1
Mechanical Elements and Systems Design Quals
Jan 2010
Robert Panas
Statistics
1
1
N
i
i
x x
N
average
2
1
1
1
N
i
i
s x x
N
standard deviation
Least squares fit
, y = ax
o
x
y
as
r
s
Chance σ>s: (stress greater than limit v
alue), want σ<s chance 0.99.
o
y i
m s s
define margin of safety variable
, m (+ if safe), y is yield, i is stress of
interest
o
2 2 2
m y i
m y i
o
m
m
deviation from average
m
z
standard deviation
o
Failure is when m is negative. Ideally it is largely
positive. Question is, what %
of m<0? Calculate p(m>0)
o
2 2
0
( 0)
y i
m
m
y i
p m
o
( 0) 1 ( 0)
p m p m
Stress
Flexures
o
Loading
4
4
( )
dV d y x
q EI
dx dx
loading (q) is
derivative of shear (positive upwards)
3
3
( )
dM d y x
V EI
dx dx
shear is derivative
of moment
(positive downwards)
2
2
( ) ( )
( )
d x d y x
M x EI EI
dx dx
for small
deflections
( )
( )
dy x
x
dx
slope of beam
( )
y x
vertical deflection of beam
Shear/Moment directions
+x
+
V(x
)
+
M(x
)
+Load
+x
+
V(x
)
+
M(x
)
+Load
+y
+x
+Moment
Load
doublet
+Force Load
+
θ
2
o
Singularity functions
Moments:
doublet
unit doublet (

2) is in NEGATIVE direction for moments
(if we
assume Z axis out of page, towards reader)
, so a positive moment
M is
2
x a
M M x a
, correct + rotation shown in fig above
point loads: delta
distributed load: sum of
step functions
higher polynomials: use
ramp, parabolic, etc…
when integrating, bring
exponent down for
everything above 0
o
Solving process
Use singularity functions to
define q(x) including
reaction loads
integrate from q four
times
to get deflection
(to
V,M,θ,y)
, adding
integration constant each
time
. When go from M to
θ, multiple M(x)*1/EI
before integration
use boundary conditions to
define the four constants
and any other unknowns (reaction loads).
Common
BC, up to 8 possibl
e at edges:
shear V(0

