- Email: [email protected]
High-speed rotation induction heating in thermal clamping technology
Applied Mathematics and Computation xxx (2015) xxx–xxx
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
High-speed rotation induction heating in thermal clamping technology Lukáš Koudela ⇑, Václav Kotlan University of West Bohemia, Univerzitní 26, 306 14 Plzenˇ, Czech Republic
a r t i c l e
i n f o
Keywords: Induction heating Thermal clamping head Shrink-fit Non-linear numerical analysis Higher-order finite element method
a b s t r a c t The clamping technology based on high-speed induction heating of the fixing part of the system is proposed and analyzed. The continuous mathematical model of the clamping process is given by three partial differential equations (PDEs) describing the distributions of electromagnetic field, temperature field and field of thermoelastic displacements. Their coefficients containing the physical parameters of the materials are nonlinear temperature-dependent functions. The system is solved numerically in the hard-coupled formulation using a fully adaptive higher-order finite element method developed by the hp-FEM group and implemented in the application Agros2D. The methodology is illustrated with a practical example of fixing the shank of a high-revolution drilling tool in the clamping head. The results are verified by experiments. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Hot pressing belongs to widely spread industrial technologies used in a great number of practical applications [1,2]. Thermal induction clamping is one of the branches of the above technology, where heat is produced by induction heating. This process is mainly used for fixing shank tools in drilling machines. Its typical representatives are induction shrink fits used for clamping high-revolution tools, for example, in the automotive and aerospace industries [3–5]. The process of heating is mostly realized by gas or induction. Induction heating is characterized by an easy control of the intensity of heating and its local distribution, no chemical changes in the surface layers of the processed material and no products of combustion. The process of assembly of a shrink fit is shown in Fig. 1. At cold, the diameter of the bore of the clamping head is d1 and diameter of the tool shank is d2 > d1 (see Fig. 1 left part). First, the clamping head must be heated (by induction) as long as the dilatation of its bore allows inserting the shank of a prescribed length l into the bore (which means, that along the length of the bore d3 > d2 , Fig. 1 middle part). The system is then intensively cooled, which produces the shrink fit. At the end of the process, both the hole and shank have the same diameter d4 , for which d1 < d4 < d2 . When the shank is not considered elastic (its diameter d2 after cooling remains unchanged), then d4 ¼ d2 . The parameters of the shrink fit (relations among dimensions d1 ; d2 and l) must be derived from the mechanical torque that is to be transferred. The process of assembly is reversible. If one wants to disassemble both parts, the first step is heating them to the prescribed temperature, which must be fast enough to avoid an excessive heating and expanding the shank tool. From this viewpoint, the shank tool is usually manufactured from a special material that expands whose dilatability is substantially lower. ⇑ Corresponding author. E-mail addresses: [email protected] (L. Koudela), [email protected] (V. Kotlan). http://dx.doi.org/10.1016/j.amc.2015.01.107 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Induction clamping exhibits (in comparison with other existing clamping technologies) numerous advantages. We can mention, for example:
Quality of balance of the clamped shank tool. Minimal eccentricity due the rotation asymmetry. High clamping force and dynamic stiffness at high torque. Usability for very high speeds of rotation. The simplicity of the entire system. Low manufacturing requirements. Possibility of using clamping heads with very narrow profile for working in problematically reachable places. Repeatability of precise clamping, economy of operation and lifetime extension.
2. Formulation of the technical problem The task is to propose an induction shrink fit for fixing a high-revolution drilling tool in the head. The materials of both the tool shank and clamping head are known. The clamping head was purchased from the company Gühring [3] that belongs to one of the leading European manufacturers of this technology. Its principal dimensions are shown in Fig. 2. The first task is to propose the minimum diameter d2 of the tool shank so that the system clamping head-drilling tool is able to transfer the prescribed mechanical torque that for casual drilling is about 5 N m, for torque limitations of commonly used technologies for clamping see Fig. 3. The second task is to propose suitable parameters of the induction heating system and map its time evolution. The process of heating will be realized using a novel and highly efficient method [6,7] based on rotation of the clamping head in static magnetic field generated by a system of permanent magnets. The basic arrangement of the system (that is sufficiently long in the axial direction) is depicted in Fig. 4. The heated body rotates in a system of fixed permanent magnets. Their magnetic field induces in the rotating body eddy currents producing there the Joule losses representing the local heat sources. 3. Mathematical model First of all, it is necessary to find the interference d (see Fig. 1) such that the shrink fit transfers the required mechanical torque. Only then we will able to model the process of induction heating. [8–10]. After heating the clamping head and inserting there the drill shank of length l the pressure between both parts is given by the formula [11] 2
p¼
2
dE d2 d1 ; 2 d1 d2
ð1Þ
and this pressure must satisfy the condition 2
p
d2 2 d2
2
d1
¼
dE ra 6 ; d1 4
where E denotes the Young modulus of the corresponding material and mechanical torque M start then follows from the expression
ð2Þ
ra is its allowable mechanical stress. The starting
2
Mstart ¼ p
d1 plfr 2pd2 Elfr ; 2
ð3Þ
where f r is the coefficient of dry friction between the materials of the clamping head and tool shank, l is length of the connection.
