[SOLVED] Complex Stiffness Solution

Hi,

I'm trying to solve a differential equation of a 1 degree of freedom mechanical system with a complex stiffness:

m*d^2u/dt^2 + k*u = A*e^(i*w*t), where k is complex (k = a + i*b), i = sqrt(-1)

First, I assume a solution for u of the form u = u_mag*e^(r*t), thus d^2u/dt^2 = r^2*u_mag*e^(r*t), and I use this to solve the homogeneous equation, getting the characteristic equation for r:

m*r^2 + k = 0

Solving for r yields:

r = +/- sqrt(-k/m) = +/- sqrt( - ( a+ i*b) / m) = +/- i * sqrt(a/m + i*b/m)

So u = u_mag * e^( i * sqrt(a/m + i*b/m))

or

u = u_mag * e^(- i * sqrt(a/m + i*b/m))

At this point I am stuck. I want to use Euler's formula to rewrite u in terms of sin and cos, but I can't isolate the real and imaginary terms in the exponent, such as in the following form to which I am accustomed:

e^(s + i*t) = e^(s) * e^(i*t)

Can someone offer some insight or point out the error of my methodology?

Thanks.