research article kinematics analysis of the roller screw based...

Research ArticleKinematics Analysis of the Roller Screw Based onthe Accuracy of Meshing Point Calculation

Yanqiang Liu Jianshen Wang Hongxun Cheng and Yapeng Sun

School of Mechanical Engineering and Automation Beihang University Beijing 100191 China

Correspondence should be addressed to Jianshen Wang jasenwang928gmailcom

Received 31 October 2014 Revised 20 March 2015 Accepted 25 March 2015

Academic Editor Francesco Pellicano

Copyright copy 2015 Yanqiang Liu 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

This paper investigates themeshing behavior of the roller screw amechanical transmission device characterized by threaded rollersthat transfer a load between the nut and the screw by analyzing the meshing characteristics between screw and rollers This studyseeks to establish a more accurate mathematical model for the thread surface by creating a modeling process in which the maxradiuses of the threads are calculated more precisely The contact line distribution and the contact location were also calculated inorder to confirm the cross section of the meshing points In the research presented in this paper the actual transmission ratio isanalyzed and the study results in a new method to calculate the actual transmission ratio In this study the helical angle and thevertex angle are proven to be of great significance after a careful analysis of their influence is conducted

1 Introduction

The roller screw is a mechanism used for converting rotarymovement into linear displacement Using rollers as theintermediary between the screw and the nut avoids the slidefound in trapezoidal screw transmission and allows a muchsmaller thread than that of the ball screw which provides theroller screw with a higher load capacity a longer working lifehigher speed higher acceleration and higher transmissionprecision Thus roller screws are widely used in high loadhigh speed and high precision applications in the aerospaceindustry the roller screw is used for the design of landinggear [1] in the shipbuilding industry it is used for driving thepropeller in optical equipment it is used for precise microdisplacement [2] it is also applied in the robotics as a kindlighter rapid responding component [3] and it is alreadyapplied inmachine tools especially the high precision grinder

A number of researchers have studied some aspects ofthe roller screw such as its efficiency failure mode [4] rollermigration kinematics [5] contact kinematics [6] stiffness [7]capability and limitations [8] The roller screw is dividedinto several types based on the different structures [9] andthe selection of structure parameters [9] is analyzed theseresearches make important progress in study of roller screw

however none of them focuses on the accurate analysis ofthe meshing behavior The previous research of meshingcalculation [10ndash12] did not consider the transmission form ofthe external threadmeshing which would cause some uniqueproblems for roller screw research

This paper builds on the parametric model of thescrewnutroller surface and the meshing between themThe meshing formulas are very complex and cannot besolved with an analytical method so a numerical methodis developed to calculate the contact line and the meshingarea Using this method this study researches the regularchanges of the contact line with different thread helical anglesand vertex angles Based on the accurate calculation of thecontact line the actual transmission ratio of the roller screw iscalculated at different thread helical angles and vertex anglesLast a brief conclusion is given about the impact this researchcan have on meshing performance

2 The Structure and the Movement Principleof the Roller Screw

The basic structure of a planetary roller screw is shown inFigure 1 The main components include the screw the nutand the rollers The screw and the nut mesh separately with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 303972 10 pageshttpdxdoiorg1011552015303972

2 Mathematical Problems in Engineering

1

2

3456

(1) Nut(2) Roller(3) Screw

(4) Inner ring gear(5) Retaining plate(6) Wire ring

Figure 1 Structure of planetary roller screw

the rollers [9] which are symmetrically distributed aroundthe screw The gear system supports the rollers and keepstheir symmetric position The screw and the nut generallyuse multiple threads while the rollers use a single threadbut share the same pitch [13] The screw and the nut alwayshave triangular threads but the roller can have a triangularthread or a rounded threadWhenmadewith highmachiningaccuracy and precise assembly the triangular profile of theroller thread will lead to a line contact which can provide ahigher load capacity The vertex angle of the thread is usually90 degrees [10] but other values are also possible

During the transmission process the nut and the rollersneed to stay static in the axial direction so that there is noslide between their thread surfaces [11]This requires the axialdisplacement of the roller because revolving along the nutthread equalizes the rotation axial displacement as shown inthe following equation

Δ120575 sdot 119877119901

119903

119877119901

119899

sdot119901 sdot 119873119899

2120587=

Δ120575 sdot 119901

2120587 (1)

Δ120575 indicates the angular displacement in the roller threadwhile meshing Equation (1) can be simplified to 119873119899 =

119877119901

119904119877119901

119903+2 which is a rule that must be observed to guarantee

that the roller will not move out from under the nut duringthe transmission process

Assuming there is no slide between these componentsthe axial movement of the nut when it is fixed circumfer-entially and the screw when it is rotated for one revolutionis defined as the transmission ratio of the roller screw Thetheoretical transmission ratio of the roller screw is shownin formula (2) It can easily be seen that the transmissionratio is determined by the pitch the lead of the screw threadthe thread directions and the quotient of the screw radiusdivided by the roller radius Thus it is possible to achievehigh speed transmission or precise positioning by choosingreasonable parameters and it is also possible to obtain a widerange of transmission ratios

119894 = 119901119904 + 119896119901 sdot 119877119901

119904

119877119901

119903

(2)

In formula (2) 119896 is the roller displacement directioncoefficient 1 if the thread of the rollers takes the same helicaldirection as the screw minus1 if the thread of the rollers takes theopposite direction as the screw and 0 for plane rollers

The theoretical transmission ratio is calculated using thepitch radius based on the assumption that themeshing pointslie on the cross section (the plane is determined by the screwand the roller axes) But the thread surfaces rise and twistat different degrees which means that the meshing pointsbetween the screw and the roller actually exist mostly on onlythe one side of the cross section Considering the axial freemovement of the roller and the threads when crossing witheach other the actual transmission ratio should be calculatedusing the relative velocity as the screw and the roller conjugateacross these surfaces at the meshing location

3 A Model of the Screw and Nut Meshing withthe Roller

31 A Model of the Screw Nut and Roller Surfaces Thesurfaces of the screw the nut and the roller are all triangularthreads and they share the same modeling method whichmeans that the formulas for the thread surfacemodeling omitthe subscripts (eg 119877 is used instead of using 119877119904 119877119899 or 119877119903

when developing the surface equation for the screw the nutor the roller) As is shown in Figure 2 the coordinate system119878(119874 119909 119910 119911) is fixed with the screwnutroller at the centerof an end face The coordinate system 1198783(1198743 1199093 1199103 1199113) inwhich the thread profile is measured rises along the helicalline with its origin 119877

119901 from the axis and with its 11990931199113 planeperpendicular to the helical line

The transformationmatrix used to convert the coordinatesystem 1198783(1198743 1199093 1199103 1199113) to 119878(119874 119909 119910 119911) can be developedfrom Figure 2 resulting in the following matrix formula

3

0119879 =

[[[[[

[

cos 120579 minus cos120572 sin 120579 sin120572 sin 120579 119877119901 cos 120579

sin 120579 cos120572 cos 120579 minus sin120572 cos 120579 119877119901 sin 120579

0 sin120572 cos120572 120579119877119901 tan120572

0 0 0 1

]]]]]

]

