Analyze the motion of a damped harmonic oscillator subjected to an external force. Calculate the amplitude and phase difference between the displacement and the external force.
Guide On Rating System
Vote
A damped harmonic oscillator is a system that experiences a damping force, typically represented by the term -bẋ, where b is a damping coefficient and ẋ is the velocity of the oscillator. In addition to the damping force, an external force F(t) is also applied to the oscillator.
The equation of motion for a damped harmonic oscillator subjected to an external force can be written as:
mẍ + bẋ + kx = F(t)
Where m is the mass of the oscillator, k is the spring constant, and x is the displacement of the oscillator from its equilibrium position.
To find the amplitude of the oscillator, we can assume a solution of the form:
x(t) = A cos(ωt + φ)
Where A is the amplitude, ω is the angular frequency, and φ is the phase difference between the displacement and the external force.
Substituting this solution into the equation of motion, we get:
-mω^2A cos(ωt + φ) - bωA sin(ωt + φ) + kA cos(ωt + φ) = F(t)
Simplifying and rearranging, we obtain:
(-mω^2 + k) A cos(ωt + φ) - bωA sin(ωt + φ) = F(t)
Comparing the coefficients of cos(ωt + φ) and sin(ωt + φ), we find:
(-mω^2 + k)A = F(t) --(1)
-bωA = 0
From equation (1), we can solve for the amplitude A:
A = F(t) / (-mω^2 + k)
The phase difference φ can be found by taking the inverse tangent of the ratio of the coefficients of sin(ωt + φ) and cos(ωt + φ):
φ = atan(-bω / (-mω^2 + k))
In summary, to calculate the amplitude and phase difference between the displacement and the external force in a damped harmonic oscillator, you need to solve for the amplitude A using the equation A = F(t) / (-mω^2 + k), and find the phase difference φ using φ = atan(-bω / (-mω^2 + k)).
Got a question with my answer?
Cheapmath.com is completely free!
Asking question is free.
Getting answers is free.
+99