Question

An instrument cabinet having a mass of 250 kg is restrained by a mounting that has a lateral stiffness of 1000 kN/meter. The term lateral stiffness means an equivalent spring constant resulting from the mounting details. Earthquakes imposes a lateral acceleration within a frequency range of 0.5 to 12 Hz. Does this cabinet have a natural oscillation frequency that places it at risk for resonant failure in an earthquake? If so, what change to either the mass or the stiffness, or both, would you recommend?

Answer 1

hooke's law

F=-kx ...........................(1)

Second newton's law

F=m.a...........................(2)

simple harmonic motion

x=A sin(wt) ..................(3)

a= - A.w²sin(wt).............(4)

equaling (1) and (2)

-K.x=m.a .....................(5)

inserting (3) and (4) in (5)

-K [ A sin(wt) ]=m.[  A.w²sin(wt) ]

clearing for w

$\omega=\sqrt{\frac{K}{m}}$ .............(6)

w=2. pi.f ..................(7)

equaling 6 and 7

$2\pi.f=\sqrt{\frac{K}{m}}$

clearing for f

$f=\frac{1}{2\pi}\sqrt{\frac{K}{m}}$

$f=\frac{1}{2\pi}\sqrt{\frac{1000 000_{N/m}}{250_{kg}}}=10.065_{hz}$

Yes, this cabinet has a natural oscillation frequency that places it at risk for resonant failure in an earthquake.

1) f > 12

m=250kg

$(12_{hz}*2\pi)^2*250_{kg}=K>1421223N/m$

2) f>12

k=1000000n/m

$m=\frac{1000000_{N/m}}{(12_{hz}*2\pi)^2}<175.90_{kg}$

If is possible decrease the mass of the cabinet below of 175.90kg it could be the solution, else increase the K spring over the 1422 KN/m