MSMechanical SciencesMSMech. Sci.2191-916XCopernicus PublicationsGöttingen, Germany10.5194/ms-7-209-2016Design formulae for a concave convex arc line gear mechanismChenYangzhimeyzchen@scut.edu.cnYaoLiSchool of Mechanical and Automotive Engineering, South China University of Technology, Guangzhou, 510640, People's Republic of ChinaYangzhi Chen (meyzchen@scut.edu.cn)11November20167220921811July201617October201630October2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://ms.copernicus.org/articles/7/209/2016/ms-7-209-2016.htmlThe full text article is available as a PDF file from https://ms.copernicus.org/articles/7/209/2016/ms-7-209-2016.pdf
Line gear is a newly developed gear mechanism with point contact meshing
according to space curve meshing theory. This paper proposes a new form of
line gear with a couple of concave convex arc tooth profiles. It has four
characteristics. First, contact curve of the driving line gear is a
cylindrical spiral curve. Second, two axes of a pair of line gears are
located in the same plane with an arbitrary angle. Third, at the mesh point,
normal tooth profiles of a line gear pair are a couple of inscribed circles.
Namely, they form a couple of concave convex tooth profiles. Fourth, the
tooth profile of a driving line gear is a convex, that of a driven line gear
is a concave, and they are interchangeable. If only consider that the arcs of
teeth at meshing point are tangent, the actual tooth surfaces may interfere
outside of the meshing point. In this paper, the geometric condition of the
tooth surface for a concave convex arc line gear mechanism is derived, and
the optimal formulae of the tooth profile parameters are derived on basis of
interference proof conditions. Finally, the 3-D modeling and kinematic
simulation of line gear pairs show that the proposed line gear pairs can
perform transmission normally. The proposed method will extend the
application of line gear in the conventional power drive.
Introduction
The concave convex arc line gear (Chen and Yao, 2015) is the line gear with
arc tooth profile. The arc gear has advantages in large loading capacity. For
instance, Sun et al. (2016) investigated the contact strength of the
circular-arc-tooth-trace cylindrical gear (C-gear). The results reveal that
the C-gear is superior to spur gear and helical gear in the contact strength. The curvilinear tooth gear proposed by Zhang et al. (2016) is
superior to spur gear in contact and bending stress. The
Double Circular-Arc Helical Gear developed by Wang (2012) can be used for
heavily-loaded planetary gear reducer. The circular-arc gear
designed by Zhou et al. (2016) can be used for a high-pressure and high-speed gear
pump in the aerospace application. Because of the
advantages of arc tooth, scholars have used the arc tooth in many different
types of gear transmission. They are hyperboloidal-type normal circular-arc
gears (Chen et al., 2016), helical gear with triple circular-arc teeth (Xie
and Yang, 2014), quadruple-arc Profile Bevel Gears (Ren, 2014), a new kind of
gear transmission with circular arc tooth profiles (Chen et al., 2014), etc.
All of these belong to cylindrical gear. Arc tooth are also used in bevel
gears (Dong and Wang, 2014; Zhang et al., 2011, 2012) and worm (Zhao and Zhang, 2011).
A pair of concave convex arc line gears at the meshing point.
The concave convex arc line gear is a new form of line gear. The early line
gear is also called the space curve meshing wheel. It contains a driving
wheel, a driven wheel, driving tines and driven tines (Chen et al., 2013b). A
cantilever structure is formed by the combination of the tines and the wheel
body. Compared with the previous proposed line gear, the concave convex arc
line gear has the advantages of arc gear. It is better than the early line
gears in loading capacity, bears much lower contact and bending stress. It
can be applied to conventional power transmission with larger power. It can
be processed by the metal cutting machine tool. Compared with other arc gear,
it has the advantages of line gear. It has a large transmission ratio and
compact structure. The gear shafts of the gear pair can be crossed at any
angle. It means that the conventional gear pair must be a driven wheel
corresponding to a driving wheel. The concave convex arc line gear pair can
be designed as a driving gear and a corresponding plurality of driven gears.
