Original Article
An improved thermal estimation model of the inverted planetary roller screw mechanism
Proc IMechE Part C: J Mechanical Engineering Science 0(0) 1–17 ! IMechE 2018 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0954406218762961 journals.sagepub.com/home/pic
Guan Qiao1 , Geng Liu1, Shangjun Ma1, Zhenghong Shi2, Yawen Wang3 and Teik C Lim3
Abstract The inverted planetary roller screw mechanism has recently become competitive in the electro-mechanical actuation system due to its high load-carrying capacity and small assembly size. However, a significant amount of heat at the frictional contact interfaces and power loss inside the electrical machine can be naturally generated in a compact and high-load inverted planetary roller screw mechanism system. The conductive heat leads to the temperature rise of inverted planetary roller screw mechanism components that subsequently results in thermal drift and error as well as the actuation accuracy degradation. An analytical approach is applied to calculate the friction torque of the contact pairs and support bearings in the inverted planetary roller screw mechanism system. As the thermal load, heat generation is derived from the friction in nut-roller-screw section and bearings. Then, the heat generation and convection boundary conditions are formulated to facilitate thermal behavior analysis. Finally, using the finite element method, steady-state and transient thermal-mechanical coupling analyses are performed to estimate the temperature distribution and thermal expansion of the inverted planetary roller screw mechanism components. Computational results reveal that operating conditions of rotation speed and external load have significant influence on the thermal characteristics of the inverted planetary roller screw mechanism. This study can serve as a foundation for modeling temperature field and analyzing coupled thermal-mechanical response of inverted planetary roller screw mechanism in electro-mechanical actuation system, which can be useful in determining thermal error compensation. Keywords Inverted planetary roller screw mechanism, thermal error, friction torque, finite elements, electro-mechanical actuator Date received: 4 July 2017; accepted: 5 February 2018
Introduction The inverted planetary roller screw mechanism (IPRSM) is a transmission device that transforms rotary motion into linear motion. Its fundamental principle is analogous to the standard planetary roller screw mechanism (SPRSM) and the ball screw mechanism (BSM). However, it is commonly used in high-speed and heavy-load applications in a relatively compact external diameter. This feature makes IPRSM suitable for aero electro-mechanical actuators (EMAs) that must be highly integrated and smallsized.1 With the increase of rotation speed and external load, the sophisticated mechanism is bound to suffer massive heat generation and a considerable thermal drift in such a limited space. It is believed that 40–70% of the overall error in precision parts arises from the thermal error.2 Treated mainly as a force-transmitted device, friction and heat related issues of IPRSM are more prominent than
motion-converted mechanisms, which may degrade the positioning accuracy and lifetime. Thus, a thermal-mechanical model is needed to predict the temperature distribution and thermal expansion of the IPRSM system. It is not until the last decade that knowledge and experience have been accumulated on the theoretical
1
School of Mechanical Engineering, Shaanxi Engineering Laboratory for Transmissions and Controls, Northwestern Polytechnical University, Xi’an, China 2 Vibro-Acoustics and Sound Quality Research Laboratory, University of Cincinnati, Cincinnati, OH, USA 3 Department of Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, USA Corresponding author: Geng Liu, School of Mechanical Engineering, Shaanxi Engineering Laboratory for Transmissions and Controls, Northwestern Polytechincal University, Xi’an 710072, China. Email:
[email protected]
2 analysis and experimental research of the roller-screw technology.3–6 Jones and Velinsky7 first presented the contact behavior and kinematics between the component surfaces in the roller screw mechanism, which laid the foundation for the contact mechanics and efficiency analysis of SPRSM. Abevi et al.8 described the static behavior of IPRSM to investigate global and local deformations in different configurations. They also investigated the load distribution and axial stiffness under tensile and compressive situations. Aure´gan et al.1 investigated the tribological behavior of contact at the load transferring interfaces of SPRSM using a specific apparatus. Ma et al.9 established a thermo-mechanical model of screw shaft and discussed the temperature rise and cooling performances of different coolants. For the IPRSM system, limited research on thermal behavior and thermal-mechanical modeling has been reported to improve its design and performance. The thermal characteristics of other similar rolling mechanisms, such as BSM,10,11 rolling bearing, and feed drive system12,13 have been studied by many researchers; some are noted below. Xu et al.14,15 did comprehensive research on modeling and compensation of a ball screw system with air-cooling and liquid-cooling to improve the system positioning error. Ramesh et al.16 attempted to analyze the thermal behavior of a vertical machining center under the influence of various operating parameters. Wu and Kung17 proposed a systematic method to investigate the thermal characteristics of a feed drive system. They measured the temperature increase and thermal deformation under long-term movement of the external load. Zhao et al.18 used finite elements to simulate the temperature field and thermal error of the spindle, following which the characteristics of heat flow and thermal deformation were analyzed. Jin et al.19 established an analytical approach to calculate the heat generation of support bearing in the machine tool by considering the operating
Figure 1. IPRSM components.
Proc IMechE Part C: J Mechanical Engineering Science 0(0) conditions. Yang et al.20 made accurate thermal behavior predictions of a dual-drive servo system based on the mapping relationships of the thermal error with equilibrium time, screw temperature, and coordinate position. The thermal error was compensated by applying a feedback integration method. Ma et al.21 proposed a three-dimensional finite element model of high-speed spindles, which considered the combined effect of thermal contact resistance and the change in heat power and stiffness. The literature review shows that few comprehensive thermal models exist for the IPRSM system that can be applied to estimate its thermal characteristics. Most authors treated the thread teeth structure as a cylinder for simplicity, but the influence of this simplification still needs to be evaluated. In addition, the heat source distribution and the long internal thread of IPRSM should be considered in the thermal-mechanical analysis. Thus, this work aims at developing a comprehensive thermal estimation model that can contribute to the implementation of thermal error compensation. Firstly, an analytical approach is proposed to calculate the friction torque of nut-roller and screw-roller contacts, and support bearings in the IPRSM system. Secondly, heat generation as the thermal load with heat fluxes, along with convection boundary conditions are calculated to facilitate the thermal behavior analysis. This heat generation is derived from the friction in roller screw pairs and the bearings. Finally, a thermal finite element model is developed to investigate the temperature distribution and thermal deformation of the components with consideration of the operating conditions, such as the rotation speed and external load.
