Gear transmission is the most common transmission mechanism in the entire mechanical field. It serves as an important transmission structure for various devices and machine tools, and it is a standard measure of modern industrialization progress. It plays a pivotal role in aerospace, automotive, manufacturing, and mining industries. Common cylindrical gears mainly include straight tooth lines, helical tooth lines, herringbone tooth lines, and curved tooth lines, among which the first three belong to linear tooth lines. In recent years, as research on curved tooth line cylindrical gears has become more mature, the concept of arc toothed line gears has been emphasized. Compared with traditional linear gears, the load-bearing capacity and transmission performance of arc gears have been significantly improved, but how to achieve efficient gear machining remains difficult.
I propose an ideal model of a planar spiral toothed line cylindrical gear. Based on differential geometry knowledge, I derive the static tooth surface equation, dynamic tooth surface equation, contact line equation, and the calculation of the contact ratio of this gear, and I verify the accuracy of the static tooth surface equation through MATLAB. Based on Visual Basic 6.0 and SolidWorks secondary development, I build an automatic modeling system for planar spiral toothed line cylindrical gears. Through the gear pair generated by the modeling system, I conduct research and analysis on the gear pair contact interval in the SolidWorks Simulation module.
Based on the accurate model generated by the automatic modeling system for planar spiral toothed line cylindrical gears, I analyze the anti-bias load characteristics of straight teeth, helical teeth, and spiral teeth. Through the static analysis module in ANSYS Workbench, I analyze the force conditions of three types of gear pairs under two conditions: the ideal meshing state and the meshing state where the gear installation axis has a parallelism error. I study the variation of the maximum contact stress on the tooth surface of the three gear pairs with the number of teeth and modulus, and I verify that the planar spiral toothed line cylindrical gear pair has excellent anti-bias load characteristics.
By analyzing the forming principle of the planar spiral rack, I propose a full-tooth-width high-linear-speed continuous generating gear machining method. Through the tool machining trajectory, I derive the planar spiral tooth line equation and the tool arrangement trajectory equation. I also elaborate on the tooth groove depth machining and involute tooth profile generating gear machining principles of this gear. Based on Vericut simulation software, I verify the feasibility of the helical tooth gear machining method. Experiments prove that this gear machining method can achieve full-tooth-width high-speed milling of gears. Through MATLAB, I construct the concave and convex tooth surface models of the planar spiral gear and the gear pair model obtained from Vericut simulation for tooth surface contact experiments. I conclude that the contact characteristics of the gear pair model are as follows: the contact interval is mainly distributed in the middle 80% to 90% region of the tooth surface, and the force on both ends of the tooth is small. This effectively reduces the bending stress at the tooth root at both ends of the tooth, which is consistent with the theoretical derivation results and is conducive to improving the force on both ends of the tooth, enhancing the gear load-bearing capacity.
Based on the planar spiral toothed line cylindrical gear machining method studied above, I independently develop a dedicated gear machining device for planar spiral toothed line cylindrical gears, and I successfully machine ideal gear samples. The tooth profile of the machined sample is an involute tooth profile, and the gear tooth surface accuracy is good. Compared with hobbing and gear shaping, this method greatly saves milling time and reduces production costs.
Gear transmission has the advantages of high transmission efficiency, large transmission torque, long service life, and compact transmission structure. As the most widely used power transmission mechanism in the mechanical field, it has become one of the symbols of modern industrialization. Gear transmission plays an irreplaceable role in various national fields such as aerospace, automobile manufacturing, construction machinery, and metallurgy. Therefore, the transmission performance of gears directly affects the production quality and performance of mechanical equipment.
The transmission performance of cylindrical gears is related to many factors, among which the most important factors are the tooth profile shape and the tooth line. Taking the most widely used cylindrical gears as an example, common tooth profile shapes include involute, circular arc, and cycloid, among which the involute tooth profile is widely used due to its excellent performance and simple gear machining. In addition to the tooth profile, the tooth line also directly determines the transmission performance of cylindrical gears. Taking the commonly used involute cylindrical gears on the market as an example, there are mainly straight teeth, helical teeth, herringbone teeth, and arc teeth. Cylindrical gears with different tooth lines have their own disadvantages in transmission and gear machining manufacturing.
For straight teeth, the instantaneous contact line of the gear tooth surface should theoretically be parallel to the gear axis during the meshing transmission process. However, due to installation errors during the gear shaft installation process, and because the gear enters meshing and exits meshing simultaneously during meshing, the gear pair is prone to bias load during meshing, which is likely to cause vibration and affect transmission stability and service life.
Compared with straight teeth with basically the same parameters, the tooth surface contact line of helical teeth is longer, and the contact stress is evenly distributed. The meshing characteristics of helical gears are gradual entry into meshing and gradual separation from meshing. This contact method can avoid meshing impact and reduce noise, so it has better transmission characteristics and stronger load-bearing capacity than straight teeth. However, its contact line is not parallel to the central axis, resulting in an axial component force acting on the bearing during meshing, which shortens the service life of the bearing.
Next is the herringbone gear. On the basis of retaining the advantages of helical gears, due to its symmetrical tooth line structure, the axial component force can be eliminated to a certain extent. However, the gear machining manufacturing process of herringbone teeth is extremely complicated. The middle section needs to be retracted during the gear machining process, and the symmetry of the left and right teeth cannot be guaranteed. It is easy to produce bias load during the meshing process of the gear pair, thereby causing increased vibration of the gear pair.
At present, the applicable working conditions of gear transmission are changing toward high speed and heavy load. Compared with traditional cylindrical gears, arc gears can replace traditional gears in most harsh working conditions due to their superior transmission performance, good tooth surface load-bearing performance, and anti-bias load performance. However, because the existing gear machining methods for arc toothed line cylindrical gears have the problems of low gear machining efficiency and low gear machining accuracy, they still cannot be practically applied.
Therefore, the need to research a new type of gear with low manufacturing cost, strong load-bearing capacity, and long service life, and to develop corresponding high-efficiency and high-precision gear machining devices, is very urgent. I propose a planar spiral toothed line cylindrical gear and its full-tooth-width high-linear-speed continuous generating gear machining method. This type of gear can be rapidly formed. The meshing contact line of the gear pair is mainly distributed in the middle 80% to 90% of the tooth surface. Since the planar spiral tooth line is approximately an arc, it is basically symmetrical with respect to the middle section tooth line of the gear, which can effectively reduce bias load and has higher load-bearing capacity. At the same time, because it can perform efficient gear machining milling, it greatly improves production efficiency and reduces manufacturing costs.
There are many different methods for gear machining technology. The adoption of gear machining technology mainly depends on the required gear type, size, and precision. The following are some common gear machining technologies.
Hobbing and gear shaping: Hobbing is a gear machining method in which a specially designed hob or roller, whose shape and size match the gear tooth profile to be machined, rolls and cuts on the workpiece. Compared with traditional milling methods, the biggest feature of hobbing is continuous milling. Hobbing has the advantages of high efficiency, high precision, and good surface quality, and is suitable for large-scale production because it can complete the gear machining of one gear in a short time. Due to the precise design of the hob or roller, hobbing can achieve high-precision gear manufacturing and provide a more accurate tooth profile. Hobbing can usually obtain higher surface quality and reduce the need for subsequent gear machining processes or grinding. However, hobbing also has several shortcomings, mainly summarized as follows: first, hobbing requires controlling the surface finish and dimensional accuracy of the blank, so workpiece preprocessing is required. Second, the wear generated after long-term use of the hob will directly affect the surface accuracy of the workpiece, so the milling cutter must be replaced in time. Gear shaping is also a mainstream gear machining method at present. This method mainly completes gear cutting through a forming milling cutter. Gears obtained by gear shaping are generally suitable for occasions with high gear precision requirements and good transmission stability. Gear shaping has high requirements for operators, and it is necessary to ensure that the gear machining parameters are completely accurate and the position of the gear machining tool is correct to carry out stable production. Because the tool of gear shaping moves in a reciprocating manner, its gear machining rate cannot be greatly improved, affecting cutting efficiency.
Gear grinding: Gear grinding is a process in which the grinding wheel contacts the gear and gradually removes material from the gear surface to form an accurate tooth profile. This method can achieve higher tooth surface precision and surface quality, usually reaching grade 5 precision or higher. Gear grinding is suitable for a variety of different gear materials, including steel, cast iron, and ceramics. Higher surface quality can be obtained through gear grinding, usually reaching a Ra value below 1.6 micrometers. This smoothness can reduce friction loss and noise in gear transmission and improve transmission efficiency. In general, gear grinding is a high-precision gear machining method suitable for application fields with high requirements for gear precision and surface quality.
Gear skiving: Gear skiving is generally suitable for large-batch, high-precision, and high-quality gear production. The tools for gear skiving are generally customized special blades and cutter heads. The workpiece is fixed, and the tool performs surface skiving along the outer surface of the workpiece to form the gear tooth profile. Parts obtained by gear skiving generally do not need finishing because gear skiving can obtain very smooth surface quality. Compared with other gear machining methods, gear skiving can usually complete gear machining faster, thereby improving production efficiency.
Electrical discharge machining: Electrical discharge machining is a very precise metal gear machining method. This method uses the principle of electrical pulse discharge to remove waste material from the workpiece through the discharge between the electrode and the workpiece, finally forming the required gear. Electrical discharge machining can achieve very high precision and is suitable for manufacturing complex gear shapes. During gear machining, the electrode and the workpiece do not directly contact, avoiding the problem of tool wear. This gear machining method is suitable for materials with high hardness, such as tool steel and cemented carbide. However, the preparation of electrodes in this gear machining method requires precise processes to ensure that they are consistent with the shape of the design requirements. Reasonable setting of gear machining parameters is very critical, and different materials and shapes may require different parameters. Compared with some other gear machining methods, this method has a slower forming speed and is suitable for small-batch and high-precision production, such as molds and precision mechanical equipment.
Powder metallurgy forming: Powder metallurgy is a gear machining method that forms the required workpiece by compressing metal powder and then solidifying it into a component through processes such as sintering or hot pressing. However, this forming method requires necessary grinding operations to obtain the precise dimensions and surface quality of the gear. In some cases, subsequent heat treatment is performed to adjust the performance of the component. In addition, the gear needs to be lubricated to improve its service life and performance. Powder metallurgy gear machining provides a cost-effective method for manufacturing gears with complex shapes, especially suitable for small and medium batch production. However, it should be noted that due to shrinkage during the sintering process, the dimensional accuracy of powder metallurgy gears may be difficult to meet high requirements.
Compared with traditional straight teeth and helical teeth cylindrical gears, arc toothed cylindrical gears greatly improve transmission performance and tooth surface load-bearing capacity. Moreover, the development direction of the gear transmission field is changing toward high speed and heavy load. Arc toothed cylindrical gears can replace traditional gears in most harsh working conditions. As a new type of transmission in transmission structures, once the gear machining technology of arc toothed cylindrical gears becomes mature, their application value will greatly increase. Therefore, there is an urgent need for theoretical research related to this gear and the development of efficient gear machining methods and gear machining devices.
As a new type of cylindrical gear, curved cylindrical gears retain the superior transmission performance of traditional involute gears while avoiding the problems of straight cylindrical gears being prone to bias load, helical cylindrical gears being prone to axial component force, and herringbone tooth lines being difficult to machine. At the same time, due to the particularity of their tooth lines, they have the advantages of anti-bias load, stable transmission, and no axial component force.
Arc toothed cylindrical gears still face many problems in their development. First, the geometric parameters of the ideal model have not yet been clearly defined. Second, research on the modification and undercutting of arc toothed cylindrical gears has not yet made substantial progress. Most of the currently proposed gear machining methods for arc toothed line gears remain at the theoretical stage, and most gear machining methods have the problems of low gear machining efficiency and poor gear machining accuracy.
In terms of gear machining for arc toothed line cylindrical gears, although various methods have been proposed to machine standard arc gear models, many difficulties are still found in practice. Based on this, after summarizing the characteristics of existing cylindrical gear gear machining principles, I propose a full-tooth-width high-linear-speed continuous generating gear machining method. Experiments show that this method can achieve efficient milling of planar spiral toothed line cylindrical gears.
The research significance of this topic is that I propose for the first time a planar spiral toothed line cylindrical gear gear machining method and its gear machining device. It is also the first time that truly full-tooth-width high-speed continuous generating gear machining has been achieved, and I have developed a dedicated gear machining device for planar spiral toothed line cylindrical gears. The successful development of this gear will promote the overall development of gear manufacturing technology. Preliminary research shows that the gear machining method proposed in this paper greatly improves gear machining efficiency, shortens gear machining time, and saves production costs while ensuring transmission characteristics similar to existing arc gears. Second, the gear pair has good anti-bias load performance, long service life, and strong adaptability to installation errors. Because its teeth are approximately arc-shaped, the curvature radius of the tooth line of the concave tooth surface is slightly larger than that of the convex tooth surface, and the tolerance for shafting deformation is high. Therefore, it is not easy to cause stress concentration at both ends of the tooth, thereby enhancing the bending fatigue strength of the gear.
The main research content of this topic is as follows. First, I clarify the main geometric parameters of a planar spiral toothed line cylindrical gear. According to the forming principle of the gear tooth surface, I derive its static tooth surface equation, dynamic meshing line equation, and contact ratio calculation, and I derive various gear model parameters, clarify the geometric parameters of the planar spiral tooth line, and verify the accuracy of the relevant tooth surface equations based on MATLAB.