(3)

where 120572 is the helical angle which can be calculated with thefollowing formula

120572 = arctan(119873 sdot 119901

2120587119877119901) (4)

The vertex angle is measured from the cross sectionas shown in Figure 3 Point 119864 is on the thread surface1198641199093

1198641199103 and 1198641199113

are the coordinates on axes 1199093 1199103 and1199113 The coordinate of point 119864 in the coordinate system of1198783(1198743 1199093 1199103 1199113) is as shown in the following formula

997888119903 3 =

[[[[[

[

119877 minus 119877119901

(119877119898

minus 119877) tan120573 sin120572

(119877119898

minus 119877) tan120573 cos1205721

]]]]]

]

(5)

Mathematical Problems in Engineering 3

z(z1)

y1

z2

z3y2

y3

O1

O2(O3)

O

y120579

120572

x

Rm

x1(x2 x3)

Figure 2 Thread surface coordinate system

z2z3

y2

y3

Ez3

E

Ex3

Ey3

120572

120572

P

O3(O2)

120573

x2(x3)

Figure 3 The coordinates of the meshing point

This coordinate can be converted to 119878(119874 119909 119910 119911) by mul-tiplying the transformation matrix 3

0119879 as shown in formula

(6) Given the range in which 119877 and 120579 change formula(6) will indicate the surface of a thread which allows us

to characterize the models of the screwnutroller threadsurface Consider the following

997888119903 =3

0119879997888119903 3 =

[[[[[

[

119877 cos 120579119877 sin 120579

(119877119898

minus 119877) tan120573 + 120579119877119901 tan120572

1

]]]]]

]

(6)

119877119898 can be calculated as shown in formula (7) 120582119901 is the

quotient of the thread tooth height divided by the pitch and120582119906 is the proportion of the crest part divided by the pitchradius taken from the tooth height 120582119906 decreases with theincrease of the pitch in order to keep the max radius frombecoming too large

119877119898

= 119877119901+ 120582119901120582119906119901 cot120573 (7)

This model can be distinguished from the ones built inthe previous researches [6 9 13] by highlighting the followingaspects First we can note that the roller thread profile is astraight line instead of a circular arc as has been assumedin previous papers This straight line method reduces theprocessing difficulties found in engineering applications andcould provide excellent performance with precise machiningtechnology though curved flanks are applied to the currentactual products Second we can note that the max radiuswhich impacts the thread size and the helical angle is nolonger seen as being free from the influence of alterations tothe pitch These advantages make the model more accuratebecause they reflect the actual thread characteristics better

32 A Model of the Meshing The screwnut thread surfacemeshes externallyinternally with the roller surface the pitchdiameters are tangent the screw axis parallels the rolleraxis and the upperlower screw surface meshes with thelowerupper roller surface [12] The numerical assemblymethod calculates the surface of the roller with that of thescrewnut thus preventing both the screw and the nut frommoving and rotating while fixing the roller axis a certaindistance away from the screwnut axis which allows it torotateThe distance between the axes is the pitch radius of thescrew plus that of the roller Now observe how the meshingbetween the roller and the screw in the plane is perpendicularto their axes as shown in Figure 4The transformationmatrixfor the roller rotation is shown in the following formula

119903

0119879 =

[[[[[

[

cos120593 minus sin120593 0 119877119901

119904+ 119877119901

119903

sin120593 cos120593 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(8)


yr

xs

xr

0

M

ys

120579r120579s

RsRr

Rms

Rps

Rpr

Rmr

Figure 4 Meshing point measured in the plane perpendicular tothe axis

The roller surface as measured in the fixed coordinatesystem can be calculated as follows

997888119903 119903 =119903

0119879997888119903 0 =

[[[[[

[

119877119903 cos 120579119903 cos120593 minus 119877119903 sin 120579119903 sin120593 + 119877119901

119904+ 119877119901

119903

119877119903 cos 120579119903 sin120593 + 119877119903 sin 120579119903 cos120593119877119903 tan120573119903 minus 119877

119898

119903tan120573119903 + (120579119903 + 120593) 119877

119901

119903tan120572119903

1

]]]]]

]

(9)

Setting the screw coordinate system 119878119904(119874119904 119909119904 119910119904 119911119904) tocoincide with the fixed coordinate system along the screwsurface can now be calculated

997888119903 119904 =

[[[[[

[

119877119904 cos 120579119904119877119904 sin 120579119904

(119877119898

119904minus 119877119904) tan120573119904 + 120579119904119877

119901

119904tan120572119904

1

]]]]]

]

(10)

Similarly the nut surface is calculated as follows

997888119903 119899 =

[[[[[

[

119877119899 cos 120579119899119877119899 sin 120579119899

(119877119898

119899minus 119877119899) tan120573119899 cos120572119899 + 120579119899119877

119901

119899tan120572119899

1

]]]]]

]

(11)

The two surfaces will mesh with each other if a pointwhere their position vectors are equal exists when the normalvectors are in the opposite direction That point is themeshing point According to this principle the system ofequations for calculating the meshing point between thescrew and the roller can be listed [14]

997888119903 119904 =

997888119903 119903

997888119873119904 = 120582

997888119873119903 (120582 = 0)

(12)

The normal vectors can be solved using the positionvector as follows

997888119873119904 =

120597997888119903 119904

120597119877119904

times120597997888119903 119904

120597120579119904

=[[

[

119877119904 cos 120579119904 tan120573119904 + 119877119901

119904sin 120579119904 tan120572119904

119877119904 sin 120579119904 tan120573119904 minus 119877119901

119904cos 120579119904 tan120572119904

119877119904

]]

]

997888119873119903 =

120597997888119903 119903

120597119877119903

times120597997888119903 119903

120597120579119903

=[[

[

119877119901

119903sin (120579119903 + 120593) tan120572119903minus119877119903 cos (120579119903 + 120593) tan120573119903

minus119877119901

119903cos (120579119903 + 120593) tan120572119903minus119877119903 sin (120579119903 + 120593) tan120573119903

119877119903

]]

]

(13)Based on formulas (12) and (13) the meshing equations

for the meshing between the screw and the roller can becalculated as shown in the following formula

119877119904 cos 120579119904 minus 119877119903 cos (120579119903 + 120593) minus 119877119901

119904minus 119877119901

119903= 0

119877119904 sin 120579119904 minus 119877119903 sin (120579119903 + 120593) = 0

minus 119877119904 tan120573119904 + 119877119898

119904tan120573119904 + 119877

119898

119904120579119904 tan120572119904

minus 119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119904119877119903 cos 120579119904 tan120573119904 + 119877119903119877119901

119904sin 120579119904 tan120572119904

+ 119877119904119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119904119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119904119877119903 sin 120579119904 tan120573119904 minus 119877119903119877119901

119904cos 120579119904 tan120572119904

+ 119877119904119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119904119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0

(14)The equations of the meshing between the nut and the

roller are similar to that of the meshing between the screwand the roller as shown in the following formula


119904minus 119877119901

119903= 0

119877119899 sin 120579119899 minus 119877119903 sin (120579119903 + 120593) = 0

minus119877119899 tan120573119899 + 119877119898

119899tan120573119899 + 119877

119898

119899120579119899 tan120572119899

minus119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119899119877119903 cos 120579119899 tan120573119899 + 119877119903119877119901