Mechanical structure The IPRSM consists of an internal multi-turn threaded nut coupled with permanent magnets, a
Qiao et al. screw shaft of the same pitch, and a group of rollers in between (Figure 1). The thread profile of the nut and screw is straight and usually grinded to 90 for its best load-carrying capacity. The IPRSM integrated in the EMA is shown in Figure 2. Directly supporting the external load, the IPRSM is mounted on the bearings so that the nut is pivotally connected to the housing.8 The nut (or rotor) is considered as a driving component. When the nut is actuated by an electromagnetic torque, the rollers rotate with a complex motion of rotation and revolution, then deliver the input torque to the screw shaft for pushing and pulling the attached structure. A servo motor in parallel with the mechanism indicates the permanent magnets are attached to the nut. This built-in design allows less length for the actuator, compared to a SPRSM or BSM-based EMA in which a motor is hanging off the frame to couple the mechanism. The profile of the threaded rollers is curved to an arc shape in the axial cross section with its center at the axes. This means that the roller turns may be represented in an equivalent form as superimposed balls. The number of the balls is equal to the number of the roller turns along the generatrix.22 Thus, the axial load is transmitted through multiple contact areas at each side of the rollers. Due to the different radii of curvature in a pair of principal plane, the contacts between the nut/screw and the rollers can be described as ellipsoidon-flat contacts, whose characteristics are formulated using Hertzian theory.1 They are subjected to a different single normal force because of the load distribution that is 45 projection of the axial load. The roller is a key rolling component in the IPRSM because of the unique structure to transfer the external load. Threaded spur gears (spur gears swept with
Figure 2. IPRSM integrated in the EMA (http://exlar.com/).
3 threads) on both ends of rollers are meshed with two spur gears that are fixed at both sides of the screw to ensure a proper operation. Two carriers guarantee the rollers to be equally spaced around the screw, which is distinctly different from the BSM to avoid collision. When designing an IPRSM, some geometrically fundamental criterions should be satisfied that are similar to the SPRSM, which are: 1. The helix angle of the roller lr is equal to that of the screw ls to avert the rollers from axial migration. 2. The pitch of the nut, roller, and screw are the same, and the three parts have a relationship in the intersecting surface dn ¼ ds þ 2dr
ð1Þ
where dn , dr , and ds are the mean thread diameters of the nut, roller, and screw, respectively. 3. The number of thread turns on the nut and the screw are identical, while those of the rollers are single. 4. The ratio of teeth number kr , i.e. teeth number of the spur gear on the screw zs to that of the threaded spur gear on the roller zr , needs to be equal to the ratio of the mean thread diameter of the screw ds to the roller dr .
Heat generation and transfer The friction of mechanical components in the EMA system represents a power loss manifested in the form of heat generation. The frictional heat generation must be effectively transferred from heat sources,
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otherwise an unsatisfactory temperature condition will occur in IPRSM. There are two predominant sources for temperature elevation that are studied in this paper (Figure 1). One is the heat generated by the nut-roller-screw contacts due to the rolling/sliding friction resulting from the rotating nut, rollers, and transverse screw shaft. The other heat source comes from the support bearings, which is due to the friction between the balls and raceway grooves. The calculation of total friction torque is critical to obtain an analytical result of heat generation, which is described in the following sections.
Friction torque of nut–roller–screw contacts The rolling resistance of the roller on the nut’s or screw’s raceway is caused by miscellaneous factors, such as elastic hysteresis loss, curvature and pivoting effect, hydrodynamic lubricant resistance, and roughness. Note that the friction due to elastic hysteresis is very small compared with other factors in rolling components.23 Similarly, it has been summarized that the friction torque due to elastic hysteresis can be ignored in the SPRSM,24 and this friction torque component also applies to the IPRSM because of their structure similarity. In this study, the total friction torque of the IPRSM is calculated by an analytical approach, which is separated in the true physical friction sources.
Figure 3. Friction torque on the ith equivalent arc.
friction force in a given contact ellipse versus X and ranging from 1 to 1. axi 2 MCxi ¼ 0:1fC Qxi 2Ra Z 1 Z ðX2 X4 ÞdX 2 1
X2
ðX2 X4 ÞdX
X1
ð3Þ Curvature friction torque. For all thread teeth of the rollers subjected to a normal load from the nut and the screw, a single contact pure rolling can only happen at two lines (Figure 3). Outside of these two lines, sliding occurs in one direction; between the lines, sliding occurs in the opposite direction. This contrary sliding velocity directions in the elliptical contact area generates the curvature friction torque MC . Essentially, this is due to the local shear stress that is developed in the slip region. The curvature friction torque is defined as25 Z MC ¼
Z z dA ¼
fC z dA
ð2Þ
where z is the distance between the position of the local shear stress and the line of pure rolling, A is the contact area, fC is the local curvature friction coefficient, and is the local contact pressure. An analytical relation of MC can be further derived by considering the Hertzian distribution of the contact pressure and the load distributions of the nut– roller and screw–roller contacts26 in Figure 3. Using the appropriate sign for sliding friction force, i.e. negative for X1 5 X 5 X2 (X ¼ xa, X1 ¼ aa1 , X2 ¼ aa2 ) and positive elsewhere in the contact area, the final friction torque MCxi due to the contact race curvature on the ith equivalent arc of different contact positions is obtained in equations (3) to (5) by integrating the
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3Qxi 1 1 2 1 2 2 P axi ¼ ma þ 2 x E1 E2
ð4Þ
dr K Ra ¼ 1þ 2þK 2 cos
ð5Þ
In the above equations, Qxi and axi are the normal contact load and the semi-major axis of the projected contact ellipse with the subscript x denoted as n or s to represent the nut-roller contacts or the screw-roller contacts, respectively; and the second subscript i corresponds to the ith equivalent arc along the roller axis. The Hertzian contact radius Ra is calculated by equation (5); K is a parameter defining the race curvature radius and is usually equal to 0.03 to 0.05;25 ma is the dimensionless semi-major axis of the contact ellipse; P x is curvature sum on the nut–roller contacts or the screw–roller contacts to describe the contact between mating surfaces of revolution. Elastic moduli and Poisson’s ratios of the two contact materials are denoted by E1 , E2 and 1 , 2 , respectively. According to the knowledge of differential geometry and IPRSM geometry, the curvature sums can be derived. For the nut–roller contacts X
n ¼
4 cos 2 cos cos ln dr ds þ 2dr
ð6Þ
Qiao et al.
