Design and modeling of noncircular gear with curvature radius function

Abstract

The movement law of the noncircular gear drive is determined by the shape of the noncircular gear pitch curve. Therefore, the design of the pitch curve of the noncircular gear is very important. On the other hand, if the noncircular gear pitch curves are concave or cusp, the tooth profile of noncircular gear cannot be generated. In order to obtain more movement laws of noncircular gear and ensure the convexity of noncircular gear pitch curves, this paper designed a new noncircular gear with curvature radius function. In the tangent polar coordinate, the relationship between diameter and curvature of curve was established. The mathematical model of noncircular gear pair was deduced. The arc lengths of the noncircular gear pairs were calculated and the concavity of the driven noncircular gear was judged. A noncircular gear auxiliary design program wrote by using the MATLAB language. The tooth profile design of noncircular gear was introduced. The given examples showed that the mathematical calculation of the noncircular gear is reasonable, and this new noncircular gear can be well applied to the actual machinery.

Keywords

Noncircular gear curvature MATLAB pitch curve modeling

1. Introduction

Gear drive can transmit the movement between two parallel shafts and be widely used in mechanical transmission systems. The general gear is circular, and its transmission ratio is constant. However, some special occasions require that the movement law of mechanism is non-uniform. The traditional circular gear drive can’t achieve such a special movement law. Noncircular gear drive has the both advantages of the cam drive and circular gear drive, so it can accurately achieve non-uniform movement between the two shafts. In many special occasions, the combined mechanical mechanisms with noncircular gears can greatly simplify the complexity of the mechanism and improve the working performance of the machine. Noncircular gear mechanism has been applied in the rice transplanter, gear pump, packaging machinery, hydraulic motors and many other occasions [1, 2, 3, 4, 5], and achieved good results. Zheng et al. [6] introduced the motion law for indexing mechanism, and proposed a indexing mechanism with three types of noncircular planetary gear trains, synthesized the gear ratios and pitch surfaces of noncircular gear. Hu et al. [7] used a third-order noncircular gear mechanism to drive a gear pump. Dooner [8] used the noncircular gear to eliminate unnecessary torque and speed fluctuations. In order to reduce the occurrence of noise and vibrations of gear mechanisms, Karpov et al. [9] investigated the applicability of noncircular gears for preventing resonance oscillations in gear mechanisms.

The movement law of noncircular gear is determined by the shape of its curve. Based on the pitch curve is closed or not, the noncircular gears are divided into the closed noncircular gears and the non-closed noncircular gears. The closed noncircular gears can achieve continuous transmission, so a lot of them are used to drive machinery in practice. The commonly noncircular gears have the elliptical gear and eccentric circular gear, whose pitch curves are composed of elliptic curve and eccentric curve respectively. Liu and Ren [10] proposed a limacon noncircular gear whose pitch curve diameter function is cosine function, and calculated the center distance of the gear pair and analyzed its transmission characteristic. Zhang and Fan [11] proposed a steepest rotation method for designing the noncircular gear with fixed boundaries using calculus of variations. In summary, different closed-space curves can produce different laws of motion. Many scholars are constantly exploring new closed curves, which can be used as noncircular gear curve.

In the whole noncircular gear design process, the manufacture of gear tooth profile is a very important process. The shape of the pitch curve is very important for designing the noncircular gear profile. The pitch curve of noncircular gear is a plane closed curve. If there is a serious concave in the pitch curve, noncircular gear tooth profiles cannot be generated. Therefore, the pitch curve of noncircular gear should have the following two characteristics. 1) The pitch curve should be closed, and the first-order derivative of this curve is continuous. 2) The radius of curvature of the curve needs to be greater than zero. Based on the relationship between diameter and curvature in the pitch curve, this paper designs a new pitch curve of noncircular gear with curvature radius function. The new noncircular pitch curve keeps convex, which is beneficial to the manufacturing of noncircular gear, and can realize a variety of transmission laws.

