Question $4$

$200612007$

A test is usually required to determine the effect of vibration on products and their packaging during transportation. A transportation test rig is constructed and an electromechanical accelerometer constructed with a mass, a spring and a damper designed to measure the amplitude of vertical sinusoidal oscillation as shown in Figure Q$4.$

$($a$)$ Suppose the rotation speed $N$ can be varied from $0-300$ revolutions per minute, calculate the maximum oscillation frequency of the rig in Hz $.$

$(3$ marks$)$

$($b$)$ It is suggested to set the damping ratio of the accelerometer to $0\mathrm{.}707.$

Do you agree? Explain.

$(4$ marks$)$

$($c$)$ Estimate the minimum natural frequency required for the accelerometer in Hz $.$

$(3$ marks$)$

$($d$)$ Given that the spring stiffness $k,$ the natural frequency $f_n($in Hz $),$ the critical damping coefficient $c_c$ are respectively :

$k=5\frac{N}{m},f_n=\frac{1}{2}\sqrt[\phantom{2}]{\frac{k}{m}},$ and $c_e=2\sqrt[\phantom{2}]{mk}$

Estimate the value of the damping coefficient that satisfies the requirement in part $($c$).$

$(6$ marks$)$

$($e$)$ Within the frequency response range, the accelerometer outputs $+-1V$ for an input oscillation of $+-5mm.$ In a transportation test, $N$ is set to $120$ revolutions per minute and the output of the accelerometer is $+-2V.$ Given that the phase lag $$ in measurement is

$=ta{n}^{-1}(\frac{2(\frac{f}{f_n})}{1-(\frac{f}{f_n}{)}^2})$

where $f$ is the input frequency in Hz $.$ Draw a graph showing the input vertical displacement and the output voltage of the measurement for one complete cycle of oscillation.

$(9$ marks$)$

Figure Q$4$

$154$