Nonlinear Transcendental Equations Research Articles

Free and forced vibrations of elastically restrained cantilever with lumped oscillator

The efficiency assessment of cantilever-based energy harvesters relies on vibrational analysis, which necessitates modifications aimed at enhancing efficiency. These modifications involve manipulating the fundamental frequency to lower values and encompassing a wider range of resonances within a specified bandwidth. Consequently, this paper introduces an original analytical-numerical exploration into the vibratory response of a cantilever with a novel boundary condition involving an elastically restrained oscillator-spring arrangement. At the microbeam's tip, an oscillator is elastically confined by a linear spring, resulting in a novel set of coupled governing equations and a distinct shearing boundary condition. Microbeam equations is derived from the modified couple stress theory to capture size dependency. During free vibration analysis, a previously unreported characteristic equation is derived. This nonlinear transcendental equation is numerically solved utilizing root-solver algorithms, such as those available in MATLAB. Significantly, it is discovered that the inclusion of a lumped oscillator with an elastic support induces a minimal (new) natural frequency. Applying the extended Hamilton's principle, the effect of the lumped oscillator emerges both on the governing equations of motion and boundary conditions of the microbeam. Novelty of the paper focuses on the both characteristic equation and transmissibility by adopting the Galerkin’s modal decomposition technique. This finding carries vital implications as the efficiency of cantilever-based energy harvesters is directly contingent upon the resonance frequency. Notably, the oscillator mass and spring constant are two parameters that directly influence the vibratory response of the microbeam. In the context of forced vibrations, harmonic base excitation is considered as the input excitation, and the mechanical frequency response function is provided. The proposed system offers two distinct advantages for energy harvester systems: the creation of minimal resonance at lower values and the potential to manipulate the system's resonance toward a desired frequency spectrum.Article HighlightsModifying the boundary conditions of a cantilever beam with lumped-parameter system, can significantly change the behavior of the vibratory response.The boundary condition directly impact the resonance frequencies; which influences the maximum amount of harvestable voltage in vibration-based energy harvesters.Spring constant and mass of the lumped oscillator, are the key factors to alter the vibratory behavior and bandwidth of frequencies. Optimizing such mentioned parameters can help reaching to the maximum harvesting of energy.

Controlled random initialization of search vectors for application of gravitational search algorithm to selective harmonic elimination PWM

The prime objective of SHE PWM is to eliminate selected lower order harmonic by solving a set of non-linear transcendental equation while maintaining the fundamental component at a desired satisfactory value. In order to solve these non-linear equations different metaheuristic optimization techniques has been used in recent past. Due to faster convergence rate increased precision and higher probability to calculus based mathematical approximation techniques. In this manuscript an improved initialized Gravitational search algorithm has been proposed for a uni-polar 3-phase inverter. The algorithm is improved upon through a controlled random initialization of search agent, specifically for the purpose of solving the set of non-linear transcendental equation in SHEPWM for 3- switching angles. The simulation of the gravitational search algorithm and the proposed improved initialized GSA has been performed in MATLAB software and comparison of different performance parameter like Harmonic distortion, convergence rate etc. has been studied. The simulation results proves that a considerable improvement has been achieved by applying improved initialized GSA algorithm in terms of harmonic mitigation, convergence rate and accuracy in finding out the appropriate switching angles. Finally, the results obtained in simulation are verified through an appropriate experimental setup.

Cables of a prescribed length under concentrated or distributed loads

A general analysis of inextensible perfectly flexible cables of negligible weight and a prescribed length, loaded by an arbitrary number of concentrated loads, or by a uniformly distributed load per unit horizontal length, is presented. A catenary cable under its own weight, with or without a concentrated force, is also considered. The sags under loads and tensions in all segments of the cable are determined. The results are derived in closed-form, apart from a numerical solution of a single nonlinear algebraic equation for a suitably introduced parameter in the last step of the analysis. The study complements the treatment of cables under concentrated forces available in undergraduate textbooks, which omit the consideration of cables whose length is prescribed in advance. The general analysis may be found appealing for a senior or graduate structural mechanics course, while its specialization to cables with two or three loads only is well suited for an introductory mechanics course. Expressions for a catenary cable subjected to a concentrated load at its midpoint are also derived, with only one nonlinear transcendental equation for the introduced load parameter needing to be solved numerically. Three student projects are suggested which address the existence and uniqueness of the cable configuration under a concentrated force with a specified sag and length of the cable, the lifting of two symmetrically hung loads by an upward force, and the determination of the deformed shape and tension in a catenary cable subjected to a concentrated force at an arbitrary point of the cable.

