A mass of 1 kg is suspended from a spring of spring constant 25 Nm–1. If the undamped (or natural) frequency is times the damped frequency, calculate the damping constant in kg s–1.
Question
A mass of 1 kg is suspended from a spring of spring constant 25 Nm–1. If the undamped (or natural) frequency is times the damped frequency, calculate the damping constant in kg s–1.
Solution
The undamped (or natural) frequency (ω_n) of a mass-spring system is given by the formula:
ω_n = sqrt(k/m)
where: k = spring constant = 25 N/m m = mass = 1 kg
So, ω_n = sqrt(25/1) = 5 rad/s
The damped frequency (ω_d) is given by the relation:
ω_n = ω_d * sqrt(1 - ζ^2)
where: ζ = damping ratio
Given that ω_n = 2 * ω_d, we can substitute ω_d = ω_n / 2 into the above equation to get:
5 = (5/2) * sqrt(1 - ζ^2)
Solving for ζ^2 gives:
ζ^2 = 1 - (2/5)^2 = 1 - 4/25 = 21/25
So, ζ = sqrt(21/25) = 0.92
The damping constant (b) is related to the damping ratio by the formula:
b = 2 * ζ * sqrt(k * m)
Substituting the known values gives:
b = 2 * 0.92 * sqrt(25 * 1) = 2 * 0.92 * 5 = 9.2 kg/s
So, the damping constant is 9.2 kg/s.
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