5
and for the screw–roller contacts X
4 cos 2 cos cos ls s ¼ þ dr ds
ð7Þ
where is contact angle, which is an intersection angle between the direction of normal contact force and the vertical direction of the nut/screw axis; ln , ls are different helix angles of the nut and the screw. It should be noticed that the direction of the curvature friction torque is along the semi-major axis of the contact ellipse, and there is a contact angle between the torque direction and the roller’s rotation speed direction. Therefore, the total MC considering both sides of the load distribution is obtained by MC ¼ N cos
n0 X
ðMCni þ MCsi Þ
ð8Þ
i¼1
that adds up all the curvature friction torques on the nut-roller contacts and the screw-roller contacts of N rollers. Friction torque from spinning/sliding motion. According to the motion fundamental of IPRSM, the rollers rotate synchronously on their own axes. Each normal contact on the thread teeth of the rollers has the angle to their rotation axes, which will lead to a simultaneous motion of rolling and sliding. The spinning motion of the roller occurs as a result of the normal component !r sin (Figure 3), which causes the friction torque due to the spinning/sliding under the contact load Qxi . The surface friction shear stress at any point (x, y) in the contact is given by27 " 2 2 #1=2 3fS Qxi x y i ¼ 1 ai bi 2ai bi
ð9Þ
The spinning friction force is calculated by integrating i over the contact area from ai to ai and bi to bi . Letting q ¼ x=ai and t ¼ y=bi , then one can deduce
FSxi
3fS Qxi ¼ 2ai bi
Z
þ1
1
ffi Z þpffiffiffiffiffiffiffi 1q2
2 2 1=2 dtdq pffiffiffiffiffiffiffiffi 1 q t 1q2
ð10Þ where ai and bi are semi-major and semi-minor axes of the contact ellipse on the ith equivalent arc. Equation (10) is valid for operating conditions involving solidfilm lubrication and boundary lubrication where the spinning friction coefficient fS can be characterized as a constant. Therefore, the friction torque on the ith equivalent arc MSxi is obtained by MSxi ¼ FSxi p
3fS Qxi ¼ 2ai bi
Z
þ1
ffi Z þpffiffiffiffiffiffiffi 1q2
1
2 2 1=2 pdtdqð11Þ pffiffiffiffiffiffiffiffi 1 q t 1q2
that multiplies the force by the relative distance between the acting point (x, y) and 1=2 the origin of the contact ellipse, i.e. p ¼ x2 þ y2 . The total axial friction torque of N rollers from spinning/sliding motion on the nut and the screw is obtained by MS ¼ N sin
n0 X
ðMSni þ MSsi Þ
ð12Þ
i¼1
that adds all MSxi on all n0 contacts. Pure sliding friction torque between rollers and carriers. During the IPRSM’s operation, the driving torque is transmitted from the nut to the threaded rollers, and then from the rollers to the screw shaft through several hundred contact areas including those between the N rollers and two carriers. Friction forces between the axle journal/axle head of the rollers and the holes of the carriers should not be neglected because these cylindrical or planar joints can be worn and eventually lead to frictional heat generation and thus mechanism jamming. Together with the holes in the carriers, the roller end pins work as slip bearings with edge contacts. The rollers are subject to the tipping torque due to the normal force acting on them. This torque is balanced by the radial and tangential forces on carrier holes, while the gear mesh only guarantees the normal movement of the rollers. Assuming that each contact force on the hole is shared equally with the average contact forces on the thread teeth, the pure sliding friction torque on one cylindrical joint MPj can be obtained as MPj ¼ fj Qj rj
ð13Þ
where fj is the equivalent coefficient of friction, rj is the radius of the axle journal, and Qj is the summation of forces and is presented in the following expressions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fr 2 þ Ft 2 Qj ¼ Nð2n0 þ 2Þ
ð14Þ
Fr ¼ Fa tan
ð15Þ
Ft ¼ Fa tan lr
ð16Þ
Here, Fa , Fr , and Ft are the total axial force, radial force, and tangential force applied to all rollers, respectively. On the other hand, under the axial force, the pure sliding friction between the axle head of the rollers and the carriers is another contributor because of the roller’s spinning motion along its axle. Only one
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of the two carriers will be in contact with the axle head of the roller under a directional force. Therefore, equation (17) shows a relationship between the pure sliding friction torque on one planar joint MPh and the axial force Fa MPh ¼ fh
Fa rh þ rj 2Nð2n0 þ 1Þ
ð17Þ