Comparative analysis of NR and DE methods for multilevel inverters: An experimental case study

This paper presents an experimental performance comparison between differential evolution (DE) algorithm and Newton Raphson (NR) method for solving the optimal switching problem for multilevel inverters with the objective of improving the output AC voltage quality. The methods are used to compute optimal switching angles to control a multilevel inverter in such way that allows the elimination of low order harmonics; this control technique is called selective harmonic elimination. The proposed inverter in this article is composed of two H-bridge cells with non-equal DC voltage sources in order to generate seven voltage levels. The differential evolution (DE) optimization and the Newton Raphson (NR) algorithm is used to solve non-linear transcendental equations necessary for the (SHPWM). Computational results obtained from computer simulations presented a good agreement with the theoretical predictions. A laboratory prototype based on STM32F407 microcontroller was built in order to validate the simulation results.

A single source five-level switched-capacitor based multilevel inverter with reduced device count

Nowadays, multilevel inverters (MLIs) are widely used for integration with renewable energy (RE) resources and in industrial applications. Among different RE sources, photovoltaic (PV) energy is generally used for the generation of power. For accumulating more power from PV systems, the Fuzzy Logic Controller (FLC) technique is used to trace the maximum power (MPP). The FLC computes the reference PV voltage guarantee to ensure optimal power transmission between the PV system and the MLI's connected load. In this paper, a novel 5L (five-level) PV-based switched-capacitor MLI topology (PV-SC-MLI) has been projected with the target of reducing the number of switches and boosting the input voltage without any bulky transformer. A selected harmonic elimination pulse width modulation technique (SHEPWM) based technique is used to remove the harmonic for 5L-PV-SC-MLI. The projected technique is appropriate for all levels of MLIs, including non-equal and equal DC sources. The SHEPWM technique is used to solve the non-linear transcendental equations and obtain the optimal values of switching angles. The obtained switching value is used to reduce the total harmonic distortion (THD) of 5L-PV-SC-MLI. The THD at the output voltage of proposed inverter has been obtained value is 8.72% without filter and with filter 4.20%, which lies within the limits of IEEE 519 standards.

Solving Nonlinear Transcendental Equations by Iterative Methods with Conformable Derivatives: A General Approach

In recent years, some Newton-type schemes with noninteger derivatives have been proposed for solving nonlinear transcendental equations by using fractional derivatives (Caputo and Riemann–Liouville) and conformable derivatives. It has also been shown that the methods with conformable derivatives improve the performance of classical schemes. In this manuscript, we design point-to-point higher-order conformable Newton-type and multipoint procedures for solving nonlinear equations and propose a general technique to deduce the conformable version of any classical iterative method with integer derivatives. A convergence analysis is given and the expected orders of convergence are obtained. As far as we know, these are the first optimal conformable schemes, beyond the conformable Newton procedure, that have been developed. The numerical results support the theory and show that the new schemes improve the performance of the original methods in some aspects. Additionally, the dependence on initial guesses is analyzed, and these schemes show good stability properties.

Reduced Device Count for Self Balancing Switched-Capacitor Multilevel Inverter Integration with Renewable Energy Source

In this study, a new switched-capacitor-based seven-level inverter topology with a photovoltaic system is presented. The proposed topology requires a smaller number of devices and has the ability to self-balance the voltage across the capacitor. The proposed topology configuration is simple and has the ability to extend to higher levels of voltage. This multilevel inverter topology is suitable for low- and medium-voltage applications with photovoltaic (PV) system integration. To improve the PV system efficiency as an input of a DC–DC boost converter, a Fuzzy logic-based maximum power point controller technique is used. A PV system with a DC–DC boost converter integrates with the proposed seven-level inverter topology. The anti-predatory particle swarm optimization (APSO) technique is used to solve the non-linear transduction equations of the seven-level PV switched-capacitor-based multilevel inverter (7L−PV−SCMLI) topology. The proposed APSO is described to minimize the harmonics in the multilevel inverter (MLIs), which is a complex optimization problem involving a non-linear transcendental equation. Furthermore, APSO can be applied in order to solve non-linear transcendental equations for all symmetrical and asymmetrical MLIs that have equal and non-equal DC sources. The APSO-based selected harmonic elimination (SHE) technique obtained the best switching angle value, and the optimized obtained switching angles reduced the total harmonic distortion (THD) of 7L−PV−SCMLI.

