applied sciences Article
Hybrid Prediction Model of the Temperature Field of a Motorized Spindle Lixiu Zhang 1 , Chaoqun Li 2 , Yuhou Wu 2 , Ke Zhang 2 and Huaitao Shi 2, * 1 2
*
Test and Analysis Center, Shenyang Jianzhu University, No. 9, Hunnan East Road, Hunnan New District, Shenyang 110168, China;
[email protected] School of Mechanical Engineering, Shenyang Jianzhu University, No. 9, Hunnan East Road, Hunnan New District, Shenyang 110168, China;
[email protected] (C.L.);
[email protected] (Y.W.);
[email protected] (K.Z.) Correspondence:
[email protected]; Tel.: +86-024-24690088
Received: 26 September 2017; Accepted: 18 October 2017; Published: 22 October 2017
Abstract: The thermal characteristics of a motorized spindle are the main determinants of its performance, and influence the machining accuracy of computer numerical control machine tools. It is important to accurately predict the thermal field of a motorized spindle during its operation to improve its thermal characteristics. This paper proposes a model to predict the temperature field of a high-speed and high-precision motorized spindle under different working conditions using a finite element model and test data. The finite element model considers the influence of the parameters of the cooling system and the lubrication system, and that of environmental conditions on the coefficient of heat transfer based on test data for the surface temperature of the motorized spindle. A genetic algorithm is used to optimize the coefficient of heat transfer of the spindle, and its temperature field is predicted using a three-dimensional model that employs this optimal coefficient. A prediction model of the 170MD30 temperature field of the motorized spindle is created and simulation data for the temperature field are compared with the test data. The results show that when the speed of the spindle is 10,000 rpm, the relative mean prediction error is 1.5%, and when its speed is 15,000 rpm, the prediction error is 3.6%. Therefore, the proposed prediction model can predict the temperature field of the motorized spindle with high accuracy. Keywords: motorized spindle; temperature field; prediction model; heat transfer coefficient; finite element
1. Introduction The motorized spindle unit is crucial to numerical control (NC) machine tools and the key component to guarantee the precision of machines. Its performance directly influences the technology used and determines the development of the machine tool [1]. The motorized spindle unit is indispensable to high performance computer numerical control (CNC) machine tools. Several studies have focused on improving the spindle [2–5]. The motorized spindle incurs mechanical and electromagnetic losses because of its high speed and non-sinusoidal power supply. Most of these losses are transformed into heat and transferred to the surrounding air, cooling fluid, and machine parts. This results in uneven heat deformation of the parts of the machine, which directly affects the accuracy of the spindle and the bearing preload. As many as 75% of the overall geometrical errors in workpieces manufactured by machines are considered to be induced by the effects of temperature [6]. The temperature of the motorized spindle is an important index to evaluate its performance at high speeds [7]. High-power and high-speed motorized spindles feature a larger rise in temperature than surroundings, which is therefore important to control.
Appl. Sci. 2017, 7, 1091; doi:10.3390/app7101091
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There are two methods for controlling the rise in temperature of the motorized spindle. One involves actively controlling the cooling and lubrication systems to avoid high temperatures, and the other consists of optimizing the design of the structure of the motorized spindle [8,9]. These two methods are different, but both require the development of a thermal model for the motorized spindle to predict the temperature field. The Ellesmere Company has been developing CAD software for motor design software for the last 10 years, which includes modules for thermal error calculation [10]. However, it is challenging to effectively calculate thermal error in the motorized spindle as it is fundamentally different from an ordinary motor. The motorized spindle system contains a few subsystems, such as water cooling, lubrication, and the inverter. Thus, its heat generation and transfer processes are more complex. Finite element analysis is often used in a simplified thermal design of the motorized spindle unit [11,12]. Such thermal boundary conditions as the processing conditions of the heating and cooling environments are loaded into the model to simulate the temperature distribution of the motorized spindle. Jedrzejewski et al. [13–17] researched finite element thermal modeling of NC machine tools and created a hybrid model of a high-speed machining center spindle box using finite element modeling. This method was used to model the components of the symmetrical axis. A comparison of the results between the hybrid prediction model and the finite difference method shows that the prediction error in the hybrid model was less than 2 ◦ C. Neugebauer et al. [18] proposed a method that allows the permanent adaptation of the heat transmission coefficient to better apply convective heat transmission to time-variable thermal simulations, especially for mechanical finite element models. Holkup [19] presented a thermo-mechanical model of spindles that predicts temperature distribution and thermal growth by considering transient changes in the bearing stiffness and contact loads. Zhang et al. [20] built a model for the thermal characteristics of the high-speed motorized spindle using finite element analysis and calculated the static/transient temperature field as well as the thermal coupled field. Chen et al. [21] proposed a bearing thermo-mechanical dynamic model that considers preload methods and thermal responses, and analyzed frictional losses and the support stiffness of the bearings. The electromagnetic loss of the built-in motor with input power was investigated using electromagnetism. Based on the analyses of the boundary conditions of heat generation and heat convection, a solution procedure was designed to analyze the comprehensive thermo-mechanical dynamic behaviors of the motorized spindle. In the same year, Chen et al. [22] created a motor loss model based on electromagnetics for finite element thermal analysis. Even when considering the virtues of thermal models of the motorized spindle, where multi-field coupling and the dynamic characteristics of its thermal state can be easily considered, its nonlinear characteristics are very difficult to handle. Moreover, cooling methods for the motor and the bearings are usually implemented using a convective cooling fluid. The coefficient of heat convection that represents the relationship between the solid surface and the cooling medium is influenced by several factors. It is challenging to predict this coefficient, and this yields models with inadequate accuracy. An effective method to improve the predictive accuracy of the thermal field involves creating a model for the heat transfer coefficient using a heuristic approach. In practice, the internal temperature of the motorized spindle is important. Predictions of the temperature field of the motorized spindle based on the mechanistic model are reliable, but the accuracy of prediction can be further improved if the model takes advantage of the easily measured surface temperature of the spindle. In this paper, a hybrid model that combines the finite element model with temperature data is proposed and the dynamic temperature field of the motorized spindle is predicted. In the proposed model, a heuristic approach based on genetic algorithm is used to optimize the heat transfer coefficient. The model was verified by comparing the results of simulations with those of experiments, and yielded high prediction accuracy.
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2. Heat Generation and Heat Transfer in a Motorized Spindle Appl. Sci. 2017, 7, 1091
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The motorized spindle is a mechatronic product composed of a stator, a rotor, a spindle, a shell, aalternating water jacket, frontmotor, bearings, is built into an asynchronous current andrear its bearings, structure and is asbearing shownhousing. in FigureIt 1. The rotor and spindle are alternating current motor, andmatching, its structure is as shown 1. Theby rotor and spindle are integrated integrated through pressure and the latterinisFigure supported front and rear bearings. The through and the latter is supported by front bearings. Thegeneration stator of the stator of pressure the motormatching, is fixed on a shell through a water jacket. It isand wellrear known that heat is motor is fixed on a shell through a water jacket. It is well known that heat generation is inevitable inevitable owing to high speed of the motorized spindle. The main sources of heat generation are the owing thefrictional motorized spindle. Thefront mainand sources of heat generation are the heatingtoofhigh the speed motor,ofand heat from the rear bearings. Furthermore, theheating power of the motor, and frictional heatisfrom the front and rear bearings. Furthermore, the power supply supply of the motorized spindle an inverter that generates higher harmonics, because of which the of the motorized spindle is an inverter that generates higher harmonics, because of which the motor motor generates additional resistive losses owing to the copper in the stator and rotor, the iron in the generates losses owing tothe theskin copper in in thethe stator and rotor, the iron losses in the stator, stator, andadditional additionalresistive stray loss. Moreover, effect rotor induces greater in the and additional stray loss. Moreover, the skin effect the rotor greater losses in the copper copper at power supplies of high frequency. The in losses frominduces these high harmonic voltages and at power result supplies frequency. The temperature. losses from these harmonic and currents currents in of anhigh increase in motor Thishigh causes thermalvoltages deformation in the result in an increase in motor temperature. causes thermalperformance. deformation in the components of the components of the motorized spindle and This affects its dynamic Lubrication and cooling motorized spindle and affects its dynamic performance. Lubrication and cooling systems are designed systems are designed to prevent the motorized spindle from overheating. In general, the stator of the to prevent the motorized spindle overheating. In general, the stator of the motorized spindle is motorized spindle is cooled usingfrom cooling water pumped through a jacket. Bearings are lubricated cooled cooling water a jacket.by Bearings are lubricated with andbetween gas, where with oilusing and gas, where finepumped drops ofthrough oil are carried compressed gas into the airoilgap the fine drops of oil are carried by compressed gas into the air gap between the stator and rotor, and the stator and rotor, and the clearance of the bearing. These drops then flow out of the motorized clearance of the bearing. These drops then flow and out of motorized spindle. Therefore, the bearings spindle. Therefore, the bearings are lubricated, thethe rotor, shaft, and bearings are actively cooled. are lubricated, and the rotor, shaft, and bearings are actively cooled.
Figure 1. Structure of the motorized spindle. Figure 1. Structure of the motorized spindle.
The above discussion shows that heat transfer in the the motorized motorized spindle spindle is is complex. complex. The heat transfer and convection. The equation of transfer mode mode of ofthe themotorized motorizedspindle spindleconsists consistsmainly mainlyofofconduction conduction and convection. The equation energy conservation is as of energy conservation is follows: as follows: ∂T T C T (k T ) Q 2 υ ρ1 C p11C p1+tρ2 C p2 p 2∇ T = ∇·( k ∇ T ) + Q ∂t
(1) (1)
where PPtottot , Ptot is heat generation, V is the volume of the heat source, 1 is the density of the where QQ= VV, Ptot is heat generation, V is the volume of the heat source, ρ1 is the density of the solid, is that of the fluid,C p2Care are the heat capacities thefluid, solidrespectively, and the fluid, C p 2 capacities ρsolid, is that of the fluid, C p1 and the heat of the solid andofthe T is 2 2 p1 and the temperature thetemperature motorized spindle, υ is fluid velocity, Laplace operator, is thermal isisthe respectively, T isofthe of the motorized spindle, ∇ fluid velocity, Laplace is kthe conductivity, and Q is the rate of heat generation. operator, k is thermal conductivity, and Q is the rate of heat generation. According According to to heat heat transfer transfer theory, theory, the the relationship relationship between between the the coefficient coefficient of of convective convective heat heat transfer and temperature is transfer and temperature is q = h · ( Text − T ) (2) q h (Text T ) (2) where q is heat flux, h is the vector of the coefficient of heat transfer, and Text is the temperature of the fluid medium.
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where q is heat flux, h is the vector of the coefficient of heat transfer, and Text is the temperature of the fluid medium. Five processes contribute to heat transfer in the motorized spindle. Between the surface of the spindle shell and air is natural convection heat transfer, with the heat transfer coefficient given by [23] h1 = 9.7 (W/(m·◦ C))
(3)
Between the gas-gap and the compressed air is the forced convection heat transfer, with the heat transfer coefficient represented as: Nuλ a h2 = (4) H 0.25
where Nu = 0.239( δr ) Re0.5 , Re = u1rH , u1 = (u2a + u2t )1/2 , u1 is the speed of airflow in the air gap (m/s), u a and ut are the input speed of the compressed air and the speed of rotation of the spindle, respectively, r is the radius of the outer surface of the rotor (m), H is the dimensionality of the geometry of the air gap (m), Re is the Reynolds number, Nu is the Nusselt number, δ is the gap between the stator and the rotor (m), and λ a is the thermal conductivity of air (W/(m·◦ C)). Between the spindle end and the environmental air is the forced convective heat transfer, with the heat transfer coefficient given as: p h3 = 28 1 + 0.45ut (5) where ut is the circumferential velocity of the end of the rotor (m/s). Between the front bearing and the compressed air, and between the rear bearing and the compressed air, is the forced convection of heat. Thus, the coefficients of heat transfer are: h4 = c0 + c1 uc2 where u =
v1 A ax
2
+
ωdm 2
2 0.8
(6)
, A ax = 2dm π∆h, dm is the average diameter of the bearing (m), ∆h is
the average distance between the cage and the inner and outer rings of the bearing (m), A ax is the area of axial air flow through the bearing (m2 ), u is the average speed of air flowing through the bearing (m/s), v1 is air flow through the bearing (m3 /s), ω is the angular velocity (rad/s) of the motorized spindle, and the values of c0 , c1 , and c2 are 9.7, 5.33, and 0.8, respectively. The heat transfer coefficient for the forced convection of the cooling water and the water jacket is: Nu0 λw D 2 3 Pr 0.11 where Nu0 = 0.012( Re0.87 − 280) Pr0.4 1 + DL ( Pr ) , Re = w h5 =
(7) vD µ ,
D=
4A X ,
v is the characteristic
velocity (m/s) of the cooling water, µ is the kinematic viscosity (m2 /s) of the cooling water, D is the geometric characteristic size (m), A is the area of cross-sectional flow (m2 ), X is the circumference of Pr the section of wet flow (m), Pr is the Prandtl number of water, with Pr ≈ 1, and λw is its thermal w ◦ conductivity (W/(m· C)). The parameters of the cooling and lubrication systems as well as the motion of the motorized spindle are directly related to the heat transfer coefficients, which are dynamic and nonlinear. In traditional finite element modeling of the motorized spindle, these characteristics are ignored, which leads to inaccurate calculations of the temperature field. The above problem can be solved by optimizing heat transfer coefficients based on test data for temperature. The implementation of this exploits the relationship between temperature and the heat transfer coefficients of the motorized spindle.
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3. Hybrid Prediction Model of the Temperature Field of the Motorized Spindle 3. Hybrid Prediction Model of the Temperature Field of the Motorized Spindle In the traditional temperature field of the motorized spindle model, the heat transfer In the traditional temperature field of the motorized spindle model, the heat transfer coefficients coefficients at key positions are calculated using an empirical formula that leads to an associated at key positions are calculated using an empirical formula that leads to an associated error in the error in the calculation of the temperature field. This paper provides a precise temperature field for calculation of the temperature field. This paper provides a precise temperature field for the finite the finite element model of the motorized spindle where the heat transfer coefficients are optimized element model of the motorized spindle where the heat transfer coefficients are optimized using a using a genetic algorithm. The finite element model of the boundary conditions of the motorized genetic algorithm. The finite element model of the boundary conditions of the motorized spindle is then spindle is then used to accurately obtain the temperature field. The prediction model of the used to accurately obtain the temperature field. The prediction model of the temperature field of the temperature field of the motorized spindle is shown in Figure 2. First, the initial values of the heat motorized spindle is shown in Figure 2. First, the initial values of the heat transfer coefficients h (h1, h2, transfer coefficients h (h1, h2, h3, h4, and h5) are first calculated according to the empirical formulae h3, h4, and h5) are first calculated according to the empirical formulae shown in Formulae (3)–(7). Heat shown in Formulae (3)–(7). Heat generation rates are calculated according to losses in the motorized generation rates are calculated according to losses in the motorized spindle. The boundary conditions spindle. The boundary conditions of the rates of heat generation and the initial value of the heat of the rates of heat generation and the initial value of the heat transfer coefficients are used as initial transfer coefficients are used as initial conditions of the finite element model to calculate the conditions of the finite element model to calculate the temperature field. Second, the speed of the temperature field. Second, the speed of the motorized spindle, the temperature of the cooling water, motorized spindle, the temperature of the cooling water, its flow rate, the pressure of the compressed, its flow rate, the pressure of the compressed, and the interval of fuel supply are varied as the surface and the interval of fuel supply are varied as the surface temperature of the motorized spindle is temperature of the motorized spindle is monitored. Third, the heat transfer coefficients of the monitored. Third, the heat transfer coefficients of the motorized spindle are optimized by genetic motorized spindle are optimized by genetic manipulation. It is clear from the flowchart in Figure 3 manipulation. It is clear from the flowchart in Figure 3 that the five elements of the genetic algorithm that the five elements of the genetic algorithm are coding, initial population generation, population are coding, initial population generation, population fitness assessment, genetic manipulation, and fitness assessment, genetic manipulation, and operation parameters’ setting. operation parameters’ setting.
Initial value of heat transfer coefficient Heat loss
Temperature field of motorized Motorized spindle spindle temperature field model based on Simulated temperature finite element
Rotate speed Convergence judgment
Initial temperature of cooling water
Motorized spindle Monitored temperature on-line temperature Compressed air pressure monitoring Model
No
Genetic manipulation
Accurate heat transfer coefficient
Prediction of Motorized SpindleTemperature Field
Cooling water flow
Yes
Heat transfer coefficient optimization model
Oil supply interval
Figure 2. 2. Prediction Prediction model model of of the the temperature temperature field field of of the the motorized motorized spindle. Figure
The The objective objective function function is is given given by by
1 mm fm 1 T Tsi f m = m∑ | Tei − T si | m i=i 11 ei
(8) (8)
where m is the number of measurement positions, Tei is the monitored temperature of the where m is the number of measurement positions, Tei is the monitored temperature of the motorized motorized spindle, and Tsi is the simulation temperature of the motorized spindle. The fitness spindle, and Tsi is the simulation temperature of the motorized spindle. The fitness function is function is 1 f f it = 1 (9) f fit 1 + f m (9) 1 fm
where f m and f are the termination conditions for iterations used to optimize the heat transfer where f m and ffitfit are the ◦termination conditions for iterations used to optimize the heat transfer coefficient. When f m ≤ 0.5 C, that is, f f it ≤ 0.67, the iterations are terminated. Moreover, to render C , thatfor f fit 0.67 , the coefficient. is, engineering iterations the are maximum terminated.number Moreover, to render m 0.5 the researchWhen resultsfmore suitable applications, of iterations is the research results for engineering applications, the maximum number of iterations is limited to 100. If themore valuesuitable of the fitness function satisfies the conditions of convergence, the given heat limited 100. If the value of theoptimized. fitness function satisfies the conditions of obtains convergence, transferto coefficient is considered Otherwise, the genetic operation a set ofthe newgiven heat heat transfer coefficient is considered optimized. Otherwise, the genetic operation transfer coefficients, evaluates their fitness, and judges the convergence once again. obtains a set of new heat transfer coefficients, evaluates their fitness, and judges the convergence once again.
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Finally, the optimized heat transfer coefficients are used as the boundary conditions of the model and the temperature field of the motorized spindle is predicted. Compared with the finite element heat transfer coefficients of traditional models, those of the model developed in this work are associated with the cooling system, lubrication system, control system, and load of the motorized spindle, which have dynamic and nonlinear characteristics. As long as the key position of spindle temperature can be measured, this model can predict the temperature Appl. Sci. 2017, 1091 6 of 14 field of 7, the motorized spindle under different operating conditions.
Coding
Calculate the initial value of the heat transfer coefficient; The number of populations M=80; The maximum number of iterations N=100; Give the numerical range of the heat transfer coefficient; Output Y Populations of individual fitness evaluation;
Convergence？ N
Calculate the selection probability according to the fitness value, use the roulette to select the regenerated individual; Create a new individual according to the set crossover probability pc 0.8 and use the single point crossover method; Genetic manipulation According to the set mutation probability, pm 0.05 the binary mutation method is used to generate new individuals; New population
Figure 3. Flowchartofofgenetic genetic algorithm. algorithm. Figure 3. Flowchart
4. Temperature Field Prediction Model Application Finally, the optimized heat transfer coefficients are used as the boundary conditions of the model andAthe temperature field of the motorized spindle is predicted. finite element model of the 170MD30 motorized spindle (Luoyang Bearing Science & Technology Compared with the finite element heat transfer coefficients of COMSOL traditionalAB, models, those 23, of the Co., Ltd., Luoyang, Henan, China) was built in COMSOL( 5.2a, Tegnergatan modelStockholm, developed in thisand work are associated the cooling system, lubrication system, control Sweden), is shown in Figure 4with [24]. Table 1 lists the main parameters of the motorized spindle and Table 2 shows the material parameters of the model. In this model, the heating of the As system, and load of the motorized spindle, which have dynamic and nonlinear characteristics. bearings is considered. The angular contact rotor, and are can assembled on the long motor as theand key position of spindle temperature canball bebearing, measured, thisstator model predict the spindle while all screws, the vent, the oil hole, and other fine structures are ignored. The finite temperature field of the motorized spindle under different operating conditions.
element model had 544,565 degrees of freedom in terms of temperature. The characteristic shape function was Lagrangian (square) and the GMRES iterative solver was used. The pre-smoothing scan 4. Temperature Field Prediction Model Application type was Sor and the post-smoother scan type was Soru. The rough solver Pardis was used as well. An Aoptimization finite element of the 170MD30 motorized spindlespindle (Luoyang Bearing Science & model model for the heat transfer coefficients of the motorized was created in MATLAB (8.0.0.783, Mathworks, Natick, MA, USA, 2012) and the results were used COMSOL as boundaryAB, Technology Co., Ltd., Luoyang, Henan, China) was builtof the in iterations COMSOL( 5.2a, conditions finite element model to calculate the temperature of the Table motorized spindle. Tegnergatan 23,of the Stockholm, Sweden), and is shown in Figurefield 4 [24]. 1 lists the main
parameters of the motorized spindle and Table 2 shows the material parameters of the model. In this model, the heating of the motor and bearings is considered. The angular contact ball bearing, rotor, and stator are assembled on the spindle while all screws, the vent, the oil hole, and other fine structures are ignored. The finite element model had 544,565 degrees of freedom in terms of temperature. The characteristic shape function was Lagrangian (square) and the GMRES iterative solver was used. The pre-smoothing scan type was Sor and the post-smoother scan type was Soru. The rough solver Pardis was used as well. An optimization model for the heat transfer coefficients of the motorized spindle was created in MATLAB (8.0.0.783,Mathworks, Natick, MA ,USA, 2012)
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(a)
(b) Figure Figure 4. 4. Finite Finite element element model model of of the the 170MD30 170MD30 motorized motorized spindle. spindle. (a) (a) Geometrical Geometrical model, model, (b) (b) finite finite element element analysis analysis meshing meshing model. model. Table 1. Table 1. The main parameters parameters of the motorized motorized spindle. spindle. Parameters Parameters Rotor external diameter (mm) Rotor external diameter (mm) Rotor iron core length (mm) Rotor iron core length (mm) Stator internal diameter (mm) Stator internal diameter (mm) Stator iron(mm) core length (mm) Stator iron core length Shaft maximum diameter (mm) Shaft maximum diameter (mm) Shaft length (mm) Shaft length (mm) Motorized spindle external diameter (mm) Motorized spindle external diameter (mm) Bearing model Bearing model
ValueValue 79.4 79.4 112 112 80 80 110 110 56 56 170 170 266 266 7008C7008C
Table 2. 2. Parameters of materials materials of of the the model. model. Table Component Component
Material Material
Stator windings Stator windings Stator core Stator core Water jacket Water jacket Rotor Rotorbar bar Shaft Shaft Coolant Coolant
Copper Copper Silicon steel Silicon steel 40Cr 40Cr Cast aluminum Cast aluminum 40Cr 40Cr Water Water
Thermal Conductivity
3) Thermal Conductivity Density (g/cm 3) Density (g/cm (W/(m· (W/(m ·◦ C))℃))
8.9 8.9 7.9 7.9 7.9 7.9 2.8 2.8 7.9 7.9 1.0
1.0
400.0 400.0 35.0 35.0 60.5 60.5 – -60.5 60.5 0.6 0.6
Specific Heat
Specific Heat Capacity Capacity (J/(kg·(◦J/(kg· C)) ℃))
386.0 386.0 535.0 535.0 434.0 434.0 875.0 875.0 434.0 434.0 4200.0
4200.0
4.1. Acquiring Temperature Data 4.1. Acquiring Temperature Data The prediction of the temperature field of the motorized spindle depends on the measured The prediction of the temperature field of the motorized spindle depends on the measured temperature at critical positions. The experimental system used is shown in Figure 5, and included temperature at critical positions. The experimental system used is shown in Figure 5, and included an oil-gas lubrication system, a cooling system, a temperature test system, a loading system, and a an oil-gas lubrication system, a cooling system, a temperature test system, a loading system, and a speed control system. Air pressure, air temperature, and intervals of fuel supply could be monitored speed control system. Air pressure, air temperature, and intervals of fuel supply could be in the oil-gas The flow of cooling water and inletwater temperature oftemperature the cooling system monitored inlubrication the oil-gassystem. lubrication system. The flow of cooling and inlet of the could also be controlled and monitored, but only the outlet temperature of the cooling system could. cooling system could also be controlled and monitored, but only the outlet temperature of the Consequently, load and speed of thethe motorized spindle alsomotorized controllablespindle and testable. The cooling systemthe could. Consequently, load and speedwere of the were also controllable and testable. The temperature test system consisted of temperature sensors placed on
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temperature system consisted temperature sensors placed surface of the motorized the surface of test the motorized spindle.ofThe surface temperature couldonbethe measured under different the surface ofsurface the motorized spindle. The surface temperature couldconditions be measured under the different spindle. The temperature could be measured under different involving conditions involving the load, temperature of the cooling water, its flow rate, and the rate of flowload, of conditions involving the load, temperature of the cooling water, its flow rate, and the rate of flow of temperature cooling water, its flow rate, and the rate of flow of the lubricating oil using this the lubricatingof oilthe using this test system. the lubricating oil using this test system. test system.
Oil-gas Lubrication System Oil-gas Lubrication System
IPC IPC
State Feedback State Feedback and Control and Control Instructions Instructions
Control Valve of Control Oil Valve of Lubricating Flow Oil-gas Device Lubricating Oil Flow Oil-gas Device
Refrigeration Dryer Refrigeration Dryer
P P
State State Feedback and Feedback Control and Control Instructions Instructions
Electrical Electrical parameter parameter tester tester
Air Pressure Air Pressure Compressor Controller CompressorGasholder Controller Gasholder The Measured The Measured Torque-speed Motorized Torque-speed Transducer Motorized Spindle Transducer Spindle
Inverter Inverter
Flowmeter Flowmeter Loading Loading Machine Machine
Control Valve of Control Valve of Cooling Water Flow Cooling Water Flow
Torque-speed Torque-speed Meter Meter T T
T T
Power Supply Supply ofPower Loading of Loading Machine Machine
Water Tank Thermometer Water Tank Thermometer
Temperature Temperature Sensor Sensor
Signal acquisition Signal acquisition device device
Temperature Temperature Controller Controller
Water Cooling System Water Cooling System
Figure 5. Temperature testing system of the motorized spindle. Figure 5. Temperature Temperature testing testing system system of of the the motorized motorized spindle. spindle.
4.2. Boundary Conditions of the Model 4.2. Boundary Conditions of the the Model Model 4.2. One of the boundary conditions of the prediction model is the rate of heat generation, which is One of of the the boundary boundary conditions conditions of the the prediction prediction model model is the the rate rate of of heat heat generation, generation, which which is is One calculated based on losses in the motorized spindle. The motor losses and bearing friction losses calculated based on losses in the motorized spindle. The motor losses and bearing friction losses calculated based on losses in the motorized spindle. The motor losses and bearing friction losses could could be measured by the loss experimental device shown in Figure 6. The specific steps are as could be measured by the loss experimental device in The Figure 6. The specific are as be measured by the loss experimental device shown in shown Figure 6. specific steps are as steps follows: follows: follows:
Figure 6. Loss experimental device. Figure Figure 6. 6. Loss Loss experimental experimental device. device.