119899sin 120579119899 tan120572119899

+ 119877119899119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119899119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119899119877119903 sin 120579119899 tan120573119899 minus 119877119903119877119901

119899cos 120579119899 tan120572119899

+ 119877119899119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119899119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0

(15)In formula (14) 119877119904 119877119903 120579119904 120579119903 and 120593 are the parameters

that determine the position of themeshing point and the poseof the roller This nonlinear equations set cannot be solvedusing an analytical method but by applying a numericalmethod the solution can be calculated if given a set ofparameters


4 Analysis of the Meshing betweenRoller and Nut

The meshing points between the nut and the roller arecalculated based on the cross section which can be confirmedby theoretical analysis [6] A conclusion that the helical angleof the nut thread equals that of the roller thread can bedrawn by applying formulas (1) (4) and (7) This meansthat the nut and the roller thread surface have the samecircumferential inclination The same vertex angles of theroller and the nut threads mean that the inclination of theradius direction is also equal Thus the point which is onthe cross section and in the pitch radius shares the samenormal direction on the nut and the roller thread surfaceThispoint matches the meshing demonstrated in (15) this is themeshing point The meshing on the cross section makes themeshing characteristics relatively simple the contact radius isthe pitch radius and the contact angle is zero This relativelyfixed contactmakes the transmission between the nut and therollers stable and easy to analyze

So the remaining analysis will emphasize the meshingbetween the screw and the rollers which have an obviouseffect on the transmission process

5 Analysis of the Meshing betweenRoller and Screw

Unlike the nut the screwmesheswith rollers outside the crosssection If the screw and roller mesh with each other in thecross section that will make 120579119904 equal to 0 If 120579119904 = 0 is takeninto formula (14) it could be solved that

119877119904 = 119877119901

119904

119877119903 = 119877119901

119903

120572119904 = minus120572119903

120573119904 = minus120573119903

(16)

This means that when meshing with each other theaxial displacement generated by rotation of roller equals theaxial displacement along the screw thread but of differentdirections So there are no axial relative movements betweenroller and screw and the rotarymovement of screwwill not beconverted into linear displacementThus the roller and screwmust mesh outside the cross section

The core parameter for the meshing behavior betweenroller and screw is the helical angles which are determinedby 119901 and 119877

119901 The helical angle of screw is different from thatof roller they are mutually independent and will influencethe meshing characteristics together The vertex angle is alsoimportant parameter which affects the meshing betweenroller and screw

51 Contact Lines The meshing pointrsquos distribution largelydetermines the meshing characteristics of the roller screwbecause it is the basis for themotion of themeshing point andthe force of the transfer between the screw and the rollersThe solution to formula (15) gives us the contact line of

the meshing surface Though the continuous function of thecontact line could not be solved the numerical solutions ofa series of the meshing point can be formed by the contactcurve According to the circumferential symmetry of themeshing the mesh area is the helical surface generated whenthe contact line rotates along the axis Figure 5(a) showsthe meshing between the screw and the roller threads andFigure 5(b) shows the specific features of the meshing pointThe screw and the roller threads have helical angles of differ-ent size and directions so the helical lines cross with eachother An appropriate incline and twist make the crossingsurfaces mesh with each other The difference between thehelical angles ensures that the meshing points are located onone side of the cross section The upperbottom surface ofscrew thread meshes with the bottomupper surface of theroller thread thus forming a contact line

The contact line is a basic and an important piece ofdata for the study of meshing characteristics because itreflects some important meshing parameters such as thecontact radius and the contact angle It also has a closeconnection with the transmission ratio the edge contactand some of the other meshing performance measurementsThe contact radius 119877119904 (119877119903) and the contact angle 120579119904 (120579119903) arenot structure parameters but they are parameters used tolocate themeshing points they aremeasured in a planewhichcontains the meshing point and are vertical to the axes

Apply numerical method to calculate the locations ofmeshing points and to determine the contact line and themeshing area During the calculation the screw pitch radiusis 975mm the roller pitch radius is 325mm and the numberof screw threads is 3 The helical angle and the vertex angletake different values and the other parameters are worked outbased on the parameters already provided

The calculation results show that the surfaces of thescrew and the roller are the contact line and the meshingline is always on only one side of the cross section InFigure 6 contact lines are calculated using the condition ofthe different helical angles while other parameters remainthe same 1198601015840119860 has a minimal helical angle of 12∘ which iscalculated with a 04mm pitch and gradually increases tothe maximum of 54∘ when it comes to 119868

1015840119868 with a pitch

of 2mm the pitch of each contact line is 02mm largerthan that of its right side It can obviously be seen that thecontact lines move along the circumferential direction andthe shapes of the curves also change On a single contact linethe meshing points near the root demonstrate more obviousposition changes this fact makes the contact line longerand also increases the incline in the direction of the radiusA longer contact line means less contact force which mayhelp reduce deformation especially if it is paired with largercontact tooth thicknessThus the increasing helical angle willenhance the load capacity of the roller screw Unfortunatelya longer and more inclined contact line may also increase therisk of generating interference which means that the helicalangle should be set within a certain limited range

As is shown in Figure 7 the value of the vertex angle alsoinfluences the position and the shape of the contact line Thecontact line on the left is the smallest and each line increasesby 75∘ until it reaches the largest contact line The increase of


8

6

4

2

0

minus2

minus10 minus5 0 5 10 15

minus10

minus5

0

5

10

(a) (b)

Figure 5 Meshing between the screw and the roller

105

10

95

93 2 1 0 minus1 minus2 minus3

(mm)

120572 = 176∘

(p = 2mm)120572 = 353

(p = 04mm)Cross section

External radiusPitch radius Thread bottom

radius

120573 = 45∘

I HG FEDCB A

I998400H998400 G998400 F998400 E998400 D998400 C998400 B998400 A998400(m

m)

Figure 6The contact lines under different helical angle conditions

105

10

95


(mm)

(mm

)

Cross section

External radius

Pitch radius Thread bottom radius

120573 = 45∘120573 = 30∘ 120573 = 60∘

120572 = 176∘

(p = 2)

Figure 7 Contact lines under different vertex angle conditions

the vertex angle moves the contact line in the cross sectionand makes it shorter and closer to the radial direction At thesame time it makes it more curved which may also causeinterference

The distribution of the contact line will to some extentreflect the meshing quality For one pair of meshing threadsthemeshing quality can be evaluated based on certain aspectsof the contact lines First the distance from the contact lineto the cross section largely determines the contact radius andthe contact angle which are proven to be important factorsfor the principle of curvature [6] The length of the contactline directly affects the contact force The inclination of thecontact line is also of great importance because a steeplyinclined contact line will have a substantial circumferential

distance which increases the axial displacement and is morelikely to result in interference

The line contact could enhance the loading ability of theplanetary roller screw and however may also increase therisk of interference In this paper a function 119891int(119877 120579) is builtto indicate the interference and help evaluate the interferencerisk119891int(119877 120579) is the 119911-axis distance between the points whichshare the same location in 119909119910 plane and exist respectively onscrew and roller surface It can be calculated using formulas(9) and (10)