OPTIMIZATION OF SWITCHING ANGLES FOR SELECTIVE HARMONIC ELIMINATION IN CASCADED H-BRIDGE MULTILEVEL INVERTERS EMPLOYING ARTIFICIAL INTELLIGENCE TECHNIQUES – A MINI REVIEW

Considering present shortage of fossil fuels and discharges of ozone harming substances, power developed from Renewable Energy Sources (RES) is identified as the excellent choice for producing the electricity. The characteristic of an inverter is to transform the dc power into ac power to fulfill out the requirements of load. Despite its advantage, the presence of harmonics in the output voltage reduces both the efficiency and the performance of the inverter. Several researches have been carried out since last three decades for eliminating the harmonics. Based upon several researches, it reveals that the Selective Harmonic Elimination Pulse-width Elimination technique (SHEPWM) has proven to be the best in eliminating lower order harmonics. But when calculus based methods are used for solving the non-linear transcendental equations, this technique has shown some complications. Artificial Intelligence (AI) techniques appear to be better in solving the above said equations. This review paper provides the performance of some AI techniques used for eliminating the harmonics in inverters. Based upon the information collected from various literatures and its results, conclusion has been made.

Optimum regulation of THD profile in multilevel inverter using parameter-less AI technique for electrical vehicle application

This article proposes investigation on harmonic profile improvement using a novel fitness function in the multiphase multilevel inverter. For the proposed study in this paper, three-phase, seven-level cascaded H-bridge (CHB) multilevel inverter (MLI) is considered. Modulation of the stepped waveform output of the MLI is done using selective harmonic elimination (SHE) method. Many algorithms are proposed for solving the set of nonlinear transcendental trigonometric equations for SHE methods. Teaching learning-based optimisation (TLBO) algorithm is a parameter less optimisation technique. Due to the lack of controlling parameters, the proposed algorithm is most robust among the family of artificial intelligence (AI) techniques. In this paper, an investigation is carried out on a novel fitness function proposed for controlling total harmonic distortion (THD) for the proposed inverter. It is observed that the proposed fitness function improves THD profile below IEEE standards. It is also observed that THD profile obtained through this method is far better than THD profile obtained through various proposed methods so far. The results of THD profile from the MATLAB Simulink simulations are verified experimentally using seven-level three-phase hardware controlled by Arduino MEGA 2560 low-cost controller.

New Trigonometrically Method for Solving Non-Linear Transcendental Equations

Abstract: The transcendental equations' non-zero positive real roots can be found using a new approach presented in this research. The proposed approach is based on the union of the Newton-Raphson method and the inverse tan(x) function. To ensure the method's applicability, it is implemented in MATLAB and applied to various issues. The suggested approach is evaluated on a variety of numerical instances, the results show that our approaches are superior and more efficient than widely used methods. For both the new proposed method and the currently accessible existing methods, error calculations have been made. When compared to well-known procedures, the mistakes were quickly minimised, and the genuine root was discovered in fewer repetitions. The proposed method's convergence is studied, and it is demonstrated that it reduces to the quadratic convergent Newton-Raphson method. This method will also assist in the production of a non-zero real root of a specified nonlinear equation in the commercial software (transcendental, algebraic, and exponential).

Two Practical Methods for the Forward Kinematics of 3-3 Type Spatial and 3-RRR Planar Parallel Manipulators

The forward kinematics in parallel manipulators is a mathematically challenging issue, unlike serial manipulators. Kinematic constraint equations are non-linear transcendental equations that can be reduced to algebraic equations with appropriate transformations. For this reason, sophisticated and time-consuming methods such as the Bezout method, the Groebner bases method, and the like, are used. In this paper, we demonstrate that these equations can be solved by non-complicated mathematical methods for some special types of manipulators such as the 3-3 and 6-3 types of Stewart platforms, and the 3-RRR planar parallel manipulator. Our first method is an analytical approach that exploits the special structure of kinematic constraint equations and yields polynomials of 32nd and 16th order, as mentioned in the previous works. In the second method, an error function is defined. This error function is employed to find the most appropriate initial values for the non-linear equation solver which is used for solving kinematic constraint equations. Determining the initial values in this manner saves computation time and guarantees fast convergence to real solutions.

