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Abstract

In this article, 3D or three-dimensional finite element analysis is used to simulate and evaluate the load distribution characteristics of a planetary roller screw mechanism under thermo-mechanical coupling. The finite element model takes into account the installation modes of the planetary roller screw mechanism, which is verified by comparison with theoretical models for a certain load magnitude in four installation modes. In addition, the effects of the installation mode, load magnitude, and temperature condition on the load distribution are also systematically analyzed. The numerical results reveal a phenomenon of threads separating from the meshing, which indicates that the influence of thermo-mechanical coupling on the load distribution cannot be ignored. Furthermore, the influence of the installation mode on the screw–roller interface is larger than that on the nut–roller interface. Compared with the screw–roller interface, the temperature difference is one of the main conditions affecting the load distribution of the planetary roller screw mechanism and has a significant effect on the nut–roller interface. In addition, the influences of the screw rotational speed and the load magnitude on the load distribution on the screw–roller interface are larger than those on the nut–roller interface for the four installation modes.

Keywords

Planetary roller screw mechanism load distribution finite element analysis thermo-mechanical coupling working temperature installation mode

Introduction

There is a general trend in the aerospace industry to increase the use of electrically powered equipment. The use of electromechanical actuator (EMA) technologies for flight control systems is an important potential step toward this goal.¹ The planetary roller screw mechanism (PRSM) is a mechanical transmission device and an important component of EMAs that converts rotary to linear motion. A PRSM mainly consists of a nut, a screw, and a group of rollers to provide more contact points than a conventional ball screw mechanism. Therefore, high stiffness, high load capacity, and long lifespan are among the advantages provided by a PRSM.

In recent years, many studies have been published that support the engineering applications of PRSMs, such as kinematic modeling and analysis,^2–5 the meshing mechanism,^6,7 rolling–sliding friction,^8,9 frictional heat and thermal analysis,^10,11 and dynamics of PRSMs.¹² Importantly, the load distribution over the threads of the screw, roller, and nut is another important research field, which is the basis for studying the carrying capacity, durability and stiffness, and so on. In research on load distributions, Yang et al.¹³ took the roller as an object to investigate the load distribution and static rigidity of the PRSM. Ryś and Lisowski¹⁴ developed a model for the load distribution between components in the PRSM. They considered the deformations of the rolling components as deformations of rectangular volumes subjected to shear stresses and performed a comparison between the analytical and numerical results. Jones and Velinsky¹⁵ constructed a stiffness model via the direct stiffness method, which could also be used to predict the load distribution of the PRSM. Based on the developed model, the design sensitivities for the number of rollers and threads were studied. Ma et al.¹⁶ established a model of the load distribution considering errors when the screw shaft is subjected to tension and the nut is subjected to compression. Zhang et al.¹⁷ developed a model for the load distribution according to the relationships of the deformation compatibility and force equilibrium and introduced an improved approach to achieve uniform load distribution over the threads. Abevi et al.¹⁸ presented a hybrid model based on using bar, beam, and nonlinear spring elements, which can be used to compute the load distribution and axial stiffness under static loading. Zhdanova and Morozova¹⁹ developed a rod model with two elastic contact layers to calculate the load distribution of a PRSM. The results show that the load distribution depends on the geometry, material of the threaded elements, manufacturing accuracy, and mating friction forces.

Rolling and sliding coexist in the PRSM, which will lead to frictional heating. Furthermore, the coupling influences of either the load magnitude and frictional heat or the load magnitude and working temperature will lead to further changes in the load distribution. However, as the PRSM is a closed system with a complex structure, it is inherently difficult to model. Currently, few works exist that address the load distribution under different load magnitudes and temperature conditions. Furthermore, the relationships between the load distribution and working conditions have seldom been mentioned in published articles.

This article uses three-dimensional (3D) finite element (FE) analysis to investigate the effect of thermo-mechanical coupling on the load distribution characteristics of a PRSM. First, the FE model of a PRSM is introduced. Second, the calculation methods for heat production in the PRSM and thermal boundary conditions are proposed. Then, the model is verified by comparing the theoretical models under a certain load magnitude for four installation modes. Finally, the effects of the installation mode, load magnitude, rotational speed, and working temperature under thermo-mechanical coupling on the load distribution are systematically analyzed.