)=0,
and moment M(0

)=0 before beam starts
shear V(L+)=0
, and moment M(L+)=0, after beam ends
slope
ydot(0
)
=0, slope ydot(L)=0 if clamped guided
y(0)=0, y(L)=0 if clamped clamped, y(x)
may want to do
this as each
condition is reached during
integration (shear = 0 after
first integration, moment = 0
after 2
nd
integration, etc)
no BC on q(x), but can have
on V, M, theta, y
o
Examples:
P4.56, distributed load
o
Derivation
dx yd
compressio
n/expansion off
neutral axis (y = dist off axis)
Shear/Moment directions
Differential bending segment
3
2
2
3
2
2
1
1
d y
d
d x
ds
dy
dx
where ρ is radius of curvature, Φ is angle and s is
displacement along beam center
(neutral axis)
dx yd y
ds ds
,
strain is ε
2
1 1
y
M ydA E ydA E y dA EI
where M is moment at s,
I is moment of inertia, E is youngs modulus. Derived from forcing
internal moment (integral sigma*y*da) to equal external applied moment
E M
E y y
I
stress proportional to moment, distance from neutral
axis and momen
t of inertia
For torsion, shear stress
Tr
J
where T is torque, r is radius and J
is the polar moment of inertia
Curved flexures
Neutral axis moved to average radius of curvature
1 1 1
n
dA
A
Common relations
Common
deflection relations
Stress concentrations
L
F
F
F
w
w
w
δ
max
θ
max
EI
FL
3
3
max
EI
FL
2
2
max
EI
wL
8
4
max
EI
FL
48
3
max
EI
wL
384
5
4
max
EI
FL
192
3
max
EI
wL
384
4
max
EI
wL
6
3
max
EI
FL
32
2
max
EI
wL
24
3
max
??
2
2
max
EI
FL
EI
wL
3000
2
max
2
max
16
FL
EI
2
max
64
FL
EI
L
F
F
F
w
w
w
δ
max
θ
max
EI
FL
3
3
max
EI
FL
2
2
max
EI
wL
8
4
max
EI
FL
48
3
max
EI
wL
384
5
4
max
EI
FL
192
3
max
EI
wL
384
4
max
EI
wL
6
3
max
EI
FL
32
2
max
EI
wL
24
3
max
??
2
2
max
EI
FL
EI
wL
3000
2
max
2
max
16
FL
EI
2
max
64
FL
EI
2
16
FL
EI
4
64
d E
k
DN
2
16
FL
EI
4
64
d E
k
DN
4
o
2
1
t
b
K
a
theoretical stress concentration factor for elliptical hole where b is
half width and a is half height and width of part is infinite. Stress in height
direction
o
Usually less than 3
Thi
n walled vessel
o
If wall radius <0.05*r, thin walled.
o
2 2
2 2 2
1
i i o
r i
o i i
r p r
p
r r r
radial stress for assumption of inner radius ri nearly
same as outer radius ro, with internal pressure pi.
o
2
pd t
pressure trying to separate cylinder in half integrated over inner
surface gives p*d, hoop stress resisting is wall area (2*t) times stress, σ.
o
4
z
pd
t
longitudinal stress (z direction in cylindrical coordinates) due to caps at
end of c
ylinder
Hertzian
Contact Stresses
o
2
cylinders, check hale thesis for other choices
1 1 1
e 1 2
R R R
effective radius, combination of the two radii of each of the
surfaces, such that sphere has 2 of equal
size, cylinder has one finite and
one infi
nite radius
2 2
1 2
1 2
1 1
1
e
E E E
eff
ective youngs modulus of full interface
1 2
3 3
1 3
4
e e
F
R E
deflection between points far from contact area
1
3
3
4
e
e
FR
a
E
radius of contact area
,
a,
between two spheres with
po
isson’s ratio, E
and diameter d for given force F
1
2
3
2 2
6
3 1
2
e
e
FE
F
p
a R
maximum surface pressure, found in exact center
of contact area, 1.5x the average pressure
Max shear stress
0.31
p
at
0.48
z a
this is where yield occurs,
whe
n this stress stress is about 58% of tensile strength
Max tensile stress
1 2
3
p
at
r c
Stress/Strain Relations
5
o
Von Mises:
2
2 2 2
2 2 2
11 22 22 33 11 33 12 23 31
6
2
v
where the von mises stress indicates failure when it reaches material yield
If all shears turned into stresses (‘principle stresses’), then von mises
criterion reduces to:
2 2 2
2
11 22 22 33 11 33
2
v
o
Tresca criterion is:
1 3
t
using pr
inciple stresses, ordered from largest
(1) to smallest (3)
o
1
i i j k
T
E
general strain equation
o
lateral
axial
o
At plastic loading, υ=0.5
o
0
2
2
,
3
2
2
2
1
xy
Y
x
Y
x
Mohr’s circle
o
T
E
thermal strain
o
2 1
E
G
Deflection and Strain
Simple spring calculations
o
AE
k
l
linear
o
GJ
k
l
rotational shear
o
r
GA
k
l
linear shear
Inertias
o
1
st
moment
x
I ydA
over surface of beam, etc.
o
2
nd
moment
2
xx
I y dA
over surface of beam, etc.
, neutral plane (linear
deflection)
Parallel Axis theorem
2
new old
I I Ad
this is true when Iold is the 2
nd
moment around the center of mass
o
Polar moment
2
J r dA
, neutral axis (rotational deflection)
, only holds for
circular cross

sections. For rectangular cross

sections, is not a clear simple
expression. See Roarkes Formulas for Torsion for details.
o
Mass moment
2
m
I r dm
6
Parallel Axis theorem
2
I
mnew mold
I md
this is true when Iold is the 2
nd
moment around the center of mass
Second moment of inertias
Strain Energy
o
2
2
M
U dx
EI
for bending
o
2
2
F
U dx
EA
for tension
o
2
2
T
U dx
GJ
for torsion
o
2
2
V
U dx
GA
for shear
o
2
2 2
U dx dV
E
for generic volume
Castigliano’s Theorem
o
i
i
U
F
displacement or force calculated as partial derivative of energy with
respect to other variable
o
Can be done for fictitious force by adding in this fake force, Q, taking derivative,
then setting Q=0
o
Can enforce zero displacement with this method too
Buckli
ng
o
Equations
4
3
4
1 1
0.21 1
3 12
is long side, is short side
b b
ab
a a
a b
3 4
4
1
1 0.63 1
3 12
ab b b
a a
a b
2 3 4
2
3
1 0.6095 0.8865 1.8023 0.9100
T b b b b
ab a a a a
a b
At the midpoint of each longer side (a)
7
2
2
crit
C EI
P
l
critical
loading for be
am
2
2
/
crit
crit
P
C E
A
l k
where
l/k is the slenderness ratio
2
I
k
A
definition of k as
the radius of gyration, with I
being 2
nd
moment and A is
cross sectional area
o
Constant:
A 4 fixed fixed
B 2 fixed pinned
C 1 fixed guided
D 1 pinned pinned
E ¼ fixed free
F ¼ pinned guided
o
Regimes
When low, buckling very high, failure at yield stress
When high, buckling first f
ailure mode, failure at crit stress
Crossover when yield=crit stress. Above this, critical stress (buckling) is
failure, below, yield is failure.
This allows calculation of l/k at which buckling becomes an issue
Usually calculate for buckling failure when
l/k ratio such that crit stress <
yield/2, when around crit=yield, use intermediary, when crit>>yield, use
yield
Damping
o
Constant orifice snubber: Classically linear with velocity
o
Conventional snubber: Can made dampers that put out nearly constant force f
or
any velocity
o
Progressive snubber: Increasing force for lower velocities
Mechanical Springs
For helical spring,
4
3
8
d G
k
D N
, d is wire diameter, D is helix diameter, N number of turns,
k is linear stiffness
Light service (less than 10
4
),
average, (10
5

