research article kinematics of planetary roller screw...

Research ArticleKinematics of Planetary Roller Screw Mechanism consideringHelical Directions of Screw and Roller Threads

Shangjun Ma1 Tao Zhang12 Geng Liu1 Ruiting Tong1 and Xiaojun Fu1

1Shaanxi Engineering Laboratory for Transmissions and Controls Northwestern Polytechnical University Xirsquoan 710072 China2CALTRampD Center China Academy of Launch Vehicle Technology Beijing 100076 China

Correspondence should be addressed to Shangjun Ma msjlxy888163com

Received 12 April 2015 Accepted 5 August 2015

Academic Editor Francesco Braghin

Copyright copy 2015 Shangjun Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Based on the differential principle of thread transmission an analytical model considering helical directions between screw androller threads in planetary roller screw mechanism (PRSM) is presented in this work The model is critical for the design of PRSMwith a smaller lead and a bigger pitch to realize a higher transmission accuracyThe kinematic principle of planetary transmission isemployed to analyze the PRSM with different screw thread and roller thread directions In order to investigate the differences withdifferent screw thread and roller thread directions the numerical model is developed by using the software Adams to validate theanalytical solutions calculated by the presented model The results indicate when the helical direction of screw thread is identicalwith the direction of roller thread that the lead of PRSM is unaffected regardless of whether sliding between screw and rollersoccurs or not Only when the direction of screw thread is reverse to the direction of roller thread the design of PRSM with asmaller lead can be realized under a bigger pitch The presented models and numerical simulation method can be used to researchthe transmission accuracy of PRSM

1 Introduction

Planetary roller screw mechanism (PRSM) is used in vari-ous motion-delivery devices where power is transmitted byconverting rotarymotion to linearmotionThemain compo-nents of PRSM are the nut the screw and the rollers and thekey components for transmission are the rollers Comparedto ball screw mechanism PRSM has higher precision higherspeed heavier load and longer life though the manufacturecost of PRSM is relatively high due to its complex structureAs such the PRSM finds its applications as an actuator devicein various machineries such as machine tool [1] medicalequipment [2] port equipment of ship [3] and flight controlequipment of more-electric aircraft [4 5]

The published research on the PRSM has been mainlydedicated in the following areas efficiency and failure modesstudy [6] dynamic load testing [7] wet and dry lubricationsunder oscillatory motion [8] Besides Hojjat and Mahdi[9] analyzed the capabilities and limitations of PRSM andproved that large leads and extremely small leads can be easilyobtained in PRSM The forces acting on the rollers during

the rotation of screw have been analyzed for investigatingthe slip phenomenon Velinsky et al [10] applied the con-cept of orbital mechanics to study the kinematics and theefficiency of PRSM They have found that although slip hasto occur between the rollers and the screw in the PRSMthe overall lead of the mechanism is independent of suchslip Sokolov et al [11] developed the principles to evaluatethe wear resistance of the PRSM Jones and Velinsky [12]built a kinematic model to predict the axial migration of therollers with respect to the nut in the PRSM It is announcedthat roller migration is due to slip at the nut side which iscaused by a pitch mismatching between the spur-ring gearand effective nut-roller helical gear pair The results indicatethat the roller migration does not affect the overall lead ofthe PRSM Furthermore they have applied the principle ofconjugate surfaces to the contact kinematical modeling atthe screw-roller and nut-roller interfaces [13] It was shownthat the contact point cannot locate on the bodiesrsquo line ofcenters at the screw side Considering the curved profile ofroller thread the equations for calculating contact radii ofthe roller screw and nut bodies have also been derived

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 459462 11 pageshttpdxdoiorg1011552015459462

2 Mathematical Problems in Engineering

Recently Jones et al [14] constructed a stiffness model of theroller screw mechanism with the direct stiffness method Inaddition to the prediction of the overall stiffness of the mech-anism the load distributions across the threads of individualbodies were also calculated Rys and Lisowski [15] presentedthe computational model for predicting the load distributionbetween the elements in PRSM based on the deformations ofrolling elements as the deformations of rectangular volumessubjected to shear stresses The contact deformations ofthreads and the deformations of screw and nut cores wereconsidered by introducing a properly chosen shear modulusHowever there is no comprehensive kinematics model thatcan apply differential principles to the design or estimate thelead of PRSM when the helical directions between the screwthread and the roller thread are considered

To address the aforementionedproblems and facilitate thePRSM design a kinematic model and a comprehensive studyon the helical directions between screw thread and rollerthread based on differential principle of thread transmissionare developed in this work The mechanical structure ofPRSM is introduced followed by the motion analysis andcomputational modeling of the lead Then kinematics sim-ulations of the PRSM are performed to validate the motionanalysis which considers the helical directions between thescrew thread and the roller screw Finally the models areexamined in detail to explicitly show the relationships ofthe helical directions and the lead of the PRSM in the forceanalysis

2 Analytical Modeling

21 Mechanical Structure Figure 1 shows a transmissionmechanism of PRSM The rotary motion of the screw shaftis converted into an axial force through the operation of therollers and the nut on the shaftThe efficiency of such processismuch greater than that of a conventional threaded shaft-nutmechanism because of the introduction of the rollers whichare in rolling contacts with both the screw and the nut insteadof sliding contacts in the conventional mechanism There isno relative axial movement between the nut and the rollerssince the helix angle of the roller is the same as that of thenutTherefore the rollers and the nut are able to move axiallywith respect to the rotating screw shaft At the same time therollers and the nut remain in constant rolling contact

The roller possesses a single start thread and two spurgears at each end of the roller and ring gears fixed at each endof the nut The rollers roll around the inside of the nut as thescrew turns and one revolution of the screw causes the nutto advance one lead regardless of rolling or slipping betweenthe componentsThe grooves on the nut and screw are helicalwith multistart In order to ensure that the rollers do notbecome skewed or driven out of axial alignment between thescrew and the nut a planetary carrier and a gear pair areprovided The planetary carrier is located under each pivotend of the rollers and keeps the rollers spaced apart circum-ferentially around the screw A gear pair includes a ring gearand spur gear on the roller The ring gears time the spinningand orbit of the rollers about the screw axis by meshing gearteeth near the ends of the rollers At the same the planetary

Nut

Screw

Roller Roller gear Ring gear

Spring ringPlanetary carrier

X

YZ

Figure 1 Configuration of PRSM

Nut

Screw

Roller

Planetary carrier

Figure 2 Kinematic principle of PRSM

carriers float relative to the nut axially secured by the springringswhich alignedwith an axial groove in thewall of the nut

22 Motion Analysis

221 Angular Motion Analysis of Components Based on theprinciple of a planetary gear train the kinematic principleof PRSM is shown in Figure 2 and the angular motionsof components in PRSM are described in Figure 3 whichrepresents the axial view of the PRSM

According to the relationships of the movements andassuming no slip between the screw and the rollers the linearvelocity of contact point 119861 between the roller and the screw isdefined as V

