Displacement of a voigt model and a mass in series

[jk1]

Homework Statement

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).

Homework Equations

F = mx''
F = kx
F = ηx'

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?

 

[KataKoniK]

Hello,

Your approach seems to be on the right track. One suggestion I have is to consider the equation mx'' + ηx' + kx = Fo*cos(ωt), which includes the mass element in the equation. This is because the mass will also experience a force from the spring and damper, not just the force from the external force.

Next, you can use the method of undetermined coefficients to solve for the particular solution. This involves assuming a solution in the form of a*cos(ωt) + b*sin(ωt) and then plugging it into the equation to solve for a and b.

Once you have the particular solution, you can combine it with the general solution you already found to get the final form. This should give you an answer without the exponential term.

Hope this helps!