The minimum tooth number of line gear can reach 1 (Chen, 2014). But it can
bear lower contact stress than other conventional forms of arc gear above due
to point contact meshing.
The main objective of the work is to improve the original form of line teeth
profile, to be capable of applying in conventional power transmission. If the
line gear is only the combination of the characteristics of the space curve
meshing wheel and the characteristics of the arc gear, it cannot be processed
by the metal cutting machine tool. It also shows the importance of the gear
profile and the parameter quantization. Moreover, the tooth profile
parameters and gear parameters are quantified to provide the basis for the
analysis of gear strength. A new design method of the concave convex arc line
gear is presented in this paper, aiming to solve the problem of tooth surface
geometry interference and the optimization of the tooth profile parameters.
In order to verify the feasibility of the parametric design and tooth profile
parameters optimization of concave convex arc line gears, this paper presents
the specific example and the 3-D modeling and kinematic simulation of the line gear.
Basic design
A pair of concave convex arc line gears is comprised a driving line gear and
a driven line gear. As shown in Fig. 1a, the axes of the driving line gear
and the driven line gear are arbitrary angle intersected axes in one plane.
The angle is θ. The contact curve of the driving line gear is a
spatial cylinder spiral curve, which forms a pair of conjugated space curves
with the contact curve of the driven line gear. The contact curves of the
driving line gear and the driven line gear are located in o1-x1y1z1 and
o2-x2y2z2 respectively. The distance between o1 and o2 is the
center distance of the driving line gear and the driven line gear. As shown
in Fig. 1b, the tooth profiles of line gears are a pair of inscribed
circles at the mesh point oM. And the point oM is also the tangent
point. The point oM and the center points of the circles are located on
one straight line, which has an angle of ϕ with xM. And -xM is the
secondary normal vector of contact curve of the driving line gear at the mesh
point oM (Chen and Yao, 2015).
The contact curve of the driving line gear, namely a driving contact curve,
is set as a spatial cylinder spiral curve. And the parametric equation in
o1-x1y1z1 is as Eq. (1) (Ding et al., 2012).
xM(1)=mcostyM(1)=msintzM(1)=nπ+nt
where m is the helix radius of the cylindrical spiral curve; n is the
pitch parameter of the cylindrical spiral curve, denoting the pitch as p,
n=p2π; t is a parameter variable, t∈ [ts, te]. The value
range of t determines the length of the contact curve. ts and te are
the starting and ending values for the meshing point, respectively.
ts=-π-Δt2, te=-π+Δt2,
Δt=te-ts. The value of Δt satisfies the condition of contact
ratio (Chen et al., 2013a): ε=ΔtN12π≥ 1.
N1 and N2 are the tooth numbers of the driving line gear and the driven
line gear respectively. The value of ε needs to be preferred. And
the value of Δt can be deduced. Then the value of ts and te can
be determined. Therefore, ε, N1, m and n are the key
factors of the contact curve of the driving line gear.
The contact curve of the driven line gear is conjugated with the driving
contact curve, and the equation in o2-x2y2z2 is as Eq. (2) (Ding et al., 2012).
xM(2)=[(m-a)cosθ-(nπ+nt-b)sinθ]cost+πi12yM(2)=-[(m-a)cosθ-(nπ+nt-b)sinθ]sint+πi12zM(2)=-(m-a)sinθ-(nπ+nt-b)cosθ
where i12 is the transmission ratio, i12=N2N1. As shown
in Fig. 1, a and b are the components of the center distance from the
axis x1 and axis z1 respectively. The position parameters for the
concave convex arc line gear mechanism can be derived from Eq. (3) or (4)
(Ding et al., 2014).
a=1+i12cosθmb=(m-a)tanθa=mb=i12m
where θ is the included angle between the angular velocity vectors of
the driving and driven line gear, θ∈ [0, π]. When
θ≠π2, the values of a and b are derived from Eq. (3). When
θ=π2, the values of a and b can be obtained by Eq. (4).
