The finite element modeling and stability prediction of highspeed spindle system dynamics with spindleholdertool joints
Bo Wang^{1} , Wei Sun^{2} , Kunpeng Xu^{3} , Junyong Zhang^{4} , Bangchun Wen^{5}
^{1, 2, 5}School of Mechanical Engineering & Automation, Northeastern University, Shenyang, 110819, China
^{3}SANYHE International Holdings Co., Ltd., Shenyang, China
^{4}No. 93115 of the PLA of China, Shenyang, China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 16, Issue 3, 2014, p. 11881199.
Received 15 November 2013; received in revised form 29 December 2013; accepted 23 January 2014; published 15 May 2014
JVE Conferences
The stability of highspeed spindle system affects the surface finish and tool life directly, which is an important factor to evaluate its performance. Meanwhile, the spindle dynamics and cutting stability are affected by the structure and dynamics of spindleholdertool joints significantly. The joints are simplified as the distributionspring, and the FEM modeling process of spindle system is proposed based on the thought of parallel rotor system. Taking a vertical machining center as example, the effectiveness of the modeling method is verified. Starting from the stability evaluation criteria and different ways of getting FRF, the influence factors of unconditional and conditional stability regions are analyzed. Based on the proposed model, the influence laws of cutting stability on cutting force amplitude and speed are characterized by the threedimensional lobes, limit cutting depths and lobe intersections, which provide the theoretical basis for optimizing the processing and improving the cutting stability.
Keywords: highspeed spindle, spindleholdertool joints, parallel rotor system cutting, stability, affect factor.
1. Introduction
The connections between spindle, holder and tool are the power and precision transmission called spindleholdertool joints. The spindleholder joint is Conic Surface and the holdertool joint is cylindrical surface, the characteristic of which are high geometric accuracy, high repeated loading and high stiffness for spindle processing safely and reliably [1]. The spindleholdertool joints affect the spindle dynamic characteristic significantly. Erturk [2] discovered the spindleholdertool joints affected the modal of spindle making the inherence frequencies and modal vibration changed. Xing [3] and Zhengjia [4] researched the modeling method of spindle and joints. Therefore, the spindle system modeling method considering spindleholdertool joints characteristics should be researched for predicting the spindle dynamic effectively.
However, the researchers pay high attention to rolling bearing modeling but neglect the influence of spindleholdertool joints on spindle dynamic characteristics during the study at present. Rantatalo [6], Altintas [7] and Tu [8] explored influence of spindle dynamic characteristic on gyroscopic moment, shaft and bearing centrifugal force inducing by high speed. The angular contact ball bearing model was emphasized, but other joints were neglected. Shin [9] and Altintas [10] built the spindleholdertool joints model rigid connection. Rturk [11] considered the spindleholdertool joints concentrated spring model as bearing simplified model.
Machine tools or machining centers have the ability of resisting vibration during cutting process (including the forced vibration and selfexcited vibration) called cutting stability, or the vibration resistance. The chatter of machine tools derives from selfexcited mechanism during the manufacturing process, which affects the removal rate, process accuracy, production efficiency and tool service life greatly. The stability receives more and more attentions because of highspeed machining widely used in automotive, mold, aerospace and other precision manufacturing.
Insperger [1] created the stability of single DOF milling system by semidiscretization method. Tlusty [13] built the stability lobe diagrams during milling process by timedomain simulation. Schmitz [14] and Ahmadi [15] predicted dynamic characteristics and cutting stability of highspeed machining center by substructure. Weixiao [16] proposed limit criterion and analysis method of multiple DOF system on high inherence frequencies under high speed cutting condition.
