In the transmission process of high order elliptic gears, when the driving
wheel move a circuit, the driven wheel multiple symmetrical cycle changes will
be occurred. As the evolution of the ordinary elliptical gear, the high order
elliptic gear are adopted in occasions of more cycle variable outputting such
as gear pump, flow meter, etc (Wuxutang, 1997; Yao,
2013; Li, 2004; Wu et al.,
2008). The analytical method of high order elliptic gear transmission analysis,
mainly involves the differential geometry, coordinate transformation, the numerical
integral operation work, so the process is complicated (Fang,
2005). In order to realize visual expression in the process of the high
order elliptic gear tooth profile design, the software of MAPLE which has the
symbolic operation and numerical calculation function and geometric figure ability
is adopted and the clear oval gear tooth meshing effect and the analysis results
of dynamic movement can be obtained easily (Tan, 2002;
Huang and Hui, 2011; Liu and Meng,
2010). This method is very effective for general dynamic simulation design
of non-circular gear and the dynamic and static graphics information and other
numerical results can be obtained at the same time.
DESIGN OF HIGH ORDER ELLIPTIC GEAR TOOTH ENVELOPE SHAPE
Polar equation of high order elliptic gears
pitch curve is as follows:
where, a represents elliptic semi-major axis type, e represents the eccentricity
of the ellipses. In this study, in the design process of choosing the high order
elliptic gears driving wheel, the order number n1 is 3, driven wheel
order number n2 is 5. As it can be seen from the calculation process,
the choice and do not break the design of general.
Uniform distribution of tooth shape on the center section: The number
of teeth on the elliptical gear is set as z and the module number is set as
m. In order to ensure the uniform distribution of tooth profile on the ellipse
section curve, among the angle 2π, the total circumference length L should
be equal to the z teeth space, namely that:
According to the general curve arc length equation r = r(φ):
Make derivative on the Eq. 1:
Put Eq. 1 and 4 into Eq.
3 and adopt the MAPLE in the sum of calculated integral value, in the rectangular
area between 0 to 2π, the rectangular number k is 1000 and then enough
ellipse circumference accurate calculation results can be gotten.
||Location of tooth shape center, (a) n1 = 3 and
(b) n2 = 5
As the requirement of uniform distribution of tooth profile, the results must
be equal to L.
Therefore, the solution of equation is as follows:
rightsum (l(φ), φ = 0..2π, 1000)
The value a conforms to the requirements can be obtained. By the same token,
i can be set as 1
z , take central angle of tooth profile Ωi
as variables and solving the equation:
rightsum (l(φ), φ = 0..Ωπ, 1000)
The central Angle of tooth profile Ωi (I = 1
can be obtained. Through the differential geometry, the center curvature Q(α,
β) of the point on the curve is as follows:
Radius of curvature is as Eq. 8:
The Eq. 1 can be written in rectangular coordinate equation
form as Eq. 9:
Through the MAPLE derivation of implicit function command and through the Eq.
9, y, y can be obtained by type and if substitute them into
the Eq. 7, 8, at the same time substitute
ωi(i = 1
z) into the Eq. 9, the center point Pi(xei,
yei) of ellipse curve and radius of curvature ρi
can be obtained. And the related curve are as shown in Fig. 1.
Design of tooth profile: The basic principle of elliptic gear tooth
profile map is using the pure rolling rack cutter and elliptical gear billet,
through the rack tooth envelope the elliptical gear tooth profile. As for the
high order elliptic gears, under the condition of outer convex section curve,
the method of using enveloping drawing to form of the gear rack is effective.
As to the outside convex problem of the high order elliptic section line, the
condition is as follows:
In the design e is set as 0.04, through the method of translation and
rotation the high order elliptic gears tooth profile can be obtained.
The circumference of the circle is divided into n equal parts, i.e., the polar
angle is φ = 2π/n. The angle between the gear and rack pole diameter
section line is as Eq. 11:
Through Eq. 3 and 11, the location of
each cycle rack can be determined. If set n = 150 and perform the programmable
cycle calculation, after 30 times calculations, the cycle graphics is as shown
in Fig. 2.