Therefore, i12, θ, a, b, m and n are the key factors of
the contact curve of the driven line gear.
Parametric design formulae for the driving line gear
A driving line gear is formed by driving line gear teeth radial attachment to
a cylindrical wheel body. The gear teeth can be convex or concave, and the
gear with convex teeth is taken as an example here. As shown in Fig. 2, a
convex tooth of a driving line gear is formed by the motion of a convex
normal profile along a driving contact curve and two tooth thickness auxiliary curves.
Schematic diagram of the driving line gear.
The two tooth thickness auxiliary curves have two functions. Firstly, they
lead the movement of the auxiliary line of the gear tooth profile. Secondly,
they provide the normal tooth thickness of the gear teeth. The auxiliary
curves of tooth thickness comprise a first auxiliary curve and a second
auxiliary curve. A first auxiliary curve 2 is arranged between a contact
curve 1 and a second auxiliary curve 3, and its equation in o1-x1y1z1
is as Eq. (5).
xM11(1)=mcost-c1nsintn2+m2yM11(1)=msint+c1ncostn2+m2zM11(1)=nπ+nt-c1mn2+m2
The equation of a second auxiliary curve 3 in o1-x1y1z1 is as Eq. (6).
xM12(1)=mcost-2c1nsintn2+m2yM12(1)=msint+2c1ncostn2+m2zM12(1)=nπ+nt-2c1mn2+m2
where, 2c1 is the normal tooth thickness of the driving line gear.
The driving line gear is designed by determining the following parameters.
Parameters of a driving line gear include driving contact curve parameters,
tooth profile parameters and wheel body parameters. Driving contact curve
parameters cover m and n. Tooth profile parameters of the driving line
gear are shown in Fig. 3. The tooth profile is composed of two sections of
arcs and a section of straight line. The two sections of arcs are symmetrical
to a first auxiliary curve of tooth thickness in the plane of the normal
tooth profile, and the radius of the arc is ρ1. At the mesh point, the
angle between a straight line and -γ is ϕ. And the straight
line is a connection between the mesh point and arc center. And
γ is the secondary normal vector of contact curve of the driving
line gear. The straight line of the driving line gear tooth profile is
parallel to -γ. The distance between them is ha1.
ha1 and hf1 are the driving line gear addendum and dedendum respectively. As
shown in Fig. 2, wheel body parameters include diameter df1 and length L.
The complete driving line gear model can be established by the
cylindrical wheel body and the line gear convex teeth, in which the height of
the tooth is limited by the driving contact curve and ha1. And
hf1 is not required to be calculated. In the engagement of a couple conjugated
space curves, the contact ratio affects the curve length, indirectly
determines the length of a driving line gear. Therefore, the value of L is
related to the contact ratio. And it depends on ts and te.
Schematic diagram of the normal tooth profile of the driving line gear.
Parametric design formulae of the driven line gear
A driven line gear is formed by the driven line gear teeth radial attachment
to a conical or cylindrical wheel body. The driven line gear teeth can be
concave or convex, and the gear with concave teeth is taken as an example
here. As shown in Fig. 4, a concave tooth of a driven line gear is formed
by the motion of a concave normal profile along a driven contact curve and
two tooth thickness auxiliary curves.
For the driven line gear, two tooth thickness auxiliary curves have also two
functions. The auxiliary curves of driven line gear tooth thickness comprise
a first auxiliary curve and a second auxiliary curve. A first auxiliary curve 2
is arranged between a driven contact curve 1 and a second auxiliary curve 3, and
its equation in o2-x2y2z2 is as Eq. (7).
xM21(2)=(m-a)cosθ-nπ+nt-b-c2mn2+m2sinθcost+πi12-c2nn2+m2sint+πi12yM21(2)=-(m-a)cosθ-nπ+nt-b-c2mn2+m2sinθsint+πi12-c2nn2+m2cost+πi12zM21(2)=-(m-a)sinθ-nπ+nt-b-c2mn2+m2cosθ
The equation of a second auxiliary curve 3 of a driven line gear in
o2-x2y2z2 is as Eq. (8).
xM22(2)=(m-a)cosθ-nπ+nt-b-2c2mn2+m2sinθcost+πi12-2c2nn2+m2sint+πi12yM22(2)=-(m-a)cosθ-nπ+nt-b-2c2mn2+m2sinθsint+πi12-2c2nn2+m2cost+πi12zM22(2)=-(m-a)sinθ-nπ+nt-b-2c2mn2+m2cosθ
where, 2c2 is the normal tooth thickness of the driven line gear.
