# The Nonlinear Stability Prediction and FEM Modeling of High-Speed Spindle System with Joints Dynamic Characteristics

Authors: Bo Wang; Wei Sun; Kunpeng Xu; Junyong Zhang; Bangchun Wen
Journal: Shock and Vibration
 (2014)

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## Abstract

When predicting the nonlinear stability of high-speed spindle system, it is necessary to create an accurate model that reflects the dynamic characteristics of the whole system, including the spindle-bearing joint and spindle-holder-tool joints. In this paper, the distribution spring model of spindle-holder-tool joints was built with the consideration of its dynamic characteristics; the five-DOF dynamic model of the angle contact ball bearing was also established to study the influence of speed and preload on the spindle-bearing joint, both of which were used in the general whole complete spindle system FEM model. The rationality of the model was verified by comparison with the FRF of traditional rigid model and experiments. At last, the influences of speed and cutting force on the nonlinear stability were analyzed by amplitude spectrum, bifurcation, and Poincaré mapping. The results provided a theoretical basis and an evaluating criterion for nonlinear stability prediction and product surface quality improvement.


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## Body

## 1. Introduction

High-speed machining technology is widely used in mould, aircraft structural components, and some other volume productions industry for productivity rising and costs reducing which will lead to poor surface finish, reduced material removal rate, high noise, and tool or work piece damage simultaneously. With the processing speed becoming higher and higher, the cutting stability affected parameters are different from traditional ones, especially the structure of high-speed spindle system. The traditional spindle is rigidity which works under the first-order critical speed; however, the processing speed is getting higher at present, which not only causes the centrifugal force, the gyroscopic moment, and speed-dependent bearing stiffness to be nonignored, but also makes the joints influence amplified.The construction of high-speed spindle system becomes a critical factor to affect the dynamics and the cutting stability, especially the spindle joints including spindle-bearing joint and spindle-holder-tool joints. The structure of spindle-bearing joint is familiar and its dynamic characteristics are decided by the bearing. The spindle-holder-tool joints consist of two parts: the spindle-holder joint is a conical surface which is the part of holder inserting spindle; the holder-tool joint is a cylindrical surface which is the part of tool inserting holder.The research of spindle-bearing joint has been done for decades and some achievements have been created. Cao and Altintas [1] have modeled the spindle-bearing with the Timoshenko using the Jones bearing model in form of force. Shin [2] has presented 6×6 bearing stiffness matrix with Jacobian matrix. Chen et al. [3] have built nonlinear dynamic model of rotor system with rolling bearing.Spindle-holder-tool joints are other significant factors in influencing the high-speed spindle system dynamic characteristics besides spindle-bearing joint. Rivin [4] has proposed the six important aspects of machining tool research, in which the joint structure and dynamic related study will become of emphasis in the future. Schmitz et al. [5] have modeled and analyzed the spindle joints considering translation degree. Namazi et al. [6] have built the contact stiffness and damping model to simulate the joint surface loose by moving the spring element. Ertürk et al. [7] have assumed the radial contact between spindle and holder ignoring the taper joint surface between holder and tool. Yumei et al. [8] have taken the joint between spindle and holder of JCS-018 as example to research dynamic characteristics of fixed taper surface. Zhang et al. [9] have discussed the stress distribution between contact surfaces.The final objective of spindle joints modeling and analysis is to predict the stability of machining center. On macroscopic view, the methods of cutting stability research will be divided into two ways: online monitoring and offline predicting. Winfough [10] has developed the online method for detecting the chatter phenomenon during the process with microphones. The result of online method could only be used to confirm the stability of that process; besides, the external environment is so changeable from moment to moment that weakens the reference for other cutting processes. Therefore, the offline method with lobe diagrams has been proposed to predict the cutting process stability since the 1960s by Koenigsberger and Tlusty [11], Tobias [12], and Merrit [13]. Park et al. [14, 15] have developed stability lobe diagrams for prediction with analytical utility. Mann et al. [16] have proposed a new method to obtain stability lobe diagrams in upmilling and downmilling machining. Schmitz et al. [17] have confirmed that the maximum cutting depth on static is lower than the actual. Weixiao [18] has established the dynamic cutting force model to research the stability criterion.The stability prediction method develops from traditional two-dimensional lobes to three-dimensional lobes for improving the prediction accuracy with the increased product quality requirement and complicated working condition. Tang and Liu [19] have gotten the three-dimensional lobes of thin-walled workpiece. Abrari et al. [20] have drawn the limit stability diagram of spherical cutter at different milling position. Dai Depai [21] has proposed an equivalent linearization method to obtain a three-dimensional stability chart. Qinghua [22] has structured three- dimensional limit chart for thin-walled workpiece milling system.In this study, the FEM of high-speed spindle system was built with joints characteristics considering the high speed influence. The spindle-bearing joint was structured with angle contact ball bearing dynamic model affected by speed and preload; the spindle-holder-tool joints were simulated in distributed spring with hysteretic nonlinearity. The model was more rational than traditional rigid connection verified by experiment. Based on this, the influence of speed and cutting force at tool point on processing stability was analyzed. The nonlinear response and stability of tool point were evaluated by amplitude spectrum and bifurcation and Poincaré mapping which could provide a reliable basis for enhancing the nonlinear stability and improving the machining accuracy.


