Transfer Matrix Modeling of Systems With Noncollocated Feedback

The transfer matrix method (TMM) is a powerful modeling approach with unique strengths for modeling flexible structures under feedback control. The TMM can model distributed parameter systems without spatial discretization, ensuring that the models are protected against model spillover. Feedback and actuator dynamics can be directly included in the structural model, leading to accurate models of ac-tuator/structure interaction. The TMM can produce closed-form symbolic expressions for the infinite-dimensional closed-loop transfer functions of distributed parameter systems under feedback control. These infinite-dimensional transfer functions can be used directly for compensator design via Bode plots. However, if modern state-space control design tools are to be applied to a flexible structure model with the TMM, the TMM model must first be discretized. The infinite-dimensional nature of TMM models and the fact that they involve transcendental expressions in the Laplace variable $s$ make it difficult to apply most of the model reduction procedures in the literature. This paper presents a technique for discretizing TMM models that is an improvement upon one discretization technique from the literature that is specific to the TMM.

The transfer matrix (TM) method was applied to calculate the transmission probability (TP) for proton transfer reactions. The tunneling factors in the reaction rate constants were also evaluated using the TPs. To test this method, TPs for the Eckart potentials modeled as a guanine-cytosine base pair were calculated by the TM method and compared to TPs by the analytical solution. As a result, the errors in the TPs by the TM method were quite small. The tunneling factors for the guanine-thymine (G-T) and adenine-cytosine (A-C) mispair reactions were then evaluated by the TM method. A shoulder appears on each potential for these reactions [Odai, K.; Umesaki,K. J. Phys. Chem. A. 2021, 125, 8196-8204]. As a result, the shoulder for the G-T mispair reaction contributes significantly to the tunneling, while the shoulder for the A-C mispair reaction contributes little to the tunneling. These results are difficult to obtain with methods such as Wigner's tunneling factor.

Recently, a new type of vibration isolator called a dynamic antiresonant vibration isolator (DAVI) has been studied for torsional systems as well as translational systems. In the DAVI for torsional systems, the inertia coupling for the anti-resonance effect is generated by a planetary gear that connects control inertia and inertias at the input and output terminals. In this research, an analytical modeling approach called the transfer matrix method is applied to design a planetary gear type DAVI for complex systems such as an automotive powertrain. Unlike a previous study, inertias of the pinion gears are newly included in the transfer matrix model to improve the model accuracy in this paper. In this part 1 of series papers, the internal configuration of a planetary gear type DAVI system is described and its dynamic model is derived using the transfer matrix method. The derived transfer matrix model is then validated by comparing its frequency response with those obtained by conventional and numerical methods. Finally, the effects of different gear ratios and pinion gear inertia are examined using the derived transfer matrix model.

Based on the similarity between the electric system and the mechanical system, this paper extends the transfer matrix method of multibody system, which has the advantages of low matrix order and fast computation speed in the mechanical system, to the field of circuit analysis to establish circuit equations. First, the transfer matrix of the basic circuit elements is derived; then, the transfer matrix of the element interconnection and the active branch is derived; then take the RLC circuit and the reverse amplification circuit as examples to verify the correctness of the transfer matrix derivation; finally, the transfer matrix method is compared with the current mainstream modified nodal approach, which proves that the transfer matrix method has the advantages of low matrix order and small calculation amount in constructing circuit equation.

