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?

Correct Answer: Time period of free oscillations will increase.

Detailed Explanation:

Let us analyze the single degree of freedom (SDOF) spring-mass-damper system shown in the first image. The system consists of a mass labeled as M, supported by a spring with stiffness labeled as K and a damper with damping coefficient labeled as C connected in parallel.

For this system, the undamped natural frequency is given by:

${\omega }_n=\sqrt{\frac{K}{M}}$

The damping ratio is defined as:

$\zeta =\frac{C}{2\sqrt{KM}}$

When an additional damper with damping coefficient $C^{\text{'}}$ is added in parallel to the existing damper, the equivalent damping coefficient of the system increases to:

$C_{\mathrm{eq}}=C+C^{\text{'}}$

Consequently, the new damping ratio increases to:

${\zeta }^{\text{'}}=\frac{C+C^{\text{'}}}{2\sqrt{KM}}$

Since both $C$ and $C^{\text{'}}$ are positive, we have ${\zeta }^{\text{'}}>\zeta$. It is also given that the system still remains underdamped, which means ${\zeta }^{\text{'}}<1$.

The damped natural frequency of an underdamped system is defined by:

${\omega }_d={\omega }_n\sqrt{1-{\zeta }^2}$

As the damping ratio increases from $\zeta$ to ${\zeta }^{\text{'}}$, the term under the square root, $1-{\zeta }^2$, decreases. Since the undamped natural frequency ${\omega }_n$ depends only on $K$ and $M$ (which remain unchanged), the damped natural frequency decreases:

${\omega }_d^{\text{'}}<{\omega }_d$

The time period of free oscillations is inversely proportional to the damped natural frequency:

$T_d=\frac{2\pi }{{\omega }_d}$

Since the damped natural frequency decreases (${\omega }_d^{\text{'}}<{\omega }_d$), the time period of free oscillations must increase ($T_d^{\text{'}}>T_d$). This is always true as long as the system remains underdamped.

Why other options are incorrect:
As shown by the transmissibility formula in the second image, transmissibility depends on the excitation frequency ratio $r=\frac{\omega }{{\omega }_n}$:
- For $r<\sqrt{2}$, an increase in damping ratio reduces transmissibility.
- For $r>\sqrt{2}$, an increase in damping ratio increases transmissibility.
Since the change in transmissibility depends on the excitation frequency, neither option stating that transmissibility always increases or always decreases is correct.