Reptile Search Algorithm (RSA)-Based Selective Harmonic Elimination Technique in Packed E-Cell (PEC-9) Inverter

The multilevel inverters (MLIs) are capable of handling large quantities of power and generating high-quality output voltages. Consequently, the size of the filters is reduced, and the circuitry is simplified. As a result, they have a diverse range of uses in the industrial sector, especially in smart grids. The input voltage boosting feature is required to utilize the MLI with renewable energy. In addition, a large number of components are required to attain higher output voltage levels, which increases the cost of the circuit and weight. A variety of MLI topologies have been identified to reduce losses, device quantity, and device ratings. The selective harmonic elimination (SHE) approaches reduce distinct lower order harmonics by computing the ideal switching angles. This research presents a nine–level Packed E–Cell (PEC–9) inverter that uses selective harmonic elimination to eliminate total harmonic distortion. In order to calculate the best switching angle, the reptile search algorithm (RSA) is implemented in this paper, a nature–inspired metaheuristic algorithm inspired by the hunting behavior of the crocodile. The hunting behavior of crocodiles is implemented in two main steps: the first is encircling, which is accomplished by belly walking or high walking, and the second is hunting, which is accomplished by hunting cooperation or hunting coordination. In this technique, nonlinear transcendental equations have been solved. The simulation was run in the MATLAB R2021b software environment. The simulation results suggest that the RSA outperforms the other metaheuristic algorithms. Furthermore, the simulation result was validated on a hardware setup using DSP–TMS320F28379D in the laboratory.

Elimination of Harmonics in Multilevel Inverter Using Multi-Group Marine Predator Algorithm-Based Enhanced RNN

Multilevel inverters (MLI) are becoming more common in different power applications, such as active filters, elective vehicle drives, and dc power sources. The Multi-Group Marine Predator Algorithm (MGMPA) is introduced in this study for resolving transcendental nonlinear equations utilizing an MLI in a selective harmonic elimination (SHE) approach. Its applicability and superiority over various SHE approaches utilized in recent research may be attributed to its high accuracy, high likelihood of convergence, and improved output voltage quality. For the entire modulation index, the optimum switching angles (SA) from Marine Predator Algorithm (MPA) is utilized to control a three-phase 11-level MLI employing cascaded H-bridge (CHB) architecture to regulate the vital element and eliminate the harmonics. The limitation of SHE is that it is difficult to find solutions for nonlinear equations. As a result, specific optimization approaches must be used. Artificial Intelligence (AI) algorithms can handle such a nonlinear transcendental equation successfully, although their time consumption as well as convergence abilities vary. Here, recurrent neural network (RNN) is considered where the hidden neurons are tuned by MGMPA with the intention of harmonic distortion parameter (HDP) minimization, thus called as enhanced recurrent neural network (ERNN). The method’s resilience and consistency are demonstrated by simulation and analytical findings. The MGMPA method is more effective and appropriate than various algorithms including the MPA, Harris Hawks optimization (HHO), and Whale optimization algorithm (WOA), according to simulation data.

A New Root-Finding Method for Univariate Non-Linear Transcendental Equations With Quadratic Convergence

This paper presents a new algorithm to find a non-zero real root of the non-linear transcendental equations. The proposed method is based on the combination of the inverse sine series and Newton-Raphson method. Implementation of the proposed method in MATLAB is applied to different problems to ensure the methods applicability. Error calculation has been done for available existing methods and the proposed method. The suggested method is evaluated using a number of numerical examples and the results indicate that the proposed method is effective than well-knownmethods. The proposed method’s convergence is discussed, and it is shown to be quadraticallyconvergence.

Proposed Section Methods to Find Root of the Single Variable Non-Linear Algebraic and Transcendental Equations

The main purpose of numerical analysis is to develop a numerical algorithm to find approximate values. The main aim of this paper is to present Proposed Section Method-I and Proposed Section Method-II (Proposed Section Methods). These method makes some slight difference in well-known bisection method to solve single variable nonlinear equation. In well-known Bisection Method, the interval in which the root of the equation lies is bisected using Arithmetic Mean (AM). These Proposed Section Method splits the interval using Geometric Mean (GM) and Harmonic Mean (HM). These method gives the roots of the equation which are closed to the root given by well-known Bisection Method. In order to analyze the results some examples related with algebraic and transcendental functions are considered. It was concluded that if the root of the single variable non-linear equation exists by all these methods then the solutions are very close to each other.

