http://hdl.handle.net/2005/5 2013-05-03T04:14:59Z 2013-05-03T04:14:59Z Sarkar, Korak http://hdl.handle.net/2005/1832 2012-12-04T05:59:07Z 2012-12-03T18:30:00Z

Title: Closed-form Solutions For Rotating And Non-rotating Beams : An Inverse Problem Approach Authors: Sarkar, Korak Abstract: Rotating Euler-Bernoulli beams and non-homogeneous Timoshenko beams are widely used to model important engineering structures. Hence the vibration analyses of these beams are an important problem from a structural dynamics point of view. The governing differential equations of both these type of beams do not yield any simple closed form solutions, hence we look for the inverse problem approach in determining the beam property variations given certain solutions. Firstly, we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams. Secondly, we study the free vibration of rotating Euler-Bernoulli beams, under cantilever boundary condition. For certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation. It is found that there are an infinite number of rotating beams, with various mass per unit length variations and flexural stiffness distributions, which share the same fundamental frequency and mode shape. The derived flexural stiffness polynomial functions are used as test functions for rotating beam numerical codes. They are also used to design rotating cantilever beams which may be required to vibrate with a particular frequency. Thirdly, we study the free vibration of non-homogeneous Timoshenko beams, under fixed-fixed and fixed-hinged boundary conditions. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, there exists a fundamental closed form solution to the coupled second order governing differential equations. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions, which share the same fundamental frequency and mode shape. They can be used to design non-homogeneous Timoshenko beams which may be required for certain engineering applications.

2012-12-03T18:30:00Z Mimani, Akhilesh http://hdl.handle.net/2005/1931 2013-02-20T07:43:56Z 2013-02-19T18:30:00Z

Title: 1-D And 3-D Analysis Of Multi-Port Muffler Configurations With Emphasis On Elliptical Cylindrical Chamber  Authors: Mimani, Akhilesh Abstract: The flow-reversal elliptical cylindrical end chamber mufflers of short length are used often in the modern day automotive exhaust systems. The conventional 1-D axial plane wave theory is not able to predict their acoustical attenuation performance in view of the fact that the chamber length is not enough for the evanescent 3-D modes generated at the junctions to decay sufficiently for frequencies below the cut-off frequency. Also, due to the large area expansion ratio at the inlet, the first few higher order modes get cut on even in the low frequency regime. This necessitates a 3-D FEM or 3-D BEM analysis, which is cumbersome and time consuming. Therefore, an ingenious 1-D transverse plane wave theory is developed by considering plane wave propagation along the major-axis of the elliptical section, whereby a 2-port axially short elliptical and circular chamber muffler is characterized by means of the transfer matrix [T] or impedance matrix [Z]. Two different approaches are followed: (1) a numerical scheme such as the Matrizant approach, and (2) an analytical approach based upon the Frobenius series solution of the Webster’s equation governing the transverse plane wave propagation. The convective effects of mean flow are neglected; however the dissipative effects at the ports are taken into account. The TL predicted by this 1-D transverse plane wave analysis is compared with that obtained by means of the 3-D analytical approach and numerical (FEM/BEM) methods. An excellent agreement is observed between this simplified 1-D approach and the 3-D approaches at least up to the cut-on frequency of the (1, 1) even mode in the case of elliptical cylindrical chambers, or the (1, 0) mode in the case of circular cylindrical chambers, thereby validating this 1-D transverse plane wave theory. The acoustical attenuation characteristics of such short chamber mufflers for various configurations are discussed, qualitatively as well as quantitatively. Moreover, the Frobenius series solution enables one to obtain non-dimensional frequencies for determining the resonance peak and trough in the TL graph. The use of this theory is, however, limited to configurations in which both the ports are located along the major axis in the case of elliptical chambers and along