Research Article Dynamics Analysis of Unbalanced Motorized...

Research ArticleDynamics Analysis of Unbalanced Motorized Spindles Supportedon Ball Bearings

Junfeng Liu,1 Tao Lai,1 and Xiaoan Chen2

1College of Mechatronic Engineering and Automation, National University of Defense Technology, Changsha 410073, China2The State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China

Correspondence should be addressed to Junfeng Liu; [email protected]

Received 21 January 2016; Revised 1 August 2016; Accepted 16 October 2016

Academic Editor: Gianluca Gatti

Copyright © 2016 Junfeng Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents an improved dynamic model for unbalanced high speed motorized spindles. The proposed model includes aHertz contact force model which takes into the internal clearance and an unbalanced electromagnetic force model based on theenergy of the air magnetic field. The nonlinear characteristic of the model is analysed by Lyapunov stability theory and numericalanalysis to study the dynamic properties of the spindle system. Finally, a dynamic operating test is carried out on a DX100A-24000/20-type motorized spindle. The good agreement between the numerical solutions and the experimental data indicatesthat the proposed model is capable of accurately predicting the dynamic properties of motorized spindles. The influence of theunbalanced magnetic force on the system is studied, and the sensitivities of the system parameters to the critical speed of thesystem are obtained. These conclusions are useful for the dynamic design of high speed motorized spindles.

1. Introduction

Due to the complexity of the reliability and performanceproblems caused by profound high speed effects [1–3], themachine tools capable of achieving the high speed cuttingwere first introduced commercially in 1980s, which lagsbehind the theory of high speed metal cutting by 50 years[4]. Motorized spindles are the core components of CNCmachine tools and greatly improve the cutting speed, whilethe inevitable unbalance of the spindle system caused byunexpected errors during manufacturing and installationprocess strongly influences the machining productivity andfinish quality of workpieces in high speed operating condi-tions [5–11]. Therefore, it is necessary to deeply study thedynamic properties of unbalanced machine tool motorizedspindle systems. With the low-friction and the conveniencein commodity standardization, ball bearings are widely usedin high speed motorized spindles [12]. Bearing clearanceconcerns the contact state between the balls and the innerand outer rings and affects the vibration response of therotor system [13]. Harsha presented a model for studyingthe structural vibrations of ball bearing-rotor systems due toradial internal clearance [14]. Sopanen andMikkola proposed

a dynamic model of a ball bearing-rotor system with sixdegrees of freedom to study the effects of the bearing clear-ance on the natural frequencies and the vibration responsesof the system [15, 16]. Based on it, Gao found that appropriatenegative internal clearance helps to reduce the vibrationresponses of the spindle and improve the stability of thesystem [17, 18]. Bearing preload could keep the bearingin a negative internal clearance state by eliminating theclearance, but too large preload will produce much frictionalheat, leading to the rise of temperature and the decline ofbearing life [19]. With the use of the Transfer Matrix Method,Jiang and Mao investigated the variable optimum preloadfor a machine tool spindle [20]. Alfares and Elsharkawyconstructed a mathematical model based on five degrees ofdynamic system to study the influences of axial preload ofangular ball bearings on the vibration behavior of a grindingmachine spindle and calculated the optimum initial axialpreload for the system [21].

Series of researches on the dynamic performance and theoptimal design method of the unbalanced rotor-ball bearingsystem of motorized spindles were deeply discussed in aboveworks. Motorized spindles are equipped with built-in motorsand their output ends are directly connected with tools, and

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 2787524, 10 pageshttp://dx.doi.org/10.1155/2016/2787524

2 Shock and Vibration

Rear bearings

Cutting tool

Front bearings

Rotor

Spindle

Stator

Figure 1: A typical high speed motorized spindle system.

mb1

k1 c1

mb2

k2 c2

kbx1 cbx1 kby1 cby1 e kbx2 cbx2 kby2 cby2

mr

Figure 2: The simplified model of a motorized spindle system.

the rotor of the system is acted upon by an additional unbal-ancedmagnetic force except the unbalanced centrifugal force.The unbalanced magnetic force is caused by the nonuniformair gap between the stator and the rotor in operation andaffects the dynamic characteristics of the system. In view ofthis fact, this paper proposes the matrix expression of themagnetic stiffness based on the gap magnetic field energy.Then a dynamic model for high speed motorized spindles isestablished with the consideration of unbalanced magneticforce, centrifugal force, and nonlinear Hertz contact force.The vibration response and critical speed of the systemare systematically calculated and experimentally validated.Finally, the sensitivities of the magnetic, mechanical, andbearing parameters to the critical speed are presented toinstruct the dynamic design of unbalanced high speedmotor-ized spindles.

