This document discusses the geometrical parameters and aerodynamic characteristics of fuselage bodies of revolution. It defines key parameters such as fuselage length, maximum diameter, aspect ratio, and nose and tail tapering. It describes three flow modes around bodies of revolution: attached flow at low angles of attack, separated flow at moderate angles, and non-symmetric vortex shedding at high angles. It provides equations for calculating aerodynamic forces and moments on a body of revolution based on surface pressure distribution.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Lesson 17: The Mean Value Theorem and the shape of curvesMatthew Leingang
- The document discusses a math class lecture on March 14, 2008 that covered topics including the Mean Value Theorem, Rolle's Theorem, and using derivatives to determine if a function is increasing or decreasing on an interval.
- It provides announcements about an upcoming midterm being graded, problem sessions, and office hours. It also announces Pi day contests happening at 3:14 PM and 4 PM to recite digits of Pi and eat pie.
- The outline previews that the lecture will cover the Mean Value Theorem, Rolle's Theorem, why the MVT is useful, and using derivatives to sketch graphs and test for extremities. It also introduces the mathematician Pierre de Fermat.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The document discusses standards 8, 10, and 11 which involve calculating geometric measurements like perimeter, area, volume, and surface area of common shapes. It then provides examples of calculating the lateral area, surface area, and volume of right cones given different attributes like height, radius, and slant height. It gives 5 problems as examples to work through.
The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
The Mean Value Theorem is the Most Important Theorem in Calculus. It allows us to relate information about the derivative of a function to information about the function itself.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
Lesson 17: The Mean Value Theorem and the shape of curvesMatthew Leingang
- The document discusses a math class lecture on March 14, 2008 that covered topics including the Mean Value Theorem, Rolle's Theorem, and using derivatives to determine if a function is increasing or decreasing on an interval.
- It provides announcements about an upcoming midterm being graded, problem sessions, and office hours. It also announces Pi day contests happening at 3:14 PM and 4 PM to recite digits of Pi and eat pie.
- The outline previews that the lecture will cover the Mean Value Theorem, Rolle's Theorem, why the MVT is useful, and using derivatives to sketch graphs and test for extremities. It also introduces the mathematician Pierre de Fermat.
The Mean Value Theorem is the most important theorem in calculus. It is the first theorem which allows us to infer information about a function from information about its derivative. From the MVT we can derive tests for the monotonicity (increase or decrease) and concavity of a function.
The document discusses standards 8, 10, and 11 which involve calculating geometric measurements like perimeter, area, volume, and surface area of common shapes. It then provides examples of calculating the lateral area, surface area, and volume of right cones given different attributes like height, radius, and slant height. It gives 5 problems as examples to work through.
The document is a lecture on derivatives and the shapes of curves. It covers the mean value theorem, testing for monotonicity using the first derivative test, and finding intervals of monotonicity. It also discusses concavity and the second derivative test. Examples are provided to demonstrate how to find the intervals where a function is increasing or decreasing using the first derivative test.
This document discusses the concept of the derivative and differentiation. It begins by explaining how the slope of a curve changes at different points, unlike the constant slope of a line. It then defines the derivative of a function f at a point x0 as the limit of the difference quotient as h approaches 0. If this limit exists, then f is said to be differentiable at x0. The derivative f'(x) then represents the slope of the curve y=f(x) at each point x and is a measure of how steeply the curve is rising or falling at that point. Several examples are provided to illustrate how to compute derivatives using this limit definition.
We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.
This document contains a mathematics exam for high school students in Greece. It is divided into 4 sections with multiple questions in each section. The questions cover topics related to functions, limits, derivatives, and integrals. Some questions ask students to prove statements, find domains of functions, determine if functions are injective or have critical points. The document is 3 pages long and aims to test students' understanding of key concepts in calculus and mathematical analysis.
Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval with equal values at the endpoints, then the derivative is 0 for at least one value in the interval. The mean value theorems - Lagrange's and Cauchy's - generalize this idea, relating the average rate of change over an interval to the instantaneous rate at a point within the interval. Examples are provided to illustrate the theorems and exceptions that can occur when their conditions are not fully met.
1. The document presents a technique called "rotated and scaled Alamouti coding" that improves upon ordinary repetition-based retransmission for 2x2 MIMO systems.
2. It introduces the concept of "scaled repetition" which maps symbols differently during retransmission compared to ordinary repetition, improving performance. For 4-PAM modulation, scaled repetition maps symbols by scaling them and compensating to remain within the symbol set.
3. Simulation results show that the maximum transmission rate achieved with scaled repetition is only slightly smaller than the channel capacity, whereas ordinary repetition performs significantly worse when SNR is not low. The rotated and scaled Alamouti code can be decoded with reasonable complexity unlike codes like the Golden code.
This document defines formulas for calculating geometric properties of common shapes including the area of a trapezium, circumference and area of a circle, surface area of a cylinder, sphere, and the volumes of right prisms, cylinders, cones, spheres, and right pyramids. It provides the formulas in both English and Malay.
This document contains a mathematics exam with 4 problems (Themes A, B, C, D) involving functions, derivatives, monotonicity, convexity, extrema, asymptotes and limits.
Theme A involves properties of differentiable functions, the definition of the derivative, and Rolle's theorem. Theme B analyzes the monotonicity, convexity, asymptotes and graph of a given function.