Fig. 1. Principle of thermal clamping – shank tool is inserted into heated clamping head and shrink fit is created by subsequent cooling of system.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Fig. 2. Axial cut through clamping head with principal dimensions in mm.
Fig. 3. Transferrable torque for several clamping head diameters.
Fig. 4. Basic arrangement of induction heating.
From the physical viewpoint, the process represents a triply coupled nonlinear and non-stationary problem characterized by a mutual interaction of three physical fields: magnetic field, temperature field and field of thermoelastic displacements. These fields (more or less influencing one another) are described by partial differential equations (PDEs) whose coefficients containing various material parameters are temperature-dependent functions. Fig. 5 describes the mutual dependencies of solved physical fields realized through their material parameters. But as the solution to the problem in the hard-coupled formulation is extremely complicated, we will accept several permissible simplifying assumptions: The influence of the displacement vector u on the geometry of the problem is neglected. Solution of magneto-thermoelastic process is in the linear region of the Hooke law, i.e. the dependence of the Young modulus E and the stress of the r is linear. Material parameters m (Poisson factor) and E are independent on temperature T.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Fig. 5. Mathematical model.
3.1. Magnetic field The first partial differential equation, describing the distribution of the magnetic vector potential in the system, may be written in general in the form [12]
1 @A curl curl A H c þ c v curl A ¼ J ext ; @t l
ð4Þ
where l stands for the magnetic permeability, c is the electrical conductivity, v is the local velocity of rotation, and H c is the coercive force. In the solved problem there are no external power sources (J ext ¼ 0). The magnetic field may be considered in steady state (the transient due to switching on the motor being extremely fast and having practically no influence on the evolution of the temperature), which leads to a simplified equation in the form
1 curl curl A H c cv curl A ¼ 0:
l
ð5Þ
The relevant physical parameters of materials involved in the system (magnetic permeability, electric conductivity) are nonlinear functions of temperature T. Moreover, the magnetic permeability also considered as non-linear variable depending on the module of magnetic flux density B. The solution of (5) provides the distribution of magnetic vector potential A. This represents the basis for finding the volumetric Joule losses wJ as is depicted in Fig. 5. These losses produced by the induced currents in the heated clamping head are given by the formula
pJ ¼
jJ eddy j2
c
;
ð6Þ
where
J eddy ¼ cv curl A:
ð7Þ
3.2. Heat transfer The temperature field in the system is described by the heat transfer equation in the form [13]
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
L. Koudela, V. Kotlan / Applied Mathematics and Computation xxx (2015) xxx–xxx
div ðk grad T Þ ¼ qcp
@T þ v grad T pJ ; @t
5
ð8Þ
where k is the thermal conductivity, q is the specific mass, and cp denotes the specific heat at a constant pressure. The symbol pJ denotes the time average internal sources of heat represented by the volumetric Joule losses. All parameters of materials included in the system are again non-linear temperature-dependent functions. The output of the heat transfer equation is the temperature T, again as a function of position r and time t. 3.3. Field of thermoelastic displacement The field of thermoelastic displacement is described by the Lamé non-isothermic equation in the form [14]
ðu þ wÞ grad ðdiv uÞ þ w Du ð3u þ 2wÞ aT grad T þ f ¼ 0;
ð9Þ
where u and w are coefficients given by formulas
u¼
mE ; ð1 þ mÞð1 2mÞ
w¼
E : 2 ð1 þ mÞ
ð10Þ
Here E denotes the modulus of elasticity and m the Poisson coefficient of the contraction. Finally, u represents the vector of the displacement, aT the coefficient of the thermal dilatability (considered again as temperature dependent function), and f the vector of internal volumetric gravitational and Lorentz forces (in comparison with the thermoelastic forces of the thermoelastic origin, however, these forces are very small and may be neglected). The distribution of the displacements then provides the distribution of deformations and radial stresses between the driving and driven parts that make it possible to determine the mechanical connection parameters, especially the maximum transferable torque. According to Fig. 1, there must be a corresponding component of the displacement u of all points along the bore of the clamping head higher than the value of d. All Eqs. (5), (8) and (9) must be supplemented with correct boundary conditions to obtain the accurate solution. In the case of magnetic field, the sufficiently distant artificial boundary is characterized by the Dirichlet condition A ¼ 0. The model of temperature field is supplemented with the boundary condition respecting generally both convection and radiation. Finally, in the field of thermoelastic displacements the fixed and free coordinates are defined. 