the same diameter for circular chambers. The method of cascading the [T] matrices of the 2-port elements cannot be used to analyze a network arrangement of 2-port elements owing to the non-unique direction of wave propagation in such a network of acoustic elements. Although, a few papers are found in the literature reporting the analysis of a network of 2-port acoustic elements, no work is seen on the analysis of a network of multi-port elements having more than two external ports. Therefore, a generalized algorithm is proposed for analyzing a general network arrangement of linear multi-port acoustic elements having N inlet ports and M outlet ports. Each of these multi-port elements constituting the network may be interconnected to each other in an arbitrary manner. By appropriate book-keeping of the equations obtained by the [Z] matrix characterizing each of the multi-port and 2-port elements along with the junction laws (which imply the equality of acoustic pressure and conservativeness of mass velocity at a multi-port junction), an overall connectivity matrix is obtained, whereupon a global [Z] matrix is obtained which characterizes the entire network. Generalized expressions are derived for the evaluation of acoustic performance evaluation parameters such as transmission loss (TL) and insertion loss (IL) for a multiple inlet and multiple outlet (MIMO) system. Some of the characteristic properties of a general multi-port element are also studied in this chapter. The 1-D axial and transverse plane wave analysis is used to characterize axially long and short chambers, respectively, in terms of the [Z] matrix. Different network arrangements of multi-port elements are constructed, wherein the TL performance of such MIMO networks obtained on the basis of either the 1-D axial or 1-D transverse plane wave theory are compared with 3-D FEA carried on a commercial software. The versatility of this algorithm is that it can deal with more than two external or terminal ports, i.e., one can have multiple inlets and outlets in a complicated acoustic network. A generalized approach/algorithm is presented to characterize rigid wall reactive multi-port chamber mufflers of different geometries by means of a 3-D analytical formulation based upon the modal expansion and the uniform piston-driven model. The geometries analyzed here are rectangular plenum chambers, circular cylindrical chamber mufflers with and without a pass tube, elliptical cylindrical chamber mufflers, spherical and hemispherical chambers, conical chamber mufflers with and without a co-axial pass tube and sectoral cylindrical chamber mufflers of circular and elliptical cross-section as well as sectoral conical chamber mufflers. Computer codes or subroutines have been developed wherein by choosing appropriate mode functions in the generalized pressure response function, one can characterize a multi-port chamber muffler of any of the aforementioned separable geometrical shapes in terms of the [Z] matrix, subsequent to which the TL performance of these chambers is evaluated in terms of the scattering matrix [S] parameters by making use of the relations between [Z] and [S] matrices derived earlier. Interestingly, the [Z] matrix approach combined with the uniform piston-driven model is indeed ideally suited for the 3-D analytical formulation inasmuch as regardless of the number of ports, one deals with only one area discontinuity at a time, thereby making the analysis convenient for a multi-port muffler configuration with arbitrary location of ports. The TL characteristics of SISO chambers corresponding to each of the aforementioned geometries (especially the elliptical cylindrical chamber) are analyzed in detail with respect to the effect of chamber dimensions (chamber length and transverse dimensions), and relative angular and axial location of ports. Furthermore, the analysis of SIDO (i.e., single inlet and double outlet) chamber mufflers is given special consideration. In particular, we examine (1) the effect of additional outlet port (second outlet port), (2) variation in the relative angular or axial location of the additional or second outlet port (keeping the location of the inlet port and the outlet ports of the original SISO chamber to be constant) and (3) the effect of interchanging the location of the inlet and outlet ports on the TL performance of these mufflers. Thus, design guidelines are developed for the optimal location of the inlet and outlet ports keeping in mind