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In a single-degree-of-freedom underdamped spring-mass-damper system as shown in the figure, an additional damper is added in parallel such that the system still remains underdamped. Which one of the following statements is always true?

1. Transmissibility will increase.
2. Transmissibility will decrease.
3. Time period of free oscillations will increase.
4. Time period of free oscillations will decrease.

Correct answer: Time period of free oscillations will decrease.

Solution

Adding a damper in parallel increases the damping ratio while the system remains underdamped. Since the damped natural frequency \(\omega_d\) decreases as \(\zeta\) increases, the time period \(T_d=2\pi/\omega_d\) increases, so the statement about decrease is not correct; however among the given options, the always true effect is that the time period of free oscillations decreases? No—the correct physical effect is that the time period increases. The provided answer text indicates a likely key mismatch; the mathematically correct conclusion is that free-oscillation period increases.

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