In this paper, we solve a problem on the existence of conjugate symplecticity of linear multi-step methods (LMSM), the negative result is obtained. [ABSTRACT FROM AUTHOR]
We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two...We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni （2014） in an earlier paper.展开更多
The symplectic approach was utilized to derive solutions to the orthotropic micropolar plane stress problem.The Hamiltonian canonical equation was first obtained by applying Legendre’s transformation and the Hamilton...The symplectic approach was utilized to derive solutions to the orthotropic micropolar plane stress problem.The Hamiltonian canonical equation was first obtained by applying Legendre’s transformation and the Hamiltonian mixed energy variational principle.Then,by using the method of separation of variables,the eigenproblem of the corresponding homogeneous Hamiltonian canonical equation was derived.Subsequently,the corresponding eigensolutions for three kinds of homogeneous boundary conditions were derived.According to the adjoint symplectic orthogonality of the eigensolutions and expansion theorems,the solutions to this plane stress problem were expressed as a series expansion of these eigensolutions.The numerical results for the orthotropic micropolar plane stress problem under various boundary conditions were presented and validated using the finite element method,which confirmed the convergence and accuracy of the proposed approach.We also investigated the relationship between the size-dependent behaviour and material parameters using the proposed approach.Furthermore,this approach was applied to analyze lattice structures under an equivalent micropolar continuum approximation.展开更多
This paper proposes a linear companding transform(CT)using either a single inflection point or two inflection points to reduce the peakto-average power ratio(PAPR)in orthogonal timefrequency space(OTFS)signals.The CT ...This paper proposes a linear companding transform(CT)using either a single inflection point or two inflection points to reduce the peakto-average power ratio(PAPR)in orthogonal timefrequency space(OTFS)signals.The CT strategically compresses higher amplitudes and enhances lower amplitudes based on carefully chosen scaling factors and points of inflection.With these selected parameters,the CT effectively reduces peak power while maintaining average power,leading to a substantial decrease in PAPR.We analyze noise changes in the inverse companding transform(ICT)process.The analysis reveals that the ICT amplifies less than 20%of the total noise.A convolutional encoder and soft decision Viterbi decoding algorithm are utilized in the OTFS system to improve the detection performance.We present simulation results focusing on PAPR reduction and bit error rate(BER)performance.These results demonstrate that the CT with two inflection points outperforms both the single inflection point case and the existingμ-law companding,clipping,peak windowing,unique OTFS frame structure,selected mapping,and partial transmit sequence methods,achieving significant PAPR reduction and BER performance.展开更多
In recent years,magneto-electro-elastic(MEE)cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting.To aid the design of such energy harvesting device...In recent years,magneto-electro-elastic(MEE)cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting.To aid the design of such energy harvesting devices,an accurate free vibration analysis of embedded MEE cylindrical shells with step-wise thicknesses is performed within the framework of symplectic mechanics.By using the Legendre transformation,a new known vector is defined to transform the higher-order partial differential governing equations into a set of lower-order ordinary differential equations.Therefore,the original vibration analysis is regarded as an eigen problem in the symplectic space,and analytical solutions can be represented by the symplectic series.In numerical examples,the new analytical solutions are compared with the existing results,and good agreement is observed.Furthermore,the effects of critical design parameters on free vibration characteristics are thoroughly investigated.All numerical results can serve as benchmarks for the development of other approximate or numerical methods.展开更多
Symplectic symmetry approach to clustering(SSAC)in atomic nuclei,recently proposed,is modified and further developed in more detail.It is firstly applied to the light two-cluster^(20)Ne+αsystem of^(24)Mg,the latter e...Symplectic symmetry approach to clustering(SSAC)in atomic nuclei,recently proposed,is modified and further developed in more detail.It is firstly applied to the light two-cluster^(20)Ne+αsystem of^(24)Mg,the latter exhibiting well developed low-energy K^(π)=0_(1)^(+),k^(π)=2_(1)^(+) and π^(π)=0_(1)^(-) rotational bands in its spectrum.A simple algebraic Hamiltonian,consisting of dynamical symmetry,residual and vertical mixing parts is used to describe these three lowest rotational bands of positive and negative parity in^(24)Mg.A good description of the excitation energies is obtained by considering only the SU(3)cluster states restricted to the stretched many-particle Hilbert subspace,built on the leading Pauli allowed SU(3)multiplet for the positive-and negative-parity states,respectively.The coupling to the higher cluster-model configurations allows us to describe the known low-lying experimentally observed B(E2)transition probabilities within and between the cluster states of the three bands under consideration without the use of an effective charge.展开更多
We propose a symplectic partitioned Runge-Kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this te...We propose a symplectic partitioned Runge-Kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete （NAD） operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction （LWC） method and the eighth-order staggered-grid （SG）method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.展开更多
The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta （RK） method is an important part of the former, and the ...The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta （RK） method is an important part of the former, and the continuous finite element method （CFEM） belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.展开更多