the broadband attenuation characteristics for a single inlet and multiple outlet (SIMO) system. The non-dimensional limits up to which a flow-reversal elliptical (or circular) cylindrical end chamber having an end-inlet and end-outlet configuration is acoustically short (so that the 1-D transverse plane wave theory is applicable) and the limits beyond which it is acoustically long (so that the 1-D axial plane wave theory is applicable) is determined in terms of the ratio or equivalently, in terms of the ratio. Towards this end, two different configurations of the elliptical cylindrical chamber are considered, namely, (1) End-Offset Inlet (located along the major-axis of the ellipse) and End-Centered Outlet (2) End-Offset Inlet and End-Offset Outlet (both the ports located on the major-axis of the ellipse and at equal offset distance from the center). The former configuration is analyzed using 3-D FEA simulations (on SYSNOISE) while the 3-D analytical uniform piston-driven model is used to analyze the latter configuration. The existence of the higher order evanescent modes in the axially long reversal chamber at low frequency (before the cut-on frequency of the (1, 1) even mode or (1, 0) mode) causes a shift in the resonance peak predicted by the 1-D axial plane wave theory and 3-D analytical approach. Thus, the 1-D axial plane wave analysis is corrected by introducing appropriate end correction due to the modified or effective length of the elliptical cylindrical chamber. An empirical formulae has been developed to obtain the average non-dimensional end correction for the aforementioned configurations as functions of the expansion ratio, (i.e., ), minor-axis to major-axis ratio, (i.e., ) and the center-offset distance ratio, (i.e., ). The intermediate limits between which the chamber is neither short nor long (acoustically) has also been obtained. Furthermore, an ingenious method (Quasi 1-D approach) of combining the 1-D transverse plane wave model with the 1-D axial plane wave model using the [Z] matrix is also proposed for the end-offset inlet and end-centered outlet configuration. A 3-D analytical procedure has also been developed which also enables one to determine the end-correction in axially long 2-port flow-reversal end chamber mufflers for different geometries such as rectangular, circular and elliptical cylindrical as well as conical chambers, a priori to the computation of TL. Using this novel analytical technique, we determine the end correction for arbitrary locations on the two end ports on the end face of an axially long flow-reversal end chamber. The applicability of this method is also demonstrated for determination of the end corrections for the 2-port circular cylindrical chamber configuration without and with a pass tube, elliptical cylindrical chambers as well as rectangular and conical chambers.

2013-02-19T18:30:00Z Tewari, Sumit http://hdl.handle.net/2005/1908 2013-01-31T09:45:20Z 2013-01-30T18:30:00Z

Title: Study Of Liquid Fuel Film Transport And Its Effect On Cold Start Hydrocarbon Emissions In A Carburetted Engine Authors: Tewari, Sumit Abstract: The present work is concerned with fundamental studies on the liquid fuel transport in the intake manifold of small carburetted engines. This work is motivated by the need for development of technologies to meet the stringent cold-start emission norms that are to be prescribed for two-wheelers in particular. More specifically, visualization studies conducted in a transparent manifold made of quartz in a small four-stroke 110-cc two-wheeler engine have shown the presence of gasoline films on the walls of the inlet manifold under cold start conditions. Advanced Laser diagnostic techniques such as Planar Laser Induced Fluorescence (PLIF) have been utilized to measure the thickness of the fuel films. The Sauter Mean Diameter for the fuel droplets at the carburettor exit is measured using Laser Shadowgraphy technique. It is observed that the films are present both at idling conditions and under load. This large amount of liquid fuel entering the engine leads to incomplete combustion and higher emissions of unburned hydrocarbons. A detailed analysis of the effects of heating the inlet manifold has been performed. The potential of this manifold heating strategy in reducing hydrocarbon emissions has been assessed and found to be promising. In addition, a need of proper control of the fuel exiting the carburettor is shown to reduce emissions and increase fuel efficiency.

