傅里叶节曲线非圆齿轮传动仿真及参数影响研究

Jun.2013机 床 与 液 压Hydromechatronics Engineering Vo1.41 No.12DOI：10.3969/j.issn.1001-3881.2013.12.015Modeling and Simulation of Fourier Section Curves ofNon-circular Gear and Study on Impact of Parametersxu Gaohuan .CHEN Jianneng. Zhejiang Water Conservancy and Hydropower Colege，Hangzhou 310018，China；2.Zhqiang Sci-tech University，Hangzhou 310018，ChinaAbstract：The nonun.fOrm transmission of non-circular gear has played a unique role in a numberof institutions.Fourier non-circular gear is one expression of its pitch curve.Fourier function peri-odicaly change gears the cyclical variable speed drive engagement point to the diameter，throughthe establishment of the non-circular gear in Fourier mathematical model，this study analyzed thecharacteristics of the transmission。

Key words：Fourier section cuwe，non-circular gears，transmission characteristicsSpecial requirements of non-uniform transmis-sion can be achieved by non-circular gear.Non-cir-cular gear pair can be designed and manufacturedprecisely due to the requirements of movement. Mo-tion accuracy is higher than other mechanical system，non-circular gear section curve can be unidirectionalcontinuous rotation，variable motion is stable and pe-riodic， gear structure is compact， transmission issmooth and it is easy to achieve balance1-3].Thecommon non-circular gear is eccentric circle and ovalgear.Both noncircular gear and pitch curve expres-sion are simple，it is easy to design and manufacture，but the adjustment parameters are comparatively lim-ited。

The eccentric circle gear is the spur gears cen-ter which is deviated from gear center.The eccentric-ity of two gears is the same，which composed of apair of eccentric gear.In the initial position，the axisand the rotational center of the two gears in thestraight line and the eccentric direction is consistent。

The eccentric amount of this non-circular gear is themain parameter and the speed could be cadjusting this parameter.Meshing gap wilhanged byReceived：2013-O1-26XU Gaohuan，PhD.E-mail：6787135### qq.cornchange atits rotation cycle which will cause vibration and speedbeating.Too much eccentric will cause detachmentdrive E4]，which cant be used to precisely drive，and the variable speed range of eccentric gear is lim-ited and it is not conducive to further optimization。

Rotation center of oval gear is the focal point ofthe ellipse.The engagement point is in a straight linewhich through the center of rotation.The engagementof two gears in any position，neither separation willnot be cut， smooth transmission， the transmissioncharacteristics superior to eccentric circle gear.Theparameters of oval gear pitch curve are the major andminor axis length and they could be changed to adjustthe shape of pitch curve，the gear ratio and transmis-sion characteristics[4].Since eliptical expressionis dificult to precisely control the shape of the detailsof the location，the gear ratio of transmission withspecial requirements cant be achieved。

Fourier noncircular gear transmission mechanismis proposed in this paper.This kind of gear has someadvantages such as continuous drive，smooth move-ment and it could provide more adjustable parametersto easily optimize the mechanical properties[5].Thetransmission characteristics of the Fourier non.circu。

1ar gear will be studied and a theoretical basis for thedesign of a new type of non·-uniform transmission sys·tern will be provided in this paper。

70 Hydromechatronics Engineering1.The basic model of Fourier noncirculargear pitch curveMathematical model of the Fourier section curvecould be expressed asllows：rl( 1)uoa1COS(n1 1)b1 sin(n1 1)a2cos(2n1 1)bEsin(2n1 1) (1)Where，the parameters are：active non.circulargear polar radius 1，the pitch curve parameters a0、a1、b1、a2、b2，the active non-circular gear order n1 andthe active non-circular gear angular displacement 1。

1.1.The diameter relationship of Fourier noncir-cular gear and center distanceDriven gear is conjugate non-circular gear ofdrive gear，meshing process must meet a necessaryand sumcient condition：The closed condition of thenon-circular gear is that the ratio of the rotation num-ber of the two non-circular gear is N or/1/N(N is apositive integer).Take the value as 1，generaly onerotation of the capstan and the driven gear is also justone rotation.The driven gear section curve could beexpressed：，2( 2)a-r1( 1) (2)Where，a is the center distance of conjugatenoncircular gears，r1( 1)is the polar radius of ac-tive non-circular gear，r2( 2)is the polar radius ofthe driven non-circular gear。

According to the engagement theory，the curvelength of meshing gear transmission section has thefollowing relationship：△ 2r2( 2)△ lr1( 1) (3)The instantaneous angle△ 1 is the capstan inthe angle 1，the instantaneous rotation angle is△ 2when the driven gear has the angle 2。

△ 2 △ l2丌 2r： ∑ △ ∑睨 1/2r l1/2r( )r1( 1 (4)- N x 2cr(5)△ 1、r1( 1)can be obtained according to spe-cific institutions， in order to ensure normal geartransmission，the center distance of a is obtained fromthis form ula'。

