Dynamics of the Planetary Roller Screw mechanism is independent of this slip. Sokolov et al.

[3] derive some

of the same kinematic relationships as Ref. [2] and present some
Mechanism additional geometric relationships. Ryakhovsky et al. [4] discuss
more geometric relationships, including the necessary spur/ring gear
ratio to attempt to eliminate slip at the nut/roller interface.
Matthew H. Jones Jones and Velinsky [5] investigate the nature of the contact
Department of Mechanical and Aerospace Engineering, between the load transferring surfaces in the RSM, i.e., between
University of California–Davis, the screw and roller threads and between the nut and roller
threads. This was the first work to take a fundamental approach to
One Shields Avenue,
understanding the kinematics of the contact between these compo-
Davis, CA 95616 nents; it allows for detailed analysis of aspects such as contact
e-mail: mhjones12@gmail.com mechanics, friction, lubrication, and wear. Also, the geometry of
the contact is necessary to accurately account for the proper sys-
Steven A. Velinsky1 tem kinematics for dynamic analysis. These same authors exam-
Fellow ASME ined pitch mismatch and its effects in the RSM [6] and also used
Department of Mechanical and Aerospace Engineering, the direct approach to determine the stiffness of the RSM [7].
Various other aspects of the RSM have been studied such as
University of California–Davis,
research on efficiency and failure modes [8], dynamical load test-
One Shields Avenue, ing [9], force, slip, and lead properties [10], a calculation method
Davis, CA 95616 for the elastic elements [11], principles for evaluating wear resist-
e-mail: savelinsky@ucdavis.edu ance [12], a study on the effects of wet and dry lubrications under
oscillatory motion [13], an investigation into the static rigidity
Ty A. Lasky and thread load distribution [14], and a study on the heat gener-
Department of Mechanical and Aerospace Engineering, ated by friction in the RSM and its effects [15].
In previous work, the general dynamic equations of motion of
University of California–Davis,
this mechanism have not been developed. As such, the current
One Shields Avenue, paper develops the dynamic equations of motion for the main
Davis, CA 95616 components in the PRSM, i.e., the screw, rollers, and nut bodies.
e-mail: talasky@ucdavis.edu The equations of motion are derived by using Lagrange’s Method
with viscous friction [16]. While the equations allow for an under-
standing of transient operation, RSMs are mostly employed in
steady-state operation. As such, herein, the steady-state angular
This paper develops the dynamic equations of motion for the plan-
velocities and screw/roller slip velocities of the mechanism will
etary roller screw mechanism (PRSM) accounting for the screw,
be derived and used to understand the sensitivities of some design
rollers, and nut bodies. First, the linear and angular velocities
parameters. The intent is for the current work to allow for optimal
and accelerations of the components are derived. Then, their
design of this invaluable mechanism in the future.
angular momentums are presented. Next, the slip velocities at the
contacts are derived in order to determine the direction of
the forces of friction. The equations of motion are derived through Dynamics
the use of Lagrange’s Method with viscous friction. The steady- In the derivation herein, the PRSM is assumed to be under a