2. Dynamic Model of Unbalanced MotorizedSpindle System

Figure 1 shows a typical motorized spindle system whichincludes a stator, a spindle shaft (rotor), a cutting tool, atool holder, and bearing components. The system can besimplified to three disc elements. Considering the motionsof the discs along the vertical and horizontal directions,the spindle-bearing system is regarded as a system with sixdegrees of freedom, as shown in Figure 2. Specifically, thefront and rear bearings are treated as springs and dampersin parallel connection, and the magnetic force is treated assprings between the stator and the rotor.

2.1. Model of Unbalance Magnetic Force. The gap eccentricityof the spindle is shown as in Figure 3, o is the inner circlecenter of stator (the origin of coordinates), 𝑜1 is the outercircle center of rotor, and |𝑜𝑜1| = (𝑥2 +𝑦2)0.5. 𝑐 is the centroidof rotor, 𝑒 is the mass eccentricity of rotor, 𝛼 is the anglebetween the 𝑥-axis and the circumferential position where

y

xo

x1

c

e

Φo1

Figure 3: Air gap eccentricity of the rotor in motorized spindles.

the air gapwidth is equal to𝛿 and𝛽 is the angle between the𝑥-axis and the circumferential position where the air gap widthis the least.

The built-in motor in high speed machine tool spindlesystem is generally a three-phase AC induction motor, so theenergy of the gap magnetic field of the motor is [22]

𝑊 = 𝑅𝑔𝐿𝑔2 ∫2𝜋0Λ 0 ∞∑𝑛=0

𝜀𝑛cos𝑛 (𝛼 − 𝛽)⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2 𝑑𝛼.

(1)

Specifically, the largest component, which is 𝑛 = 0,plays a dominant role; other components, which are 𝑛 = 0,take different proportion, respectively. Because the effectiverelative eccentricity 𝜀 is far less than 1, expand the high orderterms in (1) and take the first three terms, and (1) can beconverted into

𝑊 = 𝑅𝑔𝐿𝑔Λ 02 ∫2𝜋0(1 + 𝑥2 + 𝑦22𝜎2 + 𝑥𝜎 cos𝛼 + 𝑦𝜎 sin𝛼

+ 𝑥2 − 𝑦22𝜎2 cos 2𝛼 + 𝑥𝑦𝜎2 sin 2𝛼)⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2 𝑑𝛼 ,

(2)

where 𝑅𝑔 is the radius of the inner circle of stator; 𝐿𝑔 iseffective length of rotor; Λ 0 is the magnetic conduction ofuniformair gap, andΛ 0 = 𝜇0/𝑘𝜇𝛿0, 𝜇0 is the air permeability,𝑘𝜇 is the saturation, and 𝑘𝜇 = 1 + 𝛿Fe/𝑘0𝛿0, 𝑘0 is thecalculation coefficient of uniform air gap, 𝛿Fe is the equivalentair gap of ferromagnetic materials, 𝛿0 is the width of uniformair gap, 𝜎 = 𝑘𝜇𝛿0, 𝜔𝑓 is the speed of rotating magnetic field,and 𝜔𝑓 = 2𝜋𝑓/𝑝, 𝑓 is the current frequency, 𝑝 is the numberof pole-pairs, and 𝐹sm and 𝐹rm are the amplitudes of three-phase resultant magnetomotive forces of the stator and therotor, respectively;𝜑1+𝜑2 is the phase angle inwhich the rotor

Shock and Vibration 3

current lags behind the stator current, and 𝜑1 = 𝑥1𝑠/𝑟1, 𝜑2 =𝑥2𝑠/𝑟2, 𝑥1𝑠 and 𝑥2𝑠 are the leakage reactance of per phasewinding of the stator and the rotor, respectively; 𝑟1 and 𝑟2are the resistances of per phase winding of the stator and therotor, respectively.

Equation (2) can be expressed by

W = 12V𝑇K𝑒V + V𝑇K𝑒, (3)

where V = {𝑥, 𝑦}𝑇 and K𝑒 is the magnetic stiffness matrix ofthe gap magnetic field:

𝐾11 = 𝑅𝑔𝐿𝑔Λ 02𝜎2 ∫2𝜋0{[1 + cos (2𝛼)]

⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2} 𝑑𝛼

𝐾12 = 𝐾21 = 𝑅𝑔𝐿𝑔Λ 02𝜎2 ∫2𝜋0{[sin (2𝛼)]

⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2} 𝑑𝛼

𝐾22 = 𝑅𝑔𝐿𝑔Λ 02𝜎2 ∫2𝜋0{[1 − cos (2𝛼)]

⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2} 𝑑𝛼.

(4)

K𝑒 is the vector of the gap magnetic field energy:

𝐾1 = 𝑅𝑔𝐿𝑔Λ 02𝜎2 ∫2𝜋0{cos𝛼 ⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)

+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2} 𝑑𝛼𝐾2 = 𝑅𝑔𝐿𝑔Λ 02𝜎2 ∫2𝜋

0{sin𝛼 ⋅ [𝐹sm cos (𝜔𝑓𝑡 − 𝑝𝛼)

+ 𝐹rm cos (𝜔𝑓𝑡 − 𝑝𝛼 − 𝜑1 − 𝜑2)]2} 𝑑𝛼.