2. Relationship between diameter and curvature of curve

2.1 Tangent polar coordinates of the plane convex curve

As shown in Fig. 1, the oval curve on the plane $O x y$ is $C$ , which is a smooth closed convex curve. The tangent of the oval curve at point $m(x$ , $y$ ) is $t$ . A straight line On is perpendicular to the tangent $t$ and the intersection point of line On and $t$ is $n$ . The distance between the tangent line $t$ and the original point $O$ is $p$ . The distance between the original point $O$ and the point $m$ is $r$ . The angle between the axis Ox and the straight line On is $\theta$ . The angle between the axis Ox and the straight line Om is $\varphi$ . In the polar system, the oval curve equation is $r=r(\varphi)$ . For the smooth closed convex curve, $p$ is the single valued function of $\theta$ . Referring to the geometric relation in Fig. 1, the function $p$ can be expressed

$\displaystyle p(\theta)=r(\varphi)\sin\mu$ (1)

where $\mu$ is the angle between the polar diameter of the oval curve and the tangent of the point $C$ . The relation between $\theta$ , $\varphi$ and $\mu$ is as the follows

$\displaystyle\theta=\varphi+\mu-90$ (2)

The coordinate of $n$ is ( $\theta$ , $P$ ), so the equation of the oval curve also can be given

$\displaystyle p(\theta)=x\sin{\rm-}y\cos\theta$ (3)

All tangents form a single parameter family of straight lines, Eq. (3) is the equation of their straight family, $\theta$ is the parameter. The oval curve $t$ can be seen as the envelope of this tangent family. According to the envelope method, we can get the parameter equation of tangent

$\displaystyle\begin{cases}{x=P(\theta)\sin\theta+{P}^{\prime}(\theta)\cos% \theta}\\ {y=-P(\theta)\cos\theta+{P}^{\prime}(\theta)\sin\theta}\\ \end{cases}$ (4)

If $p$ and $\theta$ is known, it is possible to determine the only point ( $x$ , $y$ ) on the oval curve $C$ , and vice versa. The equation ( $\theta$ , $p(\theta)$ ) introduced above is called the tangent polar coordinate of the oval curve $C$ .

Figure 1.

Tangent of curve in the polar coordinate system.

2.2 The radius and curvature of oval curve

The pitch curve of noncircular gear is not circular, so the radius of curvature at each point is different. The pitch curve of a planar noncircular gear is a closed curve in the plane. The tooth profile of noncircular gear is generated by the pitch curve. In order to ensure the generation of tooth profile, the pitch curve should be a plane oval line. Assume that $\rho(\theta)$ and $p(\theta)$ are the curvature function and the tangent polar coordinate function of the planar oval, respectively. The derivative of Eq. (4) is

$\displaystyle\begin{cases}{{x}^{\prime}=-(P(\theta)+{P}^{\prime\prime}(\theta)% )\sin\theta}\\ {{y}^{\prime}=(P(\theta)+{P}^{\prime\prime}(\theta))\cos\theta}\\ \end{cases}$ (5)

The arc length of the oval curve is defined as $s$ , which is a monotonic decreasing function of $\theta(0\leqslant\theta\leqslant 2\pi)$ .

$\displaystyle\rho(\theta)=\frac{{\rm d}s}{{\rm d}\theta}=\sqrt{({x}^{\prime})^% {2}+({y}^{\prime})^{2}}=P(\theta)+{P}^{\prime\prime}(\theta)$ (6)

The above Eq. (6) is a second-order linear differential equation with an unknown parameter. According to advanced mathematics related knowledge, its general solution can be expressed as

$\displaystyle p(\theta)=C_{1}\cos(\theta)+C_{2}\sin(\theta)+\int_{0}^{\theta}{% \rho(s)}\sin(\theta-s){\rm d}s$ (7)

Figure 2.

The schematic flow chart for the proposed algorithm.

3. The noncircular gear with curvature radius function

3.1 Mathematical model of noncircular gear pair

In order to facilitate the manufacture of noncircular gears and obtain full rotation, the noncircular gear curve should be a closed oval curve. The necessary and sufficient condition for the function of a closed oval curve curvature radius is that the period of $\rho(\theta)$ is $2\pi$ , $\rho(\theta)$ is continuous and greater than zero, and

$\displaystyle\int_{0}^{2\pi}{\rho(\theta)}\cos(s){\rm d}s=\int_{0}^{2\pi}{\rho% (\theta)}\sin(s){\rm d}s=0$ (8)

If $\rho(\theta)$ satisfies the Eq. (8), the equation of the pitch curve of noncircular curve can be obtained by substituting $\rho(\theta)$ into the Eq. (7). In order to facilitate the design of driven noncircular gears which conjugated with the driving noncircular gear, the upper tangent coordinate equation is converted into polar coordinate equation [12]