state angular velocities and screw/roller slip velocities are also typical mode of operation, i.e., converting rotational motion into
derived. An example demonstrates the magnitude of the slip veloc- linear motion. Specifically, it is assumed that the screw is the
ity of the PRSM as a function of both the screw lead and the screw input entity, which is allowed to rotate about its axis and not trans-
and nut contact angles. By allowing full dynamic simulation, the late, and the nut is the output entity, which is allowed to translate
developed analysis can be used for much improved PRSM system axially and not rotate. In addition, as in Ref. [5], the roller screw
design. [DOI: 10.1115/1.4030082] is loaded such that the surfaces in contact are, (1) the top of the
screw thread, (2) the bottom of the roller thread on the screw side,
Introduction (3) the top of the roller thread on the nut side, and (4) the bottom
of the nut thread.
The roller screw mechanism (RSM) [1] is a mechanical trans-
For all following formulations, three simplifying assumptions are
mission device for converting rotary motion to linear motion or
made: (1) the mass and rotational inertia of each roller are equal, (2)
vice versa (see Fig. 1).2 There are two primary types of RSMs—
the rollers have the same angular velocity about their center axis, and
the PRSM and the recirculating RSM. The analysis in this paper is
(3) the potential energies in the mechanism are ignored. In addition,
applied to the PRSM. Compared to ball screw mechanisms, RSMs
the carrier plate enforces that all rollers (denoted by the jth roller)
are capable of higher loads, longer life, higher speeds, higher
have the same orbital angular velocity (i.e., h_R ¼ h_Rj for all j).
accelerations, and finer leads, though RSMs typically have lower
All vectors are relative to the world-fixed coordinate system,
efficiency. Thus, RSMs are used in a variety of high-load and
n^1 n^2 n^3 , which is fixed at the base of the center axis of the screw.
high-speed applications.
To allow for a clearer representation, unless otherwise noted, the
Despite the importance and usefulness of the RSM, there has been
vectors are expressed in coordinate system c^1j c^2j c^3j which is fixed
little fundamental research to support its engineering application. For
at the same point as the world-fixed coordinate system but
purposes of brevity, the literature review to follow is quite concise.
allowed to rotate such that c^1j is always pointing towards the cen-
These authors have studied various aspects in previous work and the
ter axis of the jth roller (see Fig. 2). The coordinate systems are
interested reader is referred to that work for more thorough literature
related by the transformation matrix T j as
review. Velinsky et al. [2] examine the effects of slip at the screw/ 8 9 2 38 9 8 9
roller interface on the kinematics of the PRSM. Their results show < n^1 = ChRj ShRj 0 < c^1j = < c^1j =
that some slip must always occur and that the lead of the overall n^2 ¼ 4 ShRj ChRj 0 5 c^2j ¼ T j c^2j (1)
: ; : ; : ;
n^3 0 0 1 c3j c3j
1
Corresponding author.
2
www.rollvis.com
n^3 and c^3j are equal and thus interchangeable. hRj is the orbital
Manuscript received November 3, 2014; final manuscript received March 2, angle of the jth roller about n^3 . In Eq. (1) and throughout the
2015; published online August 18, 2015. Assoc. Editor: James Schmiedeler. paper, C and S denote cosine and sine, respectively.

