Traditional Transfer Matrix Method Research Articles

A general and stable approach to modeling and coupling multilayered acoustical systems with various types of layers

In this article, a general method is proposed to model layered acoustical systems with two-by-two transfer matrices, and further, to solve for their acoustic absorption, reflection, and transmission coefficients. Since the proposed method uses the matrix representation of various layers and interfaces, as in the Transfer Matrix Method (TMM), the equation system can be established efficiently. However, the traditional TMM can become unstable when there is a large disparity between the magnitudes of the waves traveling in opposite directions within the layers (i.e., at higher frequencies, for a thick layer, or for extreme parameter values). In such cases, the contribution of the most attenuated wave can be masked by numerical errors and can induce instability when solving the system. Therefore, in the proposed method, the accuracy of the wave attenuation terms is ensured by decomposing each layer’s transfer matrix and reformulating the equation system. This method can be used to couple different layer types in a general way and is easy to assemble and implement with numerical code. The predicted acoustic properties of layered systems calculated using the proposed method have been validated by comparison with those predicted by previously-existing methods.

A general stable approach to modeling and coupling multilayered systems with various types of layers

In this article, a general method is proposed to model layered systems with two-by-two transfer matrices, and further, to solve for the acoustic absorption, reflection, and transmission coefficients. Since the proposed method uses the matrix representation of various layers and interfaces from the Transfer Matrix Method (TMM), the equation system can be established efficiently. However, the traditional TMM can lose stability when there is a large disparity between the magnitudes of the waves traveling in opposite directions within the layers (i.e., at higher frequencies, for a thick layer, or for extreme parameter values). In such cases, the contribution of the most attenuated wave can be masked by numerical errors and can induce instability when solving the system. Therefore, in the proposed method, to stabilize the calculated acoustic properties of the system, the principle is to ensure the accuracy of the wave attenuation terms by decomposing each layer's transfer matrix and reformulating the equation system. This method can couple different layer types in a general way and is easy to assemble and implement with numerical code. The predicated acoustic properties of layered systems calculated using the proposed method have been validated by comparison with those predicted by other existing methods.

Enhancing Dynamic Bandwidth of Amplified Piezoelectric Actuators by a Hybrid Lever and Bridge-Type Compliant Mechanism

Ongoing interests in high-speed precision actuation continuously sparks great attention on developing fast amplified piezoelectric actuators (APAs) with compliant mechanisms. A new type of APA with enhanced resonance frequency is herein reported based on a hybrid compliant amplifying mechanism. A two-stage displacement flexure amplifier is proposed by synthesizing the lever-type and semi bridge-type compliant mechanisms in a compact configuration, promising to a well tradeoff between the displacement amplification ratio and dynamic bandwidth. The static and dynamic performances are experimentally evaluated. The resonance frequency of 2.1 kHz, displacement amplification ratio of 6, and step response time of around 0.4 ms are realized with a compact size of 50 mm × 44 mm × 7 mm. Another contribution of this paper is to develop a comprehensive two-port dynamic stiffness model to predict the static and dynamic behaviors of the compliant amplifier. The modeling approach presented here differs from previous studies in that it enables the traditional transfer matrix method to formulate both the kinetostatics and dynamics of compliant mechanisms including serial-parallel branches and rigid bodies.

Dynamic modeling and experimental study of a complex fluid-conveying pipeline system with series and parallel structures

In this paper, the transfer matrix method is improved and the dynamic model of a complex fluid-conveying pipeline system with series and parallel structures based on the improved transfer matrix method (ITMM) is developed. Firstly, the structural coupling between parallel pipelines is realized by deducing the deformation coordination equation and expanding the matrix dimension. Then the time domain information is converted into frequency domain information by Laplace transforms. Finally, an efficient ITMM is derived according to the field transfer relationship. Compared with the traditional transfer matrix method, the ITMM can consider both fluid-structure interaction and structural coupling between two parallel liquid-filled pipes in a complex fluid-conveying pipeline system. And the model constructed in this paper is closer to the real aero-engine pipeline system. In addition, two auxiliary verification models (FEM-Beam, and FEM-Solid) are established. Furthermore, a self-designed pipeline system modal test rig is built, and the effectiveness of the dynamic model is verified by comparing the modal results of simulation and experiment. Finally, the effects of different excitation and damping on the dynamic characteristics of the complex fluid-conveying pipeline system are analyzed. The modeling method proposed in this paper can provide positive guidance for the vibration analysis and modal test of the specific aero-engine pipelines with complex series and parallel structures.