FEA model

Model assumptions

As shown in Figure 1, the main components of a PRSM are the screw, nut, and rollers. In this article, the components are created and assembled using the 3D software package SolidWorks 2016, and the 3D FE model is developed using the commercial FE package ABAQUS/Standard in ABAQUS 6.10. Because only the threads of the screw, roller, and nut are the main carrying structures, in the 3D FE model, the spur gears on the roller are simplified, and only the threads are reserved. The 3D FE model is simplified based on the symmetric structure of the PRSM. The following assumptions are made as part of the FEA:

The heat flux density is constant (which is generated by frictional heat due to moving nut) for a given load magnitude and screw rotational speed.

The heat convection coefficient is constant during the working process.

The influences of heat conduction on the lubricant and thermal deterioration are negligible.

Component materials are assumed to be linear elastic isotropic.

Figure 1.

3D FE model of the PRSM.

Elements and boundary conditions

To obtain a higher computational accuracy and save computational time, the model is meshed using a combination of hexahedral and wedge (triangular pyramid) elements to provide a smooth transition between the thread shank and the threaded section.²⁰ To reduce the model size in terms of the number of elements, the model is divided into fine and coarsely meshed zones. The thread contact regions are discretized with finer elements and different mesh densities to ensure appropriate load transfer, while other zones have coarse elements. To obtain the influence of heat production on the load distribution, the transient heat transfer analysis based on indirect coupled thermal stress method is performed. C3D8T elements (eight-node trilinear displacement and temperature elements) are used for thermal analysis to obtain the thermal load of the nodes. Then, the structural stress analysis is performed by element conversion (the corresponding element type is C3D8I, which is eight-node trilinear displacement with incompatible modes). In addition, only the loads on each thread are extracted to study the load distribution characteristics under thermo-mechanical coupling, and thermal deformation and thermal stress are not involved.

Based on the relative movement relationships of the PRSM, the mechanical constraints applied to the PRSM are as follows: (1) as the nut and roller only move in the axial direction, the radial translational and rotational degrees of freedoms of the nut and the roller are constrained; (2) the axial translational degree of freedom of the screw is released due to axial deformation when the load applied on the nut, and the mechanical constraints of the screw are imposed on fixed end according to the installation modes, as shown in Figure 2; and (3) symmetry constraints are imposed on the planes on both sides of the sector.

Figure 2.

Installation modes for a PRSM.

The load constraints in the FE model are as follows: (1) contact interactions are applied to the interfaces between the screw and roller and those between the roller and nut to realize load transfer on each thread of the roller; (2) to facilitate analysis at different load magnitudes, a reference point (i.e. the nodes on the nut flange are coupled to a node) is established at the axis of the nut, which can be used to transfer the distribution load applied on the nut flange to a concentrated load; (3) the load can be further divided into more steps to avoid any rigid body displacement. Generally, a smaller axial load is applied to the reference point to guarantee thread contact at the first load step, and the load is gradually applied on the flange in later load steps; (4) the heat flux density is the thermal load imposed on the thread’s outside surface and is constant for a given load magnitude or rotational speed; and (5) the convection coefficients are applied on the thread’s outside surface of the screw, roller, and nut, respectively.

The contact settings are as follows: (1) the contact type is surface-to-surface; (2) a contact pair algorithm is used to define the contact model and interaction problems; (3) the finite-sliding and surface-to-surface contacts are selected in the sliding formulation and discretization method; and (4) the default option is selected for contact control in the software, and the iteration step is changed automatically.

Installation modes

There are four installation modes for a PRSM based on the engineering applications. As shown in Figure 2,¹⁷ when the PRSM bears an applied load from different installation modes, the load distribution will vary, and the deformations of the components will change correspondingly. Therefore, the installation modes must be considered in the developed model.

Simulations of the installation modes can be achieved using mechanical and load constraints. For example, as shown in Figure 2(a) and (d), the upper side of the screw can be constrained, which makes the screw withstand tension when a load is applied on the upper side of the nut flange. Similarly, as shown in Figure 2(b) and (c), the upper side of the screw can be constrained, which makes the screw withstand compression when the load is applied to the lower side of the nut flange.

In addition, the numbering rule for the thread number is consistent between the four installation modes in the FE model. That is, the threads on the screw–roller and nut–roller interfaces are numbered in consecutive order starting from the fixed end of the screw to the other end.

