Complex Eigenvalues Research Articles

Boundary value problem for a system of partial differential equations with the Dzhrbashyan–Nersesyan fractional differentiation operators

A boundary value problem in a rectangular domain for a system of partial differential equations with the Dzhrbashyan–Nersesyan fractional differentiation operators with constant coefficients is studied in the case when the matrix coefficients of the system have complex eigenvalues. Existence and uniqueness theorems for the solution to the boundary value problem under study are proved. The solution is constructed explicitly in terms of the Wright function of the matrix argument.

Geometry-dependent skin effects in reciprocal photonic crystals

Abstract Skin effect that all eigenmodes within a frequency range become edge states is dictated by the topological properties of complex eigenvalues unique in non-Hermitian systems. The prevailing attempts to realize such a fascinating effect are confined to either one-dimensional or nonreciprocal systems exhibiting asymmetric couplings. Here, inspired by a recent model Hamiltonian theory, we propose a realistic reciprocal two-dimensional (2D) photonic crystal (PhC) system that shows the desired skin effect. Specifically, we establish a routine for designing such non-Hermitian systems via revealing the inherent connections between the nontrivial eigenvalue topology of order-2 exceptional points (EPs) and the skin effects. Guided by the proposed strategy, we successfully design a 2D PhC that possesses the EPs with nonzero eigenvalue winding numbers. The spectral area along a specific wavevector direction is then formed by leveraging the symmetry of the macroscopic geometry and the unit cell. The projected-band-structure calculations are performed to demonstrate that the desired skin effect exists at the specific crystalline interfaces. We finally employ time-domain simulations to vividly illustrate this phenomenon by exciting a pulse at the center of a finite-sized PhC. Our results form a solid basis for further experimental confirmations and applications of the skin effect.

Deep Gaussian Process for the Approximation of a Quadratic Eigenvalue Problem: Application to Friction-Induced Vibration

Despite numerous works over the past two decades, friction-induced vibrations, especially braking noises, are a major issue for transportation manufacturers as well as for the scientific community. To study these fugitive phenomena, the engineers need numerical methods to efficiently predict the mode coupling instabilities in a multiparametric context. The objective of this paper is to approximate the unstable frequencies and the associated damping rates extracted from a complex eigenvalue analysis under variability. To achieve this, a deep Gaussian process is considered to fit the non-linear and non-stationary evolutions of the real and imaginary parts of complex eigenvalues. The current challenge is to build an efficient surrogate modelling, considering a small training set. A discussion about the sample distribution density effect, the training set size and the kernel function choice is proposed. The results are compared to those of a Gaussian process and a deep neural network. A focus is made on several deceptive predictions of surrogate models, although the better settings were well chosen in theory. Finally, the deep Gaussian process is investigated in a multiparametric analysis to identify the best number of hidden layers and neurons, allowing a precise approximation of the behaviours of complex eigensolutions.

Investigation of negative-ion resonances using a subspace-projected multiconfigurational electron propagator perturbed with a complex absorbing potential.

The transient negative-ion resonances found in scattering experiments are important intermediates in many chemical processes. These metastable states correspond to the continuum part of the Hamiltonian of the projectile-target composite system. Usual bound-state electronic structure methods are not applicable for these. In this work, we develop a subspace-projection method in connection with an electron propagator (EP) defined in terms of a complete-active-space self-consistent-field initial state. The target Hamiltonian (Ĥ) is perturbed by a complex absorbing potential (CAP) for the analytical continuation of the spectrum of Ĥ to complex eigenvalues associated with the continuum states. The resonance is identified as a pole of the EP, which is stable with respect to variations in the strength of the CAP. The projection into a small subspace reduces the size of the complex matrices to be diagonalized, minimizes the computational cost, and affords some insight into the orbitals that are likely to play some role in the capture of the projectile. Two molecular (Πg2N2 - and 2Π CO-) and an atomic shaperesonance (2P Be-) are investigated using this method. The position and width of the resonances are in good agreement with the previously reported values.

Suppression of Brake Noise and Vibration Using Aramid and Zylon Fibers: Experimental and Numerical Study.