Schematic diagram of the driven line gear.
Parameters of a driven line gear also include driven contact curve
parameters, tooth profile parameters and wheel body parameters. The driven
contact curve parameters cover m, n, θ, a and b. Tooth profile
parameters of a driven line gear are shown in Fig. 6. The tooth profile is
composed of two sections of arcs and a section of straight line. The two
sections of arcs are symmetrical to the first auxiliary curve of tooth
thickness in the plane of the normal tooth profile, and the radius of the arc
is ρ2. At the mesh point, the angle between a straight line and -γ
is ϕ. And the straight line is a connection between the
mesh point and arc center. And γ is the secondary normal vector
of contact curve of the driving line gear. The straight line of the driven
line gear tooth profile is parallel to -γ, and the distance
between them is hf2. da2 and hf2 are the driven line gear
addendum and dedendum respectively. Wheel body parameters include diameter
da2 and length L, as shown in Fig. 5. da2 is the outside
diameter of the driven line gear at the normal tooth profile of the middle
section of the meshing curve. The complete driven line gear model can be
established by the wheel body and the line gear concave teeth, in which the
depth of the tooth is limited by the driven contact curve and hf2. And
ha2 is not required to be calculated. But when θ=π2,
the value of da2 is not representative of the size of the wheel body. At
this time, it is necessary to determine the value of ha2. In the
transmission of a couple of conjugated curves, the length of a driven line
gear tooth depends on the length of a driving line gear tooth. Therefore, the
length of the driven line gear tooth is equal to the length of the driving line gear tooth.
Rotary surface of the driven line gear.
Schematic diagram of the normal tooth profile of the driven line gear.
The optimal formulae of the tooth profile parameters of a concave convex arc line gear mechanismInterference proof condition
As shown in Fig. 7a, if only consider that the tooth profile arcs are
tangent at meshing point, the actual surfaces of the driving line gear tooth
and driven line gear tooth may interfere with each other outside of meshing
point, resulting in abnormal engagement. The tooth surface can be considered
as a set of tooth profiles on normal plane of each point at the contact curve
of the driving line gear. The relationship between the driving line gear
tooth profile and the driven line gear tooth profile on the normal plane of
meshing point is shown in Fig. 7b–d.
Coordinate system oE-xEyEzE is established, which is obtained by the
coordinate system oM1-xM1yM1zM1 moving a distance parallel
to ρ1 on the plane xM1oM1yM1. The origin of system
oE-xEyEzE is located at the center of arc of the driving line gear
tooth profile, as shown in Fig. 7b and c. Therefore, the
Interference proof condition of teeth surfaces is that the distance between oE
and any point of driven line gear tooth profile arc is not less than ρ1.
Transformation of the coordinate systems
The origin of the coordinate system oM1-xM1yM1zM1 is located on
the contact curve of the driving line gear, and the directions of each
coordinate axis are -γ, β and α. Where,
α, β and γ are the tangent vector,
normal vector and secondly normal vector of the contact curve of the driving
line gear respectively. The equation of the driven line gear tooth surface is
obtained through the coordinate transformation of oE-xEyEzE and
o2-x2y2z2, which is as Eq. (9).
XYZ1=Me2xyz1
The equation of Me2 is as Eq. (10).