Heat production in the PRSM

The major source of heat generation in the mechanism results from frictional torques in a PRSM. In this article, the frictional torques generated by elastic hysteresis, the roller’s spinning sliding motions, the viscosity of the lubricant, and the differential sliding of the rollers are taken into consideration¹⁰

$H_{P R S M} = 0.12 π f_{0} η_{0} n_{s s} M_{P R S M}$ (1)

where $H_{PRSM}$ denotes the energy loss (W), $f_{0}$ is a factor related to the nut type and lubrication method, $η_{0}$ is the kinematic viscosity of the lubricant, $n_{ss}$ is the screw rotation velocity, and $M_{PRSM}$ is the total frictional torque of the PRSM.

Four frictional torques are considered in this article:^11,13

The frictional torque generated by elastic hysteresis, $M_{fx}$ .

The frictional torque generated by the spinning sliding of the rollers, $M_{kx}$ .

The frictional torque generated by the lubricant viscosity, $M_{lx}$ .

The frictional torque generated by the differential sliding of the rollers, $M_{dx}$ .

Therefore, the total frictional torque $M_{PRSM}$ can be defined as

$M_{PRSM} = M_{fx} + M_{kx} + M_{lx} + M_{dx}$ (2)

where x = s denotes the screw–roller interface and x = n denotes the nut–roller interface.

The frictional torque generated by elastic hysteresis

The frictional torque generated by elastic hysteresis on the screw–roller or nut–roller interface is equal to the summation of the frictional torques on each roller.

On the screw–roller interface

$M_{fs} = N_{0} \sum_{i = 1}^{τ} \frac{3}{8} γ B_{s} m_{bs} {(\frac{3 {E'}_{s}}{2 \sum ρ_{s}})}^{1 / 3} {N_{i}}^{4 / 3}$ (3)

and on nut–roller interface

$M_{fn} = N_{0} \sum_{i = 1}^{τ} \frac{3}{8} γ B_{n} m_{bn} {(\frac{3 {E'}_{n}}{2 \sum ρ_{n}})}^{1 / 3} {N_{i}}^{4 / 3}$ (4)

where $N_{0}$ is the number of rollers; $τ$ is the thread number of the roller; $γ$ is the energy loss coefficient; $B_{s}$ and $B_{n}$ denote the curvature coefficient of the screw–roller and nut–roller interfaces, respectively; $m_{bs}$ and $m_{bn}$ denote the dimensionless semi-minor coefficients of the contact ellipse on the screw–roller and nut–roller interfaces, respectively; $\sum ρ_{s}$ and $\sum ρ_{n}$ denote the curvature sums of the screw–roller and nut–roller interfaces, respectively; and $E'_{s}$ and $E'_{n}$ denote the effective Young’s moduli of the screw–roller and nut–roller interfaces, respectively.

The frictional torque generated by the spinning sliding of the rollers

The frictional torque generated by spinning sliding on the screw–roller or nut–roller interface is equal to the summation of the frictional torques on each roller. On the screw–roller interface, the component of the sliding frictional torque in the axial direction is calculated with a double integral, which can be expressed as

$M_{ks} = \sum_{i = 1}^{τ} \int_{- b_{si}}^{b_{si}} \int_{- a_{si} {(1 - y^{2} / {b_{si}}^{2})}^{1 / 2}}^{a_{si} {(1 - y^{2} / {b_{si}}^{2})}^{1 / 2}} N_{0} \cos β f_{h} \frac{3 N_{i}}{2 π a_{si} b_{si}} (1 - \frac{x_{2}}{{a_{si}}^{2}} - \frac{y^{2}}{{b_{si}}^{2}}) {(x^{2} + y^{2})}^{1 / 2} dxdy$ (5)

Similarly, on the nut–roller interface

$M_{kn} = \sum_{i = 1}^{τ} \int_{- b_{ni}}^{b_{ni}} \int_{- a_{ni} {(1 - y^{2} / {b_{ni}}^{2})}^{1 / 2}}^{a_{ni} {(1 - y^{2} / {b_{ni}}^{2})}^{1 / 2}} N_{0} \cos β f_{h} \frac{3 N_{i}}{2 π a_{ni} b_{ni}} (1 - \frac{x_{2}}{{a_{ni}}^{2}} - \frac{y^{2}}{{b_{ni}}^{2}}) {(x^{2} + y^{2})}^{1 / 2} dxdy$ (6)

where $f_{h}$ is the sliding coefficient of friction; $a_{s}$ and $b_{s}$ are the semi-major and semi-minor radii of the contact ellipse on the screw–roller interface, respectively; $a_{n}$ and $b_{n}$ are the semi-major and semi-minor radii of the contact ellipse on the nut–roller interface, respectively; and $β$ is the contact angle.

