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Jan 23, 2016
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only one joint between 2 bodies.
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REVIEW OF COMPLEX NUMBERS
1) z = r ei
exponential form
where, i = 1
Euler’s identity : e i
= cos i sin (EU)
z = r ( cos + i sin ) = r cos + i r sin
z = x + i y orthogonal form
2)
x y
x
y
Real
Imaginary
r
z
Complex numbers
may be used to
express 2D vectors.
’
A
B
r
Real axis
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AB = r ei
BA = - AB = - r ei
or,
BA = r ei ’
= r ei ( + )
Let
z1 = r1 1ie
z2 = r2 2ie
3)
z1 z2 = ( r1 cos1 + i r1 sin1 ) ( r2 cos2 + i r2 sin2 )
= ( r1 cos1 r2 cos2 ) + i (r1 sin1 r2 sin2 )
4) z1 z2 = r1 r2 ) θθ i( 21 e
5) 2
1
z
z =
2
1
r
r ) θθ i( 21 e
6) i ei
= ei ( / 2 )
ei
= ei( + / 2 )
form to be used in writing
the loop closure equations
eiθ i e
iθ
1 Re
eiθ Unit vector
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z Complex conjugate of z .
7) 1z = r1 1ie
= r1 cos1 – i r1 sin1
8) z1 z2 = 1z 2z
9) d
d e
i = i e
i [
dt
d e
i = ( i e
i ) (
dt
d ) ]
10) dt
dz1 = dt
d( r1 1i
e
)
= ( dt
dr1 ) ( 1ie
) + ( r1 ) ( i 1ie
dt
d 1 )
11) ei
+ e-i
= 2cos
12) ei
- e-i
= 2 i sin
13) ei0
= 1
ei ( /2)
= i
ei
= -1
ei (3 /2)
= -i
Proof by
Euler’s theorem
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CHAPTER 5
FORCE ANALYSIS IN MACHINERY
BASIC CONCEPTS :
Force : Action of one body on another. Force is a vector
quantity which has :
- Magnitude
- Direction
- Line of Action (LA)
- Point of application.
Mechanics of deformable bodies Point of application is
important, since we are interested in deformations and
stresses.
Rigid (i.e., undeformable) body mechanics Point of
application is not important. one can “ slide ” a force
along its LA.
LA
F
( For rigid body mechanics ,
e.g., force analysis )
ijF
Force applied by body i on body j.
jiF
Force applied by body j on body i.
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Newton’s third law ijF
= - jiF
and LA’s are the
same.
Representations of forces in 2D :
Imaginary (y)
Fy
F
Fx
Real (x)
F
= Fx i
+ Fy j
= Fei
= F <
F = 22 )()( yx FF F
x = F cos
= ATAN2( F
x / F , F
y / F ) F
y = F sin
Note that if
F
= F <
then
- F
= -F < = F < +
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Moment of a force about a point :
F
r
r
A
AM
= r
F
= r
F
= …….. = -Fd k
or,
AM
= r
F
= ( r < r ) ( F < F ) = rF sin(F - r ) k
Couple : Two, equal, opposite and non-collinear forces.
A . B .
d
F
r
- F
AM
= BM
= ….. = M
M
= r
F
= r
F
= …. = - Fd k
x
y
r
x
y
d
F
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