Aeronautical Engineer’s Data Book - part 2 potx

18 Table 2.9 Area (A) Unit sq.in sq.ft sq.yd sq.mile cm 2 dm 2 m 2 a ha km 2 1 square inch 1 - – – 6.452 0.06452 – - – - 1 square foot 144 1 0.1111 - 929 9.29 0.0929 – - – 1 square yard 1296 9 1 – 8361 83.61 0.8361 – – – 1 square mile – – – 1 – – – – 259 2.59 1 cm 2 0.155 – – – 1 0.01 – – – – 1 dm 2 15.5 0.1076 0.01196 – 100 1 0.01 – – – 1 m 2 1550 10.76 1.196 – 10 000 100 1 0.01 – – 1 are (a) – 1076 119.6 – – 10 000 100 1 0.01 – 1 hectare (ha) – – – – – – 10 000 100 1 0.01 1 km 2 – – – 0.3861 – – – 10 000 100 1 19 Fundamental dimensions and units 2.3.14 Length and area Comparative lengths in USCS and SI units are: 1 ft = 0.3048 m 1 in = 25.4 mm 1 statute mile = 1609.3 m 1 nautical mile = 1853.2 m The basic unit of area is square feet (ft 2 ) or square inches (in 2 or sq.in). In SI it is m 2 . See Table 2.9. Small dimensions are measured in ‘micro- measurements’ (see Figure 2.8). The microinch (µin) is the commonly used unit for small measures of distance: 1 microinch = 10 –6 inches = 25.4 micrometers (micron ) Oil filter mesh 450µin Diameter of a hair: 2000µin Smoke particle 120µin A smooth-machined ‘mating’ –32µin 1 micron (µm) = 39.37µin A fine ‘lapped’ with peaks within 1µin surface with peaks 16 surface Fig. 2.8 Micromeasurements 2.3.15 Viscosity Dynamic viscosity (µ) is measured in lbf.s/ft 2 or, in the SI system, in N s/m 2 or pascal seconds (Pa s). 1 lbf.s/ft 2 = 4.882 kgf.s/m 2 = 4.882 Pa s 1Pas = 1Ns/m 2 = 1 kg/m s A common unit of viscosity is the centipoise (cP). See Table 2.10. 20 Aeronautical Engineer’s Data Book Table 2.10 Dynamic viscosity (␮) Unit lbf-s/ft 2 Centipoise Poise kgf/m s 1 lb (force)-s 1 4.788 4.788 4.882 per ft 2 ϫ 10 4 ϫ 10 2 1 centipoise 2.089 1 10 –2 1.020 ϫ 10 –5 ϫ 10 –4 1 poise 2.089 100 1 1.020 ϫ 10 –3 ϫ 10 –2 1 N-s per m 2 0.2048 9.807 98.07 1 ϫ 10 3 Kinematic viscosity (␯) is a function of dynamic viscosity. Kinematic viscosity = dynamic viscosity/ density, i.e. ␯ = µ/␳ The basic unit is ft 2 /s. Other units such as Saybolt Seconds Universal (SSU) are also used. 1 m 2 /s = 10.7639 ft 2 /s = 5.58001 ϫ 10 6 in 2 /h 1 stoke (St) = 100 centistokes (cSt) = 10 –4 m 2 /s 1 St > 0.00226 (SSU) – 1.95/(SSU) for 32 < SSU < 100 seconds 1 St  0.00220 (SSU) – 1.35/(SSU) for SSU > 100 seconds 2.4 Consistency of units Within any system of units, the consistency of units forms a ‘quick check’ of the validity of equations. The units must match on both sides. Example: To check kinematic viscosity (␯) = dynamic viscosity (µ) ᎏᎏᎏ = µ ϫ 1/␳ density (␳) ft 2 lbf.s ft 4 ᎏ = ᎏ ϫ ᎏ s ft 2 lbf.s 2 ft 2 s.ft 4 ft 2 Cancelling gives ᎏ = ᎏ = ᎏ s s 2 .ft 2 s OK, units match.             ᎏᎏ 21 Fundamental dimensions and units 2.5 Foolproof conversions: using unity brackets When converting between units it is easy to make mistakes by dividing by a conversion factor instead of multiplying, or vice versa. The best way to avoid this is by using the technique of unity brackets. A unity bracket is a term, consisting of a numerator and denominator in different units, which has a value of unity. 