EGR 1101: Unit 9 Lecture #1

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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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EGR

1101: Unit
9

Lecture #1

Applications of Derivatives:
Electric
Circuits


(Section 8.4 of Rattan/Klingbeil text)


Review: Some Derivative Rules






















where
a
,
c
,
n
, and


are
constants.

0
)
(

c
dt
d
)
sin(
))
(cos(
)
cos(
))
(sin(
t
t
dt
d
t
t
dt
d









1
)
(


n
n
nt
t
dt
d
at
at
ae
e
dt
d

)
(
Two
New Derivative Rules



(
Product rule
)

dt
dg
dg
df
dt
df

dt
df
t
g
dt
dg
t
f
t
g
t
f
dt
d
)
(
)
(
))
(
)
(
(



If
f(g)

is a function of
g

and
g(t)

is a function of
t
,



(Chain rule)

Today’s Examples

1.
Voltage, current, & power

2.
Current & voltage in an inductor

3.
Current & voltage in an inductor
(graphical and working backwards)

4.
Current & voltage in a capacitor

5.
Current & voltage in a capacitor
(graphical and working backwards)

Review: Some
Electrical Quantities

Quantity

Symbol

Unit

Symbol for
Unit

Voltage

V

or
v(t)

volt

V

Current

I

or
i(t)

ampere

A

Charge

Q

or
q(t)

coulomb

C

Energy

W

or
w
(
t
)

joule

J

Power

P

or
p(t)

watt

W

Time

t

second

s

Review: More
Electrical Quantities

Quantity

Symbol

Unit

Symbol for
Unit

Resistance

R

ohm



Inductance

L

henry

H

Capacitance

C

farad

F

Voltage
-
versus
-
Current Relations


For resistors,

R
t
i
t
v


)
(
)
(
dt
di
L
t
v

)
(
dt
dv
C
t
i

)
(

For inductors,



For capacitors,

EGR

1101: Unit
9

Lecture
#
2


Applications of Derivatives:
Beams


(Section 8.5
of Rattan/Klingbeil text)


Some Beam Terminology


Types of beams


Simply supported


Cantilever


Types of load on a beam


Concentrated


Distributed

More Beam Terminology


In addition to the type of beam and load,
a beam’s behavior also depends on its
geometry and the material it is made of.


Its geometry is summarized in a quantity
called the
second moment of area

(
I
)
.


Its material is summarized in a quantity
called the
modulus of elasticity

(
E
).


The product of these two (
E
I
) is called
the
flexural rigidity
.

Some Beam Quantities

Quantity

Symbol

Unit

Modulus of
elasticity

E

lb/in
2

or
N/m
2


Second moment of
area

I

in
4

or m
4

Flexural rigidity

EI

lb

in
2
or
N

m
2

Deflection

y(x)

in or m

Slope


(x)

radians

Excellent Online Resource


University of Wisconsin’s online lessons
on Strength of Materials:
http://www3.uwstout.edu/faculty/
scotta/upload/Foley
-
StaticsStrengths.pdf



See especially Topic 4 (Beams) and
Topic 8.2 (Stress on Incline Planes).

Review


Given a function
f
(x), the function’s local
maxima

occur at values of
x

where



and



Its local
minima

occur at values of
x

where



and



0

dx
df
0
2
2

dx
f
d
0

dx
df
0
2
2

dx
f
d
Today’s Examples

1.
Deflection of a cantilever beam with
end load

2.
Deflection of a simply supported beam
with central load

3.
Deflection of a simply supported beam
with distributed load

4.
Maximum stress under axial loading