119861in Figure 3 Since the rotational direction of nut

is prohibited the linear velocity of contact point 119860 betweenthe roller and the nut is zero The linear and orbital speeds ofthe roller center point 119874 are V

119900= V1198612 and 120596

119898 respectively

The linear speed of the roller center can be further expressedas V119900

= V1198612 = 120596

1199041198891199044 where 120596

119904and 119889

119904are the angular

velocity and effective diameter of the screw respectively Onthe other hand V

119900can also be shown as V

119900= 1205961198981198891198982

Mathematical Problems in Engineering 3

120596n

120596s

dm

ds

120593nm

120593m

120593slide

120596H

C

F

B

A

O o120596r

dr

D

E

dn

B

120596m

Figure 3 Axial view of PRSM

where 119889119898denotes orbital diameter of roller Accordingly the

relationship between the orbital speed of center point 119874 120596119898

and angular velocity of screw 120596119904can be written as

120596119898

=119889119904

2119889119898

120596119904=

119889119904

2 (119889119904+ 119889119903)120596119904=

119896

2 (119896 + 1)120596119904 (1)

where 119896 = 119889119904119889119903

Then the nut is fixed in the rotational direction assumingthat a roller travels from an initial point 119860 to a final point119865 with one revolution of the screw To analyze the angularmotion of the components120593

119898and120593119903(not shown in Figure 3)

are defined as orbital angle and rotational angle of the rollerand 120593

119899119898denotes the angular arc of contact of screw with

roller The pure sliding angle 120593slide must be zero thereforethe angular arc of contact of roller equals the angular arc ofcontact of nut 1006704119860119864 that is

120593119903119889119903

2=

120593119898119889119899

2 (2)

where 119889119903and 119889

119899are the effective diameter of the roller and

the nut respectivelyConsidering the relationships between the effective diam-

eters of the screw and the roller and between the roller and thenut (2) can be rewritten as

120593119903

120593119898

=119889119899

119889119903

=119889119904+ 2119889119903

119889119903

= 119896 + 2 (3)

Combining (1) and (3) and using the relationship120593119903120593119898

=

120596119903120596119898 the rotational speed of the roller 120596

119903can be given as

120596119903=

119896 (119896 + 2)

2 (119896 + 1)120596119904 (4)

The relationships of helix angles of the screw the rollerand the nut in terms of pitch starts and effective diametersare given in the following [10]

tan 120582119904=

119899119904119901

120587119889119904

tan 120582119903=

119899119903119901

120587119889119903

tan 120582119899=

119899119899119901

120587119889119899

(5)

where 120582119904 120582119903 and 120582

119899are the helix angles of the screw the

roller and the nut respectively 119899119904 119899119903 and 119899

119899are the start of

the screw the roller and the nut respectively 119901 is the pitchThe helix angles of the roller and the nut are equal that

is 120582119903= 120582119899and combining 119899

119904= 120596119903120596119898

= 119889119899119889119903and 119889

119904119889119903=

119911119899119911119903minus 2 = 119899

119904minus 2 = 119899

119899minus 2 where 119911

119899is the tooth number

of ring gears and 119911119903is the tooth number of gears near the

ends of rollers respectively The relationship can be obtainedas follows

119889119904= (119899119904minus 2) 119889

119903 (6)

Equation (6) is utilized to calculate the starts if 119889119904and 119889

119903

are given On the other hand the starts of the screw have to bemore than or equal to 3 that is 119899

119904= 119899119899ge 3 Considering the

angular motion of the PRSM is quite similar to the motion ofa planetary gear train the relationships between the angularvelocities can be shown as follows

120596119904minus 1205961199011

120596119903minus 1205961199011

= minus119889119903

119889119904

120596119904minus 1205961199012

120596119899minus 1205961199012

= minus119889119903

119889119904

119911119899

119911119903

(7)

where 1205961199011 1205961199012

are the angular velocities of the planetarycarriers on the left side and right side respectively Thetransmission ratio between the screw and the roller is definedas 119894119904119903and that between the roller and the nut is 119894

119899119903

In order to ensure pure rolling of the rollers inside the nutthe angular velocity of the planetary carrier on the left sidemust be equal to that on the right side which can bewritten as

1205961199011

= 1205961199012

= 120596119867 (8)

The transmission ratios of components are shown as

119894119904119903

=120596119903minus 120596119867

120596119904minus 120596119867

= minus119889119904

119889119903

119894119899119903

=120596119899minus 120596119867

120596119904minus 120596119867

= minus119889119904

119889119899

(9)

Utilizing the relationship for the starts of nut that is 119899119899=

(119889119904+ 2119889119903)119889119903= 119889119899119889119903 and (9) it yields

120596119867

=119889119904

119889119904+ 119889119899

120596119904=

119899119904minus 2

2119899119904minus 2

120596119904 (10)


The relationships of angular velocities between the screwthe roller and the planetary carrier are described as

120596119904minus 120596119867

=119889119899

119889119904

120596119867

=119899119904120596119904

2119899119904minus 2

(11)

120596119903minus 120596119867

= minus119889119904

119889119903

(120596119904minus 120596119867) =

minus119899119904

2+ 2119899119904

2119899119904minus 2

120596119904 (12)

222 Helical Direction and Parameter Relationships on NutSide As aforementioned the roller rolls on the inner surfaceof the nutThe helix angles of the two components are identi-calThe roller gearmeshes with the ring gear No slip betweenthe roller and the nut is allowed however there is always slipbetween the screw and the roller in the axial direction [10]Accordingly for the case with roller-screw slip the angularmotion of the components can be decomposed into two com-ponents that is the relative motion without rotational slipand the relative motion with pure rotational slip As shown inFigure 3 1006704119860119865 denotes the angular arc on the surface of the nutthat is in contact with the roller within one revolution of thescrew 1006704119861119863 denotes the angular arc on the surface of the screwthat is in contact with the roller assuming that no slip occursbetween the screw and the roller 1006704119863119862 denotes the angular arcof pure sliding motion between the screw and the roller

Because the roller and the nut have different leads andeffective diameters we assume that the axial displacement ofroller relative to nut 119871

1 can be decomposed into two com-

ponents the axial displacement of a rotationally constrainedroller relative to a rotating screw 119871

1119903and the axial movement

of a rotating roller relative to a fixed nut 1198711119888 The simple

relationship yields

1198711= 1198711119903

+ 1198711119888 (13)

Based on the relative movement of the components 1198711119888

is equal to axial displacement of the nut 1198711119899

in which it ishypothesized that the nut rotates an angle 120593

119898with angular

velocity 120596119899 that is 1006704119860119864 but the direction of 119871

1119888is reversed to

1198711119899 that is

1198711119888

= minus1198711119899 (14)

As discussed above the 1198711119899is influenced by the angle 120593

119898

and the lead of the nut 119899119899119901 where 119899

119899and 119901 are the starts and

pitch of the nut respectively Similarly the axial displacementof the roller relative to the nut 119871