1.2.Transmission ratio of Fourier noncirculargearAccording to the characteristics of the non·-cireu-lar gear，two gear at any moment，there is a pointwhere a relative speed is always equal to zero，whichis located in the connection line of the two non.circu-lar the gear rotates center，and meet the conditionsthat the meshing pitch curves are equal[1-3，6]。

Therefore，the transmission ratio is：r1( 1)uoa1COS(nl 1)b1 sin(n1 1)a2cos(2n1 1)b2sin(2nl 1) (6)2∑ 。

l1/2r/'12 r1( 1)a-r1( 1) N×2rr2( 2) a～rl( 1)r1( 1) rl( 1)(7)(8)Then the speed N could be determined。a canbe obtained and the transmission ratio can be deter。

mined according to the capstans section curve expres-sion.It still complies with the basic gear meshinglaw：① non-circular gear pitch curve is closed；②Th e perimeter of the pitch curve is Z 7r .m is themodulus. is the number of teeth。

2.M eshing simulation of Fourier noncircu-lar gear2.1.M eshing simulation of Fourier noncirculargear pitch curveAccording to the engagement theory of the Fou-rier noncircular gear，we adopt MATLAB software toanalyze the impact of various parameters on the trans-mission characteristics。

Th e parameters of the initial value are as fol-lows：n12，a30，ao20，a11.525u21.688，b10.310，b20.310Th e Fourier section curve is expressed as：r1( 1)aoa1COS(n1 1)b1 sin(n1 1)a2cos(2n1 1)b2sin(2n1 1)2520151050. 5. 1O. 15. 2O. 25. 20 .1O O 10 20Fig.1 the simulation of Fouriergear pitch curve30 40noncircular(9)XU Gaohuan，et al：Modeling and Simulation of Fourier Section Curves of Non-circularGear and Study on Impact of Parameters2.2.The impact of the parameters on the shapeof the curve and the transmission ratioIf one parameter is changed and the other keepthe same，the results are shown as folows：As shown in Fig.2(a)，with a0 increasing，thepitch curve become bigger，the center distance in-creases and the contour gear pair increases；centerdistance equal increments along with a0，i.e. eachadditional 5 mm。center distance increases 15 mm，there are equidistant lines between the driven gear re-tation centers。

mission ratio CUlWes become gently， two peaks isgradually reduced，maximum value is reduced，theminimum of transmission ratio is gradually increased，the average value is gradually reduced。

1005OO50. 1O03.53.02.52.01.51.O. 5O 0 50 100 150 200(a)gear Pitch Curve0 50 100 150 200 250 300 350 400(b)The Curve of Transmission RatioFig.2 The impact of the parameter a0 on the shape ofthe gear pitch curve and the transmission ratioAs shown in Fig.3(a)，with a1 increasing，thefolower gear section curve gradually becomes highand narrow，drive gear gradually plump and movingright.the center distance increases slowly。

Fig.3(b)shown，with a1 increasing，the curvesbetween peaks of transmission ratio are gradualy gen-tle.The curve maximum changed litter，the slope ofoutside the two peaks increased. The minimum oftransmission ratio is gradually decreased. The aver-age variation is smal1. Five curves go through twopoints with equal values。

806040200. 20. 40- 60- 803.53.02.52.O1.51.O. 50 0 50 1O0 1 50(a)gear Pitch Curve0 50 100 l50 200 250 300 350 400(b)The Curve of Transmission RatioFig.3 The impact of the parameter al on the shape ofthe gear pitch curve and the tran smission ratioAs shown in Fig.4(a)，with a2 increasing，thegear pair does not occur movement along the X-axisdirection and the y-axis direction.The capstan in theY-axis direction becomes gradually fiat and appearsconcave phenomenon.Th e driven gear becomes tact-1ess and appears recessed phenomenon，the centerdistance increasing slowly。

As shown in Fig.4(b)，with a2 is increasing，the slope of the two peaks and the troughs of thetransmission ratio are increased gradually.The maxi-mum are gradually increasing generaly and the mini-mum is gradually reduced.The average transmissionincreased slowly。

As Fig.5(a)shows，with b1 increasing，thegear pair is not obvious movement along the -axisdirection，but the capstan gradually shift along the Yaxis，and appears concave phenomenon，the drivengear around its center of rotation becomes dexal andsharp，and appears concave phenomenon.The centerdistance is slowly increasing。

As shown in Fig.5(b)，with b1 increasing，thegentle of transmission ratio curve is geting worse，thegap between the two-peaks gradually is increasing，maximum is gradually increased.and the minimumvalue is gradually reduced。

The average transmissionThe transmission ratio of72 Hydromechatronics Engineeringthe starting and ending points of the curve intersect atone point.Five curves are crossing into three pointswith equal values。

