The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation metho...The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.展开更多
For the proportional directional valve controlled by two proportional solenoids, the normal control method(NCM) energizes only one solenoid at a time. The performance of the valve is greatly influenced by the nonlinea...For the proportional directional valve controlled by two proportional solenoids, the normal control method(NCM) energizes only one solenoid at a time. The performance of the valve is greatly influenced by the nonlinearity of the proportional solenoid, such as dead zone and low force gain with a small current, and this effect cannot be eliminated by a simple dead-zone current compensation. To avoid this disadvantage, we propose the differential control method(DCM). By employing DCM, the controller outputs differential signals to simultaneously energize both solenoids of the proportional valve, and the operating point is found by analyzing the force output of the two solenoids to make a minimum variation of the current force gain. The comparisons of the valve response characteristics are made between NCM and DCM by nonlinear dynamic simulation and experiments. Simulation and experimental results show that by using DCM, the frequency response of the valve is greatly enhanced, especially when the input is small, which means that the dynamic characteristics of the proportional valve are improved.展开更多
Based on the Boltzmann's superposition principles of linear viscoelastic materials and the von Karman's hypotheses of thin plates with large deflections, a mathematical model for quasi-static problems of visco...Based on the Boltzmann's superposition principles of linear viscoelastic materials and the von Karman's hypotheses of thin plates with large deflections, a mathematical model for quasi-static problems of viscoelastic thin plates was given. By the Galerkin method in spatial domain, the original integro-partial-differential system could be transformed into an integral system. The latter further was reduced to a differential system by using the new method for temporal domain presented in this paper. Numerical results show that compared with the ordinary finite difference method, the new method in this paper is simpler to operate and has some advantages, such as, no storage and quicker computational speed etc.展开更多
Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems' objective functions have no derivatives. According to former research studies, some sear...Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems' objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843% of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360°, because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05°. Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71% less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The展开更多
In this paper, a new elimination of finite differential equations has been discussed. It applies the numerical direct iteration to obtain the residual equations, in which the number of unknowns has been reduced greatl...In this paper, a new elimination of finite differential equations has been discussed. It applies the numerical direct iteration to obtain the residual equations, in which the number of unknowns has been reduced greatly. The solution process is simple and efficient, and the solution is exact展开更多
Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain condition...Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.展开更多
This paper considers a kind of seventh order nonlinear differential equations with a deviating argument.By means of the Lyapunov direct method,some sufficient conditions are established to show the instability of the ...This paper considers a kind of seventh order nonlinear differential equations with a deviating argument.By means of the Lyapunov direct method,some sufficient conditions are established to show the instability of the zero solution to the equation.Our result is new and complements the corresponding result of [5].展开更多
文摘The behavior of beams with variable stiffness subjected to the action of variable loadings (impulse or harmonic) is analyzed in this paper using the successive approximation method. This successive approximation method is a technique for numerical integration of partial differential equations involving both the space and time, with well-known initial conditions on time and boundary conditions on the space. This technique, although having been applied to beams with constant stiffness, is new for the case of beams with variable stiffness, and it aims to use a quadratic parabola (in time) to approximate the solutions of the differential equations of dynamics. The spatial part is studied using the successive approximation method of the partial differential equations obtained, in order to transform them into a system of time-dependent ordinary differential equations. Thus, the integration algorithm using this technique is established and applied to examples of beams with variable stiffness, under variable loading, and with the different cases of supports chosen in the literature. We have thus calculated the cases of beams with constant or variable rigidity with articulated or embedded supports, subjected to the action of an instantaneous impulse and harmonic loads distributed over its entire length. In order to justify the robustness of the successive approximation method considered in this work, an example of an articulated beam with constant stiffness subjected to a distributed harmonic load was calculated analytically, and the results obtained compared to those found numerically for various steps (spatial h and temporal τ ¯ ) of calculus, and the difference between the values obtained by the two methods was small. For example for ( h=1/8 , τ ¯ =1/ 64 ), the difference between these values is 17%.
