TWI646441B - Design method and product of preset fourth-order transmission error of face gear - Google Patents

Abstract

一種面齒輪之預設四階傳動誤差的設計方法及其產品，該設計方法是應用座標轉換理論、齒輪嚙合原理及微分幾何學建立面齒輪與配對小齒輪的齒面位置向量、齒面單位法向量、齒面主曲率與主方向數學模式。接著建立齒面接觸分析與傳動誤差函數斜率之數學模式，並以此模式建立具十一條非線性方程式之非線性聯立方程組。該方程組的解當中具有研磨嚙合小齒輪的砂輪設計參數，以此設計參數來決定砂輪的輪廓，便可製得出具有預設的四階運動誤差的面齒輪，其降噪及抑振能力佳，且能提升傳輸品質。A design method and product for a preset fourth-order transmission error of a face gear, which is to establish a tooth surface position vector of a face gear and a paired pinion using a coordinate transformation theory, a gear meshing principle and a differential geometry, and a tooth surface unit method Vector, main curvature of the tooth surface and mathematical mode of the main direction. Then the mathematical model of the tooth surface contact analysis and the slope of the transmission error function is established, and the nonlinear simultaneous equations with eleven nonlinear equations are established in this mode. The solution of the equation group has the grinding wheel design parameters of the grinding meshing pinion. By using the design parameters to determine the contour of the grinding wheel, the surface gear with the preset fourth-order motion error can be obtained, and the noise reduction and vibration suppression capability can be obtained. Good, and can improve the transmission quality.

Description

面齒輪之預設四階傳動誤差的設計方法及其產品Design method and product of preset fourth-order transmission error of face gear

本發明是有關於一種齒輪的設計方法及其產品，特別是指一種面齒輪之預設四階傳動誤差的設計方法及其產品。 The invention relates to a gear design method and a product thereof, in particular to a design method and a product of a preset fourth-order transmission error of a face gear.

齒輪機構主要用途是傳遞兩軸間之運動與動力。理論上，除了非勻速比的非圓齒輪外，其餘的勻速比齒輪機構，其被動齒輪之轉速與主動齒輪之轉速總是希望為一固定比例之關係。然而在實務上，由於存在不可避免的製造與裝配誤差，被動齒輪的真實轉速往往無法與期望轉速相符，而是存在著傳動誤差。傳動誤差若為直線型誤差，則嚙合齒面對將互相撞擊，使齒輪機構產生強烈的振動與噪音。為了消除直線型傳動誤差對系統的不良影響，Litvin提出應用一個預先設計的二階傳動誤差(Second-Order Transmission Error)來吸收直線型傳動誤差，使得機構運動的誤差曲線由不連續變成連續，因而可大大地降低系統的振動與噪音。要讓齒輪組具有預設的二階傳動誤差，可透過改變刀具幾何或是改變刀具與被創成齒面間之相對運動來對共軛齒面家以修形而達 成。然而二階傳動誤差削弱了太多的齒根強度，且運動曲線也不夠平滑，而且在抑振及降低噪音上仍有改善空間。 The main purpose of the gear mechanism is to transmit the motion and power between the two shafts. Theoretically, except for the non-circular gears of non-uniform speed ratio, the other constant speed ratio gear mechanism, the rotational speed of the driven gear and the rotational speed of the driving gear are always expected to be a fixed ratio relationship. However, in practice, due to the inevitable manufacturing and assembly errors, the actual speed of the driven gear often cannot match the expected speed, but there is a transmission error. If the transmission error is a linear error, the meshing teeth face will collide with each other, causing the gear mechanism to generate strong vibration and noise. In order to eliminate the adverse effects of linear transmission errors on the system, Litvin proposes to apply a pre-designed Second-Order Transmission Error to absorb the linear transmission error, so that the error curve of the mechanism motion becomes discontinuous and continuous. Greatly reduce the vibration and noise of the system. In order for the gear set to have a preset second-order transmission error, the conjugate tooth surface can be modified by changing the tool geometry or changing the relative motion between the tool and the created tooth surface. to make. However, the second-order transmission error weakens too much root strength, and the motion curve is not smooth enough, and there is still room for improvement in vibration suppression and noise reduction.

因此，本發明之目的，即在提供一種面齒輪之預設四階傳動誤差的設計方法。 Accordingly, it is an object of the present invention to provide a method of designing a preset fourth-order transmission error for a face gear.

於是，本發明面齒輪之預設四階傳動誤差的設計方法，是將欲設計齒輪之基礎參數θ₁,u₁,θ₃,u₃,ψ_s1,φ_s1,ψ_s3,φ_s3,代入下列非線性方程式之非線性聯立方程組中： Therefore, the design method of the preset fourth-order transmission error of the surface gear of the present invention is to substitute the basic parameters θ ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , φ _s1 , ψ _s3 , φ _{s3 of the} gear to be designed. In the nonlinear simultaneous equations of the following nonlinear equations:

|n _f ^(F)|=|n _f ^(p)|=1 | n _f ^{( F )} |=| n _f ^{( p )} |=1

接著由該非線性聯立方程組的解求得研磨嚙合小齒輪的砂輪設計參數ε_a、ε_b及h_a，最後以此設計參數來決定砂輪的輪廓，並以該砂輪加工出具有預設的四階傳動誤差的面齒輪。 Then, the grinding wheel design parameters ε _a , ε _b and h _{a of} the grinding meshing pinion are obtained by the solution of the nonlinear simultaneous equations. Finally, the design parameters are used to determine the contour of the grinding wheel, and the grinding wheel is machined with presets. Face gear with fourth-order transmission error.

因此，本發明之另一目的，即在提供一種由該設計方法製成的產品。 Accordingly, it is another object of the present invention to provide a product made by the design method.

於是，本發明由該設計方法製成的產品，為一具有預設的四階傳動誤差的面齒輪。 Thus, the product made by the design method of the present invention is a face gear having a preset fourth-order transmission error.

本發明之功效在於：由該設計方法所設計出的面齒輪具有預設的四階傳動誤差，其降噪及抑振能力優於二階傳動誤差，也不像二階傳動誤差那樣削弱了太多的齒根強度，此外，四階傳動誤差所組成的運動曲線(Motion Curve)也比二階傳動誤差函數所組成的運動曲線更為平滑，從而能提升傳輸品質。 The effect of the invention is that the face gear designed by the design method has a preset fourth-order transmission error, and the noise reduction and vibration suppression capability is better than the second-order transmission error, and does not weaken too much like the second-order transmission error. The root strength, in addition, the motion curve composed of the fourth-order transmission error is also smoother than the motion curve composed of the second-order transmission error function, thereby improving the transmission quality.

