If the natural frequency of a damped oscillator is 5 rad/s and the damping ratio

If the natural frequency of a damped oscillator is 5 rad/s and the damping ratio is 0.2, what is the damped frequency?

If the natural frequency of a damped oscillator is 5 rad/s and the damping ratio is 0.2, what is the damped frequency?

Step 1: Identify the natural frequency (ω_n) of the damped oscillator, which is given as 5 rad/s.
Step 2: Identify the damping ratio (ζ), which is given as 0.2.
Step 3: Use the formula for damped frequency (ω_d): ω_d = ω_n * √(1 - ζ^2).
Step 4: Calculate ζ^2: 0.2^2 = 0.04.
Step 5: Calculate (1 - ζ^2): 1 - 0.04 = 0.96.
Step 6: Take the square root of 0.96: √(0.96) ≈ 0.98.
Step 7: Multiply the natural frequency by the square root: ω_d = 5 * 0.98 ≈ 4.8 rad/s.

Natural Frequency – The frequency at which a system oscillates when not subjected to any external force.
Damping Ratio – A measure of how oscillations in a system decay after a disturbance.
Damped Frequency – The frequency of oscillation of a damped system, which is lower than the natural frequency due to the effects of damping.