4. Numerical solution The numerical solution is carried out with respect to all mentioned nonlinearities (saturation curve and temperature dependences of material properties). The computations are performed by own code Agros2D [15,16] based on a fully adaptive higher-order finite element method [17–20] and tightly cooperating with the library Hermes [21,22] containing the most advanced procedures for numerical processing. Both codes written in C++ are freely distributable under the GNU General Public License. The most important and in some cases quite unique features of the codes follow: Fully automatic hp-adaptivity. When adaptivity is required, in every iteration step the solution is compared with the reference solution (realized on an approximately twice finer mesh), and the distribution of error is then used for selection of candidates for adaptivity [23,24]. Each physical field can be solved on quite a different mesh that best corresponds to its typical features. In non-stationary processes every mesh can change in time, in accordance with the real evolution of the corresponding physical quantities. Easy treatment of the hanging nodes appearing on the boundaries of subdomains whose elements have to be refined. These nodes bring about a considerable increase of the number of the degrees of freedom (DOFs) [25]. Curved elements able to replace curvilinear parts of any boundary by a system of circular or elliptic arcs. These elements mostly allow reaching highly accurate results near the curvilinear boundaries with very low numbers of the DOFs. In the course of solution, attention is paid to all important numerical aspects (mainly the convergence and accuracy of the results). The computation of one example takes about 10 min on a top-quality PC. 5. Illustrative example The arrangement of the system is shown in Fig. 6. The clamping head is made of special tool steel, that allows acceptable dilatations of the material for inserting the shank of the appropriate tool. The magnetic field is generated by permanent magnets placed in magnetic circuit and the rotation of the clamping head in this magnetic field induces in the head the eddy currents and leads to the heating of thermal clamping head. All material parameters of the clamping head are temperature dependent functions that are overtaken from the MPDB database. Two of them are shown in Fig. 7(a) and (b).
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Magnetic Circuit
Thermal Insulation Permanent Magnet Clamping Head
R17.125
Air
R16.125 1.5
19.75
R13.5
y
10 R4
x
20
Fig. 6. Basic arrangement.
Fig. 7. Selected material characteristics used in numerical simulation; (a) dependence of specific heat on temperature, (b) dependence of electrical conductivity on temperature (non-linearity across two physical fields).
5.1. Numerical simulation results In this subsection we present some results of the problem solution. In Fig. 8(a)–(c) we show different meshes used in Agros2D for each of the coupled physical fields. The number of elements in case of magnetic field must be slightly greater due to the small depth of penetration of eddy currents. The numerical solution of the problem makes use of all advanced techniques implemented in Agros2D – multimesh technology, hanging nodes and also curvilinear elements (see Fig. 8(d)). Next three Figs. 9–11 show some important features of the non-linear solvers, such as the dependence of the convergence on the damping factor (DF), elapsed time for the same values of DF and time evolution of the damping factor in case of automatic variant. A really tough problem is to find the generalized coefficient a of the convective heat transfer. Although its value may be determined from several recommended formulae or from the similarity theory, it has to be carefully verify by measurements. For example, Fig. 12 shows (for n ¼ 9000 rpm) the time evolution of the temperature of the clamping head for its different values.
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Fig. 8. Discretization mesh (a) magnetic field with 61,280 elements, (b) temperature field with 1568 elements, (c) field of thermoelastic displacements with 1280 elements, (d) detail of used mesh curvilinear element.
Fig. 9. Convergence of solution vs DF.
Fig. 10. Elapsed time for several used methods in non-linear solvers.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Fig. 11. Evolution of damping factor in case of Newton solver with automatic damping.
Fig. 12. Time evolution of temperature for several values of coefficients a.