A Modified Transfer Matrix Method for Modal Analysis of Stepped Rotor Assembly Applied in the Turbomolecular Pump

An accurate modal analysis of the stepped rotor assembly is significant in the design process of rotors. The transfer matrix method (TMM) is widely used in the modal analysis of the rotor. However, the influences of the step and the assembling modes are not considered in the traditional TMM. In this paper, a modified transfer matrix method is presented for the modal calculation of the rotor in the turbomolecular pump. The stiffness reduction of stepped segments and the effect of assembled components are considered in this modified method. First of all, the traditional transfer matrix model is built. Then, the stepped segments of the rotor are changed into conical segments and the lumped mass model of conical segments is calculated. Next, the conicities of conical segments are analyzed and relations between conicities and dimensions of the rotor are discovered. Finally, the stiffness factors which can describe the effects of different assembling modes are introduced and optimized. The optimized stiffness factors are analyzed and explained from the perspective of contact stress. The modal of the stepped rotor assembly could be computed with high accuracy by using this method.

An analytical solution for vertical coils near a multi-layered metallic pipe in Pulsed Eddy Current Testing

Analytical model for Pulsed Eddy Current Testing (PECT) of a multi-layered metallic pipe has not been solved, which limits the development of PECT technology. In this paper, the analytical model for vertical coils near a multi-layered metallic pipe is proposed. However, the analytical solution is hard to solve due to the large number of undetermined coefficients. Combined with the global matrix method, an improved transfer matrix across a conductor is established, which overcomes unsolvable problem in the traditional transfer matrix method. The induced voltage of pickup coil calculated based on the analytical model is verified by Finite Element Method (FEM) and experiments. Furthermore, the influence of several working conditions of pipelines on the PECT signal is studied based on the analytical model, providing guidance for engineering applications.

Characteristics of defect states in periodic railway track structure

To overcome the ill-conditioned matrix problem of the traditional transfer matrix method, the Floquet transform method and supercell technology are used to study the defect states of the periodic track structure. By solving the equations of the supercell directly, the propagation characteristics of elastic waves in the track structure with defects are analyzed. The existence of defects destroys the periodicity of track structure, thus resulting in the formation of defect states within the band gaps. Moreover, the elastic wave is localized near the defect position at the frequency of the defect state. The formation mechanism of the defect state in track structure can be explained by the local resonance at the defect. With the expansion of the defect range, the number of local resonance modes that can be formed near the defect increases, thus generating multiple defect states. Furthermore, the defect state enhances the vibration of the structure adjacent to the defect. Therefore, the vibration transmission coefficient in a finite-length range can be used to detect the defect characteristics in the track structure, and the defect degree can be evaluated by the peak frequency of the vibration transmission coefficient within the band gap.

The use of the transfer matrix method to predict the effective fluid properties of acoustical systems

The transfer matrix method (TMM) is a common method for the modelling of acoustical systems. Traditionally, this method requires each unique layer within a system to be defined by a transfer matrix and then for each matrix to be multiplied together in the sequential order of the system. Whilst the resulting matrix can be used to find the effective material properties of the modelled system, the resulting analytical expressions of these properties are often unwieldy for use. Here, a simplified approach is proposed to obtain simple analytical expressions in the low frequency regime for the effective properties of acoustical systems based on the components of the TMM and inspired by the Champoux and Stinson model. It was shown that the proposed approximation of TMM matches the effective fluid properties of a cylindrical rigid tortuous pore derived with the Champoux and Stinson model. Using this approach, analytical expressions for the effective fluid properties of a waveguide of constant cross section, side-loaded by an arbitrary number of Helmholtz resonators, were derived. These expressions were validated against the traditional transfer matrix method and with numerical computation. The result of this work offers a validated general approach that provides simple analytical low frequency approximations of the acoustical properties of media which consist of complicated networks of pores or side-branches.

