AIAA Design Engineers Guide P1

- Differential equations Complex quantities Vectors Some standard series Matrix operations Determinants Curve fitting Small-term approximations... 4-2 AAA DESIGN ENGINEERS GUIDE of terms

Section 3 Conversion Factors

- Section 4 Structural Elements Section 5 Mechanical Design

Section 6 Electrical/Electronics Section 7 Aircraft Design ©

.” Section 8 Earth, Sea, and Solar System

Section 9.Materials and Specifications

eae

1.1 2.1

34

4.1 9.1 6.1 7.1 8.1 9.1

- Differential equations Complex quantities Vectors

Some standard series Matrix operations Determinants Curve fitting Small-term approximations

4-2 AAA DESIGN ENGINEERS GUIDE

of terms is inifinite

Logarithms

M

log,N

The quadratic equation

lí b2 - 4ac =0 { the roots are real and equal

1-4 — AlAA DESIGN ENGINEERS GUIDE

Algebra-cont

The cubic equations Any cubic equation y* + py? +qy+r=0

may be reduced to the form x* + ax+b=0 by substituting for y the

+ (a3/27)<0, these formulas givé the roots in impractical form for

numerical computation (In this case, a is negative) Compute the

where the upper or lower signs describe b as positive or negative

root of the equation then becomes—

|

where the upper or lower sign describes b as positive or negative

When (bÊ⁄4) + (a° /27) =0, the roots become—

ME PENT RNB AN

where the upper or lower signs describe b as positive or negative

The binomial equation

When x” =a, the rn roots of this equation become—

y Signs of functions

Functions of several angles

sin

cos

sin cos

sin nw = 2s¡in (n — 1)œcos œ — sin (n - 2)œ CoS nœ =2COS (n - †1)œcoS œ — COS (n - 2)œ

Trigonometry-cont

Power of functions

Sin“a:= ⁄4(Ccos4œ — 4cos2o + 3) _ cos“œ= 1⁄4(cos4a + 4coS2œ + 3)

Functions: sum or difference of two angles

Sin (œ ;+ 8) = sinzcos8 + cosœsin/8

tan

1 = tanatang

Sums,, differences, and products of two functions

COSq +: cos/ = 2cos1⁄2 (a + 8)cos1⁄ (œ — 8}

COSa —:coS8 = ~ 2sin1⁄ (œ+8)sin1⁄2 (œ— 8)

sin(z+8)

tanœ + tan8=———————

COSœcos2

sinÊœ sin28 = sin (œ+ 8)sin (œ— 8)

Right triangle solution

Given any two sides, or one side and any acute angle (a), find

the remaining parts:

P,(x,,¥1) is a known point on line J Axa - By; + C

Conic where e = 1; lastus rectum, a

Parabola A: (y- k)? = a(x-h) Wertex (h,k), axis | OX

x?=ay Vertex (0,0), axis along OY

Distance from vertex to focus: “a

y = acos(bx + C’) = asin(bx +c), where c=c’ + >

y = msinbx + ncosbx = asin(bx + Cc),

y=— (et 2+e-* 4)

2 Helix

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 rotation about the axis

X= a cos@

y=a sing z=ké

Direction cosine of a line (cosines of the angles a,3.4 which the line

or any parallel line makes with the coordinate axes! are related by

cos*a + cos28 + cos2+ = 1

If COSo:cos8:cos+ = a:b:c, then

Vette te

Direction cosines of the line joining P,(x,, ¥1, Z,) and Po(Xo, Yo, 29):

cos a:cos :COS %y = X¿ — X1:Va — Y+:Z¿ — Z\

Angle (8) between two lines, whose direction angles are a,,8,,7;

and Qo, Bo, Yo:

cos 8=Ccos a, COS a2 + Cos 6, cos By +C0s +; COS 2 Equation of a plane is of the first degree in x, y, and z—

Equations of a straight line through the point P,(x,, y;, Z,) with

direction cosines proportional to a, b, and c:

(wnere esc~ 'u lies between ~ > and +=)

nth derivative of cerfain functions

|afojdx = a|fs)dx a any constant

ư? +1 n+1

=Inu+C: u any function of x

= \f,(u)du + \fo(uydu + \f,(uidu yrs)

‹

` _—~

U

Equations of first order and first degree: Mdx + Nơy = 0

Variables separable: X, Y,dx + X; Y›dy = Ô:

M+ N can be written so that x and y occur only as y+x: if every term

in M and N have the same degree in x and y

Make equation exact by multiplying by an integrating factor

u(x y)—a form readily recognized in many cases

Properties of Complex Quantities

If z,Z, 2, represent complex quantities, then Sum or difference: Z, +Z5 =(X; =X2)+ MV; +Y2) Product: 2, -Z> = yfp[Ccos(@, +6) -ssin(4, +6)}

where k takes in succession the values 0,1.2,3, 9-1

If z,=Z>, then x, =X» andy, =Yyo

Periodicity: 2 = r(cos + /sin8) = rï cos (6 + 2kz) + /sin ( ~ 2kz ) ]

sV = (saji - (Sb)J ~ (s€)k

where sV has the same direction as V and its magnitude is s times the magnitude of V