2.205 lb kg e.g.  ᎏ  or  ᎏ  are unity kg 2.205 lb brackets as are 25.4 mm in atmosphere ᎏᎏ or ᎏᎏ or ᎏᎏ in 25.4 mm 101 325 Pa Remember that, as the value of the term inside the bracket is unity, it has no effect on any term that it multiplies. Example: Convert the density of titanium 6 Al 4 V; ␳ = 0.16 lb/in 3 to kg/m 3 0.16 lb Step 1: State the initial value: ␳ = ᎏ in 3 Step 2: Apply the ‘weight’ unity bracket: 0.16 lb kg ␳ = ᎏ ᎏ in 3 2.205 lb Step 3: Then apply the ‘dimension’ unity brackets (cubed): 3 0.16 lb kg 3 in ␳ = ᎏ ᎏ ᎏᎏ in 3 2.205 lb 25.4 mm   3 1000 mm m ᎏᎏ  22 Aeronautical Engineer’s Data Book Step 4: Expand and cancel*: 0.16 lb kg in 3 ␳ = ᎏ  ᎏ  ᎏᎏ 3  in 3 2.205 lb (25.4) 3 mm  3 (1000) 3 mm 3 m 0.16 kg (1000) 3 ␳ = ᎏᎏ 3 2.205 (25.4) 3 m ␳ = 4428.02 kg/m 3 Answer *Take care to use the correct algebraic rules for the expansion, e.g. (a.b) N = a N .b N not a.b N 1000 mm 3 (1000) 3 (mm) 3 e.g.  ᎏᎏ  expands to ᎏᎏ m (m) 3 Unity brackets can be used for all unit conver- sions provided you follow the rules for algebra correctly. 2.6 Imperial–metric conversions See Table 2.11. 2.7 Dimensional analysis 2.7.1 Dimensional analysis (DA) – what is it? DA is a technique based on the idea that one physical quantity is related to others in a precise mathematical way. It is used in aeronautics for: • Checking the validity of equations. • Finding the arrangement of variables in a formula. • Helping to tackle problems that do not possess a compete theoretical solution – particularly those involving fluid mechanics. 2.7.2 Primary and secondary quantities Primary quantities are quantities which are absolutely independent of each other. They are: 23 Fundamental dimensions and units Table 2.11 Imperial-metric conversions Fraction Decimal Millimetre Fraction Decimal Millimetre (in) (in) (mm) (in) (in) (mm) 1/64 0.01562 0.39687 33/64 0.51562 13.09687 1/32 0.03125 0.79375 17/32 0.53125 13.49375 3/64 0.04687 1.19062 35/64 0.54687 13.89062 1/16 0.06250 1.58750 9/16 0.56250 14.28750 5/64 0.07812 1.98437 37/64 0.57812 14.68437 3/32 0.09375 2.38125 19/32 0.59375 15.08125 7/64 0.10937 2.77812 39/64 0.60937 15.47812 1/8 0.12500 3.17500 5/8 0.62500 15.87500 9/64 0.14062 3.57187 41/64 0.64062 16.27187 5/32 0.15625 3.96875 21/32 0.65625 16.66875 11/64 0.17187 4.36562 43/64 0.67187 17.06562 3/16 0.18750 4.76250 11/16 0.68750 17.46250 13/64 0.20312 5.15937 45/64 0.70312 17.85937 7/32 0.21875 5.55625 23/32 0.71875 18.25625 15/64 0.23437 5.95312 47/64 0.73437 18.65312 1/4 0.25000 6.35000 3/4 0.75000 19.05000 17/64 0.26562 6.74687 49/64 0.76562 19.44687 9/32 0.28125 7.14375 25/32 0.78125 19.84375 19/64 0.29687 5.54062 51/64 0.79687 20.24062 15/16 0.31250 7.93750 13/16 0.81250 20.63750 21/64 0.32812 8.33437 53/64 0.82812 21.03437 11/32 0.34375 8.73125 27/32 