Aramid pulp/fiber is the most vital ingredient of brake friction material (FM) formulation. It is perpetually added to achieve quality brake pads/shoes and improve the overall friction and wear performance. Additionally, novel Zylon fibers have a superior property to aramid fibers. However, no studies give insights on their influence on brake noise and vibration (NV) performance. In the current work, a series of six different types of eco-friendly brake pads was developed. The first five contain aramid pulp, aramid short fibers, and Zylon fibers of different sizes (1, 3, and 6 mm) as the theme ingredients (3 wt %) by keeping the parent composition identical. Additionally, one more pad was developed that contains no aramid/Zylon fibers (i.e., reference pad). The pads were characterized for physical and mechanical properties. The damping and natural frequencies of pads were measured experimentally and numerically. All brake pads were evaluated for detailed NV performance by following the SAE J 2521 test schedule. In addition, numerical simulation was performed to validate the experimental brake squeal results. Results revealed that aramid/Zylon fiber-based pads improved the porosity, damping, and compressibility. Overall, brake noise and vibrations were improved for aramid/Zylon fiber-based pads by 1.2–1.5 dBA and 20–25%, respectively, compared to the reference pad. The complex eigenvalue analysis (CEA) proved that squeal was mainly influenced by the damping and density of the pad materials. Thus, aramid/Zylon fiber-based pads can effectively suppress the instability of the brake system and reduce the brake squeal propensity.

Accurate closed-form eigensolutions of three-dimensional panel flutter with arbitrary homogeneous boundary conditions

Highly accurate closed-form eigensolutions for flutter of three-dimensional (3D) panel with arbitrary combinations of simply supported (S), glide (G), clamped (C) and free (F) boundary conditions (BCs), such as cantilever panels, are achieved according to the linear thin plate theory and the first-order piston theory as well as the complex modal analysis, and all solutions are in a simple and explicit form. The iterative Separation-of-Variable (iSOV) method proposed by the present authors is employed to obtain the highly accurate eigensolutions. The flutter mechanism is studied with the benefit of eigenvalue properties from mathematical senses. The effects of boundary conditions, chord-thickness ratios, aerodynamic damping, aspect ratios and in-plane loads on flutter properties are examined. The results are compared with those of Kantorovich method and Galerkin method, and also coincide well with analytical solutions in literature, verifying the accuracy of the present closed-form results. It is revealed that, (A) the flutter characteristics are dominated by the cross section properties of panels in the direction of stream flow; (B) two types of flutter, called coupled-mode flutter and zero-frequency flutter which includes zero-frequency single-mode flutter and buckling, are observed; (C) boundary conditions and in-plane loads can affect both flutter boundary and flutter type; (D) the flutter behavior of 3D panel is similar to that of the two-dimensional (2D) panel if the aspect ratio is up to a certain value; (E) four to six modes should be used in the Galerkin method for accurate eigensolutions, and the results converge to that of Kantorovich method which uses the same mode functions in the direction perpendicular to the stream flow. The present analysis method can be used as a reference for other stability issues characterized by complex eigenvalues, and the highly closed-form solutions are useful in parameter designs and can also be taken as benchmarks for the validation of numerical methods.

Periodic Solutions and Stability Analysis for Two-Coupled-Oscillator Structure in Optics of Chiral Molecules

Chirality is an indispensable geometric property in the world that has become invariably interlocked with life. The main goal of this paper is to study the nonlinear dynamic behavior and periodic vibration characteristic of a two-coupled-oscillator model in the optics of chiral molecules. We systematically discuss the stability and local dynamic behavior of the system with two pairs of identical conjugate complex eigenvalues. In particular, the existence and number of periodic solutions are investigated by establishing the curvilinear coordinate and constructing a Poincaré map to improve the Melnikov function. Then, we verify the accuracy of the theoretical analysis by numerical simulations, and take a comprehensive look at the nonlinear response of multiple periodic motion under certain conditions. The results might be of important significance for the vibration control, safety stability and design optimization for chiral molecules.