## 2. Modeling

The high-speed spindle system dynamic model is the foundation and premise of dynamic response and cutting stability, which includes three parts: spindle-bearing modeling, spindle-holder-tool joints modeling, and the complete spindle system model with all the above.


### 2.1. Bearing Joint Modeling

At present, the hybrid ceramic ball bearing is used to satisfy the requirement of stiffness and rotational speed in high-speed spindle widely, as shown in Figure1. The dynamic loading between the rolling element and raceway is produced because of the rolling revolution and rotation, the centrifugal force, and the gyroscopic moment, the distribution of which will be changed under the high speed condition. The displacement and load properties of bearing even the dynamic characteristics of spindle system with the bearing will be affected when the dynamic loading reaches a significant level.Figure 1
The structure of ceramic ball bearing.The force distribution of one rolling element is shown in Figure2, which will make pure rolling according to the raceway control theory when the speed increases; then the gyroscopic moment of ball is balanced with friction of outer ring; that is, λij=0,λoj=2°. Each rolling element is a balance system; that is,
(1)Qijsinαij-Qojsinαoj+2MgjDcos⁡αoj=0,Qijcos⁡αij-Qojcos⁡αoj+2MgjDsinαoj+Fcj=0,
where Qij,Qoj are normal contact load between rolling element and inner or outer ring, Qnj=Knjδnj1.5(n=i,o),αij,αoj are contact angle between rolling element and inner or outer ring, Fcj is centrifugal force of rolling element, Mgj is gyroscopic moment of rolling element, and D is rolling element diameter.Figure 2
The forces on the roller.The forces between spindle unit and inner ring of bearing are axial forceFx, radial forces Fy and Fz, and torque My and Mz; even considering the forces between rolling element and inner ring of bearing, the equilibrium equation of inner ring is
(2)Fx-∑j=1NbQijsinαij=0,Fy-∑j=1NbQijcos⁡αijcos⁡ψj=0,Fz-∑j=1NbQijcos⁡αijsinψj=0,My-∑j=1NbQijcos⁡αijRisinψj=0,Mz-∑j=1NbQijsinαijRicos⁡ψj=0,
where Ri is radius of inner ring curvature center and ψj is position angle.With the high speed processing condition, the deformation equation of one supported bearing in matrix form is(3){FxFyFzMyMz}=(kδxδxkδxδykδxδzkδxθykδxθzkδyδxkδyδykδyδzkδyθykδyθzkδzδxkδzδykδzδzkδzθykδzθzkθyδxkθyδykθyδzkθyθykθyθzkθzδxkθzδykθzδzkθzθykθzθz){δxδyδzθyθz}
simplified as
(4)Fb=Kbδb,
where Fb is the spindle-bearing force vector, Kb is bearing dynamic stiffness matrix, and δb is bearing deformation vector.The relationship between load and deformation of five-DOF ball bearing is listed in (3), which could be used in dynamics equation of system to realize the whole spindle-bearing system model in load-deformation way or could get stiffness matrix Kb used in the FEM model directly.