Second, I conduct research on the ideal geometric model modeling of planar spiral toothed line cylindrical gears. Based on Visual Basic 6.0 and SolidWorks secondary development technology, I develop a parametric modeling system for planar spiral toothed line cylindrical gears. After inputting basic parameters, automatic modeling can be completed and a three-dimensional model can be generated. This method not only improves the efficiency and accuracy of modeling, but also lays a solid foundation for the anti-bias load analysis of the teeth of planar spiral gears.
Third, I analyze the anti-bias load characteristics of planar spiral toothed line cylindrical gears. Based on the ANSYS Workbench finite element analysis module, I analyze the tooth surface force conditions of straight teeth, helical teeth, and spiral teeth under the ideal meshing state and the meshing state where the gear installation axis has a parallelism error, and I analyze the influence of the number of gear teeth and modulus on the maximum contact stress of the tooth surface of the three gear pairs. This reduces the risk of bias load caused by gear manufacturing errors and installation errors, improves the transmission performance of the gear pair during meshing, and improves the load-bearing performance of planar spiral gears.
Fourth, I propose an efficient gear machining method for planar spiral toothed line cylindrical gears. I propose a rapid forming method for planar spiral toothed line cylindrical gears: the full-tooth-width high-linear-speed continuous generating gear machining method. A large-diameter rotating cutter head is used, and several tools are arranged on the cutter head. The tools are arranged in a planar spiral line. The pitch of the planar spiral line is equal to the gear tooth pitch, and the spiral line pitch is $S = \pi m$. The cutter head and the gear blank rotate in a linked manner according to a specified numerical relationship. When the cutter head rotates one revolution, the gear blank continuously rotates one tooth, realizing full-tooth-width continuous rapid milling of the gear blank. The tooth line of the machined gear after unfolding along the pitch cylinder is a planar spiral line.
Fifth, I develop a dedicated gear machining device for planar spiral toothed line cylindrical gears. Because this gear machining method involves the linkage control of multiple rotating axes, the existing CNC machine tools cannot match the gear machining of this type of gear. Therefore, based on the forming principle of spiral teeth, the gear machining method, and the Vericut simulation results, I independently develop a planar spiral toothed line cylindrical gear gear machining device and conduct trial cutting experiments on gear blanks, verifying that this gear has high gear machining efficiency and good tooth surface accuracy.
At present, the existing gear machining methods for arc toothed cylindrical gears have problems such as low gear manufacturing efficiency. After analyzing the technical characteristics of all arc gear gear machining principles, I propose a rapid forming method and efficient gear machining device for planar spiral toothed line cylindrical gears, namely the full-tooth-width high-linear-speed continuous generating gear machining method. Research shows that this new type of cylindrical gear has excellent anti-bias load characteristics and improves its tooth surface load-bearing capacity. The proposed gear machining method and device can complete the rapid forming of planar spiral gears.
The meshing theory of straight teeth and helical teeth cylindrical gears is basically mature. However, the current research on the forming principle and geometric parameters of curved tooth line cylindrical gears is relatively insufficient. I propose a new type of curved tooth line cylindrical gear, namely the planar spiral toothed line cylindrical gear. Through differential geometry and spatial transformation theory, I calculate the tooth surface equation, dynamic meshing line equation, and tooth surface contact ratio of the planar spiral gear. Based on SolidWorks secondary development and Visual Basic 6.0 software, I independently develop a planar spiral toothed line cylindrical gear modeling system and study its meshing performance.
The planar spiral gear features mainly include a convex tooth surface, a concave tooth surface, and a planar spiral tooth line. The tooth profile at the middle section of the tooth width is an involute. The tooth line of the planar spiral toothed line cylindrical gear is a planar spiral tooth line. This special tooth line mainly depends on its special gear machining principle and method. The advantages of this tooth shape are mainly in two aspects. First, the curvature radius of the tooth line of the concave tooth surface is slightly larger than that of the convex tooth surface, which can make the contact interval of the gear pair meshing mainly distributed in the middle position of the tooth surface. This contact method can effectively reduce the stress concentration at the tooth root at both ends of the tooth and improve the contact state at both ends of the tooth. Second, because the tooth is arc-shaped, during the meshing process of the concave and convex tooth surfaces, it can effectively reduce the bias load of the tooth caused by errors generated by gear manufacturing and manual installation.
The planar spiral gear has five main parameters, namely: modulus $m$, number of teeth $Z$, pressure angle $\alpha$, tooth width $B$, and spiral line curvature radius $R_T$ at the middle line of the tooth width. Since the tooth lines on both sides of the middle line of the tooth width of the planar spiral toothed line cylindrical gear are basically symmetrical, it can effectively reduce the axial component force generated during meshing. Because the tooth is arc-shaped, the contact ratio of the gear pair during meshing is greater than that of ordinary straight cylindrical gears, improving the transmission stability of the gear pair.
The distribution of the planar spiral toothed line cylindrical gear on the pitch cylinder is shown below. At the same development angle, the arc tooth radii of the concave tooth surface and the convex tooth surface have the relationship $R_W – R_N = \frac{\pi m}{2}$, where $m$ is the gear modulus.
$$x_{DP} = R_T \pm \frac{\pi m}{4}\cos(\theta) – \frac{\pi m z}{2}\theta$$
$$y_{DP} = R_T \pm \frac{\pi m}{4}\sin(\theta)$$
In the equation, the “$\pm$” signs correspond to the tooth line equations of the concave and convex tooth surfaces, respectively. $R_T$ is the machining tool radius, and $\theta$ is the independent variable, representing the angle between each point on the tooth line and the line $O_D$ and $X_D$. According to the direction of the planar spiral line, it is divided into left-hand and right-hand planar spiral toothed line cylindrical gears. Similar to helical gears, a pair of left-hand and right-hand planar spiral toothed line cylindrical gear pairs is required for meshing. In the cross-section of the pitch surface of the planar spiral gear, the circumferential tooth thickness and circumferential tooth groove width at the middle section of the gear are equal and equal to half of the tooth pitch, that is, $S_C = P_C = \frac{\pi m}{2}$. At other sections, the circumferential tooth groove width is greater than the circumferential tooth thickness, which can ensure that the tooth surface contact force of the gear pair is mainly distributed in the middle position of the tooth surface, reducing the transmission bias load of the gear pair caused by gear manufacturing errors and installation errors.
The key to studying the model characteristics of planar spiral gears is the tooth surface equation, and the basic element of the tooth surface equation is the end face tooth profile equation. A standard involute cylindrical gear pair correct meshing diagram is considered. $G_1$ and $G_2$ are the involutes of the correct meshing of the gear pair, $k$ is the meshing point at the pitch circle, and $m-n$ is the involute tooth profile meshing line. A coordinate system $X_1O_1Y_1$ is established with gear 1 as the coordinate origin, and a coordinate system $X_2O_2Y_2$ is established with gear 2 as the coordinate origin.
Then the expression of the involute $G_1$ equation of gear 1 in the coordinate system $X_1O_1Y_1$ is:
$$r_j^{(1)} = x_j i_1 + y_j j_1 + z_j k_1$$
$$x_j = R_{b1}\cos u_1 + R_{b1}u_1\sin u_1$$
$$y_j = R_{b1}\sin u_1 – R_{b1}u_1\cos u_1$$
$$z_j = 0$$
$$u_1 = \alpha_k – \theta_1$$
In the equation, $R_{b1}$ is the base circle radius, $\alpha_1$ is the rolling angle, $\theta_1$ is the development angle, and $\theta_1 = \tan\alpha_k – \alpha_k$, with $\theta_1 \approx 0.0149$.
From the above equation, the expression of the involute $G_2$ equation of gear 2 in the coordinate system $X_2O_2Y_2$ is:
$$r_j^{(2)} = x_j i_2 + y_j j_2 + z_j k_2$$
$$x_j = R_{b2}\cos u_2 + R_{b2}u_2\sin u_2$$
$$y_j = R_{b2}\sin u_2 – R_{b2}u_2\cos u_2$$
$$z_j = 0$$
$$u_2 = \alpha_k – \theta_2$$
In the equation, $R_{b2}$ is the base circle radius, $\alpha_2$ is the rolling angle, $\theta_2$ is the development angle, and $\theta_2 = \tan\alpha_k – \alpha_k$. The pressure angle at the gear pitch circle is $20^\circ$, and $\theta_2 \approx 0.0149$.
The three-dimensional mathematical model of the planar spiral gear is the basis for analyzing the model characteristics and transmission characteristics of the planar spiral gear. Therefore, relevant coordinate systems are established with the cutter head center and the gear blank center.
The tool revolution surface equation is:
$$x_D = R_T \pm \frac{\pi m}{4}\sin(\alpha)\cos(\theta)$$
$$y_D = R_T \pm \frac{\pi m}{4}\sin(\alpha)\sin(\theta)$$
$$z_D = u\cos(\alpha)$$
After depth machining, the tooth profile equation is obtained by rotating the tool revolution surface equation by $\mu$ in the direction of $\omega_C$ in the coordinate system $O_CX_CY_CZ_C$:
$$\mu = \frac{\theta_T – \theta_k}{2Z}$$
The transformation matrix $M_{CD}$ from $O_DX_DY_DZ_D$ to $O_CX_CY_CZ_C$ is:
$$M_{CD} = \begin{bmatrix} 1 & 0 & 0 & -R_C \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}^T$$
The transformation matrix for rotating by $\mu$ in the direction of $\omega_C$ is:
$$M_C = \begin{bmatrix} \cos\mu & -\sin\mu & 0 & 0 \\ \sin\mu & \cos\mu & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
Then the tooth surface equation after depth cutting is:
$$x = \cos\mu\left(\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right) – \sin\mu(u\cos\alpha – R_C)$$
$$y = \sin\mu\left(\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right) + \cos\mu(u\cos\alpha – R_C)$$
$$z = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin\theta$$
During generating gear machining, the tool performs generating motion milling on the tooth profile at $\theta$. At this time, the tool position angle is $\theta_T = \theta$, and $k = 0$, that is, $\mu = 0$. Therefore, during generating gear machining, the tool revolution surface is still used as the meshing surface, and its vector expression is:
$$r_D = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta\, i + \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin\theta\, j + u\cos\alpha\, k$$
Then:
$$r_u = \frac{\partial r}{\partial u} = \sin\alpha\cos\theta\, i + \sin\alpha\sin\theta\, j + \cos\alpha\, k$$
$$r_\theta = \frac{\partial r}{\partial\theta} = -\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin\theta\, i + \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta\, j$$
The normal vector of the tool revolution surface is:
$$n_D = r_u \times r_\theta = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\left(\cos\theta\cos\alpha\, i + \sin\theta\cos\alpha\, j \mp \sin\alpha\, k\right)$$
Then its unit vector is:
$$e_D = \cos\theta\cos\alpha\, i + \sin\theta\cos\alpha\, j \mp \sin\alpha\, k$$
Assuming that the relative speed of the gear blank and the tool at the contact point is $v_{12}$, the relationship is obtained:
$$n_D \cdot v_{12} = \Phi = 0$$
Assuming that there is a spatial contact point $M$ between the gear blank and the tool, the vector expressions from points $O_D$ and $O_1$ to the spatial contact point $M$ are:
$$O_D M = r_1, \quad O_1 M = r_2, \quad O_1 O_2 = \varepsilon$$
And the following relationship is satisfied:
$$r_2 = r_1 + \varepsilon$$
$$\varepsilon = -(R_T – R_C)i + R_C k$$
$$r_1 = (x_D – R_T + R_C)i + y_D j + z_D k$$
At the meshing point, the speeds of the tool and the gear are $V_1$ and $V_2$, respectively. Then:
$$v_1 = \omega_D \times r_1 = \omega_D k \times (x_D i + y_D j + z_D k) = \omega_D x_D j – \omega_D y_D i$$
The vector $V_1$ is parallel to the vector $r_\theta$. Combined with the definition of the normal vector, $v_1 \parallel n_D(e_D)$. Therefore, the contact condition becomes:
$$e_D \cdot v_2 = \Phi = 0$$
$V_2$ satisfies:
$$v_2 = \omega_C \times (r_2 + R_C i) = \omega_C \times (r_1 + \varepsilon + R_C i) = \omega_C k \times (x_D i + y_D j + z_D k) = -\omega_C y_D i + \omega_C x_D j$$
Then $u$ should satisfy:
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos\theta – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
The contact line equation is:
$$x_D = R_T \pm \frac{\pi m}{4}\sin\alpha\cos\theta$$
$$y_D = R_T \pm \frac{\pi m}{4}\sin\alpha\sin\theta$$
$$z_D = u\cos\alpha$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos\theta – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
The transformation matrix $M_{1C}$ from $O_CX_CY_CZ_C$ to $O_1X_1Y_1Z_1$ is:
$$M_{1C} = \begin{bmatrix} \cos\varphi & \sin\varphi & 0 & R_C\cos\varphi \\ -\sin\varphi & \cos\varphi & 0 & R_C\sin\varphi \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
Therefore, the contact line expression is:
$$x_1 = \cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right] – \sin\varphi(u\cos\alpha – R_C)$$
$$y_1 = -\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right] – \cos\varphi(u\cos\alpha – R_C)$$
$$z_1 = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin\theta$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos\theta – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
Then the tooth surface equation is:
$$x_1 = \cos\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] – \sin\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$y_1 = \sin\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] + \cos\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos\theta – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$z_1 = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin\theta$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos\theta – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
In the equation, $\mu = \frac{\theta_T – \theta_k}{2Z}$, where $\theta_T$ does not affect the tooth surface shape and only changes the relative position of the overall tooth surface. Different $k$ values represent different tooth surfaces, with $k = 0, 1, 2, \ldots, Z-1$. The tooth surface is determined by parameters such as $R_T$ (main tool installation radius), $R_C$ (gear blank pitch circle radius), $\theta$ (rotation angle), $\varphi$ (generating rotation angle), $m$ (modulus), and $Z$ (number of teeth). The value range of $\theta$ is determined by the tooth width $B$ and $R_T$. The value range of $\varphi$ has been studied previously, as shown below:
When $R_b \geq R_a$ (for standard gears, $z \leq 41$):
$$\frac{2\left[Z – \left(\frac{\pi}{2} + h_a^*\right)\tan\alpha\right]}{Z\cos\alpha} \leq \varphi \leq \frac{2\left(Z + h_a^*\right)\tan\alpha}{Z}$$
When $R_b < R_a$ (for standard gears, $z > 41$):
$$\frac{\sqrt{Z^2}{2}\left[\left(\frac{\pi}{2} + h_a^*\right)\tan\alpha – \sqrt{1 – \left(\frac{Z\cos\alpha}{Z + 2h_a^*}\right)^2}\right]}{Z} \leq \varphi \leq \frac{\sqrt{Z^2}{2}\left[\left(\frac{\pi}{2} + h_a^*\right)\tan\alpha + \sqrt{1 – \left(\frac{Z\cos\alpha}{Z + 2h_a^*}\right)^2}\right]}{Z}$$
where the base circle $R_b$ and the root circle $R_f$. The gear addendum coefficient is $h_a^*$, and the clearance coefficient is $c^*$.