1119903 is influenced by the

rotational angle of the roller 120593119903and the lead of the roller 119899

119903119901

where 119899119903is equal to one Therefore axial displacements 119871

1119903

and 1198711119899can be written as

1198711119903

=120593119903

2120587119901

1198711119899

= plusmn120593119898

2120587119899119899119901

(15)

where the negative sign indicates that the helical directionsare identical between the roller and the nut and positive signdenotes that the helical directions are reversed between theroller and the nut

Substituting (14) and (15) into (13) it can be rearranged as

1198711=

120593119903

2120587119901 ∓

120593119898

2120587119899119899119901 (16)

It iswell known in the PRSM that there is no relative axialdisplacement between the nut and the roller that is 119871

1= 0

Therefore symbol ldquominusrdquo should be chosen in (16) on the otherhand the helical directions should be identical between theroller and the nut Substituting (3) into (16) yields

119899119899= 119896 + 2 (17)

223 Helical Direction and Parameter Relationships on ScrewSide Considering the slip between the screw and the rollersimilarly the angular motions of the PRSM can be decom-posed into two components which are the motion withoutslip and motion with pure sliding [10] Therefore the axialdisplacement of the roller relative to the screw 119871

2 is the sum

of two components that is pure rolling which is generatedon arc 1006704119861119863 and pure sliding on arc 1006704119863119862 Define 119871

2119903and 119871

2119904

lowast

as the axial displacement components of the rollermentionedabove respectively That is

1198712= 1198712119903

+ 1198712119904

lowast (18)

where

1198712119903

= 1198711119903

=120593119903

2120587119901 (19)

1198712119904

lowast= minus1198712119904 (20)

where 1198712119904is the axial displacement of screw relative to roller

1198712119904is influenced by angular arc of contact of screwwith roller

120593119899119898 pure sliding angle 120593slide and lead of the screw 119899

119904119901 where

119899119904denotes the starts of the screw The axial displacement of

screw relative to the roller is expressed as

1198712119904

= ∓120593119899119898

+ 120593slide2120587

119899119904119901 (21)

where the negative sign denotes that the helical directionsof the screw and the roller are identical and positive signindicates that the helical directions of the screw and the rollerare reversed

Based on the geometry relationship as shown in Figure 3one may get

120593119899119898

+ 120593slide + 120593119898

= 2120587 (22)

Substituting (19) (21) and (22) into (18) and assumingthat the pure sliding angle is equal to zero the axial displace-ment of the roller relative to the screw can be represented as

1198712=

120593119903

2120587119901 ∓

120593119898

2120587119899119904119901 plusmn 119899119904119901 (23)

While the relative sliding occurs between the roller andthe screw for one revolution of the screw the pure slidingangle 120593slide is a variable and 120593

119888and 120593

119903are also the variables

in (3) and (23)Therefore the axial displacement of the roller


relative to the screw will generate a higher fluctuation duringthe screw rotation Assuming there is no slip between thescrew and the rollers in the rotational direction or in orderto avoid pure sliding phenomenon the first two terms in (23)have to be zeroes so the negative sign should be used that is

120593119903

2120587119901 minus

120593119898

2120587119899119904119901 = 0

1198712= plusmn119899119904119901

(24)

where the positive sign denotes that the helical directions ofthe screw and the roller are identical and the negative signindicates that the helical directions of the screw and the rollerare reversed

Because there is no relative axial movement between thenut and the rollers the axial displacement of the roller relativeto the screw is equal to the axial displacement of the nut 119871

119899

In other words the lead 119899119904119901 can be expressed as a function of

the axial displacement of the nut Thus it is stated as

119871119899=

120596119904119905

2120587119899119904119901 (25)

where 119905 denotes the operating time of the screwFurthermore the axial speed of the nut is calculated by

differentiating the displacement of the nut with respect totime as shown in the following

V119899=

119889119871119899

119889119905=

120596119904

2120587119899119904119901 (26)

23 Lead of PRSM considering Helical Directions of the ScrewThread and the Roller Thread Based on the analyses ofSection 22 it is known that the helical direction of the rollerthread must be identical with the helical direction of thenut thread and the helical direction of the screw thread isidentical with or reversed to the helical direction of the rollerthreadTherefore the following research is focused on lead ofPRSM considering helical directions of the screw thread andthe roller thread

231 Identical Helical Directions of the Screw Thread andRoller Thread For the case in which the helical direction ofscrew thread is identical with that of roller thread the lead ofthe PRSM can be written as

1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904minus 1)] (27)

where 1198710denotes the lead of the PRSM

Substituting (4) (6) and (10) into (27) the lead of PRSM1198710 can be represented as

1198710= 119899119904119901 (28)

Equation (28) indicates that the lead of PRSM is equal tothe lead of the screw or the lead of the nut which means thatthe lead of PRSM is only determined by the starts and thepitch of the screw or the nut

232 Reverse Helical Directions of the Screw Thread and theRoller Thread For the case in which the helical direction ofthe screw thread is reversed to that of the roller thread thelead of the PRSM can be expressed as

1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904+ 1)] (29)

Similarly substituting (4) (6) and (10) into (29) the leadof PRSM 119871

0 can be represented as

1198710=

119899119904119901

119899119904minus 1

(30)

Also the lead of PRSM is only determined by the startsand the pitch of the screw Only when the helical direction ofscrew thread is reversed to the helical direction of the rollerthread can the design of a bigger pitch and a smaller lead berealized with the same parameters of the starts and the pitch

As indicated by (28) and (30) regardless of whether slipoccurs between the screw and the roller or not the lead ofthe PRSM is a constant due to the fact that the lead of thescrew is not changed If the slip occurs however the slidingof rollers can cause undesirable moments and heat due tofriction The frictional heat is directly related to efficiencyand energy loss in the PRSM and the high temperature fromthe heat will cause deterioration of lubrication and eventuallyleads to mechanical failure of the PRSM

The leads of the PRSM can be calculated by using (28) and(30) For example when the starts and the pitch of the screware 5 and 05mm the leads of identical helical directionsof the screw thread and the roller thread case and that ofthe reverse case are 25mm and 0625mm respectively Theformer is four times the latter Obviously in order to obtainhigher transmission accuracy the reverse helical directioncan be applied in the practical PRSM structure If a smallerpitch especially can be obtained in machining then thesmallest lead of the PRSM can be further realized Thereforea higher transmission accuracy can be obtained if the reversehelical directions of the screw thread and the roller thread areapplied to the PRSM

Furthermore the PRSM is an accuracy transmissionwhich achieves the smallest lead by introduction of threaddirections however compared to the conventional ball screwthe small lead is extremely difficult to reach due to therequirements of carrying capacity and transmission accuracyand the design difficulty of the return tube

3 Numerical Modeling of PRSM

31 Kinematics Model of PRSM A model of kinematicssimulation (as shown in Figure 4) of PRSM is developed witha software MSCAdams with an original CAD geometry ofPRSM converted from Solidworks software The parametersof thread pair and gear pair are shown in Tables 1 and 2