本發明之其他的特徵及功效，將於參照圖式的實施方式中清楚地呈現，其中：圖1至圖9皆為示意圖，說明本發明面齒輪之預設四階傳動誤差的設計方法之推導過程；圖10是一曲線圖，說明本實施例中的實驗例1；及圖11至圖17皆為示意圖，說明本實驗例1。 Other features and effects of the present invention will be apparent from the following description of the drawings, wherein: FIG. 1 to FIG. 9 are schematic diagrams illustrating the derivation of the design method of the preset fourth-order transmission error of the surface gear of the present invention. Fig. 10 is a graph showing Experimental Example 1 in the present embodiment; and Figs. 11 to 17 are schematic views showing the experimental example 1.

本發明面齒輪之預設四階傳動誤差的設計方法之一實施例，是先建立面齒輪的齒面數學模型。本實施例的面齒輪是透過一漸開線成型器所生成的，該漸開線成型器是由一個虛擬齒條刀具產生的，圖1顯示了該虛擬齒條刀具齒面的輪廓。由於齒面是一平 面，因此其輪廓為一直線。該平面及直線分別表示為Σ_r及L_r。取座標系S_r(o_r；x_r；y_r；z_r)與虛擬齒條刀具固連，該虛擬齒條刀具的齒Σ_r在該座標系S_r(o_r；x_r；y_r；z_r)中被量測到的位置座標可表示為： One embodiment of the design method of the preset fourth-order transmission error of the face gear of the present invention is to first establish a tooth surface mathematical model of the face gear. The face gear of this embodiment is produced by an involute profiler which is produced by a virtual rack tool, and Figure 1 shows the profile of the tooth surface of the virtual rack tool. Since the tooth surface is a plane, its contour is a straight line. The plane and the straight line are denoted as Σ _r and L _{r , respectively} . The coordinate system S _r (o _r ;x _r ;y _r ;z _r ) is fixed to the virtual rack tool, and the tooth Σ _{r of} the virtual rack tool is in the coordinate system S _r (o _r ;x _r ;y _r The measured position coordinates in ;z _r ) can be expressed as:

Σ_r的單位法線向量於S_r(o_r；x_r；y_r；z_r)可表示為： The unit normal vector of Σ _r is expressed as S _r (o _r ;x _r ;y _r ;z _r ) as:

虛擬齒條刀具的齒Σ_r是用來產生漸開線成型器的齒Σ_s。圖2顯示了產生Σ_s的座標系。將該漸開線成型器與座標系S_r(o_r；x_r；y_r；z_r)固連，虛擬齒條刀具的齒面Σ_r具有線性位移ρ_sψ_s，且漸開線成型器的齒面Σ_s具有一角度位移ψ_s。在該漸開線成型器上，齒面Σ_r由曲面族構成。而的在座標系S_r(o_r；x_r；y_r；z_r)的位置向量可表示為： The tooth Σ _r of the virtual rack tool is used to produce the Σ _s of the involute shaper. Figure 2 shows the coordinate system that produces Σ _s . The involute shaper is fixed to the coordinate system S _r (o _r ;x _r ;y _r ;z _r ), and the tooth surface Σ _{r of the} virtual rack tool has a linear displacement ρ _s ψ _s and is involute forming The tooth surface Σ _{s of the device} has an angular displacement ψ _s . On the involute shaper, the tooth surface Σ _{r is} made up of a curved family Composition. and The position vector of the coordinate system S _r (o _r ;x _r ;y _r ;z _r ) can be expressed as:

其中 among them

Σ_r在座標系S_r(o_r；x_r；y_r；z_r)中的單位法線向量為： The unit normal vector of Σ _r in the coordinate system S _r (o _r ;x _r ;y _r ;z _r ) is:

依據嚙合方程式，曲面族之包絡面存在的必要條件為： According to the meshing equation, the surface family The necessary conditions for the existence of the envelope surface are:

將第(3)式及第(4)式代入第(5)式中，簡化嚙合方程式為： Substituting equations (3) and (4) into equation (5), the simplified meshing equation is:

第(6)式中的參數可由ψ_s表示如下： The parameter in the equation (6) can be expressed by ψ _s as follows:

根據齒輪理論(Litvin,1989)，曲面族的包絡面為漸開線成型器的齒面Σ_s，以v(ψ_s)代入v後重新改寫r _s ^(r)(w,v,)，以將Σ_s在S_s(o_s；x_s；y_s；z_s)中的位置向量簡化如下： According to the gear theory (Litvin, 1989), the surface family The envelope surface is the tooth surface Σ _s of the involute shaper, and v (ψ _s ) is substituted into v and then rewritten r _s ^{( r )} ( w , v , ), to simplify the position vector of Σ _s in S _s (o _s ;x _s ;y _s ;z _s ) as follows:

Σ_s在S_s(o_s；x_s；y_s；z_s)中的單位法向量為： The unit normal vector of Σ _s in S _s (o _s ;x _s ;y _s ;z _s ) is:

漸開線成型器的齒面Σ_s是用來產生面齒輪的齒面Σ _F。圖3顯示了產生Σ_F的座標系，將座標系S_F(o_F；x_F；y_F；z_F)固連面齒輪。漸開線成型器具有一角位移量φ_s，且面齒輪的齒面Σ_F具有一角位移量φ_F，φ_s與φ_F間的關係為：φ _F =(N _s /N _F )φ _s =i _sF φ _s (10) The tooth surface Σ _s of the involute profiler is the tooth surface Σ _F used to create the face gear. Figure 3 shows the coordinate system that produces Σ _F , with the coordinate system S _F (o _F ; x _F ; y _F ; z _F ) fixed-surface gear. The involute profiler has an angular displacement φ _s , and the tooth surface Σ _{F of the} face gear has an angular displacement φ _F , and the relationship between φ _s and φ _F is: φ _F = ( N _s / N _F ) φ _s = i _sF φ _s (10)

