Answer

Solved Step by Step with Explanation: Determine the coefficients of spring.

To determine the coefficients of the spring (k) and damper (b) in the mechanical system, we can use the steady-state response and the initial slope of the curve.

A is the amplitude of the steady state response

ζ is the damping ratio

ζ = (1 / (2 * sqrt(k * m))) * slope

where m is the mass of the system, and in this case, we assume it to be 1 (arbitrary units).

T = 1 / ωd

From the given data, we can observe that the time period (T) is approximately 2020 seconds. Thus:

Substituting the values we have:

ωn = (1 / 2020) * sqrt(1 - ζ^2)

k = [(1 / 2020) * sqrt(1 - ζ^2)]^2

b = 2 * ζ * [(1 / 2020) * sqrt(1 - ζ^2)]

Using the initial slope of 0.1, we have:

0.1 = (1 / (2 * sqrt(k)))

k = 25

Next, let's calculate the damped frequency ωd:

ωn = ωd * sqrt (1 - ζ^2)

ωn = (1 / 2020) * sqrt (1 - (1 / (2 * sqrt(k)))^2)

ωn = (1 / 2020) * sqrt (99/100)

ωn = (1 / 2020) * (sqrt (99) / sqrt (100))

b = 2 * ζ * ωn

b = 2 * (1 / (2 * sqrt(k))) * [(1 / 2020) * (sqrt(99) / 10)]

Next, we calculate b:

b = 2 * (1 / (2 * sqrt(k))) * [(1 / 2020) * (sqrt(99) / 10)]

b = 1 / 10

Therefore, the coefficients of the spring and damper are: