Circular Shaft Research Articles

Effect of the rotation on a non-homogeneous infinite cylinder of orthotropic material with external magnetic field

The present investigation is concerned with a study effect of magnetic field and non-homogenous on the elastic stresses in rotating orthotropic infinite circular cylinder. A certain boundary conditions closed form stress fields solutions are obtained for rotating orthotropic cylinder under initial magnetic field with constant thickness for three cases: (1) Solid cylinder, (2) Cylinder with a circular hole at the center, (3) Cylinder mounted on a circular rigid shaft. Analytical expressions for the components of the displacement and stress fields in different cases are obtained. The effect of rotation and magnetic field and non-homogeneity on the displacement and stress fields are studied. Numerical results are illustrated graphically for each case. The effects of rotating and magnetic field and non-homogeneity are discussed.

Evaluation of the contact strength and durability of plain bearings with different types of shaft lobing

The generalized cumulative model of wear is employed to solve the tribocontact problem for plain bearings where journals have small lobing (ovality, trilobing, tetralobing). The case of mixed-area contact interaction between the shaft and bush is examined. The study investigates the effect of lobing both on variations in the maximum contact pressures during a full revolution of the shaft and on bearing durability. Regularities concerning this effect are determined. It was found that lobing causes a significant increase in the contact pressures depending on the extent of shaft out-of-roundedness and type of lobing. Specifically, ovality increases the bearing durability by 1.32 times; compared to the durability of a bearing with a circular section shaft, trilobing increases the bearing durability by 1.25 times, whereas tetralobing decreases the bearing durability even by 1.2 times.

Analysis of lateral earth pressure on a vertical circular shaft considering the 3D arching effect

The lateral earth pressure on a vertical circular shaft is investigated using both experiments and numerical analyses. The study focused on quantifying the magnitude and distribution of the lateral earth pressure, which was measured by considering the three-dimensional arching effect. A framework for determining distribution of the earth pressure based on centrifuge model tests and 2D FE analysis is introduced. The FE modelling techniques and the constitutive relationships of the soil are presented in detail. Parametric analyses showed that the arching effect on the lateral earth pressure is highly dependent on the diameter and height of the shaft, the internal friction angle and the cohesion value of the soil, the end-bearing conditions and the flexural modulus of the shaft. The study found that when the arching effect is considered, the lateral earth pressure on a vertical circular shaft is approximately 80% less than that calculated using Rankine’s theory. The study also found that the arching effect of the soil is more significant for flexible vertical shafts than for rigid vertical shafts embedded in weathered soil.

Fracture development around deep underground excavations: Insights from FDEM modelling

Over the past twenty years, there has been a growing interest in the development of numerical models that can realistically capture the progressive failure of rock masses. In particular, the investigation of damage development around underground excavations represents a key issue in several rock engineering applications, including tunnelling, mining, drilling, hydroelectric power generation, and the deep geological disposal of nuclear waste. The goal of this paper is to show the effectiveness of a hybrid finite-discrete element method (FDEM) code to simulate the fracturing mechanisms associated with the excavation of underground openings in brittle rock formations. A brief review of the current state-of-the-art modelling approaches is initially provided, including the description of selecting continuum- and discontinuum-based techniques. Then, the influence of a number of factors, including mechanical and in situ stress anisotropy, as well as excavation geometry, on the simulated damage is analysed for three different geomechanical scenarios. Firstly, the fracture nucleation and growth process under isotropic rock mass conditions is simulated for a circular shaft. Secondly, the influence of mechanical anisotropy on the development of an excavation damaged zone (EDZ) around a tunnel excavated in a layered rock formation is considered. Finally, the interaction mechanisms between two large caverns of an underground hydroelectric power station are investigated, with particular emphasis on the rock mass response sensitivity to the pillar width and excavation sequence. Overall, the numerical results indicate that FDEM simulations can provide unique geomechanical insights in cases where an explicit consideration of fracture and fragmentation processes is of paramount importance.

