, solved:(2)KV=∂FV∂δVFV,KH=∂FH∂δHFH,Kγ=∂Mγ∂γMγ.On the basis of the computing method above and MATLAB software, taking the Schneeberger MRB35 linear rolling guideway as an example, the parameters of the guideway measure are shown in Table1.Table 1
MRB35 rolling guideway model parameters.
Parameter Items Values The length of the sliderL6 103 mm The diameter of the rollerDa 4.5 mm The length of the rollerL 5.1 mm The mass of the sliderM 2.2 kg The moment of inertia aroundx-axis of the slider Jx 3.927626 × 10−3 kg m2 The moment of inertia aroundy-axis of the slider Jy 5.761594 × 10−3 kg m2 The moment of inertia aroundz-axis of the slider Jz 2.467506 × 10−3 kg m2 The number of the rollersZL 21 The horizontal distance between each rollere 32 mm The vertical distance between loaded point and upper rollersh1 23.5 mm The vertical distance between each rollerh0 10.7 mm Tight deformation amountΔ 0.00072 mm Contact angleβ 45°Figure3 shows the relation between the vertical bearing load and vertical direction deformation while the load is within the bearing capacity. In Figure 3, the line marked with red five-pointed stars represents computing data by theoretical model, and the line marked with blue points represents the experimental data provided by the guideway manufacturer. As is shown in the figure, the relation between the loading and deformation is moderate linear and it can be approximately treated as a linear relation. In the simplified model, it can be replaced by a linear model.Figure 3
The relationship of vertical stress and deformation of MRB35 guideway.


