A. Buscarino, Carlo Famoso, Luigi Fortuna, Giuseppe La Spina
Abstract On September 22, 2022, a spacecraft, designed by the Double Asteroid Redirection Test (DART) team, successfully attempted to deflect the orbit of the asteroid Dimorphos, which together with Didymos, constitutes a binary system of near-Earth asteroids orbiting around the Sun. The effect of the impact of the spacecraft was to shorten the orbit of Dimorphos of about 33 min with respect to the original one. In this communication, a simple nonlinear circuit emulator based on a mathematical model allowing the emulation of the DART mission behavior is presented. The modeling is approached referring to the Kepler problem that leads to a highly nonlinear dynamical model. The problem is approached numerically, by using appropriate integration algorithms for both the two-body and three-body formulations of the problem, and experimentally, by means of an analog/digital electronic circuit emulator of the system that allows us to realize faster and qualitative more efficient experiments.
{"title":"NASA DART mission: A preliminary mathematical dynamical model and its nonlinear circuit emulation","authors":"A. Buscarino, Carlo Famoso, Luigi Fortuna, Giuseppe La Spina","doi":"10.1515/nleng-2022-0314","DOIUrl":"https://doi.org/10.1515/nleng-2022-0314","url":null,"abstract":"Abstract On September 22, 2022, a spacecraft, designed by the Double Asteroid Redirection Test (DART) team, successfully attempted to deflect the orbit of the asteroid Dimorphos, which together with Didymos, constitutes a binary system of near-Earth asteroids orbiting around the Sun. The effect of the impact of the spacecraft was to shorten the orbit of Dimorphos of about 33 min with respect to the original one. In this communication, a simple nonlinear circuit emulator based on a mathematical model allowing the emulation of the DART mission behavior is presented. The modeling is approached referring to the Kepler problem that leads to a highly nonlinear dynamical model. The problem is approached numerically, by using appropriate integration algorithms for both the two-body and three-body formulations of the problem, and experimentally, by means of an analog/digital electronic circuit emulator of the system that allows us to realize faster and qualitative more efficient experiments.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"17 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75515876","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article deals with the problem of finding a pricing formula for weather derivatives based on temperature dynamics through an uncertain differential equation. Weather-related derivatives are being employed more frequently in alternative risk portfolios with multiple asset classes. We first propose an uncertain process that uses data from the past to describe how the temperature has changed. Despite this, pricing these assets is difficult since it necessitates an incomplete market framework. The volatility is described by a truncated Fourier series, and we provide a novel technique for calculating this constant using Monte Carlo simulations. With this approach, the risk is assumed to have a fixed market price.
{"title":"Pricing weather derivatives in an uncertain environment","authors":"Zulfiqar Ali, J. Hussain, Z. Bano","doi":"10.1515/nleng-2022-0291","DOIUrl":"https://doi.org/10.1515/nleng-2022-0291","url":null,"abstract":"Abstract This article deals with the problem of finding a pricing formula for weather derivatives based on temperature dynamics through an uncertain differential equation. Weather-related derivatives are being employed more frequently in alternative risk portfolios with multiple asset classes. We first propose an uncertain process that uses data from the past to describe how the temperature has changed. Despite this, pricing these assets is difficult since it necessitates an incomplete market framework. The volatility is described by a truncated Fourier series, and we provide a novel technique for calculating this constant using Monte Carlo simulations. With this approach, the risk is assumed to have a fixed market price.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"23 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78705350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, we consider the flow of a Burgers’ fluid of transient electro-osmotic type in a small tube with a circular cross-section. Analytical results are found for the transient velocity and, electric potential profile by solving the Navier–Stokes and the linearized Poisson–Boltzmann equations. Moreover, these equations are solved with the help of the integral transform method. We consider cases in which the velocity of the fluid changes with time and those in which the velocity of the fluid does not change with time. Furthermore, special results for classical fluids such as Newtonian, second grade, Maxwell, and Oldroyd-B fluids are obtained as the particular cases of the outcomes of this work and that these results actually strengthen the results of the solution. This study of the nonlinear problem of Burgers’ fluid in a specified physical model will help us to understand the behavior of blood clotting and the block of any kind of problem in which this type of fluid is used. The solution of the complex velocity profile of Burgers’ fluid will help us in the future to solve the problems in which this transient electro-osmotic type of small tube is involved. At the end, numerical results are shown graphically that can help us to understand the complex behavior of the Burgers’ fluid, and also the analysis of the Burgers’ fluid shows dissimilarity with other fluids that are considered in this work.
