- Email: [email protected]
Dynamic characteristics of spindle with water-lubricated hydrostatic bearings for ultra-precision machine tools
Journal Pre-proof Dynamic characteristics of spindle with water-lubricated hydrostatic bearings for ultraprecision machine tools Dmytro Fedorynenko, Rei Kirigaya, Yohichi Nakao PII:
S0141-6359(20)30104-5
DOI:
https://doi.org/10.1016/j.precisioneng.2020.02.003
Reference:
PRE 7094
To appear in:
Precision Engineering
Received Date: 5 January 2020 Accepted Date: 15 January 2020
Please cite this article as: Fedorynenko D, Kirigaya R, Nakao Y, Dynamic characteristics of spindle with water-lubricated hydrostatic bearings for ultra-precision machine tools, Precision Engineering (2020), doi: https://doi.org/10.1016/j.precisioneng.2020.02.003. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Inc.
Dynamic Characteristics of Spindle with Water-lubricated Hydrostatic Bearings for Ultra-precision Machine Tools
Dmytro Fedorynenko1, Rei Kirigaya2 and Yohichi Nakao2*
1
Department of Aerospace Engineering Tohoku University, Sendai, JAPAN
2
Department of Mechanical Engineering Kanagawa University, Yokohama, JAPAN
* Corresponding author. Postal address: Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686 Yokohama, Japan E-mail address: [email protected] Phone : +81-45-481-5661 ext. 3489 Fax
: +81-45-491-7915
1
Abstract A key component of ultra-precision machine tools is the spindle. The motivation for this study was to improve machining accuracies in precision cutting and grinding by pursuing improvements in the spindle characteristics by designing a sophisticated spindle with water-lubricated hydrostatic bearings. The static bearing stiffness of the developed spindle was investigated in previous studies. In addition to the static bearing stiffness, the dynamic characteristics regarding bearing stiffness also affect significantly on the machining results. In this study, dynamic characteristics of the developed spindle with water-lubricated hydrostatic bearings were investigated via simulations and experiments. Not only bearing dynamics but also rotor dynamics were considered in this study. In the simulation studies, the spindle dynamic characteristics were analysed based on the transfer matrix method. A spindle rotor supported with hydrostatic bearings was represented by discrete sections of the rotor. The mathematical model of transverse linear vibrations of the spindle rotor was derived with distributed parameters for these discretized rotor sections. As a result of the analysis on the amplitude-frequency characteristic, radial displacements of the rotor due to bearing displacement and bending deformation were defined. Then, the frequency characteristics were represented with Nyquist plots. Resonant frequencies and amplitudes formation in the transverse vibration of the rotor were determined. The influence of rotor bending deformations on spindle compliance was assessed. Furthermore, the study examined the influences of the supply pressure of the lubricating fluid, radial clearance and journal diameter of the hydrostatic bearings on the amplitude of the rotor vibration, and the resonance frequency of the system.
2
Furthermore, the dynamic characteristics of the spindle were examined experimentally. The simulation results were in good agreement with the actual spindle dynamics obtained experimentally. The influence of the structural parameters of the rotor and the operating parameters of the bearings on the spindle dynamic characteristics was also determined. It was verified that the amplitude of the vibration of the rotor overhang part was dominantly affected not by bearing stiffness but by bending stiffness of the bearing journal of the front bearing and the length of the rotor overhang. Then it was verified that the resultant displacement of the rotor in the radial direction due to the influence of the bearing characteristics and the structural effect of the rotor is significantly small. Practical recommendations to improve the spindle design in terms of the dynamic characteristics of the spindle with water-lubricated hydrostatic bearings were also derived.
Keywords Spindle, Hydrostatic bearings, Machine tools, Dynamic characteristics, Transfer matrix method, Dynamic compliance, Resonant frequency, Vibration amplitude
3
1. Introduction Precision machine tools are quite indispensable to the manufacture of precision parts. For instance, ultra-precision machining using precision machine tools is widely used for manufacturing precision parts for the electronic, medical, automobile, optical equipment, and other various leading-edge industries. The ultra-precision machine tools are used, for instance, for the cutting operations using a single-crystal diamond cutting tool. A critical component of the precision machine tools that determines the quality of machining is the spindle. Spindles with hydrostatic or aerostatic bearings are widely adopted for ultra-precision machine tools. Previous study [1] found that hydrostatic bearings for the machine tool spindle increase the machining reliability and productivity. Accordingly, hydrostatic and aerostatic bearings are the most common bearings for ultra-precision machine tools [2]. In general, aerostatic bearings are widely used for the spindle of the ultraprecision machine tools. A benefit of the aerostatic bearings is in its low viscosity, causing less heat generation during high-speed operation of the spindle. In contrast, the low damping and stiffness characteristics because of air compressibility are a disadvantage of aerostatic bearings. Oil hydrostatic bearings are advantageous for designing ultra-precision spindles with high stiffness and damping characteristics. However, oil has high viscosity that leads to larger heat generation due to the viscous friction during spindle rotation and, as a result, to significant thermal deformations of the spindle. Accordingly, it degrades machining accuracy. On the other hand, water is incompressible fluid that has relatively low viscosity in comparison with oil. Moreover, water has a higher specific heat and thermal conductivity. Thus, it is considered that
4
higher thermal stability can be achieved by using water as lubricating fluid simultaneously providing high bearing stiffness as well. To cope with the disadvantage of both the aerostatic and oil hydrostatic bearings, water-lubricated hydrostatic bearings has been studied for spindle applications. A spindle driven by water flow power was studied prior to designing the spindle with water-lubricated hydrostatic bearings [3]. A spindle with water-lubricated hydrostatic bearings was then designed [4]. Static stiffness of the bearings [5], the patterns of spindle temperature field change [6] and the thermal stability of the spindles with different rated rotational speeds were investigated [7, 8]. In the turning operation with a constant depth of cut condition, the actual depth of cut is however slightly changed, in the strict sense, due to the spindle runout even hydrostatic bearings support spindle rotor. As a result, the resultant cutting force acting on the spindle is not necessarily constant. In this case, a principal frequency of the cutting force is basically determined by the spindle rotational speed. For example, the principal frequency becomes 50 Hz, if the spindle rotational speed is 3,000 min-1. In addition, the cutting force is also changed, if a control of the depth of cut is synchronized with the spindle rotation using a fast tool servo system. The small variation in the cutting force affects a spindle motion that depends on the dynamic characteristics of the spindle. The resultant spindle motion due to the time variance cutting force affects a quality of the finished surface. Accordingly, not only static characteristics but also dynamic characteristics of the bearings and spindle rotor are of crucial importance in the consideration of the structural design for spindles. The dynamic characteristics can be represented in terms of the dynamic stiffness, damping, vibration modes, and frequency characteristics of the overall system of the spindle. Dynamic analysis of the spindle, including the influence
5
of external forces, such as the cutting forces, introduces useful information for determining the structure of the spindle in the design process. Therefore, the influence of the time-varying external forces on the spindle dynamic characteristics is considered in this paper. Various experimental and theoretical approaches have been applied to consider dynamic characteristics of machine tool spindles [9]. Harmonic and impulse testing methods can be possible methods based on the experimental dynamic analysis. Experimental evaluation of dynamic coefficients [12] is also the promising method to study spindle dynamics. However, the experimental methods require special facilities and time-consuming tasks. Furthermore, the experimental approaches cannot be applied before or during the design of a new spindle. To avoid these disadvantages of the experimental procedures, theoretical methods, such as those based on the finite element [14] and transfer matrix methods [13], are widely used for the dynamic analysis of the spindle. If the concentric condition of the spindle shaft, namely no-displacement of the shaft from the ideal center axis can be assumed, exact solutions without linearization procedure for the hydrostatic bearings can be introduced [16]. However, the spindle shaft can be supported at a certain offcenter position in many actual operations. The finite element method is a versatile numerical analysis tool for investigating the dynamic analysis of mechanical system. A diversity of commercialized computeraided engineering systems implementing the finite element method can be used for dynamic analysis of spindles. However, the main difficulty of using commercial computer-aided engineering software arises in description of the elastic-dissipative properties of the hydrostatic bearings of the spindle to be analysed [15].
6
To avoid this difficulty, the analysis using discrete beam model representing the spindle shaft is one of the most promising ways for the finite element analysis of spindles with hydrostatic bearings. A finite element model of hydrostatic bearings characterized by 36 linear coefficients of stiffness and damping was thus proposed [10]. It should be noted that introducing a large number of different elements of the spindle shaft is needed for spindle analysis based on the special beam finite elements. This leads to considerable difficulties in formalization of the calculation method and its implementation on a computer to carry out the analysis. In contrast, the transfer matrix method is widely used for the purposes of analysis of spindle linear dynamics [11]. The main advantage of the method is convenience and the ease formalizing the mathematical model. Accordingly, the extensive capabilities of the transfer matrix method can be used to consider the elastic-dissipative properties of hydrostatic bearings and spindle structures, including couplings and chucks connected at the ends of the spindle shaft as needed. In general, the spindle for ultra-precision machine tools is driven under the condition of small external forces. Accordingly, the amplitudes of vibration in both the radial and axial directions of the spindle are extremely small. In general, superior stiffness and dumping characteristics of the oil or waterlubricated hydrostatic bearings can be obtained as discussed in this paper. An aim of this study is to investigate the dynamic characteristics of the developed spindle with water-lubricated hydrostatic bearings via theory and experiments. Some of the important design guidelines for spindle design with hydrostatic bearings are derived through the investigations discussed in this paper.
7
In the rest of this paper, we present an explanation of the studied spindle supported with water-lubricated hydrostatic bearings followed by a mathematical model and an analysis procedure. Experimental results are then given. Finally, the dynamic characteristics of the spindle obtained via simulations and experiments are discussed.
Nomenclature A
amplitude of the rotor oscillations
A, B, C, D
complex functions of elasticity
A0
amplitude of the rotor oscillations when f=0
Ab
amplitude of the rotor oscillations in the bearing
Аlc
local resonance amplitude
Ar
resonant amplitude of the spindle
Cu, Cφ, Ccu, Ccφ complex modules of elasticity in the rotor cross-section с0
bearing stiffness coefficient
cav
average dynamic stiffness
cb
distributed stiffness coefficient in the bearing
cu, cφ, ccu, ccφ
stiffness coefficients in the rotor cross-section
Db
bearing journal diameter
e
eccentricity in the bearing
E
Young's modulus
F
external force
Fe
force due to elongation of viscoelastic environment
f
oscillation frequency
fr
resonant frequency of the spindle
8
h0
bearing damping coefficient
hb
distributed damping coefficient in the bearing
hm
damping coefficient in the rotor characterized internal damping
hu, hφ, hcu, hcφ
damping coefficients in the rotor cross-section
i
sections or cross-sections number
J
axial moment of inertia
Jµ
axial moment of inertia of the concentrated mass
K1, K2, K3, K4
bearing coefficients
k
cross-section number where external force F acts
ks
spindle static compliance
Lb
bearing length
Lr
overall rotor length
l
rotor length
la
axial land width
lc
length of the rotor overhang area
lτ
inter-recess land width
M
bending moment
Mex
external bending moment
m
rotor mass
N
load vector
Nk
load vector in cross-section number k
n
the end cross-section of the rotor
nr
rotation speed of the rotor in rev/s
P
general transition matrix
9
Pk
matrix which is equal the product of general transition matrices Pi from i=k+1 to i=n
PΣ
matrix which is equal the product of all general transition matrices Pi from i=1 to i=n
pr
recess pressure
ps
supply pressure
Q
transverse force
Qex
external transverse force
Rh, γ
flow factors
R0
restrictor impedance
r
number of recesses
S
matrix of the cross-section
t
time
U
transition matrix of the section
u
rotor displacement in the horizontal plane
WFk
frequency transfer functions of the spindle
Z
vector
z
axial coordinate of the rotor
β
gauge pressure ratio when e=0
δ0
bearing clearance when e=0
ε
elongation
φ
rotation angle at bending
η
dynamic viscosity coefficient
ϕ
phase of oscillations
10
λ
complex coefficient of elasticity
µ
concentrated mass
ω
oscillation frequency in rad/s
ωn
bearing resonant frequency in rad/s
bearing damping ratio
2. Schematics of investigated spindle supported with water-lubricated hydrostatic bearings 2.1 Spindle structure The investigated
spindle with
water-lubricated
hydrostatic bearings
is
schematically presented in Fig. 1. The spindle was designed for the turning operation using a single-point diamond cutting tool. The spindle has four recessed journal bearings and four recessed double-sided thrust bearings. The hydrostatic bearings have short pipe restrictors [21] for each recess. A displacement pump supplies pressurized water to the bearings. Parameters of the short pipe restrictors are determined to achieve the highest static bearing stiffness for the given conditions in both the radial and axial directions. In our series of simulations, the gap of the radial bearing was δ0=12 µm. The rotor is made of SUS30 stainless steel [5]. Diamond-like carbon is plated onto the outer surfaces of the rotor and the inner surfaces of the casing to prevent undesirable damage during spindle operation. The dimensions of the rotor are given in Fig. 2.