Theme C proves properties of a continuous, monotonically increasing function and finds extrema of related functions. Theme D proves properties of a power function and its relation to a given line, defines a new function, and proves monotonicity and existence of a single real root for a polynomial equation.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
Handout for the course Abstract Argumentation and Interfaces to Argumentative...Federico Cerutti
This document provides an overview of abstract argumentation frameworks and semantics. It begins with definitions of Dung's argumentation framework (AF), including concepts like conflict-free sets, acceptable arguments, and admissible sets. It then covers properties that argumentation semantics can satisfy, like being conflict-free or reinstating acceptable arguments. Several semantics are defined, like complete, grounded, preferred and stable extensions. The document also discusses labelling-based representations of semantics and computational properties of decision problems for different semantics. In the second half, it outlines implementations, ranking-based semantics, argumentation schemes, semantic web argumentation, and natural language interfaces for argumentation systems.
The document defines and discusses various concepts related to functions, including:
- A function assigns exactly one output element to each input element. Functions can be represented graphically.
- Key properties of functions include being one-to-one (injective), onto (surjective), and bijective (both one-to-one and onto).
- Important functions discussed include the identity function I(x)=x, the floor function ⌊x⌋ which returns the largest integer less than or equal to x, and the ceiling function ⌈x⌉ which returns the smallest integer greater than or equal to x.
The document contains 50 multiple choice questions testing mathematical concepts such as algebra, geometry, statistics, and trigonometry. The questions cover a wide range of topics including: simplifying expressions, solving equations, finding values based on graphs/tables, properties of shapes, percentages, and probability.
This document provides a summary of a 2 hour mathematics enrichment session on lines and planes in 3-dimensions. It includes 10 problems involving calculating angles between lines and planes using trigonometric ratios. The problems include diagrams of prisms, pyramids and cuboids with given measurements. The document concludes with answers to the 10 problems.
Wing and fuselage structural optimization considering alternative materialmm_amini
This thesis analyzes the potential weight benefits of alternative material systems for aircraft structures. It develops a computational framework to model the wing and fuselage of jet transports using a skin-stringer-frame configuration. The framework performs stress analysis and optimization to calculate structural weight. Various carbon fiber reinforced plastic and aluminum alloy properties are analyzed to identify the most beneficial for weight reduction. The results indicate that enhancing open hole compression strength provides the most benefit for CFRP, while fatigue strength is most critical for aluminum. Current CFRP minimum gauge limits potential weight savings from some material enhancements, especially on small aircraft.
This document summarizes the integration project for the aft fuselage of an aircraft. It outlines the team members and their roles. It then describes the final design of the primary and secondary structures, including dimensions and materials. It details the positioning of aircraft systems and the development of manufacturing and assembly plans. Ergonomics considerations are also discussed. In total, the project covers the integration of primary and secondary structures, systems installation, manufacturing process planning, and ergonomics analysis to complete the preliminary design of the aft fuselage section.
This document discusses the features of wing flow at subsonic speeds (M∞ < 1). For unswept high-aspect ratio wings, the flow features are determined by overflow from the lower surface to the upper surface at the wing tips, creating a spanwise flow. This induces a downwash behind the wing and results in induced drag. For optimum wings, the circulation distribution should follow an elliptical law to minimize induced drag. For swept wings, there is an additional curvature of flow lines caused by spanwise flow, affecting pressure distribution across the wing.
The document summarizes the structural analysis and undercarriage design for an aircraft. It describes:
1) The wing structure including two main spars, ribs, stringers and reinforced areas around the engine mounts. Fuel will be stored between the spars.
2) The fuselage structure with frames, stringers, and reinforced floor beams to support the landing gear.
3) Calculations of stresses on structural components like wing spars and cargo floor beams to ensure they can withstand the loads with a safety margin. Dimensioning of components is based on withstanding stresses and similarities to other aircraft.
Optimizationof fuselage shape for better pressurization and drag reductioneSAT Journals
Abstract
The fuselage of any aircraft is essentially to accommodate the payload. It is normally not as streamlined as the wing. Cabin pressurization has been a major concern in the manufacturing of aircrafts. Generally, a cylindrical shape is preferred from a pressurization point of view as it has a higher strength and weighs less too. On the other hand, a sphere is considered as the best pressure vessel among all the shapes, but, sphere being a bluff body is not suitable for carrying payloads. On this note, a cylinder is considered to be better than a sphere to carry the payload and mainly to achieve a streamlined flow. In this paper, the shape chosen is a combination of the sphere and the cylinder to achieve optimum results for pressurization as well as a better streamlined flow. Our prime aim is to convert this bluff body into something more efficient and useful, rather than only for carrying the payload. We have focused basically on two details viz. 1) Better Pressurization and 2) to assist in minimizing the drag, thereby increasing the overall lift of the aircraft and hence increasing the fuel efficiency. The proposed fuselage structure was designed in CATIA V5 software and structural analyses were done in Auto-Desk Multi-Physics software. As a result, a better structural load capacity was found. A load of 10 N/mm2 was applied on both the bodies under consideration (cylinder and ellipse) having the same material, surface area, volume and weight. For the proposed elliptical design, 78% reduction in the minimum stress value and 10% reduction in the maximum stress value were noticed.
Keywords: Fuselage, Lifting Fuselage, Drag Reduction, Pressurization, Hoop Stress, Multi body design, Toroidal Shells, Multi-cylinder, Channel Propeller Configuration, Carbon Fiber, Graphite Fiber, Stabilization and Carbonization.
