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#### skatenerd

##### Active member

- Oct 3, 2012

- 114

The problem is to consider an undamped (no friction) forced mass-spring system. The forcing is given by $$F(t)=F_o\cos{\omega_ft}$$

The general ODE for this would be $$\ddot{x}+(0)\dot{x}+\omega_o^2x=f_o\cos{\omega_ft}$$

The problem requires us to consider the resonant case (\(\omega_o=\omega_f\)) and the non-resonant case (\(\omega_o\neq\omega_f\)).

To acquire these two solutions, I began by using Euler's Identity to rewrite our ODE into one using exponentials rather than trigonometric functions on the right side, which was an encouraged step by the professor.

Using \(x+iy=z\), I turned

$$\ddot{x}+\omega_o^2x=f_o\cos{\omega_ft}$$

into

$$\ddot{z}+\omega_o^2z=f_oe^{i\omega_ft}$$

(the \(y\) part of the identity is just with a sine function on the right hand side and the variable is changed to \(y\), with the whole thing multiplied by \(i\).)

Solving the differential equations for the resonant case gave

$$z(t)=C_1e^{-i\omega{t}}+C_2e^{i\omega{t}}-\frac{if_o}{2\omega}e^{i\omega{t}}$$

and the non-resonant case

$$z(t)=C_1e^{-i\omega_o{t}}+C_2e^{i\omega_o{t}}+\frac{f_o}{ \omega_o^2- \omega_f^2}e^{i\omega_f{t}}$$

Now I have kind of a simple problem for the amount of stuff I just typed out...but I'm not quite sure how to find the solution in terms of \(x(t)\).

I know that I need to take the real part of \(z(t)\), but I am somewhat new to working with complex functions like this so I don't really know how to do that.