where fh is the coefficient of sliding friction on the planar joint, and rh is the radius of the axle head on the roller. Thus, the final pure sliding friction torque between the rollers and carriers (considering the two generic joints) is calculated as MP ¼ 2NMPj þ NMPh
ð18Þ
Friction torque generated by lubricant viscosity. To decrease IPRSM’s frictional heat and enhance its life span, grease or oil lubrication is selected and filled in the space of housing. The rollers are subjected to hydrodynamic rolling force due to Poiseuille flow of the lubricant contact. Considering the elastohydrodynamic (EHD) lubrication condition, this force on a given thread tooth FEx can be computed by28 FEx ¼ 2:86Eft Rxx 2 kx 0:348 G0:022 U0:66 W0:47
ð19Þ
where E is the reduced elastic modulus of two elements in contact, ft is a thermal correction factor including thermal effects in the inlet zone of the EHD contact, and kx is the radii ratio between equivalent radius Ryx and Rxx ; Ryx is defined as the equivalent radius of curvature in the y direction (perpendicular to the rolling direction), and Rxx is the equivalent radius in the rolling direction of either the nut-roller contacts or the screw-roller contacts. The respective classical dimensionless material, speed, and load parameters are denoted by G, U, and W, which are further derived using the equations below29 G ¼ Ep
ð20Þ
U¼
0 x ERxx
ð21Þ
W¼
Qxi ERxx 2
ð22Þ
The related parameters in equations (21) and (23) on different contact sides are highlighted below
ft ¼
70=99 1 þ 86=209L
kr ðkr þ 2Þ ds dr þ dr þ 4ðkr þ 1Þ 2
nn 60
s ¼
nn kr ðkr þ 2Þ dr 2ðkr þ 1Þ 60
0 x L ¼ kl
ð24Þ
ð25Þ
2
ð26Þ
where nn is the nut’s rotation speed, kr the ratio of teeth number, is the temperature-viscosity coefficient, and kl is the thermal conductivity of the lubricant at reference temperature and atmospheric pressure. Then, the friction torque generated by lubricant viscosity on the ith equivalent arc of a nut-roller contact and a screw-roller contact are calculated by MEni ¼ FEni
MEsi ¼ FEsi
ds þ dr 2
ds 2
ð27Þ
ð28Þ
For the N loaded rollers, the friction torque generated by lubricant viscosity ME is obtained by ME ¼ N
n0 X
ðMEni þ MEsi Þ
ð29Þ
i¼1
that sums up all the friction torques on the nut-roller and screw-roller contacts. Preload torque. For a certain actuation direction, only one side of the thread tooth works because of the existing backlash. The IPRSM is assembled with axial play, which varies between 0.02 mm and a maximum of 0.1 mm, depending on different sizes. However, in this study, backlash elimination can be achieved by fitting oversize rollers in the one-piece nut, which maintains high-load capacities and improves positioning accuracy. Preload torque Mpr due to preload Fpr is calculated from the nominal preload, according to a preliminary efficiency30 Mpr ¼
1
n ¼
Fpr Ln 1 1 pr
ð30Þ
ð23Þ
Here, p is the pressure-viscosity coefficient, 0 is the dynamic viscosity of the lubricant, x is the average rolling velocity on either the nut-roller contacts or the screw-roller contacts, and L corresponds to the dimensionless thermal loading parameter.
pr ¼
1 1þ
dn Ln
pr
ð31Þ
where pr refers to the preliminary direct efficiency, pr is the practical coefficient of friction, and Ln is
Qiao et al.
7
the lead of the nut. The preload Fpr is dependent on the maximum axial load Fmax , as follows9 Fpr ¼
Fmax 2:83
ð32Þ
Therefore, M1 , the total friction torque on nutroller-screw contacts, can be summed by all the introduced five friction torque components M1 ¼ MC þ MS þ MP þ ME þ Mpr
kinematic replenishment/starvation reduction factor
rs is calculated by
rs ¼
1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eKRS ðg nn ÞðdþDÞ KZ =2ðDdÞ
where KRS is the replenishment/starvation constant of 6 108 , KZ is the bearing type related geometric constant adopted as 3.1, d is the bearing bore diameter, and D is the bearing outside diameter.
ð33Þ Sliding friction torque. Sliding friction in rolling contacts is always present, even the support bearing is a generic rolling bearing. The sliding friction torque of the support bearing Mslid can be calculated using31
Friction torque of support bearings The friction torque in the support bearings determines the amount of heat generation, which depends on factors such as load, operating speed, and lubricant properties. The major resistance of the support bearing is the rolling and sliding friction between the rolling balls and raceways. To accurately calculate the friction torque of the rolling support bearing, the model proposed by SKF31 is used, in which the following sections and their tribological effects must be considered.