### 2.2. Spindle-Holder-Tool Joints Modeling

In this section, the spindle-holder-tool joints model will be built with FEM and the dynamic characteristics will be simulated with stiffness and damping matrices. The structure of spindle-holder-tool is double parallel rotor system which consists of two parts: one parallel rotor is spindle and holder; the other one is holder and tool shown in Figure3.Figure 3
The cutaway view of spindle system.The nodes are numbered after the rotor system becomes discrete; then a transition section with random information is added at the end of outer rotor and the inner rotor numbers will be continuous. The mass and stiffness matrices of the whole system are assembled with the ones of each unit excluding the transition section.Assumingp units in outer rotor and q units in inner rotor, there are p+q+1 units in the parallel rotor system and N nodes, N=p+1+1+1+q+1. The mass and stiffness matrices of each unit are 12×12 orders, which consist of two nodes and each node owns six degrees of freedom. The assembly of adjacent units is shown in Figure 4. The mass and stiffness matrices of the whole system are 6N×6N orders, from which the transition section matrices will be removed. The final complete mass and stiffness matrices of parallel rotor are shown in Figure 5.Figure 4
Schemes of adjacent unit assembly.Figure 5
Schemes of parallel rotors assembly.The simulated method of joint dynamic characteristics is illustrated by taking the spindle-holder joint as an example. The model of spindle and holder joint simulated by distribution spring is shown in Figure6.The distribution spring model of spindle-holder joint.
(a)
Structure
(b)
FEMAssuming the spindle and holder joint is divided inton stepped shafts, in the joint there are n+1 nodes, i,i+1,…,i+n, on the spindle and j,j+1,…j+n on the holder, respectively. The coupling forces on the spindle and holder are
(5)ΔFi=KcoupXi+CcoupX˙i-KcoupXj-CcoupX˙jΔFj=KcoupXj+CcoupX˙j-KcoupXi-CcoupX˙iΔFi+1=KcoupXi+1+CcoupX˙i+1-KcoupXj+1-CcoupX˙j+1ΔFj+1=KcoupXj+1+CcoupX˙j+1-KcoupXi+1-CcoupX˙i+1⋮ΔFi+n=KcoupXi+n+CcoupX˙i+n-KcoupXj+n-CcoupX˙j+nΔFj+n=KcoupXj+n+CcoupX˙j+n-KcoupXi+n-CcoupX˙i+n,
where Kcoup and Ccoup are coupling stiffness and damping matrices between spindle and holder.The stiffness and damping matrices of spindle-holder joint expressed withKc and Cc are
(6)Kc=i⋮j[Kcoup-Kcoup⋱-KcoupKcoup],kkkkkkkikkk⋯kkkkjCc=i⋮j[Ccoup-Ccoup⋱-CcoupCcoup].kkkkkkkikkk⋯kkkkjThe dynamic characteristics of distribution spring are shown in (8) and the assembly method of stiffness on joint is shown in Figure 7:
(7)Kcoup=[0000000kyykyz0000kzykzz000000kθ00000000000000],Ccoup=[0000000cyycyz0000czyczz000000cθ00000000000000].Figure 7
Schematic diagram of joint stiffness assembly.


### 2.3. High-Speed Spindle Modeling

The spindle system consists of HCS150 motorized spindle from GMN, HSK-E50 holder, and R245 milling cutter. The FEM model of high-speed spindle system is discretized with different inner or outer diameters, bearing locations, spindle-holder taper joint, and holder-tool cylindrical joint shown in Figure8. The bolts and lubrication are neglected. The five supporting bearings are installed in two pairs and the middle is independent.Figure 8
The model of high-speed spindle system.The dots in Figure9 represent mass point; the small sections represent unit. There are 52 units and 53 nodes in the FEM including 2 transition sections. The external forces on the right consist of spindle-bearing forces, spindle-holder-tool joints hysteretic forces, and cutting force on the tool point. The dynamic equation of high-speed spindle system could be expressed as
(8)MX¨+CX˙+KX=R(X)+H(X,X˙)+P(t)+G,C=D+Ω·J+KH,K=Ks+KCX=[q1,q2,…,qn]T,(n=51)qi=[xi,yi,zi,θxi,θyi,θzi]T,(i=1,2,…,51),
where M is mass matrix; C is damping matrix; K is stiffness matrix; D is damping matrix; J is gyroscopic matrix; KH is coupling stiffness matrix of joints; Ks is stiffness matrix of spindle; KC is centrifugal stiffness matrix of spindle; R is supporting bearing force vector; H is hysteretic force vector of spindle-holder-tool; P is cutting force vector; G is gravity vector; and X is displacement vector.Figure 9
The test system.


