Aeronautical Engineer Data Book Episode 3 ppt

ᎏ  ␾ ␾ ␾ ␾ ᎏ ᎏᎏ ᎏᎏ ᎏᎏ  ᎏ   ᎏ   ᎏ    31 Fundamental dimensions and units d dB dA ᎏᎏ (A ϫ B)= A ϫ ᎏᎏ + ᎏᎏ ϫ B dt dt dt d = – ᎏᎏ (B ϫ A) dt Gradient The gradient (grad) of a scalar field ␾ (x, y, z) is ∂ ∂ ∂ ∂ i ᎏᎏ + j ᎏᎏ + k ᎏ ∂  ∂ grad ␾ = ٌ ␾ = x y z ∂ j ᎏ ᎏ ∂ i + ᎏ ᎏ ∂ ∂ ᎏ = y ᎏ k ∂ ∂ x z Divergence The divergence (div) of a vector V = V(x, y, z) = V x (x, y, z) i + V y (x, y, z) j + V z (x, y, z)k ∂ + ᎏ ᎏ ∂ + ᎏ ᎏ ∂ V Vdiv ٌ ᎏ = · V x V y V ∂x ∂y ∂z Curl Curl (rotation) is: i j k z ∂ ∂ ∂ curl V = ٌϫV = ∂ ∂ ∂x y z V x V y V z ∂ ᎏ ᎏ – ∂ ᎏ V z V ∂y ∂z ∂ ᎏ ᎏ – ∂ ᎏ V x V ∂z ∂x i + j y z = ∂ ᎏ ᎏ – ∂ ᎏ V V y ∂x ∂y k x + 2.8.8 Differentiation Rules for differentiation: y, u and v are functions of x; a, b, c and n are constants. d du dv ᎏᎏ (au ± bv) = a ᎏᎏ ± b ᎏᎏ dx dx dx d (uv) dv du ᎏᎏ = u ᎏᎏ + v ᎏᎏ dx dx dx d u 1 du u dv ᎏᎏ ᎏᎏ = ᎏᎏ ᎏᎏ – ᎏᎏ ᎏᎏ dx v v dx v 2 dx  ᎏ  ᎏ ᎏ ᎏ  ᎏ ᎏ 32 Aeronautical Engineer’s Data Book d ᎏ x ᎏ d (u n ) = nu n–1 u d ᎏ x d ᎏ d 1 ᎏ x du ᎏ d ᎏ u n ᎏ x ᎏ d n+1 ᎏ u = –, n u d = 1 / , d u x ᎏᎏ d ᎏ x d ᎏ if dx ᎏ ᎏ d ≠ 0 u d ᎏ x ᎏ d u f (u) = f’(u) d ᎏ x d ᎏ  x d ᎏ x ᎏ d f(t)dt = f(x) a  b d ᎏ x ᎏ d f(t)dt = – f(x) x  b f(x, t)dt =  b a a f ᎏ ∂ ∂x d ᎏ x ᎏ d dt  v f(x, t)dt =  u u v ∂ v ᎏ x d ᎏ f ᎏ dt + f (x, v) ∂ d d ᎏ x ᎏ d x u ᎏ x d ᎏ – f (x, u) d Higher derivatives d ᎏ 2 y dx 2 d ᎏ x d ᎏ  y d = Second derivatives = ᎏ x ᎏ d = f"(x) = y" 2 d ᎏ 2 dx 2 d ᎏ x d ᎏ  d  2 + f'(u) u u ᎏ d ᎏ f(u) = f "(u) 2 x Derivatives of exponentials and logarithms d (ax + b) n = na(ax + b) n–1 ᎏ x ᎏ d d ᎏ x ᎏ d e ax = ae ax d ᎏ x ᎏ d ln ax = ᎏ x 1 ᎏ , ax > 0 ᎏ ᎏ 33 Fundamental dimensions and units d ᎏ x ᎏ d a u = a u ln a u d ᎏ x d ᎏ d u d ᎏ x d ᎏ ᎏ x ᎏ d log a u = log a e 1 ᎏᎏ u Derivatives of trigonometric functions in radians d d sin x = cos x, cos x = – sin x ᎏ x ᎏ d ᎏ x ᎏ d d ᎏ x ᎏ d tan x = sec 2 x = 1 + tan 2 x d ᎏ x ᎏ d cot x = –cosec 2 x d sin x ᎏ x ᎏ d ᎏ x ᎏ c sec x = 2 os = sec x tan x d cos x ᎏ x ᎏ d ᎏ x ᎏ s cosec x = – in 2 = – cosec x cot x d d arcsin x = – ᎏ x ᎏ d ᎏ x ᎏ d arccos x 1 = ᎏᎏ for angles in the 2 ) 1/2 (1 – x first quadrant. Derivatives of hyperbolic functions d d sinh x = cosh x, cosh x = sinh x ᎏ x ᎏ d ᎏ x ᎏ d d d tanh x = sech 2 cosh x = – cosech 2 ᎏ x ᎏ d x, ᎏ x ᎏ d d 1 (arcsinh x) = ᎏ 2 +1) 1/2 ᎏ x ᎏ d , (x d ±1 (arccosh x) = ᎏ 1) 1/2 ᎏ x ᎏ d 2 (x – x ᎏ ᎏᎏ ᎏᎏ ᎏ ᎏ ᎏ x 34 Aeronautical Engineer’s Data Book Partial derivatives Let