(5)

2.2. Hertz Contact Force Model of Ball Bearings. In the rotor-ball bearing system of a high speed motorized spindle, theinner and outer races of the bearings are fixed to the rotatingshaft and the bearing housing, respectively. Balls can beconsidered to make pure rolling movement between theinner race and the outer race with equal spacing. Thereforefor the ball bearing in any time, the elastic deformationassociated with the contact point at the jth ball depends onthe displacement of rotor and the initial bearing clearance,given by

Δ 𝑗 = 𝑥𝑏 sin 𝜃𝑗 + 𝑦𝑏 cos 𝜃𝑗 − 𝛾, (6)

where 𝑥𝑏 and 𝑦𝑏 are the displacements at the bearing along𝑥- and 𝑦-directions, respectively; 𝜃𝑗 is the azimuth of the jthball, and 𝜃𝑗 = (2𝜋/𝑁𝑏)(𝑗 − 1) + (𝑑𝑖/(𝑑𝑖 + 𝑑𝑜))𝜔𝑡, 𝑁𝑏 is thenumber of balls, 𝑑𝑖 is the inner diameter of inner race, and𝑑𝑜 is the outer diameter of outer race; 𝛾 is the initial bearingclearance, which can be determined according to the initialbearing preload [15].

Considering that the contact force is nonnegative, thelocal Hertz contact force and deflection relationship forbearing can be written as [18]

𝐹𝜃𝑗 = {{{𝑘𝑏 ⋅ Δ 𝑗2/3; Δ 𝑗 > 00; Δ 𝑗 ≤ 0. (7)

𝑘𝑏 is contact stiffness which is given as

𝑘𝑏 = { 1(𝑘−1𝑖 )2/3 + (𝑘−1𝑜 )2/3}

3/2

, (8)

where 𝑘𝑖 = 𝑘𝑜 = (2√2/3)⋅(𝐸/(1−𝜇))(Σ𝜌)−1/2 ⋅(𝛿∗)−3/2,𝐸 is theelasticmodulus, and 𝜇 is the Poisson ratio.The curvature sumΣ𝜌 and deformation coefficient 𝛿∗ are obtained from Harris[23].

Similar to [24], the contact damping can be written as

𝑐𝑏 = 10−5 ⋅ 𝑘𝑏. (9)

By summing the local contact forces, the total restoringforce components of the ball bearing along the 𝑥- and 𝑦-directions are given by

𝐹𝑏𝑥 = −𝑐𝑏��𝑏 −𝑁𝑏∑𝑗=1

𝐹𝜃𝑗 sin 𝜃𝑗;

𝐹𝑏𝑦 = −𝑐𝑏��𝑏 −𝑁𝑏∑𝑗=1

𝐹𝜃𝑗 cos 𝜃𝑗.(10)

2.3. Motional Differential Equations and Stability Analysisof the Spindle System. As the middle part of the spindlesystem is simplified to a disc element, the gap eccentricity ofthe motor is considered as uniform. Without regard for theinfluence of gyroscopic moment, the differential equationsgoverning motions of the motorized spindle systems can beobtained with Newton second law:

𝑚𝑏1��𝑏1 + 𝑐1 (��𝑏1 − ��𝑟) + 𝑘1 (𝑥𝑏1 − 𝑥𝑟) = −𝑁1 ⋅ 𝐹𝑏𝑥1𝑚𝑏1��𝑏1 + 𝑐1 (��𝑏1 − ��𝑐) + 𝑘1 (𝑦𝑏1 − 𝑦𝑟) = −𝑚𝑏1 ⋅ 𝑔− 𝑁1 ⋅ 𝐹𝑏𝑦1

𝑚𝑟��𝑟 + 𝑐1 (��𝑟 − ��𝑏1) + 𝑐2 (��𝑟 − ��𝑏2) + 𝑘1 (𝑥𝑟 − 𝑥𝑏1)+ 𝑘2 (𝑥𝑟 − 𝑥𝑏2) + 𝐾11 ⋅ 𝑥𝑟 + 𝐾12 ⋅ 𝑦𝑟 = 𝐹𝑐𝑥 − 𝐾1

𝑚𝑟��𝑟 + 𝑐1 (��𝑟 − ��𝑏1) + 𝑐2 (��𝑟 − ��𝑏2) + 𝑘1 (𝑦𝑟 − 𝑦𝑏1)+ 𝑘2 (𝑦𝑟 − 𝑦𝑏2) + 𝐾21 ⋅ 𝑥𝑟 + 𝐾22 ⋅ 𝑦𝑟 = 𝐹𝑐𝑦 − 𝑚𝑟 ⋅ 𝑔− 𝐾2