Study on sound insulation performance of double-layer perforated panel under normal incidence waves

Based on the single-hole model, the sound insulation performance of four perforated panel structures were studied, including single-layer perforated panel, double-layer perforated panel, double-layer through-hole panel and double-layer resonant panel. Based on the plane wave theory, a modified transfer matrix method (MTM) was developed to improve the accuracy of traditional transfer matrix method (TM). For MTM, acoustic length corrections were used to modify the transfer matrix to consider the influence of multi-dimensional waves. Then MTM was used to predict the transmission loss (TL) of the four perforated panel structures, and the results were compared with those of the finite element method (FEM). The comparison showed good agreement in the effective frequency range. Using the MTM, the effects of dimensions on the TL of the four perforated panel structures were studied. The double-layer perforated panel can significantly increase the TL at the medium and high frequencies, and the double-layer resonant panel structure can increase the TL at the low frequency because of the resonance. In addition, the TL amplitude is influenced by the opening ratio, which decreases rapidly with the increasing of opening ratio. The thickness of the panel and the spacing between the two layers can mainly change the width of the high absorption region. The parameters of the resonant neck can change the resonant frequencies and pass-by frequencies. Finally, the predictions of MTM are verified with the experimental results of double-layer perforated panel sample. The method in this paper can be used to design the perforated panel structure for sound insulation.

High-Performance Ultrafast Humidity Sensor Based on Microknot Resonator-Assisted Mach-Zehnder for Monitoring Human Breath.

Monitoring the dynamic humidity requires sensors with fast response and anti-electromagnetic interference, especially for human respiration. Here, an ultrafast fiber-optic breath sensor based on the humidity-sensitive characteristics of gelatin film is proposed and experimentally demonstrated. The sensor consists of a microknot resonator superimposed on a Mach-Zehnder (MZ) interferometer produced by a tapered single-mode fiber, which has an ultrafast response (84 ms) and recovery time (29 ms) and a large dynamic transmission range. The humidity in dynamic ambient causes changes in the refractive index of gelatin coating, which could trigger spectral intensity transients that can be explicitly distinguished between the two states. The sensing principle is analyzed using the traditional transfer-matrix analysis method. The influence of coating thickness on the sensor's trigger threshold is further investigated. Experiments on monitoring breath patterns indicate that the proposed breath sensor has high repeatability, reliability, and validity, which enable many other potential applications such as food processing, health monitoring, and other biomedical applications.

Vibro-acoustic behavior of double-walled cylindrical shells with general boundary conditions

Vibro-acoustic behaviour of submerged double-walled cylindrical shells with general boundary conditions is analyzed by a precise transfer matrix method (PTMM), which is proposed by combining traditional transfer matrix method with precise integration method. Field transfer matrixes of both inner shell and outer shell are obtained accurately by PTMM respectively. The annular plates are simulated by two methods and exist in the form of point transfer matrixes. The dynamic models of double-walled cylindrical shells are established by assembling whole transfer matrixes. Based on the Helmholtz equation, the interlayer and external fluid loads are described to solve the Neumann boundary. The vibro-acoustic response of the double-walled cylindrical shell can be obtained by determining sound pressure coefficients. The free vibration and sound radiation of analytical models are comparing with numerical results and experimental results to verify the present method. The effects of the general boundary constraint stiffnesses on the free vibration and sound radiation of double-walled cylindrical shells are also discussed.

Solving the one-dimensional Ising chain via mathematical induction: an intuitive approach to the transfer matrix

The aim of this work is to present a formulation to solve the one-dimensional Ising model using the elementary technique of mathematical induction. This formulation is physically clear and leads to the same partition function form as the transfer matrix method, which is a common subject in the introductory courses of statistical mechanics. In this way our formulation is a useful tool to complement the traditional more abstract transfer matrix method. The method can be straightforwardly generalised to other short-range chains, coupled chains and is also computationally friendly. These two approaches provide a more complete understanding of the system, and therefore our work can be of broad interest for undergraduate teaching in statistical mechanics.