f(x, y) be a function of the two variables x and y. The partial deriva- tive of f with respect to x, keeping y constant is: f (x + h, y) – f (x, y) = lim ᎏᎏᎏ ∂ ∂x h→0 h Similarly the partial derivative of f with respect to y, keeping x constant, is f ᎏ ∂ ᎏ ᎏ v ∂ ᎏ ∂ ∂y k→0 k Chain rule for partial derivatives To change variables from (x, y) to (u, v) where u = u(x, y), v = v(x, y), both x = x(u, v) and y(u, v) exist and f(x, y) = f [x(u, v), y(u, v)] = F(u, v). f ᎏ ᎏ u F ∂ ∂ ∂ ∂ F x ∂ ∂ ᎏ ∂ ᎏ = ᎏ ∂ ∂ f (x, y + k) – f (x, y) = lim ᎏᎏᎏ f ᎏ ∂ ᎏ f ᎏ y ∂v ∂v ∂v ∂y f ᎏ y ᎏ f ᎏ x ᎏ ᎏ + ᎏ ᎏ , ∂u ∂x ∂u ∂y + = f ᎏ ∂ ∂x ᎏ x ∂ ᎏ u = ∂ ᎏ v ∂ ᎏ ᎏ x ∂ ᎏ ᎏ u ∂ ᎏ F v F ∂ + ∂ ∂ , f ᎏ ∂ ∂y ᎏ y ∂ ᎏ u = ∂ ᎏ v ∂ ᎏ ᎏ y ∂ ᎏ ᎏ u ∂ ᎏ F v F + ∂ ∂ ∂ 2.8.9 Integration f(x) F(x) = ∫f(x)dx a+1 x a ≠ –1 x a ᎏ 1 ᎏ a e ᎏ + –1 ln | x | kx k , kx e a x a > 0, a ≠ 1 a x ᎏ a ᎏ ln , ln x x ln x – x sin x –cos x cos x sin x tan x ln | sec x | cot x ln | sin x | sec x ln | sec x + tan x | = ln | 1 ᎏ 1 ᎏ tan ᎏ (x + ᎏ π)| 2 2 ᎏᎏ ᎏᎏ 2 35 Fundamental dimensions and units ᎏ 2 1 ᎏ x |ln | tancosec x sin 2 ᎏ 2 1 ᎏ ᎏ 2 1 ᎏ (x – sin 2x)x 2 ᎏ 2 1 ᎏ ᎏ 2 1 ᎏ (x + sin 2x) cos x sec 2 x tan x sinh x cosh x cosh x sinh x tanh x ln cosh x sech x 2 arctan e x ᎏ 2 1 ᎏ cosech x ln | tanh x| sech 2 x tanh x ᎏ 2 + xa 1 ᎏ 2 ᎏ a 1 ᎏ ᎏ a x ᎏ , a ≠ 0arctan 1 a –x  – a ≠ aln ᎏ a ᎏ 2 ᎏ x ᎏ a , + x –a ᎏ 2 ᎏ a 1 – x 1 a ≠ 0ln ᎏ a ᎏ x + 1 x ᎏ a ᎏ 2 , a ≠ 0arcsin 2 ) 1/2 (a 2 – x ᎏ | ᎏ | , a  ln [x + (x 2 – a 2 ) 1/2 ] 1 2 ) 1/2 (x 2 – a ᎏ a x ᎏ , a ≠ 0arccosh 2.8.10 Matrices A matrix which has an array of m ϫ n numbers arranged in m rows and n columns is called an m ϫ n matrix. It is denoted by:  a 11 a 12 a 1n a 2n . . . a 21 a 22 . . .  . . . a m1 a m2 a mn   36 Aeronautical Engineer’s Data Book Square matrix This is a matrix having the same number of rows and columns. a 11 a 12 a 13 a 21 a 22 a 23 is a square matrix of order 3 ϫ a 31 a 32 a 33 3. Diagonal matrix This is a square matrix in which all the elements are zero except those in the leading diagonal.  a 11 0 0 0  0 a 22 is a diagonal matrix of order 3 0 0 a 33 ϫ 3. Unit matrix This is a diagonal matrix with the elements in the leading diagonal all equal to 1. All other elements are 0. The unit matrix is denoted by I. 1 0 0 0 1 0 I =  0 0 1 Addition of matrices Two matrices may be added provided that they are of the same order. This is done by adding the corresponding elements in each matrix. a 11 a 12 a 