First, the loading machine and motorized spindle are connected, power supply to the First, the theloading loading machine motorized spindle are connected, powertosupply to the First, machine andand motorized spindle are connected, power supply the motorized motorized spindle is cut off, and it is rotated synchronously with the loading machine. At this point, motorized spindle is cut and it is rotated synchronously with machine. the loading this point, spindle is cut off, and it isoff, rotated synchronously with the loading At machine. this point,Atthe spindle the spindle exhibits only frictional loss, and the input power PJ1 of the loading machine is tested. the spindle exhibits only frictional loss, and the input power P J1 of the loading machine is tested. exhibits only frictional loss, and the input power PJ1 of the loading machine is tested. Second, the motorized spindle and loading machine are disconnected, where the latter has the Second, the motorized motorized spindle and loading machine are disconnected, where the latter has has the the Second, same speed as in the first step, and the input power PJ2 of the loading machine is determined. same speed speed as as in in the the first first step, step, and and the the input input power power PPJ2J2 of the loading machine is determined. same Third, the motorized spindle and loading machine are disconnected, where the former has Third, the motorized spindle and loading machine are disconnected, where the former has zero load under the same speed as in the first step, the electrical parameter tester measures the zero load under the same speed as in the first step, the electrical parameter tester measures the input voltage and current of the motorized spindle, and the input power pin of the motorized input voltage and current of the motorized spindle, and the input power pin of the motorized
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of 14 Third, the motorized spindle and loading machine are disconnected, where the former has9zero load under the same speed as in the first step, the electrical parameter tester measures the input voltage spindle is obtained. At the same time, the torque/speed transducer can measure the output torque and current of the motorized spindle, and the input power pin of the motorized spindle is obtained. At and speed of the motorized spindle, and the output power pout of the motorized spindle can be the same time, the torque/speed transducer can measure the output torque and speed of the motorized measured. The loss due to friction in the motorized spindle is spindle, and the output power pout of the motorized spindle can be measured. The loss due to friction p f PJ 1 - PJ 2 in the motorized spindle is (10) p f = PJ1 − PJ2 (10) and the loss in the motor is and the loss in the motor is p = P - P - pf (11) pe =e Pin in− Pout (11) out − p f
Usually, two-thirdsofofthe themotor motorloss lossoccurs occursatat the stator and one-third at the rotor Using Usually, two-thirds the stator and one-third at the rotor [25].[25]. Using the the above method, the bearing loss, rotor loss, and stator loss of the 170MD30 motorized spindle above method, the bearing loss, rotor loss, and stator loss of the 170MD30 motorized spindle (Luoyang (Luoyang Bearing Science & Technology Co., Ltd., Luoyang, China) were measured at Bearing Science & Technology Co., Ltd., Luoyang, Henan, China)Henan, were measured at 10,000 rpm and 10,000 rpm and 15,000 rpm as shown in Table 3. All losses were converted into heat, and the rates of 15,000 rpm as shown in Table 3. All losses were converted into heat, and the rates of heat generation of heat generation of the stator,are rotor, and bearing are calculated as theofboundary of the the stator, rotor, and bearing calculated as the boundary conditions the finite conditions element model. finite element model. Table 3. The heat rates of the motorized spindle. Table 3. The heat rates of the motorized spindle. Speed (rpm)
Speed (rpm) 10,000 10,000 15,000 15,000
Heat of Stator (w)
Heat of Stator (w) 326.68 326.68 452.67 452.67
Heat of Rotor (w)
Heat of Rotor (w) 143.32 143.32 226.33 226.33
Bearing Heat (w)
Bearing Heat (w) 77.08 77.08 151 151
In the finite element model for the temperature field of the motorized spindle, another boundary condition is the heat transfer transfer coefficient at the key positions. In this paper, the traditional boundary condition empirical formula [26–28] was used to calculate the initial value of the heat transfer coefficient. The temperature-related test data for the optimization of the heat transfer transfer coefficient were drawn from a temperature test system consisting of 31 temperature sensors placed on the surface of the motorized is shown in Figure 7. Test position 1 was theatfront spindle. The arrangement arrangementofofthe thetemperature temperaturetest test is shown in Figure 7. Test position 1 at was the bearing, position 3 at the rear bearing, position 2 was directly between the first two, and the end of front bearing, position 3 at the rear bearing, position 2 was directly between the first two, and the the spindle formed formed test position 4. There4. were 10 thermocouples from testfrom positions 1 to 3 that1were end of the spindle test position There were 10 thermocouples test positions to 3 uniformly along the direction of the circumference. Moreover, an infrared thermal sensor that were distributed uniformly distributed along the direction of the circumference. Moreover, an infrared was arranged testarranged position 4. thermal sensoratwas at test position 4.
Figure 7. Arrangement of temperature test positions. Figure 7. Arrangement of temperature test positions.
To prove the validity of the model, the temperature field of the motorized spindle was To prove the validity of the model, the temperature field of the motorized spindle was predicted predicted under the following two conditions. The test parameters when the speed of the motorized under the following two conditions. The test parameters when the speed of the motorized spindle spindle was 10,000 rpm included cooling water flow of 0.25 m3/h, an initial temperature of 12 °C of was 10,000 rpm included cooling water flow of 0.25 m3 /h, an initial temperature of 12 ◦ C of the the cooling water, compressed air inlet pressure of 0.365 MPa, a fuel supply interval of 2 min, and cooling water, compressed air inlet pressure of 0.365 MPa, a fuel supply interval of 2 min, and compressed air temperature of 8 °C. The test parameters when the speed of the motorized spindle was 15,000 rpm featured cooling water flow of 0.32 m3/h, an initial temperature of 19 °C of the cooling water, compressed air inlet pressure of 0.35 MPa, a fuel supply interval of 3 min, and
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compressed air temperature of 8 ◦ C. The test parameters when the speed of the motorized spindle was 3 ◦ 15,000 Appl. Sci.rpm 2017,featured 7, 1091 cooling water flow of 0.32 m /h, an initial temperature of 19 C of the cooling 10 of 14 water, compressed air inlet pressure of 0.35 MPa, a fuel supply interval of 3 min, and compressed air ◦ C. The initial and optimized values of the heat transfer coefficients are shown in temperature of compressed air18temperature of 18 °C. The initial and optimized values of the heat transfer Table 4. coefficients are shown in Table 4. Table 4. The heat transfer coefficients.