This document discusses the fundamentals of aircraft aerodynamics. It introduces aerodynamics as the science of air motion and forces on aircraft. Aerodynamics is divided into sections based on speed and altitude ranges, including incompressible flows, subsonic, transonic, supersonic, and hypersonic aerodynamics. The main components of an aircraft are also introduced, including the wing, fuselage, tail unit, landing gear, and power plant. Coordinate systems used in aerodynamic analysis are defined.
The document summarizes the structural and force analysis of rockets. It discusses the basic forces acting on a rocket - weight, thrust, drag and lift. It explains how the weight and center of gravity of a rocket changes as fuel is expelled. The rocket equation is presented for a variable mass system. Aerodynamic aspects like lift, drag and coefficients are defined. The location and role of the center of pressure is explained. Criteria for static stability is described. Structural analysis of the fuselage considers shape, stress distribution and failure modes. An example calculation verifies the strength of a model rocket's fuselage.
Fatigue life estimation of rear fuselage structure of an aircrafteSAT Journals
This document summarizes a study on estimating the fatigue life of the rear fuselage structure of an aircraft. The researchers created a finite element model of the rear fuselage structure in CATIA and analyzed it in MSC.PATRAN and MSC.NASTRAN to identify high stress regions. They found the maximum stress locations were at cut-out corners and rivet holes in the skin. A local model with finer meshing around the cargo door cut-out was also analyzed. Fatigue life was then estimated using Miner's rule and an S-N curve, accounting for factors like surface roughness and reliability. Damage was accumulated over the expected load cycles to predict fatigue life until crack initiation.
We discuss the ideas of monotonicity (increasing or decreasing) and concavity (up or down) of a function. Because of the Mean Value Theorem, we can determine these characteristics using derivatives.
This document contains a mathematics exam for high school students in Greece. It is divided into 4 sections with multiple questions in each section. The questions cover topics related to functions, limits, derivatives, and integrals. Some questions ask students to prove statements, find domains of functions, determine if functions are injective or have critical points. The document is 3 pages long and aims to test students' understanding of key concepts in calculus and mathematical analysis.
Rolle's theorem states that if a function is continuous on a closed interval and differentiable on the open interval with equal values at the endpoints, then the derivative is 0 for at least one value in the interval. The mean value theorems - Lagrange's and Cauchy's - generalize this idea, relating the average rate of change over an interval to the instantaneous rate at a point within the interval. Examples are provided to illustrate the theorems and exceptions that can occur when their conditions are not fully met.
1. The document presents a technique called "rotated and scaled Alamouti coding" that improves upon ordinary repetition-based retransmission for 2x2 MIMO systems.
2. It introduces the concept of "scaled repetition" which maps symbols differently during retransmission compared to ordinary repetition, improving performance. For 4-PAM modulation, scaled repetition maps symbols by scaling them and compensating to remain within the symbol set.
3. Simulation results show that the maximum transmission rate achieved with scaled repetition is only slightly smaller than the channel capacity, whereas ordinary repetition performs significantly worse when SNR is not low. The rotated and scaled Alamouti code can be decoded with reasonable complexity unlike codes like the Golden code.
This document defines formulas for calculating geometric properties of common shapes including the area of a trapezium, circumference and area of a circle, surface area of a cylinder, sphere, and the volumes of right prisms, cylinders, cones, spheres, and right pyramids. It provides the formulas in both English and Malay.
This document contains a mathematics exam with 4 problems (Themes A, B, C, D) involving functions, derivatives, monotonicity, convexity, extrema, asymptotes and limits.
Theme A involves properties of differentiable functions, the definition of the derivative, and Rolle's theorem. Theme B analyzes the monotonicity, convexity, asymptotes and graph of a given function.
Theme C proves properties of a continuous, monotonically increasing function and finds extrema of related functions. Theme D proves properties of a power function and its relation to a given line, defines a new function, and proves monotonicity and existence of a single real root for a polynomial equation.
The Mean Value Theorem is the most important theorem in calculus! It allows us to infer information about a function from information about its derivative. Such as: a function whose derivative is zero must be a constant function.
The document describes polar coordinates, which represent the location of a point P in a plane using two numbers: r, the distance from P to the origin O, and θ, the angle between the positive x-axis and the line from O to P. θ is positive for counter-clockwise angles and negative for clockwise angles. The polar coordinate (r, θ) uniquely identifies P's location. The document also provides the conversion formulas between polar coordinates (r, θ) and rectangular coordinates (x, y).
Handout for the course Abstract Argumentation and Interfaces to Argumentative...Federico Cerutti
This document provides an overview of abstract argumentation frameworks and semantics. It begins with definitions of Dung's argumentation framework (AF), including concepts like conflict-free sets, acceptable arguments, and admissible sets. It then covers properties that argumentation semantics can satisfy, like being conflict-free or reinstating acceptable arguments. Several semantics are defined, like complete, grounded, preferred and stable extensions. The document also discusses labelling-based representations of semantics and computational properties of decision problems for different semantics. In the second half, it outlines implementations, ranking-based semantics, argumentation schemes, semantic web argumentation, and natural language interfaces for argumentation systems.
The document defines and discusses various concepts related to functions, including:
- A function assigns exactly one output element to each input element. Functions can be represented graphically.