Delivering the Emirates Air Line, London, UK: design and construction of the steel main towers

This paper presents the design, fabrication and erection process of the three main towers of the Emirates Air Line cable car, which spans the River Thames in London, UK. The towers consist of geometrically complex arrangements of a circular hyperboloid shaft topped by a tapering elliptical hyperboloid head. The shafts are formed of interlocking, alternatively and differently pitched, spiralling, doubly curved, structural steel plate elements of varied typologies. The paper presents the extraordinary technical demands including the plate forming process, the complex array of loadings, the nature of the wind responses investigated including impacts of aerodynamic effects from cyclic wind loading, specific requirements for advanced connection stiffness modelling to ensure the extremely tight movement criteria, the challenges of erection over the river Thames and the details of site assembly to tight tolerances. This paper is one of a series of four which describe the delivery of the Emirates Air Line project.

Choking conditions inside plunging flow dropshafts

Severe rain events can subject our existing drainage systems to capacity. While the conveyance capacity of sewers can be increased by temporarily tolerating a full pipe flow condition, the complex dynamics of water–air mixture occurring in vertical dropshafts can largely restrict the overall conveyance capacity of drainage systems. In this study, the hydraulic performance of plunging flow dropshafts is investigated experimentally with respect to their conveyance capacity. The model dropshaft consisted of an upstream horizontal inflow pipe, a vertical circular shaft, and an outflow pipe discharging to the atmosphere. Five setups were built with Ds/Difrom 1.0 to 3.0 for the range of drop heights H/Ds= 3.0 to 21.0, where Diis the inlet pipe diameter, Dsthe shaft diameter, and H the drop height. A wide range of discharges up to Q* = 25 was tested, where Q* = Q/(gDi5)(1/2). Flow patterns recognized within the shaft (free flow, surface roller, plug flow, slug flow, and full pipe flow) help to assess the two stages of choking: the incipient and fully choked state. Criteria to predict choking occurrences were developed experimentally. The flow is considered to reach its conveyance capacity when an incipient choking state is visible. For a typical dropshaft with Ds/Di= 2, Q* was found to be approximately 5.

Development of an elliptical fitting algorithm to improve change detection capabilities with applications for deformation monitoring in circular tunnels and shafts

Terrestrial laser scanning, also known as Light Detection and Ranging (LiDAR) is an emerging technology that has many proven uses in the geotechnical engineering community including rockmass characterization, discontinuity measurement and landslide monitoring. One of the newer applications of LiDAR scanning is deformation monitoring and change detection. In tunnels, deformation is traditionally measured using a series of five or more control points installed around the diameter of the tunnel with measurements recorded at regular time intervals. LiDAR provides the ability to obtain a more complete characterization of the tunnel surface, allowing for determination of the mechanism and magnitude of tunnel deformation, as the entire surface of the tunnel is being modeled rather than a fixed set of points. This paper discusses terrestrial LiDAR scanning for deformation mapping of a surface and for cross-sectional closure measurements within an active tunnel using an elliptical fit to data for profile analysis. The methods were found to be accurate to within a few millimeters when measuring 58mm of diametric difference over an 18.3m diameter circular profile, even when some sections of the data were removed from the analysis.

The Absolute Torsion of Circular Shaft and Neutral Cross-Section

Mechanics of materials are based on the relative deformation to discuss the calculation of the circular shaft torsion deformation, But the relative torsion angle can not reflect the real deformation of a specific cross-section. According to the plane hypothesis, to discuss the absolute deformation of several common shaft withstand the simple torque, defined the neutral cross-section of torsion deformation and given the method to calculate absolute torsion angle, for the rational design of engineering components have a certain reference value.

Finite element analysis of functionally hybridized carbon/glass composite shafts

External torque on circular composite shafts produces linearly decreasing shear stresses along radial direction. The inner layers never stressed as much as outmost layer. Accommodating fiber stiffness of each layer based on these linearly decreasing stresses may be advantageous. To reduce the cost of carbon fiber-reinforced composite drive shaft, inner layers of shaft can be reinforced by hybrid fiber systems with less stiffness. To achieve that, while the top layer is still reinforced with carbon fiber, inner layers can be reinforced with the mixture of glass and carbon fiber. Without changing the fiber volume fractions, replacement of some carbon fiber with glass fiber will reduce the overall stiffness of fiber system. Mixture ratio or hybrid ratio can be calculated using linearly decreasing stress levels and the rule of mixture. These hybrid composite shafts can be manufactured by filament winding technique. During filament winding, composite shafts are wound layer by layer and filament type can be changed between the layers. The proper mixing ratio can be achieved by arranging the fiber count in each filament bundle. The relation between geometrical parameters and hybrid fiber volume fractions of composite shaft was derived. Finite element analysis on carbon fiber and functionally hybridized glass/carbon fiber shafts with same geometry was conducted. Stress, strain, and moment behavior of both shafts were compared. The possible advantages and disadvantages of hybridization were discussed.