During the meshing process of the gear pair, assuming that gear 1 rotates by an angle $\xi_1$, gear 2 rotates in the opposite direction by an angle $\xi_2$, and the transmission relationship $R_1\xi_1 = R_2\xi_2$ must be satisfied, where $R_1$ and $R_2$ are the pitch circle radii of gear 1 and gear 2, respectively.
The expression of the dynamic tooth surface equation of gear 1 in the coordinate system $O_CX_CY_CZ_C$ is:
$$x_1 = \cos\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta + \xi_1) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] – \sin\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta + \xi_1) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$y_1 = \sin\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta + \xi_1) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] + \cos\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta + \xi_1) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$z_1 = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin(\theta + \xi_1)$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos(\theta + \xi_1) – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
In the equation, $R_{C1}$ is the pitch circle radius of gear 1, and $\xi_1$ is the rotation angle of gear 1, with $-B/2 < h < B/2$.
The expression of the dynamic tooth surface equation of gear 2 in the three-dimensional coordinate system $O_CX_CY_CZ_C$ is:
$$x_2 = \cos\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] – \sin\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$y_2 = \sin\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] + \cos\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$z_2 = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin(\theta – \xi_2)$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos(\theta – \xi_2) – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
In the equation, $R_{C2}$ is the pitch circle radius of gear 2, and $\xi_2$ is the rotation angle of gear 2, with $-B/2 < h < B/2$.
During the meshing process of the gear pair, the intersection line between the dynamic tooth surface equation and the meshing plane is the contact line of the gear pair. During the meshing process of involute tooth profile $G_1$ and involute tooth profile $G_2$, there is a meshing line $m-n$, which is the common tangent of the two tooth surfaces of the gear pair and is combined with the dynamic tooth surface equation obtained in the previous section. Therefore, the meshing plane equation is:
$$\frac{y_L – R_{C1}\sin 20^\circ – R_{C1}\sin 20^\circ\sin 20^\circ}{x_L – R_{C1}\cos 20^\circ – R_{C1}\cos 20^\circ\cos 20^\circ} = \frac{R_{C1}\sin 20^\circ + a_l}{R_{C1}\cos 20^\circ}$$
In the equation, $20^\circ$ is the standard pressure angle of the gear, and $a_l$ is the center distance of the two gears.
From this, it can be concluded that the dynamic meshing line expression of gear 1 in the coordinate system $O_CX_CY_CZ_C$ is:
$$x_2 = \cos\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] – \sin\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$y_2 = \sin\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] + \cos\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$z_2 = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin(\theta – \xi_2)$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos(\theta – \xi_2) – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
$$\frac{y_L – R_{C1}\sin 20^\circ – R_{C1}\sin 20^\circ\sin 20^\circ}{x_L – R_{C1}\cos 20^\circ – R_{C1}\cos 20^\circ\cos 20^\circ} = \frac{R_{C1}\sin 20^\circ + a_l}{R_{C1}\cos 20^\circ}$$
Similarly, the dynamic meshing line expression of gear 2 in the coordinate system $O_CX_CY_CZ_C$ is:
$$x_2 = \cos\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] – \sin\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$y_2 = \sin\mu\left[\cos\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \sin\varphi(u\cos\alpha – R_C)\right] + \cos\mu\left[-\sin\varphi\left[\left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\cos(\theta – \xi_2) – R_C\right] – \cos\varphi(u\cos\alpha – R_C)\right]$$
$$z_2 = \left(R_T \pm \frac{\pi m}{4}\sin\alpha\right)\sin(\theta – \xi_2)$$
$$u = \frac{R_T \sin\alpha\left[(R_T – R_C)\cos(\theta – \xi_2) – R_T\right]}{\frac{\pi m}{4}\cos\alpha}$$
$$\frac{y_L – R_{C1}\sin 20^\circ – R_{C1}\sin 20^\circ\sin 20^\circ}{x_L – R_{C1}\cos 20^\circ – R_{C1}\cos 20^\circ\cos 20^\circ} = \frac{R_{C1}\sin 20^\circ + a_l}{R_{C1}\cos 20^\circ}$$
In the equation, $R_{C2}$ is the pitch circle radius of gear 2, and $\xi_2$ is the rotation angle of gear 2, with $-B/2 < h < B/2$, $20^\circ$ is the standard pressure angle of the gear, and $a_l$ is the center distance of the two gears.
The gear contact ratio refers to the degree of frontal mutual interference between adjacent tooth surfaces in the gear transmission system, that is, the degree of overlap between the vertices or bottoms of adjacent teeth on the tooth surfaces of two gears. The calculation of the gear contact ratio is usually determined by the geometric relationship between the tooth profile curve and the meshing line. In the gear system, the setting of the gear contact ratio will directly affect the stability, efficiency, and life of the gear transmission. If the contact ratio is insufficient, it may cause impact, noise, and increased wear of the gear during transmission, affecting the stable operation of the gear. Conversely, an excessively large contact ratio may also cause mutual interference and energy loss between gears.
The tooth line of the planar spiral gear is relatively close to that of the herringbone tooth, so the contact ratio $\varepsilon_\gamma$ of the planar spiral gear is calculated with reference to the herringbone tooth, mainly composed of the end face contact ratio $\varepsilon_\alpha$ and the axial contact ratio $\varepsilon_\beta$:
$$\varepsilon_\gamma = \varepsilon_\alpha + \varepsilon_\beta$$
The end face tooth profile of the planar spiral gear is an involute, so its $\varepsilon_\alpha$ is calculated according to the calculation method of the involute straight cylindrical gear.
$$\varepsilon_\alpha = \frac{1}{2\pi}\left[z_1(\tan\alpha_{a1} – \tan\alpha_t) + z_2(\tan\alpha_{a2} – \tan\alpha_t)\right]$$
In the equation, $z$ is the number of teeth of the gear, $\alpha_a$ is the addendum pressure angle of the gear end face, and $\alpha_t$ is the pressure angle of the gear end face.
During the actual meshing process of the gear pair, the meshing interval of the planar spiral gear transmission increases by $\Delta L$ compared with the straight tooth. Therefore, the $\varepsilon_\beta$ of the planar spiral gear is:
$$\varepsilon_\beta = \frac{\Delta L}{p_{bt}} = \frac{2B\cos\left[\arcsin\left(\frac{B}{2R_T}\right)\right] – 2B\cos\left[\arcsin\left(\frac{B}{2R_T}\right)\right]}{\pi m\cos\alpha}$$
Therefore, the $\varepsilon_\gamma$ of the planar spiral gear is:
$$\varepsilon_\gamma = \frac{1}{2\pi}\left[z_1(\tan\alpha_{a1} – \tan\alpha_t) + z_2(\tan\alpha_{a2} – \tan\alpha_t)\right] + \frac{2B\cos\left[\arcsin\left(\frac{B}{2R_T}\right)\right] – 2B\cos\left[\arcsin\left(\frac{B}{2R_T}\right)\right]}{\pi m\cos\alpha}$$
From the equation, it can be seen that the methods to increase the axial contact ratio $\varepsilon_\beta$ include increasing the tooth width $B$ and reducing the tooth line curvature radius $R_T$, but they cannot be increased or decreased arbitrarily. The interval of $R_T$ satisfies $R_T \geq 1.5B$, otherwise the tooth profile thickness at both ends of the tooth will become thinner in the circumferential direction, affecting the overall transmission of the gear.
The tooth surface equation is the key basis for studying the transmission characteristics, gear life, and transmission efficiency of planar spiral toothed line cylindrical gears, so whether the tooth surface equation is correct will directly affect the subsequent related simulation experiments. Therefore, in order to verify the accuracy of the static tooth surface equation of the planar spiral toothed line cylindrical gear calculated above, I construct a three-dimensional data model through MATLAB to three-dimensionalize the tooth surface equation of the planar spiral toothed line cylindrical gear. The relevant parameter calculation formulas for the gear tooth surface and the relevant basic gear parameters are shown in the table. Based on MATLAB, I construct a spatial three-dimensional gear tooth surface.
| Name | Symbol | Calculation Formula |
|---|---|---|
| Circumferential modulus | $m_c$ | Select standard value |
| Circumferential pressure angle | $\alpha$ | Select standard value |
| Tooth width | $B$ | $B < b$ |
| Tooth line position angle | $\beta_s$ | Related to tooth line position |
| Pitch circle diameter | $d$ | $d_1 = m \cdot z_1$ |
| Addendum | $h_a$ | $h_{a1} = h_{a2} = h_a^* m$ |
| Dedendum | $h_f$ | $h_{f1} = h_{f2} = (h_a^* + c^*) m$ |
| Whole depth | $h$ | $h_1 = h_2 = (2h_a^* + c^*) m$ |
| Addendum circle diameter | $d_a$ | $d_{a1} = (z_1 + 2h_a^*) m$ |
| Dedendum circle diameter | $d_f$ | $d_{f1} = (z_1 – 2h_a^* – c^*) m$ |
| Base circle diameter | $D_D$ | $D_{D1} = d_1 \cos\alpha$ |
| Tooth pitch | $p$ | $p = \pi m$ |
| Base pitch | $p_b$ | $p_b = p \cos\alpha$ |
| Tooth thickness | $s$ | $s = \pi m / 2$ |
| Tooth groove width | $e$ | $e = \pi m / 2$ |
| Tooth thickness at radius $r_i$ | $s_i$ | $s_i = s r_i / r – 2r_i(\text{inv}\alpha_i – \text{inv}\alpha)$ |
| Bottom clearance | $c$ | $c = c^* m$ |
| Pitch circle diameter | $d’$ | (When center distance is standard center distance $a_D$) $d’ = d$ |
| Transmission ratio | $i$ | $i_{21} = \omega_1 / \omega_2 = z_1 / z_2 = d_1′ / d_2′ = d_1 / d_2 = D_{D1} / D_{D2}$ |
From the figure, it can be seen that the tooth line of the planar spiral toothed line cylindrical gear is in the shape of a planar spiral line with a slight arc, and the middle section line of the tooth surface is an involute tooth profile. The static tooth surface equation calculated above conforms to the theoretical basis.
Based on the tooth surface equation of the planar spiral gear, I build a three-dimensional tooth surface model. However, in order to further study the gear transmission characteristics, establishing an accurate three-dimensional model of the gear is a key step. The accuracy of the planar spiral gear model will directly affect the subsequent finite element analysis, interference check, or meshing analysis. At present, the relevant technologies and software are relatively mature for straight teeth and helical teeth. However, the existing modeling technology still has shortcomings. Taking curved gear cylindrical gears as an example, especially complex curved surface gears such as planar spiral toothed line cylindrical gears, accurate models cannot be established. Therefore, based on the SolidWorks and Visual Basic 6.0 platforms, I build a planar spiral toothed line cylindrical gear modeling system. In order to simplify the cumbersome modeling steps, I realize parametric automatic modeling of planar spiral toothed line cylindrical gears. While improving gear design efficiency, this system also fills the technical gap in the field of curved cylindrical gear modeling and lays a solid foundation for gear simulation analysis.
In addition to the basic gear parameters that are the same as straight teeth and helical teeth, planar spiral gears also have some specific parameters. For example, the tooth line radius of the planar spiral gear is determined by the machining radius of the gear machining tool. The main parameters of this gear are shown in the table.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Number of teeth | 32 | Modulus | 4 |
| Pressure angle $\alpha$ / (°) | 20 | Tooth width $b$ / mm | 30 |
| Addendum coefficient $h_a^*$ | 1 | Shaft hole diameter $d$ / mm | 20 |
| Minimum blade installation radius $R_T$ / mm | 250 | Number of tools | 8 |
By combining the rich API functions in SolidWorks with Visual Basic language programming and adding COM interfaces, I write a modeling program for planar spiral toothed line cylindrical gears. This program can automatically control the SolidWorks software, eliminating cumbersome modeling steps, accelerating the product development cycle, and improving the efficiency and accuracy of modeling, as well as coping with complex design requirements.