$\displaystyle\begin{cases}r_{1}=\sqrt{x^{2}+y^{2}}\\ \varphi_{1}=ac\tan(y/x)\\ \end{cases}$ (9) $\displaystyle\begin{cases}P(\theta)=r_{1}(\varphi_{1})\sin\mu\\ \theta=\varphi_{1}+\mu-90\\ \tan\mu={r_{1}}/{({{\rm d}r_{1}}/{{\rm d}\varphi_{1}})}\\ \end{cases}$ (10)

where, $r_{1}$ is the radius of the driving noncircular gear, $\varphi_{1}$ is the angle of the driving noncircular gear. By substituting $P(\theta)$ into Eqs (9) or (10), the uniqueness of the $r_{1}$ ( $\varphi_{1}$ ) is obtained. According to the transmission principle of noncircular gear pair, the equation of pitch curve of driven noncircular gear is obtained

$\displaystyle r_{2}(\varphi_{1}){\rm=}a-r_{1}(\varphi_{1})$ (11) $\displaystyle\varphi_{2}=\int_{0}^{\varphi_{1}}{\frac{r_{1}(\varphi_{1})}{a-r_% {1}(\varphi_{1})}d\varphi_{1}}$ (12)

where, $r_{2}$ is the radius of the driven noncircular gear, $\varphi_{2}$ is the angle of the driven noncircular gear, $a$ is the center distance of two gears. The center distance $a$ can be obtained by the above equations, and finally the center distance is substituted into the Eq. (11) to obtain the pitch curve of the driven noncircular gear.

Figure 3.

Auxiliary design program of the noncircular gear.

In the computational model of noncircular gear pitch curve, the Eq. (12) is used to calculate the center distance. In Eq. (12), the exact value of the unknown parameter in the integral function cannot be obtained by exact calculation. Therefore, variable step Newton-Cotes method and numerical method are used to solve the center distance, the specific process in Fig. 2. First, an initial center distance $a_{0}$ is given, and the precision value $\delta$ is set for the iterative program. Substituting the given initial center distance $a_{0}$ into Eq. (12), the integral function is calculated by using the Newton-Cotes method, and the angle of driven gear $\varphi_{2}$ is calculated. The absolute difference between the angle $\varphi_{2}$ and $2\pi$ is compared with the precision value $\delta$ . And according to the result of the comparison, the program changes the value of the center distance to meet the requirements of the iterative program precision, and finally outputs the center distance value.

3.2 Convexity determination of driven noncircular gear

According to the mathematic model of the driving noncircular gear pitch curve, the pitch curve of the driving noncircular gear must be convex. In order to ensure the continuity of the gear transmission, the pitch curve of the driven noncircular gear should also be convex as the driving noncircular gear pitch curve. Therefore, it is necessary to judge the convexity of the driven noncircular gear pitch curve. The curvature of the driven noncircular pitch curve at each point can be calculated by the Euler-Savary formula [13]

$\displaystyle\frac{1}{\rho_{2}}=\left({\frac{1}{r_{1}}+\frac{1}{a-r_{1}}}% \right)\sin\mu-\frac{1}{\rho_{1}}>0$ (13)

3.3 Arc length calculation

For the noncircular gear pairs, the arc length of the driving noncircular gear and the driven noncircular gear pitch curve are equal because that the gear is pure rolling motion with each other. If the period is $L$ , the arc length of noncircular gear curve is

$\displaystyle L=\int_{0}^{2\pi}{\frac{{\rm d}s}{{\rm d}\theta}}{\rm d}\theta=% \int_{0}^{2\pi}{P(\theta)}{\rm d}\theta+{P}^{\prime}(\theta)\left|{{}_{0}^{2% \pi}}\right.$ (14)

The curve ${P}^{\prime}(\theta)$ is also a periodic function that changes for a period of $2\pi$ , so

$\displaystyle L=\int_{0}^{2\pi}{\frac{{\rm d}s}{{\rm d}\theta}}{\rm d}\theta=% \int_{0}^{2\pi}{P(\theta)}{\rm d}\theta$ (15)

when $P(\theta)=-P(\theta+\pi)$ , the arc length is $L=2\pi R$ .

Figure 4.

The new noncircular gear pair.

