Final answer:
To achieve an overshoot of 7% with a settling time of 8 seconds in a mass-spring-damper system, the proportional and derivative gains need to be selected appropriately. The specific values depend on the system parameters and can be found using control system design techniques.
Step-by-step explanation:
An underdamped system is characterized by quick movement towards equilibrium but with oscillation around the equilibrium point. An overdamped system moves slower towards equilibrium without oscillation, while a critically damped system moves as quickly as possible towards equilibrium without oscillating.
In order to achieve an overshoot of 7% with a settling time of 8 seconds, we need to select the proportional and derivative gains appropriately. The proportional gain affects the responsiveness of the system, while the derivative gain helps in damping the oscillations. The exact values of these gains depend on the specific system parameters and can be found using control system design techniques such as root locus analysis or PID tuning methods.
Unfortunately, I cannot provide a specific answer to the question without knowing the exact parameters of the mass-spring-damper system. However, the root locus technique can be used to analyze the stability and performance of the system and determine the appropriate gains. The root locus diagram plots the locations of the closed-loop poles as the proportional gain is varied. The controlled system's root locus diagram should show that the poles lie within the desired region to achieve the desired overshoot and settling time.