Journal of Mechanisms and Robotics Copyright V

C 2016 by ASME FEBRUARY 2016, Vol. 8 / 014503-1

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

 Here, rPo PRj , the vector locating the jth roller center of mass with
respect to the screw’s axis, is given by

rPo PRj ¼ ðn rR þ n rS Þ^
c1j (6)

and hr is the angle of the roller about the roller center axis meas-
ured relative to a fixed vertical. LR denotes the lead of the roller
and hR is the orbital angle of the roller. Here, n rR and n rS are the
nominal contact radius of the roller and screw. Also, xC is the
angular velocity of the carrier plate, which is also the orbital
velocity of the rollers, and is given by
Fig. 1 Exploded view of roller screw and the relevant
components xC ¼ h_R c^3j (7)

With these, the velocity of point PRj becomes

Velocities and Accelerations. The screw is only allowed to 8 9
> 0 >
rotate about its center axis, and thus its angular velocity xS and >
> >
>
>
> >
>
acceleration aS are < =
VPRj ¼ ðn rR þ n rS Þh_R (8)
>
> >
>
xS ¼ h_S c^3j ; aS ¼ h€S c^3j (2) >
>
>
>
>
:  LS h_ þ LS  LR h_  LR h_ > ;
S R r
2p 2p 2p
respectively, where hS is the angle of the screw measured relative
to a fixed vertical, and the dot superscript ðÞ denotes differentia- The acceleration aPRj of the center of mass of the roller PRj is
tion with respect to time. The nut is constrained from rotating and written as
thus the linear velocity VPN and acceleration aPN of the nut’s cen-
ter of mass is given by 8 9
>
> ðn rR þ n rS Þh_2R >
>
>
> >
>
>
> >
>
h_S h€S dV PRj <
€
=
VPN ¼ LS c^3j ; aPN ¼ LS c^3j (3) aPRj
¼ C
þx V PRj
¼ ðn
rR þ n
rS Þh R
2p 2p dt > >
>
> >
>
>
> LS € LS  LR € LR € >
>
where LS denotes the lead of the screw thread. >
:  hS þ hR  hr > ;
The acceleration aPRj of the jth roller can be formulated accord- 2p 2p 2p
ing to Fig. 3. Point Po moves along the center axis of the screw (9)
and is always coplanar with the center of mass of the jth roller,
PRj . The velocity VPRj of the center of mass of the jth roller can where the derivative of VPRj is taken with respect to the c^1j c^2j c^3j
then be represented by coordinate system. The angular velocity xRj and acceleration aRj
of the jth roller with respect to its axis of rotation are, respectively,
VPRj ¼ VPo þ xCj  rPo PRj (4)
xRj ¼ h_r c^3j ; aRj ¼ h€r c^3j (10)
where the velocity VPo of point Po is found to be
  Angular Momentum. For our case, the coordinates are chosen
Po LS _ LS  LR _ LR _ such that the axes are along the principal directions of the bodies,
V ¼  hS þ hR  hr c^3j (5)
2p 2p 2p thus simplifying the determination of the angular momentum. For
the screw, the angular momentum HS=Po about point Po is
therefore,