The researchers explore the affected factors of stability including tool length, tool types and rotation speed with the study deepening. Tlusty [17] discussed the common modal of spindle with long end milling tool and analyzed the influence of stability on different tool lengths with changing speeds. Smith [18] and Davies [19] studied the stability of high slenderness ratio, low radial cutting depth and workpiece mass respectively. Ismail [20], Shin [21] and Altintas [22] predicted stability of nonequal interval cutter, ball end milling cutter and face milling cutter. Schmitz [23] proposed the FRF test method of nonrotation and rotation tool. Qinghua [24] researched the influence of stability on structure parameters.
The accurate dynamic characteristics of spindle system are used to structure designing and stability predicting, which is unable to satisfy the design demand without consideration of spindleholdertool joints dynamic characteristics. Therefore, the spindleholdertool joints method was built with distributing spring model, and the process of spindle system model is proposed based on the joints model thought of parallel rotor system by FEM. The effectiveness of the model on spindle system dynamics simulating is verified by experiment on a vertical machining center. From the stability evaluation criteria, the conditional stability region is also a key to process stability. Based on the proposed model, the influence laws of cutting stability on cutting force amplitude, speed and damping ratio are characterized by the three dimensional lobes, limit cutting depths and lobe intersections, which provide the theoretical basis for optimizing the processing and improving the cutting stability.
2. Modeling method of spindleholdertool joints
The local enlarge section of spindleholdertool is shown in Fig. 1. The joints consist of two parts: the spindleholder joint is conical surface producing friction force by drawer, which is the part of holder inserting spindle, the holdertool joint is cylindrical surface positioning by screws, which is the part of tool inserting holder.
Fig. 1. Local sectional drawing of spindleholdertool joints
2.1. FEM modeling principle of parallel rotors
The parallel rotors system consists of inner rotor and outer rotor. The unit nodes are numbered continuous when the rotor system is discrete. A transition section is added at the end of outer rotor, the information of which is given randomly. The assembly process includes giving the mass and stiffness matrixes of all the units firstly, assembling all the matrixes and removing the matrixes of transition section, finally, the mass matrix $\mathbf{M}$ and stiffness matrix $\mathbf{K}$ of parallel rotors could be got.
Assuming $p$ 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 is consist of two nodes and each node owns six degree freedoms. The assembly of adjacent units is shown as Fig. 2. The mass and stiffness matrices of whole system are 6$N$×6$N$ orders, from which the transition section matrices will be removed. The final complete mass and stiffness matrices of parallel rotor are shown as Fig. 3.
Fig. 2. Schemes of adjacent unit assembly
Fig. 3. Schemes of parallel rotors assembly
2.2. Dynamic simulation of spindleholdertool joints
The spindleholdertool joints consist of double parallel rotor systems, one is spindle and holder, the other is holder and tool. The sectional drawing of complete spindle system is shown as Fig. 4. Dynamic simulation of joints is illustrated taken the spindleholder joint as example. The distributionspring model of spindleholder joint is shown as Fig. 5 (Fig. 5(a) is the toper joint structure and Fig. 5(b) is the joint FEM).
Fig. 4. Sectional drawing of spindle system
Assuming the spindle and holder joint is divided into $n$ 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:
$\mathrm{\Delta}{F}_{j}={K}_{cp}{X}_{j}+{C}_{cp}{\dot{X}}_{j}{K}_{cp}{X}_{i}{C}_{cp}{\dot{X}}_{i},$
$\mathrm{\Delta}{F}_{i+1}={K}_{cp}{X}_{i+1}+{C}_{cp}{\dot{X}}_{i+1}{K}_{cp}{X}_{j+1}{C}_{cp}{\dot{X}}_{j+1},$
$\mathrm{\Delta}{F}_{j+1}={K}_{cp}{X}_{j+1}+{C}_{cp}{\dot{X}}_{j+1}{K}_{cp}{X}_{i+1}{C}_{cp}{\dot{X}}_{i+1},$
$\vdots $
$\mathrm{\Delta}{F}_{i+n}={K}_{cp}{X}_{i+n}+{C}_{cp}{\dot{X}}_{i+n}{K}_{cp}{X}_{j+n}{C}_{cp}{\dot{X}}_{j+n},$
$\mathrm{\Delta}{F}_{j+n}={K}_{cp}{X}_{j+n}+{C}_{cp}{\dot{X}}_{j+n}{K}_{cp}{X}_{i+n}{C}_{cp}{\dot{X}}_{i+n},$