Taking the tooth groove lofted cut of the planar spiral toothed line cylindrical gear as the basic idea, the specific modeling steps are as follows.
First, I build the gear blank model: taking the center point of the gear blank as the coordinate system origin and the front reference plane as the reference plane, I draw a sketch and stretch the model. Second, I draw the involute tooth groove line: establishing a reference plane with the middle section of the gear blank, drawing and fitting the involute discrete points, that is, forming the involute tooth groove profile line. Third, I complete the involute tooth groove sketch: on the basis of step 2, I draw the addendum circle and the dedendum circle, supplement the transition curves of the tooth root and addendum, and form a complete involute tooth groove curve. Fourth, I draw the tooth groove sketches of each section: first, I establish 25 reference planes parallel to the middle reference plane along the tooth width direction. Second, according to the required tooth line equation, I calculate the position angle of the tooth groove sketch center on each reference plane relative to the coordinate origin. According to the obtained position angle, I convert it into the position offset of each reference plane sketch. Finally, I draw the involute tooth groove curve on each reference plane. Fifth, lofted cut: after the involute tooth groove curves on each reference plane are completed, I perform a lofted cut on them. Sixth, circular pattern and shaft hole drawing: after the modeling of a single tooth groove in step 5 is completed, I perform a circular pattern on this feature and draw the shaft hole, obtaining the three-dimensional accurate model of the planar spiral toothed line cylindrical gear.
When modeling a planar spiral toothed line cylindrical gear, it is first necessary to calculate the discrete points on the tooth line, then fit them into a tooth line, and then perform swept cut, circular pattern, etc. Therefore, the operation steps are very cumbersome. Therefore, in order to improve the modeling efficiency of planar spiral gears and simplify the modeling process, I independently develop a planar spiral toothed line cylindrical gear modeling system based on the Visual Basic platform.
This modeling system is mainly aimed at the drawing of planar spiral toothed line cylindrical gears. After inputting the main parameters of the planar spiral gear in the interface, the system automatically calculates important drawing data such as tooth profile parameters, and can automatically generate a three-dimensional model in SolidWorks.
In order to verify the accuracy of the system modeling, a pair of planar spiral gear pairs is generated, and the meshing interval of the gear pair is verified. The relevant parameters of the planar spiral gear are shown in the table.
Based on the planar spiral toothed line cylindrical gear system, the generated gear model is shown in the figure.
Using the planar spiral toothed line cylindrical gear modeling system, a pair of gear pairs is established. The relevant parameters of the gear pair are as follows: $z_1 = 30$, $z_2 = 60$, tooth width $B = 50$ mm, cutter head radius $R_T = 150$ mm. Using the Simulation module in SolidWorks, an interference instruction is added to the gear pair. The contact interval is shown in the figure. After the planar spiral toothed line cylindrical gear pair is subjected to force contact, the contact interval generated is in the middle 80% to 90% region of the tooth surface, which fully conforms to the results obtained from the previous theoretical analysis.
In this chapter, I determine in detail the model parameters of the planar spiral toothed line cylindrical gear. Through differential geometry principles and forming principle derivation, I obtain the tooth surface equation, dynamic meshing line, and tooth surface contact ratio of this gear. By combining SolidWorks with Visual Basic, I develop a planar spiral toothed line cylindrical gear modeling system. The emergence of this system will greatly improve the modeling efficiency and accuracy of planar spiral gears. Through the modeling system, a pair of planar spiral toothed line cylindrical gear pairs is generated. In the SolidWorks Simulation module, it is concluded that in theory, the meshing interval of the planar spiral toothed line cylindrical gear pair is distributed in the middle 80% to 90% region of the contact tooth surface of the gear pair.
Based on the ideal gear model obtained from the planar spiral toothed line cylindrical gear modeling system in the previous chapter, the meshing characteristics of the planar spiral gear are verified. However, gears often have uneven tooth surface force distribution caused by bias load due to their own manufacturing accuracy and position errors during gear installation, which affects the meshing performance of the gear pair and produces noise and vibration. Due to its special tooth line, the planar spiral gear can reduce the sensitivity to manufacturing and installation errors and has anti-bias load properties.
The main causes of common gear bias load can be summarized as gear manufacturing errors, gear installation errors, and use errors.
Manufacturing errors are mainly divided into parallelism errors of the gear shaft and tooth direction errors. Among them, the parallelism error of the gear shaft center line mentioned is the most important cause of gear bias load. The center line error is mainly composed of errors in the horizontal and vertical directions. The manifestation of these two errors is the change in the backlash value in the tooth width direction and the position change of the tooth surface contact stress concentration point. Tooth direction error refers to the distance between the minimum tooth line of the tooth and the effective tooth width section.
Gear installation error refers to the gap between the positioning surface of the gear and the positioning end of the gear shaft, which causes a fit error between the gear shaft hole and the gear shaft, thereby causing a coaxiality error between the gear pair pitch circle and the gear shaft. However, in some gear installations, bearings will be involved, such as ball bearings. However, this type of gear installation will increase the risk of installation errors, ultimately leading to bias load during the gear meshing process, thereby affecting the transmission accuracy of the gear and reducing the service life of the gear.
In theoretical analysis, both the gear support and the gear can be assumed to be absolutely rigid. Therefore, under the premise of ensuring the accuracy of the gear tooth profile, the load stress distribution on the gear tooth surface can be guaranteed to be uniform. However, in actual working conditions, due to the mechanical properties of the gear support system and the gear material itself, the tooth surface of the gear pair will produce a slight elastic deformation during the meshing process. Coupled with the two points of machining error and installation error mentioned above, the force on each point on the contact line of the gear pair is uneven, resulting in different elastic deformations.
When the gear pair produces bias load, the contact area of the two gears will also decrease, and the tooth surface contact stress will increase. This will cause excessive local stress at some positions on the gear, easily causing pitting corrosion at local positions on the gear tooth surface, damaging the internal material of the gear, affecting its force direction, and ultimately causing excessive local load, leading to tooth fracture and affecting the normal operation of the equipment.
The installation error of the gear pair can mainly be attributed to a certain degree of inclination of the two gear shafts in the horizontal and vertical directions, which causes bias load of the gear pair during the meshing process, resulting in local stress concentration on the tooth surface, destroying the internal structure of the material, and even causing tooth fracture in severe cases. In order to better highlight the anti-bias load characteristics of the planar spiral toothed line cylindrical gear, I compare and analyze the meshing state of straight teeth, helical teeth cylindrical gears, and planar spiral toothed line cylindrical gears under bias load transmission.
Taking the gear installation error in the vertical direction as an example, the tooth surface bias load of straight teeth and helical teeth is shown in the figure.
From the figure, the following conclusions can be drawn: when installation error occurs, first of all, both change from full tooth width contact to upper end load concentration and lower end non-contact. However, compared with straight teeth, helical teeth have a larger bias load interval, and the load on the upper end of the helical gear tooth is greater. The reason for this phenomenon is that the circumferential adjustment ability of straight teeth and helical teeth themselves is insufficient to change their contact state. If the gear pair is in this contact state for a long time, it will cause one end of the tooth to be affected by alternating stress for a long time and the tooth surface stress to be too large, while the other end is in a zero load state, which is easy to cause fracture at the tooth root of the upper end face of the gear tooth, reducing the service life of the gear.
When the planar spiral gear has an installation error in the vertical direction, due to the particularity of its own planar spiral line tooth line, compared with straight teeth and helical teeth cylindrical gears, this gear itself can have a greater degree of adjustment interval in the axial direction, and then through slight adjustment, its contact interval is located in the middle position of the tooth, avoiding bias load. Therefore, compared with straight teeth and helical teeth cylindrical gears, the planar spiral toothed line cylindrical gear uses the particularity of its own tooth line to make slight adjustments, avoiding the situation of excessive local stress on the tooth surface caused by bias load.
The elastic deformation generated by the planar spiral toothed line cylindrical gear pair during the meshing process is mainly caused by the bending deformation of the tooth under force and the influence of the gear material itself. The gear material itself mentioned refers to the contact deformation of the tooth surface under the influence of its material hardness during the meshing process.
Many scholars use the $30^\circ$ tangent method proposed by H. Hofer to determine the position of the dangerous section. As shown in the figure, the plane tangent to the tooth profile symmetry center line at $30^\circ$ is used as the dangerous section, and the tangent point is tangent to the fillet of the tooth root.
The bending stress of the planar spiral gear at the dangerous section is:
$$\sigma_{F0} = \frac{F_t h_{Fa} \cos\gamma}{b m \cos\alpha} = \frac{F_t}{b m} Y_{Fa}$$
In the equation, $Y_{Fa}$ is the tooth form factor, which mainly determines the tooth profile shape and is independent of the gear modulus $m$.
Considering the stress concentration caused by the transition curve at the dangerous section of the tooth root, the other stresses other than the bending stress, and the error caused by the load acting on the tooth addendum, I introduce the load coefficient $K_F$ and correct $\sigma_{F0}$, thereby obtaining the tooth root bending stress of the planar spiral gear as:
$$\sigma_F = K_F \sigma_{F0} = \frac{K F_t}{b m} Y_{Sa} Y_{Fa} = \frac{K F_t}{b m} Y_{Sa} Y_{Fa} Y_\varepsilon Y_\alpha$$
According to Hooke’s law $\sigma = E \cdot \varepsilon$ and $\varepsilon = \frac{\Delta l}{l}$, the strain of the tooth surface can be obtained as:
$$\varepsilon = \frac{\sigma_F}{E} = \frac{K F_t}{b E m} Y_{Sa} Y_{Fa} Y_\varepsilon Y_\alpha$$
And the bending deformation of the tooth is:
$$\delta_W = \frac{P}{E} \varepsilon = \frac{K F_t P}{b E m} Y_{Sa} Y_{Fa} Y_\varepsilon Y_\alpha$$
The most important of the tooth surface contact stresses mentioned here is the Hertz stress. According to the Hertz formula, the tooth surface contact stress of the planar spiral gear is:
$$\sigma_H = Z_H Z_E Z_\varepsilon \sqrt{\frac{F_t}{b R_T} \cdot \frac{1}{\cos 20^\circ \arcsin\left(\frac{R}{d}\right)}}$$
According to Hooke’s law $\sigma = E \cdot \varepsilon$ and $\varepsilon = \frac{\Delta l}{l}$, the strain of the tooth surface can be obtained as:
$$\varepsilon_H = \frac{\sigma_H}{E} = \frac{Z_H Z_E Z_\varepsilon}{b E} \sqrt{\frac{F_t}{b R_T} \cdot \frac{1}{\cos 20^\circ \arcsin\left(\frac{R}{d}\right)}}$$
And the bending deformation of the tooth is:
$$\delta_H = \frac{\sigma_H}{E} P_b = \frac{Z_H Z_E Z_\varepsilon P_b}{b E} \sqrt{\frac{F_t}{b R_T} \cdot \frac{1}{\cos 20^\circ \arcsin\left(\frac{R}{d}\right)}}$$
In the equation, $P_b$ is the base pitch of the spiral tooth line, and $E$ is the elastic modulus.
The comprehensive elastic deformation of the tooth is:
$$\delta = \delta_H + \delta_W = \frac{Z_H Z_E Z_\varepsilon P_b}{b E} \sqrt{\frac{F_t}{b R_T} \cdot \frac{1}{\cos 20^\circ \arcsin\left(\frac{R}{d}\right)}} + \frac{K F_t P}{b E m} Y_{Sa} Y_{Fa} Y_\varepsilon Y_\alpha$$
In SolidWorks, I establish mechanical models for the three gear pairs of straight teeth, helical teeth, and spiral teeth under the ideal meshing state, and then based on ANSYS Workbench software, I simulate the finite element analysis of the three gear pairs during the gear pair transmission process, and deeply analyze the load distribution uniformity of straight teeth, helical teeth, and spiral teeth during the meshing process.
The relevant parameters of the planar spiral toothed line cylindrical gear are as follows: gear modulus $m = 4$, the number of teeth of the two gears is $z_1 = 20$ mm and $z_2 = 40$ mm, the tooth line radius is $R_T = 200$ mm, and the tooth width is $B = 30$ mm.
I establish a three-dimensional model of the planar spiral toothed line cylindrical gear pair in SolidWorks and save it in .STEP format, then import it into ANSYS Workbench software, select the Static Structural module, and generate the planar spiral toothed line cylindrical gear pair model interface. The relevant parameter settings of the gear pair are shown in the table.
| Planar Spiral Gear Parameter Name | Data |
|---|---|
| Material | Alloy steel |
| Elastic modulus $E$ | $2.00 \times 10^5$ MPa |
| Poisson’s ratio $\nu$ | 0.3 |
| Density $\rho$ | 7850 kg/m³ |
After the relevant parameters are set, ANSYS Workbench accurately identifies the contact interval under the gear pair meshing state.
The mesh density of the Mech module in ANSYS directly affects the accuracy of the finite element analysis results, so mesh division becomes an important part of finite element analysis. The number of meshes is directly related to the calculation accuracy, but what follows is an increase in the calculation amount. The division method used in ANSYS Workbench is the Hex Dominant method. The key to this simulation analysis is the tooth part participating in the meshing, so the mesh of the gear pair participating in the meshing area is locally refined, which can effectively improve the accuracy of the contact force cloud diagram of the gear tooth surface. The mesh size is 0.2.