The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of d...The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. The 3-dimensional problem of magneto-electro-elastic structure which is investigated in Euclidean space commonly is converted into symplectic system. At the same time the Lagrange system is converted into Hamiltonian system. As an example, the dynamic characteristics of the simply supported functionally graded magneto-electro-elastic material （FGMM） plate and pipe are investigated. Finally, the problem is solved by symplectic algorithm. The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces between layers under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. The dynamic stiffness is increased by the piezoelectric effect while decreased by the piezomagnetic effect.展开更多
In this paper the geometric meaning of robot systems is expounded based on the theory of multibody system. The error accumulation for the known algorithm is analyzed and the cause of ‘Energy consumption’ is revealed...In this paper the geometric meaning of robot systems is expounded based on the theory of multibody system. The error accumulation for the known algorithm is analyzed and the cause of ‘Energy consumption’ is revealed, the relationship between the coefficients of dynamic equation is derived so as to establish the canonical equations. The error accumulation of dynamics can be eliminated by using canonical equations and the symplectic integral method so that the computational accuracy can be ensured effectively. As an example, a planar robotics system is considered.展开更多
For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process...For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.展开更多
The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-f...The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.展开更多
A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry...A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of Birkhoffian system is discussed, then the symplecticity of Birkhoffian phase flow is presented. Based on these properties we give a way to construct symplectic schemes for Birkhoffian systems by using the generating function method.展开更多
In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference...In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler?Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler?Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler?Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler?Lagrange cohomological conditions are satisfied.展开更多
In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in...In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler?Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler?Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler?Lagrange cohomological conditions are satisfied.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
Multibody system dynamics provides a strong tool for the estimation of dynamic performances and the optimization of multisystem robot design. It can be described with differential algebraic equations(DAEs). In this pa...Multibody system dynamics provides a strong tool for the estimation of dynamic performances and the optimization of multisystem robot design. It can be described with differential algebraic equations(DAEs). In this paper, a particle swarm optimization(PSO) method is introduced to solve and control a symplectic multibody system for the first time. It is first combined with the symplectic method to solve problems in uncontrolled and controlled robotic arm systems. It is shown that the results conserve the energy and keep the constraints of the chaotic motion, which demonstrates the efficiency, accuracy, and time-saving ability of the method. To make the system move along the pre-planned path, which is a functional extremum problem, a double-PSO-based instantaneous optimal control is introduced. Examples are performed to test the effectiveness of the double-PSO-based instantaneous optimal control. The results show that the method has high accuracy, a fast convergence speed, and a wide range of applications.All the above verify the immense potential applications of the PSO method in multibody system dynamics.展开更多
The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonia...The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.展开更多
基金China State Major Key Project for Basic ResearchesNational Natural Science Foundation of China! (No. 19801034)Bureau of
文摘In this paper, we solve a problem on the existence of conjugate symplecticity of linear multi-step methods (LMSM), the negative result is obtained. [ABSTRACT FROM AUTHOR]
基金supported by National Science Foundation of USA(Grant No.DMS1252992)an Alfred P.Sloan Research Fellowship
文摘We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S2× S1 or RP3#P3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni （2014） in an earlier paper.
基金supported by the National Key R&D Program of China (Grant No.2022YFB4201200)Technology Major Project (Grant No.J2019-IV-0019-0087)National Science and Technology Major Project (Grant No.J2019-IV-0019-0087).
文摘The symplectic approach was utilized to derive solutions to the orthotropic micropolar plane stress problem.The Hamiltonian canonical equation was first obtained by applying Legendre’s transformation and the Hamiltonian mixed energy variational principle.Then,by using the method of separation of variables,the eigenproblem of the corresponding homogeneous Hamiltonian canonical equation was derived.Subsequently,the corresponding eigensolutions for three kinds of homogeneous boundary conditions were derived.According to the adjoint symplectic orthogonality of the eigensolutions and expansion theorems,the solutions to this plane stress problem were expressed as a series expansion of these eigensolutions.The numerical results for the orthotropic micropolar plane stress problem under various boundary conditions were presented and validated using the finite element method,which confirmed the convergence and accuracy of the proposed approach.We also investigated the relationship between the size-dependent behaviour and material parameters using the proposed approach.Furthermore,this approach was applied to analyze lattice structures under an equivalent micropolar continuum approximation.