Me2=sinλ1sinδ1⋅(cosϕ1cosϕ2cosθ+sinϕ1sinϕ2)+cosδ1(sinϕ1cosϕ2cosθ-cosϕ1sinϕ2)+sinδ1cosλ1cosϕ2sinθ-sinλ1sinδ1⋅(cosϕ1sinϕ2cosθ-sinϕ1cosϕ2)-cosδ1(sinϕ1sinϕ2cosθ+cosϕ1cosϕ2)-sinδ1cosλ1sinϕ2sinθ-sinλ1sinδ1⋅cosϕ1sinθ-cosδ1sinϕ1sinθ-sinδ1cosλ1cosθasinλ1⋅sinδ1cosϕ1+acosδ1sinϕ1+bsinδ1cosλ1-x1-ρ1cosϕ-sinλ1cosδ1⋅(cosϕ1cosϕ2cosθ+sinϕ1sinϕ2)+sinδ1(sinϕ1cosϕ2cosθ-cosϕ1sinϕ2)-cosδ1cosλ1cosϕ2sinθsinλ1cosδ1⋅(cosϕ1sinϕ2cosθ-sinϕ1cosϕ2)-sinδ1(sinϕ1sinϕ2cosθ+cosϕ1cosϕ2)+cosδ1cosλ1sinϕ2sinθsinλ1cosδ1⋅cosϕ1sinθ-sinδ1sinϕ1sinθ+cosδ1cosλ1cosθ-asinλ1⋅cosδ1cosϕ1+asinδ1sinϕ1-bcosδ1cosλ1-y1-ρ1sinϕ-cosλ1(cosϕ1cosϕ2cosθ+sinϕ1sinϕ2)+sinλ1cosϕ2sinθcosλ1(cosϕ1sinϕ2cosθ-sinϕ1cosϕ2)-sinλ1sinϕ2sinθcosλ1cosϕ1sinθ-sinλ1cosθ-acosλ1cosϕ1+bsinλ1-z1001
where ϕ1=t+π; ϕ2=t+πi12; λ1 is the helix
angle of contact curve of the driving line gear,
λ1=arctannm; δ1 is the included angle of β and axis x1,
δ1=arccos((x1,y1,0)⋅(1,0,0)|(x1,y1,0)||(1,0,0)|);
x1, y1 and z1 are the values of the coordinate origin oM1 in
o1-x1y1z1.
Diagram of the tooth profiles interference of the driving line gear
and the driven line gear. (1 – the tooth profile arc of the driving line gear;
2 – the tooth profile arc of driven line gear on the normal plane of the contact
curve of the driving line gear; 3 – an interference area.)
The equation of the main tooth surface of the driven line gear
The tooth surface of the driven line gear is formed by a circular arc with a
radius of ρ2 along the three space curves. At the initial position, the
center of the circle arc is at point (xc, yc, zc), which is obtained by
coordinate transformation of (x2s, y2s, z2s), the equation is as Eq. (11).
xc=x2s+ρ2sinϕ=(m-a)cosθ-nπ+nts-bsinθcosts+πi12yc=y2s-ρ2cosϕsinλ2=-(m-a)cosθ-nπ+nts-bsinθsints+πi12zc=z2s-ρ2cosϕcosλ2=-(m-a)sinθ-nπ+nts-bcosθ
where, λ2 is the helix angle of the contact curve of driven line gear.
Taking u as a parameter variable, the equation of the arc is as Eq. (12).
x0=xc+ρ2cosuy0=yc-ρ2sinusinλ2z0=zc-ρ2sinucosλ2
The equation of the driven line gear tooth surface in o2-x2y2z2 is as Eq. (13).
x=x01+ni12ϕ2-πsinθ(m-a)cosθ-nπ+nts-bsinθcosϕ2-π-y01+ni12ϕ2-πsinθ(m-a)cosθ-nπ+nts-bsinθsinϕ2-πy=x01+ni12ϕ2-πsinθ(m-a)cosθ-nπ+nts-bsinθsinϕ2-π+y01+ni12ϕ2-πsinθ(m-a)cosθ-nπ+nts-bsinθcosϕ2-πz=z0-(m-a)sinθ-ni12ϕ2-bcosθ
Substituting Eq. (13) into Eq. (9), the equation of driven line gear tooth
surface in oE-xEyEzE is obtained.