3.4 Generation of the tooth profile

According to the above sections, the pitch curves of the noncircular gear are designed. These pitch curves keep convex, so the tooth profile can be processed by generating method. There are many researches on the principle of generating the tooth profile of noncircular gear [14, 15]. By using a standard tool gear to rolling on the pitch curve of the noncircular gear, the theoretical meshing point of the cutter tooth profile and the noncircular gear is calculated at each position. The specific process is as follows: 1) According to Eq. (14), if the tooth number of the noncircular gear Z is selected, the modulus of noncircular gear is $m{\rm=}L/{\pi Z}$ . Therefore, the basic parameters of the noncircular gear can be obtained, including basic circle, addendum circle and so on. 2) According to Eq. (10), the angle between the tangent of the pitch curve and the positive direction of the axis is

$\displaystyle\lambda(i)=\arctan\left({\frac{r_{1}(i+1)\sin[{\varphi(i+1)}]-r_{% 1}(i-1)\sin[{\varphi(i-1)}]}{r_{1}(i+1)\cos[{\varphi(i+1)}]-r_{1}(i-1)\cos[{% \varphi(i-1)}]}}\right)i=[{1,2\ldots n}]$ (16)

The position of the cutter rotating center is

$\displaystyle\begin{cases}{x_{od}(i)=r_{1}(i)\cos[\varphi(i)]+m[ha^{\ast}+x(i)% ]\cos(\lambda(i)-0.5\pi)}\\ {y_{od}(i)=r_{1}(i)\sin[\varphi(i)]+m[ha^{\ast}+x(i)]\sin(\lambda(i)-0.5\pi)}% \\ \end{cases}$ (17)

3) According to the above equations, the angle of the cutter is

$\displaystyle\gamma(i)=\varphi(i)\pm\arccos\left({\frac{r_{1}^{2}(i)+x_{od}^{2% }(i)-y_{od}^{2}(i)-(0.5mz_{d})^{2}}{2r_{1}(i)\sqrt{x_{od}^{2}(i)+y_{od}^{2}(i)% }}}\right)$ (18)

where $z_{d}$ is the tooth number of the cutter, when ${{\rm d}r_{1}(i)}/{{\rm d}\varphi}(i)>0$ , $\gamma(i)$ is positive sign, when ${{\rm d}r_{1}(i)}/{{\rm d}\varphi}(i)<0$ , $\gamma(i)$ is minus sign.

According to Eqs (16)–(18) and the numerical calculation method, the tooth profile can be generated. The noncircular gear tooth profile generation method is mature, so it does not e not necessary to explain in detail.

Figure 5.

The transmission ratio of the new noncircular gear pair.

Figure 6.

The 3D model of the new noncircular gear pair.

Figure 7.

The output speed of the new noncircular gear pair.

4. Examples and program writing

4.1 Auxiliary design program writing

According to the requirement of continuous transmission, the curvature of noncircular gear pitch curve needs to satisfy the requirement of Eq. (8), continuous and derivative existence. In this paper, we use the following equation as the curvature of noncircular gear pitch curve

$\displaystyle\rho(\theta)=R+b_{0}\sin(k\theta)+c_{0}\cos(l\theta)$ (19)

When the curvature of the noncircular gear pitch curve is known, it is possible to find the pitch curves of the driving noncircular gear and the driven noncircular gear by the mathematical model established above. In order to get the noncircular gear pitch curve more intuitively and conveniently, and analyze the transmission of the noncircular gear pair, the visual aided design program of the gear curve and its conjugate noncircular gear is written by MATLAB language [16], as shown in Fig. 3. The program can calculate the center distance of the noncircular gear under the parameters of the noncircular gear pitch curve $R$ , $b_{0}$ , $k$ , $c_{0}$ and $l$ . It can get the center distance of the noncircular gear under this parameter, get the pitch curve of the driven noncircular gear, and further calculate the curve of the transmission ratio, and judge the concavity of the pitch curve. The program can simulate the transmission process of the noncircular gear and make the design more intuitive.

4.2 Example

The setting parameters are $R=10$ , $b_{0}=1$ , $k=2$ , $c_{0}=2$ and $l=4$ , the curvature equation of noncircular gear curve is

$\displaystyle\rho(\theta)=10+\sin(2\theta)+2\cos(4\theta)$ (20)

According to the Eq. (7), the tangent polar coordinates of the driving wheel curve can be obtained

$\displaystyle p(\theta)=10-\frac{148}{15}\cos(\theta)-\frac{2}{3}\sin(\theta)(% \cos(\theta)-1)-\frac{2}{15}\cos(4\theta)$ (21)

According to the above numerical method, the center distance is $a=$ 19.997 mm. The arc length of the noncircular gear pair can be obtained by substituting the Eq. (13) into the Eq. (19). The arc length of the noncircular gear pair is 20 $\pi$ . There are two main steps in designing noncircular gear. First, the pitch curve of gear pair is calculated, and then other geometric parameters of gear are designed, especially the profile of gear, so as to ensure that the gear can correctly drive according to the required motion. According to the equation established above, the tooth profiles of two gears are generated by the Section 3.4, as shown in Fig. 4.