2.2 Rotor structure and related parameters for analysis
11
To simplify the calculation model, the spindle rotor is represented as divided sections with constant diameter di and length li for the i-th section. Then each divided section can be considered to be stepped beams as depicted in Fig. 2. The moments of inertia are slightly different depending on the divided sections. Then, the average values of the moment of inertia Ji and the section mass mi were calculated for the combined rotor areas. In the analysis, small chamfers, grooves, slots, etc., were not considered. The entire elastic rotor system is presented as a discretized model with short rotor sections. A schematic drawing of the spindle rotor is given in Fig. 2. As depicted, the spindle rotor is divided into 15 sections by 16 indexes. The hydrostatic journal bearings are located between sections 5 and 6 and 11 and 12. The bearing section with length Lbla is represented with distributed characteristics of stiffness cb and damping hb. Each section of the rotor depicted in Fig. 2 is characterized by the bending stiffness EJi and mass mi per unit length li. External harmonic force F is exerted at cross-section 16 and has an amplitude of 1 N that corresponds to the highest possible cutting force during turning with a diamond tool [18]. The damping coefficient of the spindle structure is negligibly small compared to that of the hydrostatic bearings (hm<
3. Mathematical Modelling 3.1 Model of elasticity of spindle rotor In the theoretical analysis used in this paper, discrete sections of the rotor are created so that each section has same cross section and diameter as shown in Fig. 2. As
12
a rule, the influence of longitudinal perturbations on the transverse vibrations of the rotor is negligibly small. Accordingly, only the transverse vibrations of the spindle rotor are considered in the study. The cross sections of the rotor under its dynamic motions are considered to remain flat. In addition, the centers of gravity of the discrete sections of cross-sections are assumed to be located along a line corresponding to the center of gravity of the entire rotor. It is furthermore assumed that the center of gravity of the defined rotor sections is coincident with a neutral line of the bending shaft along the entire rotor length. The spindle system considered in the paper was designed for use in ultra-precision machine tools. Accordingly, the external forces acting on the spindle and spindle rotor to disrupt its balance are sufficiently small [18]. Thus, it can be assumed that the spindle system can be represented as a linear oscillating system. In this case, the dynamic characteristics of the spindle vibrating system can be solved by calculating the transfer functions derived for given frequencies. Then, the differential equation presenting the rotor bending oscillation with dissipation effects in the hydrostatic bearings is derived [19]. In the equation, the external force F, such as cutting force during machining operations acting on the rotor, is considered. As presented in Eq. (1), the sinusoidal external force given in right-hand side in Fig. 2 is considered as a function of angular velocity, frequency characteristics are introduced. EJ ( z )
∂ 4u ∂ 2u ∂u + m ( z ) + h( z ) + c ( z )u = F ( z ) sin ωt . 4 2 ∂z ∂t ∂t
(1)
According to the structure of the studied spindle rotor depicted in Fig. 2, radial hydrostatic bearings are located between cross-sections 5 and 6 and 11 and 12. Now,
13
consider the solution of Eq. (1) to express the dynamics of the radial bearing section of the rotor with a constant cross section. External force or moment does not act on the bearing sections of the spindle, as a rule. Thus, the right-hand side of Eq. (1) is equal to zero. Thus, Eq. (1) with two variables z and t can be converted to an equation with one variable. In general, taking into account the dissipation of energy at the hydrostatic bearings, displacement of any section of the rotor is associated with time dependence, as given by Eq. (2).
u ( z , t ) = u ( z ) e jω t ,
(2)
where = √−1. Substituting Eq. (2) into Eq. (1), Eq. (3) is then derived.
u IV −
λi4 li4
u =0,
(3)
where
λi4 =
(
li4 miω 2 − c0 − jh0ω EJ i (1 + jhm )
).
Detailed procedures on determining the parameters in Eq. (3) are given in section 3.2. In this paper, the dynamic characteristics of the spindle rotor represented by Eq. (1) for discretized sections of the rotor are numerically solved by the transfer matrix method [16, 18]. In the transfer matrix method, the discretized sections are presented in matrix form. In the calculations, the variables at an end section of the rotor are first determined as boundary conditions at a given frequency with known values of displacement u, angle φ, internal force Q, and moment M. Thus, the dynamic characteristics of the
14
discretized rotor sections are characterized by the displacement u, angle φ against the internal force Q, and moment M acting on the sections [17]. Magnitudes of these variables at the i-th section of the rotor can be presented by four-dimensional vector as given in Eq. (4).
Z i = [ui φi
Mi
Qi ] . T
(4)
In the calculation, the boundary conditions are given at the left end of the rotor, the index number 1, as depicted in Fig. 2. Accordingly, the transition matrix from vector Zi-1 to vector Zi is presented as Eq. (5) [17].
Zi = Pi Z i −1+ Ni .
(5)
General transition matrix of the i-th section Pi is given as Eq. (6).
Pi = Ui Si .
(6)
In Eq. (6), the transition matrix of the i-th section Ui can be represented as Eq. (7) [17]. Ai 4 λi Di l Ui = 4 i λi EJ iCi li2 4 λi EJ3 i Bi li
li Bi Ai
λi4 EJ i Di li 4 λi EJ iCi li2
li2Ci EJ i li Bi EJ i Ai
λi4 Di li
li3 Di EJ i li2Ci EJ i . li Bi Ai
(7)
where functions Ai, Bi, Ci, and Di are given by Eqs. (8)-(11).
Ai =
Bi =
Ci =
(cosh λi + cos λi ) 2
(sinhλi + sin λi ) 2λi
(coshλi − cosλi ) 2λ2i
(8)
(9)
(10)
15
Di =
(sinh λi − sin λi ) 2λ3i
.
(11)
In addition, matrix Si in Eq. (6) is defined as Eq. (12).
1 0 Si = 0 2 µiω − Cui
0
−
0
− J µiω 2 + Cφi
1 Ccφi 1
0
0
1
1 Ccui 0 . 0 1
(12)
In Eq. (5), the load vector Ni is presented by a vector form as Eq. (13).
[
Ni = 0
0
M iex
Q exi
]
T
.
(13)
Complex modules of elasticity Cu, Cφ, Ccu, and Ccφ are defined by the elasticdissipative characteristics of the lumped parameters of concentrated mass Cu and Cφ (e.g., pulleys and gears) or the parameters of tool shank Ccu and Ccφ relative to rotor radial displacement u and angular φ⋅z displacements. The mentioned modules are determined by Eqs. (14) - (17).
Cui = cui + jhui
(14)
Cφi = cφi + jhφi
(15)
Ccui = ccui + jhcui
(16)
Ccφi = ccφi + jhcφi .
(17)
In Eqs. (14)-(17), indexes u and φ present bending radial and angular components of the complex modules of elasticity, respectively. Hence, the rotor dynamics can be determined by solving the matrix equations. With the matrix equations, the parameters of rotor sections, such as u, φ, Q, and M, can
16
be determined by the linking matrix from the initial cross-section, namely i=1, to the end section i=n. Thus, the matrix Zn is represented by Eq. (18).
Z n = PΣ Z1 + Pk N k .
(18)
Matrixes PΣ and Pk in Eq. (18) are determined by Eqs. (19)-(20). n
PΣ = ∏ Pi
(19)
i =1 n
Pk = ∏Pi .
(20)
i =k +1
Matrix Nk can be determined from Eq. (13), where i equals k. As an example, let us consider the definition of the components of the matrix Z1 for the particular rotor structure shown in Fig. 2. The matrix Z1 can be obtained using Eq. (18), provided that force Q1 and moment M1 in the initial cross-section are zero. Force Qn and moment Mn in the end cross-section equal 1 N and 0 N⋅m, respectively. Thus, by considering that the external force is acting in the end cross-section n (n=k), Eq. (18) can be given as u n P11 φ P n = 21 0 P31 1 P41
P12
P13
P22
P23
P32
P33
P42
P43
P14 u1 P24 φ1 × , P34 0 P44 0
(21)
where P11..P44 are the components of the transition matrix P. By writing in algebraic form the equations regarding the last two elements of the matrix Zn in Eq. (21), we can find the parameters u1 and φ1 from Eqs. (22)-(23).
0 = P31u1 + P32φ1
(22)
1 = P41u1 + P42φ1
(23)
17
Finally, it is possible to introduce the frequency transfer functions of the spindle to represent the relationship between the external force F and the rotor displacement u at arbitrary elements, in which the external force F acts at the cross-section k. For example, the dynamic compliance in the i-th cross-section of the rotor is given as Eqs. (24)-(25):
WFk ( jω ) =
ui ( jω ) Fk ( jω )
WFk ( jω ) = Re(ω ) + j ⋅ Im(ω ) .
(24)
(25)
From Eq. (25), the amplitude and the phase-shift regarding the rotor dynamics can be introduced as Eqs. (26) and (27), respectively. A(ω ) =
Re 2 (ω ) + Im 2 (ω )
Im(ω ) . Re(ω )
ϕ (ω ) = tan −1
(26)
(27)
Nyquist plots will be then introduced using Eqs. (26) and (27). Furthermore, determining the rotor axis deformation allows us to investigate the static and dynamic compliance over a wide frequency range. The calculation results for the spindle are presented below.
3.2 Stiffness and damping coefficients of hydrostatic bearing
The sections of the spindle rotor supported by hydrostatic bearings can be represented as a beam on a Winkler foundation with evenly distributed elasticdissipative elements. Then, the bearing characteristics represented by the Kelvin-Voigt model, as given by Eq. (28). In Eq. (28), the linearized radial stiffness and damping of the hydrostatic bearing are considered to be parallel-connected elastic and dissipative elements.
18
Fe = с0ε + h0
dε . dt
(28)
In general, the matrix determining stiffness and damping of hydrostatic bearings has a dimension of 6 × 6. The cross-coefficients in matrices determine non-collinear displacements or velocities of bearing surfaces relative to the vector of external loads. Let us assume that the bearing stiffness and damping coefficients are independent of the rotational speed of the spindle and the frequency of the external force. This assumption is fairly true for low-speed spindles with sliding bearings [16, 20]. For ultra-precision machining, the rotor offsets along both the radial and axial directions are negligibly small. In addition, the compressibility of the lubricating fluid, such as oil and water, can be neglected. In this case, the coefficients of c0 and h0 can be computed as in Eqs. (29) and (30) [16].
(1 − la Lb )β 3 ps Lb Db r 2 sin 2 (π r ) с0 = 2 2πδ 0 R0 + 1 + 2γ sin (π r )
h0 =
2 2 12 ps Lb Db r 2 Rh sin 2 (π r ) (la Lb )(Lb Db ) (1 − la Lb ) , 2 ( ) πnrδ 0 R + 1 + 2 γ sin π r 0
(29)
(30)
where
β = pr p s
(31)
R0 = 0.5β (1 − β )
(32)
ηn D Rh = r b ps 2δ 0 γ=
2
rla Lb la 1 − . πlτ Db Lb
(33)
(34)
Coefficients of the stiffness cb and damping hb for the distributed bearing sections are determined based on the rotor section length li by Eqs. (35) and (36). 19
cb = c0 li
(35)
hb = h0 li .