- Key properties of functions include being one-to-one (injective), onto (surjective), and bijective (both one-to-one and onto).
- Important functions discussed include the identity function I(x)=x, the floor function ⌊x⌋ which returns the largest integer less than or equal to x, and the ceiling function ⌈x⌉ which returns the smallest integer greater than or equal to x.
The document contains 50 multiple choice questions testing mathematical concepts such as algebra, geometry, statistics, and trigonometry. The questions cover a wide range of topics including: simplifying expressions, solving equations, finding values based on graphs/tables, properties of shapes, percentages, and probability.
This document provides a summary of a 2 hour mathematics enrichment session on lines and planes in 3-dimensions. It includes 10 problems involving calculating angles between lines and planes using trigonometric ratios. The problems include diagrams of prisms, pyramids and cuboids with given measurements. The document concludes with answers to the 10 problems.
Wing and fuselage structural optimization considering alternative materialmm_amini
This thesis analyzes the potential weight benefits of alternative material systems for aircraft structures. It develops a computational framework to model the wing and fuselage of jet transports using a skin-stringer-frame configuration. The framework performs stress analysis and optimization to calculate structural weight. Various carbon fiber reinforced plastic and aluminum alloy properties are analyzed to identify the most beneficial for weight reduction. The results indicate that enhancing open hole compression strength provides the most benefit for CFRP, while fatigue strength is most critical for aluminum. Current CFRP minimum gauge limits potential weight savings from some material enhancements, especially on small aircraft.
This document summarizes the integration project for the aft fuselage of an aircraft. It outlines the team members and their roles. It then describes the final design of the primary and secondary structures, including dimensions and materials. It details the positioning of aircraft systems and the development of manufacturing and assembly plans. Ergonomics considerations are also discussed. In total, the project covers the integration of primary and secondary structures, systems installation, manufacturing process planning, and ergonomics analysis to complete the preliminary design of the aft fuselage section.
This document discusses the features of wing flow at subsonic speeds (M∞ < 1). For unswept high-aspect ratio wings, the flow features are determined by overflow from the lower surface to the upper surface at the wing tips, creating a spanwise flow. This induces a downwash behind the wing and results in induced drag. For optimum wings, the circulation distribution should follow an elliptical law to minimize induced drag. For swept wings, there is an additional curvature of flow lines caused by spanwise flow, affecting pressure distribution across the wing.
The document summarizes the structural analysis and undercarriage design for an aircraft. It describes:
1) The wing structure including two main spars, ribs, stringers and reinforced areas around the engine mounts. Fuel will be stored between the spars.
2) The fuselage structure with frames, stringers, and reinforced floor beams to support the landing gear.
3) Calculations of stresses on structural components like wing spars and cargo floor beams to ensure they can withstand the loads with a safety margin. Dimensioning of components is based on withstanding stresses and similarities to other aircraft.
Optimizationof fuselage shape for better pressurization and drag reductioneSAT Journals
Abstract
The fuselage of any aircraft is essentially to accommodate the payload. It is normally not as streamlined as the wing. Cabin pressurization has been a major concern in the manufacturing of aircrafts. Generally, a cylindrical shape is preferred from a pressurization point of view as it has a higher strength and weighs less too. On the other hand, a sphere is considered as the best pressure vessel among all the shapes, but, sphere being a bluff body is not suitable for carrying payloads. On this note, a cylinder is considered to be better than a sphere to carry the payload and mainly to achieve a streamlined flow. In this paper, the shape chosen is a combination of the sphere and the cylinder to achieve optimum results for pressurization as well as a better streamlined flow. Our prime aim is to convert this bluff body into something more efficient and useful, rather than only for carrying the payload. We have focused basically on two details viz. 1) Better Pressurization and 2) to assist in minimizing the drag, thereby increasing the overall lift of the aircraft and hence increasing the fuel efficiency. The proposed fuselage structure was designed in CATIA V5 software and structural analyses were done in Auto-Desk Multi-Physics software. As a result, a better structural load capacity was found. A load of 10 N/mm2 was applied on both the bodies under consideration (cylinder and ellipse) having the same material, surface area, volume and weight. For the proposed elliptical design, 78% reduction in the minimum stress value and 10% reduction in the maximum stress value were noticed.
Keywords: Fuselage, Lifting Fuselage, Drag Reduction, Pressurization, Hoop Stress, Multi body design, Toroidal Shells, Multi-cylinder, Channel Propeller Configuration, Carbon Fiber, Graphite Fiber, Stabilization and Carbonization.
This document discusses the fundamentals of aircraft aerodynamics. It introduces aerodynamics as the science of air motion and forces on aircraft. Aerodynamics is divided into sections based on speed and altitude ranges, including incompressible flows, subsonic, transonic, supersonic, and hypersonic aerodynamics. The main components of an aircraft are also introduced, including the wing, fuselage, tail unit, landing gear, and power plant. Coordinate systems used in aerodynamic analysis are defined.
The document summarizes the structural and force analysis of rockets. It discusses the basic forces acting on a rocket - weight, thrust, drag and lift. It explains how the weight and center of gravity of a rocket changes as fuel is expelled. The rocket equation is presented for a variable mass system. Aerodynamic aspects like lift, drag and coefficients are defined. The location and role of the center of pressure is explained. Criteria for static stability is described. Structural analysis of the fuselage considers shape, stress distribution and failure modes. An example calculation verifies the strength of a model rocket's fuselage.