13  + b 11 b 12 b 13  a 21 a 22 a 23  b 21 b 22 b 23  a 11 + b 11 a 12 + b 12 a 13 + b 13 =  a 21 + b 21 a 23 + b 23  a 22 + b 22 Subtraction of matrices Subtraction is done in a similar way to addition except that the corresponding elements are subtracted. a 11 a 12 b 11 b 12 a 11 – b 11 a 12 –b 12  –  =  a 21 –b 21 a 22 –b 22  a 21 a 22 b 21 b 22    37 Fundamental dimensions and units Scalar multiplication A matrix may be multiplied by a number as follows: a 11 a 12 ba 11 ba 12 b  =  ba 21 ba 22  a 21 a 22 General matrix multiplication Two matrices can be multiplied together provided the number of columns in the first matrix is equal to the number of rows in the second matrix. b 11 b 12 a 11 a 12 a 13 b 21 b 22 a 21 a 22 a 23 b 31 b 32 a 11 b 11 +a 12 b 22 +a 13 b 31 a 11 b 12 +a 12 b 22 +a 13 b 32 =  a 21 b 11 +a 22 b 21 +a 23 b 31 a 21 b 12 +a 22 b 22 +a 23 b 32  If matrix A is of order (p ϫ q) and matrix B is of order (q ϫ r) then if C = AB, the order of C is (p ϫ r). Transposition of a matrix When the rows of a matrix are interchanged with its columns the matrix is said to be trans- posed. If the original matrix is denoted by A, its transpose is denoted by A' or A T .   a 11 a 21 a 11 a 12 a 13 then A T =  a If A = a 12 a 22 21 a 22 a 23 a 13 a 23 Adjoint of a matrix If A =[a ij ] is any matrix and A ij is the cofactor of a ij the matrix [A ij ] T is called the adjoint of A. Thus: a a 21 a 22 a 2n A 12 A 22 A n2 11 a 12 a 1n . adj A=  A 11 A 21 A n1 A =  . .  . . .  . . . . . . . . . . . . a n1 a n2 a mn A 1n A 2n A nn      38 Aeronautical Engineer’s Data Book Singular matrix A square matrix is singular if the determinant of its coefficients is zero. The inverse of a matrix If A is a non-singular matrix of order (n ϫ n) then its inverse is denoted by A –1 such that AA –1 = I = A –1 A. adj (A) A –1 = ᎏᎏ ∆ = det (A) ∆ A ij = cofactor of a ij a a 11 a 12 a 1n   A 11 A 21 A n1 21 a 22 a 2n A 12 A 22 A n2 . . . A –1 = ᎏ 1 ᎏ . . . If A =  . . . ∆ . . . a . . . . . . n1 a n2 a nn A 1n A 2n A nn 2.8.11 Solutions of simultaneous linear equations The set of linear equations a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2 a n1 x 1 + a n2 x 2 + + a nn x n = b n a where the as and bs are known, may be repre- sented by the single matrix equation Ax = b, where A is the (n ϫ n) matrix of coefficients, ij , and x and b are (n ϫ 1) column vectors. The solution to this matrix equation, if A is