Finite Element Investigation of Stress Intensity Factors of Multiple Semi-Elliptical Surface Cracks in Solid Shaft

Fatigue cracks nucleate at the surface of the shaft operating under cyclic load. These cracks then grow into semi-elliptical shapes while extending into the shaft. The paper presents numerical estimation of stress intensity factor (SIF) of such cracks in circular solid shaft when subjected to bending and twisting loads in elastic conditions. The finite element model is first validated with the help of results reported elsewhere for a single surface crack. Later on, multiple coplanar surface cracks are modeled and the effect of the number of cracks and the distance between them on SIF of the main crack is investigated. The results are in line with the expectations.Nomenclature

Analysis on the Solar Collector Bracket with Finite Element Method

Parabolic trough solar thermal power generation technology is the most mature technology in the world, which is also the only commercial operation of the solar thermal power generation technology. In order to the structure characteristics of the collector was accurately simulated and the result was calculated to reflect the performance of collector, the whole model of collector established. The internal force and deformation of steel under the external load was analyzed by nonlinear finite element method. For the connection problems of rectangular frame and mirror of the collector, the strength and rigidity of collector mirror were calculated by the finite element software for ANSYS, Which circular shaft for collectors steel production process complex, transportation inconvenience, put out by rectangular frame instead of round axis, ANSYS finite element software to rectangular steel collectors strength and stiffness calculation, and the standard comparison, and to provide a useful reference for large-scale production of the solar Collector.

Combustion performance of eccentrically rotated flames

Experimental/computational work highlighted the combustion performance with eccentric blockage rotation in the reactive flow passage to augment the turbulence production rates in the flame for acquiring high power to weight ratios and effective heat transfer in industrial applications. The temporal/spatial velocity gradients stimulated around the reactive stream boundaries enhanced the turbulence development in both premixed and non-premixed flames. The eccentric shapes included circular, elliptical, squared and triangular shafts as well as a triple/straight blade rotor. The design was further developed to incorporate simultaneous rotation of the eccentric blockage around its axis via a planetary gear assembly to duplicate the vortical structure. While the premixed flames responded to such cyclic action by having a mixture velocity of 16.8 m/s, the non-premixed flames acquired an increase of 341% in the largest eddy size and a flame length reduction by 42%. Increasing the Strouhal number to 0.94 and the swirl ratio to 0.78 increased the turbulent kinetic energy to 9.1 m2/s2. By controlling the drag coefficient and wake vorticity for the solid shapes, the triangular rotor enhanced the combustor outward heat flux to exhibit a heater efficiency of 36.8% in the premixed flame mode and reduced the HC and CO emissions, respectively, to 0.1% and 513 ppm for non-premixed flames. Decreasing the reactive stream cross-section favorably increased the flow shearing effects, while the elliptical shape showed the highest sensitivity to axes’ orientation with respect to the direction of rotation. Due to the reduction in peak temperatures via increasing the turbulence intensity and heat transfer rate from the flame, the NO x exhaust concentrations decreased to a minimum value around 10 ppm. The combustion efficiency of diffusion flames was optimized with the fuel port central positioning in the burner, where the planetary gear design provided a turbulence intensity of 13.7% by the triple blade rotor.

