ENGR 205 Midterm: ENGR 205 Unit 5 Study Guide

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Engineering Fundamentals
ENGR 205

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 𝑑 = 0 7.7 379-403 Convolution, the Transfer Function 𝐻(𝑠) and the Impulse Function β„Ž(𝑑) 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 β„’βˆ’1{𝐹 𝑠 𝐺(𝑠) for a given 𝐹(𝑠) and 𝐺(𝑠). 𝑑 A3. Use convolutions to find β„’ ∫0𝑓 𝑑 βˆ’ 𝑣 𝑔 𝑣 𝑑𝑣} for given 𝑓 and 𝑔. B4. Find the transfer function 𝐻(𝑠) and the impulse response β„Ž(𝑑) for a given D.E. with initial conditions 𝑦 0 = 𝑦 0 = 0. π‘Ÿ B5. (a) Find β„’ 𝑑 for π‘Ÿ > βˆ’1 and π‘Ÿ a non integer. (b) Find β„’1 {𝑠} for π‘ž < 0 and π‘ž 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 π‘˜,𝑏, and π‘š can be found, determine the equation of motion and state the damping factor 𝐴𝑒 , quasiperiod, 𝑇 = 2πœ‹ , and quasifrequency, ⁄ . 𝛽 𝑇 B8. Given a damped, free vibration system with specified information so that π‘˜ and π‘š can be found, find the equation of motion for three specified values of the damping constant 𝑏 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: 𝑒2 2 7.8 5 3 7.8 16 Answer: 3𝑑 βˆ’π‘‘ 1 3 π‘‘βˆ’1) βˆ’ π‘‘βˆ’1 𝑒 + 𝑒 + 2(𝑒 βˆ’ 𝑒 )𝑒 𝑑 βˆ’ 1 + 1 βˆ’ π‘‘βˆ’3) 3 π‘‘βˆ’3 (𝑒 βˆ’ 𝑒 )𝑒 𝑑 βˆ’ 3) 4 4 7.8 29 Answer: solve the D.E., refer to example 1, section 7.8: π‘₯ 𝑑 = cos3𝑑 βˆ’ (sin3(𝑑 βˆ’ )𝑒(𝑑 βˆ’ )) πœ‹ 2 2 3πœ‹ 3πœ‹ πœ‹ = cos3𝑑 βˆ’ (sin3𝑑cos
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