Answer to Question #182143 in Mechanical Engineering for Emmanuel Osei

Question

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.

Answer 1

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

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

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

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:

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