CN102705448A - Non-circular gear pair with Fourier function pitch curves - Google Patents

Abstract

The invention discloses a non-circular gear pair with Fourier function pitch curves, which is formed by engaging a driving non-circular gear and a driven non-circular gear, wherein the driving non-circular gear and the driven non-circular gear form a one-level or high-level non-circular gear pair; and the pitch curves of the driven non-circular gear are all the Fourier function pitch curves. The non-circular gear drive mechanism adopts the non-circular gear engagement drive of the Fourier function pitch curve, the variation of the shapes of the gear pitch curves becomes more flexible; the drive ratio and the periodic adjustment are easy; the requirement on non-uniform velocity drive is easily met, the mechanism can better move according to the movable velocity ratio required by the working, so as to meet the working requirements of the machine equipment; and the no backlash exists in the non-circular gear pair with the Fourier function pitch curves, and the drive is stable.

Description

Noncircular gear pair with Fourier's function pitch curve

Technical field

The present invention relates to a kind of noncircular gear pair, be specifically related to a kind of noncircular gear pair of the Fourier's of having function pitch curve.

Background technique

Noncircular gear pair is used for transmitting two between centers non-uniform movements, can realize the motion and the functional operation of specific (special) requirements, in machines such as farm machinery, textile manufacturing machine, papermaking equipment, cigarette machine and shuttle conveyor, is widely used.

Noncircular gear pair commonly used at present has eccentric gear, elliptic gear etc., but these non-circular gear pitch curves all are typical mathematical models, and the flexibility that pitch curve changes is relatively poor, and the velocity ratio adjustment is inconvenient, is difficult to satisfy special transmission requirement.

Summary of the invention

In order to solve the problem that exists in the background technique, the object of the present invention is to provide a kind of noncircular gear pair of the Fourier's of having function pitch curve, have gear pitch curve change of shape flexibly, characteristics such as adjustment easily, cycle continuity be good.

In order to achieve the above object, the technological scheme of the present invention's employing is:

The present invention is meshing with each other by active noncircular gear and driven noncircular gear and forms; Initiatively noncircular gear and driven noncircular gear are formed the noncircular gear pair of 1 rank or high-order, and initiatively the pitch curve of noncircular gear and driven noncircular gear is Fourier's function pitch curve.

Fourier's function pitch curve of described active noncircular gear and driven noncircular gear adopts following formula to constitute respectively:

1) initiatively Fourier's function pitch curve of noncircular gear adopts formula (a):

r ₁(θ ₁)=a ₀+a ₁cos(n ₁θ ₁)+b ₁sin(n ₁θ ₁)+a ₂cos(2n ₁θ ₁)+b ₂sin(2n ₁θ ₁) (a)

In the formula, r ₁Be the utmost point footpath of active noncircular gear, a ₀Be initial pitch curve form parameter, a ₁, b ₁Be a pitch curve form parameter, a ₂, b ₂Be secondary pitch curve form parameter, n ₁Be the exponent number of active noncircular gear, θ ₁Angular displacement for the active noncircular gear.

2) Fourier's function pitch curve of driven noncircular gear adopts formula (b):

r ₂=d-r ₁(θ ₁) （b）

In the formula, r ₂Be the utmost point footpath of driven noncircular gear, d is the centre distance of active noncircular gear and driven noncircular gear, and wherein d also satisfies formula (c):

${\mathrm{\&theta;}}_2=\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="120604172330.png,120604172333.png"\; state="\{\{state\}\}">n</figure-callout>}}_2}={\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}{\mathrm{d\&theta;}}_1---\left(c\right)$

In the formula, n ₂Be the exponent number of driven noncircular gear, θ ₂Angular displacement for driven noncircular gear.

Described active noncircular gear and driven noncircular gear are 1 rank; Or initiatively noncircular gear is 2 rank, and driven noncircular gear is 3 rank.

The beneficial effect that the present invention has is:

Noncircular gear driving mechanism of the present invention adopts Fourier's function pitch curve noncircular gear engagement driving, the flexibility that can improve gear pitch curve change of shape; Velocity ratio, periodicity adjustment are easily; Satisfy non-at the uniform velocity transmission requirement easily, can make mechanism better according to the required variable ratio motion of work, to satisfy the job requirement of machinery equipment.The no sideshake of Fourier's function pitch curve noncircular gear pair engagement, stable drive.