2013-01-30T18:30:00Z Anil Chandra, A R http://hdl.handle.net/2005/1964 2013-04-04T07:08:27Z 2013-04-03T18:30:00Z

Title: Microstructure And Mechanical Properties Of Consolidated Magnesium Chips Authors: Anil Chandra, A R Abstract: Development of sustainable manufacturing and conservation of primary materials are the key challenges to environmental degradation and climate change. Recycling of primary materials is one of the approaches suggested for sustainable green manufacturing. In the present study, an attempt has been made to encompass both these concepts, i.e. recycling of waste machined chips of magnesium and development of sustainable manufacturing process. Chips generated during machining operations are of significant importance; they dissipate the heat from the work-piece and control the quality of the finished products. In recent years researchers have shown that by controlled machining it is possible to tailor size, shape and microstructure of chips and this has added new dimensions to the utility of these machined chips. Chips in the form of thin strips, rods, very fine powders with varying aspect ratio have been successfully machined with grain structure having nano size (~80nm) to submicron size. Consolidation of such machined chips and subsequent fabrication of products is of great interest from the point of view of sustainable manufacturing. Consolidation of machined chips by cold compaction followed by hot extrusion was proposed and has been termed as solid state recycling (SSR). This alternative method of manufacturing using machined chips circumvents melting and casting. Although several materials have been tried by this route, magnesium appears to be the most investigated material. Being lightest among the structural materials, magnesium and its alloys have wide ranging applications especially in automotive industry. Further, magnesium melting is cumbersome and environmentally hazardous which necessitates researchers to explore methods of overcoming the melting route. In this pursuit, SSR appears to be a choice for a soft material like magnesium whose products are fabricated by conventional processing techniques which include cold compaction followed by hot extrusion. Most of the work in literature with regard to SSR of magnesium has been centered around development of new alloys and their characterisation at room and elevated temperatures. Effect of oxide contaminants has also been widely studied. However, studies on microstructural evolution during processing (i.e. microstructure prior to and after extrusion) have not been reported. Further, such studies with pure metal is important since it is possible to separate the effect of secondary phases including precipitates which are otherwise present in alloys of Mg. Hence, commercial grade pure magnesium is the material of interest in the present work. Process of consolidation includes room temperature compaction followed by hot extrusion. The aim of the present work includes: Consolidation of machined chips of magnesium into billets by cold compaction at room temperature followed by hot extrusion, Microstructural characterisation of these cold compacted billets prior to and after extrusion, Evaluation of mechanical properties after extrusion at different temperatures. Correlating the mechanical properties with microstructure. In the present study mechanical properties evaluated include: strength properties (hardness, tensile and compressive properties), and damping properties As-cast billet of pure magnesium was turned in a lathe to produce chips at ambient conditions. The chips were cold compacted into billets of 28 mm diameter at a pressure of 350 MPa and held for 30 minutes. The billets of compacted chips (referred here as CC) were later extruded at four different temperatures, viz. 250, 300, 350 and 400°C, with an extrusion ratio of 49:1. Prior to extrusion, the CC was soaked at the desired extrusion temperature for 1 hour. Here, extrusions of compacted chips are designated as CCE (chip compacted and extruded). For comparison, the as-cast billet was extruded under similar conditions and is designated as AE (as-cast and extruded). The extruded rods had a diameter of 4 mm. Microstructural characterisation was done prior to and after extrusion, which forms the first part of the thesis. The extruded rods were characterised for their room temperature strength properties in the second part of the thesis. In the third and last part, damping properties were characterised as a function of time and temperature. Microstructural changes at the end of temperature sweep tests were also examined. Optical microscopy did not reveal the grain structure of CC due to the intense strains associated with chip formation and subsequent cold compaction. However, chip boundaries were found randomly oriented and tri-junctions were found to be porous. The CC showed a relative density of 95.4% and this happens to be the highest amongst the values reported in literature for SSR