Criterion for the occurrence of the macro destruction and formation of breaking during metal deformation

When generalizing the geometrically nonlinear law of Murnaghan elasticity to plasticity, a formally mathematical criterion was introduced for deformational macrofracture (macrocrack appearance) associated with an increase in elastic and plastic anisotropy as a failure cause. The use of the double potentiality of the governing equations in stresses and their velocities made it possible to obtain the reliable information on the structure of the deviatory section of the yield surface, the existence of which is a classical hypothesis in solid mechanics. The normal vector to the surface of the deviatory section is selected from two mutually orthogonal eigenvectors of the constructed operator. There are two families of regular concave surfaces, and a section surface is formed by joining the parts of two representatives of the families at singular points. To select normal vectors, the obtained ratio for them is used for isotropy. In connection with the considered problem of a double simple shift, it is established that multiple eigenvalues appear for the both normal vectors. To unambiguously determine the normal vector at a regular point, it is necessary to exclude the presence of multiple eigenvalues for the both normal vectors at the same time. At a singular point, the appearance of a multiple eigenvalue of one of the normal vectors is still unacceptable. These two conditions are necessary and sufficient to validate the governing equations of the generalized Murnaghan model. Otherwise, a macrocrack occurs. The theoretical construction is supported by the developed software complexes.

Nonlocal multipoint problem for a differential equation of $2n$-th order with operator coefficients

In the article, the spectral properties of a multipoint problem for a differential operator equation of order $2n$ are studied. The operator of the problem has an infinite number of multiple eigenvalues. Each multiple eigenvalue corresponds to a finite set of root functions. A commutative group of transmutation operators is constructed. Each element of the group corresponds to the isospectral perturbation of the problem operator with antiperiodic conditions. The conditions for the existence and uniqueness of the solution are established for the selected family of multipoint problems, and this solution is constructed too.

Solvability and Stability of the Inverse Problem for the Quadratic Differential Pencil

The inverse spectral problem for the second-order differential pencil with quadratic dependence on the spectral parameter is studied. We obtain sufficient conditions for the global solvability of the inverse problem, prove its local solvability and stability. The problem is considered in the general case of complex-valued pencil coefficients and arbitrary eigenvalue multiplicities. Studying local solvability and stability, we take the possible splitting of multiple eigenvalues under a small perturbation of the spectrum into account. Our approach is constructive. It is based on the reduction of the non-linear inverse problem to a linear equation in the Banach space of infinite sequences. The theoretical results are illustrated by numerical examples.

Residual Life Prediction of Metro Traction Motor Bearing Based on Convolutional Neural Network

In order to solve the problem that a single type of sensor cannot fully reflect the bearing life information in the process of bearing residual life prediction of metro traction motor, a bearing residual life prediction method based on multi-information fusion and convolutional neural network is proposed. Firstly, the vibration sensor and acoustic emission sensor are used to collect the bearing life signals on the bearing fatigue life test bench. Secondly, wavelet packet decomposition is used to denoise the collected bearing life signal and extract multiple eigenvalues. On this basis, the multiple eigenvalues are normalized, and the bearing degradation trend is analyzed. Finally, the collected bearing life is divided into five stages, and the processed multiple eigenvalues are fused and input into convolutional neural network for training and recognition. The results show that the probability of predicting the stage of bearing life based on multiple eigenvalues and convolutional neural network is more than 98%.

Iterative Methods for the Computation of the Perron Vector of Adjacency Matrices

The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.

Shape Perturbation of Grushin Eigenvalues

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of $${\mathbb {R}}^N$$ R N . We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter, we prove a Rellich–Nagy-type theorem which describes the bifurcation phenomenon of multiple eigenvalues. As corollaries, we characterize the critical shapes under isovolumetric and isoperimetric perturbations in terms of overdetermined problems and we deduce a new proof of the Rellich–Pohozaev identity for the Grushin eigenvalues.

Axial Piston Pump Fault Diagnosis Method Based on Symmetrical Polar Coordinate Image and Fuzzy C-Means Clustering Algorithm

In this paper, a fault diagnosis method based on symmetric polar coordinate image and Fuzzy C-Means clustering algorithm is proposed to solve the problem that the fault signal of axial piston pump is not intuitive under the time-domain waveform diagram. In this paper, the sampled vibration signals of axial piston pump were denoised firstly by the combination of ensemble empirical mode decomposition and Pearson correlation coefficient. Secondly, the data, after noise reduction, was converted into images, called snowflake images, according to symmetric polar coordinate method. Different fault types of axial piston pump can be identified by observing the snowflake images. After that, in order to evaluate the research results objectively, the obtained images were converted into Gray-Level Cooccurrence Matrixes. Their multiple eigenvalues were extracted, and the eigenvectors consisting of multiple eigenvalues were classified by Fuzzy C-Means clustering algorithm. Finally, according to the accuracy of classification results, the feasibility of applying the symmetric polar coordinate method to axial piston pump fault diagnosis has been validated.