The driving wheel of the planar spiral gear pair rotates around its axis, and the gear shaft hole is used as the Supports of the gear pair. Cylindrical Support conditions are applied at the shaft hole position. Among them, Supports and Stress are necessary conditions for ANSYS solution, which are applied through the unit degrees of freedom. A Rotational Velocity condition is applied in the circumferential direction of the driving wheel with a value of 10 rad/s, and a torque load is added to the driven wheel with a value of 100 N·m, in the opposite direction to the driving wheel.
After completing the setting of relevant parameters such as loads and constraints, I add the Equivalent Stress module in Solution to simulate and calculate the tooth surface contact stress of the gear. According to the obtained equivalent stress cloud diagram, I can analyze the stress distribution of the overall tooth surface of the planar spiral gear pair during the meshing process without bias load.
From the analysis of the figure, it can be seen that under the no-bias load state, the maximum contact stress of the planar spiral toothed line cylindrical gear pair is 47.186 MPa. However, in fact, the gear pair meshing is always in a dynamic meshing state, so the contact line position of the gear pair shown changes with the change of the meshing interval. As the position of the contact line on the tooth surface changes continuously, the corresponding tooth surface contact stress will also change accordingly.
I establish a straight tooth and helical tooth cylindrical gear pair model in the ideal meshing state in SolidWorks, and then according to the operation steps mentioned in the previous section, I add constraint conditions, apply torque, and perform finite element analysis to obtain the stress cloud diagram.
From the above stress cloud diagram analysis, it can be seen that under the same conditions as spiral teeth, the maximum tooth surface contact stress values of straight teeth and helical teeth cylindrical gears are 110.1 MPa and 72.351 MPa, respectively. Under the same meshing conditions, material, and applicable working conditions, the tooth surface contact stress of the planar spiral toothed line cylindrical gear pair is smaller than that of straight teeth and helical teeth cylindrical gear pairs. The contact ratio of the gear pair is directly related to its contact interval. The larger the contact interval, the higher the contact ratio. From the figure, it can be seen that the contact ratio of the planar spiral toothed line cylindrical gear pair is much larger than that of straight teeth and smaller than that of helical teeth cylindrical gear pairs.
In the actual production process, the reasons for the parallelism error between the gear installation axes are mainly summarized as the following two points: first, the parallelism error generated by the gear shaft installation during the installation process; second, the gear will generate huge instantaneous contact stress during the actual meshing process, so the tooth surface will produce elastic deformation under the influence of the material after being subjected to force, which will cause the gear installation axis to tilt and thus generate a parallelism error. Generally speaking, there are two parallelism errors in the direction of the gear installation axis, namely horizontal and vertical directions, but in the actual meshing process, it is a composite situation of horizontal and vertical non-parallelism. Now assume that one is a horizontal offset of 0.1 degrees and the other is a vertical offset of 0.1 degrees.
Based on ANSYS software, I perform finite element analysis on the meshing of planar spiral toothed line cylindrical gear pairs, helical tooth cylindrical gear pairs, and straight tooth cylindrical gear pairs. I select gears with a transmission ratio of 2:1, numbers of teeth of 20:40, 30:60, and 40:80, the gear material is set to Alloy Steel, and the modulus is 3, 4, 5, and 6. After adding constraint conditions to the driving wheel, the other conditions remain consistent, and I analyze the change of tooth surface contact stress under different modulus.
Taking the existence of a vertical direction error as an example, in the case of 20:40, under different modulus conditions, the finite element analysis results of the maximum contact stress on the tooth surface of straight teeth, helical teeth, and spiral teeth are shown in the figure.
Taking the existence of a horizontal direction error as an example, in the case of 20:40, under different modulus conditions, the finite element analysis results of the maximum contact stress on the tooth surface of straight teeth, helical teeth, and spiral teeth are shown in the figure.
The results of the maximum tooth root stress obtained by vertical offset of 0.1 and horizontal offset of 0.1 degrees are shown in the table. The data are drawn into a curve graph.
| Gear Modulus | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|
| Straight tooth cylindrical gear | Measured maximum stress / MPa (vertical offset 0.1°) | 220.79 | 168.57 | 126.95 | 105.5 |
| Measured maximum stress / MPa (horizontal offset 0.1°) | 152.85 | 148.14 | 134.35 | 102.26 | |
| Helical tooth cylindrical gear | Measured maximum stress / MPa (vertical offset 0.1°) | 420.77 | 339.9 | 260.97 | 214.63 |
| Measured maximum stress / MPa (horizontal offset 0.1°) | 306.01 | 294.58 | 272.83 | 200.64 | |
| Planar spiral toothed line cylindrical gear | Measured maximum stress / MPa (vertical offset 0.1°) | 147.85 | 112.88 | 98.856 | 71.361 |
| Measured maximum stress / MPa (horizontal offset 0.1°) | 135.27 | 118.43 | 88.45 | 66.603 |
Using ANSYS Workbench, I perform finite element analysis on the meshing of straight teeth, helical teeth, and planar spiral toothed line cylindrical gear pairs. A pair of gear pairs is meshed, the gear material is also alloy steel, constraints are applied, the external variables are controlled to be the same, the contact surface of the gear meshing surface is selected for analysis, the friction coefficient is 0.1, and the mesh density of the gear meshing contact surface part is adjusted to good, and the mesh density of other parts is defaulted to the middle value for mesh division, and finite element analysis is performed.
When the modulus of both gears is 3 and the horizontal offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 3 and the vertical offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 4 and the horizontal offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 4 and the vertical offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 5 and the horizontal offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 5 and the vertical offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 6 and the horizontal offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
When the modulus of both gears is 6 and the vertical offset is 0.1°, static analysis is performed, and the results obtained are shown in the figure.
All the experimental results obtained are summarized in the table. The experimental method of vertical offset of 0.1 degrees is the same as above, and the results are summarized in the table.
| Modulus | Gear Type | Maximum Tooth Surface Contact Stress / MPa (20:40) | Maximum Tooth Surface Contact Stress / MPa (30:60) | Maximum Tooth Surface Contact Stress / MPa (40:80) |
|---|---|---|---|---|
| 3 | Straight tooth | 152.85 | 142.04 | 127.28 |
| Helical tooth | 306.01 | 289.99 | 256.33 | |
| Spiral tooth | 135.27 | 113.51 | 102.89 | |
| 4 | Straight tooth | 148.14 | 117.17 | 73.885 |
| Helical tooth | 294.58 | 233.18 | 143.87 | |
| Spiral tooth | 118.43 | 89.03 | 52.804 | |
| 5 | Straight tooth | 134.35 | 75.46 | 51.246 |
| Helical tooth | 272.83 | 153.18 | 111.18 | |
| Spiral tooth | 88.45 | 56.159 | 33.789 | |
| 6 | Straight tooth | 102.26 | 47.624 | 21.02 |
| Helical tooth | 200.64 | 98.778 | 64.461 | |
| Spiral tooth | 66.603 | 36.463 | 21.405 |
| Modulus | Gear Type | Maximum Tooth Surface Contact Stress / MPa (20:40) | Maximum Tooth Surface Contact Stress / MPa (30:60) | Maximum Tooth Surface Contact Stress / MPa (40:80) |
|---|---|---|---|---|
| 3 | Straight tooth | 220.79 | 202.48 | 159.1 |
| Helical tooth | 420.77 | 389.01 | 330.53 | |
| Spiral tooth | 147.85 | 138.74 | 127.39 | |
| 4 | Straight tooth | 168.57 | 149.39 | 125.6 |
| Helical tooth | 339.9 | 306.04 | 278.46 | |
| Spiral tooth | 112.88 | 108.81 | 90.94 | |
| 5 | Straight tooth | 126.95 | 88.037 | 70.956 |
| Helical tooth | 260.97 | 185.43 | 148.24 | |
| Spiral tooth | 98.856 | 44.729 | 25.417 | |
| 6 | Straight tooth | 105.5 | 62.506 | 35.033 |
| Helical tooth | 214.63 | 122.81 | 76.547 | |
| Spiral tooth | 71.361 | 52.091 | 29.967 |
Taking the gear pair with modulus 3 as an example, I draw the experimental data curve graphs of horizontal offset 0.1 degrees and vertical offset 0.1 degrees, respectively.
I perform static analysis on the straight tooth, helical tooth, and planar spiral gear pairs under the conditions of horizontal offset of 0.1 degrees and vertical offset of 0.1 degrees of the installation axis for the above six situations:
1. The maximum tooth surface contact stress of straight teeth, helical teeth, and planar spiral gears all decreases with the increase of modulus. The maximum tooth surface contact stress of planar spiral gears is 20% to 30% smaller than that of straight teeth and 40% to 50% smaller than that of helical teeth. The planar spiral toothed line cylindrical gear has better anti-bias load capacity.
2. Under the condition of a certain modulus, the maximum tooth surface contact stress of planar spiral gears, straight teeth, and helical teeth all decreases with the increase of the number of teeth. The maximum tooth surface contact stress of planar spiral gears is about 30% smaller than that of straight teeth and about 50% smaller than that of helical teeth. Experiments prove that planar spiral gears have better anti-bias load capacity.
3. Compared with straight teeth and helical teeth cylindrical gears, the planar spiral toothed line cylindrical gear has more uniform tooth surface contact stress distribution and longer service life when the modulus, number of teeth, material, and force conditions are the same.
By studying the gear pairs of straight teeth, helical teeth, and planar spiral toothed line cylindrical gears under the ideal meshing state, according to the experimental results of straight teeth, helical teeth, and spiral teeth, the contact stress of the planar spiral gear is the smallest, followed by the helical gear, and finally the straight gear. The tooth surface stress distribution of the planar spiral toothed line cylindrical gear is more uniform. In addition, when the installation axis has a parallelism error, the tooth surface load of the planar spiral toothed line cylindrical gear is much smaller than that of straight teeth and helical teeth, indicating that the planar spiral toothed line cylindrical gear has excellent anti-bias load characteristics compared with straight teeth and helical teeth. At the same time, I also summarize the variation trend of the maximum contact stress on the tooth surface under different numbers of teeth and modulus.
Through a series of gear meshing mechanisms, data model calculations, and anti-bias load analysis, it can be seen that the planar spiral gear has the advantages of stable transmission, strong tooth surface load-bearing capacity, and anti-bias load characteristics. However, due to the particularity of its tooth line, there is no suitable machine tool for gear machining in China, and the main gear machining method for arc gears abroad, rotary cutter head gear machining, cannot be applied to the mass manufacturing and gear machining of arc toothed line cylindrical gears due to its slow milling efficiency. Therefore, I propose a new and efficient gear machining method: the full-tooth-width high-linear-speed continuous generating gear machining method.
According to the meshing relationship between the planar spiral gear and the planar spiral line rack, the gear machining method of the planar spiral toothed line cylindrical gear can be analyzed. The planar spiral gear and the rack can mesh smoothly, so the gear is compared to a gear blank, and the rack is compared to a tool. The meshing surface of the two is a conjugate surface. The meshing transmission of this model is equal to the generating motion gear machining in actual gear machining.
Taking the machining rack as an example, according to the gear machining method of arc toothed cylindrical gears: a blade is installed on a large cutter head, and the distance from the center line of the blade to the center of the cutter head is $R_T$. When the cutter head rotates, the tool trajectory is a circle. If the rack is stationary in the radial direction of the cutter head and the rack continuously feeds along the axial direction of the cutter head, the machined tooth groove is arc-shaped. This gear machining method requires retracting the tool after gear machining each tooth groove, then indexing the rack along the radial direction of the cutter head, and then gear machining the next tooth groove, which greatly limits the gear machining efficiency.
Now I optimize the above gear machining method. While the cutter head rotates, the rack moves linearly along the radial direction of the cutter head at a certain speed $v$. The planar spiral tooth line is the trajectory formed by the tool on the rack. The movement of the tool relative to the rack is circular motion while doing linear motion along the radial direction, so the trajectory formed by the tool on the rack is a planar spiral line with a certain pitch. When the speed relationship satisfies that the rack moves one tooth pitch per revolution of the cutter head, that is:
$$v = \frac{\pi m}{2\pi} \omega_D$$
In the equation, $\omega_D$ is the cutter head rotation speed, and $m$ is the modulus. When the blade rotates one revolution and cuts into the tooth width range again, the next tooth groove is machined, thereby realizing full-tooth-width continuous gear machining, which can effectively improve gear machining efficiency. At the same time, the cutter head can always maintain high-speed rotation during gear machining, ensuring tooth surface quality.
In the figure, $B$ is the tooth width, $a$ is the distance from the center of the tool to the right edge of the rack when the tool cuts into the rack, and $b$ is the distance from the center of the tool to the right edge of the rack when the tool cuts to the middle line of the tooth width of the rack. The trajectory formed by the tool on the rack is a planar spiral line, which is a curve that changes uniformly along the translation direction of the rack (negative direction of the $X_D$ axis) on the basis of the circular trajectory of the tool. The planar spiral line trajectory equation is:
$$x_D = R_T \cos(\theta) + \frac{m}{2}(2\pi – \theta)$$
$$z_D = R_T \sin(\theta)$$
Then the forming line equation is:
$$x_D = R_T \cos(\theta) – \frac{m}{2}\theta$$
$$z_D = R_T \sin(\theta)$$
When using the above gear machining method to machine cylindrical gears, there are the following differences from rack gear machining:
1) Gear machining is divided into depth machining and generating gear machining. During depth cutting, the speed $v$ of the rack movement is converted into the angular speed $\omega_C$ of the cylindrical workpiece. At the same time, because the tooth groove cannot be machined in one pass, the gear blank needs to perform a feed motion along the axial direction of the cutter head while rotating. After the cylindrical gear blank rotates one revolution, the already machined tooth groove is machined again to achieve depth machining.
