At the manufacturing plant where you work, there is a machine (mass 1000 kg) on
ID: 1717944 • Letter: A
Question
At the manufacturing plant where you work, there is a machine (mass 1000 kg) on a platform (see figure). The platform is 1.5 m high and supported by four identical hollow cylindrical legs with outer diameters of 4 cm, wall thicknesses of 0.5 cm, elastic module of 200 GPa, endurance limit of 200 MPa, damping ratio of 0.05, and effective length K_et = 2. You have figured out how to speed up the process and estimate that the new (higher) force will have a magnitude of 5,000 N and a frequency of 75 Hz. You need to check that the platform can survive the increased loading. Specifically, the loading in the legs must remain below the endurance limit and they cannot buckle. The critical load for buckling is What is the maximum stress in each leg? Is it safe to speed the manufacturing process up?Explanation / Answer
Machine Mass = 1000kg
height of the cylinders = h = 1.5m
Hollow cylinder outer diameter = 4 cm
cylinder inner diamter = 3 cm
elasic modulii = E = 200 GPa
endurance limit = 200 Mpa
Dmaping ratio = 0.05
Effective length = Kel = 2
Magnitude of force = 5000N
Frequency f = 75 Hz
Moment of inertia = I = pie x D^4 -d^4 / 64
= 3.14 x 0.04^4 - 0.03^4 /64
= 8.595 x 10^-8 m^4
Critical load = Pcr = Pie ^2 ExI /( Kel L )^2
3.14 ^2 x 200x 10 ^9 x 8.595 x 10^-8 / (2 x 1.5)^2
= 18829.1 N
Maximum Stress = Load / Area of cross section
18829 / 4 x pie /4 X 0.04^2 - 0.03^2
= 8566 x 10^3 KN
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