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?

The correct answer is: Time period of free oscillations will increase.

Detailed Explanation:

Consider a single degree of freedom (SDOF) spring-mass-damper system with mass $M$, spring stiffness $K$, and initial damping coefficient $C$, as shown in the diagram.

The undamped natural frequency of the system is given by:
${\omega }_n=\sqrt{\frac{K}{M}}$
where:
$K$ is the stiffness of the spring, and
$M$ is the mass.

When an additional damper is added in parallel, the equivalent damping coefficient increases:
$C_{\mathrm{eq}}=C+C_{\text{additional}}>C$

The damping ratio ($\zeta$) is given by:
$\zeta =\frac{C_{\mathrm{eq}}}{2\sqrt{KM}}$

Since the equivalent damping coefficient $C_{\mathrm{eq}}$ increases while the mass and stiffness remain constant, the damping ratio $\zeta$ also increases. The problem states that the system remains underdamped, meaning $\zeta <1$.

The frequency of damped free oscillations (${\omega }_d$) is:
${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$

As $\zeta$ increases, the factor $\sqrt{1-{\zeta }^2}$ decreases. Consequently, the damped natural frequency ${\omega }_d$ decreases.

The time period of free oscillations ($T_d$) is defined as:
$T_d=\frac{2\pi }{{\omega }_d}=\frac{2\pi }{{\omega }_n\sqrt{1-{\zeta }^2}}$

Since ${\omega }_d$ decreases, the time period of free oscillations $T_d$ must increase.