The frictional torque generated by the lubricant viscosity

The frictional torque generated by the lubricant viscosity is equal to the summation of the frictional torque components on each contact ellipse, which is defined below.

On the screw–roller interface

$M_{ls} = N_{0} \sum_{i = 1}^{τ} F_{vsi} R_{s}$ (7)

On the nut–roller interface

$M_{\ln} = N_{0} \sum_{i = 1}^{τ} F_{vni} R_{n}$ (8)

In these expressions, $F_{vs}$ and $F_{vn}$ are the hydrodynamic rolling force on the screw-roller interface and the nut-roller interface, respectively, and $R_{s}$ and $R_{n}$ are contact radius on the screw-roller interface and the nut-roller interface, respectively.

The frictional torque generated by the differential sliding of the rollers

In a PRSM, the linear velocities of the contact points are different between the rollers and the screw as well as between the rollers and the nut.⁸ Accordingly, pure rolling can only occur along two lines of contact area. At other points along the contact, sliding must occur in a direction parallel to rolling motion. Finally, differential sliding will occur.

On the screw–roller interface

$M_{ds} = N_{0} \sum_{i = 1}^{τ} F_{dsi} R_{s}$ (9)

On the nut–roller interface

$M_{dn} = N_{0} \sum_{i = 1}^{τ} F_{dni} R_{n}$ (10)

where $F_{ds}$ and $F_{dn}$ are the resistances to differential sliding on the screw–roller and nut–roller interfaces, respectively.

Heat transfer coefficient

The convective heat transfer coefficient $h$ (W/(m² °C)) is defined as

$h = N_{u} k_{fluid} / d$ (11)

where $k_{fluid}$ is the thermal conductivity of the ambient air and $d$ is the outer or inner diameter (mm) when convection occurs on the outer or inner surface, respectively. The Nusselt number, $N_{u}$ , is computed from the Reynolds number, $R_{e}$ , and the Prandtl number, $P_{r}$ , as shown below

$N_{u} = 0.133 {R_{e}}^{2 / 3} {P_{r}}^{1 / 3}$ (12)

$R_{e} = u_{liquid} d / v_{liquid}$ (13)

$P_{r} = c_{liquid} μ_{liquid} / k_{liquid}$ (14)

where $u_{liquid}$ is the velocity of the liquid, $v_{liquid}$ is the kinematic viscosity of the liquid, $c_{liquid}$ is the specific heat capacitance of the liquid, and $μ_{liquid}$ is the dynamic viscosity of the liquid.

Parameter calculation of the heat flux density and convection coefficient

The parameters of the PRSM used in the present study are listed in Table 1. The thermo-physical properties of the material are listed in Table 2. The thermo-physical properties and thermal conductivity parameters of the air calculated by equations (11)–(14) are shown in Tables 3 and 4, respectively.

Table 1.

Parameters for the PRSM used in the present study.

Parameter	Symbol	Unit	Value
Effective diameter of screw thread	d_s	mm	24
Start of screw and nut	n_s = n_n		5
Pitch	p	mm	5
Contact angle	β	°	45
Effective diameter of roller thread	d_r	mm	8
Effective diameter of nut thread	d_n	mm	40
Total thread number of roller	τ		20
Number of rollers	$N_{0}$		10
Rolling coefficient of friction	f_g		0.025
Loss coefficient of energy	γ		0.007
Young’s modulus	E_s, E_n, E_r	Pa	212 × 10⁹
Poisson’s ratio	µ_s, µ_n, µ_r		0.29

PRSM: planetary roller screw mechanism.

Table 2.

Thermo-physical properties of the material (GCr15).

Parameter	Density (kg/m³)	Specific heat capacity (kJ/(kg °C))	Thermal conductivity (W/(m °C))	Linear expansion coefficient (1/°C)
Value	7810	553	36.72	13.6 × 10⁻⁶

Table 3.