A computational framework for evaluating residual stress distributions from diffraction-based lattice strain data

Modern high energy X-ray and neutron diffraction capabilities can provide lattice strain measurements from spatially-resolved diffraction volumes over the full domain of mechanical components with complex, three dimensional geometries. Each measurement is an average of strains in those crystals within one diffraction volume whose orientations lie along a common crystallographic fiber. Presented here is a computational framework for determining residual stress fields that fully utilizes these data while simultaneously satisfying equilibrium locally over the domain of a component and traction conditions over its surface. Key attributes of the framework are the definition of two stress fields, a continuum field and a crystal scale field, and a solution methodology for minimizing the difference between these fields. The continuum field satisfies the constraints from equilibrium and surface tractions; the crystal scale field is derived from the lattice strain measurements. To effectively handle the distinct demands required of the two fields, the framework combines a traditional, element-based discretization for the continuum stress field with an element-free discretization for the crystal scale stress field. The classical example of a residual stress field imparted by an interference fit of a disk onto a circular shaft is used to demonstrate the framework. Using lattice strain data generated by high energy X-ray diffraction, residual stress distributions are determined for two different cases, one having a two-dimensional and the other a three-dimensional stress field.

Pure NOTES Rectosigmoid Resection: Transgastric Endoscopic IMA Dissection and Transanal Rectal Mobilization in Animal Models

We report a pure natural orifice translumenal endoscopic surgery (NOTES(®); American Society for Gastrointestinal Endoscopy [Oak Brook, IL] and the Society of American Gastrointestinal and Endoscopic Surgeons [Los Angeles, CA]) rectosigmoidectomy in animal models using transgastric endoscopic inferior mesenteric artery (IMA) dissection and transanal rectal mobilization. Ten live animals (2 pigs weighing 35-40 kg each and 8 dogs weighing 25-30 kg each) were used. A gastrotomy was made using a needle-knife puncture and the balloon dilatation technique or following the creation of a submucosal tunnel. A circular stapler shaft was transanally inserted up to the sigmoid colon for spatial orientation and traction of the mesocolon. The IMA was endoscopically dissected using a Coagrasper™ (Olympus, Tokyo, Japan) and then clipped. Endoscopic division of the sigmoid mesocolon was conducted laterally toward the marginal artery. Transanal full-thickness circumferential rectal and mesorectal dissections were performed, and a colorectal anastomosis was performed using a circular stapler with a single stapling technique. During the transanal approach, the gastrotomy was closed using four endoscopic clips. Endoscopic dissection of the IMA was successful in all cases, but minor bleedings occurred in 3 cases. The mean time from dissection and clipping to division of the IMA was 36.7 minutes (range, 25-45 minutes). The mean operation time was 180.5 minutes (range, 145-210 minutes). There were no intraoperative complications or hemodynamic instability. The mean length of the resected specimen was 11.2 cm (range, 9-17 cm). A pure NOTES approach to rectosigmoid resection using transgastric endoscopic IMA dissection is technically feasible in animal models.

The effect of arching pressure on a vertical circular shaft

The lateral earth pressure of vertical circular shaft is investigated by using experimental tests and a theoretical analysis. The emphasis was quantifying on the magnitude and distribution of a lateral earth pressure measured by three-dimensional arching effect. A framework for determining the earth pressure distribution on the basis of both centrifuge model and full-scale field test results is introduced. Through these studies, it is found that the lateral earth pressure decreases within active displacement (δh/H: 0.2×10−2, δh: horizontal displacement, H: Height of vertical shaft) of wall at deep excavation and the arching effect is more significant for deep excavation than for shallow excavation. It is also found that the lateral earth pressure acting on a vertical circular shaft considering arching effect is 80% smaller than that calculated by Rankine theory, and thus represents a significant improvement in the prediction of realistic earth pressure for vertical circular shaft subjected to arching pressure.

Posthumous Body Manipulation in the Classic Period Mixtequilla: Reevaluating the Human Remains of Ossuary I from El Zapotal, Veracruz

Our interdisciplinary study provides new information and interpretations of Ossuary I, a large assemblage of human bones associated with the ceremonial center dedicated to a life-size clay figure of a splendidly attired human skeleton (often identified as Mictlantecuhtli in the archaeological literature) at the site of El Zapotal in south-central Veracruz-Recovered in 1971, this assemblage has been interpreted as a ritual deposit of women who died during childbirth, whose bodies were dedicated in later Aztec lore to the Tlazolteotl goddess. The present paper provides new insights into the depositional sequence, the type and number of individuals within the assemblage, the sex and age profile of the mostly female cohort, the distribution of artificial head shapes as an ethnic marker, and evidence of perimortem violence and postmortem processing in the form of flaying. Our evidence indicates that Ossuary I represents the slow accumulation of loose bones and limb segments of partially skinned individuals in a circular shaft. Postdating the functioning of the Death God adoratorio and showing fluctuations in the patterns of pre-depositional body treatment, the assemblage expresses the Late Classic period ritual practice of flaying both males and females in Veracruz. In later stages, the ossuary was used again for a female cult, consistent with the original interpretation of women who died during childbirth.

