Calculation of geometric dimensions of straight bevel gears
Straight bevel gears are used to transmit motion and power between two intersecting shafts, typically at a 90° angle. They offer smooth transmission and compact design, making them widely used in automotive, machine tool, and construction machinery applications. The tooth profile of straight bevel gears tapers from the large end to the small end, and their geometric dimensions differ significantly from those of cylindrical gears. Parameters such as the taper pitch, pitch angle, and tooth width must be considered. These interrelated parameters require precise calculation in accordance with national standards and design specifications to ensure proper meshing and transmission performance.
The basic parameters of spur bevel gears include module, pressure angle, number of teeth, pitch cone angle, and taper pitch. The module and number of teeth are the basis for calculating other dimensions. Module (m) is the fundamental parameter of spur bevel gears and refers to the module at the large end. Standard module series are used and selected based on the gear’s load capacity and transmission requirements. A larger module increases the gear’s size and strength. The pressure angle (α) is the angle between the tooth profile curve and the pitch circle tangent on the pitch cone. The standard pressure angle is 20°. The pressure angle affects the gear’s tooth profile and transmission efficiency. The number of teeth (z1, z2) represents the number of teeth on the driving and driven gears, respectively. The transmission ratio, i = z2 / z1, is determined based on the required transmission ratio. To ensure smooth transmission, the number of teeth on the pinion is generally no less than 17. The number of teeth on the large gear is determined based on the transmission ratio. The pitch cone angle (δ1, δ2) is the angle between the axis of the gear and the generatrix of the pitch cone. The sum of the pitch cone angles of the two gears is equal to the intersection angle of the two axes (usually 90°), that is, δ1 + δ2 = 90°. The calculation formula is tanδ1 = z1 /z2, tanδ2 = z2 /z1. The pitch cone angle determines the taper of the gear and is the key to calculating other cone sizes.
The main geometric dimensions of spur bevel gears include the pitch circle diameter, addendum diameter, root diameter, taper pitch, and tooth width. These dimensions are calculated based on the large end. The pitch circle diameters (d1, d2) refer to the diameters of the pitch circle at the large end of the gear. The calculation formulas are d1 = m × z1 and d2 = m × z2. The pitch circle diameters serve as the basis for calculating gear geometric dimensions. The addendum diameters (da1, da2) are the diameters of the circle where the addendum of the gear tooth lies. The calculation formulas are da1 = d1 + 2m × cosδ1 and da2 = d2 + 2m × cosδ2. The addendum diameters determine the gear’s outer profile. The root diameters (df1, df2) are the diameters of the circle where the root of the gear tooth lies. The calculation formulas are df1 = d1 – 2.4m × cosδ1 and df2 = d2 – 2.4m × cosδ2. The root diameters affect the gear’s root strength. The taper pitch (R) is the distance from the apex of the pitch cone to the gear’s larger end. It’s calculated as R = d1 / (2sinδ1) = d2 / (2sinδ2). This reflects the size of the gear and is a crucial parameter for determining its installation position. The tooth width (b) is the width of the gear along the generatrix of the pitch cone. Generally, b ≤ R/3 is used to ensure proper meshing and strength. Too large a tooth width results in a sharper tooth profile at the smaller end, reducing strength; too small a tooth width compromises transmission stability.
The tooth height parameters of straight bevel gears include addendum, root, and total tooth depth, all of which are calculated based on the large end. Addendum (ha) is the distance from the pitch circle to the addendum circle, with a standard addendum height of ha = m. Root (hf) is the distance from the pitch circle to the root circle, with a standard root height of hf = 1.2 m. Total tooth height (h) is the sum of the addendum and root heights, i.e., h = ha + hf = 2.2 m. It should be noted that since the tooth profile of straight bevel gears gradually decreases from the large end to the small end, their tooth height also varies with position, with the large end having the largest tooth height and the small end having the smallest tooth height. The above formula calculates the tooth height parameters for the large end. In addition, the top clearance (c) of a straight bevel gear refers to the radial distance between the addendum circle of one gear and the root circle of the other gear in a pair of meshing gears. The standard top clearance c = 0.2m. The existence of the top clearance can avoid interference between the addendum and the root of the gear when the gears are meshing, and at the same time provide storage space for lubricating oil.
When calculating the geometric dimensions of spur bevel gears, gear displacement must also be considered. Gear displacement can improve transmission performance, such as preventing undercutting, increasing tooth root strength, and matching center distances. Spur bevel gear displacements are categorized as height displacement and angular displacement. Height displacement refers to when the sum of the two gear displacement coefficients is zero ( x1 + x2 = 0) and is applicable when the center distance is equal to the standard center distance. Angular displacement refers to when the sum of the two gear displacement coefficients is not equal to zero ( x1 + x2 ≠ 0) and is applicable when the center distance needs to be changed. After displacement, the formulas for calculating the addendum and root heights of the gears must be modified: addendum height ha = m × (1 + x – σ), and root height hf = m × (1.2 – x), where σ is the addendum reduction factor, σ = x1 + x2. The selection of the modification coefficient needs to be determined according to the number of teeth, transmission ratio and strength requirements of the gear. Generally, the small gear takes positive modification and the large gear takes negative modification to improve the root strength of the small gear and balance the life of the two gears.
After the geometric dimensions of spur bevel gears are calculated, necessary verification calculations are required to ensure the gear’s transmission performance and strength. First, the gear contact ratio is verified. The greater the contact ratio, the smoother the transmission. The contact ratio of spur bevel gears should generally be greater than 1.2. Next, the tooth root bending strength and tooth surface contact strength are verified. Based on parameters such as the gear material, heat treatment method, transmitted power, and speed, calculations are performed according to relevant formulas to ensure that the gears will not suffer from failures such as root fracture, tooth surface bonding, and pitting during use. Furthermore, the gear’s installation dimensions, such as center distance and axis spacing, must be checked to ensure compatibility with other components. Only through precise geometric dimension calculations and necessary verification calculations can a spur bevel gear be designed to meet the requirements and ensure safe and reliable operation in mechanical transmission. While the geometric dimension calculations for spur bevel gears are complex, understanding the relationships between the various parameters and strictly following the formulas and standards can yield accurate results, providing a reliable basis for gear design and manufacture.
