Question

Consider harmonic oscillators with mass m, spring constant k, and damping coefficient b. For the values specified,

a. write the second-order differential equation and the corresponding first-order system.
b. find the eigenvalues and eigenvectors of the linear system.
c. classify the oscillator (as underdamped, overdamped, critically damped, or undamped) and, when appropriate, give the natural period.
d. sketch the phase portrait of the associated linear system and include the solution curve for the given initial condition.

Answer 1

a) attached below

b) Eigen values : λ1 = -1 and λ2 = - 1/2

Eigenvector : V1 = - y1 and V2 = ( - 1/2 ) y2

c) since the Oscillator is > 0 hence it is Overdamped

Explanation:

Lets take : m = 2 , k = 1 and b = 3

a) Second -order differential equation and the first order system

attached below

b) determine the eigenvalues and eigenvectors of the Linear system

Eigen values : λ1 = -1 and λ2 = - 1/2

Eigenvector : V1 = - y1 and V2 = ( - 1/2 ) y2

attached below is the detailed solution

c) The oscillator is classified as OVERDAMPED

To determine if the oscillator is damped, underdamped, or overdamped we will apply the relation below

b^2 - 4km

( 3 )^2 - 4(1)(2) = 1

since the Oscillator is > 0 hence it is Overdamped

d) Attached below is the phase portrait of the associated linear system

y(0) = 0

v(0) = 0