[SOLVED] Complex Stiffness Solution

jfortiv · #1 $Old$ October 9th, 2009, 12:49 PM

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.

shawsend · #2 $Old$ October 9th, 2009, 04:06 PM

If I write it as:

$(D^2+\frac{k}{m})u=\frac{a}{m} e^{i\omega t}$

and apply the operator $(D^2+w^2)$ to both sides to annihilate the right side, and work through it, I get:

$u(t)=c_1 e^{i\sqrt{\frac{k}{m}} t}+c_2 e^{-i\sqrt{\frac{k}{m}}t}+