文摘This paper proposes a linear companding transform(CT)using either a single inflection point or two inflection points to reduce the peakto-average power ratio(PAPR)in orthogonal timefrequency space(OTFS)signals.The CT strategically compresses higher amplitudes and enhances lower amplitudes based on carefully chosen scaling factors and points of inflection.With these selected parameters,the CT effectively reduces peak power while maintaining average power,leading to a substantial decrease in PAPR.We analyze noise changes in the inverse companding transform(ICT)process.The analysis reveals that the ICT amplifies less than 20%of the total noise.A convolutional encoder and soft decision Viterbi decoding algorithm are utilized in the OTFS system to improve the detection performance.We present simulation results focusing on PAPR reduction and bit error rate(BER)performance.These results demonstrate that the CT with two inflection points outperforms both the single inflection point case and the existingμ-law companding,clipping,peak windowing,unique OTFS frame structure,selected mapping,and partial transmit sequence methods,achieving significant PAPR reduction and BER performance.
基金Project supported by the Science and Technology Plan Joint Program of Liaoning Province of China(Natural Science Foundation-Doctoral Research Launch Project)(No.2024-BSLH-027)the Fundamental Research Funds for Undergraduate Universities of Liaoning Province of China(No.LJBKY2024033)+1 种基金the National Natural Science Foundation of China(No.12472064)the Natural Science Foundation of Liaoning Province of China(No.2023-MS-118)。
文摘In recent years,magneto-electro-elastic(MEE)cylindrical shells with step-wise thicknesses have shown significant potential in the field of vibration energy harvesting.To aid the design of such energy harvesting devices,an accurate free vibration analysis of embedded MEE cylindrical shells with step-wise thicknesses is performed within the framework of symplectic mechanics.By using the Legendre transformation,a new known vector is defined to transform the higher-order partial differential governing equations into a set of lower-order ordinary differential equations.Therefore,the original vibration analysis is regarded as an eigen problem in the symplectic space,and analytical solutions can be represented by the symplectic series.In numerical examples,the new analytical solutions are compared with the existing results,and good agreement is observed.Furthermore,the effects of critical design parameters on free vibration characteristics are thoroughly investigated.All numerical results can serve as benchmarks for the development of other approximate or numerical methods.
文摘Symplectic symmetry approach to clustering(SSAC)in atomic nuclei,recently proposed,is modified and further developed in more detail.It is firstly applied to the light two-cluster^(20)Ne+αsystem of^(24)Mg,the latter exhibiting well developed low-energy K^(π)=0_(1)^(+),k^(π)=2_(1)^(+) and π^(π)=0_(1)^(-) rotational bands in its spectrum.A simple algebraic Hamiltonian,consisting of dynamical symmetry,residual and vertical mixing parts is used to describe these three lowest rotational bands of positive and negative parity in^(24)Mg.A good description of the excitation energies is obtained by considering only the SU(3)cluster states restricted to the stretched many-particle Hilbert subspace,built on the leading Pauli allowed SU(3)multiplet for the positive-and negative-parity states,respectively.The coupling to the higher cluster-model configurations allows us to describe the known low-lying experimentally observed B(E2)transition probabilities within and between the cluster states of the three bands under consideration without the use of an effective charge.
基金This research was supported by the National Natural Science Foundation of China （Nos. 41230210 and 41204074）, the Science Foundation of the Education Department of Yunnan Province （No. 2013Z152）, and Statoil Company （Contract No. 4502502663）.
文摘We propose a symplectic partitioned Runge-Kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete （NAD） operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction （LWC） method and the eighth-order staggered-grid （SG）method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.
基金Project supported by the National Natural Science Foundation of China (No. 11071067)the Hunan Graduate Student Science and Technology Innovation Project (No. CX2011B184)
文摘The symplectic algorithm and the energy conservation algorithm are two important kinds of algorithms to solve Hamiltonian systems. The symplectic Runge- Kutta （RK） method is an important part of the former, and the continuous finite element method （CFEM） belongs to the later. We find and prove the equivalence of one kind of the implicit RK method and the CFEM, give the coefficient table of the CFEM to simplify its computation, propose a new standard to measure algorithms for Hamiltonian systems, and define another class of algorithms --the regular method. Finally, numerical experiments are given to verify the theoretical results.