Table 1

The parameters of the transplanting mechanism

The equation of noncircular	The equation of noncircular	The angle	The angle	The length
gear 2-1	gear 3-1	$\alpha$ ( ${}^{\circ}$ )	$\beta$ ( ${}^{\circ}$ )	$l$ (mm)
$P(\theta)=35+5\cos(\theta)$	$P(\theta)=35+4.5\cos(\theta)$	270	114	164

Figure 8.

The transmission ratio of the new noncircular gear pair.

The transmission ratio of this new noncircular gear pair is a very important performance index. According to the above non circular gear auxiliary design program, different noncircular gear input parameters can obtain different non circular gear transmission characteristics. If the parameters of noncircular gear are $R=10$ , $l$ , $k=2$ , $c_{0}=2$ and $l=4$ , the transmission ratio curve of this noncircular gear was obtained, as shown in Fig. 5. From the Fig. 5, the transmission ratio curve of the noncircular gear pair is continuous. The maximum transmission ratio of this noncircular gear is 1.2, which located at rotation angle 258 ${}^{\circ}$ . The minimum transmission ratio of this noncircular gear is 0.83, which located at rotation angle 136 ${}^{\circ}$ . From the transmission ratio curve, it shows that the noncircular gear can achieve various transmission characteristics, and achieve a specific motion law.

In order to verify the correctness of the above noncircular gear transmission characteristics, the 3D model of this noncircular gear has been established. ADMAS is a software that can analyze the kinematics and dynamics of this mechanism. The three-dimensional model of this noncircular gear is imported into the ADMAS. As shown in Fig. 6, the Noncircular gear models are added with constraints and motors. When the speed of the driving wheel is 30 deg/sec, the driven gear can get a changing speed, as show in Fig. 7. Figure 7 shows that the output speed has some fluctuation, which caused by the tooth profile modeling error, the model installation error and so on, but the output speed has a certain regularity. By comparing Figs 6 and 7, The trend of two curves is roughly the same. Therefore, the mathematical modeling of the new noncircular gear is correct.

Figure 9.

The transmission ratio of the new noncircular gear pair.

4.3 Application

As show in Fig. 8, it is a typical vegetable transplanting mechanism, which can transplant seedlings into the particular field. The vegetable transplanting mechanism is mainly composed by the planetary frame1, the central noncircular gear 2-1, the primary noncircular driven gear 2-2, the second noncircular driving gear 3-1, the second noncircular driven gear 3-2 and the arm 4. When the planetary frame1 rotates a week, the end of the arm 4 will form a closed plane curve, which is the required by the vegetable transplanting mechanism.

The new noncircular gear pairs are used to drive the transplanting mechanism in this paper. The parameters of the transplanting mechanism are shown in Table 1. The angle between the planetary frame1 and the horizontal axis is $\alpha$ . The angle between the planetary frame1 and the arm 4 is $\beta$ . The length of the arm 4 is $l$ . Under these parameters, the end of the arm 4 forms the trajectories as shown in Fig. 9. This trajectory is good for transplanting mechanisms base on Refs [17].

5. Conclusion

With the purpose of guaranteeing the convexity of a noncircular gear, a new type of noncircular gear pair was designed basis on the curvature function of the noncircular gear pitch curve in this paper. The specific work is as follows

(1) (1)

In the plane tangent polar coordinates, the relation between the curvature of the pitch curve and the radial diameter was established. The equation of the noncircular gear pitch curve was derived, and the calculation formula of the concavity and convexity and the arc length was further established.

(2)

An example design was carried out through a given parameter, and an auxiliary design program was written in MATLAB language to study the transmission characteristics of the noncircular gear.

Footnotes

Acknowledgments

The study is supported by Visiting Engineer Program by Shaoxing Public Technology Research Project (No. 2017B70076), Zhejiang Provincial Department of Education’s Visiting Engineer Project (No. FG2014114), Science and Technology Innovation Project of College Students in Zhejiang , Science and Technology Innovation project of College Students in Shaoxing.

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