## 2.1. Bearing Joint Modeling

At present, the hybrid ceramic ball bearing is used to satisfy the requirement of stiffness and rotational speed in high-speed spindle widely, as shown in Figure1. The dynamic loading between the rolling element and raceway is produced because of the rolling revolution and rotation, the centrifugal force, and the gyroscopic moment, the distribution of which will be changed under the high speed condition. The displacement and load properties of bearing even the dynamic characteristics of spindle system with the bearing will be affected when the dynamic loading reaches a significant level.Figure 1
The structure of ceramic ball bearing.The force distribution of one rolling element is shown in Figure2, which will make pure rolling according to the raceway control theory when the speed increases; then the gyroscopic moment of ball is balanced with friction of outer ring; that is, λij=0,λoj=2°. Each rolling element is a balance system; that is,
(1)Qijsinαij-Qojsinαoj+2MgjDcos⁡αoj=0,Qijcos⁡αij-Qojcos⁡αoj+2MgjDsinαoj+Fcj=0,
where Qij,Qoj are normal contact load between rolling element and inner or outer ring, Qnj=Knjδnj1.5(n=i,o),αij,αoj are contact angle between rolling element and inner or outer ring, Fcj is centrifugal force of rolling element, Mgj is gyroscopic moment of rolling element, and D is rolling element diameter.Figure 2
The forces on the roller.The forces between spindle unit and inner ring of bearing are axial forceFx, radial forces Fy and Fz, and torque My and Mz; even considering the forces between rolling element and inner ring of bearing, the equilibrium equation of inner ring is
(2)Fx-∑j=1NbQijsinαij=0,Fy-∑j=1NbQijcos⁡αijcos⁡ψj=0,Fz-∑j=1NbQijcos⁡αijsinψj=0,My-∑j=1NbQijcos⁡αijRisinψj=0,Mz-∑j=1NbQijsinαijRicos⁡ψj=0,
where Ri is radius of inner ring curvature center and ψj is position angle.With the high speed processing condition, the deformation equation of one supported bearing in matrix form is(3){FxFyFzMyMz}=(kδxδxkδxδykδxδzkδxθykδxθzkδyδxkδyδykδyδzkδyθykδyθzkδzδxkδzδykδzδzkδzθykδzθzkθyδxkθyδykθyδzkθyθykθyθzkθzδxkθzδykθzδzkθzθykθzθz){δxδyδzθyθz}
simplified as
(4)Fb=Kbδb,
where Fb is the spindle-bearing force vector, Kb is bearing dynamic stiffness matrix, and δb is bearing deformation vector.The relationship between load and deformation of five-DOF ball bearing is listed in (3), which could be used in dynamics equation of system to realize the whole spindle-bearing system model in load-deformation way or could get stiffness matrix Kb used in the FEM model directly.


## 2.2. Spindle-Holder-Tool Joints Modeling

In this section, the spindle-holder-tool joints model will be built with FEM and the dynamic characteristics will be simulated with stiffness and damping matrices. The structure of spindle-holder-tool is double parallel rotor system which consists of two parts: one parallel rotor is spindle and holder; the other one is holder and tool shown in Figure3.Figure 3
The cutaway view of spindle system.The nodes are numbered after the rotor system becomes discrete; then a transition section with random information is added at the end of outer rotor and the inner rotor numbers will be continuous. The mass and stiffness matrices of the whole system are assembled with the ones of each unit excluding the transition section.Assumingp units in outer rotor and q units in inner rotor, there are p+q+1 units in the parallel rotor system and N nodes, N=p+1+1+1+q+1. The mass and stiffness matrices of each unit are 12×12 orders, which consist of two nodes and each node owns six degrees of freedom. The assembly of adjacent units is shown in Figure 4. The mass and stiffness matrices of the whole system are 6N×6N orders, from which the transition section matrices will be removed. The final complete mass and stiffness matrices of parallel rotor are shown in Figure 5.Figure 4
Schemes of adjacent unit assembly.Figure 5
Schemes of parallel rotors assembly.The simulated method of joint dynamic characteristics is illustrated by taking the spindle-holder joint as an example. The model of spindle and holder joint simulated by distribution spring is shown in Figure6.The distribution spring model of spindle-holder joint.
(a)
Structure
(b)
FEMAssuming the spindle and holder joint is divided inton stepped shafts, in the joint there are n+1 nodes, i,i+1,…,i+n, on the spindle and j,j+1,…j+n on the holder, respectively. The coupling forces on the spindle and holder are
(5)ΔFi=KcoupXi+CcoupX˙i-KcoupXj-CcoupX˙jΔFj=KcoupXj+CcoupX˙j-KcoupXi-CcoupX˙iΔFi+1=KcoupXi+1+CcoupX˙i+1-KcoupXj+1-CcoupX˙j+1ΔFj+1=KcoupXj+1+CcoupX˙j+1-KcoupXi+1-CcoupX˙i+1⋮ΔFi+n=KcoupXi+n+CcoupX˙i+n-KcoupXj+n-CcoupX˙j+nΔFj+n=KcoupXj+n+CcoupX˙j+n-KcoupXi+n-CcoupX˙i+n,
where Kcoup and Ccoup are coupling stiffness and damping matrices between spindle and holder.The stiffness and damping matrices of spindle-holder joint expressed withKc and Cc are
(6)Kc=i⋮j[Kcoup-Kcoup⋱-KcoupKcoup],kkkkkkkikkk⋯kkkkjCc=i⋮j[Ccoup-Ccoup⋱-CcoupCcoup].kkkkkkkikkk⋯kkkkjThe dynamic characteristics of distribution spring are shown in (8) and the assembly method of stiffness on joint is shown in Figure 7:
(7)Kcoup=[0000000kyykyz0000kzykzz000000kθ00000000000000],Ccoup=[0000000cyycyz0000czyczz000000cθ00000000000000].Figure 7
Schematic diagram of joint stiffness assembly.


