A kinematically admissible continuous velocity field was proposed for the analysis of three-dimensional forging.The linear yield criterion expressed by geometric midline of error triangle between Tresca and Twin shear...A kinematically admissible continuous velocity field was proposed for the analysis of three-dimensional forging.The linear yield criterion expressed by geometric midline of error triangle between Tresca and Twin shear stress yield loci on the π-plane,called GM yield criterion for short,was firstly applied to analysis of the velocity field for the forging.The analytical solution of the forging force with the effects of external zone and bulging parameter is obtained by strain rate inner product.Compression tests of pure lead are performed to compare the calculated results with the measured ones.The results show that the calculated total pressures are higher than the measured ones whilst the relative error is no more than 9.5%.It is implied that the velocity field is reasonable and the geometric midline yield criterion is available.The solution is still an upper-bound one.展开更多
Currently, for some complex plastic deformations, the analytical solution can not be obtained by using Mises yield criterion, because Mises yield criterion is nine dimensions, the velocity field is complex, and the so...Currently, for some complex plastic deformations, the analytical solution can not be obtained by using Mises yield criterion, because Mises yield criterion is nine dimensions, the velocity field is complex, and the solving methods are not innovative. Corresponding solutions of these problems are that yield criterion is linearized to reduce the variable numbers, and the velocity field and the solving methods are reasonably simplified, respectively. In this paper, a new linear yield criterion--mean yield（MY） criterion and inner-product of strain rate vector are used to analytically solve 3D forging taking into account bugling of sides. The velocity field is expressed as a vector in three dimensions, and rotation and divergence are applied to confirm that the velocity field is kinematically admissible. Then, the corresponding strain rate tensor of the velocity field is transformed into principal one by making the determinant of coefficients of the tensor cubic equation be zero. By using MY criterion, the plastic power is term by term integrated and summed according to inner-product of strain rate vector. An upper bound analytical solution is obtained for the forging, and verified by a pure lead press test. The test result turns out that the total pressure calculated by MY criterion is higher by 2.5%-15% than measuring value. In addition, a measuring formula of bulging parameter （a） is proposed, but the values of a measured by the formula are lower than those optimized by the golden section search. The total pressure calculated by MY criterion is compared with the ones by twin shear, Trasca yield, and Mises yield criterion. The comparing result shows that the total pressure calculated by MY criterion is slightly higher than the mean value of that by twin shear and Trasca yield criterion, and lower than that by Mises yield criterion, but more close to that by Mises yield criterion compared with that by other two. The proposed analytical solving methods can be effectively used to other complex plastic deformation, simplifying the solving process and obtaining the reasonable results.展开更多
This technical brief proposes a new approach to multi-dimensional linear time invariant discrete systems within the unity shifted unit circle which is denoted in the form of characteristic equation. The character...This technical brief proposes a new approach to multi-dimensional linear time invariant discrete systems within the unity shifted unit circle which is denoted in the form of characteristic equation. The characteristic equation of multi–dimensional linear system is modified into an equivalent one- dimensional characteristic equation. Further formation of stability in the left of the z-plane, the roots of the characteristic equation f(z) =0 should lie within the shifted unit circle. Using the coefficients of the unity shifted one dimensional equivalent characteristic equation by applying minimal shifting of coefficients either left or right and elimination of coefficient method to two triangular matrixes are formed. A single square matrix is formed by adding the two triangular matrices. This matrix is used for testing the sufficient condition by proposed Jury’s inner determinant concept. Further one more indispensable condition is suggested to show the applicability of the proposed scheme. The proposed method of construction of square matrix consumes less arithmetic operation like shifting and eliminating of coefficients when compare to the construction of square matrix by Jury’s and Hurwitz matrix method.展开更多
This paper proposes a method to ascertain the stability of two dimensional linear time invariant discrete system within the shifted unit circle which is represented by the form of characteristic equation. Further an e...This paper proposes a method to ascertain the stability of two dimensional linear time invariant discrete system within the shifted unit circle which is represented by the form of characteristic equation. Further an equivalent single dimensional characteristic equation is formed from the two dimensional characteristic equation then the stability formulation in the left half of Z-plane, where the roots of characteristic equation f(Z) = 0 should lie within the shifted unit circle. The coefficient of the unit shifted characteristic equation is suitably arranged in the form of matrix and the inner determinants are evaluated using proposed Jury’s concept. The proposed stability technique is simple and direct. It reduces the computational cost. An illustrative example shows the applicability of the proposed scheme.展开更多
基金Project(50474015)supported by the National Natural Science Foundation of China
文摘A kinematically admissible continuous velocity field was proposed for the analysis of three-dimensional forging.The linear yield criterion expressed by geometric midline of error triangle between Tresca and Twin shear stress yield loci on the π-plane,called GM yield criterion for short,was firstly applied to analysis of the velocity field for the forging.The analytical solution of the forging force with the effects of external zone and bulging parameter is obtained by strain rate inner product.Compression tests of pure lead are performed to compare the calculated results with the measured ones.The results show that the calculated total pressures are higher than the measured ones whilst the relative error is no more than 9.5%.It is implied that the velocity field is reasonable and the geometric midline yield criterion is available.The solution is still an upper-bound one.