The aim of this study is performing free vibration analysis of segmented Timoshenko beams on two parameter elastic foundation by using the transfer matrix method (TMM). The Pasternak foundation model which has an incompressible shear layer of vertical elements attached to the Winkler springs was considered. The transfer matrix formulations which are based on closed-form solutions of equations of motion of Timoshenko beams on Pasternak foundation were obtained. The natural frequencies were calculated by equating the determinant of global transfer matrix of structure to zero after the reduction according to boundary conditions. The mode shapes were plotted by normalising the state vectors at the ends. Firstly, the natural frequencies that obtained by using the proposed approach were validated by data in literature for a simply supported beam where a very good agreement was observed. Then, three-segmented beam models having various boundary conditions namely simple-simple (S-S), simple-fixed (S-F), fixed-simple (F-S) and fixed-fixed (F-F) were considered for numerical analysis. For the segmented beam models, the natural frequencies that calculated by using the TMM were compared to the results of finite element method (FEM) from SAP2000 by ignoring effects of shear layer of elastic foundation. The effects of shear layer as well as stiffness of Winkler springs on the natural frequencies of the segmented beam model were revealed for S-S, S-F, F-S and F-F boundary conditions, respectively. The mode shapes of the segmented beam model were presented. The results show that TMM can be used as an effective tool for free vibration analysis of multi-segmented Timoshenko beams on Pasternak foundation.

In This search, the phenomena of resonance in series piping systems with oscillating valve have been studied. The one-dimensional flow, unsteady state flow, and partial differential equations have been solved by using transfer matrix method. The details of the transfer matrix method, the derivation of the field matrices and point matrices, and procedures for determining the natural frequencies and frequency response of piping system are then presented. A computer program (Resonance) was developed to calculate the pressure and discharge oscillations at the valve, the phase angle between the pressure head and relative gate opening of the valve, the phase angle between the discharge and the relative gate opening of the valve and also the pressure and discharge oscillations at certain sections of the pipes. To verify the transfer matrix method, the results have been compared with impedance method and agreement has been found.

First Page

Using boundary conditions of electromagnetic waves , a new transfer matrix method that study electromagnetic waves in thin films and a dispersive formula of 1D photonic crystal are derived. Using the formula reflection and refraction of light is studied, the forbidden band of 1D photonic crystal is studied. The transfer matrix method and the classic characteristic matrix method are compared. The transfer matrix method can not only solve the problem that characteristic matrix method can solve, the transfer matrix method can also solve the problem that characteristic matrix method can not solve. Therefore the transfer matrix method is a more fundamental and wider range of application.

The transfer matrix method together with a digital computer form the foundation of the dynamic analysis of rotor-bearing systems. The properties of each segment of the rotating shaft are expressed in simple matrix form and the overall dynamic behaviour is then obtained by successive multiplication of the element matrices. The main drawback associated with this method is the numerical instability in calculating natural frequencies for complex systems. The finite element method, on the other hand, uses the element stiffness and mass matrices to form the global equation of motion for the complete system. This avoids the numerical problems of the transfer matrix method at the expense of the computer memory requirements. The new method described in this paper combines the transfer matrix and finite element techniques to form a powerful algorithm for vibration analysis of rotor-bearing systems. It is shown that the accuracy improves significantly when the transfer matrix for each shaft segment is obtained from finite element techniques. The accuracy and efficiency of the hybrid method are compared with the transfer matrix method for a simply supported uniform rotating shaft where an analytical solution for the critical speeds and mode shapes is available. The method is then applied to a flexibly supported uniform shaft and a non-uniform shaft with a large disc to show the capability of the method for finding the critical speeds of complex rotor-bearing systems.