## 2.3. High-Speed Spindle Modeling

The spindle system consists of HCS150 motorized spindle from GMN, HSK-E50 holder, and R245 milling cutter. The FEM model of high-speed spindle system is discretized with different inner or outer diameters, bearing locations, spindle-holder taper joint, and holder-tool cylindrical joint shown in Figure8. The bolts and lubrication are neglected. The five supporting bearings are installed in two pairs and the middle is independent.Figure 8
The model of high-speed spindle system.The dots in Figure9 represent mass point; the small sections represent unit. There are 52 units and 53 nodes in the FEM including 2 transition sections. The external forces on the right consist of spindle-bearing forces, spindle-holder-tool joints hysteretic forces, and cutting force on the tool point. The dynamic equation of high-speed spindle system could be expressed as
(8)MX¨+CX˙+KX=R(X)+H(X,X˙)+P(t)+G,C=D+Ω·J+KH,K=Ks+KCX=[q1,q2,…,qn]T,(n=51)qi=[xi,yi,zi,θxi,θyi,θzi]T,(i=1,2,…,51),
where M is mass matrix; C is damping matrix; K is stiffness matrix; D is damping matrix; J is gyroscopic matrix; KH is coupling stiffness matrix of joints; Ks is stiffness matrix of spindle; KC is centrifugal stiffness matrix of spindle; R is supporting bearing force vector; H is hysteretic force vector of spindle-holder-tool; P is cutting force vector; G is gravity vector; and X is displacement vector.Figure 9
The test system.