A new mechanism for friction-induced vibration and noise

For years, friction-induced vibration and noise (FIVN) has puzzled many researchers in academia and industry. Several mechanisms have been proposed for explaining its occurrence and quantifying its frequencies, notably for automotive brake squeal, clutch squeal, and even rail corrugation. However, due to the complex and complicated nature of FIVN, there is not yet one fundamental mechanism that can explain all phenomena of FIVN. Based on experimental results obtained on a simple test structure and corresponding numerical validation using both complex eigenvalue analysis (CEA) and transient dynamic analysis (TDA), this study attempts to propose a new fundamental mechanism for FIVN, which is the repeated cycles of partial detachment and then reattachment of the contact surfaces. Since friction is ubiquitous and FIVN is very common, the insight into FIVN reported in this paper is highly significant and will help establish effective means to control FIVN in engineering and daily life.

VR helicity density and its application in turbomachinery tip leakage flows

Helicity is an important quantity that represents the topological interpretation of vortices; however, helicity is not a Galilean invariant. In this study, VR helicity density (HVR) is derived via taking the dot product of vorticity with the unit real eigen vector of the velocity gradient tensor when the complex eigenvalues exist. The analytical solution of HVR is derived to resolve it in a local pointwise manner, and the Galilean invariance of HVR is proved. Tip leakage flow structures in a direct numerical simulation of a tip leakage flow model and a delayed detached eddy simulation of a low-speed large-scale axial compressor rotor are extracted using helicity, eigen helicity density and HVR methods. Results show that the utilization of HVR permits the identification and accentuation of concentrated vortices. Vortices identified by HVR appear in more connective states. As in the case of helicity, the sign of HVR distinguishes between primary and secondary vortices, while eigen helicity density fails. The normalized HVR is superior to the normalized helicity density in locating the vortex axis, especially for the induced vortex structures. Hence, HVR is a strong candidate to replace the helicity density, especially when Galilean invariance is required.

Dynamic analysis of electrorheological fluid sandwich cylindrical shells with functionally graded face sheets using a semi-analytical approach

The present research is performed to calculate the natural frequencies, loss factors, and associated mode shapes of a sandwich cylinder with moderately thick functionally graded (FG) face sheets and an electrorheological (ER) fluid core. Each face sheet is assumed to be made of FG materials, and its displacement field is estimated based on the first-order shear deformation theory, like the ER constrained layer. A suitable displacement continuity condition is considered between layers. The ER fluid used in the central middle is analyzed in the pre-yield area and considered electric field dependent. Hamilton’s principle is used to acquire the motion equations related to each layer and appropriate boundary conditions. A computational code is implemented based on the combination of trigonometric expansion and the generalized differential quadrature (GDQ) methods to extract the complex eigenvalue problem for the vibration and damping of the assumed structure. After carrying out a comprehensive verification study, some novel numerical results are obtained. In the parametric study, the effects of the applied electric field, the FG power-law index, various boundary conditions, and structure geometry to control the natural frequencies and modal loss factors, as well as mode shapes, are investigated.

Optimal disc brake design for reducing squeal instability using slip-dependent complex eigenvalue analysis

This paper proposes an improved disc brake system optimization method for squeal instability reduction using slip-dependent eigenvalue results. Although complex eigenvalue analysis is widely used for minimizing brake squeal instability, conventional optimization approaches still have the limitation of not being able to reflect slip rate-varying squeal instability characteristics. While relative angular velocity between the pad and disc declines due to braking, disc brake system instability gradually increases up to a specific peak velocity point and decreases until the vehicle stops, which means a maximum instability point exists during the braking process. Therefore, instability optimization should target the prevention of a maximum value during a braking scenario. The proposed optimization formulation is conducted considering maximum instability during full braking. To obtain braking time profiles, a model-based design method is employed and utilized instead of full finite element transient dynamic analysis to reduce computational cost. Kriging surrogate modeling is also used for solving the optimization problem and better express the multi-variable squeal problem. The proposed optimal design method produces minimal squeal instability during the full vehicle braking time range. The effectiveness of the proposed disc brake optimal design is demonstrated via acceleration power value comparison of the structure acceleration with that derived by conventional optimization approach.

