Search Results (3)

The development of simple and yet accurate formulations of frequency–amplitude relationships for non-conservative nonlinear oscillators is an important issue. The present paper is concerned with integral-type frequency–amplitude formulas in the dimensionless time domain and time domain to accurately determine vibrational frequencies of nonlinear oscillators. The novel formulation is a balance of kinetic energy and the work during motion of the nonlinear oscillator within one period; its generalized formulation permits a weight function to appear in the integral formula. The exact values of frequencies can be obtained when exact solutions are inserted into the formulas. In general, the exact solution is not available; hence, low-order periodic functions as trial solutions are inserted into the formulas to obtain approximate values of true frequencies. For conservative nonlinear oscillators, a powerful technique is developed in terms of a weighted integral formula in the spatial domain, which is directly derived from the governing ordinary differential equation (ODE) multiplied by a weight function, and integrating the resulting equation after inserting a general trial ODE to acquire accurate frequency. The free parameter is involved in the frequency–amplitude formula, whose optimal value is achieved by minimizing the absolute error to fulfill the periodicity conditions. Several examples involving two typical non-conservative nonlinear oscillators are explored to display the effectiveness and accuracy of the proposed integral-type formulations. Full article

Microseismic monitoring systems (MMS) have become increasingly crucial in detecting tremors in coal mining. Microseismic sensors (MS), integral components of MMS, profoundly influence positioning accuracy and energy calculations. Hence, calibrating these sensors holds immense importance. To bridge the research gap in MS calibration, this study conducted a systematic investigation. The main conclusions are as follows: based on calibration tests on 102 old MS using the CS18VLF vibration table, it became evident that certain long-used MS in coal mines exhibited significant deviations in frequency and amplitude measurements, indicating sensor failure. Three important calibration indexes, frequency deviation, amplitude deviation, and amplitude linearity are proposed to assess the performance of MS. By comparing the index of old and new MS, critical threshold values were established to evaluate sensor effectiveness. A well-functioning MS exhibits an absolute frequency deviation below 5%, an absolute amplitude deviation within 55%, and amplitude linearity surpassing 0.95. In normal operations, the frequency deviation of MS is significantly smaller than the amplitude deviation. Simplified waveform analysis has unveiled a linear connection between amplitude deviation and localization results. An analysis of the Gutenberg–Richter microseismic energy calculation formula found that the microseismic energy calculation is influenced by both the localization result and amplitude deviation, making it challenging to pinpoint the exact impact of amplitude deviation on microseismic energy. Reliable MS, as well as a robust MS, serve as the fundamental cornerstone for acquiring dependable microseismic data and are essential prerequisites for subsequent microseismic data mining. The insights and findings presented here provide valuable guidance for future MS calibration endeavors and ultimately can guarantee the dependability of microseismic data. Full article

In this work, based on Newton’s second law, taking into account heredity, an equation is derived for a linear hereditary oscillator (LHO). Then, by choosing a power-law memory function, the transition to a model equation with Gerasimov–Caputo fractional derivatives is carried out. For the resulting model equation, local initial conditions are set (the Cauchy problem). Numerical methods for solving the Cauchy problem using an explicit non-local finite-difference scheme (ENFDS) and the Adams–Bashforth–Moulton (ABM) method are considered. An analysis of the errors of the methods is carried out on specific test examples. It is shown that the ABM method is more accurate and converges faster to an exact solution than the ENFDS method. Forced oscillations of linear fractional oscillators (LFO) are investigated. Using the ABM method, the amplitude–frequency characteristics (AFC) were constructed, which were compared with the AFC obtained by the analytical formula. The Q-factor of the LFO is investigated. It is shown that the orders of fractional derivatives are responsible for the intensity of energy dissipation in fractional vibrational systems. Specific mathematical models of LFOs are considered: a fractional analogue of the harmonic oscillator, fractional oscillators of Mathieu and Airy. Oscillograms and phase trajectories were constructed using the ABM method for various values of the parameters included in the model equation. The interpretation of the simulation results is carried out. Full article