6
), severe (>10
6
)
Torsion springs

helical springs with different ends
Materials
Yield= 0.2% deflection from linear
UTS = maximum engineering stress observed
o
UTS (
Su
)
~3
.5
HB
[MPa] for steel, where HB is brinell hardness number
Failure Regimes
Boundary Conditions
Stress
σ
Slenderness Ratio
l/k
Yield
Buckle
σ
y
Transition
Buckling
Yield
Stress
σ
Slenderness Ratio
l/k
Yield
Buckle
σ
y
Transition
Buckling
Yield
C=
4
2
1
1
1/4
1/4
C=
4
2
1
1
1/4
1/4
8
Low cyle (<1000
)
Infinite life region (>10
6
) for steels
below about 0.5 yield
Fracture stress

stress value at which material fails
True strain:
1
ln( )
i
o
l
i
o
l
l
dl
l l
so
ln( 1)
true eng
In true stress/strain, fracture stress> UTS
Plastic stress
0
m
Plastic deformation in cold working moves elastic limit up. Return path not to 0, but to
new equilibrium with yield stress being a greater value

larger elastic limit, same E,
reduced amount of effort r
equired after this to reach UTS,
less ductility
Manufacturing Processes
o
casting
Sand casting
Shell casting
Investment casting
o
Powder metallurgy
o
Hot working processes
Rolling
Forging
Extrusion
o
Cold Working
Drawing
Blanking
F
orming
Breaks grains

get smaller
o
Annealing
Heated above
transition temp, relieve residual stresses, lower yield, greater
ductility
Grain growth
o
Quenching

can change crystal structure of steel this way
o
Plastics
Thermoplastic

can reheat/reshape
Thermoset

once set, cannot be changed, chemically set
Friction
o
Tef
lon about 0.1
o
Ice 0.1
o
Steel/aluminum 0.25
o
Brake pads 0.5
o
Plastic 0.1

0.2
Cost
o
Steel
1x $0.5/lb
o
Alloy steel
2x
o
Stainless steel
3x
o
Tool steel
13x
o
Aluminum
4x
9
o
Brass
2x
Screws, Fastener, and the Design of Nonpermanent Joints
Thread standards
o
Pitch

distance b
tw adjacent threads
o
Minor diameter

di
a of bot
of threads
o
Major diameter

dia
of top of thread
o
Lead is distance nut moves along screw axis
when it is rotated 360 degrees
1
2
m m
m
Fd fd
T
d fL
torque required to raise a load
with F load, dm is mean diameter,
L
is height gained
from one rotation, f is coefficient of friction,
o
Self locking if tan(λ)
<μ
, where λ is lead angle
tanλ = L/(πd
m
)
, μ is friction coefficient
o
Reduces to:
m
L
d
With no friction,
2
T FL
, torque T to turn 1 rotation is energy required to apply force
F over length L
First engaged thread carries about 40% load, second=25%, third = 18%,…seventh = 0
,
want about 1

1.5 D wort
h of thread holding screw
For nut, first thread takes all load at first, then yields until about 3 threads take load
50% of energy in joint into bolt head friction, 40% into thread friction, 10% into elastic
deformation
S
tiffness
:
o
Material stiffness found
through assuming pressure cone of about 30deg off
vertical
Assuming this angle and diameter about 1.5 times hole,
tan
tan
2ln
tan
w w
w w
Ed
k
l d d d d
l d d d d
where d is diameter of sc
rew hole,
dw
is diameter of
washer at top of conic section, l is depth of screw hole
thro
ugh material
Preload
o
Proof strength

