Final answer:
The rise time for a system with a damping ratio of 0.6 and a natural frequency of oscillation of 8 rad/sec cannot be calculated without a specific formula for rise time about the natural frequency and the damping ratio. Without the formula, generic calculations can only provide an estimate. The rise time is related to the period it takes for an underdamped system's response to reach a certain percentage of its final value during transient conditions.
Step-by-step explanation:
The student asks for the rise time of a system with a damping ratio of 0.6 and a natural frequency of oscillation of 8 rad/sec. The rise time is a parameter that indicates the time it takes for a system's response to go from 10% to 90% of its steady-state value (or from 0 to 100% in some definitions) during the transient phase of its step response. To calculate the rise time for an underdamped system, a specific formula related to the natural frequency (ωn) and the damping ratio (ζ) must be used. However, the exact formula was not provided in the details, and typically, such formulas are found in control systems or vibrations textbooks. For underdamped systems, which this system is due to its damping ratio being less than 1, the rise time is often approximated using rise time ≈ (1.8 to 2.2) / (ωd), where ωd is the damped natural frequency. Unfortunately, without a specific formula, an accurate calculation cannot be provided.
We can discuss that the natural frequency is the frequency at which a system oscillates under free vibration and that damping reduces the amplitude of oscillation over time. The damping ratio describes the extent of damping about critical damping. A damping ratio lower than 1, like 0.6 in this case, indicates an underdamped system that will oscillate with decreasing amplitude over time. However, this damping ratio doesn't make the system come to rest at a single equilibrium point without crossing it, unlike an overdamped system. These concepts are integral to understanding how damping affects a system's oscillatory motion and response to external forces, particularly in forced oscillations and resonance situations.