Fatigue life estimation of rear fuselage structure of an aircrafteSAT Journals
This document summarizes a study on estimating the fatigue life of the rear fuselage structure of an aircraft. The researchers created a finite element model of the rear fuselage structure in CATIA and analyzed it in MSC.PATRAN and MSC.NASTRAN to identify high stress regions. They found the maximum stress locations were at cut-out corners and rivet holes in the skin. A local model with finer meshing around the cargo door cut-out was also analyzed. Fatigue life was then estimated using Miner's rule and an S-N curve, accounting for factors like surface roughness and reliability. Damage was accumulated over the expected load cycles to predict fatigue life until crack initiation.
Fatigue Analysis of a Pressurized Aircraft Fuselage Modification using Hyperw...Altair
Fatigue Analyses of modifications on pressurized aircraft fuselages are both necessary and tedious. Using the Hyperworks software suite and StressCheck, RUAG has developed a fatigue analysis method which streamlines the process from the creation of the spectrum up to the detailed analysis of selected fastener holes and delivers results quickly and efficiently.
This method was then used to certify the installation of two large windows in the floor of a single engine turboprop A/C for aerial survey applications.
Speakers
David Schmid, Manager Structural Analysis, RUAG Schweiz AG
The document discusses the major components of aircraft, including the fuselage, wings, empennage, landing gear, and power plant. It describes the construction and design of aircraft fuselages, including open truss, monocoque, and semi-monocoque structures. It also briefly discusses wings, the empennage, landing gear, and factors considered in aircraft component design like fatigue life and material selection.
The document discusses the history and impact of climate change over the past century. It notes that global temperatures and sea levels have risen significantly, with extreme weather events like hurricanes also increasing. The causes are attributed to human activities like burning fossil fuels that release greenhouse gases and trap heat in the lower atmosphere. Major impacts are expected to continue and worsen if emissions are not reduced substantially in the coming decades.
Rapid Optimization of Composites - HyperSizer Express and FEAAswin John
Save your company valuable time!
HyperSizer Express is a must have tool for the composite design engineer. Learn more about this extremely fast and easy-to-use software that optimizes ply boundaries, determines optimal ply orientations, and sequences plies based on a weighted objective of mass minimization and manufacturability.
It performs automatic iterations with FEA models, recommends when and where to add core to the laminates, and maintains positive margins for a wide variety of failure methods. The best part, it’s all done in minutes!
The document discusses the aerodynamic characteristics of bodies of revolution like fuselages. It provides formulas to calculate:
1) The lift coefficient of different parts (nose, cylindrical, rear) of a body of revolution, accounting for factors like cross-sectional area changes, Mach number, boundary layers, etc.
2) The derivative of the lift coefficient of different parts, which depends on parameters like nose shape, aspect ratios, Mach number.
3) The aerodynamic moment of different parts and the coordinate of the aerodynamic center, calculated using formulas based on the elongated body theory.
This document discusses the aerodynamic characteristics of lifting surfaces like wings. It examines how factors like aspect ratio, sweep angle, and taper ratio impact the lift coefficient and its derivative with respect to angle of attack. Higher aspect ratio, lower sweep angle, and lower taper ratio generally result in a higher lift coefficient derivative. The lift coefficient relationship becomes non-linear for wings with very small aspect ratios, less than 1. Formulas are provided to calculate the lift coefficient, its derivative, and the angle of zero lift based on the wing geometry.
The document discusses the aerodynamic properties and geometry of aircraft wings. It describes how wings are formed by airfoils and outlines several key airfoil parameters like chord, thickness, camber, and their ratios. It also discusses wing planform characteristics such as span, taper ratio, sweep, and mean aerodynamic chord. Proper selection of wing geometry parameters like aspect ratio, taper, and twist can optimize an aircraft's aerodynamic qualities including drag, stability, and load distribution.
IJCER (www.ijceronline.com) International Journal of computational Engineeri...ijceronline
1. The document introduces the concept of a "Total Prime Graph", which is a graph that admits a special type of labeling called a "Total Prime Labeling".
2. Some properties of Total Prime Labelings are studied, and it is proved that paths, stars, bistars, combs, even cycles, helm graphs, and certain wheel graphs are Total Prime Graphs. However, odd cycles are proved to not be Total Prime Graphs.
3. The labeling must satisfy two conditions - the labels of adjacent vertices and incident edges of high degree vertices must be relatively prime. Several examples and theorems demonstrating Total Prime Graphs are provided.
The document outlines the contents of a Mathematics-I course including ordinary differential equations, linear differential equations, mean value theorems, functions of several variables, curvature, applications of integration, multiple integrals, series and sequences, vector differentiation and integration. It provides textbooks and references for the course and outlines 12 lectures covering applications of integration such as length, volume, surface area and multiple integrals including change of order and change of variables in double and triple integrals.
The document discusses Legendre polynomials, which are special functions that arise in solutions to Laplace's equation in spherical coordinates. Some key points:
1) Legendre polynomials Pn(cosθ) are a set of orthogonal polynomials that satisfy Legendre's differential equation.
2) Pn(cosθ) can be defined using a generating function or by taking partial derivatives of 1/r.
3) Important properties of Legendre polynomials include P0(t)=1, Pn(1)=1, Pn(-1)=(-1)n, and a recurrence relation involving Pn+1, Pn, and their derivatives.