From integrability to chaos in quantum Liouvillians

The dynamics of open quantum systems can be described by a Liouvillian, which in the Markovian approximation fulfills the Lindblad master equation. We present a family of integrable many-body Liouvillians based on Richardson-Gaudin models with a complex structure of the jump operators. Making use of this new region of integrability, we study the transition to chaos in terms of a two-parameter Liouvillian. The transition is characterized by the spectral statistics of the complex eigenvalues of the Liouvillian operators using the nearest neighbor spacing distribution and by the ratios between eigenvalue distances.

Analisis Dinamik Model Matematika Penyebaran COVID-19 Pada Populasi SEIR

This study discusses the dynamic analysis of the mathematical model of the spread of COVID-19 using daily cases in Indonesia which are classified into four variables, namely, Susceptible (S), Exposed (E), Infected (I), and Recovered (R) which are then analyzed dynamically by calculate the equilibrium point and look for stability properties. The two equilibrium points of this model are the disease-free equilibrium point and the endemic equilibrium point . Then, it is linearized around the equilibrium point using the given parameters. Linearization around the equilibrium point produces four eigenvalues, one of which is positive. Linearization around the equilibrium point yields two negative real eigenvalues and a pair of complex eigenvalues with negative real parts. Phase portraits and numerical simulations have shown that all variables S, E, I, and R will be asymptotically stable locally towards the equilibrium point, namely the endemic equilibrium point . Thus, based on the dynamic analysis obtained, it is shown that the disease-free equilibrium point is unstable and the endemic equilibrium point is locally asymptotically stable.

Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model

We investigate different types of complex soliton solutions with regard to their stability against linear perturbations. In the Bullough-Dodd scalar field theory we find linearly stable complex PT-symmetric solutions and linearly unstable solutions for which the PT-symmetry is broken. Both types of solutions have real energies. The auxiliary Sturm-Liouville eigenvalue equation in the stability analysis for the PT-symmetric solutions can be solved exactly by supersymmetrically mapping it to an isospectral partner system involving a shifted and scaled inverse cosh-squared potential. We identify exactly one shape mode in form of a bound state solution and scattering states which when used as linear perturbations leave the solutions stable. The auxiliary problem for the solutions with broken PT-symmetry involves a complex shifted and scaled inverse sin-squared potential. The corresponding bound and scattering state solutions have complex eigenvalues, such that when used as linear perturbations for the corresponding soliton solutions lead to their decay or blow up as time evolves.

A comprehensive damping study of variable stiffness composite rectangular/skew laminates reinforcement with curvilinear fibers by higher-order shear flexible model

In this work, a comprehensive investigation of curvilinear fiber reinforcement on damping of rectangular and skew composite plates is computationally estimated using higher-order shear flexible model. A set of governing equilibrium equations developed here in form of eigenvalue analysis is solved by adopting the Q-R algorithm. The damping factors associated with different vibrational modes are evaluated from the complex eigenvalues. The proposed model is validated against the available analytical and experimental results. The damping capability of laminated rectangular/skew composite plates is thoroughly analyzed by varying the curvilinear fiber path angles in the layers, lay-up orientations, structural boundary conditions, skew angle of the laminate, and nature of the material. Results reveal that the damping pretending to the curvilinear fibers plate is better than the conventional composite laminate; it varies significantly according to the variation in center and edge of the curvilinear fiber angles. It is also noted that the damping increases with the skew angle of the plate. The study conducted here shows the suitability of such variable stiffness composite structure for safe design under dynamic/impact loading situation.

Optical properties and conductivity of biased GaAs quantum dots

We report the intraband optical absorption influence on the tunneling currents in biased gallium arsenide quantum dots. The energy levels and tunneling times are calculated using a complex eigenvalue formalism through an axisymmetrical two-dimensional finite element method. The absorption coefficient and the relative change of the refractive index are evaluated by taking into account a variable intraband relaxation time. A field-invariable relaxation, as it was often assumed by previous theoretical models, cannot be justifiable even for moderate electric fields. On the one hand, our work shows that the electron escape probability and implicitly the tunneling current may be controlled by the resonant optical field. Optically switching the QDs from lower to higher conductivity states is shown to be feasible. On the other hand, at high values of the electric field the tunneling-related electronic linewidth broadening blue-shifts and enlarges the absorption peaks, thus reducing the resonant nature of the intraband transitions.

Exceptional points and pseudo-Hermiticity in real potential scattering

We employ a recently-developed transfer-matrix formulation of scattering theory in two dimensions to study a class of scattering setups modeled by real potentials. The transfer matrix for these potentials is related to the time-evolution operator for an associated pseudo-Hermitian Hamiltonian operator \widehat{\mathbf{H}}𝐇̂ which develops an exceptional point for a discrete set of incident wavenumbers. We use the spectral properties of this operator to determine the transfer matrix of these potentials and solve their scattering problem. We apply our general results to explore the scattering of waves by a waveguide of finite length in two dimensions, where the source of the incident wave and the detectors measuring the scattered wave are positioned at spatial infinities while the interior of the waveguide, which is filled with an inactive material, forms a finite rectangular region of the space. The study of this model allows us to elucidate the physical meaning and implications of the presence of the real and complex eigenvalues of \widehat{\mathbf{H}}𝐇̂ and its exceptional points. Our results reveal the relevance of the concepts of pseudo-Hermitian operator and exceptional point in the standard quantum mechanics of closed systems where the potentials are required to be real.

Experimental and Numerical Study on the Squeal of a Rotary Compressor

This study predicts the squeal instability of a rotary compressor using a full-size finite element model. A complex eigenvalue extraction method was used to study the tendency toward squeal of the rotary compressor during working. The numerical results showed that the unstable vibration frequency of the rotary compressor was 6,908 Hz, which corresponded to the frequency of the rotary compressor squeal. The squealing vibration of the rotary compressor was also measured, with results showing that the measured squeal frequency was in good agreement with that predicted by the finite element model. Transient dynamic analysis results showed that when squeal occurred, the main frequency of the normal acceleration of the upper flange was the same as that of the tangential acceleration. This phenomenon showed that the coupling of the normal vibration and tangential vibration of the upper flange was a main reason for the compressor squeal. Also, the greater the friction coefficient, the higher the likelihood of a squeal occurrence.

A dynamical approach to efficient eigenvalue estimation in general multiagent networks

We propose a method to efficiently estimate the eigenvalues of any arbitrary (potentially weighted or directed) network of interacting dynamical agents in the presence of control inputs from dynamical observations. These observations are discrete, temporal measurements of the evolution of the aggregated outputs from a subset of agents (potentially one) during a finite time horizon. Notably, we do not require knowledge of which agents contribute to our measurements. We propose an efficient algorithm to exactly recover the (potentially complex) eigenvalues corresponding to network modes which are observable from the output measurements. The length of the sequence of measurements required by our method to generate a full reconstruction of the observable eigenvalue spectrum is at most three times the number of agents in the network, but in practice fewer are required, dependent on the number of observable network modes. The proposed technique can be applied to networks of non-autonomous multiagent systems with arbitrary dynamics in both continuous- and discrete-time. Finally, we illustrate our results with numerical simulations.

Friction-Excited Oscillation of Air Conditioner Rotary Compressors: Measurements and Numerical Simulations

The oscillation of a rotary compressor was measured and analyzed in the frequency and time domains. The harmonic wavelet transform was used to dissolve the oscillation signal into a series of single-frequency components. A power spectrum analysis of the single-frequency components shows that there are many vibration components whose frequencies are one, double, triple, quadruple, and even tens of times the rotating frequency. An envelope spectrum analysis shows that some single-frequency components originate from the friction-excited oscillation of the compressor. A full-size mode-coupling model of the rotary compressor was established to forecast the friction-excited oscillation of the compressor using the transient dynamics method and the complex eigenvalue method. The measurement results are consistent with the predictive results. A sensitivity analysis of the main parameters shows that the coefficient of friction has a major impact on the development tendency of the friction-excited oscillation of rotary compressors.