Question

Problem 5: A measurement system can be modeled by using the following second-order ODE: j(t)2)4y(t)= 8U (t) - (a) Determine the natural frequency, damping ratio, and sensitivity of the system. (b) If the system is used to measure a harmonic signal U(t) 5sin(5t), calculate the magnitude ratio and phase shift (c) What is the range of the signal frequencies that this device can measure so that the resulting dynamic error is within ± 20 %?

Answer 1

a) Comparing the given second order differential equation with the general form:

dxо + ayхo bot а?. х +ат- аz dt2 dt

We have a₂ = 1 and a₁ = 2 and a₀ = 4 and b₀ = 8

Then natural frequency =

= 2rad/s ao wn a2

Damping ratio =

0.5 2Vaga 2/1(4)

Sensitivity =

К - 2 о 1 з19

b) The magnitude ratio is given as:

(1-(2)(2( w 2 wn Wn

V1-(3))(2(0.5)()

M 0.17

Phase shift:

tan1(2 Wn w

-25.5

c) Dynamic error is to be within

E20%

$-0.2 \leq M(\omega) - 1\leq 0.2$

0.8 M(w) 1.2

0.8 V(1 (2)(2( 1.2 Wn wn

On solving the above equation we get:

(Range of frequencies for the dynamic error to be within 20%)

1.414w2.36