Determine the response of the damped mass spring system under a hammer-blow. The instantforce is modeled with a unit impulse at time t=2. Assume y(0)=0=y'(0). Thus, the ODE isy' + 3y' + 2y = 8(t – 2)

Name: Determine the response of the damped mass spring system under a hamme
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Description: Determine the response of the damped mass spring system under a hammer-blow. The instantforce is modeled with a unit impulse at time t=2. Assume y(0)=0=y'(0). Thus, the ODE isy' + 3y' + 2y = 8(t – 2)

Answered: У' +Зу' + 2у%3 8(t - 2)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Determine the response of the damped mass spring system under a hammer-blow. The instant
force is modeled with a unit impulse at time t=2. Assume y(0)=0=y'(0). Thus, the ODE is
y' + 3y' + 2y = 8(t – 2)

Expert Solution

Step 1

The objective is to find $y'' + 3 y' + 2 y = ρ (t - 2)$

Then first find the homogenous part,

$y'' + 3 y' + 2 y = 0$

Take $m = \frac{d}{d t}$ ,

$m^{2} + 3 m + 2 = 0 m^{2} + 2 m + m + 2 = 0 m (m + 2) + 1 (m + 2) = 0 (m + 1) (m + 2) = 0$

Hence, $y_{c} = c_{1} e^{- t} + c_{2} e^{- 2 t}$ .

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