Applying inverse method for heat transfer of high speed spindle

 ISSN 2354-0575
 APPLYING INVERSE METHOD FOR HEAT TRANSFER OF HIGH SPEED SPINDLE
 Than Van The1, Ngo Thi Thao2
 1 Feng Chia University
 2 Hung Yen University of Technology and Education
 Received: 15/4/2016
 Revised: 02/6/2016
 Accepted for publication: 10/6/2016
Abstract:
 This paper presents an inverse method for finding quantity of heat sources in high speed spindle. 
A commercial ANSYS software and Conjugate Gradient optimization method are used to construct the 
proposed inverse method. Simulations for different number of measurementpoints andlocations are 
performed. These results show that excellent estimation on heat generation can be obtained through using 
only two measurement points. The current methodology will provide a useful tool to investigate the complex 
heat transfer process in the high speed spindle.
Keywords: Inverse method, High speed spindle,Conjugate Gradient Method, ANSYS Software.
1. Introduction obtain heat generated in bearings by using inverse 
 When working under high speed, friction heat transfer method. Recently, inverse method 
of bearings (and power loss of motorized) in the for estimating heat generation and interface 
spindle will generate heat which leads to increase temperature in ultrasonic welding [12, 13] and 
temperature and further cause thermal deformation temperature-dependent thermophysical properties 
in the spindle. Therefore,understanding temperature of material [14] was successfully studied.In this 
distribution in the spindle can give useful study, an inverse method is proposed to predict heat 
information for predicting and controlling thermal sources (heat generated by bearings)in the spindle. 
error. To estimatetemperature fieldin the spindle, A combination of Mechanical Ansys Parametric 
the heat sources are indispensable. Palmgren[1] Design Language (MAPDL) and Conjugate 
gave an empirical formula to estimate total bearing GradientMethod (CGM) is applied to find the 
friction torque that was then used for calculating unknown heat sources.
bearing heat generation. By adding the spinning 
friction moments to formula, Harris presented a 2. Thermal model of the high speed spindle
new form for calculating heat generation of bearing 2.1 The high speed spindle structure
in [2]. Palmgren’s model has achieved popular A direct driver spindle with 24000 rpm 
acceptance as an accurate method. Bossmanns and maximum speed, namely, TD30 is investigated in 
Tu [3] proposed a model to determine quantitative this study. To design the complex spindle, CATIA 
heat source of the built-in motor and the bearings. software is used to draw the spindle as shown in 
Based on data from coast test, they established Fig. 1. To simplify the spindle model, small nuts, 
empirical equations which are function of preload holes, and small structures are omitted. Material 
and rotational speed for calculating bearing heat properties of each part in the spindle are listed 
generated. Moorthy[4] introduced an improved in Table 1. A finite element (FE) model of the
analytical model for estimation of heat generation spindle is established in MAPDL. Because of the 
in angular contact ball bearings of high speed symmetric spindle, a small partition of the spindle 
spindle. Through literature review, it can be (10) is considered instead of entire spindle. Meshed 
seen that none of researches were investigated to model of the spindle is displayed in Fig.2.
 Fig. 1. High speed spindle structure
Khoa học & Công nghệ - Số 10/Tháng 6 - 2016 Journal of Science and Technology 23
ISSN 2354-0575
 Fig. 2. Meshed model of the spindle in ANSYS
 Table 1. Material properties includes force and free convection. The convection 
 Shaft Housing Inner/ Balls Other coefficient is defined by 
 Outer parts hN= uakdir (4)
 Material SCM SCM440 SUJ2 Ce- S45C in which d is the equivalent diameter of the rotation 
 name 415 ramic bodies/cylinder or size of small gap; Nu is the 
 Density 7850 7800 7830 3200 7830 average Nusselt number. Nu is calculated for 
 (kg/m3) different kind of convection as listed in Table2.
 Thermal 46.6 43 46 30 47 Table 2. Nusselt number for convection boundary 
 conduc- condition
 tivity
 Convection Equations for calculating Nu
 W/(m.K) boundary 
 Specific 460 450 470 850 480 condition 
 heat Free convection 16/ 2
 0 . 387RaD
 kJ/(kg.K) of horizontal Nu =+06.
 91/ 6 82/ 7
 Expansion 12.5 11.8 12.5 3.0 12.8 cylinder [9] *410+ ./559 Pr
 7A^h
 coefficient Convection 12/
 -6 o Nu = 0 .(6366 Re.Pr)
 (x10 / C) around rotation 
 shaft [10]
2.2. Heat transfer coefficients 0.241
 Forced Ta2
 The spindle is assembled from many parts, convection in Nu = 0 . 409
 cmFg
this leads to create a lot of joint between these rotating annuli 
 4 ##2 7
parts. Hence, the contact heat transfer at joint of [11] forT10 aF/ g 10
spindle parts must be considered. Thermal contact 
resistance between the balls and outer/inner rings 3. Inverse method
is given by [5] Two unknown heat generations, which 
 1 r 1 r
 Rbr = Ke,,+ Ke (1) contain heat generation at front and rear bearings, 
 2rakball 2 2rakring 2
 ``jjare regards as:
 w = qq12 (5)
 The contact between outer rings and housing 6@ 
through small air gap, so the thermal contact Solution of an inverse problem is obtained 
resistance coefficient is determined as [6, 7]: when the object function is minimized with respect 
 hring hTgapr--()inghTr$$a h to each of unknown parameters. The object function 
 Rhr = + (2)
 kSring kSair has been defined to solve this inverse problem as:
 tf
 M
 Thermal contact conductance of negative 2
 JTw = # / (,txii,)zT- mi(,tx,)zdi t (6)
 i = 1
assembly of inner ring and shaft is computedas[8]: ^h t = 0 6@
 ]Z k tan i p 09. 4
 ]h = 11. 3 where T(t,x ,z ) is the estimated temperature on 
 ] v ` H j i i
 ] a k the housing surface at the measured locations 
 ] forplastic deformation ( } $ 1
 ] ) determined from the solution of the direct problem 
 [ 09. 4 (3)
 ] k tan i p 2 by using an updated estimation for the unknown 
 ]h = 15. 5
 ] v d E tan i n quantity w. In order to minimize the objective 
 ] a k
 ] forplastic deformation ()} < 1 function, the CGM is chosen in this study.The 
 \
 The heat transfer convection in spindle algorithm of proposed inverse method as follows:
24 Khoa học & Công nghệ - Số 10/Tháng 6 - 2016 Journal of Science and Technology
 ISSN 2354-0575
 Step 1: Set index of step k =1 and give initial results as knowing temperatures at easy measuring 
 ()1 ()1 ()1
 w = qq1 2 positions (T and T ).