(36)
4. Experiments The experimental apparatus for the measurement of the dynamic characteristics of the spindle is shown in Figs. 3(a) and (b). In the experiments, sinusoidal forces with various frequencies were applied to the end of a rotor in the radial direction. The measurements were carried out in a horizontal plane as depicted in Fig.3(b). The forces were generated by the driving force of a piezo actuator (Matsusada Precision PZ20-85, Max. displacement: 45 µm with feedback control, Natural frequency: about 10 KHz) placed perpendicular to the rotor. The amplitude of the forces was controlled to be 1 N regardless of the frequencies of the applied sinusoidal forces. For the purpose of controlling and monitoring the applied forces, a load cell (Kyowa LMB-A-50N, Repeatability: < 0.3 %RO, Natural frequency: about 40 KHz) was used to measure the actual applied forces. As depicted in Fig. 3, the load cell was installed between the rotor and the piezo actuator. The frequencies of applied radial forces were changed in the ranges from 0 Hz to 1,000 Hz. The resultant sum of the radial displacement and deformation of the rotor was measured by a capacitance sensor (Nano Technology, 212S system, Resolution: 5 nm, Frequency range: 200 Hz) and a laser sensor (Ono Sokki LV-2100, Resolution: 1.5 nm, Linearity: ±0.1 %FS, Frequency range: 50 kHz) set at the opposite side of the rotor in a horizontal plane. In the series of the experiments, the water supply pressure for the hydrostatic bearings was set to 2.5 MPa. The experimental rig used in the measurements uses a contact type of the excitation facility. The limitation of the experimental rig used in the measurements is in 20
difficulty of measuring the spindle dynamics during spindle rotation. In contrast to the contact type of the excitation method, a non-contact type of the equipment has been developed and tested to investigate spindle dynamics [24]. Because of the non-contact feature, the developed excitation equipment can be used to measure spindle dynamics during spindle rotation, achieving measurement during actual machining conditions. Investigations of our spindle during actual machining conditions will be considered in our future work. In the experiments, a constant initial load was applied to the spindle shaft by the piezo actuator. The output of the displacement sensor was then defined to be zero. The tested spindle was placed in a temperature-controlled room. The temperature was set to be 20 ℃. During experiments, the room temperature was controlled within 20±0.5 ℃ with periods of several tens of minutes. In addition, required time for each measurement was at most 10 s. Accordingly, the influence of thermal deformation of the spindle and other rig on the measurements was negligibly small. Prior to the measurements of the dynamic characteristics of the spindle, the dynamic characteristics of the additional components, including the base plate and supporting block of the load cell were measured in a preliminary test. It was then verified that the additional components had a natural frequency around 360 Hz. In a series of experiments, the temperature of water supplied to the spindle was controlled by a chiller system. The water temperature was set to 20 ℃ by the chiller system. Accordingly, the influence of temperature change on the measured spindle dynamics was negligibly small.
5. Results and Discussion
21
The dynamic characteristics of the spindle were obtained, as presented in Fig. 4 by the transform matrix method [22, 23]. Figure 4 shows the spindle frequency response characteristics. The results indicate the sum of the resultant displacement of the rotor at the end, namely cross-section number 16 in Fig. 2, due to the external radial forces of F. The elastic system of the spindle has a resonant frequency of 6,106 Hz with amplitude Ar=156.2 nm, which is caused by the first natural frequency of the bending oscillation of the rotor. As the graph shows, the amplitude of oscillations, at crosssection 16, first decreases with increasing frequency to a magnitude of Аlc=26 nm, which is approximately 73% of the amplitude at a frequency of 0 Hz. Then, it increases monotonically to the resonant value of Аr. A decrease in the amplitude of the rotor vibration is observed in the frequency range up to about 1,000 Hz. This phenomenon is due to the influence of dissipative properties of hydrostatic bearings, that is, the high damping effects of the water-lubricated hydrostatic bearings. To investigate the dissipative properties of the journal bearings of the spindle, the displacement of the bearing is investigated with a simplified model. In the calculation, the spindle rotor is assumed to be a rigid body, which excludes the influence of rotor deformation. Furthermore, we assumed that the external force acts at a middle position of the journal bearings. Accordingly, the influence of the moment effect due to the external force acting on an end rotor section on the displacement was not considered. Because of the simplified model, the amplitude of rotor displacement differs from the simulation results presented in Fig. 4. However, the dumping characteristics of the bearings can be investigated.
22
The amplitude-frequency characteristic was obtained as represented in Fig. 5. In our investigation, a simplified model was obtained by the following dependency [16] when F=1 N:
Ab =
1 c0
K12 + K 22 , K32 + K 42
(37)
where
ω K1 = 1 − ωn
K2 = 2ϑ
2
ω ωn
(38)
(39)
2
2 2 ω 2 ω πnr h0 K 3 = 1 − − 2ϑ + ωn ωn c0
(40)
2 ω ω 1 − K 4 = 4ϑ ω n ω n
(41)
ω n = 2 c0 m
(42)
ϑ=
h0ωn . 2c0
(43)
The frequency response of the bearing calculated by the simplified model is given in Fig. 5. As described above, the amplitude of the rotor displacement presented in Fig. 5 differs from that shown in Fig. 4, because the external force is considered to act at the middle of the journal bearings in the simplified model. However, the obtained frequency response is a monotonically decreasing function of the frequency, which has the same pattern as the amplitude-frequency characteristic of the spindle in the frequency range from 0 to 1,000 Hz, as presented in Fig. 4. The influence of the bearing
23
displacement and structural deformation on the resultant amplitude is further investigated in this paper. In Fig. 6, the experimental results are presented. The sinusoidal external load and resultant displacement are depicted in Fig. 6(a). The frequency characteristics are given in Fig. 6(b). As is seen from Fig. 6 (a), at a frequency of 280 Hz, the phase lag between the external force and rotor displacement is small. The tendency of phase lag is also observed up to 1,000 Hz, as given in the Nyquist plot described below. In addition, the rotor displacement has a steady tendency to decrease in both the theoretical and experimental data, as given in Fig. 6(b), which is due to the influence of bearing damping on spindle dynamics at low frequencies. As shown in Fig. 6(b), the amplitude obtained experimentally has a small peak at a frequency of 360 Hz, which is caused by the influence of a natural frequency of the experimental apparatus verified in the preliminary experiments as explained in section 4. Nyquist plots derived from the theoretical analysis are depicted in Fig. 7. These plots allow more detailed analysis of the spindle dynamic characteristics. As given in Fig. 7(a), the Nyquist plot has a local resonance with amplitude Alc=25.2 nm at a frequency of 1,000 Hz with formation of the looping features. This is due to the influence of high damping of hydrostatic bearings obtained in the low-frequency region, as already described. The local resonance is accompanied by the abrupt change in the phase near a frequency of 1,000 Hz as presented in Fig. 7(b). As is seen from the Nyquist plot, rotor displacement Аr at the main resonance frequency (fr=6,105 Hz) increases by almost 4.5 times compared to the static compliance ks of the system. The fundamental frequency is determined by the rated rotational speed of the spindle. The rated rotational speed of the spindle is 3,000 min-1.
24
Accordingly, the fundamental frequency of the spindle becomes 50 Hz. In general, the fundamental frequency component plays an important role for the dominant dynamic characteristics. A main resonance frequency (fr=6,105 Hz) is much higher than the normal operational frequencies of the spindle. Furthermore, a damping characteristic of the hydrostatic bearings is dominant in the operational frequency range. These desirable dynamic characteristics achieve better machining characteristics due to the damping effect. Here, we describe how the spindle rotor deforms along the axial direction in the simulation. In Fig. 8, the sum of rotor displacements due to the sinusoidal external forces of amplitude 1 N with different frequencies is illustrated by the centerline of the rotor. As is seen in the graph, in the operating frequency range (f≤50 Hz), the amplitude A does not change significantly due to static characteristics, namely f=0 Hz. In fact, the amplitude of the rotor at f=50 Hz differs up to 9 % from the static mode, f=0 Hz. Thus, we can focus on the analysis of the rotor amplitude determined by the static characteristics, as described in the rest of this paper. A significant increase in the amplitudes of the vibration at the rotor overhang sections is observed in the first resonant frequency. In contrast, the amplitude between both journal bearings remains extremely small. According to the simulation results, it is established that the average dynamic stiffness cav of the rotor between bearings equals 493 N/µm in the frequency range up to the first rotor resonant frequency. The average dynamic stiffness cav is calculated between the cross-sections 4 and 13 illustrated in Fig. 2 as
= ∑!" #$
⁄9 , where Csi is the average dynamic stiffness of the rotor
section i in the frequency range up to the first resonant frequency. In addition, the
25
minimum dynamic stiffness of the rotor is observed at the overhang area and it is about 30 N/µm in the operating frequency ranges up to 50 Hz, as presented in Fig. 4. As clearly observed in Fig. 8, a main reason for the decreased dynamic stiffness in the overhang area is due to rotor bending. This suggests that the rotor structure, in particular the overhang area, has to be carefully designed so that the resultant actual displacement will be less than an acceptable magnitude. The influence of the supply pressure ps and clearance δ0 on the rotor amplitude A0 is shown in Fig. 9(a) and 9(b), respectively. As is seen in Fig. 9(a), an increase in the supply pressure ps contributes to a reduction in the amplitude due to the displacement A0 at the end of the rotor, where the cutting force is active. The reduction is up to 23% within the investigated range in Fig. 9(a). Furthermore, the first resonant frequency fr changes depending on the supply pressure ps; however, the magnitude of the change in the resonant frequency is negligible. As shown in Fig. 9(b), the decrease in the clearance δ0 of the bearings reduces the amplitude A0 of the static displacement as well. In addition, the reduction of the bearing clearance δ0 increases the resonant frequency fr of the bending vibration system. Figure 10 shows the influence of the diameter of bearing journals Db on the amplitude A0 and resonance frequency fr. The calculation was made with the following assumptions: Lb=Db, la=0.1Db, and Lr=250 mm. These are defined in Fig. 2 for of Db ranging from 30 mm to 70 mm. As is seen from Fig. 10(a), the diameter of bearing journals Db significantly affects both rotor amplitude A0 and its resonance frequency fr, unlike the influence of the parameters δ0 and ps, as already depicted in Fig. 9. A main reason is that the increase in the journal diameter contributes to a significant change in the rotor structural
26
characteristics, such as the bending stiffness, mass, and length (when keeping assumption of Lb=Db) of bearing section as well as bearing stiffness and damping. The dynamic characteristics presented in this paper indicate the sum of the influence of the bearing compliance and the bending deformation. Here, we consider the relationship between them and their dependence on the diameter of the rotor. In Fig. 10(b), the displacement of the rotor end position according to cutting force F is depicted with respect to the cause of the bearing compliance and bending deformation separately. In the calculations, the amplitude determined by the displacement of the rotor due to the bearing compliance was introduced by specifying an infinite structural stiffness of the rotor. This result is presented by curve 1 in Fig. 10(b). In contrast, the amplitude due to the bending deformation of the rotor was determined by specifying the infinite bearing stiffness and damping, which is given by curve 2. Thus, curve 1 shows the influence of the bearing parameters and curve 2 represents the influence of rotor bending stiffness on the rotor amplitude. The error (up to 15%) between the sum of the amplitudes of curves 1 and 2 in Fig. 10(b) and the solid curve in Fig. 10(a) arises due to the impossibility of using infinite values for Young's modulus, stiffness, and damping in the bearings in the model. As is seen in the comparison of both curves in Fig. 10(b), the increased spindle diameter contributes significantly to a decrease in the amplitude of the rotor due to the influence of bearing compliance. Our investigation verified that the rotor amplitude A0 is significantly affected by the front bearing journal diameter, its compliance, and length lc of the rotor overhang area. In contrast, the rotor displacement in the area between bearings occurs mainly due to bearing compliance, as given in Fig. 8.
27
In the presented paper, the dynamic characteristics of the spindle with waterlubricated hydrostatic bearings are investigated. Now, how the oil hydrostatic bearings and the water-lubricated hydrostatic bearings are different in terms of the dynamic characteristics is discussed. The use of oil as the lubricating fluid leads to change lubrication characteristics such as the thermal and chemical stabilities, viscosity and density. According to the mathematical model presented in chapter 3, the fluid viscosity has influence on the bearing damping coefficient h0, in terms of the spindle dynamics. Specifically, the change of the bearing lubrication from water to oil improves the damping characteristics, consequently decreases the oscillation amplitudes of the rotor. However, in the operating range determined by the rated spindle speeds (up to 3,000 min-1), the use of viscous oil lubrication does not benefit significantly in terms of reducing the amplitude of the rotor oscillations. For example, for the tested spindle, the use of the oil with viscosity ten times over than water leads to the decrease in the amplitude of rotor oscillations by about 20% at a maximum in the mentioned range of speeds in contrast with water-lubricated bearings. However, a better thermal stability of the spindle can be obtained by using the water as the lubricating fluid [8].
