This article delves into the applications of fractional calculus, an extension of classical calculus that introduces non-integer derivative orders. The primary focus of this research is to present a methodology for simulating fractional differential equations and to explore the effects of fractional order in two well-known dynamic systems: the Van der Pol and Duffing systems. These systems are known for their nonlinear characteristics and, in certain cases, exhibit complex and rich dynamic behaviors. Initially, the Grunwald-Letnikov definition of fractional derivatives is introduced, followed by the general numerical solution for a fractional differential equation. The Van der Pol system is modified by the inclusion of a fractional time derivative of order q, reducing the integer order of the system to 1+q, while the Duffing system is modified in terms of viscous damping, by the add of a fractional damping, which is now related to the fractional variation in displacement. The dynamics of the systems are characterized using classical methods of nonlinear dynamics, such as time-response, Poincaré sections, bifurcation diagrams and fast Fourier transform, as well as more advanced approaches, such as the continuous wavelet transform (CWT) and the Hilbert-Huang transform (HHT).

In this paper, we propose the development of a low-cost acquisition system using Raspberry-Pi 3 and MEMS accelerometers to measure vibrations in mechanical systems. The main objective is to assemble a signals acquisition system easy to handle, of low cost and good accuracy for teaching and industrial purposes. The central idea is that the signals are acquired and processed in the Raspberry-Pi. The Python language and numerical libraries (scipy, numpy and matplotlib) are used for implementation of acquisition and signal processing. This paper proposes the study of vibrations in time and frequency domain. The acquisition system proposed can be easy implemented and the results obtained had good precision (time and frequency domain) and agree with the literature.

Abstract This paper presents a methodology for the measurement of strain by means of strain-gauges, where the data acquisition is done by using an Arduino Uno board, due to the low-cost and the easy manipulation of this microcontrollers. The measured signal conditioning is performed by means of a Wheatstone Bridge and then discretized by an Analog-Digital Converter (ADC) external to the Arduino. For the validation of the proposed measurement system, experiments are performed on a cantilever beam and on a cantilever-supported shaft, where the experimental results are compared with those obtained analytically and by simulation using the finite element method. The results obtained are in agreement with the literature and demonstrate that the proposed system has satisfactory accuracy.

Abstract In many engineering applications, there are systems that exhibit a series of collisions in a certain amount of time, for example the impact of floating ice in ships and the rubbing between stationary and rotary parts in turbomachinery. In many of these situations, it is of vital importance to know how the impacts affect the overall system behavior for the correct functioning or the prediction of faults in the system. This can be done by proposing models that can represent the system when it is subjected to sudden impacts in its oscillation movement. However, due to the nonlinear and non-smooth characteristic of the impact phenomenon, the implementation of impacts in mechanical systems may be a challenging task. In this manner, this paper aims in proposing a vibro-impact model of two adjacent portal frame structures, being one of them driven by an unbalanced DC motor positioned on top of it. The portal frames were modeled as continuous beams by means of the Euler-Bernoulli beam theory. In order to model the impact between the structures, three different models were considered: using the Hertz theory of elastic bodies to model the contact force using a nonlinear damping; using a spring and a damper in parallel, viscoelastic or spring-dashpot system, to model the force; and considering the impact to be instantaneous and using the coefficient of restitution to adjust the velocity after the contact. To validate the impact models, an experimental procedure was performed and the measurements compared with the simulations.

Abstract In this note, some remarks on the modelling of a multi-degree-of-freedom system, a three-story shear-building structure, is presented. The modelling of a mechanical system is one of the most important aspects in engineering and physics, since is crucial for the understanding of the system. Besides, the model, if properly made, can be also useful to predict some phenomenon that might appear on the system itself and is unknown to researchers. In many situations, the system studied is simplified as a physical model, where the main characteristics regarding the dynamics of the system are defined, such as inertia and stiffness. This facilitates the obtainment of the mathematical model, which is used to obtain quantitative results that can be checked in some experimental procedure. This work presents the modeling of a shear-building structure, where the equations of motion and the natural frequencies of oscillation are obtained. The numeric values are validated by the comparison with experimental results from a previous work.

This paper is aimed at undergraduate students of physics, engineering and mathematics, where a methodology for mechanical vibration analysis of a multi degree of freedom (DOF) excited by an harmonic force in the time and frequency domain is presented. The Arduino microcontroller is used as an acquisition system and low-cost MEMS accelerometers for the instrumentation of the system. System of multi DOF are studied in a great number of problems in mechanical sciences, however its experimental study is not always present in the courses due to the high costs and complexity. These problems are overpassed with the study proposed in this work. Besides, the application presented has an interface with several disciplines in undergraduation and graduation level. The method proposed can be easy implemented and the results obtained had good precision and are in agreement with the literature.

Neste estudo, utiliza-se a plataforma Arduino em conjunto com sensores, como acelerômetro, giroscópio e ultrassom, para medição de vibrações em sistemas mecânicos. O principal objetivo deste trabalho é a montagem de um sistema para aquisição de sinais de fácil manuseio, baixo custo e boa precisão para fins didáticos. Utiliza-se também a linguagem Python e suas bibliotecas numéricas para processamento dos sinais. Neste trabalho propõem-se o estudo de vibrações em uma viga, em que se mede a sua posição, velocidade e aceleração. Uma montagem experimental foi implementada. Os resultados obtidos são comparados com modelos analíticos e de simulação computacional através de elementos finitos. Os resultados estão em concordância com a literatura.

In this paper, we use the Arduino platform together with sensors as accelerometer, gyroscope and ultrasound, to measure vibrations in mechanical systems. The main objective is to assemble a signals acquisition system easy to handle, of low cost and good accuracy for teaching purposes. It is also used the Python language and its numerical libraries for signal processing. This paper proposes the study of vibrations of a beam, which is measured by position, velocity and acceleration. An experimental setup was implemented. The results obtained are compared with analytical models and computer simulations using finite elements. The results are in agreement with the literature.