119891int (119877 120579) = minus119877 tan120573119904 + 120579119877119898

119904tan120572119904 minus 119877119903 tan120573119903

minus (120579 + 120593) 119877119898

119903tan120572119903 + 119877

119898

119904tan120573119904 + 119877

119898

119903tan120573119903

(17)

In formula (17)

119877119903 =radic(119877119901

119904 + 119877119901

119903 )2+ 1198772 minus 2119877 (119877

119901

119904 + 119877119901

119903 ) cos 120579

120579 + 120593 = 120587 minus arctan(119877 sin 120579

119877119903

)

(18)

If the value of 119891int(119877 120579) is positive it means there is nointerference 119891int(119877 120579) lt 0 means that interference and theinterference area are shown in Figure 8 of the different helicalangles and Figure 9 of different vertex angles

It can be seen from Figure 8 that the interference area isvarying with different helical angles When the thread has asmaller helical angle of 120572 = 34

∘ the interference will onlyprobably take place on the very edge of the thread and couldbe avoided by modifying the root or addendum Along withthe increasing of helical angle the interference area spreadsto almost the whole breadth of thread which will disable thethreads meshing

It can be seen in Figure 9 that just like the helical angle achanging vertex angle also has influence on the interferencearea and a larger vertex angle benefits avoiding interference


105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘

Figure 8 Meshing interference area under different helical angleconditions

105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm

Figure 9 Meshing interference area under different vertex angleconditions

Based on the analysis of the contact line of planetary rollerscrew it can be concluded that the straight line thread rollerscrew can be applied in condition of small helical angle andrelatively large vertex angle Under these conditions the linecontact will increase the contact points and enhance the loadcapacity With the helical angle increasing the line contactplanetary roller screw will face the risk of interference andneed to be modified

52 Meshing Points In the meshing process the contact linewill create an uncertain meshing radius and is more sensitiveto manufacturing and installation errors so the profile of thethread surface needs to bemodifiedThemodified profile cantake many shapes and will make the surface more complexbut the purpose and the principle of modifying the threadsurface are the same The modified profile must retain theindispensable contact points and avoid other contact pointsThe meshing point between the nut and the roller is on thecross section and in the pitch radius these locations canguarantee a relatively static interaction between the roller andthe nut so they are chosen as the points to be retained whilethe other points on the contact line must be eliminated Forthe surface most areas except the line along pitch diametersare modified to be of a certain shape to avoid unnecessarycontact but the specific shape does not affect the meshingpoint calculation So the calculation for solving the meshingpoint of the modified surface can be simplified by using thehelical line on the pitch diameter as the roller to solve (14)The helical line is separate from the roller thread surface soit has the same normal vectors as is on the surface In thisway the meshing points of the modified thread surface canbe calculated

The meshing points between the roller and the screwcan still be calculated when the helical angle changes as isshown in Figure 8 As the meshing points change from therightmost point which has the smallest helical angle of 12∘

to the leftmost point with the maximal helical angle of 146∘the meshing points demonstrate a trend that moves awayfrom the pitch radius toward the thread crest where the tooththickness is thinner and away from the cross section justas the contact lines do The meshing points would even gobeyond the boundary of the thread surface if the helical angleis too large It can be seen that the contact radius will increaseif the contact angle became larger and similar regularitiescan be found among the meshing points with different vertexangles as is shown in Figure 9 A smaller contact angle ishelpful in order to get a meshing point away from the edgeof the thread so a smaller helical angle and a larger vertexangle improve the results when the distance is taken from thethread edge in accordance with the evaluation standard Thelocations of the meshing points still affect other performancefactors of the roller screw which makes the selection ofparameters far more complex than this process suggests Therest of this paper will demonstrate how the meshing pointlocation affects the transmission ratio which is one of themost important performance parameters of the roller screw

6 The Actual Transmission Ratio of anAsymmetric Meshing

The transmission ratio is one of the basic parameters ofthe roller screw which directly influences the selectionof different working conditions There exists a differencebetween the actual and the theoretical transmission ratiobecause the meshing points are considered as if they werelying on the cross section when calculating the theoreticaltransmission ratio however it can be found that the meshingpoints between the screw and the roller always have a certaindistance from the cross section which indicates that theactual transmission ratio is not likely to be the same as thetheoretical transmission ratio

The actual transmission ratio can be defined as the nutaxial displacement for one rotation of the screw and can becalculated using the ratio of the nut axial linear velocity andthe screw angular velocityThe nut axial linear velocity equalsthe axial component of the screw velocity which is indicatedby V119911 The relative velocity between the roller and the screwat the meshing point can be calculated using the followingformula

997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)

The relationship of the angular velocity to the screw andthe roller can be easily obtained as follows

120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)

The relative velocity at the meshing point must satisfy thefollowing equations

997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)


By taking formulas (13) and (19) into (20) V119911 can becalculated as follows

V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)

The actual transmission ratio 119894119886 is the quotient V119911 dividedby 120596119904

119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)

The actual transmission ratio can be calculated usingthe solutions to (15) as stated above because the helicalangle is an important parameter that affects transmissionperformance and the transmission ratio Though 120572119904 is notexplicitly stated in formula (22) 119877119904 119877119903 120579119904 120579119903 120593 and 119901119904 areall influenced by 120572119904 (Figure 12)

As is shown in Figure 10 the actual transmission ratioalmost proportionally increases with the growth of the helicalangle as does the theoretical transmission ratio The rateof increase observed in the actual transmission ratio islarger because as the meshing point deviates from the crosssection it may increase the actual contact radius It can beseen that a larger helical angle is helpful for enhancing thetransmission ratio but as has been proven above it willalso push the meshing location to the thinner thread edgeand cause interference problems which may be harmful Sothe selection of the helical angle must take several factorsinto account and must satisfy the main requirements asdetermined by the different application situations (Figure 13)

As has been illustrated previously the growth of thevertex angles will make the meshing point closer to the crosssectionThis means that there will be a smaller contact radiusand that decreases the actual transmission ratio as is shownin Figure 11 Altering the vertex angle will not change thetheoretical transmission ratio which can be seen in formula(4) however the actual transmission ratio can be lower thanthe theoretical one if there is a large vertex angle This isbecause the meshing points are quite close to the pitch radiuswhen there is a large vertex angle as is shown in Figure 9 sothe screw contact radius is almost the pitch radius (119877119904 = 119877

119901

119904)

however a contact angle must still exist So in the trianglethat is composed of 119877119904 119877119903 and 119877

119901

119904+ 119877119901

119903 the roller contact

radius 119877119903 is larger than 119877119901

119903and 119877119904119877119903 is relatively small This

may cause a relatively smaller transmission ratio which iswhy there is a lower actual transmission ratio in Figure 11

Obviously the actual transmission ratio is different fromthe theoretical transmission ratio because the contact radiusdiffers from the pitch radius In most cases the actual radiusis larger because of the deviation between the meshing pointand the cross section

105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius

Figure 10 Meshing points under different helical angle conditions

105

10

95


(mm)

Cross section External radius

Pitch radiusThread bottom radius

(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘

Figure 11 Meshing points under different vertex angle conditions

0 5 10 150

5

10

15

20

25

30

35

40

45

Helical angle of screw (deg)

