Based on the Hamiltonian method of elastic mechanics, this paper establishes an axisymmetric analytical solution for the vibration of a two-dimensional dodecagonal quasicrystal (QC) cylinders. First, we construct the corresponding Hamiltonian system using the variational principle and rigorously derive analytical solutions for cylindrical vibration under various boundary conditions. In the numerical section, we employ multiple examples to calculate the phonon and phason field frequencies of quasicrystal materials at different size ratios, thereby revealing their vibrational characteristics. We also illustrate the variation patterns of mode shapes corresponding to different frequencies under various boundary constraints. To further validate the reliability of the presented method, we compare our results with existing numerical solutions and finite element data. The findings confirm that the present approach effectively addresses free vibration problems in cylindrical coordinate systems.

In this paper, the symplectic superposition method (SSM) is applied for the first time to the elasticity problem of quasicrystals (QCs), and a symplectic superposition framework applicable to all two-dimensional (2D) in-plane vibration problems of QC plates is presented. The benchmark analytical solution for the in-plane vibration of a 2D dodecagonal QC rectangular thin plate is elegantly expressed by introducing two dimensionless parameters with clear physical significance. Specifically, the Hamiltonian governing equations are rigorously derived via a Legendre transformation. Subsequently, by employing separation of variables and eigenfunction expansion, two sub-problems are formulated and solved. Through suitably superposing the solutions of the two subproblems, the final desired analytical solution is obtained. Remarkably, this method requires no additional assumptions, in contrast to conventional semi-inverse methods. Finally, comparison with phonon-field data validates both the symplectic framework and the analytical solution. The numerical results provide benchmarks for future research, and the obtained mode shapes for phonon and phason fields help deepen understanding of the elastic behavior and physical properties of QCs.

This paper presents a unified analytical framework for the free vibration, buckling, and bending of two-dimensional (2D) decagonal quasicrystal (QC) plates on two-parameter elastic foundations. The governing equations are derived based on Mindlin plate theory, reformulated as a Hamiltonian system, and solved analytically using the symplectic approach (SA). In contrast to traditional semi-inverse methods, the SA introduces dual variables to convert the higher-order differential equations into an equivalent first-order Hamiltonian form, eliminating the need for additional potential functions and simplifying the analytical procedure. The effects of elastic foundations, boundary conditions, geometric parameters, and phason-phonon coupling constants on the mechanical responses are systematically examined. The results show that the phonon field predominantly governs the plate behavior, while the elastic foundation markedly enhances structural stiffness. Boundary conditions and geometric parameters also exert significant influences on vibration frequencies, buckling stability, and bending deflections. These findings provide valuable guidance for the design and performance regulation of QC plates, and the SA employed in this study offers a general framework applicable to plate and shell structures made of other types of QCs.

In this paper, we give the affirmative answers to open problems about structured pseudospectra, proposed by R. Ferro and J. A. Virtanen in [J. Comput. Appl. Math. 322 (2017) 18-24]. Our primary contribution establishes the equivalence between the structured pseudospectra and unstructured pseudospectra for double-structured persymmetric Hankel and symmetric Toeplitz matrices in the complex case. Furthermore, we extend our analysis to analogous problems for centrosymmetric, Hermitian, skew-Hermitian, and circulant double-structured block matrices. The double structures are that the blocks of the given matrix are the same structure as the block matrix.

This paper studies the pseudospectral inclusion property of upper triangular bounded operator matrices in Hilbert spaces. It is proven that the pseudospectra of upper triangular bounded operator matrices are contained within the closure of the quadratic numerical range. Our result extends the inclusion relationship between the spectra of block operator matrices and the quadratic numerical range, as well as the inclusion relationship between the pseudospectra and the numerical range, to the pseudospectral case. Under appropriate conditions, we characterize the distribution range of the pseudospectra of upper triangular bounded operator matrices, and provide an example to illustrate the validity of the conclusions.

