$\frac{\text{A}\text{n}\text{n}\text{a}\text{m}\text{i}\text{c}\text{A}\text{n}\text{a}\text{l}\text{y}\text{s}\text{i}\text{s}(4)}{\frac{2005}{2006}}$

istion $4$ The mechanical model of an ultrasound piezoelectric sensor is shown in Figure Q$4.$ Mass m is

$0\mathrm{.}001$ kg $,$ the stiffness of the spring $k$ is $6000k\frac{N}{m}m$ and the coefficient of the damper $c$ is $10\frac{N}{m}{s}^{-1}.$

The effective sensing surface area of The effective sensing surface area of the sensor is $300m{m}^2.$

$($a$)$ Determine the natural frequency $($in Hz $)$ of the sensor$.$

$(3$ marks$)$

$($b$)$ Given that the critical damping coefficient $c_c$ is

$c_c=+2\sqrt[\phantom{2}]{mk}$

Determine the damping ratio $$ of the sensor$.$

$(3$ marks$)$

$($c$)$ Hence, estimate the frequency response range of the sensor $($in Hz $).$

$(3$ marks$)$

$($d$)$ It is suggested to widen the frequency response range by reducing the stiffness of the sensor material. Do you agree? Explain.

$(3$ marks$)$

$($a$)$ The electro$-$mechanical coupling effect of the sensor is modelled by$,$

where

$Q(t)=Hx(t)$

$Q(t)$ is the quantity of charge

$H$ is the piezoelectric constant

$x(t)$ is the displacement

For static pressure measurement, the sensor is connected to a charge amplifier and the gain being set to $1\mathrm{.}00\frac{V}{p}C.$ After digitization, a computer is programmed to acquire and convert the amplifier output to a numerical value of pressure in Pascal $($ Pa $).$

$($i$)$ Draw the block diagram of the system and state clearly all quantities, sensitivities $($if known$),$ and units.

$(10$ marks$)$

$($ii$)$ Given that $H=10p\frac{C}{}m,$ what is the value of the scale factor being used in the computer programme?

$(3$ marks$)$