Based on the detailed experiments, the correct dependence of a on revolutions n is depicted in Fig. 13. The curve is practically linear. With the relevant values of this coefficient we performed all remaining computations. Fig. 14 shows, for example, the dependences of the average calculated and measured temperatures in the head on time for three different values of n. The curves exhibit a very good accordance. Fig. 15 shows the time evolutions of the average temperature (for n ¼ 8618 rpm for constant electric conductivity
c ¼ 1:5 106 [S/m] and for its real nonlinear temperature dependence. It is clear that the real nonlinear curve provides much better results. Fig. 16 shows the dependences of displacement on diameter for several revolutions, that were already measured. The dependences are for acceptable time 60 s and the limitations are set according to the real shanks of drill elements. The values of transferable torque for both shanks were calculated using term (3), for values of E ¼ 2:1 1011 Pa; f r ¼ 0:55 and l ¼ 54 103 m. Shank 1 (d ¼ 3:5 106 m) provides torque 0:48 N m. In case of shank 2 (d ¼ 5:5 106 m) torque is
Fig. 13. Measured dependence of coefficient a on revolutions n used for numerical solution.
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Fig. 14. Time evolution of temperature for three different values of n.
Fig. 15. Temperature on time in case of non-linear and linear electric conductivity.
1:19 N m, that not satisfies the request for casual drilling. In order to reach torque 5 N m it is needful to have a drill element with minimal value of d ¼ 11:3 106 m. For this value of displacement the rotation induction heating is suitable in case of minimal revolutions 8618 rpm, see Fig. 16. 5.2. Experimental measurement Experimental measurement was carried out for the thermoelastic clamping head used for the conventional induction heating. For rotating induction heating it was necessary to design an appropriate magnetic circuit according to the available permanent magnets and geometrical dimensions of the clamping head. The circuit was designed for 4 permanent magnets, which corresponds to an air gap of thickness 2 mm in the base of the head. The clamping head was fixed with an interface on the shaft of a 2-pole high-speed asynchronous engine placed on a stand with the designed magnetic circuit, see Fig. 17. The asynchronous engine was supplied from the frequency converter and it was possible to adjust the speed of rotation individually for the solved range of revolutions per minute. To prevent the heat loss and avoid the unwanted heating of nearest area, especially heating of permanent magnets, the thermal insulation was used. Magnets and the entire magnetic circuit was filled with self-adhesive textile thermal insulation. Between the flange on the shaft and the clamping head, there was inserted the combined thermal insulation composed of textile and solid material.
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Fig. 16. Dependence of displacement on clamping head diameter for time 60 s with depicted limitations for two available shanks.
Fig. 17. Device for experimental measurement.
The temperature was measured by the optical pyrometer Omega that allows logging the results to PC using the USB interface. It allowed to measure temperature in the entire range during rotational movement of the clamping head. The precision of the contactless measurement was randomly controlled by thermocouples immediately after the rotational movement has stopped. The measurement procedure was repeated several times for every engine revolutions to approve the obtained results. Between two subsequent measurement, the entire experimental device was cooled by a fan. The dilatation of the inner radius of the clamping head was checked by the possibility of inserting the drill shank into the clamping head. 6. Conclusion The results presented in this paper show the possibility of using the rotation induction heating like an alternative form of heating in some applications. The described illustrative example of clamping is solved as a hard-coupled problem Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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incorporating a nonlinear interaction of the thee physical fields: magnetic field, temperature field and field of thermoelastic displacement. It is solved using the Agros2D code supplemented with own procedures and scripts. The presented results show really good agreement with the realized measurements. Acknowledgement The financial support from the Project P102/11/0498 (The Grant Agency of the Czech Republic) is highly acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