Optimisation of FMS operations by using multiple criteria decision-making

This paper aims at presenting the topic of optimising production in flexible manufacturing systems (FMSs). Research is directed towards FMSs study to find out their behaviour and performance as well, and if possible, even before their physical realisation, and to scientifically establish the models of FMS optimisation, with application in the circular concrete shafts. Mathematical models will contain a number of parameters and optimisation criteria, objective function, performance indicators, average optimal synthesis and real time.

長尺な軸のフライス加工における工作物挙動の解析と加工条件最適化 (第1報)

The productivity of circular shafts with a straight groove and/or a plane surface on outside surface of workpiece was much improved by the development of CNC lathe having a milling spindle. However, for milling of shafts with a large length and a small diameter, workpiece deflection induced by cutting forces causes deterioration of surface quality and chatter vibration. This investigation deals with an analysis of workpiece behavior and optimization of the machining conditions for ball-end milling a shaft with a small flexural rigidity in the direction of perpendicular to the shaft axis. In this paper, to apply a previously developed cutting model to milling of shafts, geometric quantities such as contact region between the cutting edge and the workpiece surface, and distribution of undeformed chip thickness along the cutting edge are analyzed. Cutting forces are calculated by using the cutting model and an energy method. Then, the workpiece deflection is analyzed on the basis of calculated results of cutting forces, and the variation of the above quantities with tool rotational angle are compared with experimental results at various tool locations on the workpiece. It is found that both of the cutting forces and the workpiece deflection take each maximum value at a location where the tool axis reaches close to the perpendicular plane passed the axis of the workpiece. Through the comparison of the results obtained by analysis of resonance frequency of the workpiece and that obtained by impact test using impulse hammer it is shown that the possibility of avoiding the occurrence of chatter vibration.

Torsional Vibrations of a Conic Shaft with Opposite Tapers Carrying Arbitrary Concentrated Elements