max safe load
o
Supposed to fail just under head
o
first thread engaged to nut
o
i
T KFd
torque
T
to preload
Fi
relation, K roughly 0.2 (when lubrication about f
= 0.15),
d is mean diameter,
o
Want preload
, F
about 90% proof load for permanent bolts
o
Preload
, F
about 70% proof load for removable bolts
Pins/Keys
Model of a bolted joint
10
o
Compressive stresses from interference fit:
2
E
p
R
, R is nominal pin size, d is
the difference in pin and hole radii, p is pressure
Want threads to fail (shear) just as shaft breaks
Welding, Brazing, Bonding, and the Design of Permanent Joints
Welding
o
Often case that weld metal is strongest part of joint
Resistance Welding (spot welding)
Adhesive Joint
o
Can be more resistant to failure
than mechanical joints
Mismatch of thermal expansion coefficients can lead to stresses
Dowel pin
Spiral pin

rolled cylinder of material, expands in hole
Spring pin

slotted cylinder, also expands in hole
rivets
Bearings
Types
o
Sliding
o
Roller/ball
o
Magneti
c
o
Flexural
o
Hydrostatic
Single pad
Opposed pad
Journal
Rotary thrust
Conical journal/thrust
o
Aerostatic
Rolling Contact Bearings
Components
o
Outer ring

surface either circular arc or gothic
, gothic allows 4 pt contact,
but more wear/friction
o
Inner ring

surface either circular arc or gothic
o
Balls/rolling element
o
Separator

omitted in cheap bearings
11
Ball Types
o
B,C,D,E
Single row, deep groove take radial and some thrust
‘Conrad’
, most
common
o
B,C deep groove has little moment resistance, does not resist angular
misalignment or thrust greatly
o
D 4 point contact good at resisting moment
and thrust
o
E angular contact good at resisting thrust in one direction
o
F double row instant centers are far apart, so very good at resisting moments
and thrust, also able to handle thermal growth of axle
o
G
full freedom in angular alignment

no moment resi
stance, moderate thrust
resistance, excellent radial loading
Roller Types:
o
H
Cylindrical roller
no thrust
, very good at radial, much higher than ball
bearing
o
I Double Cylindrical roller, same as H, just twice as much load
o
J Tapered roller
–
heavy axial
thrust, good with both radial+thrust loading in
one direction, advantages of ball and straight rollers
o
K Needle roller (d)

take no thrust, good with radial loading
o
L
Spherical roller
(b,c)
–
self aligning, good with heavy loads and
misalignment, increase
surface area with load
Bearing Life
o
Measure is number of hours at standard rotational speed or number of
revolutions, rating is that at which 10% have failed, called L
10
o
Ideal failure is metal fatigue in race
o
1/
a
FL K
, F is load, L is li
fe (# of revolutions), a is constant= 3 for ball
bearings, 10/3 for roller bearings
o
Manufacturer chooses standard life L
10
= 10
6
then
rate each bearing with the
bearing load C
10
such that 10% fail by that lifetime. Gives value of constant
1/1/
10 10
a a
C L K FL
, then can choose new load or lifetime (F or L)
o
0
0
b
x x
x
R e
For reliabilities R other than 90% (10% failure), where
x=L/L
10
,
θ is std dev of distribution, x0 is guaranteed or minimum value of the
variate, b is shape parameter of
skewness
.
Weibull distribution
L
50
= 5*L
10
for 50% reliability
L
5
= 0.62*L
10
for 95% reliability
L
1
= 0.21*L
10
for 99% reliability
o
a
i i
D F L
Damage D occurs as sum of load to the a power times number
of revolutions, sum up all segments
1/
a
a
i i
tot
i
F L
F
L
equivalent load due to damage theory
A B C D E F G H I J
K L
Types of Bearings
12
One way to c
heck where this lies
:
,
calculate K for bearing, using Ftot,
gives an L. calc x = L/L
10
, then determine R for that point, gives
chance of failure at that point.
o
When bearing is
mis

aligned, life drops
o
Friction around 0.01
Tapered Roller Bearings
o
components
Cone (inner ring)
Cup (outer ring)
Tapered rollers
Cage (space retainer)
o
0.47*
r
a
F
F
K
, induced thrust load Fa generated by radial load Fr, geometry
constant K
use
two bearing sets pointed opposite directions to cancel out
Mounting and Enclosure
o
Mounting plan: often axially constrain only one set of bea
rings, so not
overconstrained.
o
Duplexing

pairs of angular contact ball bearings set such that when clamped
togethe
r, system preloaded
o
Preloaded desired to remove internal clearances, increase fatigue life, decrease
shaft slope
Inner or outer interference fit
Buying preshrunk outer ring
Ensure all of ball is under compression
Heavy preload is 5% of max bearing load
Med
ium load is 3%
Light load is 1%
o
Seals should be used to keep out grease/dirt
Felt (rubbing on surface) cheap, good for low speed
Rubber seal (rubbing on surface) good for low speed
Labyrinth
–
non contact, good for high speed, no direct path through
gap
Lu
brication and Journal Bearings
o
Revolute pair