6. Conclusions In this paper, the dynamic characteristics of a spindle with water-lubricated hydrostatic bearings were investigated via simulations and experiments. In particular, the characteristics of transverse vibration due to the influences of structural characteristics of the spindle rotor and water-lubricated hydrostatic journal bearings were considered.
28
In the simulation studies, the spindle rotor was modelled using discretised sections. Then, the dynamic characteristics of the spindle were derived with the transfer matrix method. The simulation results were compared with the experimental results. The results from the considerations in this paper verified that the tested spindle with water-lubricated hydrostatic bearings has the characteristics of high dynamic stiffness within the operating range of the spindle. For example, the average dynamic stiffness of the rotor between bearings equals 493 N/µm, and the minimum stiffness of the rotor at the overhang part is 30 N/µm. We verified that the rotor vibration amplitude corresponds to a static deformation of the elastic axis when the spindle speed is less than its rated operating speed. The main resonant frequency of the system was observed to be 6,106 Hz. The resonant frequencies greatly exceed the range of operating speeds (up to 50 Hz) of the spindle. The structural parameters of the rotor have the major influence on the main resonant frequency. Accordingly, the influence of the water-lubricated bearing is not significant. This fact proves that water-lubricated hydrostatic bearings are advantageous in terms of not only static stiffness but also dynamic stiffness, including damping characteristics higher than those of aerostatic bearings. The difference in the fluid viscosity between water and oil affects the damping characteristics. However, the difference in the resultant oscillation amplitude of the rotor is not significant. The amplitude of vibrations in the rotor overhang area was determined primarily by bending stiffness of the bearing journal of the front bearing, its compliance, and the length of the rotor overhang area. The results suggest that to reduce rotor radial vibration, it is important to increase the bending stiffness of the overhang area so that radial vibration can be fully reduced to an acceptable level for finishing cut operations.
29
In addition, it is also effective to increase the pump pressure and to decrease the bearing clearance to reduce the radial vibration.
Acknowledgments The presented results are a part of project “Increasing of Precision Spindles Efficiency with Hydrostatic Bearings,” funded by the Matsumae International Foundation and Grant-in-Aid for Scientific Research (C) of the Japan Society for the Promotion of Science.
30
References
[1]
Perovic B., Hydrostatische Führungen und Lager: Grundlagen, Berechnung und Auslegung von Hydraulikplänen, Berlin: Springer-Verlag; 2012 (in German).
[2]
Wardle F., Ultra Precision Bearings. 1st ed. Woodhead Publishing; 2015.
[3]
Nakao Y., Mimura M., Spindle Motor Driven by Fluid Energy for Ultra-precision Machine Tool, Proceedings of ASPE 2002 Annual Meeting, pp. 215-218.
[4]
Nakao Y., Mimura M., Kobayashi F., Water Energy Drive Spindle Supported by Water Hydrostatic Bearing for Ultra-Precision Machine Tool, Proceedings of ASPE 2003 Annual Meeting, pp. 199-202.
[5]
Nakao Y., Suzuki K., Yamada K., Nagasaka K., Feasibility Study on Design of Spindle Supported by High-Stiffness Water Hydrostatic Thrust Bearing, International Journal of Automation Technology, 8(4); 2014, pp. 530-538.
[6]
Hayashi A., Nakao Y., Measurement and Evaluation of Temperature Change of Water Driven Spindle, Proceedings of ASME 2016 International Mechanical Engineering Congress & Exposition; 2016, p. V002T02A007.
31
[7]
Tsubasa Y., Hayashi A., Nakao Y., Fundamental Study on Thermal Stability of Micro Milling Spindle Supported by Water Hydrostatic Bearings under Spindle Rotation, Proceedings of 32th ASPE Annual Meeting; 2017.
[8] Nakao Y., Kirigaya R., Fedorynenko D., Hayashi A., Suzuki K., Thermal Characteristics of Spindle Supported with Water-Lubricated Hydrostatic Bearings, International Journal of Automation Technology, 13(5), 2019, pp.602-609.
[9]
Budak E., Machining Stability and Machine Tool Dynamics, Materials of 11th International Research/Expert Conference TMT, Tunisia; 2007.
[10] Savin L.A., Solomin O.V., Modeling of Rotary Systems with Bearings of Fluid Friction, Mashinostroenie; 2006 (in Russian).
[11] Danilchenko Y., Kuznetsov Y., Precision Spindles with Rolling Bearings. Ekonomichna Dumka; 2003 (in Ukrainian).
[12] Cheng F., Ji W., Simultaneous Identification of Bearing Dynamic Coefficients in a Water-gas Lubricated Hydrostatic Spindle with a Big Thrust Disc, Journal of Mechanical Science and Technology 30 (9); 2016, pp. 4095-4107.
[13] Hou Y., Wang S., Han Z., The Dynamics Modelling and Simulation for Coupled Double-rotor Spindle System of High Speed Grinder, Przegląd Elektrotechniczny 89; 2013, pp. 25-28.
32
[14] Liu D., Zhang H., Tao Z., Su Y., Finite Element Analysis of High-Speed Motorized Spindle Based on ANSYS, The Open Mechanical Engineering Journal 5(1); 2011, pp. 1-10.
[15] Chen W., Sun Y., Liang Y., Bai Q., Zhang P., Liu H., Hydrostatic Spindle Dynamic Design System and Its Verification. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 228(1); 2014, pp. 149-155.
[16] Rowe W. B., Hydrostatic, Aerostatic and Hybrid Bearing Design, Waltham: Butterworth-Heinemann Press; 2012.
[17] Ivovic V., Transition Matrices in Dynamics of Elastic Systems, Mashinostroenie; 1981 (in Russian).
[18] Masuda M., Maeda Y., Nishiguchi T., Sawa M., Ikawa N., Diamond Turning of an Aluminum Alloy for Mirror, CIRP Annals 38(1); 1989, pp. 111-114.
[19] Strutynskyi V., Fedorynenko D., Statistical Dynamics of Spindle Units with Hydrostatic Bearings, Publishing house “Aspect-polygraph”; 2011 (in Ukrainian). [20] Waumans T., Vleugels P., Peirs J., Al-Bender F., Rotordynamic Behaviour of a Micro-turbine Rotor on Air Bearings: Modelling Techniques and Experimental Verification, Proceedings of ISMA; 2006, pp. 181-197.
33
[21] Nakao Y., Nakatsugawa S., Komori M., Suzuki K., Design of Short-Pipe Restrictor of Hydrostatic Thrust Bearings, Proceedings of ASME 2012 International Mechanical Congress and Exposition; 2012.
[22] Fedorynenko D., Sapon S., Spindle Hydrostatic Bearings. ChNUT; 2016 (in Ukrainian).
[24] Fedorynenko D., Sapon S., Boyko S., Accuracy of Spindle Units with Hydrostatic Bearings, Acta Mechanica et Automatica 10 (2); 2016, pp. 117-124.
[24] Matsubara A., Yamazaki T., Ikenaga S., Non-Contact Measurement of Spindle Stiffness by Using Magnetic Loading Device, International Journal of Machine Tools and Manufacture, 71, August, 2013, pp. 20-25.
34
List of captions for illustrations
Fig. 1. Structural diagram of water-lubricated hydrostatic spindle
Fig. 2. Calculation scheme for spindle rotor
Fig. 3. Experimental rig for measurement of spindle dynamic characteristics (a) Main components for measurement (b) Experimental setup
Fig. 4. Amplitude-frequency characteristic of spindle obtained by simulations
Fig. 5. Amplitude-frequency characteristic of journal hydrostatic bearing
Fig. 6. Spindle dynamics obtained via experiments (a) Applied force and displacement at 280 Hz (b) Amplitude-frequency characteristics (square symbols are from experimental data; solid curve is from theory)
Fig. 7. Nyquist plot obtained via simulation (a) Frequency range 0-10,000 Hz (b) Local resonance at 1,000 Hz
Fig. 8. Sum of rotor displacements
35
Fig. 9. Influence of bearing parameters on dynamic characteristics (a) Influence of supply pressure (b) Influence of bearing clearance
Fig. 10. Influence of journal diameter on dynamic characteristics (ps= 2.5 MPa) (a) Overall amplitude (b) Contribution of bearing and rotor parameters to overall amplitude
Table 1 Constant parameters representing spindle for analysis
36
Fig. 1. Structural diagram of water-lubricated hydrostatic spindle
Fig. 2. Calculation scheme for spindle rotor
Piezo actuator
Water hydrostatic spindle
Load cell t F
Oscilloscope
Displacement sensor
Fig. 3. Experimental rig for measurement of spindle dynamic characteristics
160
Ar
Amplitude [nm]
140 120 100 80
1000 Hz
60 40 20 0
Alc 0
Frequency [Hz]
10000
Fig. 4. Amplitude-frequency characteristic of spindle obtained by simulations
Amplitude [nm]
2
1 200 Hz
0
0
Frequency [Hz]
1000
Fig. 5. Amplitude-frequency characteristic of journal hydrostatic bearing
40 Force
30
Force [N]
1
20 10
0
0 -10
-1
-20 -30
Displacement
-2 0
0.005
0.01
0.015
Displacement [nm]
2
-40 0.02
Time [s]
(a) Applied force and displacement at 280 Hz
Amplitude [nm]
40 35 100 Hz 30 25 20
0
Frequency [Hz]
1000
(b) Amplitude-frequency characteristics (square symbols are from experimental data; solid curve is from theory)
Fig. 6. Spindle dynamics obtained via experiments
ks
9795 Hz
Re [nm]
0 Hz
0 -80
-60
-40
-20
-20
0
20
40
60
80
100
Alc
-40 1000 Hz
5600 Hz
-60 6604 Hz
-80 -100 -120 -140
6106 Hz
-160
Ar
-180
-Im [nm]
(a) Frequency range 0-10,000 Hz -2
Re [nm] 35
20
-2.5
Alc
1000 Hz
-3 600 Hz
2000 Hz
-3.5 -4
5 nm
-4.5 400 Hz
3200 Hz
-5
-Im [nm]
(b) Local resonance at 1,000 Hz
Fig. 7. Nyquist plot obtained via simulation
Fig. 8. Sum of rotor displacements
6135
fr 40
6120
A0 35
6105
Frequency [Hz]
Amplitude [nm]
45
0.5 MPa 30
6090
1.5
Pressure [MPa]
3.5
(a) Influence of supply pressure 6400
fr
6300
40
A0
6200
35 6100 30
6000
3 µm 25
6
Frequency [Hz]
Amplitude [nm]
45
5900
Clearance [µm]
18
(b) Influence of bearing clearance
Fig. 9. Influence of bearing parameters on dynamic characteristics
80
7000
Amplitude [nm]
6000 60 50
5000
A0
40 4000 30
Frequency [Hz]
fr
70
10 mm
20
3000
30 Journal diameter [mm] 70 (a) Overall amplitude
Amplitude [nm]
40 Curve 1 (bearing stiffness and damping)
30 20
Curve 2 (bending deformation)
10
10 mm 0
30
Journal diameter [mm]
70
(b) Contribution of bearing and rotor parameters to overall amplitude
Fig. 10. Influence of journal diameter on dynamic characteristics (ps=2.5 MPa)
Table 1 Constant parameters representing spindle for analysis
Cross-
EJ × 103
l × 10-2
m/l
cb × 109
hb × 107
(N⋅m2)
(m)
(kg m-1)
(N⋅m-2)
(N⋅s m-2)
1
1.14
1.50
2.00
2
2.30
3.24
3.00
3
7.95
1.11
5.70
4
61.24
0.25
15.70
5
61.24
4.50
15.70
9.62
1.50
6
61.24
0.25
15.70
7
25.10
0.90
10.00
8
644.10
1.50
53.30
9
25.10
0.90
10.00
10
61.24
0.25
15.70
11
61.24
4.50
15.70
9.62
1.50
12
61.24
0.25
15.70
13
7.95
1.11
5.70
14
2.30
3.24
3.00
15
1.14
1.15
2.00
section number
Highlights
1. Water-lubricated hydrostatic bearings show preferable damping characteristics 2. Amplitude of the vibration of the rotor overhang part is dominantly affected not by bearing stiffness but by bending stiffness of the shaft 3. Natural frequency of the hydrostatic bearings can be increased by increasing supply pressure 4. Natural frequency of the hydrostatic bearings can be increased by decreasing bearing clearance
Conflict of Interest
Department of Mechanical Engineering Faculty of Engineering Kanagawa University
September 13, 2019
The authors have no conflicts of interest directly relevant to the content of this article.