where, ${\mathbf{K}}_{coup}$ and ${\mathbf{C}}_{coup}$ are coupling stiffness and damping matrices at spindleholder joint, ${\mathbf{X}}_{p}$ and ${\mathbf{X}}_{q}$ are the displacement vectors of spindle and holder at the joint, $\mathrm{\Delta}{\mathbf{F}}_{i}$ and $\mathrm{\Delta}{\mathbf{F}}_{q}$ are the coupling forces at spindleholder joint, $p$ and $q$ are node numbers at spindleholder joint.
The stiffness and damping matrices of spindleholder joint expressed with ${\mathbf{K}}_{c}$ and ${\mathbf{C}}_{c}$ are:
Fig. 5. The distributionspring model of spindleholder joint
a) Structure
b) FEM
The stiffness characteristic of distributionspring is shown in Eq. (2) and the damping characteristic is shown in Eq. (3). The assembly method of stiffness on joint is shown in Fig. 6.
Fig. 6. Stiffness assembly theory of spindleholder joint
2.3. Modeling process of spindleholdertool joints
The spindleholdertool joints modeling should be combined with the system modeling, which is not independent exit. Based on the parallel rotors idea, the joints are built with distributed spring model. The parameters of spring numbers, density, stiffness and damping are set, the spring numbers and density of which could be amendment with experiment. The modeling process of spindleholdertool joints is shown as Fig. 7.
Step 1: building the FEM model of spindle, holder and tool respectively. Firstly, the different sections are divided according to inner and outer diameters, secondly, two transition sections are added at the end of spindle and holder respectively, finally, the nodes of units are numbered continuously including the transition sections.
Step 2: creating the spindleholdertool joints model. The spring density, number, stiffness and damping are chosen primary. Then the distributionspring model is created with the divided units of joints using the spring stiffness and damping. The complete spindle system model will be got assembling the joints and spindle, holder, tool model.
Step 3: the modal test and dynamic simulation. The FRF at tool point of the built model is got though modal analysis, which is verified by modal test. The number and density of distributionspring will be corrected through the test and simulation results.
Step 4: the model application. The dynamic and stability prediction will be done with the complete spindle model considering the spindleholdertool joints.
Fig. 7. Modeling process of spindleholdertool joints
2.4. FEM of complete spindle system
The highspeed spindle system is established with isoparametric beam element, each unit of which exits 12DOF including 3 translational DOFs and 3 rotational DOFs on each node. The unit mass, stiffness and damping matrixes are assembled into the dynamic equation of system considering bearing force and spindleholdertool joints as follow:
$\mathbf{c}=\mathbf{D}+{n}_{\mathrm{\Omega}}\mathbf{J}+{\mathbf{k}}_{H},\mathbf{k}={\mathbf{k}}_{s}+{\mathbf{k}}_{C},\mathbf{X}={\left({\mathbf{q}}_{1},{\mathbf{q}}_{2},\dots ,{\mathbf{q}}_{53}\right)}^{T},$
${\mathbf{q}}_{i}={({x}_{i},{y}_{i},{z}_{i},{\theta}_{xi},{\theta}_{yi},{\theta}_{zi})}^{T},i=\mathrm{1,2},\cdots ,53,$
where, $\mathbf{m}$ is mass matrix; $\mathbf{c}$ is damping matrix; $\mathbf{k}$ is stiffness matrix; $\mathbf{D}$ is damping matrix; $\mathbf{J}$ is gyroscopic matrix; ${\mathbf{k}}_{H}$ is coupling stiffness matrix of joints; ${\mathbf{k}}_{s}$ is stiffness matrix of spindle; ${\mathbf{k}}_{C}$ is centrifugal stiffness matrix of spindle; ${\mathbf{F}}_{R}$ is supporting bearing force vector; ${\mathbf{F}}_{H}$ is hysteretic force vector of spindleholdertool; ${\mathbf{F}}_{P}$ is cutting force vector; $\mathbf{G}$ is gravity vector; $\mathbf{X}$ is displacement vector; ${\mathbf{q}}_{i}$ is displacement vector of node $i$; ${x}_{i}$, ${y}_{i}$, ${z}_{i}$, ${\theta}_{xi}$, ${\theta}_{yi}$, ${\theta}_{zi}$ are 3 translational and 3 rotational DOFs of node $i$.
The FEM model of highspeed spindle system in the freestate is shown as Fig. 8. There are 66 units and 66 nodes in the FEM including 2 transition sections. The structures of spindle system are shown in Table 1 and Table 2. The dynamic parameters of system are: elasticity modulus $E=$2.06×10^{11} Pa, material density $\rho =$7850 kg/m^{3}. The FEM of test object is built with 5distributionspring connection, 10distributionspring connection and rigid connection respectively.
Fig. 8. The model of high speed spindle system
Table 1. The size of motorized spindle, mm
No.

