Question

The pressure tap on the surface of a heat exchanger tube is connected via flexible tubing to a pressure transducer. Both the tubing and the transducer behave as second-order systems. The natural frequencies are 30 rad/ s for the tubing and 6280 rad/ s for the transducer. The damping ratios are 0.45 for the tubing and 0.70 for the transducer.

(a) Determine the magnitude ratio and the phase lag for the system when subjected to a sinusoidal forcing having a 100 Hz (200π rad/s) frequency.
(b) Is the tubing or the transducer acceptable for the system? If not, why?

Answer 1

Tube MR = = 2.2851 x 10⁻³, Phase lag = 2.467°, Transducer MR = 1, Phase lag = 8.049°, b. System is acceptable, explained further

Step-by-step explanation:

a. Magnification Ratio = 1/[√{(1 – (W/W(n))²)² + (2εW/W(n))²}]

For tubing, W(n) = 30rad/s, ε = 0.45, W = 2π x 100 = 628 rad/s

M.R = 1/√[{1 – (628/30)²}² + (2 x 0.45 x 628/30)²]]

= 1/√(191147.7262 + 354.9456)

= 2.2851 x 10⁻³

Phase lag = tan-1 [(2εW/W(n))/1 – (W/W(n))2²]

= tan⁻¹ (18.84/437.204) = 2.467 °

For the transducer, W(n) = 6280 rad/s, W = 628 rad/s, ε= 0.70

M.R = 1/√[{1 – (628/6280)²}² + (2 x 0.7 x 628/6280)²}]

= 1/√[{1 – (628/6280)²)² + (2 x 0.7 x 628/6280)²]

= 1/√[(99/100)² + 0.142] = 1

Phase lag = tan⁻¹ [(2εW/W(n))/1 – (W/W(n))2]

Phase lag = tan⁻¹ [(2 x 0.7 x 628/6280)/1 – (628/6280)²]

Phase lag = tan⁻¹ (0.14/0.99) = 8.049°

The system is working far away from resonance point, so it can be used without any problem

b. (30 x 0.45) (tube) < (6280 x 0.7) (transducer)

This means that the transducer will come to a vibration- less state much quicker than the tube, hence, the pressure measuring system will be quick and accurate. It shows the good suitability of the taken system to the working system.