2) During generating gear machining, on the basis of the original motion relationship, an additional rotation speed and a linear motion along the radial direction of the cutter head need to be added to the gear blank, that is, the generating motion, which will be specifically explained in the following section.
3) In order to achieve continuous gear machining, when the tool cuts out from the gear blank and rotates one revolution to cut into the tooth groove again, it needs to cut the next tooth groove. Then the speed relationship is required to satisfy that the cutter head rotates one revolution and the gear blank rotates one tooth pitch, that is, $\frac{\omega_C}{\omega_D} = \frac{1}{z}$, where $\omega_C$ is the gear blank rotation speed and $z$ is the number of teeth of the gear blank. At this time, when the tool rotates one revolution and cuts into the upper end of the gear blank again, the blade performs depth machining of the next tooth groove, which can achieve continuous gear machining. When the tool installation radius $R_T$ is large, the cutter head rotates continuously at high speed, improving the tool cutting linear speed, which can effectively improve the tooth surface gear machining quality.
The currently widely used hobbing and gear shaping methods for machining cylindrical gear teeth cannot achieve full-tooth-width continuous gear machining of gears. In order to achieve full-tooth-width meshing of cylindrical gears, the gear machining uses a large-diameter cutter head. A machining tool bar is arranged on the cutter head, and the distance from the tool bar to the center of the cutter head is $R_T$. The cutter head rotates around the center axis of the cutter head at an angular speed $\omega_D$, and the gear blank rotates around the center axis at an angular speed $\omega_C$. The cutter head and the gear blank are linked in two axes, and the numerical relationship of the two-axis linkage is:
$$\frac{\omega_C}{\omega_D} = \frac{1}{Z}$$
That is, the cutter head rotates one revolution, and the gear blank rotates one tooth pitch. At this time, the tool increases the linear motion along the radial direction of the circular motion relative to the gear tooth on the basis of the circular motion. After the two motions are combined, a planar spiral tooth groove can be machined on the gear blank. Combined with the generating motion, the planar spiral toothed line cylindrical gear can be quickly machined. The polar coordinate equation of the circular motion is:
$$r_1 = R_T(\theta)$$
The polar coordinate equation of the linear motion is:
$$r_2 = \frac{\pi m}{2\pi}\theta$$
Then the polar coordinate equation of the planar spiral line is:
$$r = R_T \pm \frac{\pi m}{2\pi}\theta$$
In the equation, $\theta$ is the cutter head rotation angle and is the independent variable. Each time the gear machining cutter head rotates one revolution, the tool cuts out the next tooth groove in sequence, which can achieve continuous gear cutting. When the tool installation radius $R_T$ is large, the cutter head rotates continuously, and the tool cutting linear speed is large, which can effectively improve the tooth surface gear machining quality.
The cutting radius of the cutting tool used is the distance between the tool center and the cutter head center $O_D$. The cutting radius of the tool cannot be too small. Because the cutter head is fixed relative to the tool during the rapid gear machining process, it is not in a translational state, so the tool development angle will be generated during the cutting process. Then in the actual gear machining process, if the tool cutting radius is too small, the tool rake face development angle is too large, which has little effect on the normal tooth width of the gear and the tooth groove profile at the middle section of the gear, but the circumferential tooth groove at both ends will become wider, eventually causing the tooth to appear “thick in the middle and narrow at both ends”. This tooth structure has low tooth surface strength and poor transmission stability. Therefore, it is necessary to expand the cutting radius of the tool, which can be understood as the cutter head radius, to offset the development angle problem generated during the gear blank gear machining process.
If only one tool is installed, due to its large cutting amount during the entire gear machining process, problems such as thermal deformation and damage to the tool are caused, and it is not easy to ensure the gear machining efficiency and gear machining accuracy of the gear. Therefore, on the basis of single-tool gear machining, multiple machining tools are arranged on the cutter head, and the arrangement trajectory is a planar spiral line. Multi-tool gear machining can effectively reduce the cutting amount of each tool. The tool tip center of the machining tool and the cutter head center are strictly controlled at the radius of the installation point on the planar spiral line. The added tools are installed at a distance in sequence. In the longitudinal height, they are installed in a form of increasing layer by layer. The tool with the smallest installation radius is the main cutting tool, that is, the tooth groove forming tool, and the remaining tools are auxiliary milling tools, which are used for pre-milling the workpiece tooth groove, thereby reducing the milling amount and cutting force of the main milling tool, extending the service life of the tool, and ensuring the gear machining accuracy of the gear tooth surface.
Multiple tools are arranged on the large cutter head according to the planar spiral line, ensuring that the planar spiral line pitch is $P = \pi m$. Except for the tool with the smallest installation radius (installation radius $R_T$), which is the main cutting tool, the installation height of the remaining tools is lower than that of the main cutting tool. They are auxiliary cutting tools, used for pre-cutting the workpiece tooth groove, reducing the cutting amount and cutting force of the main cutting tool, and not affecting the tooth groove forming.
The polar coordinate equation of the tool installation planar spiral line is:
$$x_{dk} = R_T + \frac{m}{4}\beta$$
$$R_{ib} = R_T – \frac{m}{4}\beta$$
In the equation, $R_T$ is the initial radius of the cutter head spiral line, and $\beta$ is the planar spiral line development angle. When the cutter head rotates 90°, $\beta = \frac{\pi}{2}$. For the second tool coordinate in the second equation of formula (4-8), it is $\left(R_T + \frac{\pi m}{4}, 0\right)$. For formula (4-2), at this time $\theta = \frac{\pi}{2}$, $\theta_k = 0$, $k = 0$, that is, $x_D = R_T + \frac{\pi m}{4}$, $z_D = 0$. The tool position is the same as the tooth groove position, and tooth groove cutting can be achieved. The remaining tools are arranged according to the same principle.
The tools are arranged on the cutter head according to the formula. $R_T$ is the installation radius of the main tool, $\theta$ is the planar spiral development angle, and $\theta$ is the independent variable. When $\theta = 2\pi$, $R_T = R – m\pi$. It can be seen that the planar spiral pitch is equal to the gear blank tooth pitch.
Since the planar spiral development angle at the $K$-th tool bar is:
$$\theta_K = \frac{2\pi K}{n}$$
Then the $K$-th tool is installed at:
$$x_{dk} = R_T – \frac{K\pi m}{2n}\cos\left(\frac{K\pi}{n}\right)$$
$$z_{ib} = R_T \sin\left(\frac{K\pi}{n}\right)$$
In the equation, $K$ is 0 to $n$, $m$ is the gear modulus, and $n$ is the number of tools.
The main and auxiliary cutting tools have the same shape. The cutting blade is composed of an outer blade and an inner blade. The outer blade generates the concave tooth surface, and the inner blade generates the convex tooth surface. The curvature radius of the inner blade is smaller than that of the outer blade. The pitch curvature radii of the outer blade and the inner blade are $R_{ob}$ and $R_{ib}$, respectively.
$$R_{ob} = R_T + \frac{\pi m}{4}$$
$$R_{ib} = R_T – \frac{\pi m}{4}$$
In the equation, $R_T$ is the installation radius of the main cutting tool, and $m$ is the modulus.
In order to further reduce the cutting force of the tool during the milling gear machining, the height $h$ of the tool bar in the axial direction of the cutter head is changed. Assuming that the height of the first tool bar is $h_0$, the relative axial height of the next machining tool bar and the previous machining tool bar is $h_{n-1} = h_n + 0.1$. During cutting, the tool 1 with the lowest height is allowed to mill first. On the basis of the tooth groove trace already milled by tool 1, tool 2 performs further depth cutting. This arrangement can reduce the cutting force of a single blade, improve the cutting smoothness, and improve the gear surface accuracy.
This paper adopts the full-tooth-width continuous high-speed gear cutting technology. The tool bar is in a planar spiral line on the large cutter head, and the spiral line pitch is equal to the gear tooth pitch. The cutter head rotates one revolution, and the workpiece rotates one tooth in equal proportion. Depth cutting and tooth surface generating gear machining are carried out continuously. For the first time, full-tooth-width continuous rapid gear machining of gears is achieved. In terms of gear machining efficiency, the gear machining efficiency of gears of the same size is 4 to 5 times that of hobbing gear machining and 5 to 6 times that of gear shaping gear machining, greatly reducing gear manufacturing time and reducing gear production costs.
Based on the full-tooth-width high-linear-speed continuous generating gear machining method proposed above, I build an initial model of the gear machining device in SolidWorks. This model is mainly composed of the device main body, gear machining cutter head, CNC rotary table, and other related components. The device main body mainly includes a base, X drive shaft, and Y drive shaft. Among them, the X drive shaft is mainly responsible for the left and right movement interval of the gear blank in the X-axis direction during the generating gear machining of the gear. The Y drive shaft is responsible for the feed of the gear blank in the Y-axis direction during the gear tooth groove depth cutting process until the tooth groove cutting is completed. The radial adjustment device is mainly responsible for adjusting the radial distance between the gear machining tool and the center of the gear machining cutter head, ensuring that the layout trajectory of the gear machining tool on the cutter head surface is a planar spiral line, and providing fixed support for the tool in the radial direction of the cutter head, ensuring the stability of the tool during gear machining, thereby improving the gear machining accuracy of the gear tooth surface.
The planar spiral gear is machined by a rotary, multi-blade, large-diameter cutter head. I design a full-tooth-width high-linear-speed continuous generating gear machining method. The high-speed rotating tool plane is tangent to the cylindrical surface of the gear blank. The cutter head rotates around the axis $Y$ at an angular speed $\omega_D$ as the main motion of high-speed rotation. The gear blank rotates around the axis $Z$ at an angular speed $\omega_C$. In order to ensure that the tool accurately machines the tooth groove part, it is necessary to ensure that the angular speed $\omega_C$ of the gear blank and the angular speed $\omega_D$ of the cutter head satisfy: $\frac{\omega_C}{\omega_D} = \frac{1}{Z}$. In order to complete the depth machining of the tooth groove, the gear blank continuously feeds along the $Y$ direction at a speed $V_{yc}$ while rotating at the angular speed $\omega_D$, realizing full-tooth-width continuous cutting of the gear.
This gear machining method adopts full-tooth-width, layer-by-layer progressive continuous cutting. When the tooth groove depth is cut, after the tooth surface of the first layer is completely machined, the gear blank completes the feed to the second layer in the Y-axis, and the gear blank performs the circumferential indexing gear machining of the second layer, and so on, layer-by-layer feed and layer-by-layer gear machining until all tooth surfaces are machined.
After the gear tooth groove cutting is completed, generating milling is performed between the tool and the gear blank to machine the involute tooth profile. The specific gear machining steps are as follows: on the basis of the original motion of the gear machining cutter head, the feed motion $V_{yc}$ is zero, the gear blank adjusts the angular speed $(\omega_C – \omega_s)$ to rotate clockwise around $Z$, and additionally moves linearly along the positive direction of the $X$ axis at a speed $V_{xc}$, ensuring $V_{xc} = \omega_s r$, where $\omega_s$ is the additional rotation speed of the gear blank, and $r$ is the pitch circle radius of the gear blank. After the above gear machining steps are completed, the generating gear machining of the right tooth surface is completed.
The gear machining principle of the left tooth surface of the planar spiral toothed line cylindrical gear is similar to that of the right tooth surface. Except that the gear blank rotates clockwise around the axis $Z$ at an angular speed $(\omega_C + \omega_s)$ and moves linearly along the negative direction of the $X$ axis at a speed $V_{xc}$, the movement directions of the other components remain unchanged.
In order to verify the feasibility of the proposed full-tooth-width high-linear-speed continuous generating gear machining method, I build a device model based on Vericut software for gear simulation gear machining to verify the feasibility of this gear machining method. Based on the gear machining device structure proposed in the previous section, I build it in SolidWorks, save it as a “.mcn” file, and import it into Vericut software. In the software Vericut, I convert the control system to Maz640mtpro, insert the CNC program, and the program interface instruction is “CGTECH_MACRO “GearModeOnOff””” 1″. In the process of setting the tool file, I select the option “milling cutter” and import the model file into the tool. It is worth noting that in the process of inserting the model file, the unit should be selected as inches. After the settings are completed, the simulation gear machining of the planar spiral toothed line cylindrical gear model is carried out. The specific gear machining steps are shown in the figure. The machined gear model is compared and analyzed with the theoretical model generated in the independently developed planar spiral toothed line cylindrical gear modeling system, which confirms the feasibility of this gear machining method.
The coordinates after the simulation model is imported are determined by the WCS coordinate system in SolidWorks, so it needs to be determined in SolidWorks in advance. This paper uses SolidWorks to build the models of each component of the gear machining device and saves them as STEP.A203 files. Since Vericut cannot recognize STEP files, they are saved as .STL type files and then imported into Vericut 9.0.