Tran

smisi

son

ratio

(mm

)

Actual transmission ratioIdeal transmision ratio

Figure 12 Transmission ratios under different helical angle condi-tions

7 Conclusion

The research presented in this paper has developed a methodfor modeling the roller screw meshing behavior accuratelyand has analyzed the numerically calculated data usedto evaluate the meshing characteristics The mathematicalmodel established herein can reflect the actual roller screwby taking several factors which are ignored in the analyticmodel into consideration due to their actual significance Ashas been demonstrated the contact lines and the meshingpoints can be calculated and the actual transmission ratiocan be analyzed by applying a new calculation method Thehelical angle and the vertex angle are proven to be of greatsignificance for the actual meshing characteristicsThey havemanifested effects on the position of the meshing points and


20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9

Vertex angle of screw (deg)

Tran

smisi

son

ratio

(mm

)


Figure 13 Transmission ratios under different vertex angle condi-tions

will directly impact the tooth thickness and the tooth heightBased on this analysis a larger helical angle will enhance theload capacity and the transmission ratio of the roller screwthough it may cause interference problems A larger vertexangle will increase the load capacity significantly and helpsavoid interference however it may increase the contact forceand decrease the transmission ratioThe helical angle and thevertex angle are relatively independent parameters that canbe used together to make up for the disadvantages they eachcause separately Thus different values could be selected inorder to meet the requirements of different applications andto maximize the benefits of the roller screw for different uses

Nomenclature

Given Geometric Parameters

119877119901

119903 Pitch radius of the roller (325mm)

119877119901

119904 Pitch radius of the screw (975mm)

119877119901

119899 Pitch radius of the nut (1625mm)

119873119899 Number of threads on the nut (5)119873119904 Number of threads on the screw (3)

119901 Pitch (04mmsim6mm)119901119904 Lead of screw (12mmsim18mm)120573119903 120573119904 120573119899 Half-vertex-angle of the thread of roller

screw and nut (20∘ sim75∘)120582119901 Quotient of the thread tooth height

divided by the pitch (06sim1)120582119906 Proportion of the crest part to the tooth

height (035sim065)

Intermediate Parameters for Calculation

120572119903 120572119904 120572119899 Helical angle of roller screw and nut

119877119898 Maximal radius (distance from the point

where the upper surface of the threadmeets the lower one to the axis)

120573119903 120573119904 120573119899 Half thread vertex angle of roller screwand nut

120593 The initial position angle of the roller997888119903 119903

997888119903 119904

997888119903 119899 Mesh point position of the vector for the

roller screw and nut997888119873119903

997888119873119904

997888119873119899 Normal vector of the roller screw and nut

surface at the mesh point119877119903 119877119904 119877119899 Distance between the mesh points to the

axis of the roller screw and nut120579 Contact angle (the angle between 997888

119903 andthe cross section)

997888V 119904119903 Relative velocity between the roller andthe screw

120596119903 120596119904 Angular velocity of the roller and thescrew

Performance Parameters

119894119886 Actual transmission ratio119894 Theoretical transmission ratio

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was supported by Project 51305010 supported byNational Natural Science Foundation of China and ProjectYCSJ-03-2013-06 supported by theGraduate Innovative Prac-tices Foundation of Beihang University

References

[1] W M Waide ldquoAircraft landing gear with integrated extensionretraction and leveling featurerdquo US Patent 7942366 2011

[2] M S Worthington T A Beets J H Beno et al ldquoDesign anddevelopment of a high-precision high-payload telescope dual-drive systemrdquo in Ground-Based and Airborne Telescopes III vol7733 of Proceedings of SPIE July 2010

[3] T Buschmann S Lohmeier and H Ulbrich ldquoHumanoid robotLola design and walking controlrdquo Journal of Physiology Parisvol 103 no 3-5 pp 141ndash148 2009

[4] P C Lemor ldquoThe roller screw an efficient and reliable mechan-ical component of electro-mechanical actuatorsrdquo in Proceedingsof the 31st Intersociety EnergyConversion EngineeringConference(IECEC rsquo96) pp 215ndash220 August 1996

[5] M H Jones and S A Velinsky ldquoKinematics of roller migrationin the planetary roller screwmechanismrdquo Journal of MechanicalDesign Transactions of the ASME vol 134 no 6 Article ID061006 2012

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


[7] 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

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

[9] G Liu and S Ma ldquoNew development and key technology ofplanetary roller screwrdquo Journal of Mechanical Transmission vol36 no 5 pp 103ndash108 2012 (Chinese)

[10] Y Zhao J Ni and L Lv ldquoMeshing calculation of roller screwrdquoJournal of Machine Design vol 3 no 20 pp 34ndash36 2003(Chinese)

[11] R Su ldquoPrinciple analysis of roller screwrdquo Journal of SichuanInstitute of Technology vol 17 no 3 pp 33ndash36 1998 (Chinese)

[12] Y Q Liu and J S Wang ldquoSimulation of crossing threaded plan-etary roller screw engagementrdquo Advanced Materials Researchvol 889 pp 518ndash526 2014

[13] ZWei and J Yang ldquoOptimized analysis on structural parameterfor planetary roller screwrdquo Journal of Mechanical Transmissionvol 35 no 6 pp 44ndash47 2011 (Chinese)

[14] F L Litvin and A Fuentes Gear Geometry and Applied TheoryCambridge University Press Cambridge UK 2004

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


1

2

3456

(1) Nut(2) Roller(3) Screw

(4) Inner ring gear(5) Retaining plate(6) Wire ring

Figure 1 Structure of planetary roller screw

the rollers [9] which are symmetrically distributed aroundthe screw The gear system supports the rollers and keepstheir symmetric position The screw and the nut generallyuse multiple threads while the rollers use a single threadbut share the same pitch [13] The screw and the nut alwayshave triangular threads but the roller can have a triangularthread or a rounded threadWhenmadewith highmachiningaccuracy and precise assembly the triangular profile of theroller thread will lead to a line contact which can provide ahigher load capacity The vertex angle of the thread is usually90 degrees [10] but other values are also possible

During the transmission process the nut and the rollersneed to stay static in the axial direction so that there is noslide between their thread surfaces [11]This requires the axialdisplacement of the roller because revolving along the nutthread equalizes the rotation axial displacement as shown inthe following equation

Δ120575 sdot 119877119901

119903

119877119901

119899

sdot119901 sdot 119873119899

2120587=

Δ120575 sdot 119901

2120587 (1)

Δ120575 indicates the angular displacement in the roller threadwhile meshing Equation (1) can be simplified to 119873119899 =

119877119901

119904119877119901

119903+2 which is a rule that must be observed to guarantee

that the roller will not move out from under the nut duringthe transmission process

Assuming there is no slide between these componentsthe axial movement of the nut when it is fixed circumfer-entially and the screw when it is rotated for one revolutionis defined as the transmission ratio of the roller screw Thetheoretical transmission ratio of the roller screw is shownin formula (2) It can easily be seen that the transmissionratio is determined by the pitch the lead of the screw threadthe thread directions and the quotient of the screw radiusdivided by the roller radius Thus it is possible to achievehigh speed transmission or precise positioning by choosingreasonable parameters and it is also possible to obtain a widerange of transmission ratios