The purpose of this paper is to present the exact solution for free torsional vibrations of a linearly tapered circular shaft carrying a number of concentrated elements. First of all, the equation of motion for free torsional vibration of a conic shaft is transformed into a Bessel equation, and, based on which, the exact displacement function in terms of Bessel functions is obtained. Next, the equations for compatibility of deformations and equilibrium of torsional moments at each attaching point (including the shaft ends) between the concentrated elements and the conic shaft with positive and negative tapers are derived. From the last equations, a characteristic equation of the form <svg style="vertical-align:-2.30685pt;width:78.125px;" id="M1" height="15.05" version="1.1" viewBox="0 0 78.125 15.05" width="78.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.112)"><path id="x5B" d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,5.927,12.112)"><path id="x1D43B" d="M865 650q-1 -4 -4 -14t-4 -14q-62 -5 -77 -19.5t-29 -82.5l-74 -394q-12 -61 -0.5 -77t75.5 -21l-6 -28h-273l8 28q64 5 82 21t29 76l36 198h-380l-37 -197q-11 -64 0.5 -78.5t79.5 -19.5l-6 -28h-268l6 28q60 6 75.5 21.5t26.5 76.5l75 394q13 66 2 81.5t-77 20.5l8 28&#xA;h263l-6 -28q-58 -5 -75.5 -21t-30.5 -81l-26 -153h377l29 153q12 67 2 81t-74 21l5 28h268z" /></g><g transform="matrix(.017,-0,0,-.017,20.835,12.112)"><path id="x5D" d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" /></g><g transform="matrix(.017,-0,0,-.017,26.7,12.112)"><path id="x7B" d="M293 -169q-145 0 -145 128q0 35 4 106t3 104q0 17 -2 28t-10 25.5t-28 23t-51 10.5v29q31 2 51 10.5t28 23t10 25.5t2 28q0 31 -4 101t-3 104q0 132 145 132v-28q-51 -7 -67.5 -30t-16.5 -63q0 -32 6.5 -99.5t6.5 -100.5q0 -97 -83 -115v-4q83 -20 83 -117&#xA;q0 -34 -6.5 -99.5t-6.5 -96.5q0 -41 16.5 -65.5t67.5 -31.5v-28z" /></g><g transform="matrix(.017,-0,0,-.017,32.598,12.112)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5&#xA;t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.017,-0,0,-.017,44.582,12.112)"><path id="x7D" d="M283 255q-31 -2 -51 -10.5t-28 -23.5t-10 -27t-2 -30q0 -28 4 -97.5t4 -107.5q0 -128 -146 -128v28q50 7 67.5 31t17.5 63q0 33 -7 100t-7 98q0 96 84 117v4q-84 20 -84 115q0 34 7 102.5t7 98.5q0 40 -17.5 63t-67.5 30v28q28 0 50 -4t45.5 -16.5t37 -40t13.5 -68.5&#xA;q0 -37 -4 -106.5t-4 -98.5q0 -18 2 -30t10 -27t28 -23.5t51 -10.5v-29z" /></g><g transform="matrix(.017,-0,0,-.017,55.189,12.112)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,69.893,12.112)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg> is obtained. Then, the natural frequencies of the torsional shaft are determined from the determinant equation <svg style="vertical-align:-2.22495pt;width:51.549999px;" id="M2" height="14.85" version="1.1" viewBox="0 0 51.549999 14.85" width="51.549999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.012)"><path id="x7C" d="M162 -163h-61v866h61v-866z" /></g><g transform="matrix(.017,-0,0,-.017,4.533,12.012)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,19.441,12.012)"><use xlink:href="#x7C"/></g><g transform="matrix(.017,-0,0,-.017,28.62,12.012)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,43.324,12.012)"><use xlink:href="#x30"/></g> </svg>, and, corresponding to each natural frequency, the column vector for the integration constants, <svg style="vertical-align:-2.30685pt;width:23.9px;" id="M3" height="15.05" version="1.1" viewBox="0 0 23.9 15.05" width="23.9" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.112)"><use xlink:href="#x7B"/></g><g transform="matrix(.017,-0,0,-.017,5.961,12.112)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,17.945,12.112)"><use xlink:href="#x7D"/></g> </svg>, is obtained from the equation <svg style="vertical-align:-2.30685pt;width:78.125px;" id="M4" height="15.05" version="1.1" viewBox="0 0 78.125 15.05" width="78.125" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.112)"><use xlink:href="#x5B"/></g><g transform="matrix(.017,-0,0,-.017,5.927,12.112)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,20.835,12.112)"><use xlink:href="#x5D"/></g><g transform="matrix(.017,-0,0,-.017,26.7,12.112)"><use xlink:href="#x7B"/></g><g transform="matrix(.017,-0,0,-.017,32.598,12.112)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,44.582,12.112)"><use xlink:href="#x7D"/></g><g transform="matrix(.017,-0,0,-.017,55.189,12.112)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,69.893,12.112)"><use xlink:href="#x30"/></g> </svg>. Substitution of the last integration constants into the associated displacement functions gives the corresponding mode shape of the entire conic shaft. To confirm the reliability of the presented theory, all numerical results obtained from the exact method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.

Boundary Conditions for Torsional Circular Shaft with Two-Dimensional Dodecagonal Quasicrystals

Through generalizing the method of a decay analysis technique determining the interior solution developed by Gregory and Wan, a set of necessary conditions on the end-data of torsional circular shaft in two-dimensional dodecagonal quasicrystals (2D dodecagonal QCs) for the existence of a rapidly decaying solution is established. By accurate solutions for auxiliary regular state, using the reciprocal theorem, these necessary conditions for the end-data to induce only a decaying elastostatic state (boundary layer solution) will be translated into appropriate boundary conditions for the torsional circular shaft in 2D dodecagonal QCs. The results of the present paper enable us to establish a set of boundary conditions.