Mslid ¼ Gsl sl
ð34Þ
where ish is the inlet shear heating reduction factor,
rs is the kinematic replenishment/starvation reduction factor, Grr stands for the effect of the external load on the rolling resistance, and is decided by the bearing mean diameter dm and the bearing load, and g is the kinematic viscosity of the base oil in the grease lubrication at the operating temperature. For four-point contact ball bearings, the rolling friction variable Grr is decided by Grr ¼ R1 dm
1:97
0:54 Frr þ R3 d4m n2n þ R2 Fa
ð35Þ
where R1 , R2 , and R3 are geometric constants for rolling friction torque and are dependent on the type of bearing, namely 4.78 107, 2.42, and 1.4 1012, respectively. The respective radial and axial loads on the bearing are denoted by Frr and Fa . Some of the oil close to the contact area inlet is rejected and generates a reverse flow, which shears the lubricant, thus generating heat. This influence is considered in the friction model by introducing the inlet shear heating reduction factor ish . Besides, lubricant may not have enough time to replenish the raceways when the viscosity or the operating speed are high, thereby resulting in a ‘‘kinematic starvation’’ effect. For the grease lubrication method, the
ð37Þ
where sl represents the sliding friction coefficient; Gsl expresses the influence of the external load on the sliding resistance, which also depends on both the bearing mean diameter dm and the bearing load. Similarly, the sliding friction variable Gsl of the four-point contact ball bearing is presented by Gsl ¼ S1 dm 0:26 ½ðFrr þ S3 dm 4 nn 2 Þ4=3 þ S2 Fa 4=3
Rolling friction torque. There are several sources of rolling friction losses in the rolling process. In the SKF method, the rolling friction torque Mroll of the rolling bearing can be calculated by Mroll ¼ ish rs Grr ðg nn Þ0:6
ð36Þ
ð38Þ
where S1 , S2 , and S3 are geometric constants for sliding friction torque and depend on the type of bearing, namely 1.2 102, 0.9, and 1.4 1012, respectively. For the sliding friction coefficient, it is affected by the macro-sliding and the micro-sliding. In mixed and full-film lubrication conditions, the following expression gives the total friction coefficient coming from asperity contacts and from shearing the lubricant sl ¼ 0:15’bl þ 0:04ð1 ’bl Þ
ð39Þ
where ’bl is a weighting factor for the influence of asperity and lubricant shearing. Effects of grease fill on friction. The rolling bearing can generate a higher frictional value than the calculated value when it is lubricated with grease. This difference in frictional values is due to an increase of the operating temperature. The time it takes to decrease the friction depends on the nut’s speed and how long the grease could be distributed within the free space. The effect can be considered by multiplying the rolling friction torque by a grease factor fg of 2 to 4, depending on the bearing series. Thus, the friction torque model of the support bearing is the combination of the above torque losses, which is presented as M2 ¼ fg Mroll þ Mslid
ð40Þ
including the effects of grease fill. Finally, by quantifying the different components of the different
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sections, the total friction torque M of the IPRSM system can be obtained by M ¼ M1 þ 2M2
ð41Þ
which consists of the nut–roller–screw section and the bearing section in Figure 1.
Heat generation of IPRSM Simple physics defines power as equal to force times velocity or torque times speed. Therefore, the power loss due to friction between components may be calculated from the following relationship H¼
Mn 9550
ð42Þ
where H is the power in watts, which is also the IPRSM-generated heat, n is the rotation speed of parts in r/min, and M is the total friction torque of the IPRSM in N mm, which is caused by the preload and external load. Equation (42) can also be divided into the two different heat sources in this study, which are the heat from the nut–roller–screw section H1 and that from one bearing H2 H1 ¼ H2 ¼
M 1 nn 9550
ð43Þ
M 2 nn 9550
ð44Þ
After obtaining the heat generation, the thermal boundary condition of the IPRSM model can be further derived to conduct a thermal-mechanical analysis in section ‘‘Thermal characteristics’’. Table 1 shows the component parameters used to calculate the friction torque and heat generation of the IPRSM system.
Convective heat transfer coefficient The main purpose of studying convective heat transfer is to predict the convective heat transfer coefficient h, which is defined in equation (45). The convective heat transfer in the IPRSM is a generic forced convection because the fluid motion involved in the process is induced by some external means.32 It includes the high-speed rotary motion between the outer/inner surfaces of the nut and the ambient air, as well as the repeated linear motion between the screw shaft surface and the ambient air. These complex convection heat transfer processes, which happen on the component surfaces can be regarded as the laminar flow over a flat plate because of the direction of the two motions and the value of Reynolds number ReL . Nusselt number correlation for heat transfer is shown by the following equations
Table 1. Parameters of the IPRSM system. Parameter Nut Mean thread diameter (mm) Lead (mm) Helix angle ( ) Elastic modulus (GPa) Poisson’s ratio Roller Mean thread diameter (mm) Contact angle ( ) Ratio of teeth number Helix angle ( ) Number of equivalent arc Number Radius of the axle journal (mm) Radius of the axle head (mm) Elastic modulus (GPa) Poisson’s ratio Screw shaft Mean thread diameter (mm) Lead (mm) Helix angle ( ) Elastic modulus (GPa) Poisson’s ratio Bearing Contact angle ( ) Bore diameter (mm) Outside diameter (mm) Mean diameter (mm) Width (mm)
h¼
NuL kf L
NuL ¼ 0:664ReL 1=2 Pr1=3 ReL ¼ Pr ¼
1 L
cp kf
Symbol
Value
dn Ln n En nn
20 2.54 2.315 212 0.29
dr kr r n0 N rj rh Er nr
5 45 2 4.622 15 6 2 2.225 212 0.29
ds Ls s Es ns
10 2.54 4.622 212 0.29
– d D dm –
35 25 52 38.5 15
ð45Þ ð46Þ ð47Þ ð48Þ
Here, NuL is the average Nusselt number over a plate of the characteristic length L, which is computed from the Reynolds number for the length L and the dimensionless Prandtl number Pr; 1 is the airflow velocity, is the kinematic viscosity of air, is the dynamic viscosity of air, cp is the specific heat of air, and kf refers to the thermal conductivity of air. It should be noted that equation (46) is valid for ReL 45 105 , Pr 50:5. Thus, the nut outer surface convection coefficient hout , the nut inner surface
Qiao et al.
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convection coefficient hinn , and the screw shaft outer surface convection coefficient hrod are derived as hout
hinn
2 60nn rout Lout 1=2 ¼ 0:664 1=3 cp kf Lout kf
ð49Þ
2 60nn rinn Linn 1=2 ¼ 0:664 1=3 cp kf Linn kf
ð50Þ
hrod ¼ 0:664
cp 1=3 60nn Ln Lrod 1=2 kf Lrod kf ð51Þ
where rout is the radius of the nut outer surface, rinn is the radius of the nut inner surface, Lout is the characteristic length of the nut outer surface, Linn is the characteristic length of the nut inner surface, and Lrod is the characteristic length of the screw shaft outer surface.
Thermal characteristics The heat generated in the IPRSM system flows through components mainly by conduction and convection. The thermal analysis of the IPRSM components is not easy because the heat transfer coefficient of air cannot be obtained precisely due to the geometric complexity of the thread structure. However, the quantitative analysis of the temperature distribution in the EMA system and the thermal expansion of the components are regularly required in the IPRSM design.