R.D. Cook, Finite Element Modeling for Stress Analysis, Wiley, New York, USA, 1995. V. Rudnev, D. Loveless, R.L. Cook, M. Black, Handbook of Induction Heating, CRC Press, Boca Raton, USA, 2002. Gühring s.r.o., cited 4.3.2014. URL:. Bilz Tool Company Inc, cited 4.3.2014. URL: . Haimer GmbH, cited 4.3.2014. URL: . P. Karban, F. Mach, I. Dolezel, Modeling of rotational induction heating of nonmagnetic cylindrical billets, Appl. Math. Comput. 219 (13) (2013) 7170– 7180. P. Kus, P. Karban, I. Dolezel, F. Mach, Optimization of the system for induction heating of nonmagnetic cylindrical billets in rotating magnetic field produced by permanent magnets, COMPUTING 95 (1) (2013) 537–552. M. Kutz, Mechanical Engineers’ Handbook, A Wiley-Interscience publication, Wiley, 1998. URL: . J. Carvill, Mechanical Engineer’s Data Handbook, Referex Engineering, Butterworth-Heinemann, 1994. URL: . M. Fogiel, Research, E. Association, The Handbook of Mechanical Engineering, Handbook Series, Research & Education Association, 1999. URL: . K. Grote, E. Antonsson, Springer Handbook of Mechanical Engineering, vol. 10, Springer, 2009. URL: . C. Oatley, Electric and Magnetic Fields: An Introduction, Electronics Texts for Engineers and Scientists, Cambridge University Press, 1976. URL: . J.P. Holman, Heat Transfer, McGraw Hill, New York, 2002. 2002. B.A. Boley, J.H. Weiner, Theory of Thermal Stresses, Wiley, New York, USA, 1960. P. Karban, et al., Agros2D – multiplatform application for the solution of PDEs. 2009–2015. URL:
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
High-speed rotation induction heating in thermal clamping technology Lukáš Koudela ⇑, Václav Kotlan University of West Bohemia, Univerzitní 26, 306 14 Plzenˇ, Czech Republic
a r t i c l e
i n f o
Keywords: Induction heating Thermal clamping head Shrink-fit Non-linear numerical analysis Higher-order finite element method
a b s t r a c t The clamping technology based on high-speed induction heating of the fixing part of the system is proposed and analyzed. The continuous mathematical model of the clamping process is given by three partial differential equations (PDEs) describing the distributions of electromagnetic field, temperature field and field of thermoelastic displacements. Their coefficients containing the physical parameters of the materials are nonlinear temperature-dependent functions. The system is solved numerically in the hard-coupled formulation using a fully adaptive higher-order finite element method developed by the hp-FEM group and implemented in the application Agros2D. The methodology is illustrated with a practical example of fixing the shank of a high-revolution drilling tool in the clamping head. The results are verified by experiments. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Hot pressing belongs to widely spread industrial technologies used in a great number of practical applications [1,2]. Thermal induction clamping is one of the branches of the above technology, where heat is produced by induction heating. This process is mainly used for fixing shank tools in drilling machines. Its typical representatives are induction shrink fits used for clamping high-revolution tools, for example, in the automotive and aerospace industries [3–5]. The process of heating is mostly realized by gas or induction. Induction heating is characterized by an easy control of the intensity of heating and its local distribution, no chemical changes in the surface layers of the processed material and no products of combustion. The process of assembly of a shrink fit is shown in Fig. 1. At cold, the diameter of the bore of the clamping head is d1 and diameter of the tool shank is d2 > d1 (see Fig. 1 left part). First, the clamping head must be heated (by induction) as long as the dilatation of its bore allows inserting the shank of a prescribed length l into the bore (which means, that along the length of the bore d3 > d2 , Fig. 1 middle part). The system is then intensively cooled, which produces the shrink fit. At the end of the process, both the hole and shank have the same diameter d4 , for which d1 < d4 < d2 . When the shank is not considered elastic (its diameter d2 after cooling remains unchanged), then d4 ¼ d2 . The parameters of the shrink fit (relations among dimensions d1 ; d2 and l) must be derived from the mechanical torque that is to be transferred. The process of assembly is reversible. If one wants to disassemble both parts, the first step is heating them to the prescribed temperature, which must be fast enough to avoid an excessive heating and expanding the shank tool. From this viewpoint, the shank tool is usually manufactured from a special material that expands whose dilatability is substantially lower. ⇑ Corresponding author. E-mail addresses: [email protected] (L. Koudela), [email protected] (V. Kotlan). http://dx.doi.org/10.1016/j.amc.2015.01.107 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Induction clamping exhibits (in comparison with other existing clamping technologies) numerous advantages. We can mention, for example:
Quality of balance of the clamped shank tool. Minimal eccentricity due the rotation asymmetry. High clamping force and dynamic stiffness at high torque. Usability for very high speeds of rotation. The simplicity of the entire system. Low manufacturing requirements. Possibility of using clamping heads with very narrow profile for working in problematically reachable places. Repeatability of precise clamping, economy of operation and lifetime extension.