Parameter optimization
From Fig. 7c, d and Eq. (9), the interference proof
condition of tooth surfaces between a driving line gear and a driven line
gear is described as Eqs. (14) and (15).
X<-ρ12--ρ1sinϕ-ha12Y=-ρ1sinϕ-ha1Z=0X<-ρ12--ρ1sinϕ+ha12Y=-ρ1sinϕ+ha1Z=0
Considering of interference proof, the tooth profile parameters and the wheel
body parameters for the driving line gear and driven line gear can be
optimized by combining Eqs. (14) and (15). And, ρ2=kρ1. Then, the
results can be drawn as follows.
If ϕ∈ [π6, 2π9], Then,
ρ1=1.1c1cosϕ, k∈ (0, 14), or
ρ1=1.2c1cosϕ, k∈ (0, 15), or
ρ1=1.4c1cosϕ, k∈ (0, 17),or
ρ1=1.6c1cosϕ, k∈ (0, 111).
If ϕ∈ [2π9, π4],
Then, ρ1=1.4c1cosϕ, k∈ (0, 17), or
ρ1=1.6c1cosϕ, k∈ (0, 111).
In the design of line gears, the teeth clearance should also be considered.
In the meshing process of the convex and concave teeth, the height of the top
of the convex tooth is a priority. As shown in Fig. 3, the addendum of the
driving line gear can be estimated by as Eq. (16).
ha1=ha*⋅ρ1(1-sinϕ)
where, ha* is a coefficient of addendum, and its value is within a range
of 0.8–0.97. This range is obtained by maintaining the profile shape of the
convex tooth profile, according to simulation results.
And, a design formula of the dedendum of the driven line gear is obtained as Eq. (17).
hf2=hf*⋅ha1
where, hf* is a coefficient of dedendum, and its value is within a range
of 1.4–2. The range of the value can make the driven gear tooth root and
the driving gear tooth top not interfere with each other in the meshing
process, according to simulation results.
The size of the wheel is designed to satisfy the in-Eq. (18).
df12+da22cos(π-θ)<aθ≠π2df12+ha2<mθ=π2
The concave convex arc line gear pair design needs to consider three
non-interfering aspects. They are the non-interference of teeth surfaces, the
non-interference of the tooth top and the tooth bottom, and the
non-interference of the wheel body.
Design examples
Take three pairs of line gears as design examples, the values of various
parameters are shown as Table 1. The values of θ in Table 1 are
2π3, π2, π, respectively. The value of ϕ
is π6.
By the parameters in Table 1, three sets of line gear mechanisms are
obtained, as shown in Figs. 8–10 respectively. From
Table 1 and the figures, three pairs of line gear mechanisms have the same
driving line gear. And the 3-D kinematic simulations on them were carried out
smoothly, without interference. On one hand, it verifies the feasibility of
the design formulae of parameters. On the other hand, compared with the
traditional gear pair, the novel line gear mechanism has a better replacement.
Compared with other circular arc gears, the concave convex arc line gears can
drive with two arbitrary intersecting axes. In the design of the concave
convex arc line gear pair, the same driving wheel can be engaged with
different driven wheels to drive with a variable crossed axes angle. Due to
concave convex arc tooth profile, the concave convex arc line gear pair is
better than the early form of line gear in the bearing contact stress. Due to
point contact engagement, it bears lower contact stress than the other
conventional forms of arc gear pair with line contact engagement. The tooth
number of concave convex arc line gear can reach 1. The concave convex arc
line gear pair has large transmission ratio and compact structure.
Conclusions
In this paper, a design method for a concave convex curve line gear mechanism
is presented. And the design formulae of parameters of the tooth profile are
deduced on the basis of interference proof condition proposed. Conclusions
can be drawn as follows.