Scalar product of two vectors V, Vo:

V,-V2 = IV, 1 1V>1 coso where o is the angle between V, anc V>

V,.V;=V¿.V,; V,.V,=!V,/°: (Vy-V2)V2=V Vị—-V¿-Vạ (V, = V5): (V3 - Va) = Vy -V3-V,- Va - Vo V3- VeVi

In plane V,-V>=@,4)-5,02: in space V,-V2=4,.4,

+b,bo-C,Co

The scalar product of two vectors V,-V2 a scalar quantity may be

represented by the work done by a constant force of magnitude

IV, ! on a unit particle moving through a distance !V2! where o is the angle between the line of force and the direction of motion

1-16 AIAA DESIGN ENGINEERS GUIDE

Some Standard Series

Two matrices A and B can be added if the number of rows in A

A=B=C

/=1.2, n Multiplying a matrix or vector by a scalar implies multiplication of each element by the scalar If

B=yA, then 6, =a, for all elements

Two matrices, A and B, can be multiplied if the number of columns

in A equals rows in B For A of order mx n (m rows and n columns) and B of order nxp, the product of two matrices C = AB will be a matrix of order mxp elements—

lái

ket Thus c, is the scalar product of the /’th row vector of Á and the j th column vector of B

Matrix multiplication is associative: A(BC) = (AB)C

The distributive law for multiplication and addition holds as in the

case of scalars:

(A+B)C=AC+BC C(A+B)=CA+CB

For some applications, the term-by-term product of two matrices A

and B of identical order is defined as C= A-B where c, = a,b;

For a polynomial function fit by the method of least squares, obtain

For a straight-line fit by the method of least squares, obtain the

values by and b, by solving thé normal equations:

nbạ + by ox, =Ly,

2_

bạ>x, + b x2 = Yx,y,

Solutions for these normai equations:

For an exponential curve fit by the method of least squares, obtain

the vaiues loga and logb by fitting a straight line to the set of ordered

Pairs {x,.logy,)}

For a power-function fit by the method of least squares, obtain the

values loga and 5 by fitting a straight line to the set or ordered pairs

{ (logx; logy,)}

Small-Term Approximations

This section lists some first approximations derived by neglecting all

powers but the first of the small positive or negative quantity, x =s

The expression in brackets gives the next term beyond that used

and by means of it the accuracy of the approximation can be

The following expressions may be approximated by 1 - s (s a small

positive or negative quantity and n any number):

Plane areas Solids Shells

Section 2 from Weight Engineers Handbook, Revised 1976 © Copyright

1976 by the Society of Allied Weight Engineers Used with permission

HTM HTM TAM

THM

eee eee

mm

mm

mm mm7mm

uw www

2-

SECTION PROPERTIES Plane Areas—cont

PERS RACE te

2-11 SECTION PROPERTIES

AIAA DESIGN ENGINEERS GUIDE

2-12

Plane Areas—cont Plane Areas—cont

Xin

OE

he heey EE is

2-23 SECTION PROPERTIES

2-2/ SECTION PROPERTIES

Length Area Volume Plane angle Solid angle Mass Density Time Speed Force Pressure Energy, work, heat Power

Thermal conductivity Absolute or dynamic viscosity Inductance

Kinematic viscosity Capacitance Electric resistance Electric resistivity, reciprocal conductivity

Magnetic vector Magnetomotive force Magnetic flux

Magnetic field strength Machine Screws-tap drill and clearance drill sizes

Decimal equivalents of drill size Standard gauges

nt =

rl Cua

3-9 CONVERSION FACTORS

ani6 umoys

$azis

y9/1-I 91/6L

¿9G' 9u/6

rm mmm

wa

a aaa

=

=

= t-

Độ Me

a

Section 4

STRUCTURAL ELEMENTS

Bending moment vertical shear and deflection of beams of

uniform cross section under various conditions of loading

Nomenclature

= modulus of elasticity Ib/in.é

= moment of inertia in

= length of beam in

= maximum bending moment |b-in

= bending moment at any section {b-tn

concentrated loads Ib

reactions Ib

= maximum vertical shear !b

= vertical shear at any section Ib

= uniform load per unit of length Ib/in

= total uniformoad on beam !b

= distance from support to any Section in

= Maximum deflectign in

From Handbook of Engineering Fundamentals, 3rd Edition by O.W

Esbach and M Souders Copyright 1976, John Wiley and Sons, New

York, NY Used with permission

Simple beam—two equal concentrated loads at equal

distances from supports

(at center of span)

Simple beam—load increasing uniformly from supports to

center of span

W

R.=R;=—

2 v.=w( > - 92) (wnenx< =)

Y= 39261

From Handbook of Engineering Fundamentals 3rd Edition by O.W

York, NY Used with permission

T = ior

York NY Useg with permission

From Handbook of Engineering Fundamentals, 3rd Edition by O.W

Esbach and M Souøers Copyright 1976, John Wiley and Sons, New

York, NY Used with permission

iio ce

oom mmm

York NY Used with permission

Torsion Formulas—Solid and Tubular Sections

Nomenclature

F., =ultimate shear strength, kips/in.?

a