The presence of a leak within a single line system alters the dynamic behavior of the system when subjected to transient pressure fluctuations. This change in dynamic behavior is seen in the time-domain as an increased rate of energy loss (damping rate) of the pressure response and in the frequency-domain as an attenuation in the magnitudes of the eigenfrequencies (harmonics) of the system. Two recent developments in the literature on leak detection, based on exploiting the time- and frequency-domain behavior are the transient damping method, and the frequency response method respectively. These methods are both utilize linearized approximations of the original nonlinear water hammer equations. The frequency-domain methods are essentially based on the transfer matrix method (TMM) of modeling the transient pipeline system, whereas the transient damping method utilizes a modal damping method (MDM) of modeling the system. The methods adopt the same approach to linearizing the original nonlinear water hammer equations, however their main difference lies in the manner in which they incorporate the internal boundary condition associated with the leak. These different approaches give rise to different linear systems. The focus of this paper is the comparative analysis of these different systems and the nature of how their dynamics change for different leak parameter values. As is convenient for linear systems, the TMM and the MDM are analyzed in the Laplace-domain, where this analysis takes the form of deriving the Laplace-domain pressure response functions for each method and studying the dependency of the pole locations of these functions on the leak parameters. The Laplace-domain is a useful way to consider and compare the methods as it is shown that the location of the poles of the pressure response functions are directly related to the magnitude of the eigenfrequencies (as used by the frequency-domain methods to estimate the leak properties) and are also directly related to the modal damping rates (as used by the transient damping method to estimate the leak properties). Both methods behave similarly for small leaks, however, for large leaks vastly different behaviors are observed. The TMM yields a solution that is consistent with the underlying equations for the asymptotically large leak case, but the MDM does not, therefore a difference in the behaviors is attributed to limitations on the MDM. Additionally, from the study of the TMM, insight is given into a contention that has existed in the literature on the impact that a leak has on the frequency response of a system. Through studying the structure of the poles for the TMM, it is shown that a leak only effects the eigenfrequencies of a system and that no new eigenfrequencies are introduced as a result of the leak.

This paper presents a new technique to analyze dynamic behaviour of a sucker rod string used in the oil well industry. The main concept of the proposed approach is to replace the actual complicated system by a simpler one. This needs the output or one-time accurate solutions of the actual system. This problem is solved using D‘Alembert's solution of systems equation and the adaptive filter matrix method developed by the authors. The developed computer program performs the calculations in very short real time with reasonable accuracy, and without extensive hardware and software requirements. It has been proven that it is possible to use the transfer matrix to calculate load-displacement relations a hundred or more times in one stroke. This could be very advantageous in building new well-head controllers and other tools to control the oil wells. Introduction Rod pumping is the oldest and still the most common method of artificial lift used extensively in the oil well industry(1,2). The literature devoted to the analysis of this system is very extensive. There are different computer programs aimed at the analysis and troubleshooting of rod pumping systems(3–5), however, they require fully equipped computer hardware to perform the calculations. These calculations usually take an amount of time that is longer than the period of pumping (one stroke) and cannot be directly used to control the well in real time. In this paper, a transfer matrix method has been proposed to model the dynamic behaviour of the rod of the sucker string. The main concept of the method is to replace the solution of the mathematical model by a simple matrix operation in which the bottomhole force-displacement values are obtained as the product of the vector of the data at the polished rod end by a transfer matrix. This is an extremely fast mathematical operation, which can be performed hundreds of times during the period of one stroke and does not require any sophisticated software. The system transfer matrix for any given oil well is defined only once in advance using the filter matrix method(6–8). The matrix does not change with time and replaces the mathematical model of the analyzed system. Using this technique, the calculations of the bottomhole values can be performed very fast and in a very simple way. These calculations are easy to implement in one microprocessor. In the studies presented here, we found that to create the system transfer matrix, it is enough to replace the real system by a two-segment rod. The simplified model is solved using D‘Alembert's method(9). Using the equivalent model, the transfer matrix is created. Mathematical Modelling To analyze the performance of the oil well, the force and displacement at the polished rod are measured using a dynamometer. Then these data are used as the boundary conditions for the calculation of forces and displacements at the bottom of the rod string that defines the conditions of the pump, effectiveness of pumping, production rate, etc.