non-singular, may be written as x = A –1 b which leads to a solution given by Cramer’s rule: x i = det D i /det Ai = 1, 2, , n where det D i is the determinant obtained from det A by replacing the elements of a ki of the ith column by the elements b k (k = 1, 2, , n). Note that this rule is obtained by using A –1 = (det A) –1 adj A and so again is of practical use only when n ≤ 4.   ␾ ␾ x 39 Fundamental dimensions and units If det A = 0 but det D i ≠ 0 for some i then the equations are inconsistent: for example, x + y = 2, x + y = 3 has no solution. 2.8.12 Ordinary differential equations A differential equation is a relation between a function and its derivatives. The order of the highest derivative appearing is the order of the differential equation. Equations involving only one independent variable are ordinary differential equations, whereas those involv- ing more than one are partial differential equations. If the equation involves no products of the function with its derivatives or itself nor of derivatives with each other, then it is linear. Otherwise it is non-linear. A linear differential equation of order n has the form: d n–1 ᎏ x d ᎏ ᎏ 1 ᎏ d y y x n – d where P i (i = 0, 1. , n) F may be functions of x or constants, and P 0 ≠ 0. First order differential equations ᎏ n d ᎏ n P 0 y dx + P 1 + + P n–1 + P n y = F Form Type Method ᎏ y d ᎏ d = f ᎏ x y ᎏ ᎏ x y ᎏ homo- substitute u = dy ᎏ ) ᎏ g geneous ∫ (y ᎏ x d ᎏ y = f(x)g(y) separable d = ∫ f(x)dx + C note that roots of g(y) = 0 are also solutions ∂ ᎏ ∂ ∂ ᎏ ∂ g(x, y) = f andput ᎏ x ᎏ y ᎏ x d ᎏ y d + f(x, y) = 0 exact = g and solve these ᎏ x ∂ ᎏ g = ∂ equations for ␾ ␾ (x, y) = constant is the solution f ᎏ ∂ y and ᎏ ∂ 40 Aeronautical Engineer’s Data Book dy ᎏ dx ᎏ + f(x)y linear Multiply through by p(x) = exp(∫ x f(t)dt) = g(x) giving: p(x)y = ∫ x g(s)p(s)ds + C Second order (linear) equations These are of the form: d 2 y dy P 0 (x) ᎏᎏ + P 1 (x) ᎏᎏ + P 2 (x)y = F(x) dx 2 dx When P 0 , P 1 , P 2 are constants and f(x) = 0, the solution is found from the roots of the auxiliary equation: P 0 m 2 + P 1 m + P 2 = 0 There are three other cases: (i) Roots m = ␣ and ␤ are real and ␣ ≠ ␤ y(x) = Ae ␣ x + Be ␤ x (ii) Double roots: ␣ = ␤ ␣ x y(x) = (A + Bx)e (iii) Roots are complex: m = k ± il y(x) = (A cos lx + B sin lx)e kx 2.8.13 Laplace transforms If f(t) is defined for all t in 0 ≤ t < ∞, then L[f(t)] = F(s) =  ∞ e –st f(t)dt 0 is called the Laplace transform of f(t). The two functions of f(t), F(s) are known as a transform pair, and f(t) = L –1 [F(s)] is called the inverse transform of F(s). Function Transform f(t), g(t) F(s), G(s) c 1 f(t) + c 2 g(t) c 1 F(s) + c 2 G(s) [...]