HS=Po ¼ HS=PS ¼ IS3 h_S c^3i (11)

where IS3 denotes the moment of inertia of the screw, and PS is
the center of mass of the screw. Since it is constrained from rotat-
ing, the nut’s angular momentum HN=Po about point Po is

Fig. 3 Velocity of center of mass of roller (a) in-plane and (b)

Fig. 2 Coordinate systems and contact location axially

014503-2 / Vol. 8, FEBRUARY 2016 Transactions of the ASME

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

 HN=Po ¼ HN=PN ¼ 0 (12)

where the point PN is the nut’s center of mass. The angular
momentum HRj=PRj of the jth roller about its center of mass is

HRj=PRj ¼ IR h_r c^3j (13)

where IR is the moment of inertia of the jth roller. The moment of

inertia of the screw and roller body can be estimated by

mS n rS2 mR n rR2
IS ¼ ; IR ¼ (14)
2 2

where mR and mS are the mass of the roller and screw,

respectively.

Slip Velocity at Screw/Roller Interface. The slip velocities

developed in Ref. [6] were pure kinematic relations assuming no
planar slip at the screw/roller interface. This section develops the
general slip velocity at the screw/roller interface under no Fig. 4 Screw/roller in-plane slip velocity diagram
assumptions of the rate of slip. This is necessary in order to deter-
mine the direction of the friction force, since this force always
opposes the relative motion between two bodies in contact. The VPSj PRS j ¼ VPSj  VPRS j (19)
screw contact (point PSj ) velocity VPSj at the jth screw/roller inter-
face (see Fig. 4) is written as
which leads to
V PSj
¼x r S PS PSjk
¼ rS ShCS h_S c^1j þ rS ChCS h_S c^2j (15) 8 9
>
> rS ShCS h_S þ rRS ShCRS h_r >
>
>
> >
>
where the contact vector rPS PSjk going from the center of mass of >
> >
>
< =
the screw, PS , to the kth screw/roller thread contact of the jth VPSj PRS j ¼ rS ChCS h_S  ðn rR þ n rS Þh_R  rRS ChCRS h_r (20)
roller, PSjk , is given by >
> >
>
>
> >
>
>
> LS _ LS  LR _ LR >
>
: hS  hR þ h_r ;
rPS PSjk ¼ rS ChCS c^1j þ rS ShCS c^2j þ ðÞ^
c3j (16) 2p 2p 2p

and rS is the radius to the contact point on the screw thread. Here,
hCS is the in-plane angle to the contact location on the screw Lagrange’s Method. The Lagrangian Method [16] is used to
thread. The third term in Eq. (16) has been left undefined since its formulate the equations of motion herein. The kinetic energy of
value is inconsequential for the following analysis. The roller the screw TS and the nut TN is given as
contact velocity VPRS j at the jth screw/roller interface is likewise
written as 1 mm L2S _2
TS ¼ IS h_2S ; TN ¼ h (21)
2 8p2 S
PRS j PRj Rj PRj PRS jk
V ¼V þx r
8 9 where mm is the moved mass, including the mass of the nut and
>
> rRS ShCRS h_r >
>
>
> >
> all attached masses. The roller has both rotational and linear
>
> >
>
>
> >
> kinetic energies, and its kinetic energy TR can be expressed as
< n =
ð r þ n
r Þ h_ þ r C h_
¼ R S R R S hCR S
r (17)
> >
>
>
>
>
>
> 1 1 h
>
> >
> TR ¼ IR h_2r þ mR ðn rR þ n rS Þ2 h_2R
>
> L S L S  L R _hR  h_r >
L R > 2 2
:  h_S þ ;   #
2p 2p 2p LS _ LS  LR _ LR _ 2
þ  hS þ hR  hr (22)
2p 2p 2p
Here, hCRS is the in-plane angle to the contact location on the
roller thread [5]. The second subscript S on the subscript R
denotes that this parameter is associated with the screw side.
rPRj PRS jk , the contact vector from the center of mass of the jth Finally, the kinetic energy TC of the carrier plate is expressed as
roller to the screw roller contact on the kth thread, is written as 1
8 9 TC ¼ IC h_2R (23)
rRS ChCRS 2
>
< >
=
rRS ShCRS
rPRj PRS jk ¼ (18) where IC is the rotational inertia of both carrier plates. The total
>
: Pðk  1Þ þ Z ‘R >
;
RS  system kinetic energy Ttotal is then given as
2