## 3. Model Validation

The modal experiment is done on VMC0540d by hammering to prove the effectiveness of modeling method for spindle-holder-tool joints and the whole spindle system. The results obtained from the experiment, distribution spring model, and rigid model will be compared by the frequency response function of tool point.The test system is set up and shown in Figure9. The main test equipment includes the data acquisition, hammer, and vibration sensors, shown in Table 1.Table 1
The instruments used in this test.
Number
Name
1
B&K 3560-C data acquisition front-end
2
PCB 8206-001 54627 modal hammer
3
B&K 4525B-001 54072 acceleration sensor
4
High-performance notebook computersThe FEM of test object is built with distribution spring connection and rigid connection, respectively. The dynamic parameters of system are elasticity modulusE=2.06×1011Pa, material density ρ=7850kg/m3, contact stiffness of spindle-holder-tool joints ksh=kht=1×108N/m, 7011AC angle contact ball bearing, clearance r0=1μm, and ball stiffness Cb=1×109N/m1.5. The structures of spindle system are shown in Tables 2 and 3.Table 2
The size of motorized spindle in mm.
Number
L
D
d
1
14.5
120
70
2
25
149.9
70
3
15
149.9
55
4
15
149.9
55
5
25
149.9
65
6
30
149.9
65
7
25
149.9
65
8
25
149.9
65
9
30
150
65
10
30
150
65
11
15
150
55
12
30
150
65
13
30
150
65
14
30
150
55
15
25
150
55
16
25
150
55
17
25
150
70
18
15
150
55
19
15
150
55
20
18
149.9
55
21
20
129.9
55
22
20
129.9
55
23
20
129.9
55
24
5
129.9
34
25
5
129.9
35
26
5
129.9
36
27
5
88
37Table 3
The size of holder and tool in mm.
Number
L
D
d
Holder
1
5
34
27
2
5
35
27
3
5
36
32
4
5
37
32
5
5
38
29
6
8
50
27
7
8
50
10
8
4
44
10
9
6
50
12
10
16
39
5
11
13
42
10
12
5
22
10
13
6
22.1
10
14
7
22.1
10
Tool
1
5
40
22
2
6
40
22.1
3
7
42
22.1
4
10
45
25
5
12
48
25The FRF curves of tool point are drawn in Figure10 and the results of the first five-order inherence frequencies are listed in Table 4 from experiment and two connection models. The experimental results are affected experimental environment, changing temperature, equipment precision, clamped situation, and some other factors, which lead to the differences between the simulations.Table 4
The comparison of simulated and experiment results.
Orderi
Experiment resultsAi/Hz
Distribution spring model
Rigid connection model
Simulated resultsBi/Hz
Errors|Ai-Bi|/Ai×100%
Simulated resultsCi/Hz
Errors|Ai-Ci|/Ai×100%
1
282
276
2.13%
—
—
2
875
945
8%
909
3.9%
3
1193
1139
4.53%
1118
6.3%
4
1489
1580
6.11%
2157
44.9%
5
2420
2382
1.57%
2988
23.5%Figure 10
The comparison of FRF at tool point.The results got from the distribution spring model are much closer to the experiment than the rigid connection whose error is less than 8%; however, the results from the rigid model even miss the first-order inherence frequency. Therefore, the distribution spring model of spindle-holder-tool joints reflects the real connection and dynamic characteristics among spindle, holder, and tool better which should be used in joints and spindle system dynamic modeling.


## 4. Model Applications

The stability is one of the most significant parameters for evaluating characteristics of spindle system and processing quality which could not be predicted accurately by traditional limit cutting depth method with the nonlinear phenomenon remarkable and speed higher especially. On the basis of the spindle system model with distribution spring connection, the effect rules of response and stability at tool point are studied on speed and cutting force amplitude with the bifurcation diagram and Poincaré mapping. For the cutting process, the periodic motion is named stability state, and other motions are named instability states including quasiperiodic motion and chaotic motion.


### 4.1. The Influence of Speed on Stability

The bifurcation diagram at tool point of high-speed spindle system iny direction from 1000 r/min to 25000 r/min is shown in Figure 11. The movement state of tool point is chaos in the beginning when the speed is round 1000 r/min, then becomes quasiperiodic motion gradually as the speed increases to 7000 r/min, and finally tends to be stable periodic motion similarly with the speed continuing to rise up. The amplitude spectrum and Poincaré mapping at tool point are shown in Figure 12 to research the stability changing laws with speed including 1200 r/min, 7800 r/min, 12600 r/min, and 20000 r/min clearly.Figure 11
Global bifurcation of tool point iny direction with different speeds.The amplitude diagram and Poincaré mapping of tool point with different speeds.
(a)
(b)The amplitude spectrum and mapping of tool point at 1000 r/min are shown in Figures12(a1)-12(b1). The rotating frequency is 16.67 Hz. There are lots of combination frequencies of inherence frequency and rotating frequency. The system is not stable because there is a similar frequency of 286.1 Hz near the first-order inherence frequency which is proved by the random points in Poincaré mapping.The amplitude spectrum and Poincaré mapping of tool point at 2000 r/min are shown in Figures12(a2)-12(b2). The rotating frequency is 32.71 Hz. The system is a quasiperiodic motion by lots of combination frequencies of inherence frequency and rotating frequency in the amplitude spectrum and a closed pattern in Poincaré mapping.The amplitude spectrum and Poincaré mapping of tool point at 7800 r/min are shown in Figures12(a3)-12(b3). The rotating frequency is 127.9 Hz. The frequencies in the amplitude spectrum are combination of rotating frequency and 15 Hz, while 293 Hz is the combination of inherent frequency and 15 Hz; therefore, the shape in Poincaré mapping is a regular pattern but not closed completely which leads to a transition form between quasiperiodic and chaotic motions.The amplitude spectrum and Poincaré mapping of tool point at 12600 r/min are shown in Figures12(a4)-12(b4). The rotating frequency and one-third frequency are shown in the amplitude spectrum. There are three separate points in the Poincaré mapping which shows that the system performs P-3 periodic motion at this speed moment.The amplitude spectrum and Poincaré mapping of tool point at 20000 r/min are shown in Figures12(a5)-12(b5). There is only the rotating frequency 325.5 Hz in the amplitude spectrum and one point in Poincaré mapping. The system motion is stable.