119894 = 119901119904 + 119896119901 sdot 119877119901

119904

119877119901

119903

(2)

In formula (2) 119896 is the roller displacement directioncoefficient 1 if the thread of the rollers takes the same helicaldirection as the screw minus1 if the thread of the rollers takes theopposite direction as the screw and 0 for plane rollers

The theoretical transmission ratio is calculated using thepitch radius based on the assumption that themeshing pointslie on the cross section (the plane is determined by the screwand the roller axes) But the thread surfaces rise and twistat different degrees which means that the meshing pointsbetween the screw and the roller actually exist mostly on onlythe one side of the cross section Considering the axial freemovement of the roller and the threads when crossing witheach other the actual transmission ratio should be calculatedusing the relative velocity as the screw and the roller conjugateacross these surfaces at the meshing location

3 A Model of the Screw and Nut Meshing withthe Roller

31 A Model of the Screw Nut and Roller Surfaces Thesurfaces of the screw the nut and the roller are all triangularthreads and they share the same modeling method whichmeans that the formulas for the thread surfacemodeling omitthe subscripts (eg 119877 is used instead of using 119877119904 119877119899 or 119877119903

when developing the surface equation for the screw the nutor the roller) As is shown in Figure 2 the coordinate system119878(119874 119909 119910 119911) is fixed with the screwnutroller at the centerof an end face The coordinate system 1198783(1198743 1199093 1199103 1199113) inwhich the thread profile is measured rises along the helicalline with its origin 119877

119901 from the axis and with its 11990931199113 planeperpendicular to the helical line

The transformationmatrix used to convert the coordinatesystem 1198783(1198743 1199093 1199103 1199113) to 119878(119874 119909 119910 119911) can be developedfrom Figure 2 resulting in the following matrix formula

3

0119879 =

[[[[[

[

cos 120579 minus cos120572 sin 120579 sin120572 sin 120579 119877119901 cos 120579

sin 120579 cos120572 cos 120579 minus sin120572 cos 120579 119877119901 sin 120579

0 sin120572 cos120572 120579119877119901 tan120572

0 0 0 1

]]]]]

]

(3)

where 120572 is the helical angle which can be calculated with thefollowing formula

120572 = arctan(119873 sdot 119901

2120587119877119901) (4)

The vertex angle is measured from the cross sectionas shown in Figure 3 Point 119864 is on the thread surface1198641199093

1198641199103 and 1198641199113

are the coordinates on axes 1199093 1199103 and1199113 The coordinate of point 119864 in the coordinate system of1198783(1198743 1199093 1199103 1199113) is as shown in the following formula

997888119903 3 =

[[[[[

[

119877 minus 119877119901

(119877119898

minus 119877) tan120573 sin120572

(119877119898

minus 119877) tan120573 cos1205721

]]]]]

]

(5)


z(z1)

y1

z2

z3y2

y3

O1

O2(O3)

O

y120579

120572

x

Rm

x1(x2 x3)


z2z3

y2

y3

Ez3

E

Ex3

Ey3

120572

120572

P

O3(O2)

120573

x2(x3)






997888119903 =3

0119879997888119903 3 =

[[[[[

[

119877 cos 120579119877 sin 120579

(119877119898

minus 119877) tan120573 + 120579119877119901 tan120572

1

]]]]]

]

(6)



119877119898

= 119877119901+ 120582119901120582119906119901 cot120573 (7)



119903

0119879 =

[[[[[

[

cos120593 minus sin120593 0 119877119901

119904+ 119877119901

119903

sin120593 cos120593 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(8)


yr

xs

xr

0

M

ys

120579r120579s

RsRr

Rms

Rps

Rpr

Rmr



997888119903 119903 =119903

0119879997888119903 0 =

[[[[[

[


119904+ 119877119901

119903


119898

119903tan120573119903 + (120579119903 + 120593) 119877

119901

119903tan120572119903

1

]]]]]

]

(9)


997888119903 119904 =

[[[[[

[

119877119904 cos 120579119904119877119904 sin 120579119904

(119877119898

119904minus 119877119904) tan120573119904 + 120579119904119877

119901

119904tan120572119904

1

]]]]]

]

(10)


997888119903 119899 =

[[[[[

[

119877119899 cos 120579119899119877119899 sin 120579119899

(119877119898

119899minus 119877119899) tan120573119899 cos120572119899 + 120579119899119877

119901

119899tan120572119899

1

]]]]]

]

(11)


997888119903 119904 =

997888119903 119903

997888119873119904 = 120582

997888119873119903 (120582 = 0)

(12)


997888119873119904 =

120597997888119903 119904

120597119877119904

times120597997888119903 119904

120597120579119904

=[[

[

119877119904 cos 120579119904 tan120573119904 + 119877119901

119904sin 120579119904 tan120572119904

119877119904 sin 120579119904 tan120573119904 minus 119877119901

119904cos 120579119904 tan120572119904

119877119904

]]

]

997888119873119903 =

120597997888119903 119903

120597119877119903

times120597997888119903 119903

120597120579119903

=[[

[

119877119901


minus119877119901


119877119903

]]

]




119904minus 119877119901

119903= 0

119877119904 sin 120579119904 minus 119877119903 sin (120579119903 + 120593) = 0

minus 119877119904 tan120573119904 + 119877119898

119904tan120573119904 + 119877

119898

119904120579119904 tan120572119904

minus 119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119904119877119903 cos 120579119904 tan120573119904 + 119877119903119877119901

119904sin 120579119904 tan120572119904

+ 119877119904119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119904119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119904119877119903 sin 120579119904 tan120573119904 minus 119877119903119877119901

119904cos 120579119904 tan120572119904

+ 119877119904119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119904119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0




119904minus 119877119901

119903= 0

119877119899 sin 120579119899 minus 119877119903 sin (120579119903 + 120593) = 0

minus119877119899 tan120573119899 + 119877119898

119899tan120573119899 + 119877

119898

119899120579119899 tan120572119899

minus119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119899119877119903 cos 120579119899 tan120573119899 + 119877119903119877119901

119899sin 120579119899 tan120572119899

+ 119877119899119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119899119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119899119877119903 sin 120579119899 tan120573119899 minus 119877119903119877119901

119899cos 120579119899 tan120572119899

+ 119877119899119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119899119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0









119877119904 = 119877119901

119904

119877119903 = 119877119901

119903

120572119904 = minus120572119903

120573119904 = minus120573119903

(16)









1015840119868 with a pitch




8

6

4

2

0

minus2


minus10

minus5

0

5

10

(a) (b)


105

10

95


(mm)

120572 = 176∘

(p = 2mm)120572 = 353



radius

120573 = 45∘

I HG FEDCB A

I998400H998400 G998400 F998400 E998400 D998400 C998400 B998400 A998400(m

m)


105

10

95


(mm)

(mm

)

Cross section

External radius


120573 = 45∘120573 = 30∘ 120573 = 60∘

120572 = 176∘

(p = 2)