Thermo-physical properties of air.

Air	Thermal conductivity k, W/(m °C)	Kinematic viscosity v, m²/s	Prandtl number P_r
20°C	2.59 × 10⁻²	15.06 × 10⁻⁶	0.703
60°C	2.90 × 10⁻²	18.97 × 10⁻⁶	0.696
100°C	3.21 × 10⁻²	23.13 × 10⁻⁶	0.688
140°C	3.49 × 10⁻²	27.80 × 10⁻⁶	0.684
180°C	3.78 × 10⁻²	32.49 × 10⁻⁶	0.681
250°C	4.27 × 10⁻²	40.61 × 10⁻⁶	0.677

Table 4.

Thermal conductivity parameters of air.

Air	Reynolds number R_e	Nusselt number N_u	Heat transfer coefficient h, W/(m² K)
20°C	100,130	255	26.42
60°C	79,492	218	25.28
100°C	65,195	190	24.42
140°C	54,243	168	23.44
180°C	46,413	151	22.85
250°C	37,133	130	22.20

The major heat release is from the nut, and the main heat losses are through the outer surfaces of the components. In the FE model, the heat generation (as shown in equation (1)) must be converted to a heat flux density on the boundary conditions for better simulation accuracy, which is given as²¹

$Q_{x} = \frac{H_{x}}{A_{x}}$ (15)

where $Q_{x}$ is the heat flux density and $A_{x}$ is the cylinder outer surface area of the screw or nut threads.

Results and discussion

Model verification

To verify the accuracy and correctness of the results calculated with the FE model, a comparison was made between the FE model and the numerical model in Zhang et al.¹⁷ Zhang et al.¹⁷ developed a model for the load distribution over the threads according to the relationships of deformation compatibility and force equilibrium. The effects of the installation mode, load magnitude, and thread form parameter on the load distribution were studied. However, the temperature was not accounted for in Zhang et al.¹⁷ Therefore, the temperature is also not involved in this FE model to obtain an equivalent comparison.

The analysis parameters are identical to those in Zhang et al.,¹⁷ and the number of elements is 1,061,417 and the nodes is 707,494. The load on each thread can be obtained through the output interface provided by the ABAQUS software, which is based on the relationship between the contact stress and the contact area. For brevity, only the comparison results for installation modes (a) and (b) are shown in Figures 3 and 4.

Figure 3.

Results comparison for installation mode (a).

Figure 4.

Results comparison for installation mode (b).

Figures 3 and 4 show that the variation tendencies of the load distribution curves calculated by the FE model are similar to those in Zhang et al.¹⁷ The maximum deviation of the two models is less than approximately 18% and appears on the nut–roller interface, especially at small roller thread numbers. This is because the load distribution on the nut–roller interface is mainly determined by the structural stiffness, which is related to external diameter of the nut. However, the external diameter of the nut was not given in Zhang et al.¹⁷ Therefore, in this article, the external diameter of the nut is referenced from a GSA Group product catalog.²² Thus, data trends are well captured, and the proposed model can be used to further study the load distribution of the PRSM.

Load distributions on the roller threads for different working temperatures

For an analysis load setting of 30,000 N and a reference temperature of 20°C, temperature change examinations for 20°C, 40°C, 60°C, 80°C, 100°C, and 120°C are conducted. The load distributions on the roller threads for the different working temperatures and installation modes are shown in Figures 5 –8.

Figure 5.

Results for installation mode (a) for different temperatures.

Figure 6.

Results for installation mode (b) for different temperatures.

Figure 7.

Results for installation mode (c) for different temperatures.

Figure 8.

Results for installation mode (d) for different temperatures.

To facilitate the derivation, the following notations are employed: the solid icons denote the nut–roller interface and the hollow icons denote the screw–roller interface.

From Figures 5 –8, the following conclusions can be drawn:

As shown in Figures 5 and 8, the varying tendency of the load distribution for installation mode (a) is similar to that of installation mode (d). For the 120°C case, on the nut–roller interface, the maximum load (the corresponding thread number is 20) increases 20% from 263.72 N for installation mode (a) to 329.70 N for installation mode (d). However, the maximum load (the corresponding thread number is 1) varies little over the screw–roller interface and only decreases 1.7% from 390.92 to 384.26 N. Similar conclusions are drawn from Figures 6 and 7. The influences of the temperature changes on the load distribution on the nut–roller interface are clearly larger than those on the screw–roller interface over the four installation modes. This is because the external diameter of the nut structure determines the stiffness (or deformation) of the nut. For the same load magnitude, the resulting deformations caused by the temperature on the screw or nut threads are the key reason behind the load distribution change.