### 4.2. The Influence of Cutting Force on Stability

The bifurcation diagram at tool point of high-speed spindle system iny direction from 10 N to 120 N is shown in Figure 13. The displacement at tool point increases with the cutting force obviously, but the stability is ruleless. The system is in quasiperiodic and chaotic states when the cutting force is lower than 50 N, in quasiperiodic and periodic states when the cutting force is between 50 N and 70 N, and back to the chaotic state again when the force is between 70 N and 80 N, and when the force continues to rise up, the motion finally is a periodic state. The amplitude spectrum and Poincaré mapping at tool point are shown in Figure 14 to research the stability changing laws with cutting force including 25 N, 40 N, 50 N, 80 N, and 100 N clearly.Figure 13
Global bifurcation of tool point iny direction with different cutting forces.The amplitude diagram and Poincaré mapping of tool point under different cutting forces.
(a)
(b)The amplitude spectrum and Poincaré mapping of tool point at 25 N are shown in Figures14(a1)-14(b1). The combination frequencies which existed in the amplitude spectrum included rotating frequency 195.3 Hz. The motion period of tool point is a closed pattern in Poincaré mapping which indicates that the system is in quasiperiodic motion.The amplitude spectrum and Poincaré mapping of tool point at 40 N are shown in Figures14(a2)-14(b2). The continuous spectrum occurs in the amplitude spectrum except the rotating and combination frequency which makes the random points in Poincaré mapping and the stability of tool point chaos.The amplitude spectrum and Poincaré mapping of tool point at 50 N are shown in Figures14(a3)-14(b3). The rotating frequency, a third of rotating frequency, and combination frequencies are in amplitude spectrum together, which leads to the 3 independent closed patterns in the Poincaré mapping. The system is in P-3 quasiperiodic motion state.The amplitude spectrum and Poincaré mapping of tool point at 80 N are shown in Figures14(a4)-14(b4). The state is similar to cutting force in 25 N which just shows different combination frequencies in amplitude spectrum and the Poincaré mapping is a different shape closed pattern which shows that the system is in quasiperiodic state.The amplitude spectrum and Poincaré mapping of tool point at 100 N are shown in Figures14(a5)-14(b5). There is only the rotating frequency 195.3 Hz in the amplitude spectrum and one point in Poincaré mapping. The system motion is stable.


## 4.1. The Influence of Speed on Stability

The bifurcation diagram at tool point of high-speed spindle system iny direction from 1000 r/min to 25000 r/min is shown in Figure 11. The movement state of tool point is chaos in the beginning when the speed is round 1000 r/min, then becomes quasiperiodic motion gradually as the speed increases to 7000 r/min, and finally tends to be stable periodic motion similarly with the speed continuing to rise up. The amplitude spectrum and Poincaré mapping at tool point are shown in Figure 12 to research the stability changing laws with speed including 1200 r/min, 7800 r/min, 12600 r/min, and 20000 r/min clearly.Figure 11
Global bifurcation of tool point iny direction with different speeds.The amplitude diagram and Poincaré mapping of tool point with different speeds.
(a)
(b)The amplitude spectrum and mapping of tool point at 1000 r/min are shown in Figures12(a1)-12(b1). The rotating frequency is 16.67 Hz. There are lots of combination frequencies of inherence frequency and rotating frequency. The system is not stable because there is a similar frequency of 286.1 Hz near the first-order inherence frequency which is proved by the random points in Poincaré mapping.The amplitude spectrum and Poincaré mapping of tool point at 2000 r/min are shown in Figures12(a2)-12(b2). The rotating frequency is 32.71 Hz. The system is a quasiperiodic motion by lots of combination frequencies of inherence frequency and rotating frequency in the amplitude spectrum and a closed pattern in Poincaré mapping.The amplitude spectrum and Poincaré mapping of tool point at 7800 r/min are shown in Figures12(a3)-12(b3). The rotating frequency is 127.9 Hz. The frequencies in the amplitude spectrum are combination of rotating frequency and 15 Hz, while 293 Hz is the combination of inherent frequency and 15 Hz; therefore, the shape in Poincaré mapping is a regular pattern but not closed completely which leads to a transition form between quasiperiodic and chaotic motions.The amplitude spectrum and Poincaré mapping of tool point at 12600 r/min are shown in Figures12(a4)-12(b4). The rotating frequency and one-third frequency are shown in the amplitude spectrum. There are three separate points in the Poincaré mapping which shows that the system performs P-3 periodic motion at this speed moment.The amplitude spectrum and Poincaré mapping of tool point at 20000 r/min are shown in Figures12(a5)-12(b5). There is only the rotating frequency 325.5 Hz in the amplitude spectrum and one point in Poincaré mapping. The system motion is stable.