At present, there is no ready-made CNC device on the market that can efficiently machine planar spiral toothed line cylindrical gears. Therefore, based on the full-tooth-width high-linear-speed continuous generating gear machining method proposed in the above sections, this paper builds a simulation gear machining device that can be used for milling planar spiral toothed line cylindrical gears. It includes two linear motion axes, including the workpiece feed Y axis and the workpiece generating X axis, and two rotation axes, including the workpiece rotation axis B axis and the cutter head rotation axis C axis. The X and Y axis guide rails can move along the X and Y directions. The device controls the gear blank to reach the point to be machined through the X and Y axes. The cutter head rotates around the C axis of the gear machining device, and the gear blank rotates around the B axis of the gear machining device. The numerical proportional relationship between the angular speed $\omega_C$ of the gear blank and the angular speed $\omega_D$ of the cutter head must satisfy a certain numerical relationship. The cutter head and the gear blank to be machined perform gear machining rotation according to the above numerical relationship. The gear machining tools are arranged on the cutter head surface in a planar spiral line. At the same time, the gear blank performs continuous feed motion along the Y axis of the gear machining device to complete depth machining. During generating gear machining, a new rotation speed and a linear motion along the radial direction of the cutter head are added to the workpiece.
According to the parameters shown in the table, I use SolidWorks to establish the gear blank and gear machining cutter head models. I import the cutter head and tools into the Vericut tool management library and determine the cutter head coordinates and tool setting point positions. The tools are arranged on the cutter head surface in a planar spiral line. The gear blank parameters are shown in the table.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Number of teeth | 30 | Modulus | 5 |
| Pressure angle $\alpha$ / (°) | 20 | Tooth width $b$ / mm | 40 |
| Addendum coefficient $h_a^*$ | 1 | Shaft hole diameter $d$ / mm | 40 |
| Minimum blade installation radius $R_T$ / mm | 200 | Number of tools | 8 |
After all milling parameters are set, the program file .txt is imported into the CNC program for simulation gear machining. During the milling process, the speed ratio of the cutter head and the gear blank is kept constant, that is, the simulation gear machining of the planar spiral toothed line cylindrical gear is completed.
The numerical proportional relationship between the angular speed $\omega_C$ of the worktable where the cutter head gear blank is located and the angular speed $\omega_D$ of the cutter head. The numerical proportional relationship between $\omega_C$ and $\omega_D$ depends on the design parameter number of teeth $Z$ of the gear to be machined. Assuming that the design parameter number of teeth of the machined gear is $Z = 30$, then the angular speed proportional relationship should satisfy $\frac{\omega_C}{\omega_D} = \frac{1}{30}$, that is, the gear machining cutter head rotates one revolution, and the machined gear rotates one tooth for continuous cutting. There is no interference or collision during the simulation process until the planar spiral gear milling is completed. Finally, the planar spiral toothed line cylindrical gear model is obtained. Each section of the gear obtained by simulation gear machining has a clear tooth profile, and the middle tooth profile section is a standard involute.
In order to verify whether the tooth profile of the gear obtained by Vericut simulation gear machining is an involute tooth profile, the simulation model is imported into SolidWorks, and a standard involute curve is generated based on the equation-driven generation in SolidWorks, and the two are compared. The end face tooth profile of the machined planar spiral gear coincides with the generated standard involute curve, so it can be proved that the end face tooth profile of the planar spiral gear obtained by simulation is an involute tooth profile.
In order to verify whether the contact line of the planar spiral toothed line cylindrical gear pair model obtained by simulation is consistent with the contact line of the theoretical model, two gear models with specified parameters are machined based on Vericut simulation software for meshing experiments. The correct meshing conditions of the gear pair are added in SolidWorks. The basic parameters of the gear pair are shown in the table:
| Gear I | Gear II | |
|---|---|---|
| Modulus $m$ | 4 | 4 |
| Number of teeth $z$ | 66 | 30 |
| Tooth thickness $B$ | 50 | 50 |
| Cutter head radius $R_T$ | 300 | 300 |
The planar spiral toothed line cylindrical gear pair obtained by simulation is assembled in SolidWorks. After adding constraint conditions, the contact surface of the gear pair is analyzed and calculated. When the gear pair is not under force, the theoretical contact line is a point contact distributed in the middle of the tooth. Under the force state, considering the elastic deformation of the tooth surface, the contact interval changes from point contact to line contact, mainly distributed in the middle 80% to 90% region of the tooth surface, and the force on both ends of the tooth is small. This effectively reduces the stress concentration at the tooth root at both ends of the tooth.
The establishment of gear tooth surface simulation can provide a basis for the digital modeling of planar spiral toothed line cylindrical gears and can guide the numerical simulation analysis of this gear. In order to further verify the contact interval of gear transmission, based on the static tooth surface equation in the previous section, I establish a pair of left-hand and right-hand planar spiral toothed line cylindrical gears with the above parameters in MATLAB and mesh them. In the figure, the blue tooth surface is the concave tooth surface, and the red tooth surface is the convex tooth surface. It can be seen that the theoretical contact interval of gear meshing is mainly distributed in the middle region of the tooth surface, which conforms to the conclusion that the tooth is slightly drum-shaped in the gear machining principle, with contact mainly at the middle line of the tooth width, and the force on both ends of the tooth is small, which can effectively reduce the stress concentration at the tooth root at both ends of the tooth, optimize the contact state at both ends of the tooth profile, and prolong the service life of the gear. This further verifies the accuracy of the static tooth surface equation and simulation results.
In this chapter, by analyzing the rack gear machining situation, I transform the planar spiral line gear gear machining into planar spiral line rack gear machining, summarize the gear machining method of the planar spiral toothed line cylindrical gear, and propose a full-tooth-width high-linear-speed continuous generating gear machining method. According to its gear machining principle and gear machining method, I establish a related device model in SolidWorks and write related programs based on Vericut for simulation gear machining to verify the feasibility of the gear machining method. I calculate the contact interval of the gear pair obtained by simulation to verify the accuracy of the tooth surface contact interval of the planar spiral gear.
The planar spiral toothed line cylindrical gear obtained by the full-tooth-width high-linear-speed continuous generating gear machining method uses a four-axis linkage and dual-rotation-axis control system. However, at this stage, there is no existing device on the market that can match the milling gear machining of planar spiral toothed line cylindrical gears. Therefore, based on the existing CNC device, I independently transform a CNC milling gear machining device for planar spiral toothed line cylindrical gears. According to the gear machining method mentioned in the above chapters, this chapter proposes a detailed design scheme for the planar spiral toothed line cylindrical gear gear machining device, successfully develops a dedicated gear machining device, and completes the trial cutting gear machining experiment of the planar spiral gear.
According to the gear machining principle and gear machining simulation of the planar spiral toothed line cylindrical gear, at least a four-axis linkage device is required to realize the gear machining of the planar spiral gear (two linear motion X axes and Z axes; two cutting spindle rotation motions). Since there is currently no relatively effective gear machining device on the market that matches its gear machining method, the team transforms the existing CNC lathe. The overall structure of the developed planar spiral toothed line cylindrical gear gear machining device is shown in the figure.
It mainly includes the following parts:
(1) Bed part (X-axis, Z-axis moving axes)
(2) Large-diameter cutter head, tools, and radial adjustment part
(3) Mobile and rotary CNC table
(4) CNC system
(5) Electrical control system
The main design points of the gear machining device are described in detail below.
1. Bed part of the gear machining device
The bed of the planar spiral toothed line cylindrical gear gear machining device is mainly composed of three main parts: the device spindle, the X-axis feed axis, and the Z-axis feed axis.
The first thing to mention is the device spindle. In the gear machining method mentioned above for the planar spiral gear, the rotation axis of the gear machining cutter head needs to be controlled to rotate, and it rotates in a linked manner with the CNC rotary milling table in a specified ratio. However, since the original spindle motor of the CNC device is a DC motor, it cannot accurately perform controllable rotation, so the device spindle driver needs to be modified. At present, the spindle power drive uses a Servomotor 180SG-M35015 servo motor. The rated power is 5.5 KW, the rated torque is 35 N·m, the rated speed is 1500 rpm, and the rated voltage is 180V. The transmission relationship between the servo motor and the device spindle is belt transmission, and the transmission ratio is 1:4. The belt transmission system usually has high transmission efficiency, can effectively transmit input power to the output end, reduce energy loss, and can provide relatively stable transmission, reduce vibration and noise, and help improve the stability and working efficiency of the mechanical system. The spindle of the planar spiral toothed line cylindrical gear gear machining device uses spindle encoder feedback, which requires highly precise position and speed control. It needs excellent dynamic performance and can quickly respond to changes in control signals. Due to the small internal space of the device power head, it is necessary to provide large torque under space-constrained conditions to drive the cutter head to rotate.
The second part is the two moving axes on the bed body. The high positioning accuracy and high transmission stability of the transmission system are the prerequisites for ensuring gear machining accuracy. In order to ensure the positioning accuracy and gear machining accuracy of the device, a rotary high-precision ball screw transmission group is used in the device spindle guidance system to ensure the accuracy of the gear blank during feeding. In order to ensure the positioning accuracy of the ball screw, the selected servo motor model is the Yaskawa series SGMJV-08ADE6S. This type of servo motor has the characteristics of fast dynamic response and high precision.
1. Large-diameter gear machining cutter head, gear machining tools, and radial adjustment part
According to the principle of the full-tooth-width high-linear-speed continuous generating gear machining method, the cutter head is provided with a total of n front-and-back through tool grooves for installing tools. Six bolt ports are added on the left and right sides of the through tool groove. The matching gasket is mainly used for fixing the bolt rod to prevent shaking during high-speed rotation gear machining. The circumferential fixing bolt is mainly used to resist the gasket to avoid shaking of the gear machining tool during the gear machining process, affecting gear machining accuracy. The radial positioning bolt is mainly responsible for adjusting the distance from the gear machining tool to the center of the cutter head, so that the 6 tools are distributed in a planar spiral line.
The cutter head itself is made of alloy steel. Alloy steel usually has high hardness and strength, which makes it perform better under high load, high temperature, or other harsh conditions. It also has high temperature resistance, wear resistance, and improved mechanical properties of the material itself. The blade material is cemented carbide, model W12Cr4V3Mo3Co5Si. Cemented carbide is a high-hardness alloy material, mainly made of hard compounds of refractory metals and bonding metals through powder metallurgy. This alloy has high hardness, good wear resistance, high strength, and good heat resistance. The hardness of tools made of cemented carbide reaches 69 to 81 HRC, and can maintain high hardness even at 1000°C. The gear machining tool bar is made of cemented carbide, which does not rust, is corrosion-resistant, has good hardness, and high strength, and can withstand strong shear force during the cutting process.
2. Digital rotary table
The CNC rotary table uses a servo motor as the drive and controls the angular speed through pulses to ensure high precision and fast response. The worktable is installed on the linear motion guide rail so that it can move flexibly in the axial direction. The drive source uses an AC SERVO MOTOR series servo motor, model SGMGV-09ADC61. The rated power is 0.85 KW, the rated voltage is 200V, and the rated torque is 5.39 N·m. This series of servo motors has no rotation error, no transmission gap, fast response speed, large transmission torque, and strong anti-interference performance. The servo driver SERVOPACK matched with the servo motor, model SGDV-7R6A01A002000, rotation accuracy: 5 arc seconds, output torque: 100 N·m, positioning accuracy: 1 μm. The main input parameters are rated voltage: 200-230v, rated frequency: 50/60Hz, rated current: 4.9A. The main output parameters are rated voltage: 0-230v, rated output frequency: 0-400Hz, rated current: 7.6A, and rated power 1KW.
The CNC control system of the planar spiral toothed line cylindrical gear gear machining device is completed based on the Guangzhou CNC development environment. This software module is committed to providing a superior human-computer interaction interface, enabling users to intuitively and easily understand system functions during operation. At the same time, it also pursues high stability of the motion control system and effectively improves the efficiency of program operation by designing software with good code portability.
During the gear machining process of the planar spiral gear, the linkage work of dual rotating axes and dual linear guide rails is involved, and the accuracy of the rotation ratio will directly affect the tooth profile accuracy of the gear. Therefore, it is necessary to ensure the positioning accuracy and gear machining accuracy of each component of the gear machining device. The device uses a planar spiral toothed line cylindrical gear CNC system. This CNC system covers multiple functions such as quantitative movement, continuous movement, circular interpolation, and multi-axis interpolation. Behind the development of this software module is the flexible use of high-performance servo control cards. The operating system and development library it supports provide a wide range of applicability for software development. Through the real-time communication of the PCI bus, the coordination between the computer and the system is successfully achieved, ensuring the stability and accuracy of the system. This motion control software module of the planar spiral toothed line cylindrical gear gear machining device is not only an innovation in the field of special device manufacturing technology but also provides strong support for the future development of the industrial automation field.
After introducing the main design points of the planar spiral toothed line cylindrical gear gear machining device, the trial cutting gear machining of the gear blank will be introduced in detail next, which is mainly divided into the following steps:
(1) Fix the gear blank on the digital rotary milling table, convert the device control button “C/S” to manual mode, and reposition the gear machining cutter head.
(2) Control the gear blank to reach the pre-cutting position through the X and Z axes, and control the spindle, that is, the gear machining cutter head, to reach the pre-cutting position.
(3) Convert the device control to “MDI” mode, enter the programming interface in the operating system, and program the cutting program.
(4) After the program is written, adjust the machine tool parameters, operate the button “program start”, and perform gear cutting until the gear cutting is completed and the tool is retracted.