machined chips. TEM images of CC revealed an average grain size of 0.75µm. Synopsis CCs were soaked at extrusion temperature and quenched to unravel the microstructure that exists prior to extrusion. Grain size and hardness measurements indicate that the material was recrystallised prior to extrusion. Bulk texture estimated from X-ray diffraction, showed weak crystallographic textures. The CC had a typical texture with c-axis aligned along the compaction direction which subsequently got randomised during soaking (pre-heating at extrusion temperature). After extrusion, the 250°C extruded AE had slightly stronger texture than CCE: with clear preference for < 1010 > and < 1120 > plane normals. High working temperatures removed such preference and made the textures randomised for both AE and CCE. In-grain misorientations and the relative presence of the twins, estimated from EBSD scans show a clear pattern for higher in-grain misorientations in CCE compared to AE. The values for AE at higher extrusion temperatures approached that of fully recrystallised magnesium. Higher twin fraction in AE was attributed to its relatively larger grain size compared to CCE. The chip boundaries that were randomly oriented before extrusion appeared aligned along the extrusion direction after extrusion. On the contrary AE had an equiaxed structure. Both longitudinal and transverse section micrographs showed pronounced chip boundaries in the 250°C extruded CCE while it was no so pronounced in the case of 400°C extruded material. Density measurements showed 98.6% relative density for 250°C extruded CCE as compared to 99.9% densification achieved in 400°C extruded CCE. Dislocation density estimated using Variance method from the peaks of the X-ray diffraction data showed higher values for CCE compared to AE. Dislocation density reduced with increase in extrusion temperature. For comparison extruded rods were annealed at 250°C for 2 hours and their dislocation density was estimated. Vickers hardness indentations were done at low load (25g) and higher load (200g). Both showed decreasing values with increase in extrusion temperature. Grain size dependent hardness variation followed the Hall-Petch relationship. CCE showed higher hardness compared to AE. Room temperature tensile test showed higher 0.2% tensile proof stress (TPS) in CCE material and obeyed the grain size dependent Hall-Petch relationship, though the strain to failure was poor. CCE extruded at 250°C showed fibrous fracture surface and was different from the rest of the CCEs with evidence of shearing at chip boundaries before fracture. Synopsis The rest of the CCEs had a typical fracture surface which was similar to AE material. Strain hardening behaviour, measured in terms of hardening exponent (n), hardening capacity (Hc) and hardening rate (θ) was quiet different for CCE compared to AE. Room temperature compression test showed different kind of failure for 250°C extruded CCE with longitudinal splitting (de-bonding at chip boundaries) and shearing at an angle to loading direction. The rest of the CCEs failed in a typical manner similar to AE material. The 0.2% compressive proof stress (CPS) as a function of grain size obeyed the Hall-Petch relationship for AE while the fit was not so good for CCE. Moreover, except 400°C extruded CCE (CPS was higher by ~22%) the rest of the CCEs had lower CPS compared to AE despite having finer grain size. This was contrary to the TPS and hardness findings wherein CCE was consistently higher compared to AE owing to grain refinement. Density measurements showed presence of 1.4%, 0.8% and 0.5% porosity in 250°, 300° and 350°C extruded CCE samples respectively. Prompted by density, hardness and TPS findings, the CPS values were back-calculated using the Hall-Petch relationship of AE. The back-calculated CPS values of CCE were higher than corresponding AE. Strength asymmetry, measured as a ratio of compressive proof stress to tensile proof stress was higher in CCE compared to AE. Damping capacity (tanφ) and dynamic modulus were determined as a function of time (tested upto 30 minutes) and temperature (from RT to 300°C) at a constant frequency (5 Hz). CCE material displayed higher tanφ during time and temperature sweep tests (by 10-15%) with CCE extruded at 250° showing the highest values. Dynamic modulus was comparable for both the materials (with less than 5% difference) though, modulus was higher in materials extruded at higher temperature. Microstructural changes were examined at the end of temperature sweep test, both at the point of loading and away from the point of loading. A significant grain growth was observed in region under the loading point (in a 3-point bending set-up) and was insignificant at regions away from the loading point. Coarsening was low in CCE material on account of suppression at chip boundaries. Microstructure of CCE and AE specimens subjected to similar heating conditions but without loading showed no such coarsening.

2013-04-03T18:30:00Z