A displacement transducer has the following characteristics

A displacement transducer has the following characteristics: m = 2.5 Ibm, c = 0.09 lbf - s/in, k = 0.6 lbf/in. Determine the frequency range for the transducer if an error of +/- 5% can be tolerated. What is the time constant for the system? Would you use the tranducer to measure the steady-state response of the following signal x(t) = 10 + 10sin(30t) + 20cos(40t) + 20sin(50t) in

Solution

solution: for given transducer will take input signal as sinusoidal wave and give output on frequency curve as magnitude

1) here equation of motion for given mass spring damper system as

mx\'\'+cx\'+kx=0

2) natural frequency of system as

wn=(k/m)^.5=(.6/2.5)^.5=.489897 rad/s

3) critical damper coefficient as

Cc=2*m*wn=2*2.5*.489897=2.4494

damping ratio=zeta=E=c/Cc=.09/2.4494=.03674

E=.03674

damping frequency as wd=wn(1-E2)^.5=.489566 rad/sec

4) bandwidth of frequecy at which signal level is measurable is

BW=dw=wn(1-2*E^2+(2(E^2-1)^2+1)^.5)^.5

on putting value we get

BW=dw=.809115 rad/sec

5)settling time for to achieve final output with tolerance is

Ts=4/(E*wn)=222.23 sec

6) time constant is time required achieve 63.2% peak time

peak time=pi/wd=6.4170 sec

time constant=.632*6.417=4.055 sec

7) given transducer can only use for periodic input and hence given input signal in second case can not be process by transducer as it has non periodic value