# Displacement of a voigt model and a mass in series

1. Feb 22, 2012

### jk1

1. The problem statement, all variables and given/known data

A Voigt model in series with a mass is given (so Wall: (mass and damper in parallel) -- mass -- Force ---> ) with a force, F = Fo*cos(ωt) exciting/driving the system.
Givens: Spring constant: K, Viscous constant: η, Mass element: m

Find a solution for established harmonic oscillations in the form a*cos(ωt) + b*sin(ωt).

2. Relevant equations

F = mx''
F = kx
F = ηx'

3. The attempt at a solution

Let x1 be the displacement of the system (I know the displacement will be the same on the damper and the spring - I don't see how the mass can have it's own displacement, so I'm assuming it

I labeled the forces on the spring and the damper both F1 and F2 respectively, and labeled the force on the mass F3.
F1 + F2 = F3 = Fo*cos(ωt)

I set kx + ηx' = Fo*cos(ωt) and solved the differential equation and got:

x(t) = Ae^(-t*k/η) + Fo*((k * cos(ωt) + η * ω * sin(ωt)) / (η^2 * ω^2 + k^2))

Then I solved using the initial condition x(0) = 0 because the voigt system shouldn't move instantaneously, and the mass shouldn't move unless the voigt model does.

A = -Fo(k / (η^2 * ω^2 + k))

But as you can see, there's an exponential term in my answer and that's not in the final form desired.

I tried finding a different differential equation via mx'' = kx + ηx' = Fo*cos(ωt) but I'm not having much luck - I'm pretty sure I need to have the forcing function involved or it won't work (so I can't just solve mx'' - ηx' -kx = 0).

Any suggestions?

Last edited: Feb 22, 2012