Aeronautical Engineer’s Data Book - part 2 potx

19 Fundamental dimensions and units 2.3.14 Length and area Comparative lengths in USCS and SI units are: Small dimensions are measured in ‘micromeasurements’ see Figure 2.8.. 2.7.2 Prim

Trang 1

18

Table 2.9 Area (A)

Trang 2

19 Fundamental dimensions and units

2.3.14 Length and area

Comparative lengths in USCS and SI units are:

Small dimensions are measured in ‘micromeasurements’ (see Figure 2.8)

The microinch (

Oil filter mesh

450 µin

Diameter of a hair: 2000 µin Smoke

with peaks within 1 µin

surface with peaks 16

surface

Fig 2.8 Micromeasurements

2.3.15 Viscosity

Dynamic viscosity (µ) is measured in lbf.s/ft2 or,

in the SI system, in N s/m2 or pascal seconds (Pa s)

1 lbf.s/ft2 = 4.882 kgf.s/m2 = 4.882 Pa s

1 Pa s = 1 N s/m2 = 1 kg/m s

A common unit of viscosity is the centipoise (cP) See Table 2.10

Trang 3

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 2 10 4 2 10 2

Trang 4

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

Example:

Convert the density of titanium 6 Al 4 V; # = 0.16 lb/in3 to kg/m3

0.16 lbStep 1: State the initial value: # = 3

Trang 5

33 �

Step 4: Expand and cancel*:

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:

Trang 6

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] = 3L : note the square brackets denoting

‘the dimension of’

Table 2.12 shows the most commonly used quantities

Trang 7

Table 2.12 Dimensional analysis quantities

LT–2 –1 –2 –1

ML–1T–1

L2T–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

Y is a dimensionless constant so, in dimensional

analysis terms, this equation becomes, after substituting primary dimensions:

Trang 8

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

Trang 9

If b2 –4ac > 0 the equation ax2

If b2 –4ac = 0 the equation ax2

If b2 –4ac < 0 the equation ax2

If and are the roots of the equation ax2

b

sum of the roots = + = – 3

a c

product of the roots = = 3

d

The equation whose roots are and is x2– ( +

Any quadratic function ax2

expressed in the form p (x + q)2

The function ax2

Trang 10

�

If ax2+ 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 ax2+ bx + c = r – p(x + q)2 the maximum value

of the function occurs when (x + q) = 0 and its value is r

For real coefficients

all roots are real if b2+ a3 ≤ 0,

one root is real if b2+ a3 > 0

At least two roots are equal if b2+ a3 = 0

Three roots are equal if a = 0 and b = 0 For b2

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

Trang 11

If x + iy = a + ib then x = a and y = b

(a + ib) + (c + id) = (a + c) = i(b + d) (a + ib) – (c + id) = (a – c) = i(b + d) (a + ib)(c + id) = (ac – bd) + i(ad + bc)

Trang 12

3 � � 3 � �

Trang 13

2.8.7 Vector algebra

Vectors have direction and magnitude and

satisfy the triangle rule for addition Quantities

such as velocity, force, and straight-line displacements may be represented by vectors Three-dimensional vectors are used to repre

sent physical quantities in space, e.g A x , A y , A z

or A x i + A y j + A zk

Vector Addition

The vector sum V of any number of vectors V1,

V2, V3 where = V1 a1i + b1 j + c 1 k, etc., is given

that is e · 33 = 0

dt

Trang 14

The gradient (grad) of a scalar field +(x, y, z) is

3 3

∂ +

Rules for differentiation: y, u and v are

functions of x; a, b, c and n are constants

Trang 15

3 3

33

Trang 16

3

x

3

1 3

= 33 for angles in the(1 – x2)1/2 first quadrant

Derivatives of hyperbolic functions

Trang 17

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:

v

∂

3

∂y k→0

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)

x

∂ 3

y

∂ 3

tan 3 (x + 3 π) |

Trang 18

33

2

3 2 1

Trang 19

� �a21 a22 – � �b21 b22 = �a21 –b21 a22 –b22�

Trang 20

� � � �

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

If matrix A is of order (p 2 q) and matrix B is

of order (q 2 r) then if C = AB, the order of C

is (p 2 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

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

Trang 21

�

Singular matrix

A square matrix is singular if the determinant

of its coefficients is zero

If A is a non-singular matrix of order (n then its inverse is denoted by A–1 such that AA

2.8.11 Solutions of simultaneous linear equations

The set of linear equations

ij , and x and b are (n 2 1) column vectors

The solution to this matrix equation, if A is non-singular, may be written as x = A–1b

which leads to a solution given by Cramer’s

rule:

x i = det D i /det A i = 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

Trang 22

+ +

x

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

dy

3)

note that roots of

g(y) = 0 are also

Trang 23

Second order (linear) equations

These are of the form:

P0(x) 33 + P1(x) 33 + P2(x)y = F(x)

When P0, P1, P2 are constants and f(x) = 0, the

solution is found from the roots of the auxiliary equation:

There are three other cases:

(i) Roots m = and are real and ≠

(ii) Double roots: = 

x

y(x) = (A + Bx)e

(iii) Roots are complex: m = k ± il

Trang 25

Fig 2.9 Basic trigonometry

Relations between trigonometric functions

A is assumed to be in the first quadrant; signs

of square roots must be chosen appropriately in other quadrants

Addition formulae

sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B 7 sin A sin B

tan A ± tan B tan(A ± B) = 3

tan A tan B

Sum and difference formulae

3 2 1

3 3213

sin A + sin B = 2 sin (A + B) cos (A – B)

3 2 1

3 3213

sin A – sin B = 2 cos (A + B) sin (A – B)

3 2 1

3 3213

cos A + cos B = 2 cos (A + B) cos (A – B)

3 1

3 313

cos A – cos B = 2 sin (A + B) sin (B – A)

Trang 26

Product formulae

3 2 3

sin A sin B = {cos(A – B) – cos(A + B)}

3 2 1 3

cos A cos B = {cos(A – B) + cos(A + B)} sin A cos B =323{sin(A – B) + sin(A + B)}

Powers of trigonometric functions

= angle that the perpendicular makes

with the x-axis

The distance between two points P(x1, y1) and

Q(x2, y2) and is given by:

�2)2 + (�)2

The equation of the line joining two points (x1,

y1) and (x2, y2) is given by:

Trang 27

The equation of the tangent at (x1, y1)

to the circle is:

The parametric form of the equation of an

ellipse is x = a cos , y = b sin, where is

the eccentric angle

Hyperbola (see Figure 2.12)

Trang 28

Định dạng
Số trang	28
Dung lượng	170,45 KB