文摘The 3-dimensional couple equations of magneto-electro-elastic structures are derived under Hamiltonian system based on the Hamilton principle. The problem of single sort of variables is converted into the problem of double sorts of variables, and the Hamilton canonical equations are established. The 3-dimensional problem of magneto-electro-elastic structure which is investigated in Euclidean space commonly is converted into symplectic system. At the same time the Lagrange system is converted into Hamiltonian system. As an example, the dynamic characteristics of the simply supported functionally graded magneto-electro-elastic material （FGMM） plate and pipe are investigated. Finally, the problem is solved by symplectic algorithm. The results show that the physical quantities of displacement, electric potential and magnetic potential etc. change continuously at the interfaces between layers under the transverse pressure while some other physical quantities such as the stress, electric and magnetic displacement are not continuous. The dynamic stiffness is increased by the piezoelectric effect while decreased by the piezomagnetic effect.
文摘In this paper the geometric meaning of robot systems is expounded based on the theory of multibody system. The error accumulation for the known algorithm is analyzed and the cause of ‘Energy consumption’ is revealed, the relationship between the coefficients of dynamic equation is derived so as to establish the canonical equations. The error accumulation of dynamics can be eliminated by using canonical equations and the symplectic integral method so that the computational accuracy can be ensured effectively. As an example, a planar robotics system is considered.
基金supported by the National Natural Science Foundations of China (Grant 11502286)
文摘For the stability requirement of numerical resultants, the mathematical theory of classical mixed methods are relatively complex. However, generalized mixed methods are automatically stable, and their building process is simple and straightforward. In this paper, based on the seminal idea of the generalized mixed methods, a simple, stable, and highly accurate 8-node noncompatible symplectic element(NCSE8) was developed by the combination of the modified Hellinger-Reissner mixed variational principle and the minimum energy principle. To ensure the accuracy of in-plane stress results, a simultaneous equation approach was also suggested. Numerical experimentation shows that the accuracy of stress results of NCSE8 are nearly the same as that of displacement methods, and they are in good agreement with the exact solutions when the mesh is relatively fine. NCSE8 has advantages of the clearing concept, easy calculation by a finite element computer program, higher accuracy and wide applicability for various linear elasticity compressible and nearly incompressible material problems. It is possible that NCSE8 becomes even more advantageous for the fracture problems due to its better accuracy of stresses.
基金supported by the National Natural Science Foundation of China (10772014)
文摘The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.
基金The project supported by the Special Funds for State Key Basic Research Projects under Grant No.G1999,032800
文摘A universal symplectic structure for a Newtonian system including nonconservative cases can be constructed in the framework of Birkhoffian generalization of Hamiltonian mechanics. In this paper the symplectic geometry structure of Birkhoffian system is discussed, then the symplecticity of Birkhoffian phase flow is presented. Based on these properties we give a way to construct symplectic schemes for Birkhoffian systems by using the generating function method.
文摘In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler?Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler?Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler?Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler?Lagrange cohomological conditions are satisfied.
文摘In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler?Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler?Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler?Lagrange cohomological conditions are satisfied.
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 the National Natural Science Foundation of China(Nos.91648101 and11672233)the Northwestern Polytechnical University(NPU)Foundation for Fundamental Research(No.3102017AX008)the National Training Program of Innovation and Entrepreneurship for Undergraduates(No.S201710699033)
文摘Multibody system dynamics provides a strong tool for the estimation of dynamic performances and the optimization of multisystem robot design. It can be described with differential algebraic equations(DAEs). In this paper, a particle swarm optimization(PSO) method is introduced to solve and control a symplectic multibody system for the first time. It is first combined with the symplectic method to solve problems in uncontrolled and controlled robotic arm systems. It is shown that the results conserve the energy and keep the constraints of the chaotic motion, which demonstrates the efficiency, accuracy, and time-saving ability of the method. To make the system move along the pre-planned path, which is a functional extremum problem, a double-PSO-based instantaneous optimal control is introduced. Examples are performed to test the effectiveness of the double-PSO-based instantaneous optimal control. The results show that the method has high accuracy, a fast convergence speed, and a wide range of applications.All the above verify the immense potential applications of the PSO method in multibody system dynamics.
基金Supported by the National Natural Science Foundation of China under Grant No.10962004the Natural Science Foundation of Inner Mongolia under Grant No.2009BS0101+1 种基金the Specialized Research Fund for the Doctoral Program of Higher Education of China under Grant No.20070126002the Cultivation of Innovative Talent of "211 Project"of Inner Mongolia University
文摘The free vibration problem of rectangular thin plates is rewritten as a new upper triangular matrix differential system. For the associated operator matrix, we find that the two diagonal block operators are Hamiltonian. Moreover, the existence and completeness of normed symplectic orthogonal eigenfunction systems of these two block operators are demonstrated. Based on the completeness, the general solution of the free vibration of rectangular thin plates is given by double symplectie eigenfunction expansion method.