## 2.3. Dynamic Model

Ignoring the mass of the roller, linear rolling guideway vibration model only takes the normal direction stiffness of the roller (the normal direction means the direction that is vertical to the contact interface) and simplifies the roller between the two contact interfaces as a spring that is vertical to the contact interface, whose stiffness isKn. It is shown in Figure 4.Figure 4
Simplified linear rolling guideway model.The solution for the stiffnessKn [9, 10] is as follows.The simplified spring normal stiffness of the single sphere roller is(3)Kn=dFdα=32khα1/2.The simplified spring normal stiffness of the single cylinder roller is(4)Kn=dQdδ=109klL8/9δ1/9orKn=KV4Zsin⁡β,where Kn means the spring normal stiffness which is simplified from the single roller; kh means the roller contact rigidity of the spherical guideway; α means the elastic deformation of the roll ball; kl means the roller contact stiffness of the roller guideway; L means the length of the roller guideway; δ means the elastic deformation of the roller guideway; KV means the vertical stiffness of the linear guideway; Z means the number of the single row roller; β means the contact angle of guideway and groove.Using the Lagrange equation, the dynamic equation of the linear guideway can be detected. Lagrange equation uses the variation of the generalized coordinates to present the virtual displacement of the particle in the particle system. This kind of system is more useful than Newton’s laws of motion while solving some problems (e.g., the small oscillation theory and rigid body dynamics).The Lagrange equation of linear rolling guideway is given in the following:(5)ddt∂T∂qj˙-∂T∂qj=Qjj=1,2,…,k,where T means the kinetic energy of the system; qj means the generalized coordinate of the system; qj˙ means the generalized velocity of the system; Qj means the generalized force which corresponds to the generalized coordinate of the system; k means the number of the generalized coordinates of the system, meaning the free degree of the system.If all the force applied on the particle is potential force, usingV to represent the potential function of the system; the generalized force corresponding to the generalized coordinate qj is(6)Qj=-∂V∂qj.DefiningL=T-V, L is named the Lagrange function or dynamic potential. Then the Lagrange function in potential field can be written as(7)ddt∂L∂qj˙-∂L∂qj=0j=1,2,…,k.The Lagrange function of the linear rolling guideway system shown in Figure4 can be established directly.The overall dynamic energyEk can be expressed as follows.(8)Ek=12Mu˙2+12Mv˙2+12Jxθ˙2+12Jyψ˙2+12Jzϕ˙2.The potential functionEp can be expressed as(9)Ep=Ep1=12Kn∑i=1ZLδ1i2+δ2i2+δ3i2+δ4i2,Ep1=12Kn∑i=1ZLu+h1ϕ+l1iψcos⁡β+v-eϕ-l1iθsin⁡β2+-u-h1ϕ-l2iψcos⁡β+v+eϕ-l2iθsin⁡β2+u+h3ϕ+l3iψcos⁡β+-v+eϕ+l3iθsin⁡β2+-u-h3ϕ-l4iψcos⁡β+-v-eϕ+l4iθsin⁡β2,ZL1=ZL2=ZL3=ZL4=ZL,l1i=l2i=l3i=l4i=ZL-2i+12ZL-1lLi=1,2,…,ZL,h3=h1+h0,where u means the displacement in X direction; v means the displacement in Y direction; θ means the angular displacement around x-axis (pitching motion); ψ means the angular displacement around y-axis (yawing motion); ϕ means angular displacement around z-axis (rolling motion); M means the mass of slider; Jx means the moment of inertia around x-axis of the slider; Jy means the moment of inertia around x-axis of the slider; Jz means the moment of inertia around y-axis of the slider; e means the moment of inertia around z-axis of the slider; h1 means Y direction distance from the center of roller 1 to the origin of coordinate; h0 means Y direction distance from the center of roller 1 to the center of roller 3; β means the angle between the normal direction of the contact interface and the horizontal X direction; Ep1 means the elastic potential energy of the normal direction spring; ZLj (j=1,2,3,4) means the number of the rollers in theith raceway; lji (j=1,2,3,4) means the z-axis coordinate of theith roller injth raceway as is shown in Figure 5; the angle between Y′ axis and y-axis is (π/2-β); lL means the length of single raceway roller.Figure 5
The position of the roller inY′Z plane.Establishing the linear guideway system dynamic function,(10)Mu¨+4ZLKncos2βu+2ZLKnh1+h3cos2βϕ=0,(11)Mv¨+4ZLKnsin2βv=0,(12)Jxθ¨+Kn∑i=1ZLl1i2+l2i2+l3i2+l4i2sin2βθ=0,(13)Jyψ¨+Kn∑i=1ZLl1i2+l2i2+l3i2+l4i2cos2βψ=0,(14)Jzϕ¨+2ZLKnh1+h3cos2βu+2ZLKnh12+h32cos2β+4ZLKne2sin2β+4ZLKnh0esin⁡βcos⁡βϕ=0.Detected from (11) the vibration frequency of the direction that vertical to y-axis of the guideway can be obtained:(15)fV=1πZLKnsin2βM.Detected from (12), the vibration frequency of the pitching motion around x-axis of the guideway can be obtained:(16)fXP=12πKn∑i=1ZLl1i2+l2i2+l3i2+l4i2sin2βJx.Detected from (4), the vibration frequency of the yawing motion around y-axis of the guideway can be obtained:(17)fYY=12πKn∑i=1ZLl1i2+l2i2+l3i2+l4i2cos2βJy.The displacement alongX direction u and the angular displacement around z-axis couple together so (10) and (14) are simultaneously solved to get the vibration frequency.(18)Mu¨+4ZLKncos2βu+2ZLKnh1+h3cos2βϕ=0,Jzϕ¨+2ZLKnh1+h3cos2βu+2ZLKnh12+h32cos2β+4ZLKne2sin2β+4ZLKnh0esin⁡βcos⁡βϕ=0.Suppose u=Uejωt and ϕ=Φejωt, while ω represents the frequency and j=-1.Then the simultaneous equation above can be written as follows.(19)c1-Mω2c2c2c3-Jzω2UΦ=0.In this expression,(20)c1=4ZLKncos2β,c2=2ZLKnh1+h3cos2β,c3=2ZLKnh12+h32cos2β+4ZLKne2sin2β+4ZLKnh0esin⁡βcos⁡β.Solve (19):(21)MJzω4-c3M+c1Jzω2+c1c3-c22=0.Therefore,(22)ω1,22=c3M+c1Jz∓c3M+c1Jz2-4MJzc1c3-c222MJz.The displacement alongX direction u and the angular displacement around z-axis couple together so we suppose that the frequency obtained is low order rolling frequency fRL and high order rolling frequency fRH:(23)fRL=ω12π,fRH=ω22π.The solution of the vibration model above is the vibration frequency of the linear rolling guideway.The parameters of the experimental guideway Schneeberger MRB35-V2 are displayed in Table1. The result of the vertical moving frequency, which is computed by the analytical method from the vibration model, is fV = 3529 Hz.


