Damage Detection of Laminated Rubber Bearings Based on the Antiresonance Frequencies of Periodic Structure and Its Sensitivity Analysis

An isolation bearing consumes most of the seismic energy of a structure and is vulnerable to destruction. The performance of isolation bearings is usually evaluated according to the global stiffness and energy dissipation capacity. However, the early minor damage in isolation bearings is difficult to identify. In this study, a damage detection scheme for the isolation bearing is proposed by focusing on the antiresonance of the quasiperiodic structure. Firstly, a laminated rubber bearing was simplified as a monocoupled periodic rubber-steel structure. The characteristic equation of the driving point antiresonance frequency of the periodic system was achieved via the dynamic stiffness method. Secondly, the sensitivity coefficient of the driving point antiresonance, which was obtained from the first-order derivative of the antiresonance frequency, with respect to the damage scaling parameter was derived using the antiresonance frequency characteristic equation. Thirdly, the optimised driving points of the antiresonance frequencies were selected by means of sensitivity analysis. Finally, from the measured changes in the antiresonance frequencies, the damage was identified by solving the sensitivity identification equation via a numerical optimisation method. The application of the proposed method to laminated rubber bearings under various damage cases demonstrates the feasibility of this method. This study has proven that changes in the shear modulus of each rubber layer can be identified accurately.

Method of lines for solving linear equations of mathematical physics with the third and first types boundary conditions

The use of the method of lines in solving multidimensional problems of mathematical physics makes it possible to eliminate the discrepancies caused by the use of the sweep method in certain coordinates. As a result, the solution of the Poisson equation, for example, is obtained without using the relaxation method. In the article, the problem on the eigenvalues and vectors of the transition matrix is solved for boundary conditions of the third and first types, used to solve a one-dimensional equation of parabolic type by the method of lines. Due to the features of boundary conditions of the third type for determining the eigenvalues, a mixed method was proposed based on the Vieta theorem and the representation of the characteristic equation in trigonometric form typical for the method of lines. To solve the eigenvector problem, a simple sweep method was used with the algebraic compliments to the transition matrix. Discontinuous solutions of a one-dimensional parabolic equation were presented for various values of complex 1 -αl; the method for solving the characteristic equation was selected based on these values. The calculation results are in good agreement with the analytical solution.

Mathematical modeling nonstationarity of biotechnological production of lactic acid: stability

The article presents the calculated ratios of indicators determining the stationary states of the lactic acid production process. Three technologies that are most often mentioned in scientific publications are identified: the technology of using strains of microorganisms to produce biomass is a technology that is extremely rarely used; the fairly common technology of using strains of microorganisms to produce lactic acid with the consumption of the main substrate (most often glucose); the promising technology of obtaining lactic acid using, in addition to the main substrate, a component that reproduces the main substrate in the synthesis process. For each technology, the equations of material balance for stationary and non-stationary conditions, a generalized differential equation for non-stationary conditions, and a characteristic equation are given. The formulas for estimating the coefficients of differential equations and the coefficients of the characteristic equation are also given. The equations for non-stationary conditions according to the last two technologies are based on the use of the Taylor series expansion of functions with the preservation of only the first terms of the expansion, i. e. deviations from stationarity in small. The characteristic equation is formed using the eigenvalues . The methodology for all three technologies is given, which allows us to assess the stability of the considered stationary state – the Hurwitz method. For all three technologies, numerical results are obtained for estimating the coefficients of the characteristic equations Pi. Tabular values of the coefficients are given, according to which stability estimates for the dilution rate of 0.1 h–1, 0.2 h–1, 0.3 h–1 are obtained using determinants according to the Hurwitz matrix. The results of numerical estimates for the stability of stationary states for all three technologies are presented. The estimates were based on the indicators of constants published in scientific studies.

Determining configuration parameters for proportionally integrated differentiating controllers by arranging the poles of the transfer function on the complex plane

This paper reports a solution to the problem of determining the configuration parameters of PID controllers when arranging the poles of the transfer function of a linear single-circuit automated control system for a predefined set of control objects. Unlike known methods in which the task to find the optimal settings of a PID controller is formed as a problem of nonlinear programming, in this work a similar problem is reduced to solving a system of linear algebraic equations. The method devised is based on the generalized Viète theorem, which establishes the relationship between the parameters and roots of the characteristic equation of the automatic control system. It is shown that for control objects with transfer functions of the first and second orders, the problem of determining the configuration parameters of PID controllers has an unambiguous solution. For control objects with transfer functions of the third and higher orders, the generated problem is reduced to solving the redefined system of linear algebraic equations that has an unambiguous solution when the Rouché–Capelli theorem condition is met. Such a condition can be met by arranging one of the roots of the characteristic equation of the system on a complex plane. At the same time, the requirements for the qualitative indicators of the system would not always be met. Therefore, alternative techniques have been proposed for determining the configuration parameters of PID controllers. The first of these defines configuration parameters as a pseudo solution to the redefined system of linear algebraic equations while the second produces a solution for which the value of the maximum residual for the system of equations is minimal. For each case, which was used to determine the settings of PID controllers, such indicators of the control process as overshooting and control time have been determined

