Question

write down the differential equation of the forced harmonic oscillator?

Answer 1

Answer:

The set up is a forced, damped oscillator governed by a differental equation of the form y'' + (γ/m)y' +ω_0²y = F_0 cos(ω_e t), where m, γ and ω_0 are the mass, damping constant and natural frequency of the oscillator, and F_0 and ω_e are the driving force amplitude and frequency. All 5 of these parameters can be altered with the sliders. The solution to the differential equation is shown, governed by the boundary conditions of the displacement and velocity at t=0, which can be altered by sliding the points Y0 and V0 up and down the y-axis.

You can display the temporal behaviour of the displacement (blue), force (green), velocity (red) and acceleration (brown) by selecting the relevant checkbox. You can also show the "response curve" of the oscillator (the steady-state amplitude as a functionof the driving frequency) and the phase difference between the steady state displacement and the driving force.

Answer 2

Explanation:

The general one degree-of-freedom motion equation for the mass-spring system has the form

mx¨+kx=F(t)

Dividing this equation by m, we get

x¨+ω2nx=F(t)/m=f(t)

where ω2n=km

For harmonic excitation, F(t)=F0cos(ωt+Δ). In standard form, then

x¨+ω2nx=f0cos(ωt+Δ)

The particular answer to this differential equation is

xp(t)=f0ω2n−ω2cos(ωt+Δ)

So it all depends on how you write your force function. Either your equation should be

x¨+ω2nx=F0mcos(ωt+Δ)