(Solved): Calculate and draw the two stage reducer under given conditions. Input Power: Pg [kW];Input speed n ...

Expert Answer

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1- Gear Calculations: a) Velocity Ratio Calculation: First, we need to calculate the velocity ratio for each stage of the reducer. Since we have a two-stage reducer, we can divide the total velocity ratio into two parts: [itop] = (i1) x (i2) where i1 is the velocity ratio of the first stage, and i2 is the velocity ratio of the second stage. We can select i1 and i2 based on the design recommendations. For example, we can choose i1 = 3 and i2 = 4, which gives a total velocity ratio of 12. b) Gear Teeth Calculation: Once we have the velocity ratio for each stage, we can calculate the number of teeth for each gear. Let's assume we have a spur gear reducer with a pressure angle of 20 degrees and a gear ratio of 1:1.5. We can use the following formula to calculate the number of teeth: N = (i x Pg x 1000) / (ng x m) where N is the number of teeth, Pg is the input power in kW, ng is the input speed in rpm, and m is the module. We can use this formula for each stage of the reducer. c) Gear Dimensions Calculation: Once we have the number of teeth for each gear, we can calculate the gear dimensions. The gear dimensions include the pitch diameter, the addendum, the dedendum, the whole depth, and the tooth thickness. We can use the following formulas to calculate these dimensions: Pitch Diameter: D = N x m Addendum: ha = m Dedendum: hf = 1.25 x m Whole Depth: h = hf + ha Tooth Thickness: s = ? x m / 2 We can use these formulas for each gear of each stage. 2- Shaft Calculations: a) Bending and Torsional Moment Calculation: Once we have the gear dimensions, we can calculate the bending and torsional moments on each shaft. We can use the following formulas to calculate these moments: Bending Moment: Mb = Fr x a Torsional Moment: Mt = (Pg x 1000) / (2? x ng) where Fr is the radial force on the gear, a is the distance between the gear and the bearing, and Pg and ng are the input power and speed, respectively. We can use these formulas for each gear of each stage. b) Shaft Pre-Sizing and Pre-Shaping: Once we have the bending and torsional moments, we can pre-size and pre-shape the shafts. We can use the formulas for the bending and torsional stress of a solid shaft to calculate the required diameter and shape: Bending Stress: ?b = Mb / Zb Torsional Stress: ?t = Mt x (d / 2) / J where Zb is the section modulus of the shaft, d is the diameter of the shaft, and J is the polar moment of inertia of the shaft. We can use these formulas for each shaft of each stage. c) Shaft Strength Check: Once we have the pre-sized and pre-shaped shafts, we need to check their strength. We can use the formulas for the bending and torsional stress of a solid shaft, as well as the formulas for the bending and torsional stress concentration factors, to check the strength of each shaft: Bending Stress Concentration Factor: Kf Torsional Stress Concentration Factor: Kt Bending Stress: ?b = (Mb x Kf) / Zb Torsional Stress: ?t = (Mt x KOnce we have calculated the bending and torsional stress, we need to compare them to the allowable stresses for the material used in the shafts. If the calculated stresses are less than the allowable stresses, then the shafts are considered to be strong enough. Otherwise, we need to resize the shafts and recalculate the stress.

3- Bearing Calculations:

a) Equivalent Radial and Axial Forces Calculation:

Once we have calculated the bending and torsional moments, we can calculate the equivalent radial and axial forces on each bearing. We can use the following formulas to calculate these forces:

Equivalent Radial Force: Fr = Mb / a

Equivalent Axial Force: Fa = (Pg x 1000) / (2? x ng)

where Mb is the bending moment on the gear, a is the distance between the gear and the bearing, Pg and ng are the input power and speed, respectively.