Question

Q3. A mass of 5kg is hung from a tension spring causing an increase in the length of the spring by 18mm. Calculate the following:

a) The period of the vibration.

b) The frequency of the vibration.

c) Determine the maximum velocity and maximum acceleration of the mass when it is pulled down a further 15 mm.

d) If a 15 kg machine is suspended from this spring and the machine exerts an out-of-balance force of 10 N at 5 Hz. Determine the amplitude of the steady state vibration.

Answer 1

a)

$mg=kx\to k=mg/x=5\cdot9.81/0.018=2725\ (N/m)$

$T=2\pi\sqrt{m/k}=2\cdot3.14\cdot\sqrt{5/2725}=0.269\ (s)$

b)

$f=1/T=1/0.269=3.72\ (Hz)$

c)

$v_{max}=A\omega=A\cdot2\pi f=0.015\cdot2\cdot3.14\cdot3.72=0.35\ (m/s)$

$a_{max}=A\omega^2=A\cdot(2\pi f)^2=0.015\cdot(2\cdot3.14\cdot3.72)^2=8.2\ (m/s^2)$

d)

$A=\frac{F}{m\cdot\sqrt{(\omega^2_0-\omega^2)^2}}=\frac{F}{m\cdot\sqrt{((2\pi f_0)^2-(2\pi f)^2)^2}}=\frac{10}{15\cdot\sqrt{({(2\cdot3.14\cdot 3.72)^2-(2\cdot3.14\cdot5)^2})^2}}=0.0015\ (m)$