Here, rRS is the radius to the contact point on the roller threads on Ttotal ¼ TS þ TN þ nR TR þ TC (24)
the screw side. P is the pitch (equal for roller, nut, and screw). ZRS
is the height of first thread contact on the screw side, and ‘R is the
length of the roller [5]. where nR is the number of rollers. Since the potential energies are
The slip velocity VPSj PRS j at the jth screw/roller interface is assumed negligible, the Lagrangian L for the system is equal to
then defined as the total kinetic energy, i.e.,

Journal of Mechanisms and Robotics FEBRUARY 2016, Vol. 8 / 014503-3

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

  
1 mm L2S _2 The force of friction FfR on the roller is given as
L ¼ Ttotal ¼ IS þ hS
2 4p2 2 3
1h i 1 rS ShCS h_S  rRS ShCRS Gh_R
þ IC þ nR mR ðn rR þ n rS Þ2 h_2R þ nR IR h_2r 6   7
2 2 6 7
 2 6 rS Ch h_S  n rR þ n rS þ rR Ch G h_R 7
FfR ¼ FfS ¼ lnR nt 6 CS S CRS 7
1 LS _ LS  LR _ LR 6 7
þ nR mR  hS þ hR  h_r (25) 4 5
2 2p 2p 2p LS _ LS  LR ðG þ 1Þ _
hS  hR
2p 2p
Incorporation of Friction. Viscous friction forces are included (32)
through the use of the generalized forces. Friction will only be
incorporated at the screw/roller interface since it is the most criti- The generalized force NR for the hR degree of freedom is given as
cal friction interface. This is because the orbital motion of the
roller is imparted by the screw purely through a friction interac- @VPRS j
NR ¼ FfR  (33)
tion. The nut/roller interface has a kinematic interface, which min- @ h_R
imizes slip, as investigated in Ref. [6], and thus the friction is which results in
ignored.
The spur/ring gear pair enforces   
NR ¼ lnR nt rS rRS G ChCS ChCRS  ShCS ShCRS þ rS ChCS ðn rR þ n rS Þ
G N  GR _   
h_r ¼ hR ¼ Gh_R (26) LS  LR ðG þ 1Þ _
GR þ LS hS  ln R n t rR2 S G2 þ ðn rR þ n rS Þ2
4p2
where GN and GR are the ring and spur gear pitch circle radii,   #
n n LS  LR ðG þ 1Þ 2 _
respectively, and G ¼ ðGN  GR Þ=GR . þ2rRS ChCRS Gð rR þ rS Þ þ hR (34)
The kinematic constraint at the nut/roller interface in Eq. (26) 2p
can be incorporated directly into the Lagrangian to give
  The equations of motion become
1 ðmm þ nR mR ÞL2S _2 1     
L ¼ IS þ hS þ IC þ nR IR G2 ðmm þ nR mR ÞL2S € LS  LR ðG þ 1Þ €
2 4p2 2 IS þ h  n m L hR ¼ NS
"   #) 4p2
S R R S
4p2
LS  LR ðG þ 1Þ 2
n n
þ nR mR ð rR þ rS Þ þ2
h_2R (35)
2p  
  LS  LR ðG þ 1Þ € 
LS  LR ðG þ 1Þ _ _  n R mR L S hS IC þ nR IR G2
 nR mR LS hS hR (27) 4p2
4p2 "   #)
LS  LR ðG þ 1Þ 2
n n
þ nR mR ð rR þ rS Þ þ 2
h€R ¼ NR (36)
The friction is modeled as viscous friction opposing the direc- 2p
tion of slip and proportional to the slip velocity at the screw/roller
interface. Incorporating the nut/roller kinematic constraint yields
The above equations can be used for accurate system simulation
the friction force FfS on the screw as
including transient operation and determination of corresponding
forces, etc. However, the steady-state solution is of particular in-
FfS ¼ lnR VPSj PRS j ¼ lnR nt
2 3 terest since most of the PRSM’s motion is typically in this mode.
rS ShCS h_S þ rRS ShCRS Gh_R The equations of motion in steady state become
6 7 
6 Fout LS
6   77 sin   lnR nt rS2 h_S þ lnR nt rS ChCS ðn rR þ n rS Þ
6 _ n n _ 7 (28) 2p
 6 rS ChCS hS þ rR þ rS þ rRS ChCRS G hR 7  
6 7
6 7 þ rS rRS G ChCS ChCRS  ShCS ShCRS h_R ¼ 0 (37)
4 5
LS _ LS  LR ðG þ 1Þ _
 hS þ hR
2p 2p   
where nt is the number of threads on the roller, and l is the coeffi- lnR nt rS rRS G ChCS ChCRS  ShCS ShCRS þ rS ChCS ðn rR þ n rS Þ
cient of friction. For external torque sin , the torque sS acting on   "
the screw and the output force Fout are given as LS  LR ðG þ 1Þ _
þ LS hS  lnR nt rR2 S G2 þ ðn rR þ n rS Þ2
4p2
sS ¼ sin c^3j ; Fout ¼ Fout c^3j (29)   #
n n LS  LR ðG þ 1Þ 2 _
þ 2rRS ChCRS Gð rR þ rS Þ þ hR ¼ 0
This gives the generalized force NS for the hS degree of freedom 2p
as
(38)
@VPSj @VPN @xN Equations (37) and (38) are two equations with four unknowns.
NS ¼ FfS  þ Fout  þ sin  (30)
@ h_S @ h_S @ h_S Thus, there can be two inputs to the system and two outputs. By
defining steady-state constants ASS , BSS , CSS , and DSS , the equa-
where xN is the nut angular velocity. Carrying out the operations tions can be put into the form
yields Fout LS
 ASS h_S þ BSS h_R ¼ sin  ; CSS h_S þ DSS h_R ¼ 0 (39)
2p
NS ¼ sin  lnR nt rS2 h_S þ lnR nt rS ChCS ðn rR þ n rS Þ
and any two outputs can be easily solved for in terms of the other
 