𝑚𝑏2��𝑏2 + 𝑐2 (��𝑏2 − ��𝑟) + 𝑘2 (𝑥𝑏2 − 𝑥𝑟) = −𝑁2 ⋅ 𝐹𝑏𝑥2𝑚𝑏2��𝑏2 + 𝑐2 (��𝑏2 − ��𝑟) + 𝑘2 (𝑦𝑏2 − 𝑦𝑟) = −𝑚𝑏2 ⋅ 𝑔− 𝑁2 ⋅ 𝐹𝑏𝑦2,

(11)

where 𝑥𝑏1 and 𝑦𝑏1, 𝑥𝑟 and 𝑦𝑟, and 𝑥𝑏2 and 𝑦𝑏2 are thedisplacements of mass elements distributed at the frontbearing, rotor, and the rear bearing along𝑥- and𝑦-directions,respectively. It should be noticed that the variable 𝑦 is definedalong the opposite direction of the gravity. 𝑚𝑏1, 𝑚𝑟, and 𝑚𝑏2are the masses of the spindle distributed at the front bearing,middle between the two sets of bearings, and the rear bearing,respectively, 𝑘1 and 𝑘2 are the bending stiffness of the spindle,𝑐1 and 𝑐2 are thematerial damping of the spindle, respectively,

and 𝑐𝑞 = 0.001/𝜔 ⋅ 𝑘𝑞, 𝑞 = 1, 2; 𝑁1 and 𝑁2 are the numberof the two sets of bearings, respectively; 𝐹c𝑥 = 𝑚𝑟𝑒𝜔2 cos𝜙;𝐹c𝑦 = 𝑚𝑟𝑒𝜔2 sin𝜙, where 𝜔 is the angular velocity of rotor,and 𝜔 = (1 − 𝑠)𝜔𝑓, s is the slip of motor, 𝜙 is the eccentricangle of rotor mass, and 𝜙 = 𝜔𝑡.

Without the influence of the unbalanced magnetic force,the values of the magnetic stiffness matrix and vector of thegap magnetic field in (11) are zero.

Equations (1), (2), (3), (6), (7), (8), (9), (10), and (11)constitute the dynamic model of the unbalanced motorizedspindle system.There are coupling items in the model due tothe magnetic stiffness; meanwhile the total restoring forces𝐹𝑏𝑥, 𝐹𝑏𝑦 are nonlinear functions of vibration amplitudes in𝑥- and 𝑦-directions, so the model has strong nonlinearcharacteristic and the nonlinear vibration theory must beused [25]. The state equations of (11) can be given as

𝐹 =

{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{

��𝑏1��𝑏1��𝑟��𝑟��𝑏2��𝑏2−1𝑚𝑏1 ⋅ [𝑐1 (��𝑏1 − ��𝑟) + 𝑘1 (𝑥𝑏1 − 𝑥𝑟) + 𝑁1 ⋅ 𝐹𝑏𝑥1]−1𝑚𝑏1 ⋅ [𝑐1 (��𝑏1 − ��𝑟) + 𝑘1 (𝑦𝑏1 − 𝑦𝑟) + 𝑚𝑏1 ⋅ 𝑔 + 𝑁1 ⋅ 𝐹𝑏𝑦1]−1𝑚𝑟 ⋅ [𝑐1 (��𝑟 − ��𝑏1) + 𝑐2 (��𝑟 − ��𝑏2) + 𝑘1 (𝑥𝑟 − 𝑥𝑏1) + 𝑘2 (𝑥𝑟 − 𝑥𝑏2) + 𝐾11 ⋅ 𝑥𝑟 + 𝐾12 ⋅ 𝑦𝑟 − 𝐹𝑐𝑥 + 𝐾1]−1𝑚𝑟 ⋅ [𝑐1 (��𝑟 − ��𝑏1) + 𝑐2 (��𝑟 − ��𝑏2) + 𝑘1 (𝑦𝑟 − 𝑦𝑏1) + 𝑘2 (𝑦𝑟 − 𝑦𝑏2) + 𝐾21 ⋅ 𝑥𝑟 + 𝐾22 ⋅ 𝑦𝑟 − 𝐹𝑐𝑦 + 𝑚𝑟 ⋅ 𝑔 + 𝐾2]−1𝑚𝑏2 ⋅ [𝑐2 (��𝑏2 − ��𝑟) + 𝑘2 (𝑥𝑏2 − 𝑥𝑟) + 𝑁2 ⋅ 𝐹𝑏𝑥2]−1𝑚𝑏2 ⋅ [𝑐2 (��𝑏2 − ��𝑟) + 𝑘2 (𝑦𝑏2 − 𝑦𝑟) + 𝑚𝑏2 ⋅ 𝑔 + 𝑁2 ⋅ 𝐹𝑏𝑦2]

}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}

. (12)

The accelerations and velocities are zero in (12) whenthe spindle system is in equilibrium. According to the firstapproximate theory of Lyapunov, the motion stability of thenonlinear system is determined by the latent root of the linearapproximation system. Expand the state equation (12) at theequilibrium point approximately, and the Jacobi matrix canbe obtained. If all Jacobi Matrix Eigen values have negativereal parts, the system is asymptotically stable. When part ofthe Jacobi Matrix Eigen values own negative real parts whilethe others own positive real parts, the system is unstable. Onthe other hand, once the value of a real part of the JacobiMatrix Eigen is zero while the others are not, the systemis in a critical state and is starting buckling, and then thecritical speed of the spindle could be obtained. Due to thestrongly nonlinear characteristic, the equationmust be solvedwith numerical analysis. The Runge-Kutta method is used tocalculate (11) to obtain the dynamic responses of the system.