其中N_s為漸開線成型器的齒數，而N_F為面齒輪的齒數。在面齒輪上，漸開線成型器的齒面Σ_s形成了一曲面族，在座標系S_F(o_F；x_F；y_F；z_F)中的位置向量為： Where N _s is the number of teeth of the involute profiler and N _F is the number of teeth of the face gear. On the face gear, the tooth surface Σ _s of the involute profiler forms a curved family , The position vector in the coordinate system S _F (o _F ; x _F ; y _F ; z _F ) is:

其中： among them:

Σs在S_F(o_F；x_F；y_F；z_F)中的單位法向量為： The unit normal vector of Σs in S _F (o _F ; x _F ; y _F ; z _F ) is:

其中： among them:

曲面族存在包絡面的必要條件為以下的嚙合方程式： Surface family The necessary condition for the existence of the envelope surface is the following meshing equation:

將第(11)式及第(12)式代入第(13)式中以解出下列嚙合方程式： Substituting equations (11) and (12) into equation (13) to solve the following meshing equation:

第(14)式中的參數w可由ψ_s及φ_s的函數取代如下： The parameter w in the equation (14) can be replaced by a function of ψ _s and φ _s as follows:

根據齒輪理論(Litvin,1989)，曲面族的包絡面為面齒輪的齒面Σ_F，將r _F ^(s)(w,,φ _s )中的w以w(ψ_s,φ_s)取代，解得Σ_F在S_F(o_F；x_F；y_F；z_F)中的位置座標為： According to the gear theory (Litvin, 1989), the surface family The envelope surface is the tooth surface Σ _F of the face gear, and r _F ^{( s )} ( w , , Φ _s) in the w to w (ψ _{s, φ s)} substitution, solve for the Σ _{F S F (o F; x F} ; y F; position coordinate in z _F) is:

其中： among them:

Σ_F在S_F(o_F；x_F；y_F；z_F)中的單位法向量可表示如下： The unit normal vector of Σ _F in S _F (o _F ; x _F ; y _F ; z _F ) can be expressed as follows:

由於面齒輪的齒面Σ_F是由漸開線成型器的齒面Σ_s所產生，因此Σ_F是被產生的曲面而Σ_s則是產生曲面。所生成曲面的主曲率及方向可透過幾何微分或運動學來決定(Litvin,1989)。由 於運動學直接由生成曲面的主曲率及方向、生成及被生成曲面的運動學參數來決定所生成曲面的主曲率及方向(Chen et al.,2001)，因此運動學相較於幾何微分較有效率。在本實施例中，面齒輪的齒面Σ_F的主曲率及方向是以運動學決定。Σ_s的第一類基本數量及第二類基本數量如下： Since the tooth surface Σ _F of the face gear is generated by the tooth surface Σ _s of the involute shaper, Σ _F is the generated curved surface and Σ _s is the curved surface. The principal curvature and direction of the resulting surface can be determined by geometric differentiation or kinematics (Litvin, 1989). Since kinematics directly determines the principal curvature and direction of the generated surface by the principal curvature and direction of the generated surface, and the kinematic parameters of the generated and generated surfaces (Chen et al., 2001), kinematics is compared with geometric differentials. Efficient. In the present embodiment, the principal curvature and direction of the tooth surface Σ _F of the face gear are determined kinematically. The first basic number of Σ _{s and} the basic number of the second category are as follows:

由於F_s=M_s=0，沿參數曲線的切線曲線為主方向，因此，Σ_s的主曲率為： Since F _s =M _s =0, the tangent curve along the parametric curve is the main direction, so the principal curvature of Σ _s is:

Σ_s在S_s(o_s；x_s；y_s；z_s)中的主方向為： The main direction of Σ _s in S _s (o _s ;x _s ;y _s ;z _s ) is:

Σ_s在S_s(o_s；x_s；y_s；z_s)中的主方向及S_s(o_s；x_s；y_s；z_s)中原點O_s的角速度為： Σ _s in _{S s (o s; x s} ; y s; z s) in the main direction and _{S s (o s; x s} ; y s; z s) of origin in O _s is the angular velocity of:

Σ_F在S_s(o_s；x_s；y_s；z_s)中的角速度及S_s(o_s；x_s；y_s；z_s)中原點O_F的線速度為： The angular velocity of Σ _F in S _s (o _s ;x _s ;y _s ;z _s ) and the linear velocity of the origin O _F in S _s (o _s ;x _s ;y _s ;z _s ) are:

在S_s(o_s；x_s；y_s；z_s)中，原點O_F到原點O_s的位置向量為： In S _s (o _s ;x _s ;y _s ;z _s ), the position vector of the origin O _F to the origin O _s is:

在S_s(o_s；x_s；y_s；z_s)中，Σ_s相對於Σ_F的相對線速度及角速度為： In S _s (o _s ;x _s ;y _s ;z _s ), the relative linear velocity and angular velocity of Σ _s relative to Σ _F are:

Σ_F在S_s(o_s；x_s；y_s；z_s)中的主曲率及方向可由下列式子得出： The principal curvature and direction of Σ _F in S _s (o _s ;x _s ;y _s ;z _s ) can be derived from the following equation:

其中： among them:

接著建立鼓狀齒小齒輪齒面的數學模型，在所提出的面齒輪傳動中，鼓狀齒小齒輪是由一鼓狀齒條刀具所產生，該鼓狀齒條刀具是由一特殊設計的砂輪所產生。該砂輪的軸向輪廓是透過四階多項式曲線設計而成，圖4顯示了該砂輪的軸向輪廓，該軸向輪廓是以L_g表示。將座標系S_g(o_g；x_g；y_g；z_g)與該砂輪固連，由於L_g為四階多項式曲線，因此其在S_g(o_g；x_g；y_g；z_g)中的數學模型為： Then, a mathematical model of the tooth surface of the toothed pinion is established. In the proposed face gear transmission, the toothed pinion is produced by a drum rack tool which is specially designed. Grinding wheel produced. The axial profile of the grinding wheel is designed through a fourth-order polynomial curve, and Figure 4 shows the axial profile of the grinding wheel, which is represented by L _g . The coordinate system S _g (o _g ; x _g ; y _g ; z _g ) is fixed to the grinding wheel. Since L _g is a fourth-order polynomial curve, it is in S _g (o _g ; x _g ; y _g ; z _g The mathematical model in ) is:

L_g的切線斜率為： The tangent slope of L _g is:

其中A、B、C、D及E為五個未知係數而需依靠三個條件來決定。第一個條件是L_g的輪廓必定與直線L_r相切於點c，換句話說，L_g在c點上的切線斜率必定等於cotα。第二個條件是L_g的輪廓一定會通過點a、b及c，點a、b及c的座標分別為(a_xg,a_yg)、(b_xg,b_yg)及(c_xg,c_yg)。第三個條件是L_g在a點上的切線斜率必定等於h_a。使用第(26)及(27)式並配合三個條件，可將五個未知係數的 線性等式表示如下： Among them, A, B, C, D and E are five unknown coefficients and depend on three conditions. The first condition is that the contour of L _g must be tangent to the line L _r to point c. In other words, the tangential slope of L _g at point c must be equal to cotα. The second condition is that the contour of L _g must pass through points a, b and c, and the coordinates of points a, b and c are (a _xg , a _yg ), (b _xg , b _yg ) and (c _xg ,c respectively). _Yg ). The third condition is that the tangential slope of L _g at point a must be equal to h _a . Using equations (26) and (27) in conjunction with three conditions, the linear equations for the five unknown coefficients can be expressed as follows:

透過線性代數理論，可得出五個未知係數為： Through linear algebra theory, five unknown coefficients can be derived:

其中： among them:

砂輪的旋轉曲面Σ_g是將軸向輪廓L_g沿旋轉軸線x_g旋轉而得。Σ_g在S_g(o_g；x_g；y_g；z_g)中的位置向量為： The rotating surface Σ _g of the grinding wheel is obtained by rotating the axial profile L _g along the rotation axis x _g . The position vector of Σ _g in S _g (o _g ; x _g ; y _g ; z _g ) is:

Σ_g在S_g(o_g；x_g；y_g；z_g)中的單位法向量為： The unit normal vector of Σ _g in S _g (o _g ; x _g ; y _g ; z _g ) is:

砂輪的旋轉曲面Σ_g是用來產生鼓狀齒條刀具的齒面Σ_c，圖5顯示了Σ_c的座標系。座標系S_c(o_c；x_c；y_c；z_c)是與鼓狀齒條刀具固連。砂輪的旋轉曲面Σ_g相對於鼓狀齒條刀具進行一拋物線運動。在該鼓狀齒條刀具上，砂輪的旋轉曲面Σ_g構成一曲面族，在S_c(o_c；x_c；y_c；z_c)中，的位置向量為： The rotating surface Σ _g of the grinding wheel is used to create the tooth surface Σ _{c of the} drum rack tool, and Fig. 5 shows the coordinate system of Σ _c . The coordinate system S _c (o _c ; x _c ; y _c ; z _c ) is fixed to the drum rack tool. The rotating surface of the grinding wheel Σ _g performs a parabolic motion with respect to the drum rack tool. On the drum rack cutter, the rotating surface Σ _{g of the} grinding wheel constitutes a curved family , in S _c (o _c ;x _c ;y _c ;z _c ), The position vector is:

其中： among them:

曲面族存在包絡面的必要條件為： Surface family The necessary conditions for the existence of the envelope surface are:

根據齒輪理論(Litvin,1989)，曲面族的包絡面為鼓狀齒條刀具的齒Σ_c，以η(θ)代入η後重新改寫r _c ^(g)(θ,u,η)，以將Σ_c在S_c(o_c；x_c；y_c；z_c)中的位置向量簡化如下： According to the gear theory (Litvin, 1989), the surface family The envelope surface is the tooth _c of the drum rack cutter, and η(θ) is substituted into η and then rewritten r _c ^{( g )} ( θ , u , η ) to set Σ _c at S _c (o _c ;x _c The position vector in ;y _c ;z _c ) is simplified as follows:

Σ_c在S_c(o_c；x_c；y_c；z_c)中的單位法向量為： The unit normal vector of Σ _c in S _c (o _c ;x _c ;y _c ;z _c ) is:

鼓狀齒條刀具的齒Σ_c是用來產生鼓狀齒小齒輪的齒面Σ_p，圖6顯示了Σ_p所生成的坐標系。將座標系S_p(o_p；x_p；y_p；z_p)與鼓狀齒小齒輪固連，鼓狀齒條刀具的齒面Σ_c具有線性位移量ρ_pψ_p，且鼓狀齒小齒輪的齒面Σ_p具有角位移量ψ_p。在鼓狀齒小齒輪上，鼓狀齒條刀具的齒面Σ_c形成一曲面族。在S_p(o_p；x_p；y_p；z_p)中，的位置向量為： The gingival _c of the drum rack tool is the tooth surface Σ _p used to generate the spur gear, and Fig. 6 shows the coordinate system generated by Σ _p . The coordinate system S _p (o _p ; x _p ; y _p ; z _p ) is fixedly connected to the drum tooth pinion, and the tooth surface Σ _{c of the} drum rack tool has a linear displacement amount ρ _p ψ _p , and the drum tooth The tooth surface Σ _p of the pinion has an angular displacement ψ _p . On the drum pinion, the tooth surface Σ _{c of the} drum rack cutter forms a curved family . In S _p (o _p ;x _p ;y _p ;z _p) , The position vector is:

其中： among them:

Σ_c在S_p(o_p；x_p；y_p；z_p)中的單位法向量為： The unit normal vector of Σ _c in S _p (o _p ;x _p ;y _p ;z _p) is:

其中： among them:

曲面族存在包絡面的必要條件為以下的嚙合方程式： Surface family The necessary condition for the existence of the envelope surface is the following meshing equation:

將第(36)式及第(37)式代入第(38)式中以解出下列嚙合方程式： Substituting the equations (36) and (37) into the equation (38) to solve the following meshing equation:

第(39)式中的參數ψ_p可由θ及u的函數取代如下： The parameter ψ _p in the equation (39) can be replaced by a function of θ and u as follows:

根據齒輪理論(Litvin,1989)，曲面族的包絡面為鼓狀齒小齒輪的齒面Σ_p，將r _p ^(c)(θ,u,)中的ψ_p以ψ_p(θ,u)取代，解得Σ_p在S_p(o_p；x_p；y_p；z_p)中的位置座標為： According to the gear theory (Litvin, 1989), the surface family The envelope surface is the tooth surface Σ _p of the drum tooth pinion, which will be r _p ^{( c )} ( θ , u , The ψ _p in ) is replaced by ψ _p (θ, u), and the position coordinates of Σ _p in S _p (o _p ; x _p ; y _p ; z _p ) are:

將(θ,u,)中的ψ_p以ψ_p(θ,u)取代，解得Σ_p在S_p(o_p；x_p；y_p；z_p)中的單位法向量座標為： will ( θ , u , The ψ _p in ) is replaced by ψ _p (θ, u), and the unit normal vector coordinates of Σ _p in S _p (o _p ; x _p ; y _p ; z _p ) are:

在本實施例中，鼓狀齒小齒輪的齒面Σ_p之主曲率及方向同樣是以運動學方法求得。Σ_p的主曲率及方向是直接以Σ_c的主曲率及方向，以及Σ_c與Σ_p的運動學參數決定。Σ_c的第一類基本數量及第二類基本數量如下： In the present embodiment, the principal curvature and direction of the tooth surface Σ _p of the drum tooth pinion are also obtained by kinematics. Main directions of curvature and Σ _p is the curvature and direction of the main directly of Σ _c, Σ _c and determines the kinematic parameters of the Σ _p. The first basic number of Σ _{c and} the basic number of the second category are as follows:

由於F_c=M_c=0，沿參數曲線的切線曲線為主方向，因此，Σ_c的主曲率為： Since F _c =M _c =0, the tangent curve along the parametric curve is the main direction, therefore, the principal curvature of Σ _c is:

Σ_c在S_c(o_c；x_c；y_c；z_c)中的主方向為： The main direction of Σ _c in S _c (o _c ;x _c ;y _c ;z _c ) is:

Σ_c在S_c(o_c；x_c；y_c；z_c)中的角速度及S_c(o_c；x_c；y_c；z_c)中原點O_c的速度為： The angular velocity of Σ _c in S _c (o _c ;x _c ;y _c ;z _c ) and the velocity of the origin O _c in S _c (o _c ;x _c ;y _c ;z _c ) are:

Σ_p在S_c(o_c；x_c；y_c；z_c)中的角速度及S_c(o_c；x_c；y_c；z_c)中原點O_p的速度為： Σ _p in _{S c (o c; x c} ; y c; z c) angular velocity and _{S c (o c; x c} ; y c; z c) speed of origin in O _p:

在S_c(o_c；x_c；y_c；z_c)中，原點O_p到原點O_c的位置向量為： In S _c (o _c ;x _c ;y _c ;z _c ), the position vector of the origin _Op to the origin O _c is:

在S_c(o_c；x_c；y_c；z_c)中，Σ_c相對於Σ_p的相對線速度及角速度為： In S _c (o _c ;x _c ;y _c ;z _c ), the relative linear velocity and angular velocity of Σ _c relative to Σ _p are:

Σ_p在S_c(o_c；x_c；y_c；z_c)中的主曲率及方向可由下列式子得出： The principal curvature and direction of Σ _p in S _c (o _c ;x _c ;y _c ;z _c ) can be derived from the following equation:

其中： among them:

接著建立齒面接觸分析及接觸橢圓分析的數學模型，齒面接觸分析(TCA)技術可被用來決定齒面上的接觸路徑、傳動誤差函式及接觸橢圓(Litvin，1994)。圖7顯示了進行TCA的座標系，將座標系S_f(o_f；x_f；y_f；z_f)固連齒輪殼體。鼓狀齒小齒輪的齒面Σ_p為驅動面，而面齒輪的齒面Σ_F為被驅動面。Σ_p及Σ_F的相對角位移量分別為σ_p及σ_F。角位移量σ_p為獨立參數，但角位移量σ_F為非獨立參數。Σ_p在S_f(o_f；x_f；y_f；z_f)中的位置向量及Σ_p在S_f(o_f；x_f；y_f；z_f)中的單位法向量為： A mathematical model of the tooth surface contact analysis and contact ellipse analysis is then established. Tooth surface contact analysis (TCA) techniques can be used to determine the contact path on the tooth surface, the drive error function, and the contact ellipse (Litvin, 1994). Figure 7 shows the coordinate system for the TCA, with the coordinate system S _f (o _f ; x _f ; y _f ; z _f ) fixed to the gear housing. The tooth surface Σ _p of the drum pinion is the driving surface, and the tooth surface Σ _F of the face gear is the driven surface. The relative angular displacements of Σ _p and Σ _F are σ _p and σ _{F , respectively} . The angular displacement σ _p is an independent parameter, but the angular displacement σ _{F is a} non-independent parameter. Σ _p in _{S f (o f; x f} ; y f; z f) position vector and Σ _p in _{S f (o f; x f} ; y f; z f) is the unit normal vector:

其中： among them:

Σ_F在S_f(o_f；x_f；y_f；z_f)中的位置向量及Σ_F在 S_f(o_f；x_f；y_f；z_f)中的單位法向量分別為： Σ _F in _{S f (o f; x f} ; y f; z f) in _{S f (; x f; y} f; o f z f) of the position vector of the unit normal vector and Σ _F are, respectively:

其中 among them

接觸點有兩個必要條件，其一是Σ_p在S_f(o_f；x_f；y_f；z_f)中的位置向量必定等於Σ_F在S_f(o_f；x_f；y_f；z_f)中的位置向量，其二是Σ_p在S_f(o_f；x_f；y_f；z_f)中的單位法向量必定等於Σ_F在S_f(o_f；x_f；y_f；z_f)中的單位法向量。這兩個接觸點的必要條件可寫成下式： There are two necessary conditions for the contact point. One is that the position vector of Σ _p in S _f (o _f ; x _f ; y _f ; z _f ) must be equal to Σ _F at S _f (o _f ; x _f ; y _f ; The position vector in z _f ), the second is that the unit normal vector of Σ _p in S _f (o _f ; x _f ; y _f ; z _f ) must be equal to Σ _F at S _f (o _f ; x _f ; y _f Unit normal vector in ;z _f ). The necessary conditions for these two points of contact can be written as follows:

第(53)式中的第一個向量方程式包含了三個獨立非線性代數純量方程式，第二個向量方程式由於單位法向量具有一為l的已定義長度，因此僅包含了兩個獨立非線性代數純量方程式。因此，第(53)式包含了五個具有六個未知參數的獨立非線性代數純量方程式。利用隱函數組定理，五個獨立純量方程式可由下列五個隱 函數決定： The first vector equation in equation (53) contains three independent nonlinear algebraic scalar equations. The second vector equation has a defined length of l because the unit normal vector, so it contains only two independent non- Linear algebraic scalar equation. Therefore, Equation (53) contains five independent nonlinear algebraic scalar equations with six unknown parameters. Using the implicit function group theorem, five independent scalar equations can be determined by the following five implicit functions:

將r_p(θ,u)中的θ以θ(σ_p)取代，u以u(σ_p)取代，並將Σ_p在S_p(o_p；x_p；y_p；z_p)中接觸路徑的位置向量以σ_p的方程式改寫為：R _p (σ _p )=r _p (θ(σ _p ),u(σ _p )) (55) Substituting θ in r _p (θ, u) with θ(σ _p ), u is replaced by u(σ _p ), and Σ _p is contacted in S _p (o _p ; x _p ; y _p ; z _p ) The position vector of the path is rewritten as the equation of σ _p as: R _p ( σ _p )= r _p ( θ ( σ _p ), u ( σ _p )) (55)

將r_F(ψ_s,φ_s)中的ψ_s以ψ_s(σ_p)取代，φ_s以φ_s(σ_p)取代，並將Σ_F在S_F(o_F；x_F；y_F；z_F)中接觸路徑的位置向量以σ_p的方程式改寫為： The _{r F (ψ s, φ s} ) are to ψ _s ψ _s (σ _p) substituted, φ _s replaced by φ _{s (σ p),} and the Σ _{F S F (o F; x F} ; y F The position vector of the contact path in ;z _F ) is rewritten as the equation of σ _p as:

傳動誤差方程式也以σ_p的方程式改寫為：△σ _F =σ _F (σ _p )-(N _p /N _F )σ _p (57) The transmission error equation is also rewritten as the equation of σ _p as: △ σ _F = σ _F ( σ _p )-( N _p / N _F ) σ _p (57)

為了產生預先設計的四階傳動誤差方程式(FOFTE)，必須去控制傳動誤差方程式的斜率，經計算，傳動誤差方程式的切線斜率為：h _△=(d△σ _F /dσ _p )=σ _F '(σ _p )-(N _p /N _F )=(dσ _F /dσ _p )-(N _p /N _F ) (58) In order to generate the pre-designed fourth-order transmission error equation (FOFTE), the slope of the transmission error equation must be controlled. After calculation, the tangent slope of the transmission error equation is: h _△ =( d △ σ _F / dσ _p )= σ _F ' ( σ _p )-( N _p / N _F )=( dσ _F / dσ _p )-( N _p / N _F ) (58)

接觸點上的單位法向量(θ,u,σ _p )必定正交於相對線速度(θ,u,σ _p )，因此，(θ,u,σ _p )及(θ,u,σ _F )的內積為零，可得： Unit normal vector at the contact point ( θ , u , σ _p ) must be orthogonal to the relative linear velocity ( θ , u , σ _p ), therefore, ( θ , u , σ _p ) and The inner product of ( θ , u , σ _F ) is zero, which gives:

由於鼓狀齒小齒輪的齒面Σ_p以σ_p的角位移量沿軸線z_p旋轉，且面齒輪的齒面Σ_F以σ_F的角位移量沿軸線z_F旋轉，因此相對線速度(θ,u,σ _p )可以下式表達： Since the drum-shaped tooth flank of the pinion gear to the angular displacement Σ _p σ _p z _p along the axis of rotation, and the surface of the gear tooth surface of Σ _F the amount of angular displacement of σ _F z _F along the axis of rotation, so the relative linear velocity ( θ , u , σ _p ) can be expressed as:

其中： among them:

將第(60)式代入第(59)式中解得： Substituting the formula (60) into the equation (59):

解出第(61)式中的(dσ _F /dσ _p )，σ_F相對於σ_p可衍生出： Solving ( dσ _F / dσ _p ) in equation (61), σ _F can be derived from σ _p :

將第(62)式代入第(58)式中，可得傳動誤差方程式的切線斜率為： Substituting the formula (62) into the equation (58), the tangent slope of the transmission error equation is:

由於彈性材質的原因，因此嚙合齒面Σ_p及Σ_F理論上的接觸點需被擴展為一個接觸區域。當Σ_p及Σ_F的二階泰勒展開式近似時，該接觸區域投影在切平面上的面積將為橢圓，該橢圓區域被稱為接觸橢圓。承載接觸點是由一系列的接觸橢圓複合而成，圖8顯示了接觸橢圓的模型。接觸橢圓的對稱中心是與理論接觸點重合，長軸2a、短軸2b及接觸橢圓的方向是透過嚙合面的彈性分析及嚙合面的主曲率和方向決定(Litvin，1994)。鼓狀齒小齒輪的齒面Σ_p及面齒輪的齒面Σ_F形成前述的嚙合面，角度ψ是由至沿逆時鐘方向測得，並可由下列函式決定： Due to the elastic material, the theoretical contact points of the meshing flank Σ _p and Σ _F need to be expanded into a contact area. When Σ _p Σ _F and the second order Taylor expansion approximation, the area of the contact region will be projected on a tangent plane of the ellipse, the elliptical area is called the contact ellipse. The bearing contact points are composited by a series of contact ellipse, and Figure 8 shows the model of the contact ellipse. The center of symmetry of the contact ellipse coincides with the theoretical contact point. The direction of the major axis 2a, the minor axis 2b, and the contact ellipse is determined by the elastic analysis of the meshing surface and the principal curvature and direction of the mating surface (Litvin, 1994). The tooth surface Σ of the drum tooth pinion _p and the tooth surface Σ _{F of the} face gear form the aforementioned meshing surface, and the angle ψ is to Measured in the direction of the counterclockwise and can be determined by the following function:

角度μ是由至方向T沿逆時鐘方向測得，並可由下列函式決定： Angle μ is made up of The direction T is measured in the counterclockwise direction and can be determined by the following function:

接觸橢圓的長軸及短軸可由下列函式決定： The long and short axes of the contact ellipse can be determined by the following functions:

其中： among them:

接著可生成預先設計的四階傳動誤差方程式，本實施例中，面齒輪的動作曲線式如圖9所示，中間的曲線便是預先設計的FOFTE。只要以下列方法決定設計參數ε_a、ε_b及h_a，該面齒輪便可產生預先設計的四階傳動誤差方程式。 Then, a pre-designed fourth-order transmission error equation can be generated. In this embodiment, the action curve of the face gear is as shown in FIG. 9, and the middle curve is a pre-designed FOFTE. As long as the design parameters ε _a , ε _b and h _a are determined in the following way, the face gear can generate a pre-designed fourth-order transmission error equation.