80604020O. 20- 40- 60. 8O3.53.02.52.Ol 51.O. 50 0 50 100 l5O(a)gear Pitch Curve0 50 100 l50 200 250 300 350 400(b)The Curve ofTransmission RatioFig.4 The impact ofthe parameter 62 on the shape ofthe gear pitch curve and the transmission ratio80604020O- 20- 40. 60. 803.53.02.52.01.51.O- 50 0 5O 1O0 150(a)gear Pitch Curve0 50 l00 150 200 250 300 350 400(b)The Curve ofTransmission RatioFig.5 Th e impact of the parameter b1 on the shape ofthe gear pitch curve and the transmission ratioAs shown in Fig.6(a)，with b2 increasing，gearpair did not occur the movement along the X-axis di-rection and the Y-axis direction，the driving gear andthe driven gear occur angular displacement aroundthe respective center，the driving gear gradually be-comes fiat， and appear concave phenomenon，thedriven gear become tactless and appear recessed phe-nomenon，the center distance increasing slowly。

As the Fig.6(b)shows，with b2 increasing，theslope of the two peaks and the troughs of the trans-mission ratio are increased gradually.The maximumare gradualy increasing generally and the minimum isgradually reduced. The average transmission in-creased slowly.Curves of transmission ratio are inter-sected at one point. Five curves are crossing intothree points with equal values。

8O60402O0. 20- 40. 60. 8O3.53.02.52.01.51.O. 50 0 5O 1O0 1 5O(a)gear Pitch Curve0 5O 1O0 150 200 250 300 350 400(b)The Curve ofTransmission RatioFig.6 Th e impact of the parameter b2 on the shape ofthe gear pitch curve and the transmission ratio3.Conclusions1、 Cycle continuity. Cycle of the polar coordi-nates function of the noncircular gear pitch curve is2叮T.and the function iS continuous smooth.Fourierfunctions are composed of the two functions sin( )and COS( ).Each of them iS 2，tr periodic and an n。

order derivable function.Therefore，it has good con。

tinuous smoothness when Fourier function fits withnoncircular gear pitch curve。

XU Gaohuan，et al：Modeling and Simulation of Fourier Section Curves of Non-circularGear and Study on Impact of Parameters 732)It is flexible and easy to adjust the shape ofthe curve.The gear pitch curve of elliptical gears aretypical mathematical models. Curve flexibility ispoor，parameter adjustment is difcult to meet agencyrequirements.Fourier pitch curve have more adjust-ment parameters(coeficient of order，phase，etc.)。

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[2] Shinn-Liang Ch，Chug-Bian T.Mathematical model and[3][4][5][6]undercuting analysis of eliptical gears generated by rackcutters[J].Mech Mach Thory，1996，31(7)：879-890。

ZHANG Rui，ZAO Fengxia，WU Xutang.Study on theDesign Method of a New-style Sinusoidal NoncircularGear[J].MACHINE DESIGN＆RESEARCH，2010，26(4)：71-74。

ZHAO Yun.Agricultural machinery analysis and synthe-sis[M].Beijing：China Machine Press，2008。

LI Ge ，LI Hui，YANG Aiqian，et a1.Kinematic analysison Fourier curve non·circular gears transplanting mecha·-nism[J].CSAE，2011，27(8)：126-131。

WU Xutang.WANG Haigni.Non-circular Ge ar and Non-unirm Drive[M].Beijing：Mechanical Industry Press，1996：105-106。

傅里叶节曲线非圆齿轮传动仿真及参数影响研究徐高欢 ，-，陈建能1.浙江水利水电学院，杭州 310018；2.浙江理工大学，杭州 310018摘要：非圆齿轮的非匀速传动在-些机构中起到 了独特的作用。傅里叶非圆齿轮是其 中-种，其节曲线表达是傅里叶函数。通过啮合点径向按傅里叶函数周期性变化实现齿轮周期性变速传动，建立傅里叶非圆齿轮传动数学模型，研究分析其传动特性。

关键词：傅里叶节曲线；非圆齿轮；传动特性中图分类号：TH12(Continued on 43 page)垃圾 CFB焚烧酸性烟气排放特性实验臧仁德重庆理工大学，重庆 400054摘要：通过垃圾 CFB焚烧实验，研究了垃圾焚烧产生烟气的生成机理及排放特性，得到了焚烧过程中床层温度、过量空气系数等参数对酸性烟气排放的影响。垃圾的CFB焚烧酸性烟气排放的实验结果表明：随床层温度的增加，烟气中NO、SO2、HC1排放浓度增加；随过量空气系数的增加，烟气中NO排放浓度开始增加迅速之后趋于平缓 ，S02排放浓度增加 ，HC1排放浓度减小；随钙硫比的增加，烟气中NO排放浓度增加 ，SO 的排放浓度减小，HC1的排放浓度减校关键词：垃圾；排放特性；CFB；过量空气系数中图分类号：X70