Question

A typical example of a physical system with under-damped characteristics is a ______

• A. U-tube manometer
• B. two tank non-interacting system
• C. CSTR with first-order reaction
• D. thermocouple kept immersed in a liquid-filled thermowell

Hint:

An under-damped system (\( \zeta \lt 1 \)) is the only class of passive systems that can exhibit overshoot and decaying oscillations in response to a step input.
Low-viscosity U-tube manometers are the classic physical example used to demonstrate this behavior.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks to identify a physical system that displays under-damped, second-order dynamic characteristics from the given options.
This is a standard topic in process dynamics and control.

Step 2: Key Formula or Approach:
A second-order system is described by the transfer function:
\[ G(s) = \frac{Y(s)}{X(s)} = \frac{K_p}{\tau^2 \cdot s^2 + 2 \cdot \zeta \cdot \tau \cdot s + 1} \]
where \( \zeta \) is the damping coefficient.
The system behavior depends on the value of \( \zeta \):
- Under-damped: \( \zeta \lt 1 \) (displays oscillatory behavior).
- Critically damped: \( \zeta = 1 \).
- Over-damped: \( \zeta \gt 1 \) (sluggish, non-oscillatory behavior).

Step 3: Detailed Explanation:

• U-Tube Manometer (A): When a pressure step is applied to a U-tube manometer filled with a low-viscosity liquid (like water), the liquid column oscillates up and down before settling at its new equilibrium position.
This oscillatory behavior is a characteristic of an under-damped second-order system (\( \zeta \lt 1 \)), where fluid inertia overcomes viscous drag.

• Two-Tank Non-Interacting System (B): This is a combination of two first-order systems in series.
The overall transfer function has two real poles, which means the system is inherently over-damped (\( \zeta \gt 1 \)) and cannot oscillate.

• Thermocouple in a Thermowell (D): This is also modeled as a system of two first-order thermal resistances in series, resulting in an over-damped, non-oscillatory response.

• CSTR with first-order reaction (C): This behaves as a first-order system, which cannot exhibit under-damped oscillations.


Step 4: Final Answer:
The U-tube manometer is a typical physical system that exhibits under-damped second-order dynamic characteristics.