32 Constraints

321 Displacement Constraints Assume that all rollers haveidentical movements in the PRSM and the steady statemotion of the screw is considered in this paper


Figure 4 The numerical model of PRSM

Table 1 Parameters of thread pair

Parameter name Symbol Unit ValueEffective diameter of screw 119889

119904mm 39

Starts 119899119904= 119899119899

5Pitch 119901 mm 5Effective diameter of roller 119889

119903mm 13

Effective diameter of nut 119889119899

mm 65

Table 2 Parameters of gear pair

Parameter name Symbol Unit ValueModule 119898 mm 1Tooth number of roller gear 119911

11990313

Tooth number of ring gear 119911119899

65Pressure angle 120572

119899

∘ 20Addendum coefficient ℎ

119886

lowast 08Clearance coefficient 119888

lowast 03Modification coefficients 119909

1198990

Tooth width of roller gear 1198871

mm 10Tooth width of ring gear 119887

2mm 10

Based on the relative movement shown in Figure 1 thedisplacement constraints applied to the PRSM are as follows(1) the moving joint is enforced between the nut and theframe which means only axial translation of the nut isreserved (2) rotating joint is imposed between the screwand the frame which only allows rotation of the screw (3)the rollers may spin and revolute therefore rotating jointsare applied between planet carriers and rollers (4) columnjoint is introduced between the planet carriers and the framebecause the planet carriers have both revolution and axialtranslation The connection relationships of components areshown in Figure 5 in detail

322 Load Constraints In order to realize the kinematictransmission the load constraints in the kinematics modelare as follows contact interactions are applied at the inter-faces between the screw and the rollers and those between therollers and the nut Similarly the contact interactions are alsoapplied at the interfaces between the spur gear of the rollersand the ring gears

The stiffness coefficient is set as 119870 = 10 times 105Nmmrigidity index is 15 damping coefficient is 50Nsdotsmm anddepth of penetration is 01mmThe coulomb friction force isconsidered in this model for describing the real contact statewhere the static friction coefficient is 120583

119904= 03 the dynamic

friction coefficient is 120583119889= 025 the static slip velocity is V

119904=

01mms the dynamic slip velocity is V119889

= 10mms elasticmodulus is 21 times 109 Pa Poissonrsquos ratio is 03 density is 78 times

103 kgm3 and axial force applied on the nut center of mass is10 kN in a direction reversed to its movement The constantrevolution speed of screw is 720∘s that is120596

119904= 125664 rads

simulation time is set to 10 s

4 Results and Discussions

41 Identical Helical Directions of Screw Thread and RollerThread When the helical direction is identical between thescrew thread and the roller thread the relationships of helicaldirection in PRSM are as follows screw is right-hand roller isright-hand and nut is right-hand The simulation results areshown in Figures 6ndash9

As Figure 6 shows the displacement curve of the nut isnearly a straight line due to the synchronized roller rotationssince all rollers are connected together by the planetarycarrier and the precise mesh between the roller gear and thering gear

Figure 7 exhibits the averaged angular velocity of theroller which is 10379664∘s that is 181159 rads which is theresultant of spinning angular velocity and revolution angularvelocity

The averaged angular velocity of the planet carrier is2606253∘s that is 45488 rads as is demonstrated inFigure 8 Based on the relative movement between the rollerand the planet carrier spinning angular velocity of the rollercan be approached that is 120596

119903= 181159 rads + 45488 rads =

226647 radsAs shown in Figure 9 the averaged axial speed of the

nut is 497138mms Comparisons of analytical solutionswithsimulation results are shown in Table 3

42 Reversed Helical Directions of Screw Thread and RollerThread When the helical direction of the screw thread isreversed to that of the roller thread the relationships of helicaldirection in PRSM are as follows screw is right-hand rolleris left-hand and nut is left-hand The simulation results areshown in Figures 10ndash13

Compared to Figure 6 a very analogous trend of axialdisplacement of the nut can be obtained as shown inFigure 10 Figures 11 and 12 showed that the averaged angularvelocity of roller is 10906219∘s that is 190349 rads and theaveraged angular velocity of planet carrier is 2728972∘s thatis 47630 rads respectively Based on the relative movementbetween the roller and the planet carrier spinning angularvelocity of the roller can be calculated as 120596

119903= 190349 rads +

47630 rads = 237979 radsAs shown in Figure 13 the averaged axial speed of the

nut is 118169mms Comparisons of analytical solutions withsimulation results are shown in Table 4


Table 3 Comparison of the analytical solutions with simulation results

Displacementof nut atpoint 119860


Angularvelocity ofroller (120596

119903)

Angular velocityof planet carrier

(120596119898)

Axial speedof nut (V

119899)

Simulation results 248601mm 498571mm 226647 rads 45488 rads 497138mmsAnalytical solutions 25mm 50mm 235620 rads 47124 rads 50mmsRelative error 05596 02858 38083 34632 05724

Connectionrelationships ofcomponents in

PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut

Between screwand frame

Between rollers andplanetary carriers

Between ring gearsand frame

Between planetarycarriers and frame

Between nut andframe

Between screw androllers

Between rollers andnut

Between rollers andring gears

Nut

Drive load

Figure 5 The connection relationships of components in PRSM

00 02 04 06 08 1005

10152025303540455055

Displacement point B of nutfor two rotations of screw

Displacement point Aof nut for one rotation

of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)

Figure 6 Axial displacement curve of the nut

The analytical solutions of angular velocity of planetcarrier angular velocity of roller axial speed of nut anddisplacement of nut can be obtained by (10) (12) (26)(28) and (30) respectively The comparisons of analyticalsolutions with simulation results are shown in Tables 3 and 4

00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)

Figure 7 Angular velocity curve of the roller

As shown in Tables 3 and 4 the analytical solutions arevery close to the simulation results with errors less than4 for identical helical direction (screw thread directionand roller thread direction) case and errors less than 6for reverse helical direction (screw thread direction and


Table 4 Comparisons of analytical solutions with simulation results




119903)

Angular velocity ofplanet carrier (120596

119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)

Figure 8 Angular velocity curve of planet carrier

00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)

Figure 9 Axial speed curve of nut

roller thread direction) case respectively The relative errorsmay originate from the following (1) The form of rollerthread is designed with rounded half-section to enhance thecarrying capacity and improve the contact characteristicsHowever the radius of the rounded half-section (the radiuscan be denoted as 119877 = 119889

1199032 sin120573 where 120573 is contact angle

of the roller thread) is decimal fraction in the numericalmodel which leads to error of meshing position betweenthe analytical model and the numerical model (2) The slipratio is a nonconstant which leads to slipping between thescrew and the roller and between the roller and the nutFurthermore the slip ratio cannot be ascertained in thenumerical model (3) The meshing clearance and impact