{"title":"Dynamical aspects of transient electro-osmotic flow of Burgers' fluid with zeta potential in cylindrical tube","authors":"N. Raza, Ahmad Kamal Khan, A. Awan, K. A. Abro","doi":"10.1515/nleng-2022-0256","DOIUrl":"https://doi.org/10.1515/nleng-2022-0256","url":null,"abstract":"Abstract In this article, we consider the flow of a Burgers’ fluid of transient electro-osmotic type in a small tube with a circular cross-section. Analytical results are found for the transient velocity and, electric potential profile by solving the Navier–Stokes and the linearized Poisson–Boltzmann equations. Moreover, these equations are solved with the help of the integral transform method. We consider cases in which the velocity of the fluid changes with time and those in which the velocity of the fluid does not change with time. Furthermore, special results for classical fluids such as Newtonian, second grade, Maxwell, and Oldroyd-B fluids are obtained as the particular cases of the outcomes of this work and that these results actually strengthen the results of the solution. This study of the nonlinear problem of Burgers’ fluid in a specified physical model will help us to understand the behavior of blood clotting and the block of any kind of problem in which this type of fluid is used. The solution of the complex velocity profile of Burgers’ fluid will help us in the future to solve the problems in which this transient electro-osmotic type of small tube is involved. At the end, numerical results are shown graphically that can help us to understand the complex behavior of the Burgers’ fluid, and also the analysis of the Burgers’ fluid shows dissimilarity with other fluids that are considered in this work.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"26 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84007749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Xu Yanbin, Zhang Jianhua, Xiongwei Wang, Mohammad Shabaz, Mohd Wazih Ahmad, Samrat Ray
Abstract To ensure the safe functioning of lifting equipment, a data mining-based optimization study of a crane failure predictive control system is provided. To diagnose lifting machinery faults, the system employs decision tree categorization. Using association rules, a correlation study of hoisting machinery defect and failure was performed. When the minimal confidence and support degree are entered, a total of 244 instances of 18 frequent itemset A9 (safety protection device) may be obtained, indicating that lifting machinery does not perform well in this category. A6 (main parts) and A9 appeared a total of 98 times, with support and confidence of 29.4 and 35.6, respectively, indicating that the main parts can detect that the safety protection device is also having problems. A7 (electrical control system) and A9 appeared a total of 67 times, with support and confidence of 20.1 and 27.3, respectively, indicating that the electrical control system can detect that the safety protection device is also having problems; the correlation between them was also quite large. The system’s feasibility and efficacy shows that it has some application value.
{"title":"Research on optimization of crane fault predictive control system based on data mining","authors":"Xu Yanbin, Zhang Jianhua, Xiongwei Wang, Mohammad Shabaz, Mohd Wazih Ahmad, Samrat Ray","doi":"10.1515/nleng-2022-0202","DOIUrl":"https://doi.org/10.1515/nleng-2022-0202","url":null,"abstract":"Abstract To ensure the safe functioning of lifting equipment, a data mining-based optimization study of a crane failure predictive control system is provided. To diagnose lifting machinery faults, the system employs decision tree categorization. Using association rules, a correlation study of hoisting machinery defect and failure was performed. When the minimal confidence and support degree are entered, a total of 244 instances of 18 frequent itemset A9 (safety protection device) may be obtained, indicating that lifting machinery does not perform well in this category. A6 (main parts) and A9 appeared a total of 98 times, with support and confidence of 29.4 and 35.6, respectively, indicating that the main parts can detect that the safety protection device is also having problems. A7 (electrical control system) and A9 appeared a total of 67 times, with support and confidence of 20.1 and 27.3, respectively, indicating that the electrical control system can detect that the safety protection device is also having problems; the correlation between them was also quite large. The system’s feasibility and efficacy shows that it has some application value.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"41 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90903709","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Via UAH tension B-spline DQM in the present research, numerical approximation of coupled Schrödinger equations in one and two dimensions is fetched. In the present research, a novel regime is generated as a fusion of a UAH tension B-spline of fourth-order and DQM to fetch the requisite weighting coefficients. To ensure the adaptability and effectiveness of the proposed regime, different numerical examples are elaborated. Present results are matched with previous results, and the elastic property is also validated for solitons. The fetched ordinary differential equations system is handled via the SSP-RK43 regime. The stability of the present method is verified via the matrix method. The robustness of the proposed regime is affirmed via error norms. The fetched results are acceptable and validated. Elasticity property via wave interaction is also covered in the present research. The present study also focuses on one very important property of physics, like elasticity, which is rarely discussed in the literature. The developed numerical regime will undoubtedly be useful in addressing various fractional partial differential equations of complex nature as well.