þ rS rRS G ChCS ChCRS  ShCS ShCRS h_R 
Fout LS
(31) two. The steady-state screw and roller angular velocities h_SS
S and
2p h_SS
R can be represented as

014503-4 / Vol. 8, FEBRUARY 2016 Transactions of the ASME

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

 DSS ð2psin  Fout LS Þ CSS ð2psin  Fout LS Þ
h_SS
S ¼ ; h_SS
R ¼
2pðASS DSS  BSS CSS Þ 2pðASS DSS  BSS CSS Þ
(40)
PSj PR j
The steady-state slip velocity VSS S at the screw/roller inter-
face can be determined by inserting the steady-state angular
velocities to yield

PSj PRS j ð2psin  Fout LS Þ

VSS ¼
2pðASS DSS  BSS CSS Þ
8 9
>
> rS ShCS DSS  rRS ShCRS GCSS >
>
>
> >
>
>
> >
>
>
>   >
>
< n n
=
 rS ChCS DSS þ rR þ rS þ rRS ChCRS G CSS
>
> >
>
>
> >
>
>
> >
>
>
> LS LS  LR ðG þ 1ÞCSS >
>
: DSS þ ;
2p 2p
Fig. 6 Steady-state slip velocity magnitude
(41)

the ratio of these angular velocities, yields an angular velocity

ratio of 2.667. This suggests that the orbital angular velocity of
Example. The equations of motion developed above can be
the roller in the dynamic system is slightly slower than the ideal
used to predict the transient operation of PRSM systems. How-
(slip-free) kinematic angular velocity, which is to be expected. It
ever, PRSM systems vary considerably based on their application
is worth pointing out that this kinematic relationship was not
area and the mechanism is often operated in the steady-state
incorporated in the dynamic model. That is, the convergence of
regime. Herein, results are used to show the relatively short tran-
the dynamic system to the ideal kinematic relationship helps to
sient period and the convergence to steady-state operation. More-
verify the accuracy of the dynamic model. Note that the dashed
over, based on the variations in PRSM systems and their associated
horizontal line in Fig. 5 is the ideal kinematic ratio. Moreover,
dynamic requirements, it is much more useful to base design of the
since the steady-state angular velocity ratio both quickly and
PRSM on its steady-state response. In the following example, the
closely approaches the ideal kinematic ratio, the use of a purely
nominal parameters for the roller screw are n rS ¼ 19.5 mm, n rR ¼ 5
kinematic model, such as in Ref. [2], for relating angular positions
mm, GR ¼ rRN , GN ¼ rN , nS ¼ 5, nR ¼ 10, nt ¼ 40, l ¼ 25 N s/m,
and velocities of the screw and roller bodies, as well as efficiency
mm ¼ 3000 kg, h_SSS ¼ 1 rad/s, and Fout ¼ 100 N. analysis, is validated.
First, the transient behavior of the mechanism has been simu-
Concerning steady-state analysis, the magnitude of the slip ve-
lated with a step torque input at the screw. Figure 5 shows a result
locity of Eq. (41) can be studied as various parameters are varied,
of this simulation. Specifically, the ratio of angular velocities is
as shown in Fig. 6. As can be seen, the magnitude of the slip ve-
shown, and steady-state is reached in approximately 0.2 s. More-
locity increases as the lead increases and as the contact angle
over, the result can be compared to the kinematics of Ref. [6].
decreases. This is expected, as the slip velocity is the sole source
With LS ¼ 5 mm and screw and nut contact angles
of dissipation in the developed model and the results correlate
bS ¼ bN ¼ 45 deg, the steady-state angular velocity ratio is
well with the efficiency results of Ref. [2]. That is, as the slip
given as
velocity magnitude increases, the efficiency decreases.
h_SS
S DSS The analysis developed herein can be used to determine the
¼ ¼ 2:669 (42) forces within the mechanism itself with the force analysis pro-
h_SS
R
CSS
vided in Ref. [17]. For purposes of brevity, that force analysis has
been omitted from this work. However, the use of force analysis
In comparison, Eq. (5) in Ref. [6], which is a kinematic rela-
in combination with this work provides significant additional
tionship that assumes no planar slip and is therefore the limit on
insight into the design of these valuable mechanisms. For exam-
ple, its use shows that the force on the spur gear per unit torque
decreases with increasing screw lead and decreasing contact
angle. As such, the use of this work has the potential to allow for
much improved and optimal design of the PRSM.

Conclusion
This paper developed the equations of motion for the main
components in the PRSM, i.e., the screw, rollers, and nut bodies.
First, the linear and angular velocities and accelerations of the
components were derived. Then, their angular momentum was
presented. Next, the slip velocities at the contacts were derived in
order to determine the direction of the forces of friction. The
equations of motion were derived through the use of Lagrange’s
Method with viscous friction. Steady-state angular velocities and
screw/roller slip velocities were also derived. The developed
theory can be used for accurate simulation of roller screw systems
through transient and steady-state motions and additionally allows
for determination of internal forces and stresses.
Fig. 5 Simulation results showing convergence of dynamic An example was developed which plotted the magnitude of the
angular velocity ratio to ideal kinematic relationship slip velocity of the PRSM as a function of both the screw lead and