 7A 1 3
 Step 2: Solve the direct problem by using 
MAPDL to obtain T(t,xi ,zi ).
 1
 Step 3: Check the stop criterion J()w f . 
Continue if not satisfied.
 Step 4: Calculate a gradient object function 
 2J()ww2J()T
 4J = and conjugation coefficient 
 <F2q122q
 M M
 rJ= //()ddk 2 ()J k-1 2 .
 j = 1 6 @ j = 1 6 @
 Step 5: Compute the direction of descent 
 kk+1 k
 PP= dJr+ . 
 Step 6: Compute the search step size: 
 tf
 M
 # / Tt(,xzii,)- Ttmi(,xz,)iiDTt(,xz,)i dt
 i = 1
 k t = 0 6@
 b = tf
 M
 2
 # / DTt(,xzii,)dt
 i = 1
 t = 0
 Step 7: Compute the new estimation 
 wwkk++11= - bkkP .
 Step 8: Set k = k + 1 and go to step 2.
4. Simulation results and discussions Fig. 3. Inverse results using T1;
 To illustrate for proposed inverse method, (a) Temperature; (b) Heat generation
suppose that the running spindle with 15000 
rpm speed for 7000 seconds under constant heat 
generation at front and rear bearings is used. The 
temperature at some locations is extracted and then 
employed as measured temperatures. 
 In order to consider possible effects of 
measurement pointnumber, inverse results of 
temperature and heat generation using given 
temperature from only one measurement point (T1 
or T3) are shown in Figs. 3&4. It can be seen that 
estimated temperature, although, agrees excellently 
with exact temperature, unknown heat sources can’t 
give correct solution. This phenomenon is occurred 
because one measurement temperature point 
doesn’t provide enough information for estimating 
two unknowns. However, the accuracy of inverse 
solution was quickly improved when two points 
of measurement were applied. Figs. 5&6 display 
a comparison of estimation and exact solution 
of temperatures and heat generations using two 
measurement temperaturesat different locations. 
According to these figures, the predicted results are 
in very good agreement with exact values for both 
cases. However, estimated heat sources using two 
measurement points T4 and T5 are tiny better than 
that using two measured temperatures at T1 and Fig. 4. Inverse results using T3;
T3. Clearly, the presented method can get reliable (a) Temperature; (b) Heat generation
Khoa học & Công nghệ - Số 10/Tháng 6 - 2016 Journal of Science and Technology 25
ISSN 2354-0575
 Fig. 7 indicates an excellent performance 
 results between inverse and exact solutions when 
 employing three measurement points. However, the 
 accuracy of case using three points is slightly higher 
 than that of using two points. From analysis of these 
 findings, one is said that the proposed method can 
 accurately estimate heat generations in high speed 
 spindle through using only two measurement 
 positions.
 Fig. 5. Inverse results using T1 and T3 ;
 (a) Temperatures; (b) Heat generation
 Fig. 7. Inverse results using T1, T2, and T3;
 (a) Temperatures; (b) Heat generation
 5. Conclusion
 An inverse method for determining 
 unknown heat generations in high speed spindle 
 are successfully applied in this study. Results show 
 that measurement point number affects accuracy 
 of numerical solution. These results lead to a 
 conclusion that two unknown heat generations in the 
 spindle can precisely be evaluated with minimum 
 two measurement points. This method may apply 
 to give a simple method to predict quantity of heat 
 Fig. 6. Inverse results using T4 and T5 ;
 (a) Temperatures; (b) Heat generation generation for different kind of spindle.
References
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 ỨNG DỤNG PHƯƠNG PHÁP NGHỊCH TRONG TRUYỀN NHIỆT 
 CỦA TRỤC CHÍNH TỐC ĐỘ CAO
Tóm tắt:
 Bài báo này trình bày một phương pháp nghịch để tìm nguồn nhiệt sinh ra trong trục chính làm 
việc với tốc độ cao. Phương pháp nghịch đề xuất được xây dựng bằng cách sử dụng phần mềm thương mại 
ANSYS và phương pháp tối ưu liên hợp Gradient. Các mô phỏng với số điểm đo và vị trí đo khác nhau được 
thực hiện. Kết quả cho thấy nguồn nhiệt có thể được dự đoán với độ chính xác cao khi chỉ cần sử dụng nhiệt 
độ đo tại 2 điểm. Phương pháp hiện tại sẽ cung cấp một công cụ hữu ích cho việc nghiên cứu quá trình 
truyền nhiệt phức tạp trong các trục chính làm việc với tốc độ cao.
Từ khóa: Phương pháp nghịch, Trục chính tốc độ cao, Phương pháp liên hợp Gradient, Phần mềm ANSYS.
Khoa học & Công nghệ - Số 10/Tháng 6 - 2016 Journal of Science and Technology 27