Description of drawings

Fig. 1 is single order and the single order noncircular gear pair schematic representation that adopts Fourier's function pitch curve.

Fig. 2 is second order and the three rank noncircular gear pair schematic representation that adopt Fourier's function pitch curve.

Fig. 3 is the comparison diagram as a result that adopts cubic spline a certain non-circular gear pitch curve to be carried out match.

Fig. 4 is the comparison diagram as a result that adopts quartic polynomial same non-circular gear pitch curve to be carried out match.

Fig. 5 is the comparison diagram as a result that adopts Fourier's function same non-circular gear pitch curve to be carried out match.

Fig. 6 is an advance and retreat search method calculation flow chart.

Among the figure: 1, noncircular gear initiatively, 2, driven noncircular gear.

Embodiment

Below in conjunction with accompanying drawing and embodiment the present invention is described further.

The present invention is meshing with each other by active noncircular gear 1 and driven noncircular gear 2 and forms; Initiatively noncircular gear 1 and driven noncircular gear 2 are formed the noncircular gear pair of 1 rank or high-order, and initiatively the pitch curve of noncircular gear 1 and driven noncircular gear 2 is Fourier's function pitch curve.

Fourier's function pitch curve of described active noncircular gear 1 and driven noncircular gear 2 pitch curves adopts following formula to constitute respectively:

1) initiatively noncircular gear 1 adopts formula (a):

r ₁(θ ₁)=a ₀+a ₁cos(n ₁θ ₁)+b ₁sin(n ₁θ ₁)+a ₂cos(2n ₁θ ₁)+b ₂sin(2n ₁θ ₁) (a)

In the formula, r ₁Be the utmost point footpath of active noncircular gear 1, a ₀Be initial pitch curve form parameter, a ₁, b ₁Be a pitch curve form parameter, a ₂, b ₂Be secondary pitch curve form parameter, n ₁Be the exponent number of active noncircular gear 1, θ ₁Angular displacement for active noncircular gear 1.

2) driven noncircular gear 2 adopts formula (b):

r ₂=d-r ₁(θ ₁) （b）

In the formula, r ₂Be the utmost point footpath of driven noncircular gear 2, d be the centre distance of noncircular gear 1 and driven noncircular gear 2 initiatively, periodicity and continuity wherein in order to guarantee the noncircular gear auxiliary driving, and d also satisfies formula (c):

${\mathrm{\&theta;}}_2=\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="120604172330.png,120604172333.png"\; state="\{\{state\}\}">n</figure-callout>}}_2}={\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}{\mathrm{d\&theta;}}_1---\left(c\right)$

In the formula, n ₂Be the exponent number of driven noncircular gear 2, θ ₂Angular displacement for driven noncircular gear 2.

Fourier's function pitch curve of described active noncircular gear and driven noncircular gear calculates through following step respectively:

1) calculates the initiatively utmost point footpath r of noncircular gear 1 according to formula (a) ₁, n in the formula (a) ₁Be the exponent number of active noncircular gear 1, a ₀Be initial pitch curve form parameter, a ₁, b ₁Be a pitch curve form parameter, a ₂, b ₂Be secondary pitch curve form parameter, θ ₁Angular displacement for active noncircular gear 1;

2) by the utmost point of the active noncircular gear 1 that obtains in step 1) footpath r ₁, be calculated as the initiatively centre distance d of noncircular gear 1 and driven noncircular gear 2 according to formula (c), concrete calculating as follows:

2.1) earlier given initial center is apart from d, institute's equal principle of the pitch curve length that meshed can be expressed as during gear transmission: Δ θ ₁R ₁=Δ θ ₂R ₂(d)

In the formula (d), Δ θ ₁For the active noncircular gear in angular displacement ₁The intermittent angle displacement at place, Δ θ ₂For driven noncircular gear in angular displacement ₂The intermittent angle displacement at place.