## 3. Dynamic Parameter Identification of Rolling Guideway

Because the contact part of the guideway has multiple modes, which can be known from the hammer-hitting experiment, numbers of groups vertical and horizontal “spring-damper” can be used as simplified equivalent model of the guideway contact part. We make a use of the results of the vibration model above and combine them with experiment to identify the dynamic parameters of the equivalent model.The experimental facility of the hammer-hitting test is shown in Figure6. The main apparatuses are listed in the following:(i)
MSC-1 Impact Hammer, 500 kg Force Transducer(ii)
YD 67D Miniature Accelerometer(iii)
DLF-3 Four-in-one Two-Channel Charge Amplifier(iv)
Four-Channel Data Collector with USB Interface INV306U, whose highest sampling frequency is 200 KHZFigure 6
The schematic diagram of hammer-hitting experiment.The measure in the four points A, B, C, and D is shown in Figure7.Figure 7
Measuring points map.The vibration spectrums of the four points A, B, C, and D are displayed in Figure8 (including the transfer function amplitude and correlation coefficient). The measured value of the vertical vibration frequency is fV = 3450 Hz, whose error is less than 3% compared with the theoretical calculation.Figure 8
The frequency response functional graph of acceleration-force of linear guideway.
(a)
Point A (b)
Point B (c)
Point C (d)
Point DAs is shown in Figure8, the slider-roller-guideway is a system with multiple modes. Find out the necessary parameters by combining them with the analytical results of the theoretical vibration model, which is the vertical vibration frequency fV and horizontal vibration frequency fH. And then identify the dynamic stiffness in both vertical and horizontal direction. The damping ratio and damping value are calculated by using the half-power method. The detail of the calculation is displayed in (24)–(29).The stiffness in vertical direction is(24)KV=4Mπ2fV2.The stiffness in horizontal direction is(25)KH=4Mπ2fH2.The damping ratio in vertical direction is(26)ξV=fV2-fV12fV.The damping value in vertical direction is(27)cV=4πMfVξV.The damping ratio in horizontal direction is(28)ξH=fH2-fH12fH.The damping value in horizontal direction is(29)cH=4πMfHξH.In these equations,M represents the mass; fV represents the vertical direction vibration frequency; fH represents the horizontal direction vibration frequency; fV1, fV2, fH1, and fH2, respectively, represent the homologous half-power frequencies in vertical and horizontal directions.The measured value of vibration frequency in vertical direction, which is detected from the experiment, isfV = 3450 Hz and the horizontal vibration frequency is fH = 1075 Hz. The results can be computed by using (24)–(29).Thus, the stiffness in vertical direction is(30)KV=1.03×109N/m.The damping value in vertical direction is(31)
c
V
=
2750
N
·
s
/
m
.The stiffness in horizontal direction is(32)KH=1.00×108N/m.The damping value in horizontal direction is(33)
c
H
=
3109
N
·
s
/
m
.