An improved Newton-Raphson method with quadratic convergence for solving nonlinear transcendental equations

This paper presents a new scheme to find a non-zero positive root of the non-linear single variable equations. The proposed method is based on the grouping of the arc sine series and Newton-Raphson method. The proposed method is implemented in MATLAB and is applied to various models of problems to ensure the methods applicability. The proposed method is studied on number of numerical examples and results signify that our method is better and more effective as comparable to renowned methods. These results are depicted through error analysis. The convergence of proposed method is proven to be quadratically convergence.

An Improved Hybrid PSO-TS Algorithm for Solving Nonlinear Equations of SHEPWM in Multilevel Inverters

Selective Harmonic Elimination Pulse Width Modulation (SHEWM) can eliminate selected low order harmonics by solving nonlinear transcendental equations to obtain the switching angles in advance and achieve precise harmonics control. However, when solving nonlinear equations, the convergence rate and solution accuracy of the existing intelligent algorithms for multilevel inverter SHEPWM will decrease with the increase of the number of switching angles. In this paper, an improved hybrid PSO-TS algorithm (IPSO-TS) is proposed for switching angle selection in PWM inverter. In the proposed IPSO-TS algorithm, both the PSO algorithm and TS algorithm are improved respectively to enhance their convergence when calculating multiple switching angles. Also, these two improved algorithms are combined to obtain the hybrid algorithm IPSO-TS, in which PSO is used to provide global search capabilities, and TS is used to increases the accuracy of solutions. Simulations are carried out using MATLAB/SIMULINK environment to demonstrate the better convergence of the proposed algorithm compared with the other existing methods. Moreover, the simulation results are further verified by experimental results.

Approximation of the Numerical Simulation of Nonlinear Transcendental Equation

Approximation of the Numerical Simulation of Nonlinear Transcendental Equation

Crystal Structure Algorithm (CryStAl) Based Selective Harmonic Elimination Modulation in a Cascaded H-Bridge Multilevel Inverter

This paper introduces an effective Selective Harmonic Elimination (SHE) modulation technique in a five, seven, and nine-level cascaded H-bridge (CHB) multilevel inverter (MLI). Minimization of the harmonics and device counts is the basis for the ongoing research in the area of MLI. Reduced harmonics and hence the lower Total Harmonic Distortion (THD), improve the output power quality. SHE is a low-frequency modulation scheme to achieve this goal. SHE techniques are used to eliminate the distinct lower-order harmonics by determining the optimum switching angles. These angles are evaluated by solving the non-linear transcendental equations using any optimization technique. For this purpose, the Crystal Structure Algorithm (CryStAl) has been used in this paper. It is a metaheuristic, nature-inspired, and highly efficient optimization technique. CryStAl is a simple and parameter-free algorithm that doesn’t require the determination of any internal parameter during the optimization process. It is based on the concept of crystal structure formation by joining the basis and lattice point. This natural occurrence can be realized in crystalline minerals in their symmetrically organized components: ions, atoms, and molecules. The concept has been utilized to solve non-linear transcendental equations. SIMULINK/MATLAB environment has been used for the simulation. The simulation result shows that the crystal structure algorithm is very effective and excels the other metaheuristic algorithm. Hardware results validate the performance.

Harmonic Elimination in Uniform Step Nine-Level Inverter Using Differential Evolution: Experimental Validation

This study presents the application of differential evolution algorithm to compute optimal switching angles for a single-phase nine-level inverter to improve the output voltage quality. The topology of the proposed inverter in this article is a simple cascade converter composed of two H-bridge cells with non-equal DC voltage sources in order to generate multiple voltage levels. Selective harmonic elimination pulse width modulation strategy is used to improve the generated AC output voltage waveform. The differential evolution optimization algorithm is used to solve non-linear transcendental equations necessary for the (SHPWM). Computational results obtained from computer simulations presented a good agreement with the theoretical predictions. A laboratory prototype based on STM32F407 microcontroller was built in order to validate the simulation results. The experimental results show the effectiveness of the proposed modulation method.