Final answer:
The peak overshoot, rise time, and peak time can be determined for an underdamped system using the damping ratio and natural frequency. The calculations involve applying specific formulas that require knowledge of control systems and vibrations.
Step-by-step explanation:
The question asks to determine the peak overshoot, rise time, and peak time for a damped oscillation system with a damping ratio (ζ) of 0.75 and a natural frequency (ωn) of 12 rad/sec. These values imply that the system is underdamped, as the damping ratio is less than one.
For an underdamped system, the following equations can be used:
- Peak overshoot (%OS) can be calculated using the damping ratio with the equation %OS = e(-ζπ/√(1-ζ2)) × 100%.
- The rise time (tr) is the time it takes for the response to go from 10% to 90% of the steady-state value.
- Peak time (tp) is the time it takes to reach the first peak of the response and can be found using tp = π/ωd, where ωd = ωn √(1-ζ2).
To calculate the numerical values for these parameters, specific formulas that involve the damping ratio and natural frequency must be applied. Since the mentioned equations rely on understanding advanced topics in control systems and vibrations, the calculation may not be straightforward without further study in these areas.