Velocity ratio-cum-transfer matrix method for the evaluation of a muffler with mean flow

The paper deals with a method for the evaluation of exhaust muffers with mean flow. A new set of variables, convective pressure and convective mass velocity, have been defined to replace the acoustic variables. An expression for attenuation (insertion loss) of a muffler has been proposed in terms of convective terminal impedances and a velocity ratio, on the lines of the one existing for acoustic filters. In order to evaluate the velocity ratio in terms of convective variables, transfer matrices for various muffler elements have been derived from the basic relations of energy, mass and momentum. Finally, the velocity ratiocum-transfer matrix method is illustrated for a typical straight-through muffler.

On an Integrated Transfer Matrix method for multiply connected mufflers

The commercial automotive mufflers are generally of a complicated shape with multiply connected parts and complex acoustic elements. The analysis of such complex mufflers has always been a great challenge. In this paper, an Integrated Transfer Matrix method has been developed to analyze complex mufflers. Integrated transfer matrix relates the state variables across the entire cross-section of the muffler shell, as one moves along the axis of the muffler, and can be partitioned appropriately in order to relate the state variables of different tubes constituting the cross-section. The paper presents a generalized one-dimensional (1-D) approach, using the transfer matrices of simple acoustic elements, which are available from the literature. The present approach is robust and flexible owing to its capability to construct an overall matrix of the muffler with the transfer matrices of individual acoustic elements and boundary conditions, which can then be used to evaluate the transmission loss, insertion loss, etc. Results from the present approach have been validated through comparisons with the available experimental and three-dimensional finite element method (FEM) based results. The results show good agreement with both measurements and FEM analysis up to the cut-off frequency. (C) 2011 Elsevier Ltd. All rights reserved.

Symmetry of one-dimensional dynamical systems in terms of transfer matrix parameters

The concept of symmetry for passive, one-dimensional dynamical systems is well understood in terms of the impedance matrix, or alternatively, the mobility matrix. In the past two decades, however, it has been established that the transfer matrix method is ideally suited for the analysis and synthesis of such systems. In this paper an investigatiob is described of what symmetry means in terms of the transfer matrix parameters of an passive element or a set of elements. One-dimensional flexural systems with 4 × 4 transfer matrices as well as acoustical and mechanical systems characterized by 2 × 2 transfer matrices are considered. It is shown that the transfer matrix of a symmetrical system, defined with respect to symmetrically oriented state variables, is involutory, and that a physically symmetrical system may not necessarily be functionally or dynamically symmetrical.

On the transfer matrix for a uniform tube with a moving medium

Abstract is not available.

Relationship Between The Impedance Matrix And The Transfer Matrix With Specific Reference To Symmetrical, Reciprocal And Conservative Systems

Impedance matrix and transfer matrix methods are often used in the analysis of linear dynamical systems. In this paper, general relationships between these matrices are derived. The properties of the impedance matrix and the transfer matrix of symmetrical systems, reciprocal systems and conservative systems are investigated. In the process, the following observations are made: (a) symmetrical systems are not a subset of reciprocal systems, as is often misunderstood; (b) the cascading of reciprocal systems again results in a reciprocal system, whereas cascading of symmetrical systems does not necessarily result in a symmetrical system; (c) the determinant of the transfer matrix, being ±1, is a property of both symmetrical systems and reciprocal systems, but this condition, however, is not sufficient to establish either the reciprocity or the symmetry of the system; (d) the impedance matrix of a conservative system is skew-Hermitian.

Matrix method for eigenstructure assignment - The multi-input case with application

The eigenvalue and eigenstructure assignment procedure has found application in a wide variety of control problems. In this paper a method for assigning eigenstructure to a linear time invariant multi-input system is proposed. The algorithm determines a matrix that has eigenvalues and eigenvectors at the desired locations. It is obtained from the knowledge of the open-loop system and the desired eigenstructure. Solution of the matrix equation, involving unknown controller gams, open-loop system matrices, and desired eigenvalues and eigenvectors, results hi the state feedback controller. The proposed algorithm requires the closed-loop eigenvalues to be different from those of the open-loop case. This apparent constraint can easily be overcome by a negligible shift in the values. Application of the procedure is illustrated through the offset control of a satellite supported, from an orbiting platform, by a flexible tether.

Matrix method for eigenvalue assignment: The single input case

The eigenvalue assignment/pole placement procedure has found application in a wide variety of control problems. The associated literature is rather extensive with a number of techniques discussed to that end. In this paper a method for assigning eigenvalues to a Linear Time Invariant (LTI) single input system is proposed. The algorithm determines a matrix, which has eigenvalues at the desired locations. It is obtained from the knowledge of the open-loop system and the desired eigenvalues. Solution of the matrix equation, involving unknown controller gains, open-loop system matrices and desired eigenvalues, results in the state feedback controller. The proposed algorithm requires the closed-loop eigenvalues to be different from those of the open-loop case. This apparent constraint is easily overcome by a negligible shift in the values. Two examples are considered to verify the proposed algorithm. The first one pertains to the in-plane libration of a Tethered Satellite System (TSS) while the second is concerned with control of the short period dynamics of a flexible airplane. Finally, the method is extended to determine the Controllability Grammian, corresponding to the specified closed-loop eigenvalues, without computing the controller gains.

