VIEWS: 7 PAGES: 32 POSTED ON: 2/8/2012
MECH103 Mechanisms and Dynamics of Machinery A review: Analytical geometry and Vector calculus TA: Zhengjian, Xu (Eric) Email: xuzj@ust.hk Tel:2358-8820 Impact Lab: Rm1205(Lift 19) How to describe the motion of an object? Shape and Location Velocity Acceleration Flow of fluid Mathematic tools for mechanics • Analytical geometry also called coordinate geometry and earlier referred to as Cartesian geometry , is the study of geometry using the principles of algebra. • Vector calculus (also called vector analysis) is concerned with multivariate real analysis of vectors in a metric space with two or more dimensions (3D) http://en.wikipedia.org/wiki/Vector_calculus http://en.wikipedia.org/wiki/Analytic_geometry Coordinate system A coordinate system is a plane (2D) or space (3D) where the origin and axes are defined so that coordinates can be measured. 2D analytical geometry 3D analytical geometry The coordinates of a point are the components of a tuple (sequence) of numbers (coordinate) used to represent the location of the point in the plane or space. In the space Rn by an n-tuple P = (r1, ..., rn) of real numbers r1, ..., rn. These numbers r1, ..., rn are called the coordinates of the point P. Cartesian coordinate system rectangular coordinate system 2D Coordinate system Other coordinate systems Polar coordinate system Cylinderical coordinate system Spherical coordinate system Others (r, θ) rθ Basic contents of Analytic geometry Important themes of analytical geometry Distance problems Line, Slope and intersection Description of geometrical shapes, curves Coordinate transformation Intergal and derivative Length, Area and volume Calculation of Distance Distance along a path compared with displacement Analytical geometry Line, slope and intersection Line y=mx+b Slope: θ Tan(θ)= Intersection on y-axis; Intersection on x-axis: Relationship between two lines In a Cartesian coordinate system, two straight lines L and M may be described by equations. L: y = ax + b, M: y = cx + d, y=cx+d Perpendicular a×c=-1 ax+b Parallel a=c Representation of a curve or surface Nonparametric equation (or implicit) Parametric equation (or explicit) Analytical geometry Conic sections ellipse hyperbola Analytical geometry Parabola Equation Focus Directrix 3-D shapes Cone: Hyperboloid: Sphere: Analytical geometry Transformation of coordinate systems y Translation y’ P(x,y) ⎧x = x + h ' ⎧x = x − h ' ⎨ ⎨ ' ⎩ y = y' + k ⎩y = y − k x’ O’ (h, k) x Rotation O ⎧ x = x // cosθ − y // sin θ ⎨ y ⎩ y = x // sin θ + y // cosθ y // P R x // θ ⎧ x = x cosθ + y sin θ // S ⎨ // T ⎩ y = − x sin θ + y cosθ θ O Q U x Important in CAD/CAM/CAE Derivative and Integral Differentiation and the derivative y = f (x) Δy dy slope = = = f ' ( x) Δx dx Δy dx, dy are the differentiations of x and y. Δx f’(x) is called the derivative of f(x) Integral b b ∫a f ' ( x)dx = ∫ df ( x) = f (b) − f (a ) a Table of derivatives and Integrals f(x) d [f(x)] / dx xn n xn-1 ex ex ln (x) 1/x sin x cos x cos x - sin x tan x sec 2x cot x - csc 2x sec x sec x tan x csc x - csc x cot x Calculation about curves and shapes Length b b L=∫ (dx) + (dy ) = ∫ 2 2 1 + ( f ' ) 2 dx a a Area b A = ∫ f ( x)dx a Volume b V = ∫ zdA a Example f ( x) = x 2 1 1 B S = ∫ 1 + ( f ' ) dx = ∫ 1 + 4 x dx 2 2 0 0 A 1 1 3 1 2 Area = ∫ (1 − x )dx = ( x − x ) 0 = 2 0 3 3 Foundation of Vector Calculus Vector and Scalar Scalar has any magnitude, Vector has its magnitude and orientation Vector space: defined by a coordinate system and unit vectors. In 3D rectangular coordinate system: Unit vector: i= (1, 0, 0) j= (0, 1, 0) k= (0, 0, 1) For example, an element of vector space Fn , x, is written: F = ( x1 , x2 ,......xn ) Length (magnitude) of F: F = ( x1 + x 2 + ..... + x n ) 2 2 2 Operation of Vector Addition and Substraction X ± Y = ( x1 ± y1 , x2 ± y2 ,......xn ± yn ) α X = (α x , α x ,...... α x ) 1 2 n 0 = ( 0 , 0 ,...... 