3. Experimental Setup and Measurements

3.1. Parameters. To validate the proposed dynamic model,a dynamic operating test was carried out on a DX100A-24000/20-typemotorized spindle.Themaximum rated speedof the spindle is 24000 rpm. Both the two sets of the bearingsare a pair in bearing combination with constant preload,and the front and rear bearings are the angular contact ballbearings of 7006CYTAHQ1 and 7005CYTASP4, respectively,from HRB-BC. Table 1 lists the other parameters of thespindle system. It deserves noting that the spindle system canoperate with the speed higher than themaximum rated speedin a short time, and the oil dripping rate remains constant inthe test to obtain the experimental data with same parametersand conditions.

3.2. Experimental Setup. A dynamic operating test is carriedout to obtain the values of vibration response at different


Table 1: Parameters of DX100A-24000/20-type motorized spindle.

Parameter Units Given value𝑚𝑏1 kg 0.212𝑚𝑏2 kg 0.201𝑚𝑟 kg 0.483𝑘1 N/mm 12100𝑑𝑖 mm 6.35/6.70𝑑𝑜 mm 6.65/7.05𝑁𝑏 18/18𝑘2 N/mm 9840𝑠 0.01𝑒 𝜇m 0.879𝑅𝑔 mm 48𝐿𝑔 mm 45Λ 0 H/mm2 1.468 × 10−9𝜎 mm 0.856𝐹sm A 463.4𝐹rm A 418.3𝜑1 ∘ 57.2𝜑2 ∘ 11.3𝑝 1𝜇 0.3Γ mm −0.0012𝛿0 mm 0.97

UA300

DX100APC

LDS

LDSy-direction

x-direction

Figure 4: Experimental setup.

speeds. Due to the sharp increase of the vibratory output ofthe spindle system at the resonant frequency, the first criticalspeed of the system can be obtained from the variation of thevibration responses of the rotor. To avoid the influence of thethermal displacement, each vibration signal with a speed wascollected as quickly as possible once the operating speed isstable, and the interval of each signal collection is more thanhalf an hour.

As shown in Figure 4, the vibration signals in 𝑥- and 𝑦-directions at the front end of rotor under work conditionswere acquired by using Laser Displacement Sensor (LDS),then transmitted and transformed through the signal collec-tor and analyser UA300, and finally transmitted to the PC tobe handled by the signal acquisition procedure compiled byVisual C++ 6.0.

Table 2: First critical speeds of DX100A-24000/20-type motorizedspindle.

Number ofpole-pairs

Calculation result (rpm) Measured result(rpm)With MS Without MS

1 30336.12 34710.15 324002 30391.23 34710.153 30487.51 34710.154 30621.24 34710.15

4. Discussion of the Calculation andExperimental Results

4.1. Validation of the Dynamic Model. Figure 5 shows theexperimental vibration signals of the front end of spindle in𝑦-direction with different speeds. When the rotational speedis 32400 rpm, the vibration value of the spindle is larger thanother values with different speeds.

Figure 6 shows the experimental vibration displacementcurve of the front end of the spindle rotor in y-direction. Itcan be found from the curve that the first critical speed of thesystem is about 32400 rpm.

To discuss the influence of the unbalancedmagnetic forceon the dynamic behaviors of the system, (12) was studiedwith andwithout themagnetic stiffness (MS).Thefirst criticalspeeds of the spindle system obtained from calculation andexperiment are listed in Table 2. As can be seen, consideringthe effect of MS, the calculated value of the first criticalspeed decreases and has an error of −6.37% compared withthe experimental value. The difference in the first criticalspeeds between the two calculations is about 14.42%. Thedifference between the calculation and the measured resultin this paper is within 7%, so the data in Table 2 can testify tothe accuracy of the proposed dynamic model for motorizedspindles. In addition, the calculated value of the first criticalspeed increases with the number of pole-pairs, which meansthe MS has a little influence on the system stability in multi-pole-pairs motors.

To study the influence of the MS more directly, theradial displacement of the rotor center is discussed andits value is (𝑥2𝑟 + 𝑦2𝑟 )0.5. Figure 7 shows the theoreticalradial displacements of the rotor center. When taking theMS into account, the vibration amplitudes in nonresonanceand resonance regions increase; meanwhile the width of thespeed region of high vibration amplitudes becomes larger.Combined with the analysis result from Table 2, it can beknown that the MS reduces the system stability.

