Calculation of geometric dimensions of cylindrical worm gear transmission
A cylindrical worm drive, consisting of a cylindrical worm and a worm wheel, is used to transmit motion and power between two intersecting axes in space, typically at a 90° angle. It offers advantages such as a high transmission ratio, compact structure, smooth transmission, and low noise, making it widely used in machine tools, lifting machinery, metallurgical equipment, and other fields. Calculating the geometric dimensions of a cylindrical worm drive is fundamental to its design and manufacture, directly impacting its performance, strength, and service life. Accurate calculations must be performed based on the basic parameters of the worm drive, in accordance with national standards and design specifications.
The basic parameters of cylindrical worm gearing are the prerequisites for geometric dimension calculations. They primarily include the module m , pressure angle α , number of worm starts z₁ , number of worm wheel teeth z₂ , worm diameter coefficient q , and lead angle γ . The module m is a fundamental parameter of worm gearing, determining the dimensions of the worm and worm wheel. The larger the module, the greater the strength of the worm and worm wheel. National standards specify a standard module series for cylindrical worm gearing (such as 1 , 1.25 , 1.6 , 2 , and 2.5 ). The pressure angle α typically refers to the axial pressure angle of the worm. The standard pressure angle is 20° . For power transmission, a 25° pressure angle can be used to improve load capacity; for indexing transmission, a 15° pressure angle can be used to improve transmission accuracy. The number of starts z₁ in a worm gear is generally 1-4 . A higher number of starts increases transmission efficiency, but also reduces the transmission ratio. Single-start worms can achieve larger transmission ratios ( i=10-80 ), but with lower efficiency. Multi-start worms have smaller transmission ratios ( i=5-10 ) and higher efficiency. The number of teeth in a worm gear, z₂ , is determined by the transmission ratio i and the number of starts z₁. z₂ = i × z₁ , and is generally between 28 and 80. Too few teeth will result in insufficient bending resistance, while too many teeth will make the worm gear bulky and prone to undercutting. The worm diameter coefficient q is the ratio of the worm pitch diameter d₁ to the module m ( q = d₁/m ). The standard specifies a series of diameter coefficients corresponding to different modules (e.g., 8 , 10 , 12 , 16 , etc.). The diameter coefficient affects the stiffness and lead angle of the worm gear. The lead angle γ is related to the number of worm heads z₁ , the module m and the pitch circle diameter d₁ . tanγ = z₁×m/d₁ = z₁/q . The larger the lead angle, the higher the efficiency of the worm transmission.
Calculating worm geometry is a crucial component of cylindrical worm gear design. This includes factors such as the pitch diameter, addendum diameter, root diameter, lead, and worm length. The pitch diameter, d₁, is a key worm parameter and is calculated using the formula d₁ = m × q . The pitch diameter directly impacts the worm’s stiffness and strength; a larger diameter increases the stiffness and strength. The addendum diameter, dₐ₁, is the diameter of the circle where the worm’s tooth tips are located. The calculation formula is dₐ₁ = d₁ + 2hₐ*m , where hₐ is the addendum height coefficient (standard value is 1 ). The root diameter, d₁ , is the diameter of the circle where the worm’s tooth roots are located. The calculation formula is d₁ = d₁ – 2 (hₐ + c)*m , where c is the head clearance coefficient (standard value is 0.2 ). The worm lead, P , is the distance the worm helix travels axially during one revolution. It’s calculated as P = z₁ × m × π . The worm length, L , must be determined based on the number of worm wheel teeth and the number of worm starts to ensure effective meshing between the worm and the worm wheel. For a single-start worm, L ≈ (11 + 0.06z₂) m ; for a multi-start worm, L ≈ (12.5 + 0.09z₂) m . Furthermore, the worm length must be at least 1.5 times the lead to avoid insufficient meshing length. Furthermore, the worm tooth width, b₁ , is generally greater than the worm length, L , to ensure that the worm can still mesh with the worm wheel during axial movement.
The worm gear’s geometric dimensions must match those of the worm to ensure proper meshing. These primarily include the pitch circle diameter, addendum diameter, root diameter, throat diameter, and wheel width. The worm gear pitch circle diameter, d₂, is a key parameter of the worm gear and is calculated using the formula d₂ = m×z₂ . Together, the worm gear pitch circle diameter and the worm gear pitch circle diameter determine the worm gear drive’s center distance , a , which is defined as (d₁ + d₂)/2 . This center distance is determined based on the installation space and transmission requirements. Standards specify a series of center distance values (e.g., 40 , 50 , 63 , 80 , etc.). The worm gear tip circle diameter, dₐ₂, is the diameter of the circle surrounding the worm gear tooth tips and is calculated using the formula dₐ₂ = d₂ + 2hₐ*m . The worm gear root diameter, d₂ , is the diameter of the circle containing the worm gear tooth roots and is calculated as d₂ = d₂ – 2 (hₐ + c)*m . The worm gear throat diameter, dₐ₂’ , is the maximum diameter of the worm gear tooth tip. For standard worm gears, dₐ₂’ = dₐ₂ . The worm gear width , b₂ , is the width of the worm gear teeth. To ensure effective engagement between the worm gear and the worm shaft, the worm gear width, b₂, should be less than the worm shaft length , L. Typically, b₂ = 0.75dₐ₁ (for single-start worms) or b₂ = 0.67dₐ₁ (for multi-start worms). Furthermore, the worm gear width must not exceed 1/3 of the pitch diameter to prevent the worm gear from being too large.
Other important geometric dimensions of cylindrical worm gear transmission include center distance, top clearance, and tooth pitch. Center distance a is the distance between the axis of the worm and the worm wheel. It is an important parameter of worm gear transmission. Its calculation formula is a = (d₁ + d₂)/2 = m (q + z₂)/2 . The size of the center distance directly affects the structural size and installation space of the worm gear transmission. When designing, it is necessary to select a suitable center distance according to the transmission requirements and installation conditions. Top clearance c is the gap between the root circle of the worm and the top circle of the worm wheel, and the gap between the root circle of the worm wheel and the top circle of the worm tooth. Its calculation formula is c = c*m = 0.2m . The function of the top clearance is to store lubricating oil to avoid interference between the root and top of the worm and worm wheel. The axial pitch pₓ₁ of the worm and the end face of the worm wheel