0 ) − X = ( − x ,− x ,...... − x ) 1 2 n Dot product of Vectors dot product, also known as the scalar product, is a binary operation which takes two vectors and returns a scalar quantity. Definition: Phycial meaning in mechanics: For example, work done by a force F within a displacement d Cross product of Vectors cross product is: a binary operation on vectors in 3D vector space. It is also known as the vector product. It results in a pseudovector (Vector) rather than in a scalar. a = a1i + a2j + a3k = [a1, a2, a3] b = b1i + b2j + b3k = [b1, b2, b3] a × b = [a2b3 − a3b2, a3b1 − a1b3, a1b2 − a2b1]. Rule of Sarrus Phycial meaning in Mechanics: Moment of a force F with a force arm d Vector calculus Important themes of vector calculus Gradient: measures the rate and direction of change in a scalar field; the gradient of a scalar field is a vector field. Curl: measures a vector field's tendency to rotate about a point; the curl of a vector field is another vector field. Divergence: measures a vector field's tendency to originate from or converge upon a given point. Stoke’s theorem Divergnece theorem These knowledges will be very useful in Thermal Analysis and Fluid Mechanics Vector Calculus: Gradient The gradient of a scalar filed is a vector field which points in the direction of the greatest rate of change of the scalar field, and whose magnitude is the greatest rate of change The gradient of a scalar function f is denoted by: In Cartesian coordinate: f = 2 x + 3 y − sin(z ) 2 In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows. Engineering Examples Temperature gradient and heat transfer Air Pressure gradient and wind Vector calculus: Curl Curl : shows a vector field’s rate of rotation about a point A vector field which has zero curl everywhere is called irrotational Apply curl on vector field F [Fx, Fy, Fz] : = = In a vector field that describes the linear velocities of each individual part of a rotating disk, the curl will have a constant value on all parts of the disk. Example for Curl z V = {v x , v y , v z } v x = −v0 sin θ v y = v0 cos θ R y v0 vz = 0 ⎡ i ⎡ ⎤ ⎡ ⎤ j k⎤ ⎢ 0 ⎥ ⎢ 0 ⎥ ⎢∂ ∂ ∂⎥ ⎢ ∇ ×V = ⎢ ⎥= 0 ⎥=⎢ 0 ⎥ ⎢ ∂x ∂y ∂z ⎥ ⎢ ∂v y ∂v ⎥ ⎢ 2v0 ⎥ ⎢ vx ⎣ vy vz ⎥ ⎢ − x⎥ ⎢ ⎦ ⎢ ∂x ∂y ⎥ ⎣ R ⎥ ⎦ ⎣ ⎦ whorl Vector calculus: Divergence Divergence measures a vector field’s tendency to originate from or converge upon a given point. The divergence of a continuously differentiable vector field F = Fx i + Fy j + Fz k is defined to be the scalar-valued function Negative for a drain Positive for a well Zero for uniform flow http://www.math.umn.edu/~nykamp/m2374/readings/divcurl/ Examples for Vector Divergence y V = {v x , v y , v z } v x = −v0 cos θ x v y = −v0 sin θ vz = 0 ∂v x ∂v y ∂v z 2v0 ∇ •V = + + =− ∂x ∂y ∂z R Important operation in Vector Calculus Operation Notation Description Domain/Range Gradient Measures the rate and Maps scalar fields to direction of change in a scalar vector fields. field. Curl Measures a vector field's Maps vector fields to tendency to rotate about a vector fields. point. Divergence Measures the magnitude of a Maps vector fields to vector field's source or sink at scalar fields. a given point. Laplacian A composition of the divergence Maps scalar fields to and gradient operations. scalar fields. http://en.wikipedia.org/wiki/Vector_calculus Fundamental theorem of Vector calculus Theorem Statement Description Gradient The line integral through a gradient (vector) theorem field equals the difference in its scalar field at the endpoints of the curve. Green's The integral of the scalar curl of a vector theorem field over some region in the plane equals the line integral of the vector field over the curve bounding the region. Stokes' The integral of the curl of a vector field theorem over a surface equals the line integral of the vector field over the curve bounding the surface. Divergenc The integral of the divergence of a vector e theorem field over some solid equals the integral of the flux through the surface bounding the solid. http://en.wikipedia.org/wiki/Vector_calculus