When gear machining gears of the same batch, first determine the gear parameters mainly including: modulus $m$, number of teeth $z$, pressure angle $\alpha$, tooth width $B$, spiral line curvature radius $R_T$ at the middle line of the tooth width. Among them, the blade angle is determined by the pressure angle $\alpha$, the tool spiral line arrangement position is determined by the modulus $m$ and the spiral line curvature radius $R_T$ at the middle line of the tooth width, and the tool clamping position of the large cutter head is adjusted. The center of the middle section of the workpiece tooth width is ensured to coincide with the pre-cutting point of the gear machining tool through the tooth width $B$ and the fixture. The speed relationship between the cutter head and the workpiece is determined by the number of teeth $Z$, and the CNC program is written. After tool setting, gear machining is carried out. During gear machining, the cutter head speed is 200 r/min, and the main cutting tool installation radius $R_T = 230$ mm, so the milling linear speed is $V_T = 289$ m/min. High milling linear speed can ensure tooth surface quality. Gear machining parameters: $m = 5$, $Z = 63$, $B = 50$ mm.
The trial cutting program for the planar spiral toothed line cylindrical gear is as follows:
; Set workpiece coordinate system
G54 G92.1
; Move to initial point
G0 Z2592 A-360
; Start gear machining cycle
M98 P1000 L20
; End gear machining
M30
; Subprogram: gear machining cycle
O1000
; Perform cutting
G91 G01 Z51840 A-7200 Y-12 F4444
G91 G01 Z51840 A-7210 X31.42 F4444
G91 G01 Z51840 A-7190 X-31.42 F4444
G91 G01 Z51840 A-7190 X-31.42 F4444
G91 G01 Z51840 A-7210 X31.42 F4444
; Return to initial point
G91 G01 Z2592 A-360 Y10 F3333
; Check cycle times
IF [I] GT 0 THEN
GOTO 1000
ENDIF
; End subprogram
M99
During the trial cutting process, depth cutting motion is first carried out. Since the numerical relationship between the rotating axes cannot be changed by setting in the original CNC system, the numerical relationship between the rotating axes is controlled by controlling the ratio of the total rotation angle of the cutter head to the total rotation angle of the CNC rotary milling table under the same number of revolutions. The numerical proportional relationship between the total angle of the gear blank and the total angle of the cutter head must satisfy the Z-fold relationship. The cutter head and the gear blank perform gear machining rotation according to the above numerical relationship. Since the material of the gear blank is aluminum alloy, it is necessary to strictly control the cutting amount of each tool not to be too large. The tooth height with modulus 5 is 11.25 mm. In the above program, the feed is divided into 12 times, and the feed amount per time is controlled not to exceed 1 mm, which can control the gear machining accuracy of the tooth surface and prolong the service life of the gear machining tool. The gear blank is continuously fed at a speed $v_{yc}$ until all tooth grooves are machined.
After the gear blank tooth groove gear machining is completed, the next step is to machine the involute tooth profile. The gear machining tool and the gear blank workpiece perform generating motion. The total rotation angle of the cutter head remains unchanged. By increasing the rotation angle of the gear blank, 10° is added/subtracted in the program. The angle of 10° is the generating angle of the gear. The cutter head rotates, and the blade forms a conical surface trajectory. The cutter head maintains the original rotation speed, and the blade conical surface is relatively stationary. The gear blank and the blade conical surface form a conjugate motion. Adding 10° completes the involute generation of the convex tooth surface, and reducing 10° completes the involute generation of the concave tooth surface. And through calculation, during generating gear machining, it is necessary to move back and forth in the X-axis direction, and the movement distance is ±31.42 mm. The generating gear machining process is shown in the figure.
Through the above gear machining device, the planar spiral gear sample was successfully trial cut. The tooth line of the planar spiral toothed line cylindrical gear is a planar spiral line. The curvature radius of the tooth line of the concave tooth surface is slightly larger than that of the convex tooth surface, forming a slightly drum-shaped tooth, which effectively reduces meshing bias load. The tooth line is basically symmetrical, which can effectively reduce the axial component force caused by helical gear transmission, and at the same time reduce the meshing impact of the gear pair and reduce noise. The tooth surface of the trial cut sample of the machined planar spiral toothed line cylindrical gear is smooth, and tool marks are almost invisible. Taking the trial cut gear sample as an example, the gear has 63 teeth, modulus 5, and tooth width 60 mm. At present, the final forming time of the straight tooth cylindrical gear with hobbing cutting is 160 min, and the final forming time of the planar spiral toothed line cylindrical gear is 35 min. Compared with hobbing, it greatly saves milling time and reduces production costs.
According to the gear machining principle and gear machining method of the planar spiral toothed line cylindrical gear proposed in the above chapters, I independently research and develop an efficient gear machining device for planar spiral toothed line cylindrical gears, and carry out trial cutting of the planar spiral toothed line cylindrical gear. Subsequently, I introduce the relevant gear machining parameters of the device and the basic parameters of the main components, and introduce the relevant program steps of trial cutting gear machining in detail. According to the tooth surface condition and tooth line condition of the machined gear, on the pitch circle section, the curvature radius of the tooth line of the concave tooth surface is slightly larger than that of the convex tooth surface, forming a slightly drum-shaped tooth, which effectively reduces meshing bias load, and the tooth line is basically symmetrical, which can effectively reduce the axial component force caused by helical gear transmission, and at the same time reduce the meshing impact of the gear pair and reduce noise. In addition, the tooth surface is smooth, no tool marks are visible, and the surface accuracy is high.
On the basis of analyzing the existing arc toothed line cylindrical gears, I propose a planar spiral toothed line cylindrical gear. Its main geometric features are: the tooth line is a planar spiral line, the tooth profile is an involute tooth profile, the curvature radius of the concave tooth line is slightly larger than that of the convex tooth line, and the tooth shape is drum-shaped.
Based on the forming principle of the planar spiral gear, differential geometry transformation, and tool spatial milling trajectory, I derive the relevant parameter equations of this gear, propose a full-tooth-width high-linear-speed continuous generating gear machining method, verify the feasibility of this gear machining method through Vericut gear machining simulation, and build a corresponding dedicated gear machining device for gears and develop a parametric modeling system for planar spiral toothed line cylindrical gears, laying a foundation for studying the meshing relationship and anti-bias load characteristics of the gear pair.
The specific achievements of this project are summarized in the following five points:
(1) I propose a geometric model of a planar spiral toothed line cylindrical gear that can be efficiently machined, derive its tooth surface equation, dynamic meshing line equation, and contact ratio calculation, and verify the accuracy of its tooth surface equation through MATLAB. On the pitch cylinder development surface of this gear, the curvature radius of the tooth line of the concave tooth surface is slightly larger than that of the convex tooth surface, so that it has excellent anti-bias load characteristics. Through MATLAB, I construct the concave and convex tooth surface models of the planar spiral gear and the gear pair model obtained from Vericut simulation for tooth surface contact experiments. I conclude that the contact characteristics of the gear pair model are as follows: the contact interval is mainly distributed in the middle 80% to 90% region of the contact tooth surface of the gear pair, reducing the force on both ends of the tooth. The gear has better bending strength and stronger load-bearing capacity.
(2) Based on SolidWorks secondary development and Visual Basic 6.0, I write a planar spiral toothed line cylindrical gear modeling system. After inputting the basic parameters of the planar spiral gear, the system can automatically generate an accurate geometric model of the planar spiral toothed line cylindrical gear. While improving the efficiency and accuracy of gear design, this system also fills the technical gap in the field of curved cylindrical gear modeling and lays a solid foundation for gear simulation analysis.
(3) I complete the anti-bias load analysis of the planar spiral toothed line cylindrical gear. According to the meshing research and analysis of straight teeth, helical teeth, and spiral teeth: under the ideal meshing state, the maximum contact stress on the tooth surface of the planar spiral gear is the smallest, followed by the helical tooth cylindrical gear, and finally the straight tooth cylindrical gear. When the gear installation axis has a parallelism error, the maximum contact stress on the tooth surface of the planar spiral toothed line cylindrical gear is much smaller than that of straight teeth and helical teeth cylindrical gear pairs, indicating that the planar spiral toothed line cylindrical gear has anti-bias load characteristics. At the same time, I also study the variation trend of the maximum contact stress on the tooth surface under different numbers of teeth and modulus: the maximum tooth surface contact stress of straight teeth, helical teeth, and planar spiral gears all decreases with the increase of modulus and number of teeth.
(4) I propose an efficient forming method for planar spiral toothed line cylindrical gears: the full-tooth-width high-linear-speed continuous generating gear machining method. I derive the planar spiral tooth line equation and the tool arrangement trajectory equation. I clarify the tooth groove depth cutting and involute tooth profile generating gear machining methods of this gear. Based on Vericut simulation software, I verify the feasibility of the planar spiral gear gear machining method. Experiments prove that this gear machining method can achieve full-tooth-width high-speed milling of spiral teeth.
(5) I successfully develop an efficient gear machining device for planar spiral toothed line cylindrical gears. On the basis of a CNC lathe, I transform it by adding two CNC rotating axes to achieve a four-axis CNC gear machining device with dual rotating axes controllable rotation, and introduce and study its overall structure and key components. I complete the trial milling of spiral tooth samples and successfully machine planar spiral toothed line cylindrical gear samples. The tooth surface of the trial cut sample of the machined planar spiral toothed line cylindrical gear is smooth, and tool marks are almost invisible. The trial cutting experiment proves that the device can achieve efficient milling of planar spiral gears.
The main innovation points proposed in this paper are the following two points:
(1) I propose a new type of toothed line cylindrical gear: the planar spiral toothed line cylindrical gear. The curvature radius of the tooth line of the concave tooth surface of this gear is slightly larger than that of the convex tooth surface, which can make the gear pair have a good contact interval, effectively reduce the stress concentration at the tooth root at both ends of the tooth, and improve the contact state at both ends of the tooth. At the same time, due to its special tooth line, this gear has anti-bias load characteristics. It has a greater degree of adjustment interval in the axial direction, and then through slight adjustment, its contact interval is located in the middle position of the tooth, avoiding bias load.
(2) I propose the full-tooth-width high-linear-speed continuous generating gear machining method and develop a dedicated gear machining device for planar spiral toothed line cylindrical gears. A rotating large cutter head is used to quickly mill gears. Multiple gear machining tools arranged in a planar spiral line on the cutter head continuously mill gear tooth grooves. The planar spiral line pitch is equal to the gear tooth pitch. The cutter head rotates one revolution, and the workpiece rotates one tooth in equal proportion. During gear machining, depth cutting and tooth surface generating gear machining are carried out continuously. Full-tooth-width continuous rapid gear machining of gears is achieved. In terms of gear machining efficiency, gear manufacturing time is greatly reduced, gear manufacturing costs are reduced, and the tooth surface of the machined planar spiral toothed line cylindrical gear sample is smooth.
The research progress of this paper is strictly implemented in accordance with the previous work arrangement. Up to now, the theoretical research of the planar spiral toothed line cylindrical gear, the construction of the gear machining device, and the gear sample have been completed. However, due to time and laboratory conditions and other aspects, there are still many dynamics-related studies of the gear model and dynamic motion characteristic studies of the gear sample that need to be continued.
1. The anti-bias load characteristic analysis of the planar spiral toothed line cylindrical gear in this paper is still in the theoretical research stage of the three-dimensional model, lacking experimental research on gear samples in actual working conditions. In order to further study the bias load state of gear samples in the actual meshing process, gear pair experiments can be carried out with the help of a reducer transmission platform, strictly controlling relevant variables, and analyzing the bias load force conditions of straight teeth, helical teeth, and planar spiral gears.
2. Regarding the research and development of the gear machining device for planar spiral toothed line cylindrical gears, the currently used Guangzhou CNC CK6180 CNC lathe cannot guarantee the error accuracy of the gear blank during gear machining due to its own spindle and rotating axis positioning accuracy and gear machining accuracy. Moreover, there is a lack of error data between the planar spiral gear sample and the standard three-dimensional model. In response to these problems, a laser detection plane can be built later to analyze the shape error of the solid model of the gear sample.
3. Based on the research results of this paper, other dynamic characteristics of the planar spiral gear can be studied later, such as: transmission characteristics of the planar spiral toothed line cylindrical gear, gear anti-wear characteristics, and optimization of its gear machining method.
In general, as a new type of gear, the planar spiral gear integrates the transmission characteristics of ordinary straight teeth and helical teeth cylindrical gears and overcomes the shortcomings of herringbone gears. This type of gear has a series of advantages such as stable transmission, high load-bearing capacity, and strong anti-bias load capacity. Therefore, the planar spiral toothed line cylindrical gear has broad application prospects in replacing traditional gears. In mechanical equipment, the application of planar spiral toothed line cylindrical gears will promote a substantial improvement in the performance of devices in various fields. This type of gear can improve production capacity and reduce energy consumption in many fields of the country, for example: automobile transmissions to improve their load-bearing capacity and service life; similarly, it can also be used to build devices, so that the stability of the transmission system operation of the device is improved. In addition, this technology can also be used in the field of wind power generation to increase the rotation speed limit of its engine and improve the transmission efficiency of its transmission system. It can also be used in the aviation industry. In the hydraulic and navigation systems inside the aircraft, gearboxes are required for transmission, and planar spiral gears can greatly improve their transmission stability. With the continuous development and progress of modern technology, gears, as a basic component in the mechanical field, have increasingly strict performance requirements, and the improvement and upgrading of gear manufacturing will also accelerate the development of the entire mechanical field.