00 02 04 06 08 100

2

4

6

8

10

12

14 Displacement point B of nutfor two rotations of screw


of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)

Figure 10 Axial displacement curve of nut

00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)

Figure 11 Angular speed curve of roller

(the contact of components is defined by using impactfunction in the numerical model and the impact is correlativetomeshing clearance) are considered in the numericalmodelwhich lead to fluctuation of simulation results

According to the results of numerical simulation theangular velocity and axial speed curves of the componentsgenerate a higher fluctuation In addition to the influence ofimpact and clearance the sliding is another important factorTherefore the analysis of the forces has been performed

When the helical direction of screw thread is identicalwith that of the roller thread as shown in Figure 14 thefriction force (equal to 120583119865

119873cos 120582119903 where 120583 is coefficient of

friction) applied on the roller thread is in the helical direction


00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)


of roller movement and the tangent force component isopposite to the movement direction Such configurationtends to slip and requires sufficient friction force to workproperly [9] On the other hand the roller rotates due tofriction force and the lack of friction force (compared withtangent force) causes slipping

In Figure 14 119865119886is axial force 119865

119905is tangential force 120583119865

119873is

friction force and 119865119873is resultant force of 119865

119886and 119865

119905

When the helical direction of screw thread is reversedto that of the roller thread the force analysis is shown inFigure 15 [9]

It is similar to Figure 14 the directions of friction force(also equal to 120583119865

119873cos 120582119903) and the tangent force component

are reversed If there is not enough friction force between thescrew and the rollers the roller has tendency to slip

Furthermore the relative displacement errors shown inTable 4 are 944 times (point119860) and 1538 times (point 119861) thecorresponding values in Table 3 In other words the reversedhelical directions of screw thread and roller thread havehigher slipping tendency than the identical helical directionsof screw thread and roller thread under the same constraintconditions

Screw thread surface

Roller thread surface

FN

120583FN

Fa

Ft

120582s120582r

Figure 14 Force analysis when thread direction is identical betweenthe screw thread and the roller thread



FN

120583FN

Fa

Ft 120582s120582r

Figure 15 Force analysis when thread direction is reversed betweenthe screw thread and the roller thread

Besides the results of Table 4 also indicate that some slipalways occurs between the roller and the screw as a resultof reversed relative movement direction between the screwand the roller The slip is closely impacted by the rotationalspeed of the screw axial load applied on the nut lubricationconditions and so on Generally the position accuracy of thePRSM is secured by applying a higher preload on the nut andoperating at high axial load and low rotational speeds

5 Conclusions

This paper develops the kinematics by analytical modelingand numerical modeling of the PRSM considering helicaldirections between screw thread and roller thread to providea method to support its design and application The majorfindings are as follows

(1) The analytical modeling considering helical direc-tions between the screw and the roller threads inPRSM is presented to realize the design of PRSMwitha smaller lead under a bigger pitch based on the dif-ferential principle of thread transmission Numericalmodeling is developed by using Adams to validate theproposed analytical solutions Besides the kinematicmodels and simulation method considering helicaldirections of screw and roller threads are available toPRSM which are beneficial to the further research ofthe PRSM


(2) The analytical solutions are close to the numericalresults with errors less than 4 and 6 when thedirection of screw thread is identical with or reversedto the direction of roller thread respectively

(3) When the helical direction is identical between thescrew thread and the roller thread the friction forceapplied on the roller thread is in the helical directionof roller movement However the tangential forcecomponent is opposite to the movement directionTherefore such case has slip tendency and requiressufficient friction force to work properly

(4) When the helical direction of the screw thread isreversed to that of the roller thread the PRSM is anaccuracy transmission which achieves the smallestlead by introduction of a bigger pitch and a smallerlead as compared to the conventional ball screwwherethe small lead is extremely difficult to reach due todesign difficulty of the return tube

Notations

1198871 Tooth width of roller gear

1198872 Tooth width of ring gear

119888lowast Clearance coefficient119889119904 Effective diameter of the screw

119889119898 Denotes orbital diameter of roller

119889119903 Effective diameter of the roller

119889119899 Effective diameter of the nut

119865119886 Axial force

119865119905 Tangential force

119865119873 Resultant force of 119865

119886and 119865

119905

ℎ119886

lowast Addendum coefficient119894119904119903 Transmission ratio between the screw

and the roller119894119899119903 Transmission ratio between the roller

and the nut119870 Stiffness coefficient1198711 Axial displacement of roller relative to

nut1198711119903 Axial displacement of roller relative to a

rotating screw1198711119888 Axial displacement of a rotating roller

relative to a fixed nut1198711119899 Axial displacement of the nut relative to

roller1198712 Axial displacement of the roller relative

to the screw1198712119903 Axial displacement component of the

roller1198712119904

lowast Axial displacement component of theroller relative to the screw

1198712119904 Axial displacement of screw relative to

roller119871119899 Axial displacement of the nut

1198710 Lead of the PRSM

119898 Module of gear pair119899119904 Start of the screw

119899119903 Start of the roller

119899119899 Start of the nut

119901 Pitch119877 Radius of rounded half-section of roller

thread119905 Operating time of the screwV119904 Static slip velocity

V119889 Dynamic slip velocity

V119861 Linear velocity of the contact point

V119900 Linear speed of the roller center point

V119899 Axial speed of the nut

119909119899 Modification coefficient

119911119899 Tooth number of ring gears

119911119903 Tooth number of gears near the ends of

rollers120572119899 Pressure angle of gear pair

120573 Contact angle120582119904 Helix angles of the screw

120582119903 Helix angles of the roller

120582119899 Helix angles of the nut

120583119904 Static friction coefficient

120583119889 The dynamic friction coefficient

120593119898 Orbital angle of the roller

120593119903 Rotational angle of the roller

120593119899119898 Angular arc of contact of screw with roller

120593slide Pure sliding angle120596119898 Orbital speeds of the roller center point

120596119904 Angular velocity of the screw

120596119903 Rotational speed of the roller

1205961199011 Angular velocities of the planetary carrier

on the left side1205961199012 Angular velocities of the planetary carrier

on the right side120596119867 Angular velocities of the planetary carriers

120596119899 Angular velocity of the nut

Conflict of Interests

The authors declare that there is no known conflict ofinterests associated with this publication and there has beenno significant financial support for this work that could haveinfluenced its outcome

Authorsrsquo Contribution

The authors confirm that the paper has been read andapproved by all named authors and that there are no otherpersons who satisfied the criteria for authorship but are notlisted The authors further confirm that the order of authorslisted in the paper has been approved by all of them

Acknowledgments

The research was supported by the National Natural ScienceFoundation of China (no 51275423) Specialized ResearchFund for the Doctoral Program of Higher Education (no20126102110019) the 111 Project (no B13044) and Funda-mental Research Funds for the Central Universities (no3102015JCS05008)


