CN203248657U - Helical-spur gear in Fibonacci helical line gear shape - Google Patents

Abstract

The utility model discloses a helical-spur gear in a Fibonacci helical line gear shape. The gear shape of the end face, from a root circle to an addendum circle, of the helical-spur gear is the Fibonacci helical line gear shape, namely the section with the pressure angle a of 17 degrees to 28 degrees is the Fibonacci helical line. According to the helical-spur gear, compared with an involute gear shape with the same reference diameter, the Fibonacci helical line gear shape has the larger curvature radius, so that the contact stress can be effectively reduced, the contact fatigue strength of the helical-spur gear is improved, therefore, the service life of the helical-spur gear is prolonged, and the bearing capacity of the helical-spur gear is improved. Compared with the involute gear shape with the same curvature radius, the Fibonacci helical line gear shape has the smaller reference circle radius, therefore, the size of the helical-spur gear is smaller, the weight is lighter, when the helical-spur gear runs at the high speed, the rotary inertia is small, the noise is low, the running is stable, and the high-speed requirement of the helical-spur gear can be well met. The overlap ratio of the helical-spur gear can be improved through the limitation of the Fibonacci helical line gear shape on the gear width, the continuous transmitting condition of the helical-spur gear is ensured, meanwhile, the meshing contact area is increased, and the bearing capacity of the helical-spur gear is improved.

Description

A kind of helical gear with Fibonacci helix profile of tooth

Technical field

The utility model relates to a kind of gear for translatory movement, especially relates to a kind of helical gear with Fibonacci helix profile of tooth.

Background technique

Along with machinery to heavy duty and high speed development direction, the bearing capacity that dwindle the parts size, improves it has become problem demanding prompt solution.As everyone knows, the radius of curvature of point of mesh is larger, and suffered contact stress is less, and the bearing capacity of gear is higher.And on the other hand, and the stationarity in order to guarantee that it runs up, the size of gear is unsuitable excessive.Yet for involute gear, because its radius of curvature is subjected to the restriction of gear size size, it has been difficult to satisfy high speed and the heavy duty requirements that day by day improves now.Therefore how working out tooth and make its radius of curvature under gear size one stable condition large as far as possible, is the key of dealing with problems.

Fibonacci sequence (Fibonacci sequence), claim again Fibonacci sequence, 0,1,1,2,3,5,8,13,21 refer to such ordered series of numbers: ..., on mathematics, can use n Fibonacci number formula about Fibonacci number φ to represent:

$F_n=\frac{{\mathrm{\φ}}^n-{(1-\mathrm{\φ})}^n}{\sqrt{5}},n\∈N---\left(1\right)$

$\mathrm{\φ}=\frac{1+\sqrt{5}}{2}=1.618043\·\·\·\·\·\·---\left(2\right)$

Classifying as with Fibonacci number is connected to form behind the quadrant forearc of radius is the Fibonacci helix.The radius of curvature of this helix is relatively large, and this provides thinking for the novel profile of tooth of studying a kind of energy and satisfying at a high speed with fully loaded transportation condition.

Summary of the invention

The purpose of this utility model is to provide a kind of helical gear with Fibonacci helix profile of tooth, the Fibonacci profile of tooth can reach larger radius of curvature in the certain situation of gear size, thereby can improve the bearing capacity of gear, improve the high speed running performance of gear.

The technical solution adopted in the utility model is:

The contrate tooth profile of cylindrical gears from the root circle to the top circle of invention is Fibonacci helix profile of tooth, and one section of namely pressure angle a=17 °～28 ° is the Fibonacci helix.

Described Fibonacci helix is determined by following formula (1);

$\left\{\begin{array}{c}x=\cos (\mathrm{\θ}+\frac{\mathrm{\π}}{2}n)F_n+{x^{\′}}_n\\ y=\sin (\mathrm{\θ}+\frac{\mathrm{\π}}{2}n)F_n+y_n^{\′}\end{array}\right.,\mathrm{\θ}\∈[0,\frac{\mathrm{\π}}{2}]---\left(1\right)$

X, y in the formula (1) is respectively arbitrfary point A(x, y on this Fibonacci helix) abscissa and ordinate value, the xoy system of coordinates is take radius of curvature as F ₀The centre of curvature of Fibonacci helix be true origin, have the center of circle 0 of the cylindrical gears of Fibonacci helix profile of tooth _nBe ° point of locating of θ=0 on the Fibonacci helix, and θ is that arbitrfary point A is around 0 _n0 _N+1The centre of curvature 0 of one section Fibonacci helix _n', with 0 _n' 0 _nThe angle that is rotated counterclockwise for benchmark, wherein, Fibonacci number columns formula F _n, Fibonacci number φ, 0 _n' coordinate x' _n, y' _nCalculated by following formula (2), (3), (4), (5) respectively;