基金Project supported by the National Natural Science Foundation of China(No.51221004)the Program for Zhejiang Leading Team of S&T Innovation(No.2010R50036)
文摘For the proportional directional valve controlled by two proportional solenoids, the normal control method(NCM) energizes only one solenoid at a time. The performance of the valve is greatly influenced by the nonlinearity of the proportional solenoid, such as dead zone and low force gain with a small current, and this effect cannot be eliminated by a simple dead-zone current compensation. To avoid this disadvantage, we propose the differential control method(DCM). By employing DCM, the controller outputs differential signals to simultaneously energize both solenoids of the proportional valve, and the operating point is found by analyzing the force output of the two solenoids to make a minimum variation of the current force gain. The comparisons of the valve response characteristics are made between NCM and DCM by nonlinear dynamic simulation and experiments. Simulation and experimental results show that by using DCM, the frequency response of the valve is greatly enhanced, especially when the input is small, which means that the dynamic characteristics of the proportional valve are improved.
文摘Based on the Boltzmann's superposition principles of linear viscoelastic materials and the von Karman's hypotheses of thin plates with large deflections, a mathematical model for quasi-static problems of viscoelastic thin plates was given. By the Galerkin method in spatial domain, the original integro-partial-differential system could be transformed into an integral system. The latter further was reduced to a differential system by using the new method for temporal domain presented in this paper. Numerical results show that compared with the ordinary finite difference method, the new method in this paper is simpler to operate and has some advantages, such as, no storage and quicker computational speed etc.
基金supported in part by the Teaching Reform Research Foundation of Shengli College in China University of Petroleum(East China)(JG201725)the Natural Science Foundation Shandong Province of China(ZR2018PEE009)the Project of Science and Technology of Shandong Universities in China(J17KA044,J17KB061)。
文摘Many effective optimization algorithms require partial derivatives of objective functions, while some optimization problems' objective functions have no derivatives. According to former research studies, some search directions are obtained using the quadratic hypothesis of objective functions. Based on derivatives, quadratic function assumptions, and directional derivatives, the computational formulas of numerical first-order partial derivatives, second-order partial derivatives, and numerical second-order mixed partial derivatives were constructed. Based on the coordinate transformation relation, a set of orthogonal vectors in the fixed coordinate system was established according to the optimization direction. A numerical algorithm was proposed, taking the second order approximation direction as an example. A large stepsize numerical algorithm based on coordinate transformation was proposed. Several algorithms were validated by an unconstrained optimization of the two-dimensional Rosenbrock objective function. The numerical second order approximation direction with the numerical mixed partial derivatives showed good results. Its calculated amount is 0.2843% of that of without second-order mixed partial derivative. In the process of rotating the local coordinate system 360°, because the objective function is more complex than the quadratic function, if the numerical direction derivative is used instead of the analytic partial derivative, the optimization direction varies with a range of 103.05°. Because theoretical error is in the numerical negative gradient direction, the calculation with the coordinate transformation is 94.71% less than the calculation without coordinate transformation. If there is no theoretical error in the numerical negative gradient direction or in the large-stepsize numerical optimization algorithm based on the coordinate transformation, the sawtooth phenomenon occurs. When each numerical mixed partial derivative takes more than one point, the optimization results cannot be improved. The
文摘In this paper, a new elimination of finite differential equations has been discussed. It applies the numerical direct iteration to obtain the residual equations, in which the number of unknowns has been reduced greatly. The solution process is simple and efficient, and the solution is exact
文摘Using direct algebraic method,exact solitary wave solutions are performed for a class of third order nonlinear dispersive disipative partial differential equations. These solutions are obtained under certain conditions for the relationship between the coefficients of the equation. The exact solitary waves of this class are rational functions of real exponentials of kink-type solutions.
文摘This paper considers a kind of seventh order nonlinear differential equations with a deviating argument.By means of the Lyapunov direct method,some sufficient conditions are established to show the instability of the zero solution to the equation.Our result is new and complements the corresponding result of [5].