Matrix method for eigenstructure assignment - The multi-input case with application

The eigenvalue and eigenstructure assignment procedure has found application in a wide variety of control problems. In this paper a method for assigning eigenstructure to a Linear time invariant multi-input system is proposed. The algorithm determines a matrix that has eigenvalues and eigenvectors at the desired locations. It is obtained from the knowledge of the open-loop system and the desired eigenstructure. solution of the matrix equation, involving unknown controller gains, open-loop system matrices, and desired eigenvalues and eigenvectors, results in the state feedback controller. The proposed algorithm requires the closed-loop eigenvalues to be different from those of the open-loop case. This apparent constraint can easily be overcome by a negligible shift in the values. Application of the procedure is illustrated through the offset control of a satellite supported, from an orbiting platform, by a flexible tether,

Matrix method for eigenvalue assignment: The single input case

The eigenvalue assignment/pole placement procedure has found application in a wide variety of control problems. The associated literature is rather extensive with a number of techniques discussed to that end. In this paper a method for assigning eigenvalues to a Linear Time Invariant (LTI) single input system is proposed. The algorithm determines a matrix, which has eigenvalues at the desired locations. It is obtained from the knowledge of the open-loop system and the desired eigenvalues. Solution of the matrix equation, involving unknown controller gains, open-loop system matrices and desired eigenvalues, results in the state feedback controller. The proposed algorithm requires the closed-loop eigenvalues to be different from those of the open-loop case. This apparent constraint is easily overcome by a negligible shift in the values. Two examples are considered to verify the proposed algorithm. The first one pertains to the in-plane libration of a Tethered Satellite System (TSS) while the second is concerned with control of the short period dynamics of a flexible airplane. Finally, the method is extended to determine the Controllability Grammian, corresponding to the specified closed-loop eigenvalues, without computing the controller gains.

A transfer matrix approach for evaluation of the response of a multi-layer infinite plate to a two-dimensional pressure excitation

A 6 X 6 transfer matrix is presented to evaluate the response of a multi-layer infinite plate to a given two-dimensional pressure excitation on one of its faces or, alternatively, to evaluate the acoustic pressure distribution excited by the normal velocity components of the radiating surfaces. It is shown that the present transfer matrix is a general case embodying the transfer matrices of normal excitation and one-dimensional pressure excitation due to an oblique incident wave. It is also shown that the present transfer matrix obeys the necessary checks to categorize the physically symmetric multi-layer plate as dynamically symmetric. Expressions are derived to obtain the wave propagation parameters, such as the transmission, absorption and reflection coefficients, in terms of the elements of the transfer matrix presented. Numerical results for transmission loss and reflection coefficients of a two-layer configuration are presented to illustrate the effect of angles of incidence, layer characteristics and ambient media.

Transfer matrix modeling of hyperbolic and parabolic ducts with incompressible mean flow

A general differential equation for the propagation of sound in a variable area duct or nozzle carrying incompressible mean flow (of low Mach number) is derived and solved for hyperbolic and parabolic shapes. Expressions for the state variables of acoustic pressure and acoustic mass velocity of the shapes are derived. Self‐consistent expressions for the four‐pole parameters are developed. The conical, exponential, catenoidal, sine, and cosine ducts are shown to be special cases of hyperbolic ducts. Finally, it is shown that if the mean flow in computing the transmission loss of the mufflers involving hyperbolic and parabolic shapes was not neglected, little practical benefit would be derived.

Velocity ratio in the analysis of linear dynamical systems

The transfer matrix method is known to be well suited for a complete analysis of a lumped as well as distributed element, one-dimensional, linear dynamical system with a marked chain topology. However, general subroutines of the type available for classical matrix methods are not available in the current literature on transfer matrix methods. In the present article, general expressions for various aspects of analysis-viz., natural frequency equation, modal vectors, forced response and filter performance—have been evaluated in terms of a single parameter, referred to as velocity ratio. Subprograms have been developed for use with the transfer matrix method for the evaluation of velocity ratio and related parameters. It is shown that a given system, branched or straight-through, can be completely analysed in terms of these basic subprograms, on a stored program digital computer. It is observed that the transfer matrix method with the velocity ratio approach has certain advantages over the existing general matrix methods in the analysis of one-dimensional systems.

Analysis of a hybrid noise control system for a duct

This letter proposes the combination of a passive muffler and an active noise control system for the control of very high‐level noise in ducts used with large industrial fans and similar equipment. The analysis of such a hybrid system is presented making use of electroacoustic analogies and the transfer matrix method. It turns out that a passive muffler upstream of the input microphone can indeed lower the acoustic pressure and, hence, the power requirement of the auxiliary source. The parameter that needs to be optimized (or maximized) for this purpose is a certain velocity ratio that can readily be evaluated in a closed form, making it more or less straightforward to synthesize the configuration of an effective passive muffler to go with the active noise control system.

Recent advances in muffler acoustics

Exhaust noise in engines has always been a major source of automotive noise. Challenges for muffler design have been constraints on size, back pressure, and, of course, the cost. Designing for sufficient insertion loss at the engine firing frequency and the first few harmonics has been the biggest challenge. Most advances in the design of efficient mufflers have resulted from linear plane wave theory, making use of the transfer matrix method. This review paper deals with evaluating approximate source characteristics required for prediction of the unmuffled intake and exhaust noise, making use of the electroacoustical analogies. In the last few years, significant advances have been made in the analysis of variable area perforated ducts, transverse plane wave analysis of short elliptical as well as circular chambers, double-tuned expansion chambers and concentric tube resonators, catalytic converters, diesel particulate filters, air cleaners, etc. The development of long strand fibrous materials that can be used in hot exhaust systems without binders has led to the use of combination mufflers in exhaust systems. Breakthroughs have been achieved in the prediction and control of breakout noise from the elliptical and circular muffler shell as well as the end plates of typical mufflers. Diesel particulate filters and inlet air cleaners have also been modeled acoustically. Some of these recent advances are the subject of this review paper.