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Nov 15, 2013 (3 years and 8 months ago)

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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)

, 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