Study the Effects of Mass in a Simple Damped Harmonic Motion of a Mass-spring System Using Simulation

The effect of mass on the behavior of oscillatory systems in a damped spring-mass system was studied using simulation. It was found that the mass affects the amplitude and displacement in the case of an undamped oscillatory system. In critically damped systems, the mass affects the displacement exponentially, and the system doesn’t oscillate. In the case of an overdamped system, there is also no oscillatory motion, and an increase in the mass was not affected, since the system gets to rest very quickly. The study shows that simulation can be a very helpful tool to study the behavior of oscillatory physical systems. Keywords: Simple harmonic motion damped mass-Spring system, simulation

Graphical Comparison of Critically Damped Versus Overdamped Free Vibration of Single Degree of Freedom Structure

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.

Vibrational Resonance in an Overdamped System with a Fractional Order Potential Nonlinearity

We investigate the vibrational resonance by the numerical simulation and theoretical analysis in an overdamped system with fractional order potential nonlinearities. The nonlinearity is a fractional power function with deflection, in which the response amplitude presents vibrational resonance phenomenon for any value of the fractional exponent. The response amplitude of vibrational resonance at low-frequency is deduced by the method of direct separation of slow and fast motions. The results derived from the theoretical analysis are in good agreement with those of numerical simulation. The response amplitude decreases with the increase of the fractional exponent for weak excitations. The amplitude of the high-frequency excitation can induce the vibrational resonance to achieve the optimal response amplitude. For the overdamped systems, the nonlinearity is the crucial and necessary condition to induce vibrational resonance. The response amplitude in the nonlinear system is usually not larger than that in the corresponding linear system. Hence, the nonlinearity is not a sufficient factor to amplify the response to the low-frequency excitation. Furthermore, the resonance may be also induced by only a single excitation acting on the nonlinear system. The theoretical analysis further proves the correctness of the numerical simulation. The results might be valuable in weak signal processing.

Time-dependent Kramers escape rate in overdamped system with power-law distribution

The probability distribution of Brownian particles moving in an overdamped complex system follows the generalized Smoluchowski equation, which can be rigorously proven that the exact time-dependent solution for this equation follows Tsallis form. Time-dependent escape rate in overdamped system with power-law distributions is then established based on the flux over population theory. The stationary state escape rate in overdamped system with power-law distribution which has been obtained before based on mean first passage time theory is recovered from time-dependent escape rate as time toward infinity.