OBRABOTKAMETALLOV Vol. 23 No. 3 2021 MATERIAL SCIENCE EQUIPMENT. INSTRUMENTS 7 1 5 The torques for the second and third cams can be expressed as follows: 2 1 2 3 1 3 ( ) ( ), ( ) ( ), Ì M Ì M φ = φ + ψ φ = φ + ψ (6) where M2 and M3 are the torques for the second and third cams; ψ1, ψ2 are the phase displacement angles for the cams on the drive shaft. The torques on the drive shaft (cam) were determined based on the phase shift magnitudes of the angles ψi and profile angles βᵢ. The total torque Mc on the drive shaft was determined as the sum of the component torques Mᵢ, of which there are three in this case: 3 1 c i i M M = = ∑ . (7) The power consumption applied to the drive shaft can be expressed as [34]: c P = M ω max , (8) where Mcmax is the maximum value of the total torque. The profile angles β were determined iteratively using the programs shown in Figs. 3 and 5. As a result, by identifying the minimum velocity value from the family of curves, we selected simple harmonic motion law with profile angles β₁ = 280° and β₂ = 80° and initiated the design of the cam mechanism according to the specified parameters [12]. Furthermore, the pressure angles were determined using the following Equation: ( ) 1 ( ) tan ( ) v e a sh s φ + δ φ = + φ . (9) To achieve this, it was necessary to find the cam’s minimum radius, Rmin. A graph was constructed in v(s) coordinates using the calculated position functions s(φ) and velocity analogs v(φ) (Fig. 6) during the ascent and descent phases. The only difference in this process was the use of numerical velocity values. The resulting scheme is shown in Fig. 7. The velocity vectors are indicated by points from A0 to As. The ascent phase is represented by points from A0 to A4, and the descent phase, from A5 to As. The minimum cam radius vector was determined to be Rmin = 90 mm. The next step in determining the design parameters of the mechanism is to determine the roller radius. This can be found using an algorithm that calculates the radii of curvature for the cam profile based on its rotation angle. The curvature radius values, ρcur, can be calculated using a general formula applicable to any cam type [23]: 2 3/2 2 2 2 2 2 , 2 cur d d d d d d ρ ρ + β ρ = ρ ρ ρ + − ρ β β (10) where p is the radius vector of the theoretical profile; β is the profile angle. Fig. 4. Listing of the program for determining the torque on the cam shaft 1 3 2 1 sin 0 0 1 0 ( 360 deg k F if if k F if if 1 1 1 1 2 1 2 1 2 1 2 1 2 3 3 1 2 3 π φ ⋅ ⋅ π ⋅ ⋅ ≤ φ ≤ φ φ φ φ ≤ φ ≤ φ + φ φ − φ + φ φ − φ + φ ⋅ − ⋅ φ + φ ≤ φ ≤ φ + φ + φ φ φ + φ + φ ) ≤ φ ≤ v(φ) = (5)
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