Figure 4. Finite element model of IPRSM.
Thermal model The finite-element thermal model of the IPRSM is built using the 10-node SOLID87 elements and the 20-node SOLID90 elements, as shown in Figure 4. In the thermal-mechanical coupling analysis, the structural elements 10-node SOLID187 and 20-node SOLID186 are used to calculate the thermal expansion of the nut and screw shaft due to the temperature rise. Mesh size is an important factor that affects the accuracy of temperature distribution and computation time when solving the finite element model. Thus, an analysis will be conducted to illustrate how to choose a compromising mesh size in the steady-state temperature distribution of the IPRSM system. The thermal-mechanical coupling analyses are carried out by using Ohio Supercomputer Center.33 Depending on the requested core and cluster of the Supercomputer, the computation time can be varying from several hours to a couple of days. To simplify the thermal model, the numerical temperature estimation model should be based on the following assumptions: 1. The frictional heat generation from the nutroller-screw section and the two bearings is a constant value at a certain external load and nut’s speed. 2. The convective heat transfer coefficient is a constant value during the operation of the mechanism, and is not related to temperature. 3. Thermal deterioration and the effectiveness of heat conduction through the lubricant are negligible. 4. The fraction of frictional dissipated energy converted into heat is equal to 1, and the heat dissipation shares equally between the two contact surfaces. 5. Radiation heat transfer is ignored.
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Thermal boundary conditions The steady-state and transient thermal characteristics of the IPRSM can be obtained using the previous derivation and thermo-physical properties of the materials and air listed in Tables 2 and 3. In the process of simulation, the heat transferred into the nut and the screw shaft, H, is converted into the heat fluxes, @ . As shown in Figure 5, they are all applied to contact surfaces, including those within the roller’s effective stroke. The heat source is moving along the stroke back and forth with a uniformly transverse velocity. Similarly, the heat generation from the motor and the two ball bearings are replaced with heat sources on the nut outer surface, according to the required electric power and the bearing power loss. In addition, the calculated convective heat transfer coefficients are also applied on the external contours of the nut and the screw shaft. Numbers 1–10 in
Table 2. Thermo-physical properties of the material. Linear Nut/roller/ Specific Thermal heat (kJ/ conductivity expansion screw Density (kg C)) (W/(m C)) coefficient (1/ C) (kg/m3) (GCr15) 552.66
36.92
1.18 105
Figure 5 are the temperature monitoring points on the nut and screw surfaces for transient thermal analysis. The calculated heat fluxes and convection coefficients are listed in Tables 4 and 5, which are the main basis of thermal prediction through the finite element method (FEM). The major heat generation comes from frictional contact interfaces, bearings, and power loss inside the electrical machine. The main heat convention happens on the nut outer surface and inner surface, as well as the screw outer surface. Five operating conditions are compared in this work to investigate the influence of rotation speed and external load on the thermal behavior of the IPRSM components.
Results and discussion In this section, the temperature distribution and thermal deformation analysis of the IPRSM system are presented by using a three-dimensional finite-element model. The steady-state analysis is first performed to examine the FEM accuracy by tuning the total Table 4. Calculated results of heat flux.
7810 Operating condition
Table 3. Thermo-physical properties of air. Temperature (ºC) 20 30 40 50
Thermal conductivity (W/(mºC)) 2
2.59 10 2.67 102 2.76 102 2.83 102
Kinematic viscosity (m2/s) 6
15.06 10 16.00 106 16.96 106 17.95 106
Figure 5. Boundary conditions: (a) nut; (b) screw.
Prandtl number 0.703 0.701 0.699 0.698
Inner surface of the ball Inner surface Outer surface bearing of the nut of the screw (W/m2) (W/m2) (W/m2)
2546.48 F ¼ 5000 N nn ¼ 500 r/min F ¼ 15,000 N 9973.71 nn ¼ 500 r/min 19,098.60 F ¼ 25,000 N nn ¼ 500 r/min 14,854.46 F ¼ 15,000 N nn ¼ 1000 r/min 24,708.92 F ¼ 15,000 N nn ¼ 2000 r/min
780.11
2507.95
1182.32
3800.97
1548.76
4979.04
2488.68
8000.71
4947.51
15,905.50
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element size. Then, the transient thermal analysis of the nut and the screw shaft is carried out, taking into consideration the moving heat source generated by nut–roller contacts. After obtaining the temperature
Table 5. Calculated results of convection coefficient. Operating condition
Outer surface Inner surface Outer surface of the nut of the nut of the screw (W/(m2ºC)) (W/(m2ºC)) (W/(m2ºC))
547.45 F ¼ 5000 N nn ¼ 500 r/min 547.45 F ¼ 15,000 N nn ¼ 500 r/min 547.45 F ¼ 25,000 N nn ¼ 500 r/min 774.21 F ¼ 15,000 N nn ¼ 1000 r/min 1094.90 F ¼ 15,000 N nn ¼ 2000 r/min
498.18
89.41
498.18
89.41
498.18
89.41
704.54
126.45
996.37
178.82
80 79
Element Node
Maximum temperature (°C)
78 77 76 75 74 73 72 71 70 5.0E+05
1.0E+06
1.5E+06
2.0E+06
2.5E+06
3.0E+06
Number
Figure 6. Maximum temperature with different global element sizes.
Figure 7. Steady-state temperature distribution of IPRSM (ºC).
distribution, the thermal error is analyzed to discuss the influence of varying operating conditions.