{"title":"Numerical approximations of CNLS equations via UAH tension B-spline DQM","authors":"Mamta Kapoor, V. Joshi","doi":"10.1515/nleng-2022-0283","DOIUrl":"https://doi.org/10.1515/nleng-2022-0283","url":null,"abstract":"Abstract Via UAH tension B-spline DQM in the present research, numerical approximation of coupled Schrödinger equations in one and two dimensions is fetched. In the present research, a novel regime is generated as a fusion of a UAH tension B-spline of fourth-order and DQM to fetch the requisite weighting coefficients. To ensure the adaptability and effectiveness of the proposed regime, different numerical examples are elaborated. Present results are matched with previous results, and the elastic property is also validated for solitons. The fetched ordinary differential equations system is handled via the SSP-RK43 regime. The stability of the present method is verified via the matrix method. The robustness of the proposed regime is affirmed via error norms. The fetched results are acceptable and validated. Elasticity property via wave interaction is also covered in the present research. The present study also focuses on one very important property of physics, like elasticity, which is rarely discussed in the literature. The developed numerical regime will undoubtedly be useful in addressing various fractional partial differential equations of complex nature as well.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"29 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74495809","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this article, the bond performance between recycled concrete and corroded steel bars is analyzed by the nonlinear numerical simulation. The result shows that the maximum bond strength between recycled concrete and steel bar decreases with the increase in steel bar corrosion rate; when the recycled concrete strength is large, the simulated maximum bond strength is in good agreement with the experimental maximum bond strength; when the recycled concrete strength is small, the simulated maximum bond strength is in relatively poor agreement with the experimental maximum bond strength, but there is still an error within the allowable range; the slip between recycled concrete and steel bar increases with the increase in steel bar corrosion rate; when the steel bar corrosion rate exceeded 5%, the bond strength decreases more rapidly; the maximum bond strength increases with the increase in specimen sizes under the same steel bar corrosion rate; the maximum bond strength decreases with the increase in steel bar diameter under the same steel bar corrosion rate.
{"title":"Nonlinear numerical simulation of bond performance between recycled concrete and corroded steel bars","authors":"Zhenfang Li, Dongshao Gao, Chuanji Wu, Guoqing Lv, Xin Liu, Haoran Zhai, Zhan-Fang Huang","doi":"10.1515/nleng-2022-0275","DOIUrl":"https://doi.org/10.1515/nleng-2022-0275","url":null,"abstract":"Abstract In this article, the bond performance between recycled concrete and corroded steel bars is analyzed by the nonlinear numerical simulation. The result shows that the maximum bond strength between recycled concrete and steel bar decreases with the increase in steel bar corrosion rate; when the recycled concrete strength is large, the simulated maximum bond strength is in good agreement with the experimental maximum bond strength; when the recycled concrete strength is small, the simulated maximum bond strength is in relatively poor agreement with the experimental maximum bond strength, but there is still an error within the allowable range; the slip between recycled concrete and steel bar increases with the increase in steel bar corrosion rate; when the steel bar corrosion rate exceeded 5%, the bond strength decreases more rapidly; the maximum bond strength increases with the increase in specimen sizes under the same steel bar corrosion rate; the maximum bond strength decreases with the increase in steel bar diameter under the same steel bar corrosion rate.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"262 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76265865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract To improve the accuracy effect of the iterative hard threshold, an improved iterative hard threshold (IHT) method is proposed. The specific contents of this method include the principle of an IHT algorithm based on compression sensing (nonlinear optimization), weighted least squares improvement, the establishment of an IHT algorithm model based on weighted least squares improvement, and the experimental research of traditional algorithms and improved algorithms on one-dimensional signal reconstruction. The results show that the improved IRLSIHT algorithm takes 8.37, 29.63, and 30.86 s when the sampling rate is 0.2, 0.5, and 0.8, respectively, and the signal-to-noise ratio is 20.11, 27.47, and 31.82 dB, respectively. Compared with the traditional IHT algorithm, it takes a long time, which is a deficiency, but the signal-to-noise ratio is the largest, and the improved algorithm improves the accuracy. It has been proven that combining the method proposed in this article with automatic control can significantly save time and increase industrial output.
