In the planetary gear train shown in the figure, the input power is transferred to two gears; the external central gear (the sun gear) and the internal gear (the ring gear). These gears rotate with angular speeds 𝜔𝑠 and 𝜔𝑟. The positive sense of rotation is considered as the clockwise rotation. The compound planets are mounted on the same shaft rotating with the same angular speed. One set of planet gears (planet 1) are meshed with the sun gear; another set of planet gears (planet 2) are meshed with the ring gear (𝜔𝑝1=𝜔𝑝2=𝜔𝑝). The arm (or carrier) (not shown) is connected the shafts of the planet gears and it moves with the same speed as their centers. Power is taken from the arm. (a) Derive a relation between the angular speed of the arm, 𝜔𝑎 and the input speeds (𝜔𝑠 and 𝜔𝑟) in terms of the number of teeth in the gears. Show your steps clearly. (b) Consider the case in which the ring is fixed. Find the tangential forces between the sun and planet 1, between the ring and planet 2, and also find the force applied by planet 2 to the arm in terms of the input power, 𝑊̇, pitch diameters, and the speeds. Show your steps clearly.
(c) Consider the case in which the ring is fixed. Develop a computer program to design for proper gearing such that the transmitted power is maximum. The design requires a bending safety factor at least 1.5 and a surface safety factor of at least 1.2 with 99 % reliability. Safe use of 1108 revolutions of the arm is required. Power is taken from an electric motor and delivered to a punch press through this transmission. The sun is rotated at 3200 pm. The arm is required to rotate at a speed between 100 and 200 rpm. The teeth have standard involute profile with 20 pressure angle. All the gears have the same module and face width. Power loss is negligibly small. Mountings correspond to good industrial practice but not high precision. Teeth are finished by average quality hobbing. HB of the steel material is between 200 and 400. There should be no interference between any meshed gear pairs. This transmission must fit within a housing having a maximum inner diameter of 470 mm. There should be 5 mm clearance between the gears and the housing. The maximum pitch line velocity should be less than 25 m/s. Use SI units. Print out your program and output file. The output file should include the following in the given order: Power transmitted, material (HB), input speed (𝑛𝑠 (rpm)), output speed (𝑛𝑎 (rpm)), module, pitch diameters of the gears (𝑑𝑠,𝑑𝑝1,𝑑𝑝2,𝑑𝑟), number of teeth of the gears (𝑁𝑠,𝑁𝑝1,𝑁𝑝2,𝑁𝑟), inner diameter of the housing, speeds of the sun (𝑛𝑠 (rpm)) and the planets (𝑛𝑝 (rpm)), the speed of the arm (𝑛𝐴 (rpm)), pitch line velocity of the sun (𝑣𝑠𝑝), tangential force between the sun and planet 1 (𝐹𝑠𝑝1), tangential force between the ring and planet 2 (𝐹𝑝𝑟), the force applied by planet 2 on the arm (𝐹𝑎𝑟𝑚), face width (𝑏), contact ratio (CR) between the sun and planet 1 and between the ring and planet 2, interference check for these pairs of gears (𝑟𝑎𝑝.𝑚𝑎𝑥 - 𝑟𝑎𝑝 and 𝑁 - 𝑁𝑚𝑖𝑛 comparisons), specify the most critical gear. Give the following quantities for the most critical gear (having the lowest safety factor): 𝐽, 𝐾𝑣, 𝐾𝑚, 𝐾𝑜, bending stress (𝑏), 𝑆𝑛′ , 𝐶𝐿, 𝐶𝐺, 𝐶𝑆, 𝑘𝑟, 𝑘𝑚𝑠, endurance limit (𝑆𝑛), safety factor against fatigue failure due to bending stresses, 𝐶𝑝, 𝐼, surface fatigue stress (𝐻), 𝑆𝑓𝑒, number of contacts that its teeth make contact during 1108 revolutions of the arm, 𝐶𝐿𝑖, 𝐶𝑅, surface fatigue strength (𝑆𝐻), safety factor against surface failure. The computer program should read the inputs from an input file or within the program and find the values of design variables without user intervention through iterations. (d) Write a report containing the analysis of the optimum design by hand calculations.
Hint: 1. The life requirement is given as the number of rotations of the driven shaft. However, the number of times a gear tooth is subjected to contact forces may be different. 2. In evaluating the 𝐾𝑣 factor, the pitch line velocity, 𝑣, can be taken as the average of the maximum pitch line velocities of the engaged gears. The same value for 𝑣 should be used for both engaged gears. Besides, the limit on the maximum allowable pitch line velocity can be checked using the average value.
İşin Yapılacağı Konum: ONLINE
Görevin Başlangıç Tarihi: 17-11-2021
Görevin Bitiş Tarihi: 21-11-2021
Kategori: Ders / Çeviri