The influence of the radial displacement of the rotorcenter on the system stability is shown as in Figure 8. Forthe sake of discussion, the relative radial displacement of therotor center 𝑒𝑐 is taken and its value is (𝑥2𝑟 + 𝑦2𝑟 )0.5/𝑒. Thelarger the value of 𝑒𝑐 is, the lower the critical speed of thesystemwill be, and the downward trend ismore gentle, whichmeans when the critical speed of the spindle system is low, alittle decrease of the critical speed would lead to more drasticwhirling motion. Meanwhile, the ratio between the linearcritical angular velocity 𝜔𝑛 and the nonlinear critical angular


−7.5

−5

−2.5

0

2.5

5

7.5

Disp

lace

men

t (

m)

0.02 0.04 0.06 0.08 0.100Time (s)

(a) 12000 rpm

0.02 0.04 0.06 0.08 0.100Time (s)

−9

−6

−3

0

3

6

9

Disp

lace

men

t (

m)

(b) 18000 rpm

0.02 0.04 0.06 0.08 0.100Time (s)

−60

−40

−20

0

20

40

60

Disp

lace

men

t (

m)

(c) 32400 rpm

0.02 0.04 0.06 0.08 0.100Time (s)

−12

−7.2

−2.4

2.4

7.2

12

Disp

lace

men

t (

m)

(d) 36000 rpm

Figure 5: Experimental vibration signals of the front end of spindle in 𝑦-direction.

0

10

20

30

40

50

Am

plitu

de (

m)

0 5 10n (krpm)

15 20 25 30 35 40

Figure 6: Experimental vibration displacement curve of the frontend of spindle in 𝑦-direction.

velocity 𝜔𝑐 gently increases with the increasing of the valueof 𝑒𝑐 and then sharply decreases. Respectively, 𝜔𝑛 presents[{𝑘1 + 𝑘2 + (𝐾11 + 𝐾12 + 𝐾21 + 𝐾22)/4}/𝑚𝑟]0.5 [26] and 𝜔𝑐 isthe angular velocity form of the first critical speed discussedin Table 2. In a significant extent of the value of 𝑒𝑐, the linearcritical speed is about 0.65 times that of the nonlinear criticalspeed.

Figure 9 shows the Poincare mapping of the rotor centerin y-direction, which is a typical period-doubling bifurcation.

0 10 20 30 40 50n (krpm)

10

20

30

40

50

60

70

(x2 r+y2 r)0.

5(

m)

Figure 7: Theoretical radial vibration displacement curves of therotor center.

When the angular velocity 𝜔𝑓 is at 3000 rad/s, there is afrequency division phenomenon in the system (Figure 9(a));with the 𝜔𝑓 increase to 3180 rad/s, the chaos phenomenonoccurs (Figure 9(b)), while the chaos disappears with theinverse bifurcation at 3300 rad/s.

4.2. Dynamic Design for Motorized Spindles. To instruct thedynamic design of the spindle system, the influence of system


20 40 60 80 1000ec

20

30

40

50

60

70

80

90

Criti

cal s

peed

(krp

m)

(a) Critical speed

20 40 60 80 1000ec

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n/

c

(b) Angular velocity ratio

Figure 8: Theoretical influence of the radial displacement of the rotor center on the system stability.

Velo

city

(mm·s−

1)

0.050 0.1−0.1 −0.05

Displacement (mm)

5

10

15

20

25

30

35

40

45

(a) 3000 rad/s

Velo

city

(mm·s−

1)

0.050 0.1−0.1 −0.05

Displacement (mm)

5

10

15

20

25

30

35

40

45

(b) 3180 rad/s

Velo

city

(mm·s−

1)

0.050 0.1−0.1 −0.05

Displacement (mm)

5

10

15

20

25

30

35

40

45

(c) 3300 rad/s

Figure 9: Poincare mapping of the rotor center in 𝑦-direction.


0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.5Magnetic parameters (times)

20

25

30

35

40

45Cr

itica

l spe

ed (k

rpm

)

(a) Magnetic parameters

With MSWithout MS

0.822

24

26

28

30

32

34

36

38

Criti

cal s

peed

(krp

m)

1 1.1 1.2 1.3 1.4 1.5 1.60.9Shaft stiffness (104 N/mm)

(b) Shaft stiffness

With MSWithout MS

25

30

35

40

Criti

cal s

peed

(krp

m)

6420 8 10−4−6−8 −2−10

Bearing clearance (m)

(c) Bearing clearance

With MSWithout MS

26

27

28

29

30

31

32

Criti

cal s

peed

(krp

m)

13 14 15 16 17 18 19 2012Ball number

(d) Ball number

Figure 10: Theoretical influence of system parameters on critical speed.

parameters on the first critical speed is discussed and thetheoretical results are shown as in Figure 10.