References

[1] G Brandenburg S Bruckl J Dormann J Heinzl and CSchmidt ldquoComparative investigation of rotary and linearmotorfeed drive systems for high precision machine toolsrdquo in Pro-ceedings of the 6th International Workshop on Advanced MotionControl pp 384ndash389 Nagoya Japan 2000

[2] Y Ohashi A De Andrade and Y Nose ldquoHemolysis in an elec-tromechanical driven pulsatile total artificial heartrdquo ArtificialOrgans vol 27 no 12 pp 1089ndash1093 2003

[3] D Tesar andGKrishnamoorthy ldquoIntelligent electromechanicalactuators to modernize ship operationsrdquo Naval Engineers Jour-nal vol 120 no 3 pp 77ndash88 2008

[4] A A Abdelhafez and A J Forsyt ldquoA review of more-electricaircraftrdquo in Proceedings of the 13th International Conference onAerospace Sciences amp Aviation Technology (ASAT-13 rsquo09) pp 1ndash13 Cairo Egypt May 2009

[5] F Claeyssen P Janker R Leletty et al ldquoNew actuators foraircraft space and military applicationsrdquo in Proceedings of the12th International Conference on New Actuators pp 324ndash330Bremen Germany 2010

[6] P C Lemor ldquoRoller screw an efficient and reliable mechanicalcomponent of electro-mechanical actuatorsrdquo in Proceedings ofthe 31st Intersociety Energy Conversion Engineering Conference(IECEC rsquo96) pp 215ndash220 Washington DC USA August 1996

[7] D E Schinstock and T A Haskew ldquoDynamic load testing ofroller screw EMAsrdquo in Proceedings of the 31st Intersociety EnergyConversion Engineering Conference (IECEC rsquo96) vol 1 pp 221ndash226 Washington DC USA 1996

[8] M Falkner T Nitschko L Supper G Traxler J V Zemannand E W Roberts ldquoRoller screw lifetime under oscillatorymotion from dry to liquid lubricationrdquo in Proceedings of the10th European Space Mechanisms and Tribology Symposium(ESMATS rsquo03) pp 297ndash301 San Sebastian Spain 2003

[9] M Y Hojjat and A Mahdi ldquoA comprehensive study oncapabilities and limitations of roller-screwwith emphasis on sliptendencyrdquo Mechanism and Machine Theory vol 44 no 10 pp1887ndash1899 2009

[10] S A Velinsky B Chu and T A Lasky ldquoKinematics andefficiency analysis of the planetary roller screw mechanismrdquoJournal of Mechanical Design vol 131 no 1 Article ID 0110168 pages 2009

[11] P A Sokolov D S Blinov O A Ryakhovskii E E Ochkasovand A Y Drobizheva ldquoPromising rotation-translation convert-ersrdquo Russian Engineering Research vol 28 no 10 pp 949ndash9562008

[12] M H Jones and S A Velinsky ldquoKinematics of roller migrationin the planetary roller screwmechanismrdquo Journal of MechanicalDesign vol 134 no 6 Article ID 061006 2012

[13] M H Jones and S A Velinsky ldquoContact kinematics in theroller screw mechanismrdquo Transactions of the ASMEmdashJournal ofMechanical Design vol 135 no 5 Article ID 51003 2013

[14] M H Jones and S A Velinsky ldquoStiffness of the roller screwmechanism by the direct methodrdquo Mechanics Based Design ofStructures and Machines vol 42 no 1 pp 17ndash34 2014

[15] J Rys and F Lisowski ldquoThe computational model of the loaddistribution between elements in planetary roller screwrdquo inProceedings of the 9th International Conference on Fracture ampStrength of Solids pp 9ndash13 Jeju Republic of Korea June 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Recently Jones et al [14] constructed a stiffness model of theroller screw mechanism with the direct stiffness method Inaddition to the prediction of the overall stiffness of the mech-anism the load distributions across the threads of individualbodies were also calculated Rys and Lisowski [15] presentedthe computational model for predicting the load distributionbetween the elements in PRSM based on the deformations ofrolling elements as the deformations of rectangular volumessubjected to shear stresses The contact deformations ofthreads and the deformations of screw and nut cores wereconsidered by introducing a properly chosen shear modulusHowever there is no comprehensive kinematics model thatcan apply differential principles to the design or estimate thelead of PRSM when the helical directions between the screwthread and the roller thread are considered

To address the aforementionedproblems and facilitate thePRSM design a kinematic model and a comprehensive studyon the helical directions between screw thread and rollerthread based on differential principle of thread transmissionare developed in this work The mechanical structure ofPRSM is introduced followed by the motion analysis andcomputational modeling of the lead Then kinematics sim-ulations of the PRSM are performed to validate the motionanalysis which considers the helical directions between thescrew thread and the roller screw Finally the models areexamined in detail to explicitly show the relationships ofthe helical directions and the lead of the PRSM in the forceanalysis

2 Analytical Modeling

21 Mechanical Structure Figure 1 shows a transmissionmechanism of PRSM The rotary motion of the screw shaftis converted into an axial force through the operation of therollers and the nut on the shaftThe efficiency of such processismuch greater than that of a conventional threaded shaft-nutmechanism because of the introduction of the rollers whichare in rolling contacts with both the screw and the nut insteadof sliding contacts in the conventional mechanism There isno relative axial movement between the nut and the rollerssince the helix angle of the roller is the same as that of thenutTherefore the rollers and the nut are able to move axiallywith respect to the rotating screw shaft At the same time therollers and the nut remain in constant rolling contact

The roller possesses a single start thread and two spurgears at each end of the roller and ring gears fixed at each endof the nut The rollers roll around the inside of the nut as thescrew turns and one revolution of the screw causes the nutto advance one lead regardless of rolling or slipping betweenthe componentsThe grooves on the nut and screw are helicalwith multistart In order to ensure that the rollers do notbecome skewed or driven out of axial alignment between thescrew and the nut a planetary carrier and a gear pair areprovided The planetary carrier is located under each pivotend of the rollers and keeps the rollers spaced apart circum-ferentially around the screw A gear pair includes a ring gearand spur gear on the roller The ring gears time the spinningand orbit of the rollers about the screw axis by meshing gearteeth near the ends of the rollers At the same the planetary

Nut

Screw

Roller Roller gear Ring gear

Spring ringPlanetary carrier

X

YZ

Figure 1 Configuration of PRSM

Nut

Screw

Roller

Planetary carrier

Figure 2 Kinematic principle of PRSM

carriers float relative to the nut axially secured by the springringswhich alignedwith an axial groove in thewall of the nut

22 Motion Analysis

221 Angular Motion Analysis of Components Based on theprinciple of a planetary gear train the kinematic principleof PRSM is shown in Figure 2 and the angular motionsof components in PRSM are described in Figure 3 whichrepresents the axial view of the PRSM

According to the relationships of the movements andassuming no slip between the screw and the rollers the linearvelocity of contact point 119861 between the roller and the screw isdefined as V

119861in Figure 3 Since the rotational direction of nut

is prohibited the linear velocity of contact point 119860 betweenthe roller and the nut is zero The linear and orbital speeds ofthe roller center point 119874 are V