(d) can get according to formula: ${\mathrm{\&Delta;\; \&theta;}}_2={\mathrm{\&Delta;\; \&theta;}}_1\frac{r_1}{d-r_1}---\left(e\right)$

2.2) under the condition of the right centre distance d of Fourier's function expression r1 of known active non-circular gear pitch curve and conjugate gears, can obtaining initiatively according to formula (d), the noncircular gear angular displacement be Δ θ ₁The time corresponding driven noncircular gear angular displacement Δ θ ₂: if initiatively the angular displacement in one week of noncircular gear rotation is divided into n part (promptly According to following formula, can obtaining initiatively earlier, the noncircular gear angular displacement does

Shi Congdong noncircular gear corresponding angles displacement θ ₂, try to achieve n driven noncircular gear angular displacement Δ θ at last altogether ₂, then all Δ θ ₂Sum is the angular displacement of driven noncircular gear: ${\mathrm{\&theta;}}_2=\underset{{\mathrm{\&theta;}}_1=0}{\overset{\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="120604172330.png,120604172333.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;\; \&theta;}}_1\frac{r_1}{d-r_1}---\left(f\right)$

Δ θ in the formula ₁, r ₁Be known quantity and maybe can try to achieve, so formula (f) is an equation that only contains a unknown quantity, it is found the solution to calculate the initiatively centre distance value d of noncircular gear 1 and driven noncircular gear 2.

2.3) formula (f) is the polynomial equation that a denominator contains unknown number, if adopt the method for direct solving equation to calculate, the amount of calculation that needs is too big, so the present invention abandons the method for direct solving equation, but has adopted the algorithm of optimization searching.Simultaneously, apart from relatively taking computer resource, influence computing time according to the mode computer center of integration; The present invention is under the prerequisite that satisfies the available accuracy requirement; The method that employing adds up is calculated, and has simplified algorithm, reduces the amount of calculation of computer: within the specific limits centre distance value d is optimized search; Obtain in certain accuracy rating and satisfy equational value, promptly satisfy condition:

$|\frac{2\mathrm{\&pi;}}{n_2}-{\mathrm{\&theta;}}_2|=|\frac{2\mathrm{\&pi;}}{n_2}-{\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}{\mathrm{d\&theta;}}_1|=|\frac{2\mathrm{\&pi;}}{n_2}-\underset{{\mathrm{\&theta;}}_1=0}{\overset{\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="120604172330.png,120604172333.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;\&theta;}}_1\frac{r_1}{d-{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="120604172330.png,120604172333.png"\; state="\{\{state\}\}">r</figure-callout>}}_1}|\&le;\mathrm{\&beta;}---\left(g\right)$

In the formula, β is a calculation accuracy.

2.4) from the above, under the condition that the pitch curve of driving wheel is confirmed, i.e. r ₁Value can confirm that if centre distance d is long, (e) can know by formula, Δ θ according to its Fourier's function ₂Value can be less, the θ that therefore finally calculates ₂Will be less than 2 π, otherwise if centre distance is too small, Δ θ ₂Value can be bigger, the θ that finally calculates ₂Will be greater than 2 π; Therefore the searching method that adopts of the present invention is the advance and retreat search methods, with the length of the centre distance search variables as one dimension, searches for to the maximum value direction along minimum value; Finish search when in searching accuracy rating, satisfying the value of formula (g), export the result then.The present invention's search method calculation procedure of advancing and retreat is following, and concrete calculation process is as shown in Figure 6:

A. given initial center is apart from d, step-length h and calculation accuracy β;

B. calculate driven noncircular gear angular displacement according to formula (e) and formula (f) ₂

C. judge the theoretical angular displacement of driven noncircular gear

With driven noncircular gear actual displacement angle θ ₂The absolute value of difference whether less than calculation accuracy β, if less than β, then obtain satisfactory result, finish to calculate, if then continue next step calculating greater than β;

D. judge the theoretical angular displacement of driven noncircular gear Whether less than driven noncircular gear actual displacement angle θ ₂, if less than, the value of then new centre distance is that the value of former centre distance adds step value, and returns step b, if greater than, then new step-length equals 1/2nd of former step-length, and new centre distance equals former centre distance and adds new step-length, and returns step b.

3), adopt formula (b) to calculate the utmost point footpath r2 of driven noncircular gear 2 according to the centre distance d of active noncircular gear that calculates 1 and driven noncircular gear 2

Embodiment 1:

Fig. 1 is active noncircular gear exponent number n ₁=1, driven noncircular gear exponent number n ₂=1 o'clock Fourier's function pitch curve noncircular gear transmission schematic representation, wherein, initial pitch curve form parameter a ₀=30, pitch curve form parameter a ₁=3.525, b ₁=3.527, secondary pitch curve form parameter a ₂=3.688, b ₂=1.59.