(a) The magnetic parameters present 𝑅𝑔𝐿𝑔Λ 0/2𝜎2 inFigure 10(a), and the unit of the abscissa is the multiple of𝑅𝑔𝐿𝑔Λ 0/2𝜎2. With the increase of the inner radius of stator𝑅𝑔 and the effective length of rotor 𝐿𝑔 and the decrease ofuniform air gap 𝛿0, the critical speed decreases nonlinearly.When the magnetic parameters are 1.5 times that of thedesign value, the critical speed is close to the minimumrated speed. So the magnetic parameters are vital to thedynamic design of the spindle system, and their effects onthe dynamic characteristic of motorized spindles cannot beignored. Besides, the increase of the uniform air gap is ofadvantage to the rotor stability, but the gap cannot be too largeand should be optimized.

(b) As shown in Figure 10(b), with the increase of the shaftstiffness, the first critical speed of the system increases, andthe difference between the two calculations decreases, whichmeans that high shaft stiffness can weaken the influence ofthe MS.

(c) Figure 10(c) shows the impact of the bearing clearance𝛾 on the critical speed. Preload makes the bearing in anegative clearance state and improves the stability of thespindle system, which also can reduce the effect of MS.

(d) Figure 10(d) shows that the critical speed rises withthe increase of ball number.This is because the increase of ballnumber leads to the improvement of the support stiffness andthe frequency of the parametric vibration for ball bearingsandmeanwhile undermines the frequencymultiplication andharmonic components of the parametric vibration, whichbenefits the stability of system. Without the consideration ofthe waviness on the race, the bearings which have more ballscan effectively improve the operational stability of the spindleand decrease the influence of MS.

Based on the results of Figures 10(b), 10(c), and 10(d), itcan be known that with the natural properties of the systemincrease, such as shaft stiffness and support stiffness, theinfluence of MS to the stability of the system decreases. Thatmeans the higher the stability of the system is, the lower theinfluence of MS to the stability of the system is.


5. Conclusions

This paper deals with the problem of analysing the dynamicbehaviors of unbalanced high speed motorized spindlessupported on ball bearings. An improved dynamic modelwhich takes into account the unbalanced magnetic force isdeveloped for the presented system. The model is studiedby using the Lyapunov nonlinear vibration theory. Then,the validation test on a DX100A-24000/20-type motorizedspindle is performed. From the calculated and experimentalresults the following conclusions are drawn.

(a) The proposed model accurately predicted the criticalspeed observed in the experiment and allows explorationof the magnetic force contribution within the context of itsassumptions, which can be used for analysing the dynamicproperties of high speed machine tool motorized spindlesystems.

(b) The unbalanced magnetic force in the machine toolmotorized spindle system increases the vibration response ofthe rotor and reduces the critical speed of the system. Theinfluence of the MS on the stability of the system decreaseswith increasing the number of pole-pairs. With the increaseof 𝑒𝑐, the critical speed of system decreases, while the ratio𝜔𝑛/𝜔𝑐 increases and then sharply decreases. The period-doubling bifurcation occurs with the angular velocity 𝜔𝑓being about 3200 rad/s.

(c) The system parameters greatly influence the criticalspeed of the system.The increase of the shaft stiffness and theball number of bearings is of advantage to the stability, but theincrease of the magnetic parameters and bearing clearancewould lead to the decrease of the critical speed of the system.If the spindle system has higher natural properties, theinfluence of MS on the stability of the system will decrease.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (nos. 51405151 and 51005259).

References

[1] M. Munawar, J. C.-S. Chen, and N. A. Mufti, “Investigationof cutting parameters effect for minimization of sur faceroughness in internal turning,” International Journal of PrecisionEngineering and Manufacturing, vol. 12, no. 1, pp. 121–127, 2011.

[2] S.-T. Chen, K.-E. Chang, W.-P. Huang, H.-Y. Yang, and X.-M. Lee, “Development of a cost-effective high-precision benchmachine tool for multi-level micro aspheric lighting-lens moldmachining,” International Journal of Precision Engineering andManufacturing, vol. 13, no. 12, pp. 2225–2231, 2012.

[3] C.-W. Lin, “Simultaneous optimal design of parameters andtolerance of bearing locations for high-speed machine toolsusing a genetic algorithm andMonte Carlo simulationmethod,”International Journal of Precision Engineering and Manufactur-ing, vol. 13, no. 11, pp. 1983–1988, 2012.

[4] T.-H. Duong, H.-C. Kim, D.-Y. Lee, S.-W. Lee, E.-S. Park, andT.-J. Je, “A theoretical deformation prediction ofmicro channelsin ultra-precisionmachining,” International Journal of PrecisionEngineering and Manufacturing, vol. 14, no. 2, pp. 173–181, 2013.

[5] S.Mohan, S.-H. Kweon, D.-M. Lee, and S.-H. Yang, “ParametricNURBS curve interpolators: a review,” International Journal ofPrecision Engineering and Manufacturing, vol. 9, no. 2, pp. 84–92, 2008.

[6] R. Kurniawan and T. J. Ko, “A study of surface texturingusing piezoelectric tool holder actuator on conventional CNCturning,” International Journal of Precision Engineering andManufacturing, vol. 14, no. 2, pp. 199–206, 2013.