Yohichi Nakao
S0141-6359(20)30104-5
DOI:
https://doi.org/10.1016/j.precisioneng.2020.02.003
Reference:
PRE 7094
To appear in:
Precision Engineering
Received Date: 5 January 2020 Accepted Date: 15 January 2020
Please cite this article as: Fedorynenko D, Kirigaya R, Nakao Y, Dynamic characteristics of spindle with water-lubricated hydrostatic bearings for ultra-precision machine tools, Precision Engineering (2020), doi: https://doi.org/10.1016/j.precisioneng.2020.02.003. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier Inc.
Dynamic Characteristics of Spindle with Water-lubricated Hydrostatic Bearings for Ultra-precision Machine Tools
Dmytro Fedorynenko1, Rei Kirigaya2 and Yohichi Nakao2*
1
Department of Aerospace Engineering Tohoku University, Sendai, JAPAN
2
Department of Mechanical Engineering Kanagawa University, Yokohama, JAPAN
* Corresponding author. Postal address: Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686 Yokohama, Japan E-mail address: [email protected] Phone : +81-45-481-5661 ext. 3489 Fax
: +81-45-491-7915
1
Abstract A key component of ultra-precision machine tools is the spindle. The motivation for this study was to improve machining accuracies in precision cutting and grinding by pursuing improvements in the spindle characteristics by designing a sophisticated spindle with water-lubricated hydrostatic bearings. The static bearing stiffness of the developed spindle was investigated in previous studies. In addition to the static bearing stiffness, the dynamic characteristics regarding bearing stiffness also affect significantly on the machining results. In this study, dynamic characteristics of the developed spindle with water-lubricated hydrostatic bearings were investigated via simulations and experiments. Not only bearing dynamics but also rotor dynamics were considered in this study. In the simulation studies, the spindle dynamic characteristics were analysed based on the transfer matrix method. A spindle rotor supported with hydrostatic bearings was represented by discrete sections of the rotor. The mathematical model of transverse linear vibrations of the spindle rotor was derived with distributed parameters for these discretized rotor sections. As a result of the analysis on the amplitude-frequency characteristic, radial displacements of the rotor due to bearing displacement and bending deformation were defined. Then, the frequency characteristics were represented with Nyquist plots. Resonant frequencies and amplitudes formation in the transverse vibration of the rotor were determined. The influence of rotor bending deformations on spindle compliance was assessed. Furthermore, the study examined the influences of the supply pressure of the lubricating fluid, radial clearance and journal diameter of the hydrostatic bearings on the amplitude of the rotor vibration, and the resonance frequency of the system.
2
Furthermore, the dynamic characteristics of the spindle were examined experimentally. The simulation results were in good agreement with the actual spindle dynamics obtained experimentally. The influence of the structural parameters of the rotor and the operating parameters of the bearings on the spindle dynamic characteristics was also determined. It was verified that the amplitude of the vibration of the rotor overhang part was dominantly affected not by bearing stiffness but by bending stiffness of the bearing journal of the front bearing and the length of the rotor overhang. Then it was verified that the resultant displacement of the rotor in the radial direction due to the influence of the bearing characteristics and the structural effect of the rotor is significantly small. Practical recommendations to improve the spindle design in terms of the dynamic characteristics of the spindle with water-lubricated hydrostatic bearings were also derived.
Keywords Spindle, Hydrostatic bearings, Machine tools, Dynamic characteristics, Transfer matrix method, Dynamic compliance, Resonant frequency, Vibration amplitude
3
1. Introduction Precision machine tools are quite indispensable to the manufacture of precision parts. For instance, ultra-precision machining using precision machine tools is widely used for manufacturing precision parts for the electronic, medical, automobile, optical equipment, and other various leading-edge industries. The ultra-precision machine tools are used, for instance, for the cutting operations using a single-crystal diamond cutting tool. A critical component of the precision machine tools that determines the quality of machining is the spindle. Spindles with hydrostatic or aerostatic bearings are widely adopted for ultra-precision machine tools. Previous study [1] found that hydrostatic bearings for the machine tool spindle increase the machining reliability and productivity. Accordingly, hydrostatic and aerostatic bearings are the most common bearings for ultra-precision machine tools [2]. In general, aerostatic bearings are widely used for the spindle of the ultraprecision machine tools. A benefit of the aerostatic bearings is in its low viscosity, causing less heat generation during high-speed operation of the spindle. In contrast, the low damping and stiffness characteristics because of air compressibility are a disadvantage of aerostatic bearings. Oil hydrostatic bearings are advantageous for designing ultra-precision spindles with high stiffness and damping characteristics. However, oil has high viscosity that leads to larger heat generation due to the viscous friction during spindle rotation and, as a result, to significant thermal deformations of the spindle. Accordingly, it degrades machining accuracy. On the other hand, water is incompressible fluid that has relatively low viscosity in comparison with oil. Moreover, water has a higher specific heat and thermal conductivity. Thus, it is considered that
4
higher thermal stability can be achieved by using water as lubricating fluid simultaneously providing high bearing stiffness as well. To cope with the disadvantage of both the aerostatic and oil hydrostatic bearings, water-lubricated hydrostatic bearings has been studied for spindle applications. A spindle driven by water flow power was studied prior to designing the spindle with water-lubricated hydrostatic bearings [3]. A spindle with water-lubricated hydrostatic bearings was then designed [4]. Static stiffness of the bearings [5], the patterns of spindle temperature field change [6] and the thermal stability of the spindles with different rated rotational speeds were investigated [7, 8]. In the turning operation with a constant depth of cut condition, the actual depth of cut is however slightly changed, in the strict sense, due to the spindle runout even hydrostatic bearings support spindle rotor. As a result, the resultant cutting force acting on the spindle is not necessarily constant. In this case, a principal frequency of the cutting force is basically determined by the spindle rotational speed. For example, the principal frequency becomes 50 Hz, if the spindle rotational speed is 3,000 min-1. In addition, the cutting force is also changed, if a control of the depth of cut is synchronized with the spindle rotation using a fast tool servo system. The small variation in the cutting force affects a spindle motion that depends on the dynamic characteristics of the spindle. The resultant spindle motion due to the time variance cutting force affects a quality of the finished surface. Accordingly, not only static characteristics but also dynamic characteristics of the bearings and spindle rotor are of crucial importance in the consideration of the structural design for spindles. The dynamic characteristics can be represented in terms of the dynamic stiffness, damping, vibration modes, and frequency characteristics of the overall system of the spindle. Dynamic analysis of the spindle, including the influence
5
of external forces, such as the cutting forces, introduces useful information for determining the structure of the spindle in the design process. Therefore, the influence of the time-varying external forces on the spindle dynamic characteristics is considered in this paper. Various experimental and theoretical approaches have been applied to consider dynamic characteristics of machine tool spindles [9]. Harmonic and impulse testing methods can be possible methods based on the experimental dynamic analysis. Experimental evaluation of dynamic coefficients [12] is also the promising method to study spindle dynamics. However, the experimental methods require special facilities and time-consuming tasks. Furthermore, the experimental approaches cannot be applied before or during the design of a new spindle. To avoid these disadvantages of the experimental procedures, theoretical methods, such as those based on the finite element [14] and transfer matrix methods [13], are widely used for the dynamic analysis of the spindle. If the concentric condition of the spindle shaft, namely no-displacement of the shaft from the ideal center axis can be assumed, exact solutions without linearization procedure for the hydrostatic bearings can be introduced [16]. However, the spindle shaft can be supported at a certain offcenter position in many actual operations. The finite element method is a versatile numerical analysis tool for investigating the dynamic analysis of mechanical system. A diversity of commercialized computeraided engineering systems implementing the finite element method can be used for dynamic analysis of spindles. However, the main difficulty of using commercial computer-aided engineering software arises in description of the elastic-dissipative properties of the hydrostatic bearings of the spindle to be analysed [15].
6
To avoid this difficulty, the analysis using discrete beam model representing the spindle shaft is one of the most promising ways for the finite element analysis of spindles with hydrostatic bearings. A finite element model of hydrostatic bearings characterized by 36 linear coefficients of stiffness and damping was thus proposed [10]. It should be noted that introducing a large number of different elements of the spindle shaft is needed for spindle analysis based on the special beam finite elements. This leads to considerable difficulties in formalization of the calculation method and its implementation on a computer to carry out the analysis. In contrast, the transfer matrix method is widely used for the purposes of analysis of spindle linear dynamics [11]. The main advantage of the method is convenience and the ease formalizing the mathematical model. Accordingly, the extensive capabilities of the transfer matrix method can be used to consider the elastic-dissipative properties of hydrostatic bearings and spindle structures, including couplings and chucks connected at the ends of the spindle shaft as needed. In general, the spindle for ultra-precision machine tools is driven under the condition of small external forces. Accordingly, the amplitudes of vibration in both the radial and axial directions of the spindle are extremely small. In general, superior stiffness and dumping characteristics of the oil or waterlubricated hydrostatic bearings can be obtained as discussed in this paper. An aim of this study is to investigate the dynamic characteristics of the developed spindle with water-lubricated hydrostatic bearings via theory and experiments. Some of the important design guidelines for spindle design with hydrostatic bearings are derived through the investigations discussed in this paper.
7
In the rest of this paper, we present an explanation of the studied spindle supported with water-lubricated hydrostatic bearings followed by a mathematical model and an analysis procedure. Experimental results are then given. Finally, the dynamic characteristics of the spindle obtained via simulations and experiments are discussed.
Nomenclature A
amplitude of the rotor oscillations
A, B, C, D
complex functions of elasticity
A0
amplitude of the rotor oscillations when f=0
Ab
amplitude of the rotor oscillations in the bearing
Аlc
local resonance amplitude
Ar
resonant amplitude of the spindle
Cu, Cφ, Ccu, Ccφ complex modules of elasticity in the rotor cross-section с0
bearing stiffness coefficient
cav
average dynamic stiffness
cb
distributed stiffness coefficient in the bearing
cu, cφ, ccu, ccφ
stiffness coefficients in the rotor cross-section
Db
bearing journal diameter
e
eccentricity in the bearing
E
Young's modulus
F
external force
Fe
force due to elongation of viscoelastic environment
f
oscillation frequency
fr
resonant frequency of the spindle
8
h0
bearing damping coefficient
hb
distributed damping coefficient in the bearing
hm
damping coefficient in the rotor characterized internal damping
hu, hφ, hcu, hcφ
damping coefficients in the rotor cross-section
i
sections or cross-sections number
J
axial moment of inertia
Jµ
axial moment of inertia of the concentrated mass
K1, K2, K3, K4
bearing coefficients
k
cross-section number where external force F acts
ks
spindle static compliance
Lb
bearing length
Lr
overall rotor length
l
rotor length
la
axial land width
lc
length of the rotor overhang area
lτ
inter-recess land width
M
bending moment
Mex
external bending moment
m
rotor mass
N
load vector
Nk
load vector in cross-section number k
n
the end cross-section of the rotor
nr
rotation speed of the rotor in rev/s
P
general transition matrix
9
Pk
matrix which is equal the product of general transition matrices Pi from i=k+1 to i=n
PΣ
matrix which is equal the product of all general transition matrices Pi from i=1 to i=n
pr
recess pressure
ps
supply pressure
Q
transverse force
Qex
external transverse force
Rh, γ
flow factors
R0
restrictor impedance
r
number of recesses
S
matrix of the cross-section
t
time
U
transition matrix of the section
u
rotor displacement in the horizontal plane
WFk
frequency transfer functions of the spindle
Z
vector
z
axial coordinate of the rotor
β
gauge pressure ratio when e=0
δ0
bearing clearance when e=0
ε
elongation
φ
rotation angle at bending
η
dynamic viscosity coefficient
ϕ
phase of oscillations
10
λ
complex coefficient of elasticity
µ
concentrated mass
ω
oscillation frequency in rad/s
ωn
bearing resonant frequency in rad/s
bearing damping ratio
2. Schematics of investigated spindle supported with water-lubricated hydrostatic bearings 2.1 Spindle structure The investigated
spindle with
water-lubricated
hydrostatic bearings
is
schematically presented in Fig. 1. The spindle was designed for the turning operation using a single-point diamond cutting tool. The spindle has four recessed journal bearings and four recessed double-sided thrust bearings. The hydrostatic bearings have short pipe restrictors [21] for each recess. A displacement pump supplies pressurized water to the bearings. Parameters of the short pipe restrictors are determined to achieve the highest static bearing stiffness for the given conditions in both the radial and axial directions. In our series of simulations, the gap of the radial bearing was δ0=12 µm. The rotor is made of SUS30 stainless steel [5]. Diamond-like carbon is plated onto the outer surfaces of the rotor and the inner surfaces of the casing to prevent undesirable damage during spindle operation. The dimensions of the rotor are given in Fig. 2.
