Process Calculation of Flangeless Cylindrical Parts
Flangeless cylindrical parts are common components used in stamping, such as bearing sleeves and container barrels. Process calculations primarily include blank size calculation, forming cycle determination, dimension calculation for each forming cycle, and forming force calculation. These calculations form the basis for developing stamping process plans, designing molds, and selecting equipment. The forming of flangeless cylindrical parts is a deep drawing process. Under the influence of the drawing force, the material undergoes plastic deformation, gradually transforming from a flat blank into a cylindrical shape. Process calculations must consider factors such as the material’s plasticity, the drawing coefficient, and mold parameters to ensure a smooth forming process without defects such as wrinkling and cracking.
Calculating the blank size is the first step in the process calculations for flangeless cylindrical parts. This calculation is based on the principle of volume invariance, which states that the volume of the material remains the same before and after drawing (ignoring thickness changes). For a flangeless cylindrical part with diameter d, height h, and thickness t, the blank diameter D is calculated as: D = √(d² + 4dh). When the material thickness t is larger (t>3mm) or when high precision is required, the effect of thickness variation must be considered, and the formula is modified to: D = √[(d + t)² + 4 (d + t)(h – 0.43t)]. For example, for a mild steel cylinder with a diameter of 50mm, a height of 100mm, and a thickness of 2mm, the blank diameter D = √(50² + 4×50×100) = √(2500 + 20000) = √22500 = 150mm. Taking the thickness into account, D = √[(50+2)² + 4×(50+2)×(100 – 0.43×2)] = √[2704 + 4×52×99.14] ≈ √[2704 + 20626] ≈ √23330 ≈ 152.7mm. The blank dimensions must be calculated accurately, with an error of ≤±1mm. Otherwise, the final height will be insufficient or material will be wasted after drawing.
The drawing coefficient is a key parameter for measuring the difficulty of forming flangeless cylindrical parts. It refers to the ratio of the diameter of the cylindrical part after each drawing to the diameter of the blank (or the previously drawn part) before drawing, i.e., m = dn / dn-1 (dn is the diameter after the nth drawing, dn-1 is the previous diameter). The initial drawing coefficient, m1, is the smallest, while subsequent drawing coefficients, such as m2 and m3, gradually increase. This is because the material undergoes work hardening and its plasticity decreases after the initial drawing. The limiting drawing coefficients of commonly used materials are: for mild steel, m1 = 0.5-0.55, m2 = 0.7-0.75, m3 = 0.75-0.8, and m4 = 0.8-0.85. For example, a blank with a diameter of 150 mm is drawn into a cylindrical part with a diameter of 50 mm. The total drawing coefficient mtotal = 50/150≈0.33, which is less than the initial limit drawing coefficient of 0.5, and multiple drawing operations are required.
The number of forming cycles is determined based on the total drawing coefficient and the limit drawing coefficients of each cycle. This is done by starting with the first drawing and gradually calculating the diameter after each drawing cycle using the limit drawing coefficient until the target diameter is reached. For example, if the target diameter is 50mm and the blank diameter is 150mm: the first drawing cycle d1 = m1 × D = 0.5 × 150 = 75mm; the second drawing cycle d2 = m2 × d1 = 0.7 × 75 = 52.5mm; the third drawing cycle d3 = m3 × d2 = 0.75 × 52.5 ≈ 39.4mm (which is less than the target diameter of 50mm. Therefore, the diameter of 52.5mm after the second drawing cycle is close to the target and can be reached by the third drawing cycle). The actual number of forming cycles is three, with the second drawing cycle reaching 52.5mm and the third drawing cycle reaching 50mm. The drawing coefficients for each cycle are 0.5, 0.7, and 0.95 (50/52.5 ≈ 0.95), respectively, all exceeding the corresponding limit drawing coefficients to ensure safe forming.
The dimensional calculations for each drawing process include diameter, height, and fillet radius. After the diameter is determined according to the drawing coefficient, the height calculation formula is: h_n = (d_{n-1}² – d_n²)/(4d_n) + 0.57 (r_n + r_{n-1}) (r_n is the fillet radius of the punch for this drawing process, and r_{n-1} is the fillet radius of the die). For example, the first drawing d1 = 75mm, the blank diameter is 150mm, the punch radius r1 = 10mm, the die radius R1 = 15mm, then h1 = (150² -75²)/(4×75) + 0.57×(10+15) = (22500-5625)/300 + 0.57×25 = 16875/300 +14.25 = 56.25+14.25 = 70.5mm; the second drawing d2 = 52.5mm, r2 = 8mm, R2 = 12mm, then h2 = (75² -52.5²)/(4×52.5) +0.57×(8+12) = (5625-2756.25)/210 +0.57×20 = 2868.75/210 +11.4≈13.66+11.4=25.06mm; for the third drawing, d3=50mm, r3=5mm, R3=8mm, then h3=(52.5² -50²)/(4×50) +0.57×(5+8)= (2756.25-2500)/200 +0.57×13=256.25/200 +7.41≈1.28+7.41=8.69mm, the total height is 70.5+25.06+8.69≈104.25mm, which is slightly larger than the target height of 100mm and needs to be adjusted during the last drawing.
The calculation of drawing force is an important part of the process calculation of flangeless cylindrical parts, which is used to select the tonnage of the press. The initial drawing force F1 = K1×π×d1×t×σb, and the subsequent drawing force Fn = Kn×π×dn×t×σb, where K1=0.5-0.8, Kn=0.6-0.9 (the larger n is, the larger the K value is), and σb is the tensile strength of the material. For example, for low carbon steel with a thickness of 2mm (σb=450MPa), the first drawing is d1=75mm, F1=0.6×3.14×75×2×450≈0.6×3.14×75×900≈0.6×211950≈127170N≈127.2kN; the second drawing is d2=52.5mm, F2=0.7×3.14×52.5×2×450≈0.7×3.14×52.5×900≈0.7×148365≈103.855kN; the third drawing is d3=50mm, F3=0.8×3.14×50×2×450≈0.8×3.14×50×900≈0.8×141300≈113.04kN. The tonnage of the press needs to be greater than 1.2-1.5 times the maximum drawing force. A 200kN press can meet the requirement.
The process calculation for flangeless cylindrical parts also requires consideration of auxiliary factors such as lubrication, annealing, and die radius. During the drawing process, lubricant (such as engine oil or drawing paste) should be applied to the contact surface between the die and the blank to reduce friction. Annealing treatment (such as 600-650°C annealing for low-carbon steel) is required between multiple drawing cycles to eliminate work hardening and restore material plasticity. The radii of the punch and die corners should be appropriately designed. For the initial drawing, the die radius R1 should be (5-10) t, and the punch radius r1 should be (0.6-1) R1. These radiuses should be gradually reduced to avoid cracking at the corners. Comprehensive calculation and optimization of process parameters can ensure the forming quality of flangeless cylindrical parts and improve production efficiency.