預先設計的四階傳動誤差方程式包括左及右兩部分，左邊的部分是由點p₁至點p₂，右邊的部分是由點p₂至點p₃。FOFTE在點p₁的斜率為零，當鼓狀齒小齒輪的齒面Σ_p與面齒輪的齒面Σ_F於點p₁接觸時，參數θ、u、ψ_s、φ_s、σ_p、△σ_F及σ_F可分別設為θ₁、u₁、ψ_s1、φ_s1、σ_p1、△σ_F1及σ_F1。圖9顯示σ_p1、△σ_F1、σ_F1及h_△還具有下列條件： The pre-designed fourth-order transmission error equation includes the left and right parts, the left part is from point p ₁ to point p ₂ , and the right part is from point p ₂ to point p ₃ . The slope of FOFTE at point p ₁ is zero. When the tooth surface Σ _{p of the} drum pinion is in contact with the tooth surface Σ _{F of the} face gear at point p ₁ , the parameters θ, u, ψ _s , φ _s , σ _p , Δσ _F and σ _F can be set to θ ₁ , u ₁ , ψ _s1 , φ _s1 , σ _p1 , Δσ _{F1 ,} and σ _{F1 , respectively} . Figure 9 shows that σ _p1 , Δσ _F1 , σ _F1 and h _△ also have the following conditions:

因此，第(53)式及第(63)式可由點p₁改寫為： Therefore, equations (53) and (63) can be rewritten by point p ₁ as:

同樣地，當鼓狀齒小齒輪的齒面Σ_p與面齒輪的齒面Σ_F於點p₃接觸時，參數θ、u、ψ_s、φ_s、σ_p、△σ_F及σ_F可分別設為θ₃、u₃、ψ_s3、φ_s3、σ_p3、△σ_F3及σ_F3。由圖9可得知σ_p3、△σ_F3及σ_F1還具有下列條件： Similarly, when the tooth surface Σ _{p of the} drum pinion is in contact with the tooth surface Σ _{F of the} face gear at point p ₃ , the parameters θ, u, ψ _s , φ _s , σ _p , Δσ _F and σ _F may be Set as θ ₃ , u ₃ , ψ _s3 , φ _s3 , σ _p3 , Δσ _{F3 ,} and σ _{F3 , respectively} . It can be _seen from Fig. 9 that σ _p3 , Δσ _F3 and σ _F1 also have the following conditions:

因此，第(53)式可由點p₃改寫為： Therefore, the equation (53) can be rewritten by the point p ₃ as:

由於|n _f ^(F)|=|n _f ^(p)|=1，故第(68)式及第(70)式分別包含了6個及5個獨立非線性代數純量方程式。換句話說，第(68)式及第(70)式構成了一組由11個非線性方程式所組成的聯立方程組，並具有8個未知參數：θ₁、u₁、θ₃、u₃、ψ_s1、ψ_s3、φ_s1及φ_s3，將砂輪輪廓L_g的三個設計參數：ε_a、ε_b及h_a加入八個未知參數中，最後，可形成由11個未知參數(θ₁,u₁,θ₃,u₃,ψ_s1,φ_s1,ψ_s3,φ_s3,ε_a,ε _b,h_a)及11條非線性方程式構成的非線性聯立方程組： Since | n _f ^{( F )} |=| n _f ^{( p )} |=1, equations (68) and (70) respectively contain six and five independent nonlinear algebraic scalar equations. In other words, equations (68) and (70) form a set of simultaneous equations consisting of 11 nonlinear equations with 8 unknown parameters: θ ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , ψ _s3 , φ _s1 and φ _s3 , the three design parameters of the grinding wheel profile L _g : ε _a , ε _b and h _{a are} added to the eight unknown parameters, and finally, 11 unknown parameters can be formed ( θ ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , φ _s1 , ψ _s3 , φ _{s3 ,} ε _a , ε _b , h _a ) and 11 nonlinear equations of nonlinear equations:

這11個未知參數可由牛頓求根法得解，這11個未知參數的初始猜測值可由下列的最佳化問題中得解： These 11 unknown parameters can be solved by Newton's root finding method. The initial guess values of these 11 unknown parameters can be solved by the following optimization problems:

第(72)式的最佳化問題可透過萬用啟發式最佳化算法(meta-heuristic optimization algorithms)得解，例如遺傳演算法(Genetic Algorithm,GA)、粒子群算法(Particle Swarm Optimization，PSO)，或差分進化算法(Differential Evolution，DE)。只要由第(71)式中的11條非線性方程式解得設計參數：ε_a、ε_b及h_a，所製得的面齒輪便可如圖9所示地精確產生預先設計的四階傳動誤差方程式。 The optimization problem of equation (72) can be solved by universal-heuristic optimization algorithms, such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO). ), or Differential Evolution (DE). As long as the design parameters are solved by the eleven nonlinear equations in equation (71): ε _a , ε _b and h _a , the resulting face gear can accurately produce a pre-designed fourth-order transmission as shown in FIG. 9 . Error equation.

[實驗例1] [Experimental Example 1]