2.2 Rotor structure and related parameters for analysis
11
To simplify the calculation model, the spindle rotor is represented as divided sections with constant diameter di and length li for the i-th section. Then each divided section can be considered to be stepped beams as depicted in Fig. 2. The moments of inertia are slightly different depending on the divided sections. Then, the average values of the moment of inertia Ji and the section mass mi were calculated for the combined rotor areas. In the analysis, small chamfers, grooves, slots, etc., were not considered. The entire elastic rotor system is presented as a discretized model with short rotor sections. A schematic drawing of the spindle rotor is given in Fig. 2. As depicted, the spindle rotor is divided into 15 sections by 16 indexes. The hydrostatic journal bearings are located between sections 5 and 6 and 11 and 12. The bearing section with length Lbla is represented with distributed characteristics of stiffness cb and damping hb. Each section of the rotor depicted in Fig. 2 is characterized by the bending stiffness EJi and mass mi per unit length li. External harmonic force F is exerted at cross-section 16 and has an amplitude of 1 N that corresponds to the highest possible cutting force during turning with a diamond tool [18]. The damping coefficient of the spindle structure is negligibly small compared to that of the hydrostatic bearings (hm<
3. Mathematical Modelling 3.1 Model of elasticity of spindle rotor In the theoretical analysis used in this paper, discrete sections of the rotor are created so that each section has same cross section and diameter as shown in Fig. 2. As
12
a rule, the influence of longitudinal perturbations on the transverse vibrations of the rotor is negligibly small. Accordingly, only the transverse vibrations of the spindle rotor are considered in the study. The cross sections of the rotor under its dynamic motions are considered to remain flat. In addition, the centers of gravity of the discrete sections of cross-sections are assumed to be located along a line corresponding to the center of gravity of the entire rotor. It is furthermore assumed that the center of gravity of the defined rotor sections is coincident with a neutral line of the bending shaft along the entire rotor length. The spindle system considered in the paper was designed for use in ultra-precision machine tools. Accordingly, the external forces acting on the spindle and spindle rotor to disrupt its balance are sufficiently small [18]. Thus, it can be assumed that the spindle system can be represented as a linear oscillating system. In this case, the dynamic characteristics of the spindle vibrating system can be solved by calculating the transfer functions derived for given frequencies. Then, the differential equation presenting the rotor bending oscillation with dissipation effects in the hydrostatic bearings is derived [19]. In the equation, the external force F, such as cutting force during machining operations acting on the rotor, is considered. As presented in Eq. (1), the sinusoidal external force given in right-hand side in Fig. 2 is considered as a function of angular velocity, frequency characteristics are introduced. EJ ( z )
∂ 4u ∂ 2u ∂u + m ( z ) + h( z ) + c ( z )u = F ( z ) sin ωt . 4 2 ∂z ∂t ∂t
(1)
According to the structure of the studied spindle rotor depicted in Fig. 2, radial hydrostatic bearings are located between cross-sections 5 and 6 and 11 and 12. Now,
13
consider the solution of Eq. (1) to express the dynamics of the radial bearing section of the rotor with a constant cross section. External force or moment does not act on the bearing sections of the spindle, as a rule. Thus, the right-hand side of Eq. (1) is equal to zero. Thus, Eq. (1) with two variables z and t can be converted to an equation with one variable. In general, taking into account the dissipation of energy at the hydrostatic bearings, displacement of any section of the rotor is associated with time dependence, as given by Eq. (2).
u ( z , t ) = u ( z ) e jω t ,
(2)
where = √−1. Substituting Eq. (2) into Eq. (1), Eq. (3) is then derived.
u IV −
λi4 li4
u =0,
(3)
where
λi4 =
(
li4 miω 2 − c0 − jh0ω EJ i (1 + jhm )
).
Detailed procedures on determining the parameters in Eq. (3) are given in section 3.2. In this paper, the dynamic characteristics of the spindle rotor represented by Eq. (1) for discretized sections of the rotor are numerically solved by the transfer matrix method [16, 18]. In the transfer matrix method, the discretized sections are presented in matrix form. In the calculations, the variables at an end section of the rotor are first determined as boundary conditions at a given frequency with known values of displacement u, angle φ, internal force Q, and moment M. Thus, the dynamic characteristics of the
14
discretized rotor sections are characterized by the displacement u, angle φ against the internal force Q, and moment M acting on the sections [17]. Magnitudes of these variables at the i-th section of the rotor can be presented by four-dimensional vector as given in Eq. (4).
Z i = [ui φi
Mi
Qi ] . T
(4)
In the calculation, the boundary conditions are given at the left end of the rotor, the index number 1, as depicted in Fig. 2. Accordingly, the transition matrix from vector Zi-1 to vector Zi is presented as Eq. (5) [17].
Zi = Pi Z i −1+ Ni .
(5)
General transition matrix of the i-th section Pi is given as Eq. (6).
Pi = Ui Si .
(6)
In Eq. (6), the transition matrix of the i-th section Ui can be represented as Eq. (7) [17]. Ai 4 λi Di l Ui = 4 i λi EJ iCi li2 4 λi EJ3 i Bi li
li Bi Ai
λi4 EJ i Di li 4 λi EJ iCi li2
li2Ci EJ i li Bi EJ i Ai
λi4 Di li
li3 Di EJ i li2Ci EJ i . li Bi Ai
(7)
where functions Ai, Bi, Ci, and Di are given by Eqs. (8)-(11).
Ai =
Bi =
Ci =
(cosh λi + cos λi ) 2
(sinhλi + sin λi ) 2λi
(coshλi − cosλi ) 2λ2i
(8)
(9)
(10)
15
Di =
(sinh λi − sin λi ) 2λ3i
.
(11)
In addition, matrix Si in Eq. (6) is defined as Eq. (12).
1 0 Si = 0 2 µiω − Cui
0
−
0
− J µiω 2 + Cφi
1 Ccφi 1
0
0
1
1 Ccui 0 . 0 1
(12)
In Eq. (5), the load vector Ni is presented by a vector form as Eq. (13).
[
Ni = 0
0
M iex
Q exi
]
T
.
(13)
Complex modules of elasticity Cu, Cφ, Ccu, and Ccφ are defined by the elasticdissipative characteristics of the lumped parameters of concentrated mass Cu and Cφ (e.g., pulleys and gears) or the parameters of tool shank Ccu and Ccφ relative to rotor radial displacement u and angular φ⋅z displacements. The mentioned modules are determined by Eqs. (14) - (17).
Cui = cui + jhui
(14)
Cφi = cφi + jhφi
(15)
Ccui = ccui + jhcui
(16)
Ccφi = ccφi + jhcφi .
(17)
In Eqs. (14)-(17), indexes u and φ present bending radial and angular components of the complex modules of elasticity, respectively. Hence, the rotor dynamics can be determined by solving the matrix equations. With the matrix equations, the parameters of rotor sections, such as u, φ, Q, and M, can
16
be determined by the linking matrix from the initial cross-section, namely i=1, to the end section i=n. Thus, the matrix Zn is represented by Eq. (18).
Z n = PΣ Z1 + Pk N k .
(18)
Matrixes PΣ and Pk in Eq. (18) are determined by Eqs. (19)-(20). n
PΣ = ∏ Pi
(19)
i =1 n
Pk = ∏Pi .
(20)
i =k +1
Matrix Nk can be determined from Eq. (13), where i equals k. As an example, let us consider the definition of the components of the matrix Z1 for the particular rotor structure shown in Fig. 2. The matrix Z1 can be obtained using Eq. (18), provided that force Q1 and moment M1 in the initial cross-section are zero. Force Qn and moment Mn in the end cross-section equal 1 N and 0 N⋅m, respectively. Thus, by considering that the external force is acting in the end cross-section n (n=k), Eq. (18) can be given as u n P11 φ P n = 21 0 P31 1 P41
P12
P13
P22
P23
P32
P33
P42
P43
P14 u1 P24 φ1 × , P34 0 P44 0
(21)
where P11..P44 are the components of the transition matrix P. By writing in algebraic form the equations regarding the last two elements of the matrix Zn in Eq. (21), we can find the parameters u1 and φ1 from Eqs. (22)-(23).
0 = P31u1 + P32φ1
(22)
1 = P41u1 + P42φ1
(23)
17
Finally, it is possible to introduce the frequency transfer functions of the spindle to represent the relationship between the external force F and the rotor displacement u at arbitrary elements, in which the external force F acts at the cross-section k. For example, the dynamic compliance in the i-th cross-section of the rotor is given as Eqs. (24)-(25):
WFk ( jω ) =
ui ( jω ) Fk ( jω )
WFk ( jω ) = Re(ω ) + j ⋅ Im(ω ) .
(24)
(25)
From Eq. (25), the amplitude and the phase-shift regarding the rotor dynamics can be introduced as Eqs. (26) and (27), respectively. A(ω ) =
Re 2 (ω ) + Im 2 (ω )
Im(ω ) . Re(ω )
ϕ (ω ) = tan −1
(26)
(27)
Nyquist plots will be then introduced using Eqs. (26) and (27). Furthermore, determining the rotor axis deformation allows us to investigate the static and dynamic compliance over a wide frequency range. The calculation results for the spindle are presented below.
3.2 Stiffness and damping coefficients of hydrostatic bearing
The sections of the spindle rotor supported by hydrostatic bearings can be represented as a beam on a Winkler foundation with evenly distributed elasticdissipative elements. Then, the bearing characteristics represented by the Kelvin-Voigt model, as given by Eq. (28). In Eq. (28), the linearized radial stiffness and damping of the hydrostatic bearing are considered to be parallel-connected elastic and dissipative elements.
18
Fe = с0ε + h0
dε . dt
(28)
In general, the matrix determining stiffness and damping of hydrostatic bearings has a dimension of 6 × 6. The cross-coefficients in matrices determine non-collinear displacements or velocities of bearing surfaces relative to the vector of external loads. Let us assume that the bearing stiffness and damping coefficients are independent of the rotational speed of the spindle and the frequency of the external force. This assumption is fairly true for low-speed spindles with sliding bearings [16, 20]. For ultra-precision machining, the rotor offsets along both the radial and axial directions are negligibly small. In addition, the compressibility of the lubricating fluid, such as oil and water, can be neglected. In this case, the coefficients of c0 and h0 can be computed as in Eqs. (29) and (30) [16].
(1 − la Lb )β 3 ps Lb Db r 2 sin 2 (π r ) с0 = 2 2πδ 0 R0 + 1 + 2γ sin (π r )
h0 =
2 2 12 ps Lb Db r 2 Rh sin 2 (π r ) (la Lb )(Lb Db ) (1 − la Lb ) , 2 ( ) πnrδ 0 R + 1 + 2 γ sin π r 0
(29)
(30)
where
β = pr p s
(31)
R0 = 0.5β (1 − β )
(32)
ηn D Rh = r b ps 2δ 0 γ=
2
rla Lb la 1 − . πlτ Db Lb
(33)
(34)
Coefficients of the stiffness cb and damping hb for the distributed bearing sections are determined based on the rotor section length li by Eqs. (35) and (36). 19
cb = c0 li
(35)
hb = h0 li .