119891int (119877 120579) = minus119877 tan120573119904 + 120579119877119898

119904tan120572119904 minus 119877119903 tan120573119903

minus (120579 + 120593) 119877119898

119903tan120572119903 + 119877

119898

119904tan120573119904 + 119877

119898

119903tan120573119903

(17)

In formula (17)

119877119903 =radic(119877119901

119904 + 119877119901

119903 )2+ 1198772 minus 2119877 (119877

119901

119904 + 119877119901

119903 ) cos 120579

120579 + 120593 = 120587 minus arctan(119877 sin 120579

119877119903

)

(18)






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘


105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm









997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)


120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)


997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)



V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










z(z1)

y1

z2

z3y2

y3

O1

O2(O3)

O

y120579

120572

x

Rm

x1(x2 x3)


z2z3

y2

y3

Ez3

E

Ex3

Ey3

120572

120572

P

O3(O2)

120573

x2(x3)






997888119903 =3

0119879997888119903 3 =

[[[[[

[

119877 cos 120579119877 sin 120579

(119877119898

minus 119877) tan120573 + 120579119877119901 tan120572

1

]]]]]

]

(6)



119877119898

= 119877119901+ 120582119901120582119906119901 cot120573 (7)



119903

0119879 =

[[[[[

[

cos120593 minus sin120593 0 119877119901

119904+ 119877119901

119903

sin120593 cos120593 0 0

0 0 1 0

0 0 0 1

]]]]]

]

(8)


yr

xs

xr

0

M

ys

120579r120579s

RsRr

Rms

Rps

Rpr

Rmr



997888119903 119903 =119903

0119879997888119903 0 =

[[[[[

[


119904+ 119877119901

119903


119898

119903tan120573119903 + (120579119903 + 120593) 119877

119901

119903tan120572119903

1

]]]]]

]

(9)


997888119903 119904 =

[[[[[

[

119877119904 cos 120579119904119877119904 sin 120579119904

(119877119898

119904minus 119877119904) tan120573119904 + 120579119904119877

119901

119904tan120572119904

1

]]]]]

]

(10)


997888119903 119899 =

[[[[[

[

119877119899 cos 120579119899119877119899 sin 120579119899

(119877119898

119899minus 119877119899) tan120573119899 cos120572119899 + 120579119899119877

119901

119899tan120572119899

1

]]]]]

]

(11)


997888119903 119904 =

997888119903 119903

997888119873119904 = 120582

997888119873119903 (120582 = 0)

(12)


997888119873119904 =

120597997888119903 119904

120597119877119904

times120597997888119903 119904

120597120579119904

=[[

[

119877119904 cos 120579119904 tan120573119904 + 119877119901

119904sin 120579119904 tan120572119904

119877119904 sin 120579119904 tan120573119904 minus 119877119901

119904cos 120579119904 tan120572119904

119877119904

]]

]

997888119873119903 =

120597997888119903 119903

120597119877119903

times120597997888119903 119903

120597120579119903

=[[

[

119877119901


minus119877119901


119877119903

]]

]




119904minus 119877119901

119903= 0

119877119904 sin 120579119904 minus 119877119903 sin (120579119903 + 120593) = 0

minus 119877119904 tan120573119904 + 119877119898

119904tan120573119904 + 119877

119898

119904120579119904 tan120572119904

minus 119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119904119877119903 cos 120579119904 tan120573119904 + 119877119903119877119901

119904sin 120579119904 tan120572119904

+ 119877119904119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119904119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119904119877119903 sin 120579119904 tan120573119904 minus 119877119903119877119901

119904cos 120579119904 tan120572119904

+ 119877119904119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119904119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0




119904minus 119877119901

119903= 0

119877119899 sin 120579119899 minus 119877119903 sin (120579119903 + 120593) = 0

minus119877119899 tan120573119899 + 119877119898

119899tan120573119899 + 119877

119898

119899120579119899 tan120572119899

minus119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119899119877119903 cos 120579119899 tan120573119899 + 119877119903119877119901

119899sin 120579119899 tan120572119899

+ 119877119899119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119899119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119899119877119903 sin 120579119899 tan120573119899 minus 119877119903119877119901

119899cos 120579119899 tan120572119899

+ 119877119899119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119899119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0









119877119904 = 119877119901

119904

119877119903 = 119877119901

119903

120572119904 = minus120572119903

120573119904 = minus120573119903

(16)









1015840119868 with a pitch




8

6

4

2

0

minus2


minus10

minus5

0

5

10

(a) (b)


105

10

95


(mm)

120572 = 176∘

(p = 2mm)120572 = 353



radius

120573 = 45∘

I HG FEDCB A

I998400H998400 G998400 F998400 E998400 D998400 C998400 B998400 A998400(m

m)


105

10

95


(mm)

(mm

)

Cross section

External radius


120573 = 45∘120573 = 30∘ 120573 = 60∘

120572 = 176∘

(p = 2)






119891int (119877 120579) = minus119877 tan120573119904 + 120579119877119898

119904tan120572119904 minus 119877119903 tan120573119903

minus (120579 + 120593) 119877119898

119903tan120572119903 + 119877

119898

119904tan120573119904 + 119877

119898

119903tan120573119903

(17)

In formula (17)

119877119903 =radic(119877119901

119904 + 119877119901

119903 )2+ 1198772 minus 2119877 (119877

119901

119904 + 119877119901

119903 ) cos 120579

120579 + 120593 = 120587 minus arctan(119877 sin 120579

119877119903

)

(18)






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘


105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm









997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)


120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)


997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)



V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










yr

xs

xr

0

M

ys

120579r120579s

RsRr

Rms

Rps

Rpr

Rmr



997888119903 119903 =119903

0119879997888119903 0 =

[[[[[

[


119904+ 119877119901

119903


119898

119903tan120573119903 + (120579119903 + 120593) 119877

119901

119903tan120572119903

1

]]]]]

]

(9)


997888119903 119904 =

[[[[[

[

119877119904 cos 120579119904119877119904 sin 120579119904

(119877119898

119904minus 119877119904) tan120573119904 + 120579119904119877

119901

119904tan120572119904

1

]]]]]

]

(10)


997888119903 119899 =

[[[[[

[

119877119899 cos 120579119899119877119899 sin 120579119899

(119877119898

119899minus 119877119899) tan120573119899 cos120572119899 + 120579119899119877

119901

119899tan120572119899

1

]]]]]

]

(11)


997888119903 119904 =

997888119903 119903

997888119873119904 = 120582

997888119873119903 (120582 = 0)

(12)


997888119873119904 =

120597997888119903 119904

120597119877119904

times120597997888119903 119904

120597120579119904

=[[

[

119877119904 cos 120579119904 tan120573119904 + 119877119901

119904sin 120579119904 tan120572119904

119877119904 sin 120579119904 tan120573119904 minus 119877119901

119904cos 120579119904 tan120572119904

119877119904

]]

]

997888119873119903 =

120597997888119903 119903

120597119877119903

times120597997888119903 119903

120597120579119903

=[[

[

119877119901


minus119877119901


119877119903

]]

]




119904minus 119877119901

119903= 0

119877119904 sin 120579119904 minus 119877119903 sin (120579119903 + 120593) = 0

minus 119877119904 tan120573119904 + 119877119898

119904tan120573119904 + 119877

119898

119904120579119904 tan120572119904

minus 119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119904119877119903 cos 120579119904 tan120573119904 + 119877119903119877119901