... 0.1 234 1 0.07806 0.01114 0.005611 0.07807 0.089212 All values at atmospheric pressure and 0°C 50 Aeronautical Engineer s Data Book 3. 3 Densities of liquids at 0°C See Table 3. 3 Table 3. 3 Densities of liquids at 0°C Liquid kg/m3 lb/ft3 Specific gravity Water Sea water Jet fuel JP 1 JP 3 JP 4 JP 5 Kerosine Alcohol Gasoline (petrol) Benzine Oil 1000 1025 800 775 785 817 820 801 720 899 890 62. 43 63. 99... 1026/kg mol 34 0.29 m/s (1116.44 ft/sec) Air pressure at sea level (p0) 760 mmHg = 1.0 132 5 ϫ 105 N/m2 = 2116.22 lb/ft2 Air temperature at sea level (T0) 15.0°C (59°F) Air density at sea level (␳0) 1.22492 kg/m3 (0.00 237 8 slug/ft3) Air dynamic viscosity at sea 1.4607 ϫ 10–5 m2/s level (µo) (1.57 23 ϫ 10–4 ft2/s) 3. 2 Weights of gases See Table 3. 2 Table 3. 2 Weights of gases Gas kg/m3 lb/ft3 Air Carbon... ASTM E380 and IEEE 268) 2 Taylor, B.N Guide for the use of the Inter­ national System of units (SI): 1995 NIST special publication No 8111 48 Aeronautical Engineer s Data Book 3 Federal Standard 37 6B: 19 93: Preferred Metric Units for general use by the Federal Government General Services Administra­ tion, Washington DC, 20406 Section 3 Symbols and notations 3. 1 Parameters and constants See Table 3. 1... ᎏ2ᎏ (B – A) Fundamental dimensions and units 43 Product formulae sin A sin B = ᎏ1ᎏ{cos(A – B) – cos(A + B)} 2 cos A cos B = ᎏ1ᎏ{cos(A – B) + cos(A + B)} 2 sin A cos B = ᎏ1ᎏ{sin(A – B) + sin(A + B)} 2 Powers of trigonometric functions sin2 A = ᎏ1ᎏ – ᎏ1ᎏ cos 2A 2 2 cos2 A = ᎏ1ᎏ + ᎏ1ᎏ cos 2A 2 2 3 sin3 A = ᎏ4ᎏsin A – ᎏ1ᎏ sin 3A 4 cos3 A = 3 cos A + ᎏ1ᎏ cos 3A 4 4 2.8.15 Co-ordinate geometry Straight-line... Table 3. 1 Important parameters and constants Planck’s constant (h) Universal gas constant (R) Stefan–Boltzmann constant (␴) Acceleration due to gravity (g) Absolute zero Volume of 1 kg mol of ideal gas at 1 atm, 0°C Avagadro’s number (N) Speed of sound at sea level (a0) 6.6260755 ϫ 10 34 J s 8 .31 4510 J/mol/K 5.67051 ϫ 10–8 W/m2 K4 9.80665 m/s2 (32 .17405 ft/s2) –2 73. 