$\mathbf{L}$

$\mathbf{D}$

$\mathbf{d}$

No.

$\mathbf{L}$

$\mathbf{D}$

$\mathbf{d}$

No.

$\mathbf{L}$

$\mathbf{D}$

$\mathbf{d}$

1

14.50

120.00

70.00

11

15.00

150.00

55.00

21

11.30

129.90

43.46

2

25.00

149.90

70.00

12

30.00

150.00

65.00

22

11.30

129.90

46.76

3

15.00

149.90

55.00

13

30.00

150.00

65.00

23

11.30

129.90

50.06

4

15.00

149.90

55.00

14

30.00

150.00

55.00

24

11.30

129.90

53.36

5

25.00

149.90

65.00

15

25.00

150.00

55.00

25

11.30

129.90

56.66

6

30.00

149.90

65.00

16

25.00

150.00

55.00

26

11.30

129.90

59.96

7

25.00

149.90

65.00

17

25.00

150.00

70.00

27

11.30

88.00

63.26

8

25.00

149.90

65.00

18

15.00

150.00

55.00

28

11.40

88.00

66.56

9

30.00

150.00

65.00

19

15.00

150.00

55.00

29

5.00

88.00

69.85

10

30.00

150.00

65.00

20

11.30

149.90

40.16

Table 2. The size of holder and tool, mm
Holder

Tool


No.

$\mathbf{L}$

$\mathbf{D}$

$\mathbf{d}$

No.

$\mathbf{L}$

$\mathbf{D}$

$\mathbf{d}$

No.