119904sin 120579119904 tan120572119904

+ 119877119904119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119904119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119904119877119903 sin 120579119904 tan120573119904 minus 119877119903119877119901

119904cos 120579119904 tan120572119904

+ 119877119904119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119904119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0




119904minus 119877119901

119903= 0

119877119899 sin 120579119899 minus 119877119903 sin (120579119903 + 120593) = 0

minus119877119899 tan120573119899 + 119877119898

119899tan120573119899 + 119877

119898

119899120579119899 tan120572119899

minus119877119903 tan120573119903+119877119898

119903tan120573119903 minus 119877

119901

119903120579119903 tan120572119903 = 0

119877119899119877119903 cos 120579119899 tan120573119899 + 119877119903119877119901

119899sin 120579119899 tan120572119899

+ 119877119899119877119903 cos (120579119903 + 120593) tan120573119903 minus 119877119899119877119901

119903sin (120579119903 + 120593) tan120572119903 = 0

119877119899119877119903 sin 120579119899 tan120573119899 minus 119877119903119877119901

119899cos 120579119899 tan120572119899

+ 119877119899119877119903 sin (120579119903 + 120593) tan120573119903 + 119877119899119877119901

119903cos (120579119903 + 120593) tan120572119903 = 0









119877119904 = 119877119901

119904

119877119903 = 119877119901

119903

120572119904 = minus120572119903

120573119904 = minus120573119903

(16)









1015840119868 with a pitch




8

6

4

2

0

minus2


minus10

minus5

0

5

10

(a) (b)


105

10

95


(mm)

120572 = 176∘

(p = 2mm)120572 = 353



radius

120573 = 45∘

I HG FEDCB A

I998400H998400 G998400 F998400 E998400 D998400 C998400 B998400 A998400(m

m)


105

10

95


(mm)

(mm

)

Cross section

External radius


120573 = 45∘120573 = 30∘ 120573 = 60∘

120572 = 176∘

(p = 2)






119891int (119877 120579) = minus119877 tan120573119904 + 120579119877119898

119904tan120572119904 minus 119877119903 tan120573119903

minus (120579 + 120593) 119877119898

119903tan120572119903 + 119877

119898

119904tan120573119904 + 119877

119898

119903tan120573119903

(17)

In formula (17)

119877119903 =radic(119877119901

119904 + 119877119901

119903 )2+ 1198772 minus 2119877 (119877

119901

119904 + 119877119901

119903 ) cos 120579

120579 + 120593 = 120587 minus arctan(119877 sin 120579

119877119903

)

(18)






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘


105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm









997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)


120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)


997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)



V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119877119904 = 119877119901

119904

119877119903 = 119877119901

119903

120572119904 = minus120572119903

120573119904 = minus120573119903

(16)









1015840119868 with a pitch




8

6

4

2

0

minus2


minus10

minus5

0

5

10

(a) (b)


105

10

95


(mm)

120572 = 176∘

(p = 2mm)120572 = 353



radius

120573 = 45∘

I HG FEDCB A

I998400H998400 G998400 F998400 E998400 D998400 C998400 B998400 A998400(m

m)


105

10

95


(mm)

(mm

)

Cross section

External radius


120573 = 45∘120573 = 30∘ 120573 = 60∘

120572 = 176∘

(p = 2)






119891int (119877 120579) = minus119877 tan120573119904 + 120579119877119898

119904tan120572119904 minus 119877119903 tan120573119903

minus (120579 + 120593) 119877119898

119903tan120572119903 + 119877

119898

119904tan120573119904 + 119877

119898

119903tan120573119903

(17)

In formula (17)

119877119903 =radic(119877119901

119904 + 119877119901

119903 )2+ 1198772 minus 2119877 (119877

119901

119904 + 119877119901

119903 ) cos 120579

120579 + 120593 = 120587 minus arctan(119877 sin 120579

119877119903

)

(18)






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘


105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm









997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)


120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)


997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)



V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










8

6

4

2

0

minus2


minus10

minus5

0

5

10

(a) (b)


105

10

95


(mm)

120572 = 176∘

(p = 2mm)120572 = 353



radius

120573 = 45∘

I HG FEDCB A

I998400H998400 G998400 F998400 E998400 D998400 C998400 B998400 A998400(m

m)


105

10

95


(mm)

(mm

)

Cross section

External radius


120573 = 45∘120573 = 30∘ 120573 = 60∘

120572 = 176∘

(p = 2)






119891int (119877 120579) = minus119877 tan120573119904 + 120579119877119898

119904tan120572119904 minus 119877119903 tan120573119903

minus (120579 + 120593) 119877119898

119903tan120572119903 + 119877

119898

119904tan120573119904 + 119877

119898

119903tan120573119903

(17)

In formula (17)

119877119903 =radic(119877119901

119904 + 119877119901

119903 )2+ 1198772 minus 2119877 (119877

119901

119904 + 119877119901

119903 ) cos 120579

120579 + 120593 = 120587 minus arctan(119877 sin 120579

119877119903

)

(18)






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘


105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm









997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)


120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)


997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)



V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 172∘120572 = 103∘ 120572 = 34∘


105

10

95


(mm)

(mm

)

120573 = 225∘ 120573 = 675∘120573 = 45∘ 120572 = 34∘

p = 004mm









997888V 119904119903 = (997888120596119903 minus

997888120596 119904) times

997888119903 119904 minus

997888120596119903 times (119877

119901

119904119894119904) + V119911119896119904 (19)


120596119904

120596119903

=2119877119901

119903

119877119901

119904

(20)


997888V 119904119903 sdot997888119873119904 =

997888V 119904119903 sdot997888119873119903 = 0 (21)



V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V119911 =119877119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos (120579119903 + 120593 minus 120579119904) +

119901119904

2120587sdot 120596119904

minus119877119901

119903

119877119904

sdot119901119904

2120587sdot 120596119903 cos 120579119904 + 119877

119901

119904tan120573119904120596119903 sin 120579119904

(22)


119894119886 =2120587V119911120596119904

=119901119904119877119903119877

119901

119904

2119877119904119877119901

119903

cos (120579119903 + 120593 minus 120579119904) + 119901119904

minus119901119904119877119901

119904

2119877119904

cos 120579119904 +120587 (119877119901

119904)2

119877119901

119903

tan120573119904 sin 120579119904

(23)




119901

119904)


119901

119904+ 119877119901






105

10

95


(mm)

(mm

)

120573 = 45∘120572 = 146∘120572 = 12∘

Cross section


radius


105

10

95


(mm)



(mm

)

120572 = 12∘

120573 = 45∘120573 = 15∘ 120573 = 75∘


0 5 10 150

5

10

15

20

25

30

35

40

45


Tran

smisi

son

ratio

(mm

)



7 Conclusion



20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20 30 40 50 60 70 80 90 100 110 1205

55

6

65

7

75

8

85

9


Tran

smisi

son

ratio

(mm

)




Nomenclature


119877119901


119877119901


119877119901






height (035sim065)







997888119903 119904



997888119873119904











Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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