2. Formulation of the technical problem The task is to propose an induction shrink fit for fixing a high-revolution drilling tool in the head. The materials of both the tool shank and clamping head are known. The clamping head was purchased from the company Gühring [3] that belongs to one of the leading European manufacturers of this technology. Its principal dimensions are shown in Fig. 2. The first task is to propose the minimum diameter d2 of the tool shank so that the system clamping head-drilling tool is able to transfer the prescribed mechanical torque that for casual drilling is about 5 N m, for torque limitations of commonly used technologies for clamping see Fig. 3. The second task is to propose suitable parameters of the induction heating system and map its time evolution. The process of heating will be realized using a novel and highly efficient method [6,7] based on rotation of the clamping head in static magnetic field generated by a system of permanent magnets. The basic arrangement of the system (that is sufficiently long in the axial direction) is depicted in Fig. 4. The heated body rotates in a system of fixed permanent magnets. Their magnetic field induces in the rotating body eddy currents producing there the Joule losses representing the local heat sources. 3. Mathematical model First of all, it is necessary to find the interference d (see Fig. 1) such that the shrink fit transfers the required mechanical torque. Only then we will able to model the process of induction heating. [8–10]. After heating the clamping head and inserting there the drill shank of length l the pressure between both parts is given by the formula [11] 2
p¼
2
dE d2 d1 ; 2 d1 d2
ð1Þ
and this pressure must satisfy the condition 2
p
d2 2 d2
2
d1
¼
dE ra 6 ; d1 4
where E denotes the Young modulus of the corresponding material and mechanical torque M start then follows from the expression
ð2Þ
ra is its allowable mechanical stress. The starting
2
Mstart ¼ p
d1 plfr 2pd2 Elfr ; 2
ð3Þ
where f r is the coefficient of dry friction between the materials of the clamping head and tool shank, l is length of the connection.
Fig. 1. Principle of thermal clamping – shank tool is inserted into heated clamping head and shrink fit is created by subsequent cooling of system.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Fig. 2. Axial cut through clamping head with principal dimensions in mm.
Fig. 3. Transferrable torque for several clamping head diameters.
Fig. 4. Basic arrangement of induction heating.
From the physical viewpoint, the process represents a triply coupled nonlinear and non-stationary problem characterized by a mutual interaction of three physical fields: magnetic field, temperature field and field of thermoelastic displacements. These fields (more or less influencing one another) are described by partial differential equations (PDEs) whose coefficients containing various material parameters are temperature-dependent functions. Fig. 5 describes the mutual dependencies of solved physical fields realized through their material parameters. But as the solution to the problem in the hard-coupled formulation is extremely complicated, we will accept several permissible simplifying assumptions: The influence of the displacement vector u on the geometry of the problem is neglected. Solution of magneto-thermoelastic process is in the linear region of the Hooke law, i.e. the dependence of the Young modulus E and the stress of the r is linear. Material parameters m (Poisson factor) and E are independent on temperature T.
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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Fig. 5. Mathematical model.
3.1. Magnetic field The first partial differential equation, describing the distribution of the magnetic vector potential in the system, may be written in general in the form [12]
1 @A curl curl A H c þ c v curl A ¼ J ext ; @t l
ð4Þ
where l stands for the magnetic permeability, c is the electrical conductivity, v is the local velocity of rotation, and H c is the coercive force. In the solved problem there are no external power sources (J ext ¼ 0). The magnetic field may be considered in steady state (the transient due to switching on the motor being extremely fast and having practically no influence on the evolution of the temperature), which leads to a simplified equation in the form
1 curl curl A H c cv curl A ¼ 0:
l
ð5Þ
The relevant physical parameters of materials involved in the system (magnetic permeability, electric conductivity) are nonlinear functions of temperature T. Moreover, the magnetic permeability also considered as non-linear variable depending on the module of magnetic flux density B. The solution of (5) provides the distribution of magnetic vector potential A. This represents the basis for finding the volumetric Joule losses wJ as is depicted in Fig. 5. These losses produced by the induced currents in the heated clamping head are given by the formula
pJ ¼
jJ eddy j2
c
;
ð6Þ
where
J eddy ¼ cv curl A:
ð7Þ
3.2. Heat transfer The temperature field in the system is described by the heat transfer equation in the form [13]
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div ðk grad T Þ ¼ qcp
@T þ v grad T pJ ; @t
5
ð8Þ
where k is the thermal conductivity, q is the specific mass, and cp denotes the specific heat at a constant pressure. The symbol pJ denotes the time average internal sources of heat represented by the volumetric Joule losses. All parameters of materials included in the system are again non-linear temperature-dependent functions. The output of the heat transfer equation is the temperature T, again as a function of position r and time t. 3.3. Field of thermoelastic displacement The field of thermoelastic displacement is described by the Lamé non-isothermic equation in the form [14]