In this paper, the symplectic elasticity approach is applied to analyze the thermoelastic behavior of a simply supported, two-dimensional hexagonal quasicrystal plate with the local thermal load applied to its top surface. Specifically, by employing the symplectic approach, general solutions for the extended displacement and stress components are derived. By expressing the local thermal load as a double Fourier series, an analytical solution with a concise, elegant expression is obtained. Three different scales of local thermal loads are applied to the surface of the quasicrystal plate, and the response of the plate is investigated, as well as the response of the same homogeneous quasicrystal plate under the local thermal load with different stress-temperature coefficients. Numerical examples illustrate how the stacking sequence influences the bending deformation of multilayered plates composed of quasicrystal and crystal materials. In addition, a systematic parametric study is conducted to assess the individual effects of phonon-phason coupling constants and load amplitude. The relative error between this analytical solution and the three-dimensional finite-element solution is within 0.5

Given a bounded positive linear operator A on a Hilbert space 𝒳 , this operator induces a semi-Hilbertian structure on 𝒳 . For a bounded linear operator T defined on the corresponding semi-Hilbertian space, we introduce a new definition of a Fredholm operator that is compatible with the semi-Hilbertian structure. Based on this definition, we proceed to define the essential spectra of T and establish their fundamental properties. Within this framework, we consider the bounded off-diagonal operator matrix 𝕋 = [ 0 M; N 0 ] acting on the semi-Hilbertian space and demonstrate that the essential spectra of 𝕋 are entirely determined by the essential spectra of the products MN and NM. Finally, an illustrative example is provided to substantiate the theoretical conclusions.

Let P and Q be two orthogonal projections in Hilbert space H. For alpha,beta E C\{0}, the lower bound and the upper bound of the pseudospectra of alpha P + beta Q are obtained. The bounds are represented by the product PQ which are independent of the choice of scalars alpha, beta. For alpha, beta E C\{0}, alpha + beta not equal xi, xi is an element of C, the bounds of the pseudospectra of alpha P + beta Q-xi PQ are also obtained in the same way. Finally, two examples are constructed to show the effectiveness of the results.

A Hamiltonian system is developed for the three-dimensional (3D) problem of two-dimensional (2D) decagonal piezoelectric quasicrystals via the variational principle. Based on the full state vector and the properties of the Hamiltonian operator matrix, the superposition principle of solutions obtains the symplectic analytical solutions of the problem under simply supported boundary conditions. Numerical examples are illustrated to display the effects of the stacking sequences and material constants on the stresses, displacements, electric potential, and electric displacements under the mechanical and electric displacement loadings. The symplectic analytical solutions presented in the article can be used as a reference for further numerical research.

The Berry-Tabor (BT) conjecture is a famous statistical inference in quantum chaos, which not only establishes the spectral fluctuations of quantum systems whose classical counterparts are integrable but can also be used to describe other wave phenomena. In this paper, the BT conjecture has been extended to L & eacute;vy plates. As predicted by the BT conjecture, level clustering is present in the spectra of L & eacute;vy plates. The consequence of level clustering is studied by introducing the distribution of nearest neighbor frequency level spacing ratios P(r similar to) , which is calculated through the analytical solution obtained by the Hamiltonian approach. Our work investigates the impact of varying foundation parameters, rotary inertia, and boundary conditions on the frequency spectra, and we find that P(r similar to) conforms to a Poisson distribution in all cases. The reason for the occurrence of the Poisson distribution in the L & eacute;vy plates is the independence between modal frequencies, which can be understood through mode functions.

A Hamiltonian system is derived for the plane elasticity problem of two-dimensional dodecagonal quasicrystals by introducing the simple state function. By using symplectic elasticity approach, the analytic solutions of the phonon and phason displacements are obtained further for the quasicrystal plates. In addition, the effectiveness of the approach is verified by comparison with the data of the finite integral transformation method.