$F_n=\frac{{\mathrm{\φ}}^{n+1}-{(1-\mathrm{\φ})}^{n+1}}{\sqrt{5}},n\∈N---\left(2\right)$

$\mathrm{\φ}=\frac{1+\sqrt{5}}{2}=1.618043\·\·\·\·\·\·---\left(3\right)$

$x_n^{\&prime;}=\left\{\begin{array}{c}\\ 0,n=0;\\ F_0-{\mathrm{<figure-callout\; id="1"\; label="F"\; filenames="130513154416.png"\; state="\{\{state\}\}">F</figure-callout>}}_1,n=\mathrm{1,2}\\ F_0-F_1+\underset{k=1}{\overset{\frac{n-1}{2}}{\mathrm{\&Sigma;}}}\left[{(-1)}^{k+1}{F}_{2k-1}\right],n\&GreaterEqual;3\end{array}\right.---\left(4\right)$

$y_n^{\&prime;}=\left\{\begin{array}{c}0,n=\mathrm{0,1}\\ \underset{k=0}{\overset{\frac{n-2}{2}}{\mathrm{\&Sigma;}}}\left[{(-1)}^{k+1}{F}_{2k}\right],n\&GreaterEqual;2\end{array}\right.---\left(5\right).$

The facewidth b of described Fibonacci helix profile of tooth satisfies formula (6); Wherein, P _xBe axial pitch, β is helix angle, m _nBe normal module, ε _βFace contact ratio.

${\mathrm{\&epsiv;}}_{\mathrm{\&beta;}}=\frac{b}{P_x}=\frac{b\sin \mathrm{\&beta;}}{\mathrm{\&pi;}{m}_n}\&GreaterEqual;1.25---\left(6\right)$

The beneficial effect that the utlity model has is:

1, compare with the involute profile of identical standard pitch diameter, the Fibonacci profile of tooth has larger radius of curvature, can effectively reduce contact stress thus, improves the contact fatigue strength of gear, thereby prolongs the working life of gear, improves gear capacity.

2, compare with the involute profile of same curvature radius, the reference radius of Fibonacci profile of tooth is less, so gear size is less, and weight is lighter, rotary inertia is little during the high-speed gear running, noise is little, smooth running, can satisfy well the high-speed requirement of gear;

3, the Fibonacci profile of tooth can improve its contact ratio by the restriction to the facewidth, guarantees the continuous meshing condition of gear, also increases simultaneously the engagement area of contact, improves gear capacity.

Description of drawings

Fig. 1 is Fibonacci helix schematic representation.

Fig. 2 is the cylindrical gears schematic representation with Fibonacci profile of tooth.

Fig. 3 is Fibonacci tooth-profile of gear mesh schematic representation.

Embodiment

Below in conjunction with drawings and Examples the utility model is further described.

As shown in Figure 2, the contrate tooth profile of cylindrical gears of the present utility model from the root circle to the top circle is Fibonacci helix profile of tooth, and one section of namely pressure angle a=17 °～28 ° is the Fibonacci helix.

As shown in Figure 1 and Figure 2, the Fibonacci helix is to be formed by connecting behind a series of quadrant forearcs of classifying radius value with Fibonacci number as.Described Fibonacci helix is calculated by following formula (1).

$\left\{\begin{array}{c}x=\cos (\mathrm{\&theta;}+\frac{\mathrm{\&pi;}}{2}n)F_n+{x^{\&prime;}}_n\\ y=\sin (\mathrm{\&theta;}+\frac{\mathrm{\&pi;}}{2}n)F_n+y_n^{\&prime;}\end{array}\right.,\mathrm{\&theta;}\&Element;[0,\frac{\mathrm{\&pi;}}{2}]---\left(1\right)$

X, y in the formula (1) is respectively arbitrfary point A(x, y on this Fibonacci helix) abscissa and ordinate value, the xoy system of coordinates is take radius of curvature as F ₀The centre of curvature of Fibonacci helix be true origin, have the center of circle 0 of the cylindrical gears of Fibonacci helix profile of tooth _nBe ° point of locating of θ=0 on the Fibonacci helix, and θ is that arbitrfary point A is around 0 _n0 _N+1The centre of curvature 0 of one section Fibonacci helix _n', with 0 _n' 0 _nThe angle that is rotated counterclockwise for benchmark, wherein, Fibonacci number columns formula F _n, Fibonacci number φ, 0 _n' coordinate x' _n, y' _nCalculated by following formula (2), (3), (4), (5) respectively;

$F_n=\frac{{\mathrm{\&phi;}}^{n+1}-{(1-\mathrm{\&phi;})}^{n+1}}{\sqrt{5}},n\&Element;N---\left(2\right)$

$\mathrm{\&phi;}=\frac{1+\sqrt{5}}{2}=1.618043\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;\&CenterDot;---\left(3\right)$