ðu þ wÞ grad ðdiv uÞ þ w Du ð3u þ 2wÞ aT grad T þ f ¼ 0;
ð9Þ
where u and w are coefficients given by formulas
u¼
mE ; ð1 þ mÞð1 2mÞ
w¼
E : 2 ð1 þ mÞ
ð10Þ
Here E denotes the modulus of elasticity and m the Poisson coefficient of the contraction. Finally, u represents the vector of the displacement, aT the coefficient of the thermal dilatability (considered again as temperature dependent function), and f the vector of internal volumetric gravitational and Lorentz forces (in comparison with the thermoelastic forces of the thermoelastic origin, however, these forces are very small and may be neglected). The distribution of the displacements then provides the distribution of deformations and radial stresses between the driving and driven parts that make it possible to determine the mechanical connection parameters, especially the maximum transferable torque. According to Fig. 1, there must be a corresponding component of the displacement u of all points along the bore of the clamping head higher than the value of d. All Eqs. (5), (8) and (9) must be supplemented with correct boundary conditions to obtain the accurate solution. In the case of magnetic field, the sufficiently distant artificial boundary is characterized by the Dirichlet condition A ¼ 0. The model of temperature field is supplemented with the boundary condition respecting generally both convection and radiation. Finally, in the field of thermoelastic displacements the fixed and free coordinates are defined. 4. Numerical solution The numerical solution is carried out with respect to all mentioned nonlinearities (saturation curve and temperature dependences of material properties). The computations are performed by own code Agros2D [15,16] based on a fully adaptive higher-order finite element method [17–20] and tightly cooperating with the library Hermes [21,22] containing the most advanced procedures for numerical processing. Both codes written in C++ are freely distributable under the GNU General Public License. The most important and in some cases quite unique features of the codes follow: Fully automatic hp-adaptivity. When adaptivity is required, in every iteration step the solution is compared with the reference solution (realized on an approximately twice finer mesh), and the distribution of error is then used for selection of candidates for adaptivity [23,24]. Each physical field can be solved on quite a different mesh that best corresponds to its typical features. In non-stationary processes every mesh can change in time, in accordance with the real evolution of the corresponding physical quantities. Easy treatment of the hanging nodes appearing on the boundaries of subdomains whose elements have to be refined. These nodes bring about a considerable increase of the number of the degrees of freedom (DOFs) [25]. Curved elements able to replace curvilinear parts of any boundary by a system of circular or elliptic arcs. These elements mostly allow reaching highly accurate results near the curvilinear boundaries with very low numbers of the DOFs. In the course of solution, attention is paid to all important numerical aspects (mainly the convergence and accuracy of the results). The computation of one example takes about 10 min on a top-quality PC. 5. Illustrative example The arrangement of the system is shown in Fig. 6. The clamping head is made of special tool steel, that allows acceptable dilatations of the material for inserting the shank of the appropriate tool. The magnetic field is generated by permanent magnets placed in magnetic circuit and the rotation of the clamping head in this magnetic field induces in the head the eddy currents and leads to the heating of thermal clamping head. All material parameters of the clamping head are temperature dependent functions that are overtaken from the MPDB database. Two of them are shown in Fig. 7(a) and (b).
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Magnetic Circuit
Thermal Insulation Permanent Magnet Clamping Head
R17.125
Air
R16.125 1.5
19.75
R13.5
y
10 R4
x
20
Fig. 6. Basic arrangement.
Fig. 7. Selected material characteristics used in numerical simulation; (a) dependence of specific heat on temperature, (b) dependence of electrical conductivity on temperature (non-linearity across two physical fields).
5.1. Numerical simulation results In this subsection we present some results of the problem solution. In Fig. 8(a)–(c) we show different meshes used in Agros2D for each of the coupled physical fields. The number of elements in case of magnetic field must be slightly greater due to the small depth of penetration of eddy currents. The numerical solution of the problem makes use of all advanced techniques implemented in Agros2D – multimesh technology, hanging nodes and also curvilinear elements (see Fig. 8(d)). Next three Figs. 9–11 show some important features of the non-linear solvers, such as the dependence of the convergence on the damping factor (DF), elapsed time for the same values of DF and time evolution of the damping factor in case of automatic variant. A really tough problem is to find the generalized coefficient a of the convective heat transfer. Although its value may be determined from several recommended formulae or from the similarity theory, it has to be carefully verify by measurements. For example, Fig. 12 shows (for n ¼ 9000 rpm) the time evolution of the temperature of the clamping head for its different values.
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Fig. 8. Discretization mesh (a) magnetic field with 61,280 elements, (b) temperature field with 1568 elements, (c) field of thermoelastic displacements with 1280 elements, (d) detail of used mesh curvilinear element.
