Question $1$:

Consider a simple structure which is idealized as a single$-$degree$-$of$-$freedom $($SDF$)$ system as shown in Figure $1.$ The mass $m$ of the structure is G$000kg$ which is supported by massless columns with combined stiffness $k=16{10}^5\frac{N}{m}.$

$G=6$

Task $1$: Determine the natural cyclic frequency $"f",$ natural circular frequency $"",$ and natural time period $"$T$"$ of the system using analytical expressions derived in class.

Task $2$: determine the change in the natural cyclic frequency $"f"$ of the system if

Case $($a$)$: the mass $m$ is doubled

Case $($b$)$: the stiffness $k$ is reduced to $50\%$ of its original value

Task $3$: Suppose that we want to reduce the natural cyclic frequency $"f"$ of this system to $85\%$ of its actual value. For this purpose, suppose that only the mass $m$ of this system is allowed to be changed. Determine the change in mass $m$ which would result in the required reduction in natural cyclic frequency.

Task $4$: Suppose that the real structure has some tendency to dissipate energy during the oscillations and we are interested in determining the damping coefficient $c$ and critical damping ratio $$ of this system. For this purpose, a free vibration response test is conducted for this system and the resulting response is shown in Figure $2$ below.

Figure $2$: Free vibration response of the SDF system measured from experiment. Determine the damping coefficieny and critical damping ratio.