When the screw (or nut) is subjected to tension, the influences of the temperature changes on the unevenly distributed loads will increase, and the larger loads move to the fixed end. In contrast, the larger loads deviate from the fixed end when the screw (or nut) is subjected to compression, and the influence of the temperature change on the load distribution is relatively limited. These behaviors are determined by the constraint conditions (as shown in Figure 2).

By comparing Figures 5 –8, it can be seen that a proper temperature rise will decrease the uneven load for installation mode (c). In particular, the load distributions for installation mode (c) are more even for the other installation modes with the same temperature increase. Therefore, installation mode (c) is a good choice for engineering applications.

Load distributions on the roller threads for different screw rotational speeds

The load distributions on the roller threads vary with the screw rotational speeds, as shown in Figures 9 –12. The analysis setting is 30,000 N, the reference temperature is 20°C, and the screw rotational speeds are 600, 700, 800, 900, and 1000 r/min. The frictional torques, heat flux densities, and heat transfer coefficients of the air for different screw rotational speeds are shown in Tables 5 and 6.

Figure 9.

Results for installation mode (a) for different rotational speeds.

Figure 10.

Results for installation mode (b) for different rotational speeds.

Figure 11.

Results for installation mode (c) for different rotational speeds.

Figure 12.

Results for installation mode (d) for different rotational speeds.

Table 5.

Heat flux density of the PRSM for different screw rotational speeds.

Rotational speed of screw (r/min)	600	700	800	900	1000
M_PRSM (N mm)	15,430	15,430	15,431	15,431	15,432
H_PRSM (W)	970	1131	1293	1455	1616
Q_s (W/m²)	96,054	112,066	128,080	144,094	160,109
Q_n (W/m²)	75,081	87,598	100,115	112,633	125,151

PRSM: planetary roller screw mechanism.

Table 6.

Heat transfer coefficient of air for different screw rotational speeds.

Heat transfer coefficient h, W/(m² K)	600 r/min	700 r/min	800 r/min	900 r/min	1000 r/min
20°C	26.42	29.28	32.01	34.62	37.14
60°C	25.28	28.01	30.62	33.12	35.53
100°C	24.42	27.06	29.58	32.00	34.33
140°C	23.44	25.98	28.40	30.72	32.95
180°C	22.85	25.32	27.68	29.94	32.12
250°C	22.20	24.60	26.89	29.09	31.21

From Figures 9 –12, the following conclusions can be drawn:

The varying tendencies of the load distribution over the four installation modes are similar to those in Figures 5 –8. As shown in Figure 9, the maximum loads (the corresponding thread number is 1) at 1000 r/min on the screw–roller interface and on the nut–roller interface for installation mode (a) are 584.66 and 464.16 N. As shown in Figure 12, the maximum loads (the corresponding thread number is 1) at 1000 r/min on the screw–roller interface and on the nut–roller interface for installation mode (d) are 583.57 and 424.38 N, respectively. It is clear that the influence of the rotational speed on the load distribution for installation mode (d) is greater than that for installation mode (a). Similarly, the influence of the rotational speed on the load distribution for installation mode (b) is greater than that for installation mode (c).

When the load is 30,000 N and the screw rotational speeds vary from 600 to 1000 r/min, the phenomenon of threads separating from the meshing occurs. When the screw rotational speed is 1000 r/min, 9 threads (Figures 10 and 11) and 10 threads (Figures 9 and 12) separated from the meshing. This is almost half of the threads that are out of the meshing, indicating that there is no contact between the roller threads and the screw or nut threads. This phenomenon also indicates that the frictional heat caused by the screw rotational speed cannot be ignored and has an important influence on the load distribution.

The influence of the installation mode has a limited effect on the number of threads that separate from the meshing but affects the position of the separating threads. As shown in Figures 9 and 12, for installation modes (a) and (d), the positions of the threads that separated from the meshing are located at the back end. Similarly, the positions of the threads that separated from the meshing are located at the fore end for installation modes (b) and (c).