$\mathbf{L}$

$\mathbf{D}$

$\mathbf{d}$

1

11.30

40.16

25.00

13

11.40

100.00

20.00

1

10.00

32.00

0.00

2

11.30

43.46

25.00

14

12.00

95.00

20.00

2

10.00

32.00

0.00

3

11.30

46.76

24.00

15

12.00

90.00

20.00

3

10.00

32.00

0.00

4

11.30

50.06

24.00

16

10.00

86.00

20.00

4

10.00

32.00

0.00

5

11.30

53.36

24.00

17

22.00

86.00

32.00

5

10.00

32.00

0.00

6

11.30

56.66

24.00

18

10.00

86.00

32.00

6

10.00

32.00

0.00

7

11.30

59.96

20.00

19

10.00

86.00

32.00

7

11.00

32.00

0.00

8

11.30

63.26

20.00

20

10.00

86.00

32.00

8

8.00

30.00

0.00

9

11.40

66.56

20.00

21

10.00

86.00

32.00

9

18.00

25.60

0.00

10

5.00

69.85

20.00

22

10.00

80.00

32.00

10

18.00

25.60

0.00

11

19.00

100.00

20.00

23

10.00

80.00

32.00

11

20.00

25.60

0.00

12

7.40

85.00

20.00

24

11.00

80.00

32.00

3. Model verification
3.1. Test object
The modal experiment is done on VMC0540d by hammering to prove the effective of modeling method for spindleholdertool joints and the whole spindle system. The results obtained from the experiment, 5distributionspring model. 10distributionspring model and rigid model will be compared by the frequency response function of tool point.
The test system is set up and shown in Fig. 9. The test object is a spindle system which consists of HCS150 motorized spindle from GMN, BBT50MEGA32D165 holder and HLXX32 end milling cutter. The main test equipments include the data acquisition, hammer and vibration sensors in Table 3.
Fig. 9. The test system
Table 3. The instruments used in this test
No.

Name

1

B&K 3560C data acquisition frontend

2

PCB 8206001 54627 modal hammer

3

B&K 4525B001 54072 acceleration sensor

4

Highperformance notebook computers

The FRFs at tool point of rigid connection joints, 5distributionspring joints and test results are show as Fig. 10(a), and the FRFs of 5distributionspring joints, 10distributionspring joints and test results are show as Fig. 10(b).
Fig. 10. Comparison of different joints models and test results
a) Different model types
b) Different spring density
The inherence frequencies of joints different forms are shown in Table 4 including rigid connection, concentrated spring model, different distributionspring models and test results. There is no rigid modal in test results. From the Table, the simulation errors of rigid model are largest than other connections from the test results, which is from 82.65 % to 158 %.
The Fig. 10 shows that the FRF of 5distributionspring joint model is closer to the test result than 10distributionspring model, and also closer than the rigid one. The Table 4 shows that the distributionspring model results are much closer to the test than the concentrated. For the spring layout form, the 5distributionspring joint model are more perfect than the 10distributionspring joints for the size of joints which is closer to the other unit in the model, which illustrates the influence of FRF on distributionspring form is obvious and the 5distributionspring joint model is verified effectively.
Table 4. Comparison of inherent frequencies of different joints models and test results
Order
Model types

Rigid modal

Elastic modal


First

Second

First

Second

Third

Fourth


Rigid connection

${\mathrm{f}}_{\mathrm{B}1}$ / Hz

315.52

502.32

1 336.4

2 579.7

2 974.0

4 133.4


$\frac{{\mathrm{f}}_{\mathrm{B}1}{\mathrm{f}}_{\mathrm{A}}}{{\mathrm{f}}_{\mathrm{A}}}\times 100$ %

–

–

143 %

158 %

142 %

82.65 %


Concentrated spring

${\mathrm{f}}_{\mathrm{B}2}$ / Hz

171.9

426.81

532.95

970.46

1 382.4

2 187.2


$\frac{{\mathrm{f}}_{\mathrm{B}2}{\mathrm{f}}_{\mathrm{A}}}{{\mathrm{f}}_{\mathrm{A}}}\times 100$ %

–

–

–3.1 %

–2.954 %

12.57 %

–3.358 %


Distribution springs

5 units

${\mathrm{f}}_{\mathrm{B}3}$ / Hz

148.56

419.67

530.95

978.33

1 212.3

2 277

$\frac{{\mathrm{f}}_{\mathrm{B}3}{\mathrm{f}}_{\mathrm{A}}}{{\mathrm{f}}_{\mathrm{A}}}\times 100$ %

–

–

–3.46 %

–2.167 %

–1.278 %

0.618 4 %


10 units

${\mathrm{f}}_{\mathrm{B}4}$ / Hz

125.4

423.92

531.75

880.61

1 120.8

2 327.1


$\frac{{\mathrm{f}}_{\mathrm{B}4}{\mathrm{f}}_{\mathrm{A}}}{{\mathrm{f}}_{\mathrm{A}}}\times 100$ %