Dynamic Analysis of Flexible Rotor Based on Transfer Symplectic Matrix

In view of the numerical instability and low accuracy of the traditional transfer matrix method in solving the high-order critical speed of the rotor system, a new idea of incorporating the finite element method into the transfer matrix is proposed. Based on the variational principle, the transfer symplectic matrix of gyro rotors suitable for all kinds of boundary conditions and supporting conditions under the Hamilton system is derived by introducing dual variables. To verify the proposed method in rotor critical speed, a numerical analysis is adopted. The simulation experiment results show that, in the calculation of high-order critical speed, especially when exceeding the sixth critical speed, the numerical accuracy of the transfer symplectic matrix method is obviously better than that of the reference method. The relative errors between the numerical solution and the exact solution are 0.0347% and 0.2228%, respectively, at the sixth critical speed. The numerical example indicates the feasibility and superiority of the method, which provides the basis for the optimal design of the rotor system.

A method for considering a distributed spring constant for studying the flexural vibration of an Euler-beam with lightweight multistage local resonators

For the traditional locally resonant beams there always attached the one-stage local resonator and result that the lower band gap the heavier the scattering ring. In order to resolve this problem, the flexural vibration band gap in an Euler beam with periodically arranged lightweight multistage local resonators was theoretically investigated using the transfer matrix method based on discretization of lumped mass. The present method considered a distributed spring constant, which showed fast convergence with less computational requirements. A finite element method was then employed to calculate the frequency response function of a finite sample simultaneously, which demonstrated that the results calculated using the proposed method were closer to the simulation results than those obtained using the traditional transfer matrix method. The study found that, under the same additional mass, the lightweight multistage structure had much lower beginning frequency than one-stage structure, and the total width of the gaps was basically the same. In addition, a simplified model of the beginning frequency of gaps was proposed, and the effect of scattering density on the model precision was further explored numerically. The results show that the lower scattering density, the more important the role of the rubber mass and the higher precision of the simplified model.

Band Structures Analysis of Elastic Waves Propagating along Thickness Direction in Periodically Laminated Piezoelectric Composites

Existing studies on elastic waves in periodically laminated piezoelectric structures mainly concerned the passive band properties, since the electrical boundaries in the considered structures cannot vary. This paper investigates the tuning of band properties of uncoupled primary and shear (P- and S-) waves along the thickness direction by actively varying the electrical field in periodically multilayered piezoelectric structures consisting of orthotropic materials. The alteration of the electrical field is realized in the multilayered unit cell here by either applying or switching four kinds of electrical boundary conditions, including the electric-open, applied electric capacitance, electric-short, and applied feedback voltage, to the constituent piezoelectric layer via the constituent electrode layers covering both its surfaces. First, the state space formalism is introduced to obtain the partial wave solution of any constituent orthotropic layer in the unit cell. Second, the traditional transfer matrix method is adopted to derive the dispersion equation of general, periodically laminated piezoelectric composites with unit cells consisting of an arbitrary number of piezoelectric layers with various boundaries and of elastic layers. Third, numerical examples are provided to verify the proposed analysis method, and to study the influences of electrode thickness as well as four electrical boundaries on the band structures. All the frequency-related dispersion curves are also illustrated by numerical examples to summarize the general dispersion characteristics of uncoupled P- and S-waves in periodically laminated piezoelectric composites. The main finding is that the innovative dispersion characteristic resulting from the negative capacitance may also be achieved via feedback control.

Calculation and analysis of critical speed of high speed motor spindle rotor system

The traditional transfer matrix method used to calculate the critical speed of high-speed spindle motor rotor system will produce large errors. Taking the 170SD30 electrospindle as the analysis object, a high-speed electric spindle dynamics theoretical model was established based on the Riccati method, using Matlab to program and calculate the kinetic parameters of the first three critical speeds and natural frequencies of the high-speed spindle. Finally, the simulation data was compared with experimental data. The results show that the Riccati transfer matrix method is 2.3% more accurate than the traditional transfer matrix method, and it verifies the accuracy and feasibility of the Riccati transfer matrix method, and is of great significance to effectively guide the analysis and design of high-performance and high-speed motor spindle.

