I need help with this please. A damped oscillation is described by following equation:d²y 1 dy 1+dt2 'b dtab dt *y = 0a) Use complex exponential function to solve this equation, namely find y(t), similar to thedamped mass-spring oscillation discussed in lecture notes. Here a, b and c are constants.b) Determine the mathematical conditions for under damped oscillation.

Answered: A damped oscillation is described by…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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I need help with this please.

A damped oscillation is described by following equation:
d²y 1 dy 1
+
dt2 'b dt
a
b dt *y = 0
a) Use complex exponential function to solve this equation, namely find y(t), similar to the
damped mass-spring oscillation discussed in lecture notes. Here a, b and c are constants.
b) Determine the mathematical conditions for under damped oscillation.

Expert Solution

Step 1

(a)

Write the given expression of damped oscillation.

$\begin{array}{rcl} a \frac{d^{2} y}{d t^{2}} + \frac{1}{b} \frac{d y}{d t} + \frac{1}{c} y & = & 0 \\ a D^{2} y + \frac{1}{b} D y + \frac{1}{c} y & = & 0 \\ a D^{2} + \frac{1}{b} D + \frac{1}{c} & = & 0 \\ D^{2} + \frac{1}{a b} D + \frac{1}{a c} & = & 0 \end{array}$

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