## 4. The Dynamic Analysis and Experimental Verification of Machine Tool Considering Contact Interface of Guideway

### 4.1. The FEM Model of Machine Tool

We conduct the research on the gantry machining center driven by a linear electric motor (as is shown in Figure9) and make use of the FEM software to analyze the dynamic property of the machining center. The guideway in the machine tool system, which is shown in Figure 10, is equivalent to the four-vertical-direction spring-damper system and eight-horizontal-direction spring-damper system. The dynamic parameters can be obtained by using the method mentioned above.Figure 9
The model of gantry machining center.x-axis: the direction along the length of longeron, y-axis: the direction along the length of crossbeam, and z-axis: vertical direction.Figure 10
The dynamically equivalent model of the guideway.After adding the material property parameters, constraints, and loadings, we apply the modal analysis to the machining center and compute the first-six-order frequencies and modes of the overall machine tool (shown in Table2).Table 2
The first-six-order frequencies and modes of machine tool computed by FEM method.
Order Frequency of FEM model (Hz) Modes Reasons 1 27.5 Crossbeam bends around the horizontalY axis Junction plate and longeron’s guideway pitching vibration 2 53.7 Crossbeam vibrating up and down. Sliding box bends around the horizontalY axis Crossbeam vibrates, sliding box, and crossbeam guideway’s rolling vibration 3 55.1 Longeron and crossbeam wagging Longeron’s flexural vibration 4 78.7 Junction Plate’s small amplitudes pitching vibration. Sliding box’s guideway pitching vibration Junction plate guideway and sliding box guideway’s pitching vibration 5 83.0 Crossbeam bends aroundZ axis. Sliding box rolls around Y axis Crossbeam’s flexural vibration 6 115.7 Crossbeam’s pitching vibration aroundY axis Junction plate guideway’s pitching vibration


### 4.2. The Overall Test

Using the hammer-hitting method, knock and pick up the signals, respectively, from these three directionsX,Y, andZ. On account of the reason that the main consideration is the influence on the machine parts during the process exerted by the dynamic property of the machining center, the acceleration sensor is attached to the bottom of the principal axis and the vibration signal is collected. Respectively, knock on eight main components, which are the column, longeron, sliding plate, crossbeam, sliding box, ram, spindle box, and the bottom of the principal axis. Test the natural frequency of the overall machine and compare the results with frequency obtained from FEM analysis. The results are shown in Table 3. The method to calculate the error is shown as follows.(34)Relative  Error=Computed  Value-Measured  ValueMeasured  Value×100%.Table 3
The first-six-order natural frequencies of overall machine tool (unit: Hz).
Order FEM model frequency Experimental measured frequency Error 1 27.5 25.8 6.6% 2 53.7 50.3 6.8% 3 55.1 55.9 1.4% 4 78.7 68.0 14.7% 5 83.0 85.6 3.0% 6 115.7 107.9 7.2%The result of the experiment indicates that the solution of the FEM analysis has a relative small error compared to the experimental result, and the simulation of FEM model well matches the practical situation.


## 4.1. The FEM Model of Machine Tool

We conduct the research on the gantry machining center driven by a linear electric motor (as is shown in Figure9) and make use of the FEM software to analyze the dynamic property of the machining center. The guideway in the machine tool system, which is shown in Figure 10, is equivalent to the four-vertical-direction spring-damper system and eight-horizontal-direction spring-damper system. The dynamic parameters can be obtained by using the method mentioned above.Figure 9
The model of gantry machining center.x-axis: the direction along the length of longeron, y-axis: the direction along the length of crossbeam, and z-axis: vertical direction.Figure 10
The dynamically equivalent model of the guideway.After adding the material property parameters, constraints, and loadings, we apply the modal analysis to the machining center and compute the first-six-order frequencies and modes of the overall machine tool (shown in Table2).Table 2
The first-six-order frequencies and modes of machine tool computed by FEM method.
Order Frequency of FEM model (Hz) Modes Reasons 1 27.5 Crossbeam bends around the horizontalY axis Junction plate and longeron’s guideway pitching vibration 2 53.7 Crossbeam vibrating up and down. Sliding box bends around the horizontalY axis Crossbeam vibrates, sliding box, and crossbeam guideway’s rolling vibration 3 55.1 Longeron and crossbeam wagging Longeron’s flexural vibration 4 78.7 Junction Plate’s small amplitudes pitching vibration. Sliding box’s guideway pitching vibration Junction plate guideway and sliding box guideway’s pitching vibration 5 83.0 Crossbeam bends aroundZ axis. Sliding box rolls around Y axis Crossbeam’s flexural vibration 6 115.7 Crossbeam’s pitching vibration aroundY axis Junction plate guideway’s pitching vibration


