Question

A single degree of freedom system has a mass, stiffness, and damping of 200 kg, 20 N/m, and 62 N-s/m respectively. For a forced oscillation system, if the excitation frequency is equal to the undamped natural frequency, then the dynamic magnification factor is  ………….(rounded off to three decimal places).
 

Hint:

For dynamic systems: 1. At resonance (\( r = 1 \)), the dynamic magnification factor simplifies to \( \frac{1}{2 \zeta} \) when damping is very low.
2. When the damping ratio (\( \zeta \)) is sufficiently high, the DMF approaches unity even at resonance.
3. Resonance occurs when the excitation frequency equals the natural frequency.

Answer 1

Step 1: Recall the formula for the dynamic magnification factor (DMF).
The dynamic magnification factor for a forced oscillation system is given by: \[ \text{DMF} = \frac{1}{\sqrt{(1 - r^2)^2 + \left( 2 \zeta r \right)^2}}, \] where: - \( r = \frac{\omega}{\omega_n} \) is the frequency ratio, - \( \zeta = \frac{c}{2 \sqrt{k m}} \) is the damping ratio, - \( \omega_n = \sqrt{\frac{k}{m}} \) is the natural frequency, - \( \omega \) is the excitation frequency. Step 2: Determine the natural frequency and damping ratio.
Given: - \( m = 200 \, \text{kg} \), - \( k = 20 \, \text{N/m} \), - \( c = 62 \, \text{N-s/m} \). The natural frequency \( \omega_n \) is: \[ \omega_n = \sqrt{\frac{k}{m}} = \sqrt{\frac{20}{200}} = \sqrt{0.1} \approx 0.316 \, \text{rad/s}. \] The damping ratio \( \zeta \) is: \[ \zeta = \frac{c}{2 \sqrt{k m}} = \frac{62}{2 \sqrt{20 \cdot 200}} = \frac{62}{2 \sqrt{4000}} = \frac{62}{2 \cdot 63.245} \approx \frac{62}{126.49} \approx 0.49. \] Step 3: Evaluate the dynamic magnification factor.
For \( r = \frac{\omega}{\omega_n} = 1 \) (since \( \omega = \omega_n \)): \[ \text{DMF} = \frac{1}{\sqrt{(1 - r^2)^2 + \left( 2 \zeta r \right)^2}} = \frac{1}{\sqrt{(1 - 1^2)^2 + \left( 2 \cdot 0.49 \cdot 1 \right)^2}}. \] Simplify: \[ \text{DMF} = \frac{1}{\sqrt{(0)^2 + (2 \cdot 0.49)^2}} = \frac{1}{\sqrt{(0.98)^2}} = \frac{1}{0.98} = 1.000. \] Conclusion: The dynamic magnification factor is \( 1.000 \).