(36)
4. Experiments The experimental apparatus for the measurement of the dynamic characteristics of the spindle is shown in Figs. 3(a) and (b). In the experiments, sinusoidal forces with various frequencies were applied to the end of a rotor in the radial direction. The measurements were carried out in a horizontal plane as depicted in Fig.3(b). The forces were generated by the driving force of a piezo actuator (Matsusada Precision PZ20-85, Max. displacement: 45 µm with feedback control, Natural frequency: about 10 KHz) placed perpendicular to the rotor. The amplitude of the forces was controlled to be 1 N regardless of the frequencies of the applied sinusoidal forces. For the purpose of controlling and monitoring the applied forces, a load cell (Kyowa LMB-A-50N, Repeatability: < 0.3 %RO, Natural frequency: about 40 KHz) was used to measure the actual applied forces. As depicted in Fig. 3, the load cell was installed between the rotor and the piezo actuator. The frequencies of applied radial forces were changed in the ranges from 0 Hz to 1,000 Hz. The resultant sum of the radial displacement and deformation of the rotor was measured by a capacitance sensor (Nano Technology, 212S system, Resolution: 5 nm, Frequency range: 200 Hz) and a laser sensor (Ono Sokki LV-2100, Resolution: 1.5 nm, Linearity: ±0.1 %FS, Frequency range: 50 kHz) set at the opposite side of the rotor in a horizontal plane. In the series of the experiments, the water supply pressure for the hydrostatic bearings was set to 2.5 MPa. The experimental rig used in the measurements uses a contact type of the excitation facility. The limitation of the experimental rig used in the measurements is in 20
difficulty of measuring the spindle dynamics during spindle rotation. In contrast to the contact type of the excitation method, a non-contact type of the equipment has been developed and tested to investigate spindle dynamics [24]. Because of the non-contact feature, the developed excitation equipment can be used to measure spindle dynamics during spindle rotation, achieving measurement during actual machining conditions. Investigations of our spindle during actual machining conditions will be considered in our future work. In the experiments, a constant initial load was applied to the spindle shaft by the piezo actuator. The output of the displacement sensor was then defined to be zero. The tested spindle was placed in a temperature-controlled room. The temperature was set to be 20 ℃. During experiments, the room temperature was controlled within 20±0.5 ℃ with periods of several tens of minutes. In addition, required time for each measurement was at most 10 s. Accordingly, the influence of thermal deformation of the spindle and other rig on the measurements was negligibly small. Prior to the measurements of the dynamic characteristics of the spindle, the dynamic characteristics of the additional components, including the base plate and supporting block of the load cell were measured in a preliminary test. It was then verified that the additional components had a natural frequency around 360 Hz. In a series of experiments, the temperature of water supplied to the spindle was controlled by a chiller system. The water temperature was set to 20 ℃ by the chiller system. Accordingly, the influence of temperature change on the measured spindle dynamics was negligibly small.
5. Results and Discussion
21
The dynamic characteristics of the spindle were obtained, as presented in Fig. 4 by the transform matrix method [22, 23]. Figure 4 shows the spindle frequency response characteristics. The results indicate the sum of the resultant displacement of the rotor at the end, namely cross-section number 16 in Fig. 2, due to the external radial forces of F. The elastic system of the spindle has a resonant frequency of 6,106 Hz with amplitude Ar=156.2 nm, which is caused by the first natural frequency of the bending oscillation of the rotor. As the graph shows, the amplitude of oscillations, at crosssection 16, first decreases with increasing frequency to a magnitude of Аlc=26 nm, which is approximately 73% of the amplitude at a frequency of 0 Hz. Then, it increases monotonically to the resonant value of Аr. A decrease in the amplitude of the rotor vibration is observed in the frequency range up to about 1,000 Hz. This phenomenon is due to the influence of dissipative properties of hydrostatic bearings, that is, the high damping effects of the water-lubricated hydrostatic bearings. To investigate the dissipative properties of the journal bearings of the spindle, the displacement of the bearing is investigated with a simplified model. In the calculation, the spindle rotor is assumed to be a rigid body, which excludes the influence of rotor deformation. Furthermore, we assumed that the external force acts at a middle position of the journal bearings. Accordingly, the influence of the moment effect due to the external force acting on an end rotor section on the displacement was not considered. Because of the simplified model, the amplitude of rotor displacement differs from the simulation results presented in Fig. 4. However, the dumping characteristics of the bearings can be investigated.
22
The amplitude-frequency characteristic was obtained as represented in Fig. 5. In our investigation, a simplified model was obtained by the following dependency [16] when F=1 N:
Ab =
1 c0
K12 + K 22 , K32 + K 42
(37)
where
ω K1 = 1 − ωn
K2 = 2ϑ
2
ω ωn
(38)
(39)
2
2 2 ω 2 ω πnr h0 K 3 = 1 − − 2ϑ + ωn ωn c0
(40)
2 ω ω 1 − K 4 = 4ϑ ω n ω n
(41)
ω n = 2 c0 m
(42)
ϑ=
h0ωn . 2c0
(43)
The frequency response of the bearing calculated by the simplified model is given in Fig. 5. As described above, the amplitude of the rotor displacement presented in Fig. 5 differs from that shown in Fig. 4, because the external force is considered to act at the middle of the journal bearings in the simplified model. However, the obtained frequency response is a monotonically decreasing function of the frequency, which has the same pattern as the amplitude-frequency characteristic of the spindle in the frequency range from 0 to 1,000 Hz, as presented in Fig. 4. The influence of the bearing
23
displacement and structural deformation on the resultant amplitude is further investigated in this paper. In Fig. 6, the experimental results are presented. The sinusoidal external load and resultant displacement are depicted in Fig. 6(a). The frequency characteristics are given in Fig. 6(b). As is seen from Fig. 6 (a), at a frequency of 280 Hz, the phase lag between the external force and rotor displacement is small. The tendency of phase lag is also observed up to 1,000 Hz, as given in the Nyquist plot described below. In addition, the rotor displacement has a steady tendency to decrease in both the theoretical and experimental data, as given in Fig. 6(b), which is due to the influence of bearing damping on spindle dynamics at low frequencies. As shown in Fig. 6(b), the amplitude obtained experimentally has a small peak at a frequency of 360 Hz, which is caused by the influence of a natural frequency of the experimental apparatus verified in the preliminary experiments as explained in section 4. Nyquist plots derived from the theoretical analysis are depicted in Fig. 7. These plots allow more detailed analysis of the spindle dynamic characteristics. As given in Fig. 7(a), the Nyquist plot has a local resonance with amplitude Alc=25.2 nm at a frequency of 1,000 Hz with formation of the looping features. This is due to the influence of high damping of hydrostatic bearings obtained in the low-frequency region, as already described. The local resonance is accompanied by the abrupt change in the phase near a frequency of 1,000 Hz as presented in Fig. 7(b). As is seen from the Nyquist plot, rotor displacement Аr at the main resonance frequency (fr=6,105 Hz) increases by almost 4.5 times compared to the static compliance ks of the system. The fundamental frequency is determined by the rated rotational speed of the spindle. The rated rotational speed of the spindle is 3,000 min-1.
24
Accordingly, the fundamental frequency of the spindle becomes 50 Hz. In general, the fundamental frequency component plays an important role for the dominant dynamic characteristics. A main resonance frequency (fr=6,105 Hz) is much higher than the normal operational frequencies of the spindle. Furthermore, a damping characteristic of the hydrostatic bearings is dominant in the operational frequency range. These desirable dynamic characteristics achieve better machining characteristics due to the damping effect. Here, we describe how the spindle rotor deforms along the axial direction in the simulation. In Fig. 8, the sum of rotor displacements due to the sinusoidal external forces of amplitude 1 N with different frequencies is illustrated by the centerline of the rotor. As is seen in the graph, in the operating frequency range (f≤50 Hz), the amplitude A does not change significantly due to static characteristics, namely f=0 Hz. In fact, the amplitude of the rotor at f=50 Hz differs up to 9 % from the static mode, f=0 Hz. Thus, we can focus on the analysis of the rotor amplitude determined by the static characteristics, as described in the rest of this paper. A significant increase in the amplitudes of the vibration at the rotor overhang sections is observed in the first resonant frequency. In contrast, the amplitude between both journal bearings remains extremely small. According to the simulation results, it is established that the average dynamic stiffness cav of the rotor between bearings equals 493 N/µm in the frequency range up to the first rotor resonant frequency. The average dynamic stiffness cav is calculated between the cross-sections 4 and 13 illustrated in Fig. 2 as
= ∑!" #$
⁄9 , where Csi is the average dynamic stiffness of the rotor
section i in the frequency range up to the first resonant frequency. In addition, the
25
minimum dynamic stiffness of the rotor is observed at the overhang area and it is about 30 N/µm in the operating frequency ranges up to 50 Hz, as presented in Fig. 4. As clearly observed in Fig. 8, a main reason for the decreased dynamic stiffness in the overhang area is due to rotor bending. This suggests that the rotor structure, in particular the overhang area, has to be carefully designed so that the resultant actual displacement will be less than an acceptable magnitude. The influence of the supply pressure ps and clearance δ0 on the rotor amplitude A0 is shown in Fig. 9(a) and 9(b), respectively. As is seen in Fig. 9(a), an increase in the supply pressure ps contributes to a reduction in the amplitude due to the displacement A0 at the end of the rotor, where the cutting force is active. The reduction is up to 23% within the investigated range in Fig. 9(a). Furthermore, the first resonant frequency fr changes depending on the supply pressure ps; however, the magnitude of the change in the resonant frequency is negligible. As shown in Fig. 9(b), the decrease in the clearance δ0 of the bearings reduces the amplitude A0 of the static displacement as well. In addition, the reduction of the bearing clearance δ0 increases the resonant frequency fr of the bending vibration system. Figure 10 shows the influence of the diameter of bearing journals Db on the amplitude A0 and resonance frequency fr. The calculation was made with the following assumptions: Lb=Db, la=0.1Db, and Lr=250 mm. These are defined in Fig. 2 for of Db ranging from 30 mm to 70 mm. As is seen from Fig. 10(a), the diameter of bearing journals Db significantly affects both rotor amplitude A0 and its resonance frequency fr, unlike the influence of the parameters δ0 and ps, as already depicted in Fig. 9. A main reason is that the increase in the journal diameter contributes to a significant change in the rotor structural
26
characteristics, such as the bending stiffness, mass, and length (when keeping assumption of Lb=Db) of bearing section as well as bearing stiffness and damping. The dynamic characteristics presented in this paper indicate the sum of the influence of the bearing compliance and the bending deformation. Here, we consider the relationship between them and their dependence on the diameter of the rotor. In Fig. 10(b), the displacement of the rotor end position according to cutting force F is depicted with respect to the cause of the bearing compliance and bending deformation separately. In the calculations, the amplitude determined by the displacement of the rotor due to the bearing compliance was introduced by specifying an infinite structural stiffness of the rotor. This result is presented by curve 1 in Fig. 10(b). In contrast, the amplitude due to the bending deformation of the rotor was determined by specifying the infinite bearing stiffness and damping, which is given by curve 2. Thus, curve 1 shows the influence of the bearing parameters and curve 2 represents the influence of rotor bending stiffness on the rotor amplitude. The error (up to 15%) between the sum of the amplitudes of curves 1 and 2 in Fig. 10(b) and the solid curve in Fig. 10(a) arises due to the impossibility of using infinite values for Young's modulus, stiffness, and damping in the bearings in the model. As is seen in the comparison of both curves in Fig. 10(b), the increased spindle diameter contributes significantly to a decrease in the amplitude of the rotor due to the influence of bearing compliance. Our investigation verified that the rotor amplitude A0 is significantly affected by the front bearing journal diameter, its compliance, and length lc of the rotor overhang area. In contrast, the rotor displacement in the area between bearings occurs mainly due to bearing compliance, as given in Fig. 8.
27
In the presented paper, the dynamic characteristics of the spindle with waterlubricated hydrostatic bearings are investigated. Now, how the oil hydrostatic bearings and the water-lubricated hydrostatic bearings are different in terms of the dynamic characteristics is discussed. The use of oil as the lubricating fluid leads to change lubrication characteristics such as the thermal and chemical stabilities, viscosity and density. According to the mathematical model presented in chapter 3, the fluid viscosity has influence on the bearing damping coefficient h0, in terms of the spindle dynamics. Specifically, the change of the bearing lubrication from water to oil improves the damping characteristics, consequently decreases the oscillation amplitudes of the rotor. However, in the operating range determined by the rated spindle speeds (up to 3,000 min-1), the use of viscous oil lubrication does not benefit significantly in terms of reducing the amplitude of the rotor oscillations. For example, for the tested spindle, the use of the oil with viscosity ten times over than water leads to the decrease in the amplitude of rotor oscillations by about 20% at a maximum in the mentioned range of speeds in contrast with water-lubricated bearings. However, a better thermal stability of the spindle can be obtained by using the water as the lubricating fluid [8].
6. Conclusions In this paper, the dynamic characteristics of a spindle with water-lubricated hydrostatic bearings were investigated via simulations and experiments. In particular, the characteristics of transverse vibration due to the influences of structural characteristics of the spindle rotor and water-lubricated hydrostatic journal bearings were considered.