–

–

–3.318 %

–11.94 %

–8.73 %

2.83 %


Test results

${\mathrm{f}}_{\mathrm{A}}$ / Hz

–

–

550.6

1 000.7

1 228.9

2 263.4

4. Influence factors and prediction method of cutting stability
4.1. Influence factors of stability
The cutting stability is evaluated through lobe diagram generally, which is characterized by cutting limit depth, as Eq. (5) [14]:
where, ${b}_{lim}$ is the maximum cutting depth without chatter, $Z$ is cutter teeth, ${K}_{f}$ is cutting force coefficient in feed direction, $G\left({\omega}_{c}\right)$ is real part of FRF, ${\omega}_{c}$ is angular frequency of negative FRF.
From the Eq. (5), the zone under ${b}_{lim}$ is unconditionally stability zone affected by $Z$, ${K}_{f}$ and $G$. The system will chatter when the cutting depth is higher than the ${b}_{lim}$ under specific speed theoretically. However, the emergence of conditionally stability zone will make the system stability with higher cutting depth. The conditionally stability zone is affected by lobe distribution, which still is controlled by FRF. Therefore, the FRF is a significant factor of cutting stability.
The FRF of system could be obtained by direct method and inherent characteristics method except modal test. The direct method is using the concept of frequency response function, which is the Laplace transformation ratio of response (or output) and excitation (or input) when the initial condition is zero in a linear constant system as Eq. (6):
where, $y\left(s\right)$ is Laplace transformation ration of response, $F\left(s\right)$ is Laplace transformation ration of excitation, $\psi \left(s\right)$ is frequency response function of system.
The inherent characteristics method begins with the essence of FRF. When the system equation is $\mathbf{m}\ddot{\mathbf{x}}+\mathbf{c}\ddot{\mathbf{x}}+\mathbf{k}\mathbf{x}=\mathbf{F}\left(t\right)\text{,}$ Laplace transformation of two sides is $\left(\mathbf{m}{s}^{2}+\mathbf{c}s+\mathbf{k}\right)\stackrel{}{\mathbf{x}}\left(s\right)=\stackrel{}{\mathbf{F}}\left(s\right)$, then there is:
where, ${\mathbf{H}}_{d}\left(s\right)$ is transfer function matrix of system, $\mathbf{F}\left(s\right)$ is Laplace transformation of excitation vector, $\stackrel{}{\mathbf{x}}\left(s\right)$ is Laplace transformation of response vector:
Take the Eq. (8) into Eq. (7), $s=j\omega $, the displacement FRF of excitation at $q$, and response at $p$ is as follow:
where, ${\phi}_{pi}$ and ${\phi}_{qi}$ are the $i$ order modal shape of $p$ and $q$, ${\xi}_{i}$ is damping ratio, ${\omega}_{i}$ is inherence angular frequency, $\omega $ is excitation angular frequency.
From the Eq. (7) to (9), there are many influence parameters on FRF including excitation $F$, response $y$, mass $m$, stiffness $k$, damping $c$, modal shape $\phi $, inherence frequency $\omega $, damping ratio $\xi $, even the spindleholdertool joints dynamic characteristics discussed in the paper above and rotational speed verified by [15]. This paper focuses on the influence of cutting stability on cutting force (excitation) and speed (process condition) with specific system.
4.2. Stability prediction process
Based on modeling the system with distributionspring joint model and solving the FRF, the lobe diagram will be drawn as follow:
Step 1: the FRF is divided into two parts: the real and imaginary part as $\varphi \left(j{\omega}_{c}\right)=G+jH$. Solve the value of ${\omega}_{c}$ to satisfy $G\left({\omega}_{c}\right)<0$ and obtain the ${b}_{lim}$ according to the ${K}_{f}$ and Eq. (5).
Step 2: positive and negative of imaginary part $H\left({\omega}_{c}\right)$ is judged by ${\omega}_{c}$. If $H\left({\omega}_{c}\right)<0$ then the phase angle of system will be $\psi =\pi +\text{arctan}(H/G)$ or $\psi =\pi \text{arctan}\left(\rightH/G\left\right)$. The rotational speed $n$ will be got according to $\epsilon =3\pi +2\psi $, $T=(l+\epsilon /2\pi )/{f}_{c}$ and $n=60/T$.
Step 3: the ${b}_{lim}$ and $n$ will be obtained in each lobe. The lobe diagram will be formed through drawing several lobes in the picture.
5. Influence rules of cutting stability
5.1. Influence of cutting force