Embodiment 2:

Fig. 2 is active noncircular gear exponent number n ₁=2, driven noncircular gear exponent number n ₂=3 o'clock Fourier's function pitch curve noncircular gear transmission schematic representation, wherein, initial pitch curve form parameter a ₀=30, pitch curve form parameter a ₁=3.525, b ₁=0.127, secondary pitch curve form parameter a ₂=0.188, b ₂=0.59.

Series expansion principle according to function in the higher mathematics; General periodic function can both well be expressed with Fourier series, and the condition of convergence of function is lower when adopting Fourier expansion, therefore becomes the condition of Fourier series more much lower than the condition that is launched into other progression functional expansion; And the pitch curve of noncircular gear is an irregular smooth non-circular curve; If place it in the polar coordinates and express, then be the nonlinear function of one-period property, so adopt Fourier's function that the pitch curve of noncircular gear is carried out match; Has extraordinary similarity; And formula (a) n can lead on rank, therefore with Fourier's function match non-circular gear pitch curve the time, than using other function to have better slickness continuously.

Fig. 3 adopts cubic spline to approach the comparison diagram as a result of a certain target non-circular gear pitch curve; Fig. 4 adopts quartic polynomial to approach the comparison diagram as a result of same target non-circular gear pitch curve; Fig. 5 is the comparison diagram as a result that adopts the same target non-circular gear pitch curve of Fourier's function to carry out match; Wherein the pitch curve of noncircular gear is marked in the heavy line feeling the pulse with the finger-tip, and fine line refers to the curve that match is come out.

Can find out that from above-mentioned three width of cloth figure Fourier's function is near the target non-circular gear pitch curve, visible noncircular gear adopts the presentation of Fourier's function can describe the pitch curve shape of noncircular gear accurately, easily.

Above-mentioned embodiment is used for the present invention that explains, rather than limits the invention, and in the protection domain of spirit of the present invention and claim, any modification and change to the present invention makes all fall into protection scope of the present invention.

Claims (3)

1. noncircular gear pair with Fourier's function pitch curve is meshing with each other by active noncircular gear (1) and driven noncircular gear (2) and forms; It is characterized in that: initiatively noncircular gear (1) and driven noncircular gear (2) are formed the noncircular gear pair of 1 rank or high-order, and initiatively the pitch curve of noncircular gear (1) and driven noncircular gear (2) is Fourier's function pitch curve.

2. a kind of noncircular gear pair with Fourier's function pitch curve according to claim 1 is characterized in that: Fourier's function pitch curve of described active noncircular gear (1) and driven noncircular gear (2) adopts following formula to constitute respectively:

1) initiatively Fourier's function pitch curve of noncircular gear (1) adopts formula (a):

r ₁(θ ₁)=a ₀+a ₁cos(n ₁θ ₁)+b ₁sin(n ₁θ ₁)+a ₂cos(2n ₁θ ₁)+b ₂sin(2n ₁θ ₁) (a)

In the formula, r ₁Be the utmost point footpath of active noncircular gear (1), a ₀Be initial pitch curve form parameter, a ₁, b ₁Be a pitch curve form parameter, a ₂, b ₂Be secondary pitch curve form parameter, n ₁Be the exponent number of active noncircular gear (1), θ ₁Angular displacement for active noncircular gear (1);

2) Fourier's function pitch curve of driven noncircular gear (2) adopts formula (b):

r ₂=d-r ₁(θ ₁) (b)

In the formula, r ₂Be the utmost point footpath of driven noncircular gear (2), d is the centre distance of active noncircular gear (1) and driven noncircular gear (2), and wherein d also satisfies formula (c):

${\mathrm{\&theta;}}_2=\frac{2\mathrm{\&pi;}}{n_2}={\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}{\mathrm{d\&theta;}}_1---\left(c\right)$

In the formula, n ₂Be the exponent number of driven noncircular gear (2), θ ₂Angular displacement for driven noncircular gear (2).

3. a kind of noncircular gear pair with Fourier's function pitch curve according to claim 1 is characterized in that: described active noncircular gear (1) and driven noncircular gear (2) are 1 rank; Or initiatively noncircular gear (1) is 2 rank, and driven noncircular gear (2) is 3 rank.

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