Aeroacoustic simulation of bluff body noise using a hybrid statistical methodCon Doolan
The document summarizes a presentation on aeroacoustic simulation of bluff body noise using a hybrid statistical method. It discusses limitations of conventional computational fluid dynamics methods for noise prediction and introduces a new statistical correction method to improve noise simulation results from unsteady Reynolds-averaged Navier Stokes simulations. Key results from applying the statistical correction to simulations of circular cylinder noise are also summarized.
Formulas Perimeter and Area 10 7,8,10,11katiavidal
The document provides formulas for calculating perimeter, area, circumference, and circle area. It defines perimeter as the distance around a shape and area as the surface covered within a perimeter. Formulas are given for calculating the perimeter and area of squares, rectangles, parallelograms, triangles, and circles. The circumference of a circle is defined as the distance around it and can be calculated using C=πd or C=2πr, where π is approximately 3.14.
Contour integrals provide a useful technique for evaluating certain integrals. This document introduces contour integrals and provides examples to illustrate their application. It explains that contour integrals can be viewed as line integrals and that Cauchy's theorem allows continuous deformation of the contour without changing the result, as long as it does not cross singularities. The key points are:
1) Contour integrals around a pole give the residue of the pole times 2πi.
2) Examples are worked out to show contour integrals give the same results as conventional integrals for integrals that cannot be expressed in terms of elementary functions.
3) Contour integrals enable evaluation of new integrals that would be difficult to evaluate otherwise, like calculating an integral of an exponential
The Newton polytope of the resultant, or resultant polytope, characterizes the resultant polynomial more precisely than total degree. The combinatorics of resultant polytopes are known in the Sylvester case [Gelfand et al.90] and up to dimension 3 [Sturmfels 94]. We extend this work by studying the combinatorial characterization of 4-dimensional resultant polytopes, which show a greater diversity and involve computational and combinatorial challenges. In particular, our experiments, based on software respol for computing resultant polytopes, establish lower bounds on the maximal number of faces. By studying mixed subdivisions, we obtain tight upper bounds on the maximal number of facets and ridges, thus arriving at the following maximal f-vector: (22,66,66,22), i.e. vector of face cardinalities. Certain general features emerge, such as the symmetry of the maximal f-vector, which are intriguing but still under investigation. We establish a result of independent interest, namely that the f-vector is maximized when the input supports are sufficiently generic, namely full dimensional and without parallel edges. Lastly, we offer a classification result of all possible 4-dimensional resultant polytopes.
This document discusses various topics related to sphere packings, lattices, spherical codes, and energy minimization on the sphere. It defines sphere packings, lattices, and spherical codes. It describes problems like finding the densest sphere packing in each dimension, determining optimal spherical codes, and minimizing potential energy on the sphere. Linear programming bounds are introduced as a technique for proving optimality of codes. Properties of positive definite kernels and Gegenbauer polynomials are also summarized.
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Theme 091011
1. SECTION 2. AERODYNAMICS OF BODY OF REVOLUTIONS
THEME 9. FUSELAGE GEOMETRICAL PARAMETERS
9.1. General
Fuselage is the main part of an airplane structure. It serves for joining of all its
parts in a whole, and also for arrangement of crew, passengers, equipment and freights.
The exterior shape of a fuselage is determined by the airplane assigning, range of
speeds of flight, arrangement of engines and other factors.
The airplane fuselages (Fig. 9.1) and engine nacelles have the shape of the body
of revolutions or close to it. For the airplanes having integral configurations the wing
passes smoothly into the fuselage and the shape of the fuselage cross-section can
essentially differ from circular. The fuselages of transport airplanes frequently have tail
unit deflected upwards. The noses of modern fighters, as a rule, are rejected downwards.
Fig. 9.1. The basic geometrical characteristics of a fuselage
The basic geometrical parameters of a fuselage are the following ones.
Length of a fuselage l f is the greatest size of a fuselage along its centerline.
The area of fuselage midsection S m . f . is the greatest area of fuselage cross-
section by a plane, perpendicular to its centerline.
The shape of a fuselage cross section essentially influences the interference
aerodynamic characteristics at the installation of a wing and tail unit. While calculating
the aerodynamic characteristics of an isolated fuselage it is approximately substituted by
a body of revolution with the equivalent area of cross sections.
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2. Maximum equivalent diameter of a fuselage d m . f . is the diameter of a circle,
which area is equal to the fuselage midsection area.
d f = d m. f . = 4 Sm. f . π . (9.1)
The distinction of a fuselage from the body of revolution is taken into account by
an addend in the aerodynamic characteristics depending on design features. Therefore
we shall consider the geometric and aerodynamic characteristics of body of revolutions,
connecting them with the specific fuselage below.
Fuselage aspect ratio λ f is the ratio of the fuselage length to its maximum
equivalent diameter,
λf = lf d f . (9.2)
In some cases, especially when the fuselage is the body of revolution, it is
possible to allocate nose (head), cylindrical (central) and rear parts (fig. 9.1) and to
introduce the appropriate geometric parameters for them. As the total fuselage length
l f = l nose + lcil + l rear , then its aspect ratio
λ f = λ nose + λ cil + λ rear , (9.3)
where λ nose , λ cil , λ rear is the aspect ratio of nose, cylindrical and rear parts,
λ nose = l nose d f , λ cil = lcil d f , λ rear = l rear d f .