It is easy to mass production. The teeth of the driving line gear and driven
line gear are attached to the wheel body radially. And this kind of concave
convex curve line gear is easy to be processed by the numerical control
machine tool, so that it is easy to mass production.
It has good replacement. For the transmission of a line gear pair with an
arbitrary angle in the same plane, only one of driving line gear needs to be
designed, which can match to many of driven line gear with different
geometric parameters or arbitrary crossed angles. Therefore, compared with
the traditional gear pairs and the other arc gear pairs, this kind of concave
convex curve line gear mechanism has a good replacement.
It has power transmission capacity. Compared with previous proposed line gear
mechanism, this kind of concave convex curve line gear can bear much greater
power transmission capacity according to Hertz contact theory, but smaller
than that of traditional concave convex surface gear pair due to its point
contact meshing principle.
It has compact structure. The least number of driving line gear teeth of this
kind of concave convex curve line gear mechanism can reach 1. Compared with
the traditional gear pair, such as spur gear pair or helical gear pair, its
structure is much more compact, that can greatly save installation space.
Compared with the traditional gear pairs and the other arc gears, the
characteristic of concave convex arc line gears is the space curve meshing
theory. It makes the line gear have good replacement and compact structure.
Compared with the early form of line gears, the characteristics of the
concave convex arc line gears are the tooth profile and the relationship
between the wheel body and line gear tooth. On the one hand, it is
strengthened in the tooth contact strength and the bending strength of the
gear tooth. This makes it available for use in the field of conventional
power transmission. And that allows the transmission with a larger power. On
the other hand, it can be processed by the metal cutting machine tool. And
that is more convenient for efficient processing of concave convex arc line
gear and the expansion of its application. The tooth profile design and tooth
formation are considered by the tool shape and the motion trajectory of the
machine tool. The appearance of special NC machine tool for the concave
convex arc line gear can greatly improve the machining efficiency.
The remainder of this work is to design and make a special-purpose numerical
control machine tool for the line gear, to facilitate the efficient
processing of gear, to expand the scope of application. We are doing these
research works on contact pressure, sliding speeds and efficiency, conducted
by newly test rig on transmission ability and dynamic performance experiments.
Notation and units
a, bDistances from o1 to axis x1 and axis z1, [mm]c1, c2Normal tooth thickness of the driving and driven line gear, [mm]da2Outside diameter of the driven line gear at the normal tooth profile of the middle section ofthe meshing curve, [mm]df1Diameter of the driving line gear wheel body, [mm]ha*Coefficient of addendum, [-]ha1, ha2Addendums of the driving and driven line gear, [mm]hf*Coefficient of dedendum, [-]hf1, hf2Dedendums of the driving and driven line gear, [mm]i12Ttransmission ratio, [-]kCoefficient associated with ρ2, [-]LLength of the driving line gear wheel body, [mm]mHelix radius of the cylindrical spiral curve, [mm]nPitch parameter of the cylindrical spiral curve, [mm]N1, N2Tooth numbers of the driving and driven line gear, [-]pPitch of the cylindrical spiral curve, [mm]tParameter variable, [-]ts ,teValues of t at initial and terminal meshing points, [-]ΔtDifference from initial meshing point to terminal meshing point, [-]uParameter variable, [-]αTangent vector of the driving contact curve, [-]βNormal vector of the driving contact curve, [-]γSecondly normal vector of the driving contact curve, [-]δ1Included angle of β and axis x1, [rad]εContact ratio, [-]θIncluded angle between the angular velocity vectors of the driving and driven line gear, [rad]λ1, λ2Helix angles of the driving and driven contact curve, [rad]ρ1, ρ2Radius of the driving and driven tooth profile arc, [mm]ϕAngle between -γ and a line passing through tooth profile arc center and oM, [rad]ϕ1, ϕ2Included angles of the driving and driven contact curves, [rad]
Acknowledgements
Funding supports from the National Natural Science Foundation of China
(No. 51175180, No. 51575191) are gratefully acknowledged.
Edited by: D. Pisla
Reviewed by: three anonymous referees
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