{"title":"The application of iterative hard threshold algorithm based on nonlinear optimal compression sensing and electronic information technology in the field of automatic control","authors":"Kun-han jiang, M. Bradha","doi":"10.1515/nleng-2022-0305","DOIUrl":"https://doi.org/10.1515/nleng-2022-0305","url":null,"abstract":"Abstract To improve the accuracy effect of the iterative hard threshold, an improved iterative hard threshold (IHT) method is proposed. The specific contents of this method include the principle of an IHT algorithm based on compression sensing (nonlinear optimization), weighted least squares improvement, the establishment of an IHT algorithm model based on weighted least squares improvement, and the experimental research of traditional algorithms and improved algorithms on one-dimensional signal reconstruction. The results show that the improved IRLSIHT algorithm takes 8.37, 29.63, and 30.86 s when the sampling rate is 0.2, 0.5, and 0.8, respectively, and the signal-to-noise ratio is 20.11, 27.47, and 31.82 dB, respectively. Compared with the traditional IHT algorithm, it takes a long time, which is a deficiency, but the signal-to-noise ratio is the largest, and the improved algorithm improves the accuracy. It has been proven that combining the method proposed in this article with automatic control can significantly save time and increase industrial output.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"22 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80998523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.
{"title":"The regularization of spectral methods for hyperbolic Volterra integrodifferential equations with fractional power elliptic operator","authors":"F. Mirzaei G., D. Rostamy","doi":"10.1515/nleng-2022-0250","DOIUrl":"https://doi.org/10.1515/nleng-2022-0250","url":null,"abstract":"Abstract In this study, a numerical approach is presented to solve the linear and nonlinear hyperbolic Volterra integrodifferential equations (HVIDEs). The regularization of a Legendre-collocation spectral method is applied for solving HVIDE of the second kind, with the time and space variables on the basis of Legendre-Gauss-Lobatto and Legendre-Gauss (LG) interpolation points, respectively. Concerning bounded domains, the provided HVIDE relation is transformed into three corresponding relations. Hence, a Legendre-collocation spectral approach is applied for solving this equation, and finally, ill-posed linear and nonlinear systems of algebraic equations are obtained; therefore different regularization methods are used to solve them. For an unbounded domain, a suitable mapping to convert the problem on a bounded domain is used and then apply the same proposed method for the bounded domain. For the two cases, the numerical results confirm the exponential convergence rate. The findings of this study are unprecedented for the regularization of the spectral method for the hyperbolic integrodifferential equation. The result in this work seems to be the first successful for the regularization of spectral method for the hyperbolic integrodifferential equation.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"59 1","pages":""},"PeriodicalIF":8.3,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91342244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In order to address the impact of reduced transmission stability and reliability caused by volume reduction on the quality of gear transmission, this article proposes a multi-objective optimization model for nonlinear dynamic load transmission errors of double helical gears. This study aims to introduce a multi-objective design method for gear transmission, using the volume and smooth reliability of helical gears as objective functions, and establish a multi-objective optimization design mathematical model for helical cylindrical gear transmission. In order to solve this multi-objective optimization problem, we utilized the optimization toolbox in the scientific calculation software MATLAB with examples. The results show that after the joint optimization design of volume and coincidence degree, it is calculated that the volume after the joint optimization design is still 2.2624 × 10 7 mm 3 , and the coincidence degree is 5.9908. After rounding, the design result is mn=3,Z1=31,β=20∘,Ψd=1.2 {m}_{{rm{n}}}=3,{Z}_{1}=31,beta ={20}^{circ },{Psi }_{{rm{d}}}=1.2 . The optimization design results show that the joint optimization design with the minimum volume and the maximum coincidence as the objective function can reduce the volume and improve the output stability of the helical gear.