Journal of Mechanisms and Robotics FEBRUARY 2016, Vol. 8 / 014503-5

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use

 the screw and nut contact angles. In addition, a dynamic simula- [6] Jones, M. H., and Velinsky, S. A., 2012, “Kinematics of Roller Migration
tion demonstrated that for a particular set of PRSM dimensions, in the Planetary Roller Screw Mechanism,” ASME J. Mech. Des., 134(6),
p. 061006.
the relationship between the steady-state angular velocities [7] Jones, M. H., and Velinsky, S. A., 2014, “Stiffness of the Roller Screw Mecha-
quickly converges to the ideal kinematic relationship. Thus, it nism by the Direct Method,” Mech. Based Des. Struct. Mach., 42(1), pp. 17–34.
may be reasonable to assume the ideal kinematic relationship [8] Lemor, P., 1996, “The Roller Screw: An Efficient and Reliable Mechanical
between these generalized coordinates for the development of Component of Electro-Mechanical Actuators,” 31st Intersociety Energy Con-
version Engineering Conference (IECEC 96), Washington, DC, Aug. 11–16,
velocities and efficiency under steady-state operation. pp. 215–220.
[9] Schinstock, D., and Haskew, T., 1996, “Dynamic Load Testing of Roller Screw
EMAs,” 31st Intersociety Energy Conversion Engineering Conference (IECEC
Acknowledgment 96), Washington, DC, Aug. 11–16, pp. 221–226.
The authors gratefully acknowledge the Division of Research, [10] Hojjat, Y., and Mahdi Agheli, M., 2009, “A Comprehensive Study on Capabil-
ities and Limitations of Roller–Screw With Emphasis on Slip Tendency,”
Innovation and System Information of the California Department Mech. Mach. Theory, 44(10), pp. 1887–1899.
of Transportation for the support of this work through the [11] Tselishchev, A. S., and Zharov, I. S., 2008, “Elastic Elements in Roller–Screw
Advanced Highway Maintenance and Construction Technology Mechanisms,” Russ. Eng. Res., 28(11), pp. 1040–1043.
(AHMCT) Research Center at the University of California, Davis. [12] Sokolov, P., Blinov, D., Ryakhovskii, O., Ochkasov, E., and Drobizheva, A. Y.,
2008, “Promising Rotation–Translation Converters,” Russ. Eng. Res., 28(10),
pp. 949–956.
References [13] Falkner, M., Nitschko, T., Supper, L., Traxler, G., Zemann, J., and Roberts, E.,
[1] Strandgren, C. B., 1954, “Screw-Threaded Mechanism,” U.S. Patent No. 2003, “Roller Screw Lifetime Under Oscillatory Motion: From Dry to Liquid
2,683,379. Lubrication,” 10th European Space Mechanisms and Tribology Symposium
[2] Velinsky, S. A., Chu, B., and Lasky, T. A., 2009, “Kinematics and Efficiency (ESMATS), San Sebastian, Spain, Sept. 24–26, pp. 297–301.
Analysis of the Planetary Roller Screw Mechanism,” ASME J. Mech. Des., [14] Yang, J., Wei, Z., Zhu, J., and Du, W., 2011, “Calculation of Load Distribution
131(1), p. 011016. of Planetary Roller Screws and Static Rigidity,” J. Huazhong Univ. Sci. Tech-
[3] Sokolov, P. A., Ryakhovsky, O. A., Blinov, D. S., and Laptev, A., 2005, nol., 39(4), pp. 1–4.
“Rbyevanbra Gkayenahyß[ jkbrjdbynjdß[ e[aybÅvjd (Kinematics [15] Ma, S., Liu, G., Tong, R., and Fu, X., 2015, “A Frictional Heat Model of Plane-
of Planetary Roller–Screw Mechanisms),” Vestn. MGTU, Mashinostr., 1, pp. tary Roller Screw Mechanism Considering Load Distribution,” Mech. Based
3–14. Des. Struct. Mach., 43(2), pp. 164–182.
[4] Ryakhovsky, O., Blinov, D., and Sokolov, P., 2002, “ffyakbÅ a,jnß [16] Crandall, S., Karnopp, D., Kurtz, E. F., Jr., and Pridmore-Brown, D., 1982,
Gkayenahyj½ jkbrjdbynjdj½ Gehelaxb (Analysis of the Operation of a Dynamics of Mechanical and Electromechanical Systems, Krieger Publishing
Planetary Roller–Screw Mechanism),” Vestn. MGTU, Mashinostr., 4, pp. Company, Malabar, FL.
52–57. [17] Jones, M. H., 2013, “Mechanics Based Design of the Planetary Roller Screw
[5] Jones, M. H., and Velinsky, S. A., 2013, “Contact Kinematics in the Roller Mechanism,” Ph.D., thesis, Mechanical and Aerospace Engineering, University
Screw Mechanism,” ASME J. Mech. Des., 135(5), p. 051003. of California, Davis, CA, p. 124.

014503-6 / Vol. 8, FEBRUARY 2016 Transactions of the ASME

Downloaded From: http://mechanismsrobotics.asmedigitalcollection.asme.org/ on 01/27/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use