The cutting force during the process is simulated by harmonic force, and the influence rule of stability is analyzed by changing the force amplitude. The threedimensional lobe diagram with changing forces is shown as Fig. 11(a) and stability lobe diagram under two different forces is shown as Fig. 11(b), both of which show the limit cutting depth reduces as the force increases and the conditional zone affected by the force.
The limit cutting depths and lobe intersections correspond to the critical speeds changing with forces are shown as Fig. 12. From the Fig. 12(a), the limit cutting depths are reduced as the force increasing, which are affected greatly when the cutting force is small or gently when it is bigger. The first intersection is the one of $l=$0 and $l=$1, then the second intersection is the one of $l=$1 and $l=$2 from the Fig. 12(b), which shows the intersections move to the left as the force increasing. Therefore, the conditional zones are changed with the forces.
Fig. 11. Stability lobes of different cutting forces
a) Threedimensional lobe
b) Lobe under different cutting forces
Fig. 12. Stability influence rules of cutting forces
a) Minimum limit cutting depth
b) Lobe intersection
5.2. Influence of rotational speed
The threedimensional lobe diagram of stability with changing rotational speeds is shown as Fig. 13(a) and stability lobe diagram under two different sweep speeds are shown as Fig. 13(b). As the rotational speed rise, the limit cutting depths increase, which illustrates the cutting stability strengthen.
Fig. 13. Stability lobes of different sweep speeds
a) Threedimensional lobe
b) Lobe under different sweep speeds
The limit cutting depths and lobe intersections correspond to the critical speeds changing with rotational speed are shown as Fig. 14. From the Fig. 14(a), the limit cutting depths are enlarged as the speed increasing, but the first order remarkably and the secondly order is slow. Fig. 14(b) shows the intersections move to the right as the speed increasing, which states the conditional zones change complicated.
Fig. 14. Stability influence rules of sweep speeds
a) Minimum limit cutting depth
b) Lobe intersection
6. Conclusions
In this paper, the dynamic model of spindleholdertool joints has been built and assembled into the complete highspeed spindle system model. From the practical viewpoint of application, the influence rules of cutting force and speed have been investigated. Based on the modeling and application results, the following conclusions can be drawn:
(1) The model of spindleholdertool joints were created with distributionspring method, the joints model was assembled into the whole spindle system model and the modeling process was established.
(2) The distributionspring method was verified by other simulation method and test, and the arrangement form of distributionspring method was not the better the denser, but its length was close to other units because of the joints model truly reflect the actual one.
(3) The novel viewpoint was proposed that the conditional zone was as significant as unconditional zone in improving the stability. From the analysis of influence rules, the stability declined as the cutting force rising, the unconditional zone was expanded as the rotational speed increasing but the conditional zone moved to the right, which implies the influence of speed on stability is complicated, so the particular speed needs particular analysis.
From the investigation, the distributionspring method is the closest arrangement form to the real spindleholdertool joints. How to apply the joints model into the nonlinear stability evaluation and influence parameters analysis may be a good research subject in the future.
Acknowledgements
This work supported by the National Natural Science Foundation of China (50905029), National Science Foundation for Postdoctoral Scientists of China (2013M541239), Research Foundation of Education Bureau of Liaoning Province, China (L2013115), Fundamental Research Funds for the Central Universities of China (N110603004).
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