A computational study of vibrating system using the constant proportional Caputo derivative operator approach

For modeling of complex systems, in signal processing, and the explorations of nonlinear dynamics and memory effects, the utilization of noninteger-order dynamics provides adaptable control over oscillation patterns and frequencies. Moreover, various waveforms result in unique sonic qualities. By manipulation of these waveforms, musicians and sound designers have the capacity to craft a wide spectrum of auditory experiences, ranging from basic tones to intricate sound scape. This paper is about study of such vibrating systems using the noninteger-order derivative operator approach. In particular, we will discuss fractional relaxation oscillator and Scott-Blair oscillator employing constant proportional Caputo fractional derivative operator. Laplace transform method and Tzou’s numerical inversion algorithm are utilized to solve these vibrating models with respective initial conditions. A thorough graphical analysis is done to discuss the control of noninteger-order parameters and the parameters involved in the definition of fractional derivative operator. Finally, useful conclusions are recorded to highlight the behavior of oscillators and influence of noninteger-order parameters, the parameters involved in the definition of fractional derivative operator and system parameters that are helpful in controlling over oscillation pattern, frequencies and shape of vibrations produced by the vibrating systems.