Dispersion curves of magneto-electro-elastic imperfect cylinders filled with fluid

Imperfect interfaces due to manufacturing errors can affect the dynamic performance of multiferroic composite cylinders, especially in the high frequency range. However, the traditional transfer matrix method will encounter numerical instability when high frequency is involved. In this paper, an alternative approach called the reverberation-ray matrix method is presented for the wave propagation of a fluid-filled multiferroic composite cylinder with imperfect interfaces, which is unconditionally numerically stable for high frequency analysis. Based on the three-dimensional magneto-electro-elastic theory and the Hamiltonian state equation, the reverberation-ray formulations are established to analyze the dispersion relations of laminate composite cylinders. The effects of interfacial imperfection as well as different material and geometric parameters on the dispersion curves of multiferroic composite cylinders are investigated. The present method could be useful for the analysis and damage detection of layered composite cylinders.

Study on Dynamic Optimization Design of Double Span Rotor Test System

The inherent vibration characteristics of the structure of a machine is modal and each modal only corresponds to one natural frequency, all kinds of rotor in various engineering need to avoid resonance through its natural frequency. The finite element method is a numerical approximation algorithm, which is used to modal analysis of rotor and divides the complex large continuum into many simpler small units instead of traditional transfer matrix method. The 3D model was built by Solid works and imported into Ansys workbench for the modal analysis and harmonic response analysis of the rotor system with and without base of the pedestal, to determine the natural frequency and principal modal of the system, the theory result was compared with the two cases, It can avoid resonance and ensure the safe operation of the rotor system.

Structural and acoustic response of a finite stiffened submarine hull

After borrowing the idea of precise integration method, a precise integration transfer matrix method (PITMM) is proposed by modifying traditional transfer matrix method. The submarine hull can be modeled as joined conicalcylindrical-spherical shells. By considering the effect of the ring-stiffeners, the field transfer matrixes of shells of revolution are obtained accurately by PITMM. After assembling the field transfer matrixes into an entire matrix, the dynamic model is established to solve the dynamic responses of the joined shell. By describing the sound pressure in fluid by modified wave superposition method (MWSM) and collocating points along the meridian line of the joined shell, finally the structural and acoustic responses of a finite stiffened submarine hull can be predicted by coupled PITMM and MWSM. The effectiveness of the present method has been verified by comparing the structural and acoustic responses of the spherical shell with existing results. Furthermore, the effects of the model truncation, stiffness and thickness on the structural and acoustic responses of the submarine hull are studied.

Reflection phase characteristics and their applications based on one-dimensional coupled-cavity photonic crystals with gradually changed thickness ofsurface layer

In this paper, we first improve the traditional transfer matrix method to adapt to one-dimensional photonic crystal consisting of arbitrary materials, and then use it to study the reflection phase characteristics of two kinds of photonic crystals, i.e., a simple periodic photonic crystal structure and a coupled-cavity asymmetric photonic crystal with gradually changed thickness of surface layer. For both of the structures, the reflectivity within photonic band gap is above 98% and hardly affected by the thickness of the surface layer. However, their reflection phases exhibit distinctly different properties. For the simple photonic crystal structure, the reflection phases of both TE and TM polarizations are sensitively dependent on the thickness of surface layer, but their phase difference is almost the same as the thickness of surface layer varies, which cannot change the polarization of reflected light. While for the coupled-cavity asymmetric photonic crystal structure, studies show that the degenerate defect modes within photonic band gap will split as the thickness of the surface layer varies. Moreover, around the splitting defect modes the reflection phases of both TE and TM polarizations, as well as their phase difference, are sensitively dependent on the thickness of surface layer, resulting in sensitive polarization change of reflected light. The physical reason is attributed to the dramatic phase change caused by the splitting of degenerate defect modes. The above reflection phase characteristics of coupled-cavity asymmetric photonic crystals have potential in lowering or even eliminating the coherence of lasers in some special application cases. As an example, we design a one-dimensional photonic crystal structure with two-dimensional periodic varying thickness of surface layer. After an oblique incident narrowband laser beam is reflected from this structure and then focused by a lens, various polarized light beams (including linear polarized light beams along different directions, left-hand (or right-hand) circular (or elliptical) polarized light beams) will exist simultaneously, whose superposition will produce optical field with random phase and polarizations in the focal region. These results can effectively reduce the coherence of lasers, which holds promise in many fields such as laser nuclear fusion.