ENGR 205 - Fall
Unit 5 Study Guide
Unit Content:
Section Pages Content
7.8 404-410 Impulses and the Dirac Delta Function
Supplemental Solving Differential Equations with Functions Involving Jump
Material Discontinuities at ๐ก = 0
7.7 379-403 Convolution, the Transfer Function ๐ป(๐ ) and the Impulse
Function โ(๐ก)
7.6 392-393 The Gamma Function
3.1 89-91 Mathematical Modeling
4.1 153-154
Mechanical Vibrations and Simple Harmonic Motion,
4.9 214 -(Figure 4.28 Undamped and Underdamped or Oscillatory Vibrations
at bottom of217
4.9 218-222 Critically Damped and Overdamped Motion
Learning objectives:
A1. Solve a given problem that results in a D.E. involving the Dirac delta function.
A2. Use the convolution theorem to find โโ1{๐น ๐ ๐บ(๐ ) for a given ๐น(๐ ) and ๐บ(๐ ).
๐ก
A3. Use convolutions to find โ โซ0๐ ๐ก โ ๐ฃ ๐ ๐ฃ ๐๐ฃ} for given ๐ and ๐.
B4. Find the transfer function ๐ป(๐ ) and the impulse response โ(๐ก) for a given D.E. with
initial conditions ๐ฆ 0 = ๐ฆ 0 = 0.
๐
B5. (a) Find โ ๐ก for ๐ > โ1 and ๐ a non integer.
(b) Find โ1 {๐ } for ๐ < 0 and ๐ a non integer.
B6. Given an undamped, free vibration system and its initial conditions, determine the
equations of motion of the mass, along with the amplitude, period, and natural
frequency.
B7. Given an underdamped, free vibration system with specified information so that ๐,๐,
and ๐ can be found, determine the equation of motion and state the damping factor
๐ด๐ , quasiperiod, ๐ = 2๐ , and quasifrequency, โ .
๐ฝ ๐
B8. Given a damped, free vibration system with specified information so that ๐ and ๐ can
be found, find the equation of motion for three specified values of the damping
constant ๐ that yield the cases: underdamped, overdamped, and critically damped
motion. Unit 5 Homework Problems
Homework 5A (7 pts)
# Section Problem Notes/Hint/Answer
1 7.8 4 Answer: ๐2
2 7.8 5
3 7.8 16 Answer:
3๐ก โ๐ก 1 3 ๐กโ1) โ ๐กโ1
๐ + ๐ + 2(๐ โ ๐ )๐ข ๐ก โ 1 +
1 โ ๐กโ3) 3 ๐กโ3
(๐ โ ๐ )๐ข ๐ก โ 3)
4
4 7.8 29 Answer: solve the D.E., refer to example 1, section 7.8:
๐ฅ ๐ก = cos3๐ก โ (sin3(๐ก โ )๐ข(๐ก โ )) ๐
2 2
3๐ 3๐ ๐
= cos3๐ก โ (sin3๐กcos

More
Less