## 4.2. The Influence of Cutting Force on Stability

The bifurcation diagram at tool point of high-speed spindle system iny direction from 10 N to 120 N is shown in Figure 13. The displacement at tool point increases with the cutting force obviously, but the stability is ruleless. The system is in quasiperiodic and chaotic states when the cutting force is lower than 50 N, in quasiperiodic and periodic states when the cutting force is between 50 N and 70 N, and back to the chaotic state again when the force is between 70 N and 80 N, and when the force continues to rise up, the motion finally is a periodic state. The amplitude spectrum and Poincaré mapping at tool point are shown in Figure 14 to research the stability changing laws with cutting force including 25 N, 40 N, 50 N, 80 N, and 100 N clearly.Figure 13
Global bifurcation of tool point iny direction with different cutting forces.The amplitude diagram and Poincaré mapping of tool point under different cutting forces.
(a)
(b)The amplitude spectrum and Poincaré mapping of tool point at 25 N are shown in Figures14(a1)-14(b1). The combination frequencies which existed in the amplitude spectrum included rotating frequency 195.3 Hz. The motion period of tool point is a closed pattern in Poincaré mapping which indicates that the system is in quasiperiodic motion.The amplitude spectrum and Poincaré mapping of tool point at 40 N are shown in Figures14(a2)-14(b2). The continuous spectrum occurs in the amplitude spectrum except the rotating and combination frequency which makes the random points in Poincaré mapping and the stability of tool point chaos.The amplitude spectrum and Poincaré mapping of tool point at 50 N are shown in Figures14(a3)-14(b3). The rotating frequency, a third of rotating frequency, and combination frequencies are in amplitude spectrum together, which leads to the 3 independent closed patterns in the Poincaré mapping. The system is in P-3 quasiperiodic motion state.The amplitude spectrum and Poincaré mapping of tool point at 80 N are shown in Figures14(a4)-14(b4). The state is similar to cutting force in 25 N which just shows different combination frequencies in amplitude spectrum and the Poincaré mapping is a different shape closed pattern which shows that the system is in quasiperiodic state.The amplitude spectrum and Poincaré mapping of tool point at 100 N are shown in Figures14(a5)-14(b5). There is only the rotating frequency 195.3 Hz in the amplitude spectrum and one point in Poincaré mapping. The system motion is stable.


## 5. Conclusions

In this paper, the dynamic model of high-speed spindle system with spindle-bearing joint and spindle-holder-tool joints has been built. From the practical viewpoint of application, the influence of speed and cutting force has been investigated. Based on the modeling and application results, the following conclusions can be drawn.(1)
A distribution spring model of the spindle-holder-tool joints was proposed and applied in the general high-speed spindle FEM model. The rationality of model was verified by comparing the distribution spring connection model, rigid connection model, and the experimental results.(2)
The influence of speed was analyzed. When the speed gradually increases, the nonlinear stability goes through chaos-quasiperiodic transition form between quasiperiodic and chaos-P3 periodic-periodic motions, illustrating that in high speed situation the spindle system is more stable.(3)
The influence of cutting force was analyzed. The spindle system is unstable either in chaos or in quasiperiodic state under low cutting force but more stable under large cutting force. Through the comparison with the actual manufacturing situation, it is found that the dynamic model built in this paper was suitable for large cutting force.The influence of nonlinear stability on high-speed spindle system includes many other aspects, such as affected parameters of chaotic motion, the stability of thin-walled workpiece, and bearing parameters. How to structure a nonlinear stability evaluation system and to analyze the effects of the parameters may be a good research subject in the future.



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*Source: 153504-2014-05-20.xml*