The uneven distribution of the loads increases with screw rotational speed. The influences of the screw rotational speed on the load distribution over the screw–roller interface are larger than those on the nut–roller interface over the four installation modes.

Load distributions on the roller threads under different working loads

The load distributions on the roller threads for varying load magnitudes are shown in Figures 13 –16. The analysis setting is 600 r/min, the reference temperature is 20°C, and the load magnitudes are 10,000, 15,000, 20,000, 25,000, and 30,000 N. The frictional torques and heat flux densities for the different load magnitudes are shown in Table 7.

Figure 13.

Results for installation mode (a) for different load magnitudes.

Figure 14.

Results for installation mode (b) for different load magnitudes.

Figure 15.

Results for installation mode (c) for different load magnitudes.

Figure 16.

Results for installation mode (d) for different load magnitudes.

Table 7.

Heat flux density of the PRSM under different load magnitudes.

Load magnitudes (N)	10,000	15,000	20,000	25,000	30,000
M_PRSM (N mm)	5132	7703	10,276	12,852	15,430
H_PRSM (W)	323	484	646	808	970
Q_s (W/m²)	31,943	47,950	63,972	80,007	96,054
Q_n (W/m²)	24,969	37,481	50,004	62,538	75,081

PRSM: planetary roller screw mechanism.

From Figures 13 –16, the following conclusions can be drawn:

Compared to Figures 9 –12, similar conclusions can be drawn. Importantly, the load distributions on the screw–roller interface and nut–roller interface are nearly the same for installation mode (c) for different load magnitudes.

When the rotational speed of the screw is 600 r/min and the load magnitude varies from 10,000 to 30,000 N, the number of threads that have separated from the meshing is always 5–7 over the four installation modes. The influences of the installation mode and load magnitude have a limited effect on the number of threads that separate from meshing, and the positions of the threads that separate from the meshing are similar to those in section “Load distributions on the roller threads for different screw rotational speeds.” Compared to section “Load distributions on the roller threads for different screw rotational speeds,” the influence of the screw rotational speed on the load distribution is larger than that of the load magnitude. This is because the frictional torque and heat flux density increase quickly with the rotational speed.

The load distribution trend for installation mode (a) with different load magnitudes is similar to that for installation mode (d); the load is mainly borne by the 12 fore threads, and nearly 40% of the load is borne by the 4 fore threads. This occurs because the four fore threads are closer to the nut flange, and the stiffness of the shaft sections is less than that of other subsequent threads. Similarly, the same results are observed for installation modes (b) and (c).

Load distributions on the roller threads for different internal and external temperatures

A discrepancy in the load distribution caused by the temperature difference between the internal and external structures must occur during motion. However, the PRSM is a closed system, which makes it difficult to study the difference directly when it is moving. Therefore, to reveal the influence of the internal and external temperatures on the load distribution, this section examines results from individually changing either the internal or external temperature to study the load distribution as a function of temperature.

Load distributions variation with internal temperature changes

For this analysis, a rotational speed of 600 r/min is used, the temperature of the outer surfaces of the nut and end surfaces of the screw and roller is 20°C, the load magnitude is 30,000 N, and the internal temperature variations on the thread surfaces are 20°C, 40°C, 60°C, 80°C, 100°C, 120°C, and 140°C. The results are shown in Figures 17 –20.

Figure 17.

Results for installation mode (a) with varying internal temperatures.

Figure 18.

Results for installation mode (b) with varying internal temperatures.

Figure 19.

Results for installation mode (c) with varying internal temperatures.

Figure 20.

Results for installation mode (d) with varying internal temperatures.

From Figures 17 –20, the following conclusions can be drawn:

The uneven load distribution increases with the internal temperature for each of the installation modes and the uneven distribution of the nut–roller interface is similar to that of the screw–roller interface.

The phenomenon of threads separating from the meshing occurs when the internal temperatures are greater than 100°C. The load distribution tendencies are similar for installation modes (a) and (d), and the numbers of threads that separated from the meshing for these two modes are 6 and 5, respectively, when the internal temperature reaches 140°C. Similarly, the load distribution tendencies are similar for installation modes (b) and (c), and the numbers of separated threads are 4 and 6, respectively.

As the internal temperature increases, the loads move to the fixed end for installation modes (a) and (d) and to the back end for installation modes (b) and (c).

