Question 6.6: A sensitive instrument that requires to be insulated from vi...
A sensitive instrument that requires to be insulated from vibration is to be installed in a laboratory where a reciprocating machine is in use. The vibrations of the floor of the laboratory may be assumed to be a simple harmonic motion having a frequency in the range 1000 to 3000 cycles per minute. The instrument is to be mounted on a small platform and supported on four springs arranged to carry equal loads. The combined mass of the instrument and supporting table is 5 kg. Calculate a suitable value for the stiffness of each spring if the amplitude of transmitted vibrations is to be less than 15% of the floor vibrations over the given frequency range. Assume that the damping is negligible.
The frequency ratio should be larger than \sqrt{2} for TR to be less than 1 . The transmission ratio is then given by
\mathrm{TR}=\frac{1}{\beta^{2}-1}
The condition that TR be less than 0.15 leads to
\beta^{2}>\frac{1}{T R}+1=\frac{1}{0.15}+1=7.67
Hence the natural frequency f_{0}<f / \sqrt{7.67}=f / 2.77. The governing value of f will be the lower limit of the range of exciting frequency, that is, equal to 1000 cycles per minute. Hence
\begin{aligned}f_{0} &=\frac{1000}{60} \times \frac{1}{2.77} \\&=6.02 \mathrm{~Hz}\end{aligned}
The total stiffness is now obtained from
\frac{1}{2 \pi} \sqrt{\frac{k}{m}}=6.02
or
\begin{aligned}k &=(6.02 \times 2 \pi)^{2} \times 5 \\&=7154 \mathrm{~N} / \mathrm{m}\end{aligned}
The stiffness of each spring is 1 / 4 \times 7154=1788 \mathrm{~N} / \mathrm{m}.