實驗例1主要是驗證前述設計方法是可行的，本實驗例1的參數是如表1所示，齒輪模數m設為10mm，壓力角α設為20度，鼓狀齒小齒輪、漸開線成型器(本實驗例中使用鉋齒刀)及面齒輪的齒數分別設為19、22及66，砂輪的拋物線動作參數λ設為0.001，砂輪半徑ρ_g設為40mm，四階傳動誤差方程式左半部的傳 動誤差量ε設為0.7，預先設計的四階傳動誤差方程式之振幅ζ設為10弧秒(arcsec)。為了能產生預先設計的四階傳動誤差方程式，三個設計參數ε_a、ε_b及h_a需於第(71)式中的11條非線性方程式中求得。前述的求解過程中，這11個未知參數θ₁,u₁,θ₃,u₃,ψ_s1,φ_s1,ψ_s3,φ_s3,ε_a,ε_b,h_a的初始猜測值可透過萬用啟發式最佳化算法對第(72)式求解而得出。此處透過差分進化算法使用一個變異因子DE/rand/bin來解此最佳化問題。該變異因子DE/rand/bin的種群(population)設為50，比例因子(scaling)設為0.6，交叉概率(cross probability)設為0.5，而最大代數(maximum number of generation)設為100，θ₁,u₁,θ₃,u₃,ψ_s1,φ_s1,ψ_s3,φ_s3,ε_a,ε_b,h_a的最低及最高上限分別設為(-0.01,0.01)、(-7,-5)、(-0.01,0.01)、(-10,-8)、(-0.4,0.4)、(-0.4,0.4)、(-0.4,0.4)、(-0.4,0.4)、(0,0.1)、(0,0.5)及(2.6,2.9)。圖10顯示了目標函數Y的收斂歷程(convergence history)，在第一次的代數中，Y的最佳值為8.43074E-3。在最後一次的代數中，Y的最佳解減為3.23733E-5。最佳化問題的解為(-8.42472E-4，-6.34222，422764E-3，-9.54734，1.89304E-1，1.98516E-1，-6.52967E-2，-7.93193E-2，0.1，0.5，2.82514)。前述的解是用來作為初始猜測值以代入此11個未知參數中，透過牛頓求根法可解得θ₁,u₁,θ₃,u₃,ψ_s1,φ_s1,ψ_s3,φ_s3,ε_a,ε_b,h_a分別為 1.20887E-15、-6.21437、5.79669E-3、-9.56288、2.00065E-1、2.00065E-1、-5.60149E-2、-7.90781E-2、3.13387E-2、3.41192E-1及2.84076，從而可決定三個設計參數。由於所有的面齒輪參數可於預先設計的四階傳動誤差方程式中得知，因此可如圖11至圖17所示地建立面齒輪方程式的實體模型。 Experimental Example 1 is mainly to verify that the above design method is feasible. The parameters of the experimental example 1 are as shown in Table 1, the gear modulus m is set to 10 mm, the pressure angle α is set to 20 degrees, the drum-shaped pinion, involute The number of teeth of the wire former (the tooth cutter in this experimental example) and the face gear are set to 19, 22, and 66, respectively. The parabolic action parameter λ of the grinding wheel is set to 0.001, the radius ρ _{g of the} grinding wheel is set to 40 mm, and the fourth-order transmission error equation is set. The transmission error amount ε of the left half is set to 0.7, and the amplitude ζ of the pre-designed fourth-order transmission error equation is set to 10 arc seconds (arcsec). In order to generate a pre-designed fourth-order transmission error equation, the three design parameters ε _a , ε _b and h _a need to be found in the eleven nonlinear equations in equation (71). In the above solution process, the initial guess values of the 11 unknown parameters θ ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , φ _s1 , ψ _s3 , φ _{s3 ,} ε _a , ε _b , h _a can be transmitted through 10,000 The heuristic optimization algorithm is used to solve the equation (72). Here, a variation factor DE/rand/bin is used to solve this optimization problem by differential evolution algorithm. The population of the mutation factor DE/rand/bin was set to 50, the scaling factor was set to 0.6, the cross probability was set to 0.5, and the maximum number of generation was set to 100, θ. ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , φ _s1 , ψ _s3 , φ _{s3 ,} ε _a , ε _b , h _a the lowest and upper upper limits are set to (-0.01, 0.01), (-7, respectively. -5), (-0.01, 0.01), (-10, -8), (-0.4, 0.4), (-0.4, 0.4), (-0.4, 0.4), (-0.4, 0.4), (0, 0.1), (0, 0.5) and (2.6, 2.9). Figure 10 shows the convergence history of the objective function Y. In the first algebra, the optimal value of Y is 8.43074E-3. In the last algebra, the best solution for Y is reduced to 3.23373E-5. The solution to the optimization problem is (-8.42472E-4, -6.34222, 422764E-3, -9.55474, 1.789304E-1, 1.98516E-1, -6.52967E-2, -7.93193E-2, 0.1, 0.5, 2.82514). The foregoing solution is used as the initial guess value to be substituted into the 11 unknown parameters. The Newton root method can be used to solve θ ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , φ _s1 , ψ _s3 , φ _{s3 . ,} ε _a , ε _b , h _a are 1.20887E-15, -6.21437, 5.79669E-3, -9.56288, 2.00065E-1, 2.00065E-1, -5.60149E-2, -7.90781E-2, 3.13387, respectively. E-2, 3.412192E-1 and 2.407076, so that three design parameters can be determined. Since all of the face gear parameters are known in the pre-designed fourth-order transmission error equation, a solid model of the face gear equation can be established as shown in FIGS. 11-17.

綜上所述，透過上述的設計方法可得出具有預設的四階運動誤差的面齒輪，其降噪及抑振能力佳，且能提升傳輸品質，故確實能達成本發明之目的。 In summary, through the above design method, a face gear having a preset fourth-order motion error can be obtained, which has good noise reduction and vibration suppression capability, and can improve transmission quality, so that the object of the present invention can be achieved.

惟以上所述者，僅為本發明之實施例而已，當不能以此限定本發明實施之範圍，凡是依本發明申請專利範圍及專利說明書內容所作之簡單的等效變化與修飾，皆仍屬本發明專利涵蓋之範圍內。 However, the above is only the embodiment of the present invention, and the scope of the invention is not limited thereto, and all the equivalent equivalent changes and modifications according to the scope of the patent application and the patent specification of the present invention are still The scope of the invention is covered.

Claims (2)

一種面齒輪之預設四階傳動誤差的設計方法是將欲設計齒輪之基礎參數θ ₁,u ₁,θ ₃,u ₃,ψ _s1,φ _s1,ψ _s3,φ _s3,代入下列非線性方程式之非線性聯立方程組中： 接著由該非線性聯立方程組的解求得研磨嚙合小齒輪的砂輪設計參數ε _a、ε _b及h _a，最後以此設計參數來決定砂輪的輪廓，並以該砂輪加工出具有預設的四階傳動誤差的面齒輪。 The design method of the preset fourth-order transmission error of the face gear is to substitute the basic parameters θ ₁ , u ₁ , θ ₃ , u ₃ , ψ _s1 , φ _s1 , ψ _s3 , φ _{s3 of the} gear to be substituted into the following nonlinear equations. In the nonlinear simultaneous equations: Then, the grinding wheel design parameters ε _a , ε _b and h _{a of} the grinding meshing pinion are obtained by the solution of the nonlinear simultaneous equations. Finally, the design parameters are used to determine the contour of the grinding wheel, and the grinding wheel is machined with presets. Face gear with fourth-order transmission error. 一種以如請求項1所述的設計方法製成的產品，該產品為一具有預設的四階傳動誤差的面齒輪。A product made by the design method of claim 1, which is a face gear having a predetermined fourth-order transmission error.