(journal bushing)

allows
rotation while constraining other motion
o
Lubrication
Hydrodynamic

su
rfaces separated by thi
n
film, fluid mechanics physics
du U R
dx h h
fluidic shear stress, τ, with relative
velocity U and gap h, dynamic viscosity μ
(10

3
water, 10

5
air),
radius R and rotational speed ω
Petroff’s equation:
2
2
N r
f
P h
, coeff of friction f; rev per sec,
N; radius r; P = W/(2rl)
13
Hydrostatic
–
pressurized air/water into gap, no need for motion of
surfaces
Elastohydrodynamic
–
between surfaces in rolling contact (gears)
Boundary
–
very thin film, few molecules, viscosity less important than
chemical composition
Oily chemicals bond to surface v
ery strongly
Solid film
–
graphite as example, extreme temperatures
Gears
Can be up to 95% efficient
o
Generally used to slow down rotation, increase torque, because velocity easy to
generate
with voltage, torque requires current
Types:
o
Spur gears, teeth
parallel to axis of rotation
Used to transmit torques between parallel axes
About 98% efficient
o
Helical gears, teeth sheared from spur gear alignment
Less noisy

more gradual interaction of teeth
Develops thrust loads and bending, fixed with double helica
l
o
Bevel gear, conical shaped surface for gears
Between intersecting shafts
Spiral bevel, like helical
o
Hypoid gear
Helical gear for non

intersecting shafts
o
Worm gear
Used when speed ratio >3 between shafts
Non

intersecting shafts
Worm gear, ~ 30

50%
efficient (sliding contact)
o
Planetary Gear
About 97% efficiency per stage
Sun in center
Ring gear outside
Planet rolls between these
TR
from sun to arm which is
holding planets:
must be higher than 2
(corresponds to 0 size
planets)
lowest practical: 3:1
highest practical: 10:1,
beyond this, sun is so
small planets hit
Usually big speed
reduction
Planetary Gear Transmission Ratios
Memorization Trick for Planetary TR
!
!
14
arm ring
sun
ring sun arm
N
N
sun is inner gear in planetary arrangement, ring
is the ou
ter ring with gears on it’s inner surface, planet moves between
these
Nomenclature
o
Pitch circle:
theoretical gear
diameter for calc
Pitch diameter
o
Pinion:
smaller gear
o
Gear: larger of the
two
o
Circular pitch
p
:
distance between
gears on pitch circle
d
p
N
Sum of tooth
thickness and
spacing between gears
o
Module
pitch
d
m
N
ratio pitch diameter to number of teeth
o
Diametral pitch
P
= 1/m
,
N
P
d
o
Addendum: radial distance from pitch circle to tip of teeth
o
Dedendum: radial distance from pitch circle to bottom of teeth
o
Clearance circle: height of other gears teeth tips
Teeth
o
Constant angular velocit
y ratio during meshing: conjugate action
o
Common form: involute profile
o
Pressure angle
cos
b
r
r
, rb is radius of clearance circle, r is radius of pitch
circle
Spur Gear
Helical Ge
a
r (double)
Spiral Bevel Gear
Worm Gear
Gear Details
15
o
Contact ratio:
t
c
q
m
p
p is circular pitch, qt is the arc of action (arc over which
gears touch)
Can be more than 1, indicating multiple teeth in contact, generally <1.2
o
To avoid teeth causing interference, must choose number of teeth carefully

too
few or too many and will hav
e problem
Forming Gears
o
Methods
Sand casting
Shell molding
Investment casting
Permanent mold casting
Die casting
Centrifugal casting
Powder metallurgy
Extrusion
Form cutters
Cold forming
Milling
Shaping
Hobbing
Designing
o
Determine necessary power, speeds
and transmission ratio
Gear relation: contact enforced so
/2/2
A A B B
d d
for gears A and B
A A B
B B A
d
d
A A
B B
d
d
from energy conservation
o
Choose type of gear
Shaft alignment
Efficiency
Size
o
Size out gear
Estimate
module
(1, 1.5, 2 mm)
length/tooth = m*
π
Choose min N
usually >12 for good fit
Set diameter
pitch
d mN
Tooth avg thickness h
2
m
h
thickness about half the length/tooth
16
Tooth length L
2.2
L m
length
roughly 2.2 times module
Tooth width b
Make wide enough to keep stress down
Assume tooth is cantilever, max stress at base
max
2
6
y
My FL
SF I bh
b<d sets upper limit on width, should be less than diameter
Calculations
o
Forces come in at the power angle, only
tangential
part of interest to power
transfer, but normal part exists
Spur and Helical Gears
o
Basic calculation:
o
Fixed