28
In the simulation studies, the spindle rotor was modelled using discretised sections. Then, the dynamic characteristics of the spindle were derived with the transfer matrix method. The simulation results were compared with the experimental results. The results from the considerations in this paper verified that the tested spindle with water-lubricated hydrostatic bearings has the characteristics of high dynamic stiffness within the operating range of the spindle. For example, the average dynamic stiffness of the rotor between bearings equals 493 N/µm, and the minimum stiffness of the rotor at the overhang part is 30 N/µm. We verified that the rotor vibration amplitude corresponds to a static deformation of the elastic axis when the spindle speed is less than its rated operating speed. The main resonant frequency of the system was observed to be 6,106 Hz. The resonant frequencies greatly exceed the range of operating speeds (up to 50 Hz) of the spindle. The structural parameters of the rotor have the major influence on the main resonant frequency. Accordingly, the influence of the water-lubricated bearing is not significant. This fact proves that water-lubricated hydrostatic bearings are advantageous in terms of not only static stiffness but also dynamic stiffness, including damping characteristics higher than those of aerostatic bearings. The difference in the fluid viscosity between water and oil affects the damping characteristics. However, the difference in the resultant oscillation amplitude of the rotor is not significant. The amplitude of vibrations in the rotor overhang area was determined primarily by bending stiffness of the bearing journal of the front bearing, its compliance, and the length of the rotor overhang area. The results suggest that to reduce rotor radial vibration, it is important to increase the bending stiffness of the overhang area so that radial vibration can be fully reduced to an acceptable level for finishing cut operations.
29
In addition, it is also effective to increase the pump pressure and to decrease the bearing clearance to reduce the radial vibration.
Acknowledgments The presented results are a part of project “Increasing of Precision Spindles Efficiency with Hydrostatic Bearings,” funded by the Matsumae International Foundation and Grant-in-Aid for Scientific Research (C) of the Japan Society for the Promotion of Science.
30
References
[1]
Perovic B., Hydrostatische Führungen und Lager: Grundlagen, Berechnung und Auslegung von Hydraulikplänen, Berlin: Springer-Verlag; 2012 (in German).
[2]
Wardle F., Ultra Precision Bearings. 1st ed. Woodhead Publishing; 2015.
[3]
Nakao Y., Mimura M., Spindle Motor Driven by Fluid Energy for Ultra-precision Machine Tool, Proceedings of ASPE 2002 Annual Meeting, pp. 215-218.
[4]
Nakao Y., Mimura M., Kobayashi F., Water Energy Drive Spindle Supported by Water Hydrostatic Bearing for Ultra-Precision Machine Tool, Proceedings of ASPE 2003 Annual Meeting, pp. 199-202.
[5]
Nakao Y., Suzuki K., Yamada K., Nagasaka K., Feasibility Study on Design of Spindle Supported by High-Stiffness Water Hydrostatic Thrust Bearing, International Journal of Automation Technology, 8(4); 2014, pp. 530-538.
[6]
Hayashi A., Nakao Y., Measurement and Evaluation of Temperature Change of Water Driven Spindle, Proceedings of ASME 2016 International Mechanical Engineering Congress & Exposition; 2016, p. V002T02A007.
31
[7]
Tsubasa Y., Hayashi A., Nakao Y., Fundamental Study on Thermal Stability of Micro Milling Spindle Supported by Water Hydrostatic Bearings under Spindle Rotation, Proceedings of 32th ASPE Annual Meeting; 2017.
[8] Nakao Y., Kirigaya R., Fedorynenko D., Hayashi A., Suzuki K., Thermal Characteristics of Spindle Supported with Water-Lubricated Hydrostatic Bearings, International Journal of Automation Technology, 13(5), 2019, pp.602-609.
[9]
Budak E., Machining Stability and Machine Tool Dynamics, Materials of 11th International Research/Expert Conference TMT, Tunisia; 2007.
[10] Savin L.A., Solomin O.V., Modeling of Rotary Systems with Bearings of Fluid Friction, Mashinostroenie; 2006 (in Russian).
[11] Danilchenko Y., Kuznetsov Y., Precision Spindles with Rolling Bearings. Ekonomichna Dumka; 2003 (in Ukrainian).
[12] Cheng F., Ji W., Simultaneous Identification of Bearing Dynamic Coefficients in a Water-gas Lubricated Hydrostatic Spindle with a Big Thrust Disc, Journal of Mechanical Science and Technology 30 (9); 2016, pp. 4095-4107.
[13] Hou Y., Wang S., Han Z., The Dynamics Modelling and Simulation for Coupled Double-rotor Spindle System of High Speed Grinder, Przegląd Elektrotechniczny 89; 2013, pp. 25-28.
32
[14] Liu D., Zhang H., Tao Z., Su Y., Finite Element Analysis of High-Speed Motorized Spindle Based on ANSYS, The Open Mechanical Engineering Journal 5(1); 2011, pp. 1-10.
[15] Chen W., Sun Y., Liang Y., Bai Q., Zhang P., Liu H., Hydrostatic Spindle Dynamic Design System and Its Verification. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 228(1); 2014, pp. 149-155.
[16] Rowe W. B., Hydrostatic, Aerostatic and Hybrid Bearing Design, Waltham: Butterworth-Heinemann Press; 2012.
[17] Ivovic V., Transition Matrices in Dynamics of Elastic Systems, Mashinostroenie; 1981 (in Russian).
[18] Masuda M., Maeda Y., Nishiguchi T., Sawa M., Ikawa N., Diamond Turning of an Aluminum Alloy for Mirror, CIRP Annals 38(1); 1989, pp. 111-114.
[19] Strutynskyi V., Fedorynenko D., Statistical Dynamics of Spindle Units with Hydrostatic Bearings, Publishing house “Aspect-polygraph”; 2011 (in Ukrainian). [20] Waumans T., Vleugels P., Peirs J., Al-Bender F., Rotordynamic Behaviour of a Micro-turbine Rotor on Air Bearings: Modelling Techniques and Experimental Verification, Proceedings of ISMA; 2006, pp. 181-197.
33
[21] Nakao Y., Nakatsugawa S., Komori M., Suzuki K., Design of Short-Pipe Restrictor of Hydrostatic Thrust Bearings, Proceedings of ASME 2012 International Mechanical Congress and Exposition; 2012.
[22] Fedorynenko D., Sapon S., Spindle Hydrostatic Bearings. ChNUT; 2016 (in Ukrainian).
[24] Fedorynenko D., Sapon S., Boyko S., Accuracy of Spindle Units with Hydrostatic Bearings, Acta Mechanica et Automatica 10 (2); 2016, pp. 117-124.
[24] Matsubara A., Yamazaki T., Ikenaga S., Non-Contact Measurement of Spindle Stiffness by Using Magnetic Loading Device, International Journal of Machine Tools and Manufacture, 71, August, 2013, pp. 20-25.
34
List of captions for illustrations
Fig. 1. Structural diagram of water-lubricated hydrostatic spindle
Fig. 2. Calculation scheme for spindle rotor
Fig. 3. Experimental rig for measurement of spindle dynamic characteristics (a) Main components for measurement (b) Experimental setup
Fig. 4. Amplitude-frequency characteristic of spindle obtained by simulations
Fig. 5. Amplitude-frequency characteristic of journal hydrostatic bearing
Fig. 6. Spindle dynamics obtained via experiments (a) Applied force and displacement at 280 Hz (b) Amplitude-frequency characteristics (square symbols are from experimental data; solid curve is from theory)
Fig. 7. Nyquist plot obtained via simulation (a) Frequency range 0-10,000 Hz (b) Local resonance at 1,000 Hz
Fig. 8. Sum of rotor displacements
35
Fig. 9. Influence of bearing parameters on dynamic characteristics (a) Influence of supply pressure (b) Influence of bearing clearance
Fig. 10. Influence of journal diameter on dynamic characteristics (ps= 2.5 MPa) (a) Overall amplitude (b) Contribution of bearing and rotor parameters to overall amplitude
Table 1 Constant parameters representing spindle for analysis
36
Fig. 1. Structural diagram of water-lubricated hydrostatic spindle
Fig. 2. Calculation scheme for spindle rotor
Piezo actuator
Water hydrostatic spindle
Load cell t F
Oscilloscope
Displacement sensor
Fig. 3. Experimental rig for measurement of spindle dynamic characteristics
160
Ar
Amplitude [nm]
140 120 100 80
1000 Hz
60 40 20 0
Alc 0
Frequency [Hz]
10000
Fig. 4. Amplitude-frequency characteristic of spindle obtained by simulations
Amplitude [nm]
2
1 200 Hz
0
0
Frequency [Hz]
1000
Fig. 5. Amplitude-frequency characteristic of journal hydrostatic bearing
40 Force
30
Force [N]
1
20 10
0
0 -10
-1
-20 -30
Displacement
-2 0
0.005
0.01
0.015
Displacement [nm]
2
-40 0.02
Time [s]
(a) Applied force and displacement at 280 Hz
Amplitude [nm]
40 35 100 Hz 30 25 20
0
Frequency [Hz]
1000
(b) Amplitude-frequency characteristics (square symbols are from experimental data; solid curve is from theory)
Fig. 6. Spindle dynamics obtained via experiments
ks
9795 Hz
Re [nm]
0 Hz
0 -80
-60
-40
-20
-20
0
20
40
60
80
100
Alc
-40 1000 Hz
5600 Hz
-60 6604 Hz
-80 -100 -120 -140
6106 Hz
-160
Ar
-180
-Im [nm]
(a) Frequency range 0-10,000 Hz -2
Re [nm] 35
20
-2.5
Alc
1000 Hz
-3 600 Hz
2000 Hz
-3.5 -4
5 nm
-4.5 400 Hz
3200 Hz
-5
-Im [nm]
(b) Local resonance at 1,000 Hz
Fig. 7. Nyquist plot obtained via simulation
Fig. 8. Sum of rotor displacements
6135
fr 40
6120
A0 35
6105
Frequency [Hz]
Amplitude [nm]
45
0.5 MPa 30
6090
1.5
Pressure [MPa]
3.5
(a) Influence of supply pressure 6400
fr
6300
40
A0
6200
35 6100 30
6000
3 µm 25
6
Frequency [Hz]
Amplitude [nm]
45
5900
Clearance [µm]
18
(b) Influence of bearing clearance
Fig. 9. Influence of bearing parameters on dynamic characteristics
80
7000
Amplitude [nm]
6000 60 50
5000
A0
40 4000 30
Frequency [Hz]
fr
70
10 mm
20
3000
30 Journal diameter [mm] 70 (a) Overall amplitude
Amplitude [nm]
40 Curve 1 (bearing stiffness and damping)
30 20
Curve 2 (bending deformation)
10
10 mm 0
30
Journal diameter [mm]
70
(b) Contribution of bearing and rotor parameters to overall amplitude
Fig. 10. Influence of journal diameter on dynamic characteristics (ps=2.5 MPa)
Table 1 Constant parameters representing spindle for analysis
Cross-
EJ × 103
l × 10-2
m/l
cb × 109
hb × 107
(N⋅m2)
(m)
(kg m-1)
(N⋅m-2)
(N⋅s m-2)
1
1.14
1.50
2.00
2
2.30
3.24
3.00
3
7.95
1.11
5.70
4
61.24
0.25
15.70
5
61.24
4.50
15.70
9.62
1.50
6
61.24
0.25
15.70
7
25.10
0.90
10.00
8
644.10
1.50
53.30
9
25.10
0.90
10.00
10
61.24
0.25
15.70
11
61.24
4.50
15.70
9.62
1.50
12
61.24
0.25
15.70
13
7.95
1.11
5.70
14
2.30
3.24
3.00
15
1.14
1.15
2.00
section number
Highlights
1. Water-lubricated hydrostatic bearings show preferable damping characteristics 2. Amplitude of the vibration of the rotor overhang part is dominantly affected not by bearing stiffness but by bending stiffness of the shaft 3. Natural frequency of the hydrostatic bearings can be increased by increasing supply pressure 4. Natural frequency of the hydrostatic bearings can be increased by decreasing bearing clearance
Conflict of Interest
Department of Mechanical Engineering Faculty of Engineering Kanagawa University
September 13, 2019
The authors have no conflicts of interest directly relevant to the content of this article.
Yohichi Nakao