Temperature distribution To get satisfactory simulation results, a test of convergence related to temperature field is needed in this study. As shown in Figure 6, with the increase of element and node numbers, the maximum temperature of steady-state analysis is converged to the value of about 75.42 C. Even setting the element number to 1,720,557 and node number to 2,791,646, the result is only changed by 0.49 C within the temperature rise tolerance of 0.5 C. Therefore, to compromise between computational accuracy and costs, a global element size of 1,043,507 (with node number of 1,826,757) is adopted in the following thermal analysis. The analysis setting for rotation speed of the nut is 500 r/min, 1000 r/min, and 2000 r/min. The external load is set to 5000 N, 15,000 N, and 25,000 N. Thus, an operating condition (OC) combination of speed and load is recommended in this study: OC1 is at 500 r/min with 5000 N, OC2 is at 500 r/min with 15,000 N, OC3 is at 500 r/min with 25,000 N, OC4 is at 1000 r/min with 15,000 N, and OC5 is at 2000 r/min with 15,000 N. The simulation time is 1200 s, and the reference temperature of the coupled analysis is 25 C. Figure 7 shows the steady-state temperature distribution of the IPRSM. The maximum temperature of the whole system is 75.421 C, occurring on the nut– roller–screw section under OC2, which may degrade the lubricating performance. Also, it can be noted the temperature field of the nut–roller–screw section is almost uniform after a period of operation time. Figure 8 shows the maximum temperature evolution of the nut under varying operating conditions. With the increase of the load and speed, the maximum balanced temperature is increased because of the enhancement of the calculated friction torque. They all show a gradually rising trend and then remain stable at a certain temperature within a period
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Figure 8. Maximum temperature variation of the nut under varying operating conditions.
Proc IMechE Part C: J Mechanical Engineering Science 0(0) section. The result also shows that the maximum temperature on the middle right part drops slightly because of the applied uniformly heat fluxes from the motor and the increased heat convection that is caused by speed change, which shows a different trend compared with the bearing section. Next, the temperature variation at ten temperature monitoring points (Figure 5) on the nut and the screw can be evaluated through the above transient thermal analysis. Figure 11 demonstrates the maximum temperature variation on different nut locations under OC2. The temperature rise of the center at the two bearings (#2) is balanced until the 100 s, and the maximum value is about 49.59 C. Also, the temperature development of the center between the roller’s effective stroke is showed in #4 line. The balanced temperature is about 43.92 C, which attributes to the moving heat source, motor heat generation, and heat convention on the nut surfaces. Similarly, the transient thermal analysis of different screw locations under OC2 is gained in Figure 12. It can be seen that the variation of the #6 line and #7 line are quite small compared to other lines. This is due to a lack of applied thermal loads there, but the thermal error can reach a high value because of the free end. For the temperature rise on the two spur gears, the left one in Figure 5 is about 8 C less than that of the right one due to the strong heat convention nearby.
Thermal deformation
Figure 9. Maximum temperature variation of the screw under varying operating conditions.
of 300 s. However, the time to be temperaturebalanced is affected by the speed of the nut. When the nut’s speed is changed from 500 r/min to 2000 r/min, the time to be balanced is increased from about 100 s to 250 s under the external load of 15,000 N. A similar trend of the temperature rise on the screw is obtained through the 1200 s simulation in Figure 9. The steady-state temperature of the screw shaft increases by almost 9 C per every 10,000 N increment of external load. A considerable temperature rise of up to 113.41 C can be seen under the OC5, which could cause a large thermal error in the IPRSM system. This high temperature rise occurs because of the continuous heat generation and deficient heat emission condition on the screw. Figure 10 depicts the nut’s steady-state temperature distribution under the five OCs. The maximum temperature occurs at the bearing section except OC1. Under the OC1, the main heat source comes from the motor compared to those from the bearings and the nut–roller–screw
Once the temperature distribution is obtained, the thermal deformation of the nut or screw shaft can be predicted. The thermal-mechanical coupling analysis employs the indirect coupling method to get a transient thermal expansion response under the different OCs, which means the above temperature distributions are loaded as thermal loads. The maximum axial deformations of the nut are plotted in Figure 13. The axial deformation is exponentially increased when the load is increased, reaching a maximum of 15.22 mm at 500 r/min and 25,000 N. However, it is interesting that the maximum axial deformation is decreased when changing the nut’s speed from 500 r/min to 2000 r/min. This indicates that the impact of the coefficients of heat transfer on the thermal deformation is more significant than that of the heat generation caused by the speed variation. Thus, a potential method to compensate the thermal error is to utilize a strong convective medium or liquid-cooling system. In reality, it is more difficult to integrate a cooling system for such a compact IPRSM structure, compared with other ball screw systems in the machine tool industry. The maximum thermal deformations of the screw grow gradually until they reach a steady state, as shown in Figure 14. All the maximum deformations occur at the rod side of the screw (free end) under the five conditions. The result shows that the maximum
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Figure 10. Steady-state temperature distribution of the nut: (a) under OC1 (500 r/min and 5000 N); (b) under OC2 (500 r/min and 15,000 N); (c) under OC3 (500 r/min and 25,000 N); (d) under OC4 (1000 r/min and 15,000 N); (e) under OC5 (1000 r/min and 15,000 N).
Figure 11. Maximum temperature variation on different nut locations under OC2.
Figure 12. Maximum temperature variation on different screw locations under OC2.
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Proc IMechE Part C: J Mechanical Engineering Science 0(0) the IPRSM components. The heat generation from the nut-roller-screw section and two bearing elements, as the thermal loads with the calculated heat fluxes, are imposed on the exterior surfaces of grooves. The steady-state and transient analyses are conducted to estimate the influence of operating conditions on thermal characteristics. Simulation results reveal that:
Figure 13. Maximum axial deformation of the nut under varying operating conditions.