摘要为解决体积减小导致传动稳定性和可靠性降低对齿轮传动质量的影响，提出了双斜齿轮非线性动载传动误差的多目标优化模型。本研究旨在引入一种齿轮传动的多目标设计方法，以螺旋齿轮的体积和光滑可靠性为目标函数，建立螺旋圆柱齿轮传动的多目标优化设计数学模型。为了解决这一多目标优化问题，我们利用科学计算软件MATLAB中的优化工具箱并举例说明。结果表明，在对体积和契合度进行联合优化设计后，计算得出，结合优化设计后的体积仍为2.2624 × 10 7 mm 3，契合度为5.9908。舍入后的设计结果为m n =3,Z 1=31， β =20°，Ψ d =1.2 {m＿}=3{{rm{n}}},{Z＿1}=31{, }beta ={20}^ {circ, }{Psi ＿}=1.2。优化设计结果表明，以最小体积和最大重合为目标函数的联合优化设计可以减小齿轮体积，提高齿轮输出稳定性。{{rm{d}}}
{"title":"Multi-objective optimization model of transmission error of nonlinear dynamic load of double helical gears","authors":"Xingling Yao","doi":"10.1515/nleng-2022-0323","DOIUrl":"https://doi.org/10.1515/nleng-2022-0323","url":null,"abstract":"Abstract In order to address the impact of reduced transmission stability and reliability caused by volume reduction on the quality of gear transmission, this article proposes a multi-objective optimization model for nonlinear dynamic load transmission errors of double helical gears. This study aims to introduce a multi-objective design method for gear transmission, using the volume and smooth reliability of helical gears as objective functions, and establish a multi-objective optimization design mathematical model for helical cylindrical gear transmission. In order to solve this multi-objective optimization problem, we utilized the optimization toolbox in the scientific calculation software MATLAB with examples. The results show that after the joint optimization design of volume and coincidence degree, it is calculated that the volume after the joint optimization design is still 2.2624 × 10 7 mm 3 , and the coincidence degree is 5.9908. After rounding, the design result is <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>m</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">n</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>3</m:mn> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>Z</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>31</m:mn> <m:mo>,</m:mo> <m:mi>β</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mn>20</m:mn> </m:mrow> <m:mrow> <m:mrow> <m:mo>∘</m:mo> </m:mrow> </m:mrow> </m:msup> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>Ψ</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"normal\">d</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>1.2</m:mn> </m:math> {m}_{{rm{n}}}=3,{Z}_{1}=31,beta ={20}^{circ },{Psi }_{{rm{d}}}=1.2 . The optimization design results show that the joint optimization design with the minimum volume and the maximum coincidence as the objective function can reduce the volume and improve the output stability of the helical gear.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134883399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A nonlinear Boussinesq equation under fractal fractional Caputo’s derivative is studied. The general series solution is calculated using the double Laplace transform with decomposition. The convergence and stability analyses of the model are investigated under Caputo’s fractal fractional derivative. For the numerical illustrations of the obtained solution, specific examples along with suitable initial conditions are considered. The single solitary wave solutions under fractal fractional derivative are attained by considering small values of time (t) left(t) . The wave propagation has a symmetrical form. The solitary wave’s amplitude diminishes over time, and its extended tail expands over a long distance. It is observed that the fractal fractional derivatives are an extremely constructive tool for studying nonlinear systems. An error analysis is also carried out for compactness.
{"title":"Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative","authors":"Obaid J. Algahtani","doi":"10.1515/nleng-2022-0338","DOIUrl":"https://doi.org/10.1515/nleng-2022-0338","url":null,"abstract":"Abstract A nonlinear Boussinesq equation under fractal fractional Caputo’s derivative is studied. The general series solution is calculated using the double Laplace transform with decomposition. The convergence and stability analyses of the model are investigated under Caputo’s fractal fractional derivative. For the numerical illustrations of the obtained solution, specific examples along with suitable initial conditions are considered. The single solitary wave solutions under fractal fractional derivative are attained by considering small values of time <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> left(t) . The wave propagation has a symmetrical form. The solitary wave’s amplitude diminishes over time, and its extended tail expands over a long distance. It is observed that the fractal fractional derivatives are an extremely constructive tool for studying nonlinear systems. An error analysis is also carried out for compactness.","PeriodicalId":37863,"journal":{"name":"Nonlinear Engineering - Modeling and Application","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135157157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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