119900= V1198612 and 120596

119898 respectively

The linear speed of the roller center can be further expressedas V119900

= V1198612 = 120596

1199041198891199044 where 120596

119904and 119889

119904are the angular

velocity and effective diameter of the screw respectively Onthe other hand V

119900can also be shown as V

119900= 1205961198981198891198982


120596n

120596s

dm

ds

120593nm

120593m

120593slide

120596H

C

F

B

A

O o120596r

dr

D

E

dn

B

120596m





120596119898

=119889119904

2119889119898

120596119904=

119889119904

2 (119889119904+ 119889119903)120596119904=

119896

2 (119896 + 1)120596119904 (1)

where 119896 = 119889119904119889119903






120593119903119889119903

2=

120593119898119889119899

2 (2)

where 119889119903and 119889




120593119903

120593119898

=119889119899

119889119903

=119889119904+ 2119889119903

119889119903

= 119896 + 2 (3)


=



120596119903=

119896 (119896 + 2)

2 (119896 + 1)120596119904 (4)


tan 120582119904=

119899119904119901

120587119889119904

tan 120582119903=

119899119903119901

120587119889119903

tan 120582119899=

119899119899119901

120587119889119899

(5)

where 120582119904 120582119903 and 120582





is 120582119903= 120582119899and combining 119899

119904= 120596119903120596119898

= 119889119899119889119903and 119889

119904119889119903=

119911119899119911119903minus 2 = 119899

119904minus 2 = 119899

119899minus 2 where 119911




119889119904= (119899119904minus 2) 119889

119903 (6)


119903




120596119904minus 1205961199011

120596119903minus 1205961199011

= minus119889119903

119889119904

120596119904minus 1205961199012

120596119899minus 1205961199012

= minus119889119903

119889119904

119911119899

119911119903

(7)

where 1205961199011 1205961199012


119899119903


1205961199011

= 1205961199012

= 120596119867 (8)


119894119904119903

=120596119903minus 120596119867

120596119904minus 120596119867

= minus119889119904

119889119903

119894119899119903

=120596119899minus 120596119867

120596119904minus 120596119867

= minus119889119904

119889119899

(9)


(119889119904+ 2119889119903)119889119903= 119889119899119889119903 and (9) it yields

120596119867

=119889119904

119889119904+ 119889119899

120596119904=

119899119904minus 2

2119899119904minus 2

120596119904 (10)



120596119904minus 120596119867

=119889119899

119889119904

120596119867

=119899119904120596119904

2119899119904minus 2

(11)

120596119903minus 120596119867

= minus119889119904

119889119903

(120596119904minus 120596119867) =

minus119899119904

2+ 2119899119904

2119899119904minus 2

120596119904 (12)







relationship yields

1198711= 1198711119903

+ 1198711119888 (13)




119898with angular



1198711119899 that is

1198711119888

= minus1198711119899 (14)


119898






119903119901


1119903


1198711119903

=120593119903

2120587119901

1198711119899

= plusmn120593119898

2120587119899119899119901

(15)



1198711=

120593119903

2120587119901 ∓

120593119898

2120587119899119899119901 (16)


1= 0


119899119899= 119896 + 2 (17)


2 is the sum


2119903and 119871

2119904

lowast


1198712= 1198712119903

+ 1198712119904

lowast (18)

where

1198712119903

= 1198711119903

=120593119903

2120587119901 (19)

1198712119904

lowast= minus1198712119904 (20)




119904119901 where



1198712119904

= ∓120593119899119898

+ 120593slide2120587

119899119904119901 (21)



120593119899119898

+ 120593slide + 120593119898

= 2120587 (22)


1198712=

120593119903

2120587119901 ∓

120593119898

2120587119899119904119901 plusmn 119899119904119901 (23)


119888and 120593





120593119903

2120587119901 minus

120593119898

2120587119899119904119901 = 0

1198712= plusmn119899119904119901

(24)



119899



119871119899=

120596119904119905

2120587119899119904119901 (25)



V119899=

119889119871119899

119889119905=

120596119904

2120587119899119904119901 (26)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904minus 1)] (27)



1198710= 119899119904119901 (28)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904+ 1)] (29)



1198710=

119899119904119901

119899119904minus 1

(30)







32 Constraints






119904mm 39

Starts 119899119904= 119899119899


119903mm 13


mm 65



11990313



119899


119886



1198990



2mm 10






119904=




119904= 125664 rads







119903= 181159 rads + 45488 rads =





119903= 190349 rads +








119903)


(120596119898)


119899)



PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut









Nut

Drive load


00 02 04 06 08 1005

10152025303540455055



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)



00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120596n

120596s

dm

ds

120593nm

120593m

120593slide

120596H

C

F

B

A

O o120596r

dr

D

E

dn

B

120596m





120596119898

=119889119904

2119889119898

120596119904=

119889119904

2 (119889119904+ 119889119903)120596119904=

119896

2 (119896 + 1)120596119904 (1)

where 119896 = 119889119904119889119903






120593119903119889119903

2=

120593119898119889119899

2 (2)

where 119889119903and 119889




120593119903

120593119898

=119889119899

119889119903

=119889119904+ 2119889119903

119889119903

= 119896 + 2 (3)


=



120596119903=

119896 (119896 + 2)

2 (119896 + 1)120596119904 (4)


tan 120582119904=

119899119904119901

120587119889119904

tan 120582119903=

119899119903119901

120587119889119903

tan 120582119899=

119899119899119901

120587119889119899

(5)

where 120582119904 120582119903 and 120582





is 120582119903= 120582119899and combining 119899

119904= 120596119903120596119898

= 119889119899119889119903and 119889

119904119889119903=

119911119899119911119903minus 2 = 119899

119904minus 2 = 119899

119899minus 2 where 119911




119889119904= (119899119904minus 2) 119889

119903 (6)


119903




120596119904minus 1205961199011

120596119903minus 1205961199011

= minus119889119903

119889119904

120596119904minus 1205961199012

120596119899minus 1205961199012

= minus119889119903

119889119904

119911119899

119911119903

(7)

where 1205961199011 1205961199012


119899119903


1205961199011

= 1205961199012

= 120596119867 (8)


119894119904119903

=120596119903minus 120596119867

120596119904minus 120596119867

= minus119889119904

119889119903

119894119899119903

=120596119899minus 120596119867

120596119904minus 120596119867

= minus119889119904

119889119899

(9)


(119889119904+ 2119889119903)119889119903= 119889119899119889119903 and (9) it yields

120596119867

=119889119904

119889119904+ 119889119899

120596119904=

119899119904minus 2

2119899119904minus 2

120596119904 (10)



120596119904minus 120596119867

=119889119899

119889119904

120596119867

=119899119904120596119904

2119899119904minus 2

(11)