Frequently nose parts of fuselages have bluntness.
In some cases, nose can have an inside channel - the engine air intake.
The rear part of a fuselage can have a blunt base.
Then, the following additional parameters for the description of a nose and rear
part are used:
Nose tapering η nose = d nose d f is the ratio of a fuselage nose diameter to its
maximum equivalent diameter.
Tapering of a rear part η rear = d base d f is the ratio of diameter of a base of a
fuselage to its maximum equivalent diameter.
The relative area of the blunt base - S base = d base d 2 .
2
f
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3. The following angles are used at presence of a cam tail part. An angle of mean
line deviation β rear , the deviation of the nose part is determined by the angle β nose
(angle β nose is considered as positive at nose part deflection downwards).
9.1. Shape of a nose part and its geometrical parameters
The conical shape of a fuselage nose:
1 1
λ nose = , tgβ 0 = ;
2 tgβ 0 2λ nose
1
Wnose = l nose S m . f . - volume of the nose part.
3
1 − η nose 1 − η nose
λ nose = , tgβ 0 = ;
2 tgβ 0 2λ nose
Wnose =
1
3
( 2
)
1 + η nose + η nose l nose S m . f . .
The parabolic shape of a fuselage nose:
x
r = 0 .5 d f x( 2 − x ) , x = , 0 ≤ x ≤ 1;
l nose
1
η nose = 0 , tgβ0 = .
λ nose
[
r = 0 .5 d f ηnose + (1 − ηnose ) x( 2 − x ) ; ]
Wnose =
1
15
( 2
)
8 + 4η nose + 3η nose l nose S m . f . .
The ogival shape formed by arcs of a circle is close to the parabolic one, which
β0
aspect ratio is equal to λ nose = 0 .5 ctg . Volume of the nose part at λ nose ≥ 2 .5 is
2
determined by expression Wnose =
2
3
( 2
)
1 + 0 .5η nose l nose S m . f . .
The elliptical (ellipsoidal) shape of the nose part.
92
4. x
r = 0 .5 d f 1 − ( x − 1) = 0 .5 d f x( 2 − x ) , x =
2
;
l nose
2
Wnose = l nose S m . f .
3
The particular case of an elliptical nose part is the hemisphere.
The shapes of rear parts are designed the same as nose ones.
THEME 10. FEATURES OF FLOW ABOUT BODYS OF
REVOLUTION
Let's consider a body of revolution, which is streamlined by undisturbed flow at
angle of attack α . The flow can be represented as a result of superposition of two flows
- longitudinal with speed V∞ cos α and transversal with speed V∞ sinα . At large angles
of attack the flow is determined by transversal flow, and at small angles of attack - by
longitudinal. The transversal flow is always subsonic at small angles of attack.
Conditionally we shall point out three flow modes about body of revolution.
1. Attached flow at small angles of attack ( α = 0 ...5 o ). At small angles of attack
the flow about cross-sections differs only by thickness and status of the boundary layer.
The laminar boundary layer of small thickness is in the nose. Further thickness of a
boundary layer gradually arises along the length of the body of revolution. Its character
varies, the boundary layer becomes turbulent.
92
5. 2. At moderate angles of attack
( α ≤ 20 o ) the axis-symmetrical character of
flow is upset. The flow about fuselage takes
place with separation of the boundary layer at
lateral areas. The separated boundary layer is
turned into two vortex bundles (Fig. 10.1, a).
The location of the point of boundary layer
separation depends on the shape and aspect
ratio of the nose part, Mach numbers and
some other factors.
3. At large angles of attack ( α > 30 o )
the disturbance of a symmetry of a vortex
system takes place. Vortex bundles separate
from the surface of the body of revolution,
not having reached its rear part. At
Fig. 10.1. The flow scheme disturbance of flow rotational symmetry the
whole system of vortexes on the upper
surface of a body of revolution (Fig. 10.1, a)
is formed. It results into formation of sizable
transversal forces and moments of yaw.
At supersonic speed M ∞ > 1 and large
angles of attack the internal (hanging) shock
waves appear because of a large positive
Fig. 10.2. Flow near a cone: a 1-cone;
gradient of pressure on leeward fuselage side
2-head shock wave;
(Fig. 10.2).
3-vortexes; 4-hanging internal shock
At that, the flow structure is similar to
waves.
the structure of the track behind the cylinder
with circular cross section. The hanging shock waves prevent from the loss of symmetry
of the fuselage vortical system. Therefore, at large supersonic speeds transversal forces
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6. and the moments of a yaw conditioned by non-symmetry of the fuselage vortical are
absent.
THEME 11. DEFINITION OF THE AERODYNAMIC
CHARACTERISTICS OF A BODY OF REVOLUTION
USING PRESSURE DISTRIBUTION
It is possible to define aerodynamic forces effecting the body of revolution
knowing the law of pressure distribution along its surface. The pressure in the given
point of the surface is determined by two factors: parameters of incoming flow and
geometrical features of the streamlined body. The angle of attack α and number M ∞
exert the essential influence onto pressure distribution along outline of the body of
revolution. If we compare pressure distribution along wing airfoil with pressure
distribution along body of revolution, then it is possible to note the following: the
rarefaction on the body of revolution is much less than rarefaction on the wing. It takes
place due to spatial flow about the body of revolution. The flow spatiality enfeebles the
influence of flow compressibility onto character of flow about the body of revolution. In
subsonic flow for the body of revolution the factor of pressure is equal
C p incomp
Cp = , (11.1)
4 2
1 − M∞
C p incomp
while at flow about the wing C p = , i.e. the factor of pressure C p of the body
2
1− M∞
of revolution depends on number M ∞ less than C p for the wing.
