Mathematical Problems in Engineering

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix , where denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.

The gearbox seal system is a critical component of the electrical multiple units drive system, with a direct impact on the safety and reliability of train operations. This study intends to analyze how structural parameters of the gearbox seal system influence the seal performance from the viewpoint of “oil–air separation and scavenge oil” by using a discrete phase model and volume of fluid. The results show that the oil–air separation performance is mainly affected by the oil droplets’ inertia force and the airflow traction force, and the scavenge oil performance is decided by the oil–air separation performance of each seal chamber. The relationships between seal chamber size, axial seal clearance width, height difference, scavenge oil hole diameter, and seal performance are analyzed, and the study also found that optimizing the stator chamber depth–width ratio of the seal system studied in this paper at 2.5, reducing the rotor chamber depth–width ratio and axial seal clearance width, and increasing the relative height difference can improve the seal performance. And increasing the scavenge oil hole diameter also enhances the seal system’s performance. The simulation results can be used as design references for gearbox seal systems.

The motivation of this study is to develop effective and economical assistive technologies for people with physical disabilities. The novelty in this manuscript is the application of the average value-based technique to accurately represent the involved biomechanics of the lower limb joints during the human gait cycle. This mathematical formulation of lower limb joints’ biomechanics forms the first objective for modeling and final exoskeleton prototype development. To account for modeling the characteristics of human locomotion, the nth-order linear differential equation with constant coefficients is considered with appropriate modification. The physical characteristics of an individual are represented by the constant coefficients (, , , and ) of the modified infinite series, which are obtained by processing experimental data collected using an optical technique. The differential terms of the infinite series are replaced by difference terms (, , and ) since the data were captured as a set of digital values. The work presented here is based on the experimental results of individuals suitably categorized according to their physical nature like age and other physical structure. The optically monitored positional values of the lower limb joints of the individual subjects while they are completing the gait cycles are used for getting values of different terms of the model. The data collected through the conduct of experiments are used for finding the values of the terms of the differential equation. The model is effectively validated through experimental results. It was determined that the representation’s accuracy fell within the 5% acceptable tolerance limit. The model is prepared for healthy as well as disabled persons, through which the disability is quantified. The resulting model can be used to develop assistive devices for people with physical disabilities. This results in the rehabilitation process and thereby helps the reintegration into society, subsequently allowing them to lead a normal life.

The present research aims to measure the segmental bioelectrical impedance (BI) of the human body at multifrequency, using a server user interface-based prototype, which provides subjects with measured data online anywhere accessed by their unique identifications. The present research measures the BI of the human leg and arm at a multifrequency range of 50–400 kHz with a developed and standard device. Recorded data can be transferred to the subject using Wi-Fi technology with their unique identifications and password. The system uses Wi-Fi interfacing for real-time data measurement and online data storage. The prototype can be used commercially. The compact size of the prototype becomes the demand of the common population. The pocket carry size makes it easy to carry anywhere for regular monitoring of the human body to prevent critical disease. The resultant data show that the mean and standard deviation of the left and right leg are 282.2 ± 14.29 and 274.80 ± 13.91 Ω, respectively. Similarly, in the case of the left arm and right arm, the mean and standard deviation are 325.41 ± 16.54 and 320.73 ± 16.07 Ω, respectively. The relative error between developed and standard devices is 3.53%. Results show that the left leg and arm impedance is always greater. However, the right leg and arm muscles are stronger than the left one, with less impedance at all frequencies.

In this paper, we propose and analyze the dynamical behaviors of two opinion formation models, one with leadership and the other without leadership. The two proposed models are formulated by fractional differential equations (FDEs) with the frame of the new generalized Hattaf fractional (GHF) derivative. The stability in the sense of Mittag–Leffler is rigorously established for both models. The convergence of agents’ opinions to the consensus opinion is fully investigated. Numerical simulations are given to illustrate the analytical results.

The accidents caused by the domino effect in industries are highly harmful. This study aims to analyze the occurrence probability of the domino effect with respect to possible explosion and fire scenarios in chemical tanks. Using the results obtained by previous studies, reviewing past accidents, and according to the equipment damage models, threshold values were used for extraction process equipment and inherent safety distances as a criterion to prevent domino accidents. According to primary scenarios and experimental equations, the escalation vector was determined for different tanks. According to the assumption that fire radius is equivalent to inherent safety distance, the fireball radius for tank 1 was calculated 535.7 m. According to the results, the DCP index of tank 3 can be considered the most critical unit. This research studies the probability of the domino effect and means to prevent them according to criteria and hazard index parameters.