By a novel approach proposed by Luo, the unconventional Hamilton-type variational principle in phase space for elastodynamics of multidegree-of-freedom system is established in this paper. It not only can fully charac...By a novel approach proposed by Luo, the unconventional Hamilton-type variational principle in phase space for elastodynamics of multidegree-of-freedom system is established in this paper. It not only can fully characterize the initial-value problem of this dynamic, but also has a natural symplectic structure. Based on this variational principle, a symplectic algorithm which is called a symplectic time-subdomain method is proposed. A non-difference scheme is constructed by applying Lagrange interpolation polynomial to the time subdomain. Furthermore, it is also proved that the presented symplectic algorithm is an unconditionally stable one. From the results of the two numerical examples of different types, it can be seen that the accuracy and the computational efficiency of the new method excel obviously those of widely used Wilson-? and Newmark-? methods. Therefore, this new algorithm is a highly efficient one with better computational performance.展开更多
This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish d...This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.展开更多
The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
With the help of the coordinate transformation technique, the symplectic dual solv- ing system is developed for multi-material wedges under antiplane deformation. A virtue of present method is that the compatibility c...With the help of the coordinate transformation technique, the symplectic dual solv- ing system is developed for multi-material wedges under antiplane deformation. A virtue of present method is that the compatibility conditions at interfaces of a multi-material wedge are expressed directly by the dual variables, therefore the governing equation of eigenvalue can be derived easily even with the increase of the material number. Then, stress singularity on multi-material wedges under antiplane deformation is investigated, and some solutions can be presented to show the validity of the method. Simultaneously, an interesting phenomenon is found and proved strictly that one of the singularities of a special five-material wedge is independent of the crack direction.展开更多
Non-self-adjoint quasi-differential expression M and its formal adjoint M＋may generate nonsymmetric ordinary differential operators. Although minimal operators T0, T＋0 generated by M, M＋are not symmetric, they form...Non-self-adjoint quasi-differential expression M and its formal adjoint M＋may generate nonsymmetric ordinary differential operators. Although minimal operators T0, T＋0 generated by M, M＋are not symmetric, they form an adjoint pair. In this paper, author studies regularly solvable operators with respect to the adjoint pair T0, T＋0 in two kinds of conditions and give their geometry description in the corresponding ways.展开更多
We study the dynamics of the cubic–quintic nonlinear Schrodinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions fr...We study the dynamics of the cubic–quintic nonlinear Schrodinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter.展开更多
According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the...According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space, which consists of original variables and their dual variables . The eigenvalue - 1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate . In general, the eigenvalue - 1 is a single eigenvalue, and the classical solution of an elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue - 1. But the eigenvalue - 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment, that is, the paradox should occur. Here the Jordan form eigenfunction vector for eigenvalue - 1 exists, and solution of the paradox on elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is obtained directly by solving this special Jordan form eigenfunction. The result shows again that the solutions of the special paradox on elastic wedge in the classical theory of elasticity are just Jordan form solutions in symplectic space under Hamiltonian system .展开更多
基金the National Natural Seienee Foundation of China(Grant Nos10172097,19902022,19672074)
文摘By a novel approach proposed by Luo, the unconventional Hamilton-type variational principle in phase space for elastodynamics of multidegree-of-freedom system is established in this paper. It not only can fully characterize the initial-value problem of this dynamic, but also has a natural symplectic structure. Based on this variational principle, a symplectic algorithm which is called a symplectic time-subdomain method is proposed. A non-difference scheme is constructed by applying Lagrange interpolation polynomial to the time subdomain. Furthermore, it is also proved that the presented symplectic algorithm is an unconditionally stable one. From the results of the two numerical examples of different types, it can be seen that the accuracy and the computational efficiency of the new method excel obviously those of widely used Wilson-? and Newmark-? methods. Therefore, this new algorithm is a highly efficient one with better computational performance.
基金supported by the National Natural Science Foundation of China (10772039 and 10632030)the National Basic Research Program of China (973 Program) (2010CB832704)
文摘This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.
基金Project supported by NSFC （11371124, 11301150） and the Natural Science Foundation of Henan Province of China （142300410134, 162300410066）.
文摘The automorphism group of G is determined, where G is a nonabelian p-group given by a central extension as 1→Zpm→G→Zp×…×Zp→1 such that its derived subgroup has order p.
基金Project supported by the National Natural Science Foundation of China (No. 10772039)the National Basic Research Program of China (Program 973, No. 2010CB832704)
文摘With the help of the coordinate transformation technique, the symplectic dual solv- ing system is developed for multi-material wedges under antiplane deformation. A virtue of present method is that the compatibility conditions at interfaces of a multi-material wedge are expressed directly by the dual variables, therefore the governing equation of eigenvalue can be derived easily even with the increase of the material number. Then, stress singularity on multi-material wedges under antiplane deformation is investigated, and some solutions can be presented to show the validity of the method. Simultaneously, an interesting phenomenon is found and proved strictly that one of the singularities of a special five-material wedge is independent of the crack direction.
文摘Non-self-adjoint quasi-differential expression M and its formal adjoint M＋may generate nonsymmetric ordinary differential operators. Although minimal operators T0, T＋0 generated by M, M＋are not symmetric, they form an adjoint pair. In this paper, author studies regularly solvable operators with respect to the adjoint pair T0, T＋0 in two kinds of conditions and give their geometry description in the corresponding ways.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11301350,11472124,and 11271158)the Doctor Start-up Fund in Liaoning Province,China(Grant No.20141050)
文摘We study the dynamics of the cubic–quintic nonlinear Schrodinger equation by the symplectic method. The behaviors of the equation are discussed with harmonically modulated initial conditions, and the contributions from the quintic term are discussed. We observe the elliptic orbit, homoclinic orbit crossing, quasirecurrence, and stochastic motion with different nonlinear parameters in this system. Numerical simulations show that the changing processes of the motion of the system and the trajectories in the phase space are various for different cubic nonlinear parameters with the increase of the quintic nonlinear parameter.
基金Foundation items: the National Natural Science Foundation of China (10172021) the Doctoral Point Foundation of Education Minitry PRC(20010141024)
文摘According to the Hellinger-Reissner variational principle and introducing proper transformation of variables , the problem on elastic wedge dissimilar materials can be led to Hamiltonian system, so the solution of the problem can be got by employing the separation of variables method and symplectic eigenfunction expansion under symplectic space, which consists of original variables and their dual variables . The eigenvalue - 1 is a special one of all symplectic eigenvalue for Hamiltonian system in polar coordinate . In general, the eigenvalue - 1 is a single eigenvalue, and the classical solution of an elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is got directly by solving the eigenfunction vector for eigenvalue - 1. But the eigenvalue - 1 becomes a double eigenvalue when the vertex angles and modulus of the materials satisfy certain definite relationships and the classical solution for the stress distribution becomes infinite at this moment, that is, the paradox should occur. Here the Jordan form eigenfunction vector for eigenvalue - 1 exists, and solution of the paradox on elastic wedge dissimilar materials subjected to a unit concentrated couple at the vertex is obtained directly by solving this special Jordan form eigenfunction. The result shows again that the solutions of the special paradox on elastic wedge in the classical theory of elasticity are just Jordan form solutions in symplectic space under Hamiltonian system .
