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Final

# MECH 262 Study Guide - Final Guide: Ramp Function, Even And Odd Functions

Department
Mechanical Engineering
Course Code
MECH 262
Professor
Farhang Daneshmand
Study Guide
Final

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5.1
CHAPTER 5
5.1 The ramp function appears as the following:
T = 0.02 sec
o = 2f
= 2(1/T)
= 2(1/0.02)
f(t) = 100t
0 001 t .
f(t) = 100t 2
001 002. . t
From Eq. 5.5,
T
on tdtntf
T
b0sin)(
2
Thus,
 
01.0
0
02.0
01.0
1)100sin()2100()100sin(100
02.02dtttdtttb
 
 
 
 
100 31831 10 31831 10
06366
3 3
. .
.
 
01.0
0
02.0
01.0
2)1002sin()2100()1002sin(100
02.02dtttdtttb
 
 
 
 
 
100 15916 10 15916 10
03183
3 3
. .
.
ao is the average over the period, Eq. 5.3
 
dttf
T
aT
o
0
1
Thus,
 
 
 
 
50 510 510
0
3 3
From Eq. 5.4,

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5.2
T
on tdtntf
T
a0cos)(
2
 
01.0
0
02.0
01.0
1)100cos()2100()100cos(100
02.02dtttdttta
 
 
 
 
100 2023 10 2026 10
0
3 3
. .
0
)1002cos()2100()1002cos(100
02.0201.0
0
02.0
01.0
2
  dtttdttta
Without actually evaluating the values for ao, a1, and a2, we could have found
that they were each equal to zero since f(t) is an odd function.
5.2 The ramp function appears as the following:
T = 0.04 sec
o = 2f
= 2(1/T)
= 2(1/0.04)
f(t) = 50t
02.00 t
f(t) = 50t 2
04.002.0 t
From Eq. 5.5,
T
on tdtntf
T
b0sin)(
2
Thus,
 
02.0
0
04.0
02.0
1)50sin()250()50sin(50
04.0
2dtttdtttb
 
6366.0
103662.6103662.650 33
 
02.0
0
04.0
02.0
2)502sin()250()502sin(50
04.0
2dtttdtttb

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5.3
 
3183.0
101831.3101831.350 33
ao is the average over the period,Eq. 5.3
 
dttf
T
aT
o
0
1
Thus,
 
02.0
0
04.0
02.0
0)0cos()250()0cos(50
04.0
1dttdtta
 
 
0
01.001.025
From Eq. 5.4,
T
on tdtntf
T
a0cos)(
2
 
02.0
0
04.0
02.0
1)50cos()250()50cos(50
04.0
2dtttdttta
 
0
10053.410053.450 33
Without actually evaluating the values for ao, a1, and a2, we could have found
that they were each equal to zero since f(t) is an odd function.