## 4.2. The Overall Test

Using the hammer-hitting method, knock and pick up the signals, respectively, from these three directionsX,Y, andZ. On account of the reason that the main consideration is the influence on the machine parts during the process exerted by the dynamic property of the machining center, the acceleration sensor is attached to the bottom of the principal axis and the vibration signal is collected. Respectively, knock on eight main components, which are the column, longeron, sliding plate, crossbeam, sliding box, ram, spindle box, and the bottom of the principal axis. Test the natural frequency of the overall machine and compare the results with frequency obtained from FEM analysis. The results are shown in Table 3. The method to calculate the error is shown as follows.(34)Relative  Error=Computed  Value-Measured  ValueMeasured  Value×100%.Table 3
The first-six-order natural frequencies of overall machine tool (unit: Hz).
Order FEM model frequency Experimental measured frequency Error 1 27.5 25.8 6.6% 2 53.7 50.3 6.8% 3 55.1 55.9 1.4% 4 78.7 68.0 14.7% 5 83.0 85.6 3.0% 6 115.7 107.9 7.2%The result of the experiment indicates that the solution of the FEM analysis has a relative small error compared to the experimental result, and the simulation of FEM model well matches the practical situation.


## 5. Conclusion

Based on the research of static property of the linear rolling guideway, this paper makes a research on the dynamic performance of the rolling guideway and the simplified model of the guideway and the identification of dynamic parameters. At the same time, the research achievement is applied to a gantry machining center driven by a linear electric motor, which solves the problem of dynamic modeling of contact interface in NC machine tool. On account of the research mentioned above, the conclusions are obtained as follows.(1)
Based on the Hertz contact theory, the relation between the bearing load and deformation of the linear rolling guideway is detected to be approximate linear function. In the simplified model it can be replaced by the linear model.(2)
Based on the Lagrange method, we conduct a research on the vibration property of the linear guideway and deduce the dynamic equations. It is explained through model analysis that the ‘Slider-Roller-Guideway’ system has multiple vibration modes called pitching vibration, yawing vibration, vertical vibration, low order rolling vibration, and high order rolling vibration, which means that it is a complex multiple modes system. Therefore, we use the multigroups of vertical and horizontal “spring-damper” system as the simplified equivalent model.(3)
The paper proposed a type of guideway contact interface modeling method combining four stages which were static stiffness model, vibration model, experimental parameter, and parameter identification. This method is effective while solving the stiffness and damping value of the guideway equivalent model. Moreover, it solves the key problem in machine tool modeling—the problem of contact interface modeling.(4)
Combined with the research and development of the practical machine tool, for the object of a type of gantry machining center, we establish an overall FEM model of the gantry machining center driven by linear motors which includes the guideway contact interface model; moreover, the results of natural frequencies and modes of the machine tool are obtained. By comparing the computed solution with the experimental result, the efficiency and accuracy of the guideway contact interface model is verified.(5)
The spring model is used to replace the roller in the guideway system to improve the comprehension of stiffness and damping in the joint. However, when the interaction in the joint is needed to consider, the spring model is not suitable for simplified modeling of the roller, and it will be discussed in the future research.



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*Source: 5658181-2016-12-15.xml*