free cantilever
, stress at base to yield
o
M = F*l
o
2
6
Fl
bh
where F is tangential load on gear tip, l is height of gear, b with
thickness of whole gear and h is width of tooth (in direction of force)
o
Set this to yield
Bevel and Worm Gears
o
Straight bevel
:
Like spur gear, up to about 1000 ft/mim (5 m/s) otherwise no
ise
too high
o
Spiral bevel
:
For higher speeds
o
Zerol bevel
:
Lower thrust loads generated
o
Hypoid gear
:
For non

intersecting shafts
o
Spiroid gear
:
Halfway between spiral bevel and worm gear
Clutches, Brakes, Couplings, Flywheels
Clutch

both sides rotate
Brake

only one side rotating, often self

locking
Self acting if will increase normal force as load is applied

holds better
Internal expanding rim clutches and brakes
o
If no spring, torque transmitted goes with speed^2
External contracting rim clutches and brak
es
o
Solenoids
o
Levers
o
Linkages with springs
o
Hydraulic
Capstan effect
o
1
2
f
P
e
P
, P1 is load, P2 is holding effort, f is coefficient of friction and phi is
angle of contact
17
Cone clutch

uses conical inner surface, slides equivalent surface
over, makes near
radially normal surface contact with only axial motion
o
About 10

15deg angle best
Must dissipate heat

kinetic to thermal
Flexible Mechanical Elements
Belts
o
Long separation
o
Have creep
o
Idler/tension pulley removes slack
o
Like gears, enforce
displacement equality on surface of pulleys
o
Efficiency of around 70

98%
If belt crossed on itself, reversing
Can use clutch to shift belt between wheels
Flat belts are made of urethane, reinforced with wire
o
Quiet
o
Efficient
at high speed
o
Large amounts of
power
V belts are smooth, made of fabric
o
Less efficient than flat
o
Can be used together for multiple drive

more compact
Timing belts have teeth
o
Cannot slip
o
No speed problems
o
High torque
All must have initial tension, or will slip (except timing)
o
Belt
weight
o
Spring
o
Weighted pulley in belt return path
Flat metal belts
o
Has minimum pulley diameter to avoid breaking
o
Diameter of pulley over belt thickness related to lifetime
Roller chains are like timing belts
o
Want to minimize angle made when going around pu
lley
o
At least 1/3 teeth should be in contact
o
Do not provide constant angular speed

angle of component leaving wheel is not
always horizontal
o
High strength
o
High maintenance
o
Efficiency about
60

80%
Wire rope
o
Regular lay

Strands and rope twisted in opposit
e directions
o
Lang lay
–
same direction, better for abrasion
o
Because of wrapping, elasticity of the rope is much higher than single wire
Pulley efficiencies can easily be around 95%
18
M
otors
Model
o
Torque
2
T rL r
where τ is shear stress, r is radius, and L is axial length
of motor, so shear stress times area times lever arm
o
Shear stress for dif technologies:
5

10 kPa
induction
30

40 kPa
high performance perm mag
130

140 kPa
large power turbine
o
Power
2
2 2
surf
P T r L rL r A v
is a product
of torque and rotational velocity w, can be rearranged to give shear stress*rotor
surface area*surface speed, or shear stress*rotor volume*rot speed
Sizing:
Optimal power transfer when impedance of
motor matched to load
Should oversize
power rate
by about 3

5 times
load to
account for efficiency
DC motor
synchronous
o
Linear relation from torque to speed
o
No torque max speed
o
No speed max torque
o
Modeled as resistor, inductor and back
emf
in series
o
Brushed motors

coils on inside
, permanent magnet on outside
75

80% efficient
Low
cost, simple speed control
o
Brushless are vice versa
85

95% efficient
Highest performance, long lifespan, low maintenance
o
Axial pole (current runs radially through disk)
o
Brushes break off, add dust to system
o
Burn out possible failure, or over torque to
gears, shaft/bearing failure, wiring
failures
Induction Motor
o
A
sychronous
Rotating magnetic field in stator
Low cost, high power, not exact speed (changes with torque)
Induces current in center which drives conductor out of field
Slip
(due to magnetic fre
q dif from rotor freq

asynchronous)
frequency
determines the amount of torque, for small deviations is proportional
Beyond given slip, torque drops off
Efficiency of about 80%
o
Synchronous
Brushless DC, but driven with sinusoidal current rather than on/off
Synchronous reluctance
o
Many reluctance / solenoid type torque actuators in series, enable semi continuous
torque
Motor Sizing Chart
Horsepower
Size
Cost
AC 1/100
3.3” dia
=
A
=
䅃‱A㔰
=
3.3” dia
=
A
=
䅃‱A㈵
=
3.3” dia
=
A
=
䅃‱A
=
4.4” dia
=
A
=
䅃‱AQ
=
5 5/8”
=
=
䅃‱A2
=
5 5/8”
=
㌰
=
䅃‱
=
5 5/8”
=
㔰
=
䑃‱
=
Q

