CN202659811U - Non-circular gear pair with Fourier function pitch curve - Google Patents

Abstract

The utility model discloses a non-circular gear pair with a Fourier function pitch curve. The non-circular gear pair is formed by a driving non-circular gear and a driven non-circular gear which are meshed with each other; the driving non-circular gear and the driven non-circular gear are formed into a 1-order or high-order non-circular gear pair; and the pitch curves of the driving non-circular gear and the driven non-circular gear are both Fourier function pitch curves. The non-circular gears with the Fourier function pitch curves are adopted for meshing and transmitting by a non-circular gear transmission mechanism provided by the utility model, so that the flexibility of the change in the shape of the gear pitch curve is increased; the transmission ratio and the periodicity are easily adjusted; the non-constant speed transmission demand is easily met; the mechanism can more effectively move according to the transmission ratio required by working, so as to meet the working demand of machinery equipment; and the non-circular gear pair with the Fourier function pitch curve is backlash-free in meshing and is stable in transmission.

Description

A kind of noncircular gear pair with Fourier's function pitch curve

Technical field

The utility model relates to a kind of noncircular gear pair, is 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 motion and the functional operation of specific (special) requirements, is widely used in the machines such as farm machinery, textile manufacturing machine, papermaking equipment, cigarette machine and shuttle conveyor.

Noncircular gear pair commonly used has eccentric gear, elliptic gear etc. at present, but these non-circular gear pitch curves all are typical mathematical models, and the flexibility that pitch curve changes is relatively poor, and gear 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 purpose of this utility model is to provide a kind of noncircular gear pair of the Fourier's of having function pitch curve, have gear pitch curve change in shape flexibly, easily adjust, the characteristics such as the cycle continuity is good.

In order to achieve the above object, the technical solution adopted in the utility model is:

The utility model is by the active noncircular gear and driven noncircular gear is intermeshing forms; Initiatively noncircular gear and driven noncircular gear form 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 respectively following formula to consist of:

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

${\mathrm{\&theta;}}_2=\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="120604173127.png,120604173130.png"\; state="\{\{state\}\}">n</figure-callout>}}_2}={\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}d{\mathrm{\&theta;}}_1---\left(c\right)$

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 utlity model has is:

The utility model non-circular gear transmission mechanism adopts Fourier's function pitch curve noncircular gear engagement driving, the flexibility that can improve gear pitch curve change in shape; Velocity ratio, periodicity are adjusted easily; Easily satisfy non-at the uniform velocity transmission requirement, can make better mechanism according to the required variable ratio motion of work, to satisfy the job requirement of machinery equipment.Fourier's function pitch curve noncircular gear pair meshes without sideshake, stable drive.

Description of drawings

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

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

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

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

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

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

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

Embodiment

The utility model is described in further detail below in conjunction with drawings and Examples.

The utility model is by active noncircular gear 1 and 2 intermeshing compositions of driven noncircular gear; Initiatively noncircular gear 1 and driven noncircular gear 2 form 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 respectively following formula to consist of:

1) initiatively noncircular gear 1 adopts following 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 Curves 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 following 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 active noncircular gear 1 and driven noncircular gear 2, periodicity and the continuity in order to guarantee the noncircular gear auxiliary driving wherein, and d also satisfies following formula (c):

${\mathrm{\&theta;}}_2=\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="120604173127.png,120604173130.png"\; state="\{\{state\}\}">n</figure-callout>}}_2}={\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}d{\mathrm{\&theta;}}_1---\left(c\right)$

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 by 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 Curves 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), specifically be calculated as follows:

2.1) first 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) at Fourier's function expression r of known active non-circular gear pitch curve ₁Under the condition of the right centre distance d of 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 ₂If: initiatively noncircular gear rotates the angular displacement in a week and is divided into n part (namely ), according to following formula, can obtaining first initiatively, the noncircular gear angular displacement is Shi Congdong noncircular gear corresponding angles displacement θ ₂, try to achieve altogether at last n driven noncircular gear angular displacement Δ θ ₂, all Δ θ then ₂Sum is the angular displacement of driven noncircular gear: ${\mathrm{\&theta;}}_2=\underset{{\mathrm{\&theta;}}_1=0}{\overset{\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="120604173127.png,120604173130.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;\&theta;}}_1\frac{r_1}{d-r_1}---\left(f\right)$

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 adopting the method for direct solution equation calculates, the amount of calculation that needs is too large, so the utility model abandons the method for direct solution equation, but has adopted the algorithm of Optimizing Search.Simultaneously, mode computer center distance according to integration relatively takies computer resource, affect computing time, the utility model is under the prerequisite that satisfies the available accuracy requirement, adopt cumulative method to calculate, simplified algorithm, reduce 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, namely satisfy condition:

$|\frac{2\mathrm{\&pi;}}{n_2}-{\mathrm{\&theta;}}_2|=|\frac{2\mathrm{\&pi;}}{n_2}-{\&Integral;}_0^{\frac{2\mathrm{\&pi;}}{n_1}}\frac{r_1\left({\mathrm{\&theta;}}_1\right)}{d-r_1\left({\mathrm{\&theta;}}_1\right)}d{\mathrm{\&theta;}}_1|=|\frac{2\mathrm{\&pi;}}{n_2}-\underset{{\mathrm{\&theta;}}_1=0}{\overset{\frac{2\mathrm{\&pi;}}{{\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="120604173127.png,120604173130.png"\; state="\{\{state\}\}">n</figure-callout>}}_1}}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;\&theta;}}_1\frac{r_1}{d-{\mathrm{<figure-callout\; id="1"\; label="r"\; filenames="120604173127.png,120604173130.png"\; state="\{\{state\}\}">r</figure-callout>}}_1}|\&le;\mathrm{\&beta;}---\left(g\right)$

In the formula, β is calculation accuracy.

2.4) from the above, under the condition that the pitch curve of driving wheel is determined, i.e. r ₁Value can determine according to its Fourier's function, if centre distance d is long, by formula (e) as can be known, Δ θ 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 larger, the θ that finally calculates ₂Will be greater than 2 π, therefore the searching method that adopts of the utility model 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, then Output rusults when in searching accuracy rating, satisfying the value of formula (g).The utility model advance and retreat search method calculation procedure is as follows, and specifically 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 ₂If, less than, then the value of 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) according to the active noncircular gear 1 that calculates and the centre distance d of driven noncircular gear 2, adopt formula (b) to calculate the utmost point footpath r of driven noncircular gear 2 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 non-circular gear drive schematic diagram, wherein, Initial Pitch Curves 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 non-circular gear drive schematic diagram, wherein, Initial Pitch Curves 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.

According to series expansion of a function principle 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 become 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-circle curve, express if place it in the polar coordinates, it then is the nonlinear function of one-period, 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 Fitting non-circular gear pitch curve the time, than using other function to have better continuous light slip.

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

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

Above-mentioned embodiment is used for the utility model of explaining; rather than the utility model limited; in the protection domain of spirit of the present utility model and claim, any modification and change to the utility model is made all fall into protection domain of the present utility model.

Claims (3)

1. noncircular gear pair with Fourier's function pitch curve is by active noncircular gear (1) and driven noncircular gear (2) intermeshing composition; It is characterized in that: initiatively noncircular gear (1) and driven noncircular gear (2) form 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 respectively following formula to consist of:

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

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

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, it 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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