Table 4. The heat transfer coefficients.
Speed Speed (rpm) (rpm) 10,000 10,000 15,000 15,000
h1h1 9.79.7 9.79.7
Initial Value Value (W/(m ·◦℃)) C)) Initial (W/(m· h2 h2 h3h3 hh44 146.8 121.4 71.4 146.8 121.4 71.4 178.4 109.1 178.4 142.5 142.5 109.1
Optimized (W/(m ·◦ C)) ℃)) Optimized (W/(m· hh55 190.0 190.0 223.9 223.9
hh11**
h2h*2*
20.0 20.0 10.8 10.8
188.4 188.4 171.3 171.3
h3 * h3*
h5* h4 * h4* h5 * 188.2 127.7 500.3 188.2 127.7 500.3 223.8 134.9 134.9 223.8 331.4 331.4
4.3. 4.3. Simulation Simulation When the speed speedof ofthe themotorized motorizedspindle spindle was 10,000 rpm, isotherm obtained When the was 10,000 rpm, thethe isotherm waswas obtained fromfrom the the temperature field prediction predictionmodel modeltotooptimize optimizethe the heat transfer coefficient as shown in Figure temperature field heat transfer coefficient as shown in Figure 8. 8. Figure 8a,b show that the isotherms changed significantly with increase in the number of iterations Figure 8a,b show that the isotherms changed significantly with increase in the number of iterations of of heat transfer coefficient 25 to 50. However, there was change a slightfor change for 75 iterations thethe heat transfer coefficient from from 25 to 50. However, there was a slight iterations to 100 as75 to 100 as shown in Figure 8c,d. This implies that the results of the optimization of the heat shown in Figure 8c,d. This implies that the results of the optimization of the heat transfer coefficienttransfer of coefficient of the motorized spindle basedalgorithm on the genetic algorithm were convergent. the motorized spindle based on the genetic were convergent.
(a)
(b)
(c)
(d)
Figure 8. Isotherm iterations, 5050 iterations, (c) 75 Figure 8. Isotherm of ofthe themotorized motorized spindle spindle at at10,000 10,000 rpm. rpm.(a)(a)2525 iterations,(b)(b) iterations, iterations, (d) 100 iterations. (c) 75 iterations, (d) 100 iterations.
Similarly, Figure 9 shows the isotherm of the motorized spindle at 15,000 rpm. Figure 9a shows the isotherm for the initial values of the heat transfer coefficient and Figure 9b the isotherm when the coefficient was optimized over 100 iterations. Comparing Figure 9a,b, the effect of the heat transfer coefficient on the temperature field of the motorized spindle appeared to be significant.
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Similarly, Figure 9 shows the isotherm of the motorized spindle at 15,000 rpm. Figure 9a shows the isotherm for the initial values of the heat transfer coefficient and Figure 9b the isotherm when the coefficient was optimized over 100 iterations. Comparing Figure 9a,b, the effect of the heat transfer coefficient on the temperature field of the motorized spindle appeared to be significant. Appl. Sci. 2017, 7, 1091 11 of 14 Appl. Sci. 2017, 7, 1091
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(a) (a)
(b) (b)
Figure 9. Isotherm of the motorized spindle at 15,000 rpm. (a) 0 iteration, (b) 100 iterations. Figure Figure9.9.Isotherm Isothermof ofthe themotorized motorizedspindle spindleatat15,000 15,000rpm. rpm.(a) (a)00iteration, iteration,(b) (b)100 100iterations. iterations.
4.4. Results and Comparison 4.4. 4.4.Results Resultsand andComparison Comparison The simulation and monitored temperatures of the motorized spindle were compared to The andand monitored temperatures of the of motorized spindle were compared to determine Thesimulation simulation monitored temperatures the motorized spindle were compared to determine the accuracy of the model. Figure 10 shows this comparison after the heat transfer the accuracythe of the model.ofFigure 10 shows this 10 comparison after the heat transfer coefficient was determine accuracy the model. Figure shows this comparison after the heat transfer coefficient was optimized for 100 iterations. Figure 10a shows the results for 10,000 rpm and Figure optimized 100optimized iterations.for Figure 10a shows Figure the results 10,000 and for Figure 10brpm for 15,000 rpm. coefficientfor was 100 iterations. 10a for shows therpm results 10,000 and Figure 10b for 15,000 rpm. From Figure 10, it is clear that the predicted temperatures were very close to the From Figure it isFrom clearFigure that the predicted temperatures were very close were to the experimental 10b for 15,00010, rpm. 10, it is clear that the predicted temperatures very close to the experimental measurements for the latest time steps. measurements for the latest time experimental measurements for steps. the latest time steps.
(a) (a)
(b) (b)
Figure 10. Comparison between the simulation temperature and experimental measurements. (a) Figure 10. Comparison between the simulation temperature and experimental measurements. (a) Spindle of 10,000 rpm, (b) Spindle speed of 15,000 rpm. and experimental measurements. Figure 10.speed Comparison between the simulation temperature Spindle speed of 10,000 rpm, (b) Spindle speed of 15,000 rpm. (a) Spindle speed of 10,000 rpm, (b) Spindle speed of 15,000 rpm.
