MatLab help! a. Which of these equations does not represent a mechanical vibrati

Question

MatLab help!

a. Which of these equations does not represent a mechanical vibration? Why not? In your comments, explain how to recognize that the equation cannot describe a mechanical vibration 1. from the graph of the solution, and also ii. from the coefficients of the original equation. (Recall how we have interpreted the coefficients m, b, and k where my + by + ky = 0.)

i. y + 3y = 0

ii. y + 4y + 29y = 0

iii. y y/36 = 0

iv. y + 2y + y = 0

v. y + 6y + 6y = 0

b. In your comments, classify the other four solutions as undamped, underdamped, criticallydamped, or overdamped.

Explanation / Answer

Part (a)

In order to determine whether or not a given second order differential equation represents mechanical vibration, we check the nature of roots of characteristic equation. If roots of characteristic equation are complex, then only a mechanial system can have vibrations. In short, we just check the descriminant of the characteristic equation to decide if it will have vibrations. So let us check the descriminants of the given systems

i. y + 3y = 0

m=1, b=0, k=3

D=b2-4mk = 0-4*1*3 = -12

Since descriminant is negative, therefore, this mechanical system will have vibrations. Also, it is underdamped.

ii. y + 4y + 29y = 0

m=1, b=4, k=29

D=b2-4mk = 16-4*1*29 = negative

Since descriminant is negative, therefore, this mechanical system will have vibrations. Also, it is underdamped.

iii. y y/36 = 0

m=1, b=0, k=-1/36

D=b2-4mk = 0-4*1*(-1/36) = 1/9

Since descriminant is positive. We will not have vibrations in this mechanical system. Also, it is overdamped.

iv. y + 2y + y = 0

m=1, b=2, k=1

D=b2-4mk = 4-4*1*1 = 0

Since descriminant is zero. We won't have mechanical vibrations in this system. And since descriminant is zero, system is critically damped.

y + 6y + 6y = 0

m=1, b=6, k=6

D=b2-4mk = 36-4*1*6 = positive

Since the descriminant is positive. System won't have vibrations. And it is a overdamped system.

Part (b)

i. y + 3y = 0 (Underdamped because D<0)

ii. y + 4y + 29y = 0 (Underdamped because D<0)

iii. y y/36 = 0 (Overdamped because D>0)

iv. y + 2y + y = 0 (Critically damped because D=0)

v. y + 6y + 6y = 0 (Overdamped because D>0)

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