Theoretically, solutions of the damped harmonic oscillator asymptotically approach equilibrium, i.e., the zero energy state, without ever reaching it exactly, and the critically damped solution approaches equilibrium faster than the underdamped or the overdamped solution. Experimentally, the systems described with this model reach equilibrium when the system's energy has dropped below some threshold corresponding to the energy resolution of the measuring apparatus. We show that one can (almost) always find an optimal underdamped solution that will reach this energy threshold sooner than all other underdamped solutions, as well as the critically damped solution, no matter how small this threshold is. We also comment on one exception to this for a particular type of initial condition, when a specific overdamped solution reaches the equilibrium state sooner than all other solutions. We experimentally confirm some of our findings.
In most vibration structural problems, the value of damping is less than unity. Such a small amount of damping may increase near or exceed unity under certain special circumstances. Critically damped and overdamped solutions are completed until the final expressions are generated and an indication provided by MATLAB as to how these expressions depend on viscous damping ratios, natural frequencies, and initial conditions. The developed equations of various damping systems, which are commonly employed in vibration analyses, are compared, with several important observations are noted. Natural frequency is of primary importance when controlling the settling time of critically damped and overdamped vibration responses. Initial conditions are also considered main factors that affect critically damped and overdamped vibration peak responses. Damping plays a crucial role in the peak response of an overdamped system. A direct relationship between the damping ratio and the peak response is observed, whereas an inverse relationship exists between the damping ratio and the settling time. Therefore, critically damped and overdamped systems exhibit an identical response in the large scale perspective, whereby they first rise and then fall. In the zoomed scale, the peak response of the overdamped system is lower than that of the critically damped system, and the latter falls faster than the former. No cyclic response is observed, and the vibration statement is abnormally used for both critically damped and overdamped systems.
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