$x_n^{\&prime;}=\left\{\begin{array}{c}\\ 0,n=0;\\ F_0-{\mathrm{<figure-callout\; id="1"\; label="F"\; filenames="130513154416.png"\; state="\{\{state\}\}">F</figure-callout>}}_1,n=\mathrm{1,2}\\ F_0-F_1+\underset{k=1}{\overset{\frac{n-1}{2}}{\mathrm{\&Sigma;}}}\left[{(-1)}^{k+1}{F}_{2k-1}\right],n\&GreaterEqual;3\end{array}\right.---\left(4\right)$

$y_n^{\&prime;}=\left\{\begin{array}{c}0,n=\mathrm{0,1}\\ \underset{k=0}{\overset{\frac{n-2}{2}}{\mathrm{\&Sigma;}}}\left[{(-1)}^{k+1}{F}_{2k}\right],n\&GreaterEqual;2\end{array}\right.---\left(5\right)$

By the arbitrarily point pressure angle α formula that is easy to get on the Fibonacci helix of the geometrical relationship among Fig. 1:

$\mathrm{\&alpha;}=\arcsin \frac{\sqrt{2}}{2}\sqrt{1-\cos \mathrm{\&theta;}}=2\mathrm{\&theta;}---\left(6\right)$

As shown in Figure 2, can get in conjunction with formula (6), get the A point that α on the Fibonacci helix=20 ° point of locating is θ=40 °, cross this point and do standard pitch circle, upwards get an addendum h _aIntercept a C at the Fibonacci helix, so that the θ that should locate=56 °, pressure angle is 28 °, gets a dedendum of the tooth h downwards _fIntercept a D at the Fibonacci helix, so that the θ that should locate=34 °, pressure angle is 17 °, intercepts

Be the Fibonacci profile of tooth.The radius of curvature ρ of arbitrfary point on the described Fibonacci profile of tooth _n, reference radius r obtains by formula (7), (8).

ρ _n＝F _n （7）

r＝2ρ _nsin20° （8）

Compare and can get with the standard pitch diameter formula (9) of involute gear, under the identical reference radius r, the radius of curvature ρ of Fibonacci profile of tooth _nLarger than involute profile, can effectively reduce contact stress thus, improve contact fatigue strength and the bearing capacity of gear.And same curvature radius ρ _nLower, the reference radius r of Fibonacci profile of tooth is less than involute gear, so under the same bearer ability, the former size of gear is less, and weight is lighter, and rotary inertia is little during the speed running, noise is little, smooth running, can satisfy well the high-speed requirement of gear.

As shown in Figure 3, the facewidth b of described Fibonacci profile of tooth satisfies formula (6).Wherein, P _xBe axial pitch, β is helix angle, m _nBe normal module, ε _βFace contact ratio.By the restriction to the facewidth, can improve its contact ratio, guarantee the continuous meshing condition of gear, the while is enlarge active surface also, improves gear capacity.

${\mathrm{\&epsiv;}}_{\mathrm{\&beta;}}=\frac{b}{P_x}=\frac{b\sin \mathrm{\&beta;}}{\mathrm{\&pi;}{m}_n}\&GreaterEqual;1.25---\left(10\right)$

At last, what need to replenish is, when the design heavy-duty gear, n gets large value, and when the design light-load gear, n gets the small value, wherein n 〉=0.

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. helical gear with Fibonacci helix profile of tooth, it is characterized in that: the contrate tooth profile of described helical gear from the root circle to the top circle is Fibonacci helix profile of tooth, and one section of namely pressure angle a=17 °～28 ° is the Fibonacci helix.

2. a kind of helical gear with Fibonacci helix profile of tooth according to claim 1 is characterized in that: described Fibonacci helix is determined by following formula (1);

X, y in the formula (1) is respectively arbitrfary point A(x, y on this Fibonacci helix) abscissa and ordinate value, the xoy system of coordinates is take radius of curvature as F ₀The centre of curvature of Fibonacci helix be true origin, have the center of circle 0 of the cylindrical gears of Fibonacci helix profile of tooth _nBe the point at θ on the Fibonacci helix=00 place, and θ is that arbitrfary point A is around 0 _n0 _N+1The centre of curvature 0 of one section Fibonacci helix _n', with 0 _n' 0 _nThe angle that is rotated counterclockwise for benchmark, wherein, Fibonacci number columns formula F _n, Fibonacci number φ, 0 _n' coordinate x' _n, y' _nCalculated by following formula (2), (3), (4), (5) respectively;

3. a kind of helical gear with Fibonacci helix profile of tooth according to claim 2, it is characterized in that: the facewidth b of described Fibonacci helix profile of tooth satisfies formula (6); Wherein, P _xBe axial pitch, β is helix angle, m _nBe normal module, ε _βFace contact ratio.

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