Many multi-beam energy harvesters appearing in the literature require custom analytical or finite-element models to compute their eigensolutions and piezoelectric coupling effects. This paper discusses the use of the transfer matrix method to derive analytical solutions to beam structures with point-wise discontinuities or with lumped inertias between members or at the tip. In this method, transfer matrices are developed for the beam’s states (deflection, slope, shear force, and bending moment) analogously to the state transition matrix of a linear system. Euler-Bernoulli beam theory is used to derive transfer matrices for the uniform beam segments, and point transfer matrices are derived to handle discontinuities in the structure. The transfer matrix method is shown to be advantageous for analyzing complex structures because the size of the transfer matrix does not grow with increasing number of components in the structure. Furthermore, the same formulation can be used for a wide range of geometries, including arbitrary combinations of beam segments — single- or multi-layered — and lumped inertias. The eigensolution of the transfer matrix is shown to produce the natural frequencies and mode shapes for these structures. Subsequently, the electromechanical coupling effects are incorporated and the base excitation problem is considered. The electromechanical equations of motion are decoupled by mode and shown to be a generalization of existing analytical models. Parametric case studies are provided for beam structures with varying piezoelectric layer coverage.

Aiming at the design and application of the interband cascade structure in long wavelength detection, E-k relation and energy band information of multi-quantum-well relaxation region were calculated by using two-band model under the envelope-function approach and transfer matrix method. The structure of relaxation region was specially optimized. On the premise of the tunneling process of photon-generated carriers in relaxation region, the electric field of absorption region was reduced to suppress generation-recombination current and tunneling current, and to improve the electrical performance of device. The quantum efficiency of prepared double-stage interband cascade photodetector was up to 20% at 10.5 mu m, which proved that the relaxation region and tunneling region had well transport of photoexcited carriers. The 50% cutoff wavelength of the device was 11.5 mu m at 80 K, which was the longest wavelength infrared detector of interband cascade structure at 80 K.

 In this paper possibility of creating a medium based on graphene-dielectric (quartz) structures with a hyperbolic shape of the dispersion relation is considered. Using the Bloch theorem and the transfer matrix method, dispersion relations are found for hyperbolic media, consisting of periodically located layers of graphene and a dielectric. The transfer matrix method gives the possibility to relate the amplitudes of the incident and reflected electromagnetic waves at the input to the layered structure with the amplitudes at the output by means of propagation and transfer matrixes. The frequency dependent optical conductivity of graphene sheet is calculated by Kubo formula which takes into account both interband and intraband transitions. It is shown that when the chemical potential of graphene is increased, the elliptic form of the dispersion curve turns into to hyperbolic. This result is due to the fact that the perpendicular component of the permittivity changes its sign when the chemical potential is changed.

Modeling methodologies based on state-of-the-art and classic theories of acoustics have been developed to provide a comprehensive toolbox, which can be used to model multilayer systems that involve acoustical and/or damping treatments, and to optimize these treatments' performance by designing their geometrical structures. The objective of this work was to understand, predict and optimize conventional sound absorbing porous media's near-field damping performance, so that automotive and aerospace industries can take full advantage of layered porous treatments' lightness and multi-functionality: i.e., absorption of airborne sound and reduction of structure-borne vibration, for noise control applications. First, acoustical models that include the Transfer Matrix Method and the Arbitrary Coefficient Method were developed to build connections between the bulk properties and acoustical properties of porous media when coupled into layered systems. Given a specified layered system consisting of a vibrating panel and a porous damping treatment, the acoustics models were then incorporated into the Near-field Damping model to predict the acoustical near-field and spatial response of the panel, based on which the near-field damping performance can be evaluated for a limp or an elastic porous layer when applied on different structures including an infinitely-extended panel, a partially-constrained panel, an aircraft fuselage-like structure and a vehicle floor pan-like structure. Furthermore, the relations between the material's microstructural details and bulk properties were established via an Air-Flow Resistivity model for porous media that are made of fibers, and the optimal fiber size that provides the largest damping for certain vibrating structures was identified. Relatively large fibers were found to be better at reducing lower frequency vibrations; fibers made of polymer were found to have manufacturing benefits over fibers made of glass to achieve equivalent optimal damping performance; and elastic fibers were found to have both manufacturing and damping advantages over limp fibers.