Load distributions variation with external temperature changes

For this analysis, a rotational speed of 600 r/min is used; the temperature variations of the outer surfaces of the nut and end surfaces of the screw and roller are 140°C, 120°C, 100°C, 80°C, 60°C, 40°C, and 20°C; the load magnitude is 30,000 N; and the internal temperature of the thread surfaces is 140°C. The results of the load distribution variations with changes to the external temperature are shown in Figures 21 –24.

Figure 21.

Results for installation mode (a) with varying external temperatures.

Figure 22.

Results for installation mode (b) with varying external temperatures.

Figure 23.

Results for installation mode (c) with varying external temperatures.

Figure 24.

Results for installation mode (d) with varying external temperatures.

From Figures 21 –24, the following conclusions can be drawn:

By comparing with Figures 17 –20, similar tendencies for the load distributions over the four installation modes are obtained. The results indicate that the difference in the external temperature is the main factor behind the variations in the load distribution, especially on the nut–roller interface.

On the screw–roller interface, the influences of the internal temperature variations on the load distribution are larger than those of the external temperature variations. Conversely, for the nut–roller interface, the influences of the external temperature variations on the load distribution are larger than those of the internal temperature variations. This occurs because the difference in the temperature caused by the internal temperature has an important influence not only on thermal deformations of the screw, roller, and nut threads but also on their structural deformations. Nevertheless, the external temperature difference mainly influences the thermal deformation of the nut, which has a slight effect on the screw and the roller threads because the structural size of the nut in the radial direction is larger than those of the screw and roller.

To obtain a uniform load distribution, the working temperature and the temperature caused by the frictional torques must be controlled for engineering applications.

Conclusion

This study uses a 3D FE model to simulate the load distribution characteristics of a PRSM under thermo-mechanical coupling conditions. From the results, some conclusions are drawn as follows:

Under the effect of thermo-mechanical coupling, the temperature changes will cause the load distribution to be more uneven when the screw (or nut) is subjected to tension and the heavier loads move to the fixed end. When the screw (or nut) is subjected to compression, the heavier loads move from the fixed end, and the influences of the temperature changes on the load distribution are minimal. In addition, the influences of the temperature changes on the load distribution on the nut–roller interface are larger than those on the screw–roller interface for the four installation modes, and installation mode (c) has a more even load distribution.

When the load magnitude and screw rotational speed are taken into account, their influences on the load distribution can be ranked from greatest to smallest as installation modes (d), (a), (b), and (c). The phenomenon of threads separating from the meshing occurs, which indicates that the frictional heat caused by the screw rotational speed cannot be ignored. The influences of the screw rotational speed and load magnitude on the load distribution on the screw–roller interface are larger than those on the nut–roller interface over the four installation modes.

The uneven load distribution increases with the internal and external temperatures over the four installation modes, and the uneven distribution of the nut–roller interface is higher than that of the screw–roller interface. The influence of the external temperature on the load distribution of the nut–roller interface is larger than that of the internal temperature. The separation of the threads from the meshing occurs at the back end for installation modes (a) and (d). In contrast, the separation of the threads from the meshing occurs at the fore end for installation modes (b) and (c).

Footnotes

Handling Editor: Jose Ramon Serrano

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

The author(s) disclosed receipt of the following financial support for the research,authorship,and/or publication of this article: The research is supported by National Natural Science Foundation of China (grant no. 51505381,51275423) and the 111 Project (grant no. B13044).

ORCID iD

Shangjun Ma

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http://www.gewindeziegler.ch/

Thermo-mechanical coupling–based finite element analysis of the load distribution of planetary roller screw mechanism

Abstract

Keywords

Introduction

FEA model

Model assumptions

Elements and boundary conditions

Installation modes

Heat production in the PRSM

The frictional torque generated by elastic hysteresis

The frictional torque generated by the spinning sliding of the rollers

The frictional torque generated by the lubricant viscosity

The frictional torque generated by the differential sliding of the rollers

Heat transfer coefficient

Parameter calculation of the heat flux density and convection coefficient

Results and discussion

Model verification

Load distributions on the roller threads for different working temperatures

Load distributions on the roller threads for different screw rotational speeds

Load distributions on the roller threads under different working loads

Load distributions on the roller threads for different internal and external temperatures

Load distributions variation with internal temperature changes

Load distributions variation with external temperature changes

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References