Abstract
In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction.
Highlights
The theory of fractional order calculus (FOC) dates back to the birth of the theory of differential calculus, but its inherent complexity delayed the application of its associated concepts
Besides the graphs containing the simulations of the angular positions for all the three cases of the simple pendulum, it is presented in Figures 7, 8 and 9 the pseudo-phase portraits for Case B, with different values of α
The obtained results point to curious and instigating aspects of the effects that arise from using fractional orders in the differential equations that represent the dynamics of the studied systems
Summary
The theory of fractional order calculus (FOC) dates back to the birth of the theory of differential calculus, but its inherent complexity delayed the application of its associated concepts. Fractional calculus is a natural extension of classical mathematics. Perhaps, this backwardness is due to the FOC’s inherent complexity and to the current lack of meaning regarding its physical and geometric interpretation. It is worth mentioning that FOC can count on an additional degree of freedom since the order of the derivatives can be arbitrary changed to match a specific behavior. This advantage may enable the FOC to represent systems with high order dynamics and complex nonlinear phenomena, making use of only a few coefficients.
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