Finding Damping Ratio of a System with Given Transfer Function

Given transfer function: G(s) = 1/(s^2 4s 4)

Damping ratio (ζ) is a measure of how much a system is damped in response to a disturbance. It is a dimensionless quantity that ranges from 0 (undamped) to 1 (critically damped) to infinity (overdamped). The damping ratio determines the behavior of the system's transient response, which is the response of the system to a sudden change in input.

To find the damping ratio of the given system, we can use the following formula:

ζ = ξ / √(1-ξ^2)

Where ξ is the natural frequency of the system (in radians per second) and can be calculated as:

ξ = ωn / s

Where ωn is the undamped natural frequency of the system and can be calculated as:

ωn = √(k / m)

Where k is the system's spring constant and m is its mass.

1. Calculate Undamped Natural Frequency

From the given transfer function, we can see that the denominator is in the form of:

s^2 + 4s + 4 = (s+2)^2

Therefore, the system can be represented by the following differential equation:

m*s'' + c*s' + k*s = 0

Where m = 1, c = 4, and k = 4

Using the formula for the undamped natural frequency, we can calculate:

ωn = √(k / m) = √(4 / 1) = 2 radians per second

2. Calculate Damping Ratio

Using the formula for the damping ratio, we can calculate:

ξ = ωn / s = 2 / 4 = 0.5

ζ = ξ / √(1-ξ^2) = 0.5 / √(1-0.5^2) = 0.707

Therefore, the damping ratio of the given system is 0.707 or approximately 0.5.

Answer: A. 0.5