Figure 14. Maximum axial deformation of the screw under varying operating conditions.
axial deformations under the different OCs are 6.81 mm, 10.32 mm, 13.52 mm, 16.07 mm, and 23.31 mm, respectively, which follows a consistent variation trend with the temperature–time relationship when changing OCs. It can also be found that the relationship of maximum axial deformations in OC3, OC4, and OC5 is distinguishing between the nut and the screw. This is because the heat generation from the bearing is mainly controlled by external load and the complicated heat source is distributed on the nut. These deformation values can be taken as the thermal error of the screw shaft and may be measured with a laser interferometer in the future.
1. With the increase of external load, the maximum balanced temperatures of the nut and the screw shaft are increased because of the friction torque’s enhancement. The time to reach balance is affected by the nut’s speed. With the speed changed from 500 r/min to 2000 r/min, the time to be balanced is increased from about 100 s to 250 s under the external load of 15,000 N. The nut’s maximum temperatures under the five OCs are located at the bearing section except OC1. The steadystate temperature of the screw shaft increases by almost 9 C per every 10,000 N increment of external load. 2. The balanced temperature of the nut center between the roller’s effective stroke is about 5.7 C less than the temperature rise at the center of two bearings, which is the result caused by the moving heat source, motor heat generation, and heat convention on the nut surfaces. For the temperature rise on the two spur gears, the left one is about 8 C less than that of the right one due to the strong heat convention there. 3. The axial thermal expansion of the nut increases exponentially with external load, therefore reaching the thermal error of 15.22 mm at 500 r/min and 25,000 N. However, the increasing nut’s speed reduces thermal errors and improves the positioning accuracy of the IPRSM, which suggests that the impact of the coefficients of heat transfer on the thermal error is more significant than that of the heat generation by changing the speed. This proposed thermal model can be used to estimate the temperature field distribution and thermal deformation of the IPRSM system. As a next step, experiments will be conducted to study the influences of varying operating conditions and cooling methods on the thermal behavior to validate the thermalmechanical model. Declaration of Conflicting Interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
Conclusions A thermal estimation model of the IPRSM is developed using the finite element method to investigate the temperature distribution and thermal deformation of
The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (Grant No. 51505381, 51275423), 111 Project of China (Grant
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No. B13044), the Fundamental Research Funds for the Central Universities (3102014ZD0035), China Postdoctoral Science Foundation funded project (2014M552483), and China Scholarship Council (CSC).
ORCID iD Guan Qiao
http://orcid.org/0000-0002-3386-5831
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Appendix
D E E1 , E2 fg fh fj ft fC fS Fa Fmax Fpr Fr Frr Ft FEx FS G Grr Gsl h hinn hout hrod H H1 H2 i kf kl kr
K KRS KZ L L Linn
Notation a a1 , a2 axi A b cp d dm dr , dn , ds
kx
semi-major axis of the contact ellipse rolling line location semi-major axis of the ith contact ellipse contact area semi-minor axis of the contact ellipse specific heat of air bearing bore diameter bearing mean diameter mean thread diameters of the roller, nut, screw bearing outside diameter reduced elastic modulus of two contact bodies elastic modulus of two materials grease factor coefficient of rolling friction on the axle head equivalent coefficient of friction on the axle journal thermal correction factor curvature friction coefficient spinning friction coefficient total axial force of all rollers maximum axial load preload total radial force of all rollers radial load on the bearing total tangential force of all rollers hydrodynamic rolling force on a thread tooth spinning friction force dimensionless material parameter geometric and load-dependent rolling frictional variable geometric and load-dependent sliding frictional variable convective heat transfer coefficient nut inner surface convection coefficient nut outer surface convection coefficient screw shaft outer surface convection coefficient IPRSM-generated heat heat from the nut–roller–screw section heat from one bearing the ith equivalent arc thermal conductivity of air thermal conductivity of the lubricant ratio of teeth number
Lout Lrod ma Mpr Mroll Mslid ME MP MPh MPj MS M1 M2 nn n0 N NuL p Pr Qj Qxi rh rinn rj rout Ra Rxx Ryx R1 , R2 , R3 S1 , S2 , S3 V
x W
ratio between equivalent radius Ryx and Rxx parameter defining race curvature radius replenishment/starvation constant bearing type related geometric constant characteristic length dimensionless thermal loading parameter characteristic length of the nut inner surface characteristic length of the nut outer surface characteristic length of the screw shaft outer surface dimensionless semi-major axis of the contact ellipse preload torque rolling friction torque of the bearing sliding friction torque of the bearing friction torque generated by lubricant viscosity pure sliding friction torque between rollers and carriers pure sliding friction torque on a planar joint pure sliding friction torque on a cylindrical joint friction torque from spinning/sliding motion friction torque of the nut–roller–screw section friction torque of the bearing nut’s rotation speed number of equivalent arc on a roller number of rollers average Nusselt number over a plate of length L distance between acting point and origin of the contact ellipse Prandtl number summation of forces on the axle journal normal contact load radius of the axle head radius of the nut inner surface radius of the axle journal radius of the nut outer surface Hertzian contact radius equivalent radii in the major axis direction equivalent radii in the minor axis direction geometric constant for rolling friction torque geometric constant for sliding friction torque dimensionless speed parameter average rolling velocity dimensionless load parameter
Qiao et al. x z zr , zs p @
pr 0 lr , ln , ls
17 denoted as n or s to represent nut–roller or screw–roller contacts distance between shear stress and the line of pure sliding teeth number of the spur gear on the roller, screw pressure–viscosity coefficient contact angle heat flux practical direct efficiency dynamic viscosity of the lubricant temperature–viscosity coefficient helix angle of the roller, nut, screw dynamic viscosity of air
pr sl 1 g !r
bl
ish
rs P
x 1 , 2
practical coefficient of friction sliding friction coefficient of the bearing airflow velocity kinematic viscosity of air viscosity of the base oil angular speed of the roller weighting factor of the sliding friction coefficient inlet shear heating reduction factor kinematic replenishment/starvation reduction factor curvature sum shear stress Poisson’s ratio of two materials