[7] S. Jee and J. Lee, “Real-time inertia compensation for multi-axis CNC machine tools,” International Journal of PrecisionEngineering and Manufacturing, vol. 13, no. 9, pp. 1655–1659,2012.

[8] E. Abele, Y. Altintas, and C. Brecher, “Machine tool spindleunits,” CIRP Annals—Manufacturing Technology, vol. 59, no. 2,pp. 781–802, 2010.

[9] J. L. Dohne, T. D. Hinnerichs, J. P. Lauffer, C.-M. Kwan, M.E. Regelbrugge, and N. Shankar, “Active chatter control in amilling machine,” in Smart Structures and Materials: Industrialand Commercial Applications of Smart Structures Technologies,vol. 3044 of Proceedings of SPIE, pp. 281–294, San Diego, Calif,USA, March 1997.

[10] B. Bossmanns and J. F. Tu, “A thermal model for high speedmotorized spindle,” International Journal of Machine Tools &Manufacture, vol. 43, no. 11, pp. 1163–1170, 1999.

[11] I. A. Zverev, I.-U. Eun, W. J. Chung, and C. M. Lee, “Simulationof spindle units running on rolling bearings,”The InternationalJournal of AdvancedManufacturing Technology, vol. 21, no. 10-11,pp. 889–895, 2003.

[12] L.-S. Law, J. H. Kim, W. Y. H. Liew, and S.-K. Lee, “An approachto monitoring the thermomechanical behavior of a spindlebearing system using acoustic emission (AE) energy,” Interna-tional Journal of Precision Engineering and Manufacturing, vol.14, no. 7, pp. 1169–1175, 2013.

[13] M. Tiwari, K. Gupta, and O. Prakash, “Dynamic response of anunbalanced rotor supported on ball bearings,” Journal of Soundand Vibration, vol. 238, no. 5, pp. 757–779, 2000.

[14] S. P. Harsha, “Nonlinear dynamic response of a balanced rotorsupported by rolling element bearings due to radial internalclearance effect,” Mechanism and Machine Theory, vol. 41, no.6, pp. 688–706, 2006.

[15] J. Sopanen and A. Mikkola, “Dynamic model of a deep-grooveball bearing including localized and distributed defects—part 1:theory,” Proceedings of the Institution of Mechanical Engineers,Part K: Journal of Multi-body Dynamics, vol. 217, no. 3, pp. 201–218, 2003.

[16] J. Sopanen and A. Mikkola, “Dynamic model of a deep-grooveball bearing including localized and distributed defects. Part2: implementation and results,” Proceedings of the Institution ofMechanical Engineers, Part K: Journal of Multi-Body Dynamics,vol. 217, no. 3, pp. 213–223, 2003.

[17] S.-H. Gao, X.-H. Long, and G. Meng, “Nonlinear responseand nonsmooth bifurcations of an unbalanced machine-toolspindle-bearing system,”Nonlinear Dynamics, vol. 54, no. 4, pp.365–377, 2008.

[18] S.-H. Gao, X.-H. Long, and G. Meng, “Nonlinear response andstability of a spindle system supported by ball bearings,”Archiveof Applied Mechanics, vol. 80, no. 9, pp. 1069–1081, 2010.


[19] X. Chen, J. Liu, Y. He, P. Zhang, and W. Shan, “Thermalproperties of high speed motorized spindle and their effects,”Journal of Mechanical Engineering, vol. 49, no. 11, pp. 135–142,2013.

[20] S. Y. Jiang and H. B. Mao, “Investigation of variable optimumpreload for a machine tool spindle,” International Journal ofMachine Tools & Manufacture, vol. 50, no. 1, pp. 19–28, 2010.

[21] M. A. Alfares and A. A. Elsharkawy, “Effects of axial preloadingof angular contact ball bearings on the dynamics of a grindingmachine spindle system,” Journal of Materials Processing Tech-nology, vol. 136, no. 1–3, pp. 48–59, 2003.

[22] J. J. Qiu, Electromechanical Analysis of Dynamics, Science Press,Beijing, China, 1992.

[23] T. A. Harris, Rolling Bearing Analysis, John Wiley & Sons, NewYork, NY, USA, 4th edition, 2001.

[24] E. Kramer, Dynamics of Rotors and Foundations, Springer, NewYork, NY, USA, 1993.

[25] K. H. Shin, “On the performance of a single degree-of-freedomhigh-static-low-dynamic stiffness magnetic vibration isolator,”International Journal of Precision Engineering and Manufactur-ing, vol. 15, no. 3, pp. 439–445, 2014.

[26] Z. Z. Li, D. Kong, Y. H. Jiao, Z. B. Chen, andM. Z. Li, “Nonlineardynamics analysis on sealed rotor system,” Journal of Vibrationand Shock, vol. 28, no. 6, pp. 159–163, 2009.

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