基金supported by National Natural Science Foundation of China (Grant No. 50474015)State Key Laboratory of Rolling and Automation(RAL) Self-determination Science Foundation of UK (Grant No. RAL_SD_2008_2)
文摘Currently, for some complex plastic deformations, the analytical solution can not be obtained by using Mises yield criterion, because Mises yield criterion is nine dimensions, the velocity field is complex, and the solving methods are not innovative. Corresponding solutions of these problems are that yield criterion is linearized to reduce the variable numbers, and the velocity field and the solving methods are reasonably simplified, respectively. In this paper, a new linear yield criterion--mean yield（MY） criterion and inner-product of strain rate vector are used to analytically solve 3D forging taking into account bugling of sides. The velocity field is expressed as a vector in three dimensions, and rotation and divergence are applied to confirm that the velocity field is kinematically admissible. Then, the corresponding strain rate tensor of the velocity field is transformed into principal one by making the determinant of coefficients of the tensor cubic equation be zero. By using MY criterion, the plastic power is term by term integrated and summed according to inner-product of strain rate vector. An upper bound analytical solution is obtained for the forging, and verified by a pure lead press test. The test result turns out that the total pressure calculated by MY criterion is higher by 2.5%-15% than measuring value. In addition, a measuring formula of bulging parameter （a） is proposed, but the values of a measured by the formula are lower than those optimized by the golden section search. The total pressure calculated by MY criterion is compared with the ones by twin shear, Trasca yield, and Mises yield criterion. The comparing result shows that the total pressure calculated by MY criterion is slightly higher than the mean value of that by twin shear and Trasca yield criterion, and lower than that by Mises yield criterion, but more close to that by Mises yield criterion compared with that by other two. The proposed analytical solving methods can be effectively used to other complex plastic deformation, simplifying the solving process and obtaining the reasonable results.
文摘This technical brief proposes a new approach to multi-dimensional linear time invariant discrete systems within the unity shifted unit circle which is denoted in the form of characteristic equation. The characteristic equation of multi–dimensional linear system is modified into an equivalent one- dimensional characteristic equation. Further formation of stability in the left of the z-plane, the roots of the characteristic equation f(z) =0 should lie within the shifted unit circle. Using the coefficients of the unity shifted one dimensional equivalent characteristic equation by applying minimal shifting of coefficients either left or right and elimination of coefficient method to two triangular matrixes are formed. A single square matrix is formed by adding the two triangular matrices. This matrix is used for testing the sufficient condition by proposed Jury’s inner determinant concept. Further one more indispensable condition is suggested to show the applicability of the proposed scheme. The proposed method of construction of square matrix consumes less arithmetic operation like shifting and eliminating of coefficients when compare to the construction of square matrix by Jury’s and Hurwitz matrix method.
文摘This paper proposes a method to ascertain the stability of two dimensional linear time invariant discrete system within the shifted unit circle which is represented by the form of characteristic equation. Further an equivalent single dimensional characteristic equation is formed from the two dimensional characteristic equation then the stability formulation in the left half of Z-plane, where the roots of characteristic equation f(Z) = 0 should lie within the shifted unit circle. The coefficient of the unit shifted characteristic equation is suitably arranged in the form of matrix and the inner determinants are evaluated using proposed Jury’s concept. The proposed stability technique is simple and direct. It reduces the computational cost. An illustrative example shows the applicability of the proposed scheme.