0.84375 21.43125 23/64 0.35937 9.12812 55/64 0.85937 21.82812 3/8 0.37500 9.52500 7/8 0.87500 22.22500 25/64 0.39062 9.92187 57/64 0.89062 22.62187 13/32 0.40625 10.31875 29/32 0.90625 23.01875 27/64 0.42187 10.71562 59/64 0.92187 23.41562 7/16 0.43750 11.11250 15/16 0.93750 23.81250 29/64 0.45312 11.50937 61/64 0.95312 24.20937 15/32 0.46875 11.90625 31/12 0.96875 24.60625 31/64 0.48437 12.30312 63/64 0.98437 25.00312 1/2 0.50000 12.70000 1 1.00000 25.40000 M Mass L Length T Time For example, velocity (v) is represented by length divided by time, and this is shown by: [v] = L ᎏ T : note the square brackets denoting ‘the dimension of’. Table 2.12 shows the most commonly used quantities. 24 Aeronautical Engineer’s Data Book Table 2.12 Dimensional analysis quantities Quantity Dimensions Mass (m) Length (l) Time (t) Area (a) Volume (V) First moment of area Second moment of area Velocity (v) Acceleration (a) Angular velocity ( ␻ ) Angular acceleration ( ␣ ) Frequency (f) Force (F) Stress {pressure}, (S{P}) Torque (T) Modulus of elasticity (E) Work (W) Power (P) Density ( ␳ ) Dynamic viscosity (µ) Kinematic viscosity ( ␰ ) M L T L L L L 2 3 3 4 LT –1 T T T LT –2 –1 –2 –1 ML MLT –2 ML –1 T –2 ML 2 T –2 ML –1 T –2 ML 2 T –2 ML 2 T –3 –3 ML –1 T –1 L 2 T –1 Hence velocity is called a secondary quantity because it can be expressed in terms of primary quantities. 2.7.3 An example of deriving formulae using DA To find the frequencies (n) of eddies behind a cylinder situated in a free stream of fluid, we can assume that n is related in some way to the diameter (d) of the cylinder, the speed (V) of the fluid stream, the fluid density ( ␳ ) and the kinematic viscosity ( ␯ ) of the fluid. i.e. n = ␾ {d,V, ␳ , ␯ } Introducing a numerical constant Y and some possible exponentials gives: c n = Y{d a ,V b , ␳ ,␯ d } Y is a dimensionless constant so, in dimensional analysis terms, this equation becomes, after substituting primary dimensions:     ᎏ  ᎏ 25 Fundamental dimensions and units T –1 = L a (LT –1 ) b (ML –3 ) c (L 2 T –1 ) d = L a L b T –b M c L –3c L 2d T –d In order for the equation to balance: For M, c must = 0 For L, a + b –3c + 2d = 0 For T, –b –d = –1 Solving for a, b, c in terms of d gives: a = –1 –d b = 1 –d Giving n = d (–1 –d) V (1 –d) ␳ 0 ␯ d Rearranging gives: nd/V = (Vd/ ␯ )X Note how dimensional analysis can give the ‘form’ of the formula but not the numerical value of the undetermined constant X which, in this case, is a compound constant containing the original constant Y and the unknown index d. 2.8 Essential mathematics 2.8.1 Basic algebra m+n a a m ϫ a n = a m Ϭ a n = a m–n (a m ) n = a mn n  a m = a m/n 1 a = a –n n a o = 1 (a n b m ) p = a np b mp a n a n ᎏ = ᎏ b n b n  ab = n  a    ϫ n  b n  a 3 n  a\b = n  b   ᎏ 26 Aeronautical Engineer’s Data Book 2.8.2 Logarithms N If N = a x then log a N = x and N = a log a log b N log a N = ᎏ log b a log(ab) = log a + log b a log  = log a – log b b log a n = n log a  1 log n  a = ᎏ log a n log a 1 = 0 log e N = 2.3026 log 10 N 2.8.3 Quadratic equations If ax 2 + bx + c = 0 –b ±  b 2 – 4ac x = ᎏᎏ 2a If b 2 –4ac > 0 the equation ax 2 + bx + c = 0 yields two real and different roots. If b 2 –4ac = 0 the equation ax 2 + bx + c = 0 yields coincident roots. If b 2 –4ac < 0 the equation ax 2 + bx + c = 0 has complex roots. If ␣ and ␤ are the roots of the equation ax 2 + bx + c = 0 then b sum of the roots = ␣ + ␤ = – ᎏ a c product of the roots = ␣␤ = ᎏ d The equation whose roots are ␣ and ␤ is x 2 – ( ␣ + ␤ )x + ␣␤ = 0. Any quadratic function ax 2 + bx + c can be expressed in the form p (x + q) 2 + r or r – p (x + q) 2 , where r, p and q are all constants. The function ax 2 + bx + c will have a maximum value if a is negative and a minimum value if a is positive.  27 Fundamental dimensions and units If ax 2 + bx + c = p(x + q) 2 + r = 0 the minimum value of the function occurs when (x + q) = 0 and its value is r. If ax 2 + bx + c = r – p(x + q) 2 the maximum value of the function occurs when (x + q) = 0 and its value is r. 2.8.4 Cubic equations x 3 + px 2 + qx + r = 0 gives y 3 + 3ay + 2b = 0 ᎏ 3 1 ᎏ x = y – p where 2 ,2b = 3 – ᎏ 3 1 ᎏ pq + r ᎏ 3 1 ᎏ 2 3a = –q – p p ᎏ 7 ᎏ 2 On setting 3 ) 1/2 ] 1/3 S = [–b + (b 2 + a and 3 ) 1/2 ] 1/3 T = [–b – (b 2 + a the three roots are x 1 = S + T – ᎏ 3 1 ᎏ p (S + T) +  3  \2 i(S – T) – ᎏ 2 1 ᎏ ᎏ 3 1 ᎏ x 2 = – p (S + T) –  3  \2 i(S – T) – ᎏ 2 1 ᎏ ᎏ 3 1 ᎏ x 3 = – p. For real coefficients all roots are real if b 2 + a 3 ≤ 0, one root is real if b 2 + a 3 > 0. At least two roots are equal if b 2 + a 3 = 0. Three roots are equal if a = 0 and b = 0. For b 2 + a 3 < 0 there are alternative expressions: ᎏ 3 1 ᎏ x x 1 = 2c cos ␪ – 3 = 2c cos ᎏ 3 1 ᎏ ᎏ 3 1 ᎏ px 2 = 2c cos ( ␪ + 2π) – ᎏ 3 1 ᎏ p ᎏ 3 1 ᎏ ᎏ 3 1 ᎏ ( ␪ + 4π) – p b where c 2 = –a and cos ␪ = – 2.8.5 Complex numbers ᎏ 3 ᎏ c If x and y are real numbers and i =  –1  then the complex number z = x + iy consists of the real part x and the imaginary part iy. z = x – iy is the conjugate of the complex number z = x + iy. [...]... a � = �ba a a ba 11 12 11 21 b 22 21 ba 12 ba 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 � a11 a 12 a13 a21 a22a23 = �� b11 b 12 b21 b 22 b31 b 32 +a �aa bb +a bb 11 11 12 22 21 11 22 21 � � +a13b31 a11b 12 +a12b 22 +a13b 32 +a23b31 a21b 12 +a22b 22 +a23b 32 If matrix A is of order... b � a a a b b b a +b a +b a +b =� a + b a + b a + b � 11 12 13 11 12 13 21 22 23 21 22 23 11 11 12 12 13 13 21 21 22 22 23 23 Subtraction of matrices Subtraction is done in a similar way to addition except that the corresponding elements are subtracted b �a a � – �b b � = �aa ––b a a b b 11 12 11 12 21 22 21 22 11 21 11 21 � a 12 –b 12 a 22 –b 22 Fundamental dimensions and units 37 Scalar multiplication... A' or AT If A = �aa aa a � then A a 11 21 12 13 22 23 T � � a11 a21 = a 12 a 22 a13 a23 Adjoint of a matrix If A =[aij] is any matrix and Aij is the cofactor of aij the matrix [Aij]T is called the adjoint of A Thus: � � � � a11 a 12 a1n a21 a 22 a2n A= an1 an2 amn A11 A21 An1 A 12 A 22 An2 adj A = A1n A2n Ann 38 Aeronautical Engineer’s Data Book Singular matrix A square matrix is... 