16°C (–459.688°F) 22.41 m3 6.0 23 ϫ... cosh at (πt)–1/2 2n tn–1/2 ᎏᎏ , 1 3 5 (2n –1)�π � n integer s+b ᎏ2 ᎏ (s + b) + a2 s–1/2 s–(n+1/2) exp(–a2/ t) 4 � ᎏᎏ (a > 0) e–a s � 2(πt3)1/2 2.8.14 Basic trigonometry Definitions (see Figure 2.9) y x sine: sin A = ᎏᎏ cosine: cos A = ᎏᎏ r r y x tangent: tan A = ᎏᎏ cotangent: cot A = ᎏᎏ x y r r secant: sec A = ᎏᎏ cosecant: cosec A = ᎏᎏ x y 42 Aeronautical Engineer s Data Book r y A x Fig 2.9 Basic trigonometry... Fig 2.11 Ellipse S P S(ae,0) x axis a Fig 2.12 Hyperbola D Directrix Directrix y axis a 46 Aeronautical Engineer s Data Book The parametric form of the equation is x = a sec␪, y = b tan␪ where ␪ s the eccenteric angle The equation of the tangent at (x1, y1) is xx1 yy1 ᎏᎏ – ᎏᎏ = 1 a2 b2 Sine Wave (see Figure 2. 13) y = a sin(bx + c) y = a cos(bx + c') = a sin(bx + c) (where c = c'+π/2) y = m sin bx +... y1) and Q(x2, y2) and is given by: PQ = ����)2 (x1 – x2)2 + (y1 – y2� The equation of the line joining two points (x1, y1) and (x2, y2) is given by: y–y x–x ᎏ 1 =ᎏ 1 ᎏ ᎏ y1 – y2 x1 – x2 44 Aeronautical Engineer s Data Book Circle General equation x2 – y2 + 2gx + 2fy + c = 0 The centre has co-ordinates (–g, –f) 2 The radius is r = ��–c g2 + f � The equation of the tangent at (x1, y1) to the circle is:... 44.9 56.12 55.56 1 1.025 0.8 0.775 0.785 0.817 0.82 0.801 0.72 0.899 0.89 3. 4 Notation: aerodynamics and fluid mechanics See Table 3. 4 Table 3. 4 Notation: aerodynamics and fluid mechanics The complexity of aeronautics means that symbols may have several meanings, depending on the context in which they are used a a' a0 a1 a2 a3 a∞ ah ay ac A A AF b b1 b2 Lift curve slope Acceleration or deceleration... (see Figure 2. 13) y = a sin(bx + c) y = a cos(bx + c') = a sin(bx + c) (where c = c'+π/2) y = m sin bx + n cos bx = a sin(bx + c) where a = ��2, c = tan–1 (n/m) m2 + n� y axis a x axis 0 c/b 2π/b Fig 2. 13 Sine wave Helix (see Figure 2.14) A helix is a curve generated by a point moving on a cylinder with the distance it transverses parallel to the axis of the cylinder being proportional to the angle of . 0°C. 50 Aeronautical Engineer s Data Book 3. 3 Densities of liquids at 0°C See Table 3. 3. Table 3. 3 Densities of liquids at 0°C Liquid kg/m 3 lb/ft 3 Specific gravity Water 1000 62. 43 1 Sea. a 13 b 21 b 22 a 21 a 22 a 23 b 31 b 32 a 11 b 11 +a 12 b 22 +a 13 b 31 a 11 b 12 +a 12 b 22 +a 13 b 32 =  a 21 b 11 +a 22 b 21 +a 23 b 31 a 21 b 12 +a 22 b 22 +a 23 b 32 .  36 Aeronautical Engineer s Data Book Square matrix This is a matrix having the same number of rows and columns. a 11 a 12 a 13 a 21 a 22 a 23 is a square matrix of order 3 ϫ a 31