6” dia x1’
=
〰
=
19
o
Hard to start
o
Linear version called sawyer motor
Hysteresis
o
Induction in solid metal core

magnetic drag
o
Low torque/inertia ratio
o
No ripple
S
tepper
o
Reluctance with rotation sensor
o
Precision positioning
Type
Advantages
Disadvantages
Applications
AC Induction
Che
a
p, High power,
long life
Rotation Slips from
Frequency
Fans and
Appliances
AC Synchronous
Rotation in

sync
with frequency,
Long life
Expensive
Clocks, turntables,
tape drives
DC Stepper
Precision
positioning,
High holding torque
Slow speed,
Requires a
controller
Positioning in
printers and floppy
drives
DC Brushless motor
Long lifespan,
low maintenance,
High efficiency
High initial co
st,
Requires a
controller
Hard drives,
CD/DVD players,
electric vehicles
DC Brushed Motor
Low initial cost,
Simple speed
control
High maintenance,
Low lifespan, Slow
speed
Treadmill
exercisers,
automotive starters
Transmissions
o
power from a motor to an actuator or mechanism
o
optimal transmission ratio most efficiently distributes power to the
motor/drivetrain and the load, which are connected and must accelerate together
o
For a motor to create a rotational velocity
motor
load
TR
J
J
n
o
For friction or belt drive
load
motor
Pulley
M
J
r
o
For a leadscrew with the optimal lead is
load
leadscrew
motor
M
J
J
20
Pump
s
Displacement
o
Constant flow machines

indep of pressure
Must therefor have safety if
exit blocked
o
Rotary dra
w fluid through motion of piston
Gear
(lobe)
pumps

two gears meshed
Screw pumps

two scre
w
s with opposing thread, moving in opposite
rotations, parallel to one another
, progressive cavity pump
Moving vane

cylinder in housing with
little gap, rotation draws fluid
o
Reciprocating
Expanding cavity on suction side, decreasing cavity on discharge side
Piston (plunger) pulls fluid in one step, pushes it out other valve in second
step
Dynamics waste a bit of energy
D
iaphragm
Peristaltic

m
edical type, squeeze tube to push fluid through tube
Hydraulic ram

passive uses some fluid from input to hammer small
amount of the fluid up a gravitational gradient

siphons energy from flow
and condenses it into small amount. Water hammer effect drives
it up
tube with one

way valve.
Dynamic
Displacement Pumps (Piston, Gear)
21
o
Pressure based, can withstand blockages
Lower efficiencies, but less
maintenance
o
Centrifugal pump

spings the water out, pressurize
Enters near rotating axis, spins out radia
l
If just flows along
axis, ‘axial’, like propeller
>80% of all pumps are centrifugal
o
Screw centrigual impeller

single blade, axially extended at inlet, like a corkscrew
Efficiency is around 50

80%, power imparted to fluid vs. power into pump
o
Efficiency reaches peak at midway
through operating range of flow rate
o
Power to fluid is
Power PQ gH Q
output power Po,
pressure P,
density
ρ
, height gained H, gravity g, flow rate Q (m
3
/s)
Design considerations
o
Cost
o
Flow rate
o
Contamination
o
Efficiency
o
Viscosity
o
Duty

cycle
o
Ene
rgy source (AC, DC, hydraulic, air, gas or diesel engine, steam, etc.)
o
Size
o
Pressure gradient
o
Maintenance
Actuators
electromagnetic rotary motor
iBNlr
lr
B
i
2
electromagnetic linear motor
iBNl
F
l
B
i
F
electromagnetic reluctance:
2 2
2
2
0
N I
μ
F A
h
Electrostatic
2
2
1
r
q
q
k
F
e
chemical (internal combustion, rocket)
piezoelectric
hydraulic linear
in
in
out
out
F
A
A
F
pneumatic (linear & rotary)
thermal actuator
conduction polymers
shape memory alloy
leadscrew, ball
screw
friction drive
rack and pinion
Centrifugal pump
22
Concepts:
Saint

Venant’s Principal
–
Several characteristic dimensions away from an effect the
effect is essentially dissipated
Design Process
o
Functional Requirements
o
Design Parameters
o
Analysis
o
References
o
Risks
o
Countermeasures
Leadscrews
/Ballscrews
Leadscrew efficiency about 40

