Answer to Question #182143 in Mechanical Engineering for Emmanuel Osei

Question #182143

Consider a small spring about 30mm long, welded to a stationary table (ground) so that it is fixed at the point with a 12mm bolt welded to the other end which is free to move. The mass of this system is about 49.2×10–³kg . The spring stiffness is measured to be k=857.8N/m. Calculate the natural frequency and period. Also determine the maximum amplitude of the response if the spring is initially deflected 10mm.


1
Expert's answer
2021-04-21T07:05:25-0400

"k = 857.8 \\frac{N}{m}"

"m = 49.2 \\times 10^-3 kg"

"c = 0.11\\frac{kg}{s}"

a) For the "f_n" :

"f_n = \\frac{1}{2\\pi} \\sqrt{\\frac{k}{m}} =\\frac{1}{2\\pi} \\sqrt{\\frac{857.8}{49.2 \\times 10^{-3}}} = 21.02Hz"

for T

"T = \\frac{1}{f_n} =0.0475 sec"

b) Maximum Amplitude :

"C = x_0 = 10"

c) Damping Ratio:

"\\xi=\\frac{c}{c_c} = \\frac{c}{2\\sqrt{km}}=\\frac{0.11}{2\\sqrt{857.8\\times49.2\\times10^{-3}}} = 0.0085"



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