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 Relations between trigonometric functions sin2 A + cos2 A = 1 sec2 A = 1 + tan2 A cosec2 A = 1 + cot2 A sin A = s cos A = c tan A = t sin A s (1 – c2)1 /2 t(1 + t2)–1 /2 cos A (1 – s2)1 /2 c (1 + t2)–1 /2 tan A s(1 – s2)1 /2 (1 – c2)1 /2/ c t A is assumed to be in the first quadrant; signs... r = |z| x2 + y� r = � 2 ␪ is called the argument and this may be written ␪ = arg z y tan ␪ = ᎏᎏ x If z1 = r (cos␪1 + i sin ␪1) and z2 = r2 (cos 2 + i sin 2) z1z2 = r1r2 [cos(␪1 + 2) + i sin(␪1 + 2) ] = r1r2Є(␪1 + 2) r1[cos(␪1 – 2) + i sin(␪1 + 2) ] z1\z2 = ᎏᎏᎏᎏ r2 r1 = ᎏᎏ Є(␪1 – 2) r2 2. 8.6 Standard series Binomial series n(n – 1) (a + x)n = an + nan–1 x + ᎏᎏ an 2 x2 2! n(n – 1)(n – 2) + ᎏᎏ an–3... = ln | tan 2 (x + 2 π) | xa x–1 ekx ax Fundamental dimensions and units cosec x ln | tan ᎏ1ᎏ x | 2 sin2 x 1 ᎏᎏ 2 cos2 x 1 ᎏᎏ 2 sec2 x tan x sinh x cosh x cosh x sinh x tanh x ln cosh x sech x 2 arctan ex cosech x ln | tanh ᎏ1ᎏ x | 2 sech2 x tanh x 1 x ᎏᎏ arctan ᎏᎏ, a a 35 1 ᎏᎏ a2 + x2 1 ᎏᎏ a2 – x2 1 ᎏᎏ (a2 – x2)1 /2 (x – ᎏ1ᎏ sin 2x) 2 (x + ᎏ1ᎏ sin 2x) 2 a≠0 � 1 a–x – ᎏᎏ ln ᎏ , a ≠ a ᎏ 2a a+x 1 x–a... with the x-axis The distance between two points P(x1, 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... 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) ∆ Aij = cofactor of aij � � � � a11 a 12 a21 a 22 If A = an1 an2 a1n A11 A21 An1 a2n A 12 A 22 An2 1 –1 A = ᎏᎏ ∆ ann A1n A2n Ann 2. 8.11 Solutions of simultaneous linear equations The set of linear equations a11x1 + a12x2 + + a1nxn = b1 a21x1 + a22x2... F(s) – � sn–r f(r–1) (0+) 0 (–t)n f(t) n r=1 a>0 s+b 2 ᎏ (s + b + a2 ) e–bt cos at 1 ᎏᎏ e–bt sinh at, a 1 2 ᎏ (s = b) + a2 1 a > 0 2 ᎏ (s + b) + a2 e–bt cosh at (πt)–1 /2 2n tn–1 /2 ᎏᎏ , 1·3·5 (2n –1)�π � n integer s+b 2 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... ln ᎏ , a ≠ a ᎏ 2a a+x 1 x–a ᎏᎏ ln ᎏᎏ, a ≠ 0 2a x+a x arcsin ᎏᎏ, a ≠ 0 |a| � ln [x + (x2 – a2)1 /2] 1 ᎏᎏ (x2 – a2)1 /2 x arccosh ᎏᎏ, a 2. 8.10 Matrices a≠0 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: � � a11 a 12 a1n a21 a 22 a2n am1 am2 amn 36 Aeronautical Engineer’s Data Book Square matrix This is a matrix having . 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 . 0.87500 22 .22 500 25 /64 0.390 62 9. 921 87 57/64 0.890 62 22. 621 87 13/ 32 0.40 625 10.31875 29 / 32 0.90 625 23 .01875 27 /64 0. 421 87 10.715 62 59/64 0. 921 87 23 .415 62 7/16 0.43750 11.1 125 0 15/16 0.93750 23 .8 125 0. 0.3 125 0 7.93750 13/16 0.8 125 0 20 .63750 21 /64 0. 328 12 8.33437 53/64 0. 828 12 21.03437 11/ 32 0.34375 8.73 125 27 / 32 0.84375 21 .43 125 23 /64 0.35937 9. 128 12 55/64 0.85937 21 . 828 12 3/8 0.37500 9. 525 00