Fig. 9. Convergence of solution vs DF.
Fig. 10. Elapsed time for several used methods in non-linear solvers.
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Fig. 11. Evolution of damping factor in case of Newton solver with automatic damping.
Fig. 12. Time evolution of temperature for several values of coefficients a.
Based on the detailed experiments, the correct dependence of a on revolutions n is depicted in Fig. 13. The curve is practically linear. With the relevant values of this coefficient we performed all remaining computations. Fig. 14 shows, for example, the dependences of the average calculated and measured temperatures in the head on time for three different values of n. The curves exhibit a very good accordance. Fig. 15 shows the time evolutions of the average temperature (for n ¼ 8618 rpm for constant electric conductivity
c ¼ 1:5 106 [S/m] and for its real nonlinear temperature dependence. It is clear that the real nonlinear curve provides much better results. Fig. 16 shows the dependences of displacement on diameter for several revolutions, that were already measured. The dependences are for acceptable time 60 s and the limitations are set according to the real shanks of drill elements. The values of transferable torque for both shanks were calculated using term (3), for values of E ¼ 2:1 1011 Pa; f r ¼ 0:55 and l ¼ 54 103 m. Shank 1 (d ¼ 3:5 106 m) provides torque 0:48 N m. In case of shank 2 (d ¼ 5:5 106 m) torque is
Fig. 13. Measured dependence of coefficient a on revolutions n used for numerical solution.
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Fig. 14. Time evolution of temperature for three different values of n.
Fig. 15. Temperature on time in case of non-linear and linear electric conductivity.
1:19 N m, that not satisfies the request for casual drilling. In order to reach torque 5 N m it is needful to have a drill element with minimal value of d ¼ 11:3 106 m. For this value of displacement the rotation induction heating is suitable in case of minimal revolutions 8618 rpm, see Fig. 16. 5.2. Experimental measurement Experimental measurement was carried out for the thermoelastic clamping head used for the conventional induction heating. For rotating induction heating it was necessary to design an appropriate magnetic circuit according to the available permanent magnets and geometrical dimensions of the clamping head. The circuit was designed for 4 permanent magnets, which corresponds to an air gap of thickness 2 mm in the base of the head. The clamping head was fixed with an interface on the shaft of a 2-pole high-speed asynchronous engine placed on a stand with the designed magnetic circuit, see Fig. 17. The asynchronous engine was supplied from the frequency converter and it was possible to adjust the speed of rotation individually for the solved range of revolutions per minute. To prevent the heat loss and avoid the unwanted heating of nearest area, especially heating of permanent magnets, the thermal insulation was used. Magnets and the entire magnetic circuit was filled with self-adhesive textile thermal insulation. Between the flange on the shaft and the clamping head, there was inserted the combined thermal insulation composed of textile and solid material.
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Fig. 16. Dependence of displacement on clamping head diameter for time 60 s with depicted limitations for two available shanks.
Fig. 17. Device for experimental measurement.
The temperature was measured by the optical pyrometer Omega that allows logging the results to PC using the USB interface. It allowed to measure temperature in the entire range during rotational movement of the clamping head. The precision of the contactless measurement was randomly controlled by thermocouples immediately after the rotational movement has stopped. The measurement procedure was repeated several times for every engine revolutions to approve the obtained results. Between two subsequent measurement, the entire experimental device was cooled by a fan. The dilatation of the inner radius of the clamping head was checked by the possibility of inserting the drill shank into the clamping head. 6. Conclusion The results presented in this paper show the possibility of using the rotation induction heating like an alternative form of heating in some applications. The described illustrative example of clamping is solved as a hard-coupled problem Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107
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incorporating a nonlinear interaction of the thee physical fields: magnetic field, temperature field and field of thermoelastic displacement. It is solved using the Agros2D code supplemented with own procedures and scripts. The presented results show really good agreement with the realized measurements. Acknowledgement The financial support from the Project P102/11/0498 (The Grant Agency of the Czech Republic) is highly acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]
R.D. Cook, Finite Element Modeling for Stress Analysis, Wiley, New York, USA, 1995. V. Rudnev, D. Loveless, R.L. Cook, M. Black, Handbook of Induction Heating, CRC Press, Boca Raton, USA, 2002. Gühring s.r.o., cited 4.3.2014. URL:
Please cite this article in press as: L. Koudela, V. Kotlan, High-speed rotation induction heating in thermal clamping technology, Appl. Math. Comput. (2015), http://dx.doi.org/10.1016/j.amc.2015.01.107