Finding Damping Ratio of a System

To find the damping ratio of a system, we need to first understand what it is and how it is related to the transfer function. The damping ratio is a measure of the rate at which the oscillations in a system decay over time. It is denoted by the Greek letter "zeta" (ζ) and is given by the equation:

ζ = c / (2 * sqrt(k * m))

where c is the damping coefficient, k is the spring constant, and m is the mass of the system. In terms of the transfer function, the damping ratio can be found by analyzing the poles of the transfer function.

Given Transfer Function

The transfer function of the system is given by:

G(s) = 1/(s^2 4s 4)

To find the damping ratio, we need to find the poles of the transfer function. The poles are the values of s that make the denominator of the transfer function equal to zero. In this case, the denominator is:

s^2 + 4s + 4 = 0

Using the quadratic formula, we can solve for s:

s = (-4 ± sqrt(16 - 4*4)) / 2

s = -2 ± j

where j is the imaginary unit. The poles of the transfer function are therefore located at s = -2 + j and s = -2 - j.

Finding Damping Ratio

To find the damping ratio, we need to determine how quickly the oscillations in the system decay over time. This can be done by analyzing the location of the poles in the complex plane. The damping ratio is given by the equation:

ζ = -Re(s) / |s|

where Re(s) is the real part of the pole and |s| is the magnitude of the pole. In this case, we have two poles, located at s = -2 + j and s = -2 - j. The real parts of these poles are both equal to -2, and the magnitudes are both equal to sqrt(5).

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

ζ = -(-2) / sqrt(5)

ζ = 0.894

Therefore, the damping ratio of the system is approximately 0.89, which is option C.