8
0%
Ballscrew efficiency can be up to
80

95%
Must have some clearance to avoid binding

results in backlash
o
Fixed with preload
At high frequency can vibrate
Can stack
ElectroMagnetic
Equations
d
d N
dt
E l
Faraday’s Law
d Ni
H l
Ampere’s Law
0
d
B s
Gauss’s Law
N
flux linkage
Li
cross domain inductance
d
V
dt
2
N
L
R
2
1
2
E LI
dE
F
dx
Fluidic Equations
2
C L
P RQ Q
A
Fluidic resistance R relates pressure drop ΔP to volumetric flow
rate Q through length of pipe L, viscosity μ and pipe cross secti
onal area A
o
8
C
for round tubes
o
35
C
for triangular tubes
23
2
2
v
F C A
Air drag force, scales with density ρ, cross sectional area A, relative
velocity v and with coefficient corresponding to aerodynamic shaping
o
1
C
for cylinder
o
1
C
for aerodynamic shapes
o
1
C
for flat plates
Error Calculation with HTM
R
r n
v v
coordinates in reference
(r)
as outputs, local frame
(n)
as inputs, add separation
between frames to local frame
rTipActual rTipIdeal
e = v v
error calculated by taking final operating
point (tip) calculation and
subtracting the no

error prediction (ideal) from the full prediction
1 0 0
0 1 0
0 0 1
0 0 0 1
x
y
z
x/y/z
R
linear translation, these numbers (x,y,z) describe the
location of
the origin of the n

frame with respect to the origin of the
reference (r

frame)
1 0 0 0
0 cos sin 0
0 sin cos 0
0 0 0 1
x x
x x
x
R
positive rotations θ around the x

axis.
cos 0 sin 0
0 0 0 0
sin 0 cos 0
0 0 0 1
y y
y y
y
R
positive rotations θ around the y

axis.
cos sin 0 0
sin cos 0 0
0 0 1 0
0 0 0 1
z z
z z
z
R
positive rotations θ around the z

axis.
Sensors
Capacitance
Hall effect se
nsor
Inclinometer
Inductive sensor
LVDT
Magnetic Scale
Magnetostricti
ve
Sensor
Mechanical switch
Piezoelectric
24
Potentiometer
Synchro (measures rotor location, like ac motor)
Ultrasonic
Velocity
o
LVT
o
Tachometer
Optical
o
Interferometer
o
O
ptical encoder
o
Laser
triangulation
o
photoelectric
Engines
Internal Combustion
o
Description:
chemical

mechanical transformer
oxidization of fuel occurs in confined space (combustion chamber)
exothermic, high pressure, temp
chamber expands, generating motion
efficiency about 20%
about 50

75 Hp/liter
o
Four stroke
Intake

Combustible mixtures are emplaced in the combustion chamber
Compression

The mixtures are placed under pressure
Combustion

The mixture is burnt, the hot mixture is expanded, pressing
on and moving parts of the
engine and performing useful work. Ignition
can either be a spark ignition or a compression ignition where ignition
relies solely on heat and pressure created by the engine in its compression
process.
Exhaust

The cooled combustion products are exhausted
o
Two stroke
Intake and compression in down stroke
Combustion and exhaust in up stroke
Simpler, smaller, lighter for given power output
Less efficient, more polluting
Gas Turbine
o
Description
Rotary, extracts energy from continuous flow of combustion gas
effi
ciency about 45

50%
high power/weight ratio
continuous flow, easy to scale up power generation
much more complex and expensive than IC
o
Brayton cycle:
isentropically compressed air
combustion at constant pressure
expansion over turbine isentropically back
to ambient pressure
25
Jet Engine
o
Expansion of gas used to generate thrust
o
Similar layout to gas turbine but exhaust forced through nozzle
Human Factors and Conversions
Person
o
Generate about 0.25 to 0.75 Hp or 200 to 600 Watts
o
1

2 kW for few seconds, peak
athletes
o
Mass about 75 kg
o
Lift about 15

30% of body mass
o
Walk at 3

5 mph
o
Max speed about 5 Hz for pedaling, other motions
Low force at this and higher
Max power at around 1

2 hz
Useful facts
o
Energy in gasoline, 45 MJ/kg
o
Internal resistance of battery start
s at about 1 ohm for solid batteries, about 0.1
ohm for car batteries
Standard to metric conversion
o
3ft
=
1 m
o
2
mph =
1
m/s
o
2.2lbs = 1kg
o
1HP
= 550 ft lbs/s
= 33000 ft lbs/min
= 746W
o
1lbf = 4.45N
o
1in = 0.0254m
o
1psi =
7000Pa
o
1mile = 1.61km
o
1gal = 4L
o
10 rpm =
1 rad/s
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