When 150 cycle calculations are finished, elliptical gear enveloping graph
is as shown in Fig. 3.
According to the feature points in the process of enveloping the tooth shape,
the outline of elliptical gear is as shown in Fig. 4.
In the calculation above, m = 1.5, z1 = 45, z2 = 75, e = 0.04.
||Design process of tooth envelope, (a) n1 = 3 and
(b) n2 = 5
Meshing of high elliptic gears: Closed n1 order elliptic
gear's pitch curve is within the scope of n~2π, r1 changes n1
cycles, r2 changes n2 cycles, closed condition of the
driven wheel section curve is Eq. 12:
If substitute Eq. 1 into the solution of high order elliptic
gears meshing center distance S. Take center distance S as distance translation
and make the transform translation of high order elliptic gears, effects of
static of elliptic gears meshing are obtained which is as shown in Fig.
||Design results complete tooth envelope, (a) n1
= 3 and (b) n2 = 5
The design program of high order elliptic gear generated by wire-cutting processing
of physical elliptical gears is shown in Fig. 6.
MOTION SIMULATION OF HIGH ORDER ELLIPTIC GEARS
The dynamic design can be intuitive to see the whole cycle and get a clear
understanding of the mechanism movement. Especially for variable transmission
ratio movement, the simulation is very necessary to determine whether the design
agencies can meet the actual demand.
Motion simulation approach: In order to get high order elliptic motion
simulation, the driving wheel of a cycle is divided into n equal parts, in each
of the driving wheel position the MAPLE is adopted as the plot tools. In order
to ensure the elliptical gear pair and maintain the correct meshing relationship,
the driven wheel angle must satisfy transmission ratio function.
||Full uniform distribution of high order elliptic gears, (a)
n1 = 3 and (b) n2 = 5
|| Meshing elliptical gear pair
|| Physical structure of elliptical gears
The whole program with loop statements can draw graphics for all position,
after running the program, all of the graphics, according to the order and the
dynamic effect of elliptic gears transmission can be seen. If change the value
of n, it is actually changed the number of frames in MAPLE mapping, so it can
founded that the driving wheel angular velocity does not equal to the movement
of elliptical gear at the same time.
Transmission relationship of high order elliptic gears: According to the curve
equation of non-circular gear driven wheel:
Substitute Eq. 1 into the Eq. 13, the
relationship between the driven wheel and driving wheel is as Eq.
Transmission ratio function is as follows:
where, n = n2/n1, through the type Eq.
14 and 15, motion simulation graphics the high order
elliptic gears meshing can be gotten.
Examples of elliptic gears dynamic motion graphic: Through the geometric
figure function of MAPLE can draw the ellipse gears meshing transmission motioned
graphics. According to the above methods, through the store display graphics
and graphics rotation transformation function of MAPLE, the dynamic simulation
of high order elliptic gears movement cab be obtained, in the Fig.
7, it give out eight locations of the a motion period. If using the loop
playback function, you can see continuous movement effect in the MAPLE environment.
||Transmission signal of elliptic gears dynamic, (a) φ1
= 0°, (b) φ1 = 45°, (c) φ1 = 90°,
(d) φ1 = 135°, (e) φ1 = 180°, (f)
φ1 = 225°, (g) φ1 = 270° and (h)
φ1 = 315°
|| Changes of driven wheel angular velocity
CHANGE RULE DRIVEN WHEEL SPEED
Through dynamic simulation results of high order elliptic gears, it can easily
get the driven wheel angular velocity change rule used numerical method, as
shown in Fig. 8. It can be seen from the diagram, when the
3 order elliptic gear of driving wheel change one cycle, the 5 order elliptic
gears of driven wheel will change three cycles accordingly. Thus it reflects
the high order elliptic gear transmission characteristics of multiple cycle
The Software of MAPLE can adopted in high order elliptic gear tooth envelope
design and motion analysis, it can get uniform distribution tooth profile design
effect of high order elliptic and the it can dynamic display the movement effect
of high order elliptic gears, at the same time. And on the basis of dynamic
simulation, the angular velocity curve of driven wheel can be obtained. Through
the analysis of process, we can find that this method can be applied in the
general design of non-circular gear.