120596119903minus 120596119867

= minus119889119904

119889119903

(120596119904minus 120596119867) =

minus119899119904

2+ 2119899119904

2119899119904minus 2

120596119904 (12)







relationship yields

1198711= 1198711119903

+ 1198711119888 (13)




119898with angular



1198711119899 that is

1198711119888

= minus1198711119899 (14)


119898






119903119901


1119903


1198711119903

=120593119903

2120587119901

1198711119899

= plusmn120593119898

2120587119899119899119901

(15)



1198711=

120593119903

2120587119901 ∓

120593119898

2120587119899119899119901 (16)


1= 0


119899119899= 119896 + 2 (17)


2 is the sum


2119903and 119871

2119904

lowast


1198712= 1198712119903

+ 1198712119904

lowast (18)

where

1198712119903

= 1198711119903

=120593119903

2120587119901 (19)

1198712119904

lowast= minus1198712119904 (20)




119904119901 where



1198712119904

= ∓120593119899119898

+ 120593slide2120587

119899119904119901 (21)



120593119899119898

+ 120593slide + 120593119898

= 2120587 (22)


1198712=

120593119903

2120587119901 ∓

120593119898

2120587119899119904119901 plusmn 119899119904119901 (23)


119888and 120593





120593119903

2120587119901 minus

120593119898

2120587119899119904119901 = 0

1198712= plusmn119899119904119901

(24)



119899



119871119899=

120596119904119905

2120587119899119904119901 (25)



V119899=

119889119871119899

119889119905=

120596119904

2120587119899119904119901 (26)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904minus 1)] (27)



1198710= 119899119904119901 (28)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904+ 1)] (29)



1198710=

119899119904119901

119899119904minus 1

(30)







32 Constraints






119904mm 39

Starts 119899119904= 119899119899


119903mm 13


mm 65



11990313



119899


119886



1198990



2mm 10






119904=




119904= 125664 rads







119903= 181159 rads + 45488 rads =





119903= 190349 rads +








119903)


(120596119898)


119899)



PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut









Nut

Drive load


00 02 04 06 08 1005

10152025303540455055



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)



00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120596119904minus 120596119867

=119889119899

119889119904

120596119867

=119899119904120596119904

2119899119904minus 2

(11)

120596119903minus 120596119867

= minus119889119904

119889119903

(120596119904minus 120596119867) =

minus119899119904

2+ 2119899119904

2119899119904minus 2

120596119904 (12)







relationship yields

1198711= 1198711119903

+ 1198711119888 (13)




119898with angular



1198711119899 that is

1198711119888

= minus1198711119899 (14)


119898






119903119901


1119903


1198711119903

=120593119903

2120587119901

1198711119899

= plusmn120593119898

2120587119899119899119901

(15)



1198711=

120593119903

2120587119901 ∓

120593119898

2120587119899119899119901 (16)


1= 0


119899119899= 119896 + 2 (17)


2 is the sum


2119903and 119871

2119904

lowast


1198712= 1198712119903

+ 1198712119904

lowast (18)

where

1198712119903

= 1198711119903

=120593119903

2120587119901 (19)

1198712119904

lowast= minus1198712119904 (20)




119904119901 where



1198712119904

= ∓120593119899119898

+ 120593slide2120587

119899119904119901 (21)



120593119899119898

+ 120593slide + 120593119898

= 2120587 (22)


1198712=

120593119903

2120587119901 ∓

120593119898

2120587119899119904119901 plusmn 119899119904119901 (23)


119888and 120593





120593119903

2120587119901 minus

120593119898

2120587119899119904119901 = 0

1198712= plusmn119899119904119901

(24)



119899



119871119899=

120596119904119905

2120587119899119904119901 (25)



V119899=

119889119871119899

119889119905=

120596119904

2120587119899119904119901 (26)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904minus 1)] (27)



1198710= 119899119904119901 (28)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904+ 1)] (29)



1198710=

119899119904119901

119899119904minus 1

(30)







32 Constraints






119904mm 39

Starts 119899119904= 119899119899


119903mm 13


mm 65



11990313



119899


119886



1198990



2mm 10






119904=




119904= 125664 rads







119903= 181159 rads + 45488 rads =





119903= 190349 rads +








119903)


(120596119898)


119899)



PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut









Nut

Drive load


00 02 04 06 08 1005

10152025303540455055



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)



00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120593119903

2120587119901 minus

120593119898

2120587119899119904119901 = 0

1198712= plusmn119899119904119901

(24)



119899



119871119899=

120596119904119905

2120587119899119904119901 (25)



V119899=

119889119871119899

119889119905=

120596119904

2120587119899119904119901 (26)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904minus 1)] (27)



1198710= 119899119904119901 (28)



1198710= 119901[119899

119904minus

120596119903

120596119904

minus120596119867

120596119904

(119899119904+ 1)] (29)



1198710=

119899119904119901

119899119904minus 1

(30)







32 Constraints






119904mm 39

Starts 119899119904= 119899119899


119903mm 13


mm 65



11990313



119899


119886



1198990



2mm 10






119904=




119904= 125664 rads







119903= 181159 rads + 45488 rads =





119903= 190349 rads +








119903)


(120596119898)


119899)



PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut









Nut

Drive load


00 02 04 06 08 1005

10152025303540455055



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)



00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119904mm 39

Starts 119899119904= 119899119899


119903mm 13


mm 65



11990313



119899


119886



1198990



2mm 10






119904=




119904= 125664 rads







119903= 181159 rads + 45488 rads =





119903= 190349 rads +








119903)


(120596119898)


119899)



PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut









Nut

Drive load


00 02 04 06 08 1005

10152025303540455055



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)



00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119903)


(120596119898)


119899)



PRSM

Column joint

Restriction joint

Rotating joint

Moving joint

Contact joint

Load applied innut









Nut

Drive load


00 02 04 06 08 1005

10152025303540455055



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)



00 02 04 06 08 10950

975

1000

1025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119903)


119898)


119899)


00 02 04 06 08 10230

240

250

260

270

280

290

300

Time (s)

Ang

ular

velo

city

of t

he p

lane

t car

rier(

∘ s)


00 02 04 06 08 1010

20

30

40

50

60

70

80

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





00 02 04 06 08 100

2

4

6

8

10

12



of screw

Time (s)

Axi

al d

ispla

cem

ent o

f the

nut

(mm

)


00 02 04 06 08 101025

1050

1075

1100

1125

1150

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)








00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00 02 04 06 08 10260

265

270

275

280

285

290

Time (s)

Ang

ular

vel

ocity

of t

he ro

ller(

∘ s)


00 02 04 06 08 100

5

10

15

20

25

30

Time (s)

Axi

al sp

eed

of th

e nut

(mm

s)





119873is


119886and 119865

119905








FN

120583FN

Fa

Ft

120582s120582r




FN

120583FN

Fa

Ft 120582s120582r



5 Conclusions







Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Notations










119886and 119865

119905

ℎ119886









roller1198712119904





































Acknowledgments



References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article kinematics of planetary roller screw...

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