Calculation of geometric dimensions of cylindrical gears
Cylindrical gears are one of the most widely used components in mechanical transmissions. Calculating their geometric dimensions is fundamental to gear design and manufacturing, directly impacting their transmission performance, strength, and service life. Cylindrical gear geometric dimensions include module, number of teeth, pressure angle, addendum diameter, root diameter, pitch diameter, tooth pitch, and tooth thickness. These parameters are strictly mathematically related and require precise calculation in accordance with national standards. Correct geometric dimension calculation ensures proper gear meshing, reduces impact, vibration, and noise during transmission, and improves transmission efficiency and reliability.
Module and number of teeth are the most fundamental parameters of cylindrical gears and form the basis for calculating other geometric dimensions. Module, denoted by the symbol “m,” is a parameter that characterizes the size of the gear teeth and is measured in millimeters. Its value is determined by the gear’s load capacity and transmission requirements. National standards specify a standard module series, and standard modules should be preferred during design. The number of teeth, denoted by the symbol “z,” refers to the number of teeth on a gear. This affects the gear’s transmission ratio and size. A larger number of teeth results in a larger gear but a smoother transmission. Once the module and number of teeth are determined, the pitch circle diameter (d) can be calculated using the formula d = m × z. The pitch circle is an imaginary circle on the gear where the tooth thickness is equal to the tooth space width. It serves as the basis for calculating other dimensions. For example, a cylindrical gear with a module m = 3 mm and a number z = 20 teeth would have a pitch circle diameter d = 3 × 20 = 60 mm.
The pressure angle, denoted by the symbol “α,” is another important parameter for cylindrical gears. It refers to the angle between the normal to the tooth profile curve and the plane perpendicular to the gear axis on the pitch circle. National standards stipulate a standard pressure angle of 20°, though angles of 15°, 25°, and so on may be used in special circumstances. The pressure angle affects the gear’s tooth profile, transmission efficiency, and load-bearing capacity. A larger pressure angle increases the root strength of the tooth, but also increases the radial force during transmission. A smaller pressure angle results in smoother transmission, but lower root strength. The addendum diameter (da) is the diameter of the circle containing the gear tooth tips. The formula for calculation is da = d + 2ha × m, where ha is the tooth height coefficient. For standard gears, the tooth height coefficient ha = 1, so da = m × (z + 2). The root diameter (df) is the diameter of the circle around the gear tooth root. It’s calculated as df = d – 2 (ha + c) × m, where c is the clearance coefficient. For standard gears, the clearance coefficient c = 0.25, so df = m × (z – 2.5). For example, for a standard gear with a module m = 3 mm and 20 teeth z = 20, its tip diameter da = 3 × (20 + 2) = 66 mm, and its root diameter df = 3 × (20 – 2.5) = 52.5 mm.
Tooth pitch, tooth thickness, and tooth space width are parameters that describe the distribution of gear teeth along the pitch circle. Pitch (p) is the arc length of the tooth profiles on the same side of two adjacent teeth on the pitch circle. It is calculated as p = π × m, where π is pi, approximately equal to 3.1416. Tooth thickness (s) is the arc length between the tooth profiles on either side of a tooth on the pitch circle. For standard gears, s = p/2 = π × m/2. Tooth space width (e) is the arc length between two adjacent teeth on the pitch circle. For standard gears, e = s = π × m/2. For example, a standard gear with a module m = 3 mm has a pitch p = 3.1416 × 3 ≈ 9.4248 mm, a tooth thickness s = 9.4248/2 ≈ 4.7124 mm, and a tooth space width e = 4.7124 mm. In addition, the tooth addendum height (ha) refers to the radial distance from the pitch circle to the tooth addendum circle, and the calculation formula is ha = ha×m = m; the tooth root height (hf) refers to the radial distance from the pitch circle to the tooth root circle, and the calculation formula is hf = (ha + c)×m = 1.25m; the total tooth height (h) refers to the sum of the tooth addendum height and the tooth root height, and the calculation formula is h = ha + hf = 2.25m.
When calculating the geometric dimensions of cylindrical gears, the gear’s modification coefficient must also be considered. For modified gears, the geometric dimension calculation differs from that for standard gears. Modified gears are manufactured by changing the relative position of the tool and the gear blank, known as modified cutting. The goal is to improve gear transmission performance, such as avoiding undercutting, increasing gear strength, and matching center distances. The pitch diameter of a modified gear remains d = m × z, but the calculation formulas for the addendum and root diameters require corrections based on the modification coefficient (x). The addendum diameter da = m × (z + 2ha + 2x), and the root diameter df = m × (z – 2ha – 2c + 2x). The selection of the modification coefficient should be determined based on the specific application requirements of the gear, through gear strength calculations and meshing condition verification. In addition, the center distance (a) of the gear refers to the sum of the pitch circle radii of the two meshing gears. For standard gears, the center distance a = (d1 + d2)/2 = m×(z1 + z2)/2; for modified gears, the center distance a’ = a + y×m, where y is the center distance variation coefficient, y = (x1 + x2) – σ, and σ is the tooth addendum reduction coefficient.
The geometrical dimension calculations for cylindrical gears must be performed strictly in accordance with national standards and design specifications. During the calculation process, attention should be paid to the consistency of units and the accuracy of numerical values to avoid miscalculation that could result in improper gear meshing or reduced transmission performance. In actual design, the calculation results must be corrected and optimized, taking into account factors such as the gear material, heat treatment method, and machining process. For example, for gears requiring quenching, the effect of deformation on the geometrical dimensions after quenching should be considered, and appropriate machining allowances should be reserved. For gears requiring higher precision, parameters such as tooth thickness and pitch must be precisely measured and corrected to ensure transmission accuracy. Through accurate geometrical dimension calculations and appropriate parameter selection, cylindrical gears with excellent performance and long service life can be designed to meet the needs of various mechanical transmissions.