Let's separately consider the pressure on a lateral surface of the body of
revolution and the pressure on the blunt base. Let's define the aerodynamic
characteristics of the body of revolution in body axes, and then by the transition
formulae we shall receive the aerodynamic characteristics in wind axes. Let's assume
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7. that an overpressure p − p∞ (Fig. 11.1) acts on an elementary site of the lateral surface
dS = r dl dϕ .
Fig. 11.1. A nose part of a body of revolution.
The function for elementary normal force dY from pressure p − p∞ effecting
onto an element of the lateral surface dS looks like this:
dY = − ( p − p∞ ) ds cos ϕ cos ϑ = − q∞ C p r dl dϕ cos ϕ cos ϑ , (11.1)
taking into account, that dl cos ϑ = dx , we shall receive
dY = − q∞C p cos ϕ r( x ) dx . (11.2)
Analogously, we shall find a function for elementary longitudinal force dX from
pressure p − p∞
dX = ( p − p∞ ) ds sinϑ = q∞ C p r dϕ dl sinϑ , (11.3)
dr
taking into account, that dl sinϑ = dr and dr = dx , we shall receive
dx
.
dX = q∞C p r r dx dϕ . (11.4)
Integrating expressions (11.2) and (11.4) by length of the body of revolution from
0 up to l f and by an arc of a circle from 0 up to 2π we shall receive the formulae for
normal and longitudinal force without the account of forces of friction
lf 2π
Y = − q∞ ∫ r( x) dx ∫ C p ( x , ϕ ) cos ϕ dϕ ; (11.5)
0 0
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8. lf 2π
.
X = q∞ ∫ r( x) r( x)dx ∫ C p ( x ,ϕ ) dϕ . (11.6)
0 0
Let's write down the expression for the elementary moment dM z from normal
and longitudinal force
dM z = − dY x + dX r cos ϕ = ( p − p∞ )ds cos ϕ ( x cos ϑ + r sin ϑ ) ;
⎛ .⎞ (11.7)
dM z = q ∞ C p cos ϕ ⎜ x + r r⎟ rdxdϕ
⎝ ⎠
also we shall define the function of the longitudinal moment from pressure as
lf 2π
⎛ . ⎞
M z = q∞
∫ ⎝ ⎠ ∫
r( x )⎜ x + r( x ) r( x )⎟ dx C p ( x ,ϕ ) cos ϕ dϕ . (11.8)
0 0
Let's pass in the formulae (11.5), (11.6) and (11.8) from forces to their factors. As
the characteristic area we shall accept the area of midsection of a body of revolution
S m . f . , and as characteristic length - length of a fuselage l f . We can write down
lf π
2
C ya ≈ C y = −
Sm. f . ∫ r( x) dx ∫ C p cos ϕ dϕ ; (11.9)
0 0
lf π
2 .
Cx =
Sm. f . ∫ r( x) r( x) dx ∫ C p dϕ , (11.10)
0 0
at α = 0 the factor of pressure C p in the last formula does not depend on the angle ϕ ,
and depends only on coordinate x and in this case
lf
2π .
C xa0 = C x0 =
Sm. f . ∫ C p r( x ) r( x )dx ; (11.11)
0
lf π
2 ⎛ .⎞
mz =
Sm. f . l f ∫ ⎝ ⎠ ∫
r⎜ x + r r⎟ dx C p cos ϕ dϕ , (11.12)
0 0
For thin bodies ( r << 1 ) it is possible not to take into account a moment from
&
longitudinal force:
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9. lf π
2
mz ≈
Sm. f . l f ∫ r x dx ∫ C p cos ϕ dϕ . (11.13)
0 0
Dimensionless coordinate of center of pressure location can be defined by
formula for an aerodynamic moment relatively to the fuselage nose M z = −Y xc . p . :
xc . p . mz m
x c . p. = =− ≈− z . (11.14)
lf Cy C yа
Let's define pressure forces which act onto the blunt base. If the blunt base is
located along the normal to an axis of the body of revolution, there is only longitudinal
force, which at small angles of attack practically does not vary on α . This force is
called the force of base drag. The force of base drag can be determined at α = 0 . In
this case, value of ( pbase − p∞ ) along the circle with radius r is a constant
( pbase − p∞ ) = const (Fig. 11.2).
Fig. 11.2. The blunt base of the fuselage.
The function for elementary longitudinal force dX from pressure ( pbase − p∞ )
on an element of the base surface dS = 2π r dr looks like this:
dX = dX base = − ( pbaase − p∞ )2π rdr = − q∞ C p base 2π rdr . (11.15)
Let's pass to the factor of force of base drag
rbase
2π
C x base = −
Sm. f . ∫ C p base r dr . (11.16)
0
Practically, pressure on the blunt base C p base ( r ) ≈ C p base = const and in this
case it is possible to consider, that
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10. C x base = − C p base S base . (11.17)
Thus, the aerodynamic characteristics of the body of revolution can be calculated,
if the pressure distribution along its surface and blunt base is known.
92