To further validate the temperature field of the prediction model of the motorized spindle, it is To further validate the temperature field of the prediction model of the motorized spindle, it is necessary to compare the temperature simulation temperatures and test temperatures of other locations on is the To further validatethe the field of the prediction model of theofmotorized spindle, necessary to compare simulation temperatures and test temperatures other locations onitthe motorized compare spindle, the and to calculate the overall error in the model.ofIn the test system, the necessary temperatures testintemperatures on the motorizedtospindle, and simulation to calculate the overall and error the model. Inother the locations test system, the temperature of water at the outlet was assumed to be consistent with that of the water jacket at the motorized spindle, andattothe calculate error model. In thethat test of system, the temperature temperature of water outlet the wasoverall assumed to in bethe consistent with the water jacket at the given location. The temperature of water at the outlet can be determined; therefore, the temperature of water at the outlet was assumed be consistent withcan that the water jacket at thethe given location. given location. The temperature ofto water at the outlet beofdetermined; therefore, temperature of the water jacket outlet could be used to determine the accuracy of prediction. The position of the The temperature of outlet water could at thebe outlet be determined; therefore, the temperature of the water of the water jacket usedcan to determine the accuracy of prediction. The position of the outlet is shown in Figure 1. The relative mean error of the predictions of the model is outlet is shown in Figure 1. The relative mean error of the predictions of the model is 1 n Tt Tt 1 n AARE 1 nTt (12) Tt 1 n aˆt AARE nt 1 Tt nt 1aˆt (12) n t 1 Tt n t 1 2 The variance s and standard deviation s of the prediction model are The variance s2 and standard deviation s of the prediction model are 1 n 1 n
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jacket outlet could be used to determine the accuracy of prediction. The position of the outlet is shown in Figure 1. The relative mean error of the predictions of the model is _ Tt − T t 1 n 1 = ∑ | aˆ t | AARE = ∑ n t=1 Tt n t =1 n
(12)
The variance s2 and standard deviation s of the prediction model are s2 =
Appl. Sci. 2017, 7, 1091
_ 2 1 n 1 n ( Tt − T t ) = ∑ a2t ∑ n t =1 n t =1
(13)
12 of 14
s
s _ 22 1 1n n2 2 1 1n n at a s = s ∑ ( T(tTt− Tt t)) = ∑ n 1t 1 n tn1t=1 t n t=
(14) (14)
Figure and measured differences in temperature at theatwater outlet.outlet. Both Figure11 11shows showsthe thepredicted predicted and measured differences in temperature the water the table figure show data pertaining to the temperature of the water foroutlet spindle Both the and tablethe and the figure show data pertaining to the temperature of thejacket wateroutlet jacket for speeds 10,000ofrpm andrpm 15,000 Also shown Figure is the11 prediction error inerror the model. spindleofspeeds 10,000 andrpm. 15,000 rpm. Also in shown in11 Figure is the prediction in the According to Formulae (12)–(14),(12)–(14), when thewhen spindle wasspeed 10,000was rpm, the relative mean prediction model. According to Formulae thespeed spindle 10,000 rpm, the relative mean − 4 −4 error was 1.5%, 2.87 ×was 10 2.87 , and standard deviation was 0.017. At 15,000 rpm, prediction error variance was 1.5%,was variance × 10the, and the standard deviation was 0.017. At 15,000 the prediction error was small,small, the relative meanmean prediction error error was 3.6%, variance was rpm, the prediction errorsimilarly was similarly the relative prediction was 3.6%, variance − 3 −3 2.0 10 × 10 , and the the standard deviation waswas 0.045. was×2.0 , and standard deviation 0.045.
(a)
(b)
Figure 11. The difference between the predicted and measured temperatures at the water outlet. (a) Figure 11. The difference between the predicted and measured temperatures at the water outlet. Spindle speed at 10,000 rpm, (b) Spindle speed at 15,000 rpm. (a) Spindle speed at 10,000 rpm, (b) Spindle speed at 15,000 rpm.
5. Conclusions 5. Conclusions In this paper, a hybrid model to predict the temperature field of a motorized spindle was In this paper, a hybrid model to predict the temperature field of a motorized spindle was proposed proposed and experimentally verified. The model, based on the finite element method and and experimentally verified. The model, based on the finite element method and temperature data temperature data collected at the surface of a motorized spindle, is dynamic, and can predict the collected at the surface of a motorized spindle, is dynamic, and can predict the characteristics of the characteristics of the temperature field distribution of the motorized spindle. The heat transfer temperature field distribution of the motorized spindle. The heat transfer coefficients were optimized coefficients were optimized based on the genetic algorithm by using surface temperature data as based on the genetic algorithm by using surface temperature data as the boundary conditions of the boundary conditions of the model. A comparison of simulation analysis with experimental the model. A comparison of simulation analysis with experimental results showed that the model results showed that the model can predict the temperature field with high accuracy. The prediction can predict the temperature field with high accuracy. The prediction of the temperature field of the of the temperature field of the motorized spindle can provide the basis for the assessment of its motorized spindle can provide the basis for the assessment of its state. In future work, a prediction state. In future work, a prediction model of the thermal distortion of the motorized spindle based model of the thermal distortion of the motorized spindle based on the temperature field prediction on the temperature field prediction model will be built to found the automatic compensation of model will be built to found the automatic compensation of thermal deformation. thermal deformation. Acknowledgments: The authors acknowledge support from the National Science Foundation (No. 51375317), the National Science Foundation (No. 51675353), the Liaoning Province Science Foundation (No. 2015020122), and the National (local) Joint Engineering Laboratory for Open Funds (sjsc-2015-6). Author Contributions: L.Z. and H.S. conceived and designed the experiments; C.L. performed the experiments; Y.W. analyzed the data; K.Z. contributed materials/analysis tools; L.Z. wrote the paper.
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Acknowledgments: The authors acknowledge support from the National Science Foundation (No. 51375317), the National Science Foundation (No. 51675353), the Liaoning Province Science Foundation (No. 2015020122), and the National (local) Joint Engineering Laboratory for Open Funds (sjsc-2015-6). Author Contributions: L.Z. and H.S. conceived and designed the experiments; C.L. performed the experiments; Y.W. analyzed the data; K.Z. contributed materials/analysis tools; L.Z. wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.
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