The Characteristic Equation of The Exceptional Jordan Algebra: Its Eigenvalues, And Their Relation With Mass Ratios For Quarks And Leptons

We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued La-grangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra J3 (O) [exceptional Jordan algebra] which in turn is related to the three fermion generations. The exceptional Jordan algebra [also known as the Albert algebra] is the finite dimensional algebra of 3x3 Hermitean matrices with octonionic entries. Its automorphism group is the exceptional Lie group F4. These matrices admit a cubic characteristic equation whose eigenvalues are real and depend on the invariant trace, determinant, and an inner product made from the Jordan matrix. Also, there is some evidence in the literature that the groups F4 and E6 could play a role in the unification of the standard model symmetries, including the Lorentz symmetry. The octonion algebra is known to correctly yield the electric charge values (0, 1/3, 2/3, 1) for standard model fermions, via the eigenvalues of a U (1) number operator, identified with U (1)em. In the present article, we use the same octonionic representation of the fermions to compute the eigenvalues of the characteristic equation of the Albert algebra, and compare the resulting eigenvalues with the known mass ratios for quarks and leptons. We find that the ratios of the eigenvalues correctly reproduce the [square root of the] known mass ratios for quarks and charged leptons. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibiting an F4 and E6 symmetry. In conjunction with the trace dynamics Lagrangian, the Jordan eigenvalues also provide a first principles theoretical derivation of the low energy value of the fine structure constant, yielding the value 1/137.04006. The Karolyhazy correction to this value gives an exact match with the measured value of the constant, after assuming a specific value for the electro-weak symmetry breaking energy scale.

Analytical modeling of the binary dynamic circuit motion

The binary dynamic circuit, which can be a design scheme for a number of technical systems is considered in the paper. The dynamic circuit is characterized by the kinetic energy of the translational motion of two masses, the potential energy of these masses’ elastic interaction and the dissipative function of energy dissipation during their motion. The free motion of a binary dynamic circuit is found according to a given initial phase state. A mathematical model of the binary dynamic circuit motion in the canonical form and the corresponding characteristic equation of the fourth degree are compiled. Analytical modeling of the binary dynamic circuit is carried out on the basis of the proposed particular solution of the complete algebraic equation of the fourth degree. A homogeneous dynamic circuit is considered and the reduced coefficients of elasticity and damping are introduced. The dependence of the reduced coefficients of elasticity and damping is established, which provides the required class of solutions to the characteristic equation. An ordered form of the analytical representation of a dynamic process is proposed in symmetric determinants, which is distinguished by the conservatism of notation with respect to the roots of the characteristic equation and phase coordinates.

The Characteristic Equation of the Exceptional Jordan Algebra: Its Eigenvalues, and their relation with the Mass Ratios of Quarks and Leptons

We have recently proposed a pre-quantum, pre-space-time theory as a matrix-valued Lagrangian dynamics on an octonionic space-time. This pre-theory offers the prospect of unifying the internal symmetries of the standard model with gravity. It can also predict the values of free parameters of the standard model, because these parameters arising in the Lagrangian are related to the algebra of the octonions which define the underlying non-commutative space-time on which the dynamical degrees of freedom evolve. These free parameters are related to the algebra $J_3(\mathbb O)$ [exceptional Jordan algebra] which in turn is related to the three fermion generations. The exceptional Jordan algebra [also known as the Albert algebra] is the finite dimensional algebra of 3x3 Hermitean matrices with octonionic entries. Its automorphism group is the exceptional Lie group $F_4$. These matrices admit a cubic characteristic equation whose eigenvalues are real and depend on the invariant trace, determinant, and an inner product made from the Jordan matrix. Also, there is some evidence in the literature that the groups $F_4$ and $E_6$ could play a role in the unification of the standard model symmetries, including the Lorentz symmetry. The octonion algebra is known to correctly yield the electric charge values (0, 1/3, 2/3, 1) for standard model fermions, via the eigenvalues of a $U(1)$ number operator, identified with $U(1)_{em}$. In the present article, we use the same octonionic representation of the fermions to compute the eigenvalues of the characteristic equation of the Albert algebra, and compare the resulting eigenvalues with the known mass ratios for quarks and leptons. We find that the ratios of the eigenvalues correctly reproduce the [square root of the] known mass ratios for quarks and charged leptons. We also propose a diagrammatic representation of the standard model bosons, Higgs and three fermion generations, in terms of the octonions, exhibiting an $F_4$ and $E_6$ symmetry. In conjunction with the trace dynamics Lagrangian, the Jordan eigenvalues also provide a first principles theoretical derivation of the low energy value of the fine structure constant, yielding the value $1/137.04006$. The Karolyhazy correction to this value gives an exact match with the measured value of the constant, after assuming a specific value for the electro-weak symmetry breaking energy scale.