In a single degree of freedom undamped spring-mass system, an additional damper is added in parallel such that the system remains underdamped. Which one of the following statements is always true?
Show Hint
- Transmissibility will decrease.
- Transmissibility will increase.
- Time period of free oscillation will decrease.
- Time period of free oscillation will increase.
The Correct Option is D
Solution and Explanation
Step 1: Analyze the frequency change.
The relationship between damped and undamped natural frequency is: \[ \omega_d = \omega_n \sqrt{1 - \zeta^2} \] Where $\zeta$ is the damping ratio. Since the system was undamped ($\zeta = 0$) and then became damped ($0 < \zeta < 1$), the factor $\sqrt{1 - \zeta^2}$ is less than 1. Therefore, $\omega_d < \omega_n$.
Step 2: Determine the effect on the time period.
The time period ($T$) is inversely proportional to frequency: \[ T = \frac{2\pi}{\omega} \] Since the frequency decreases ($\omega_d < \omega_n$), the time period must increase ($T_d > T_n$).
Step 3: Evaluating Transmissibility.
Transmissibility depends on the frequency ratio. Adding damping decreases transmissibility near resonance but increases it at very high frequency ratios ($\omega/\omega_n > \sqrt{2}$), so statements 1 and 2 are not "always" true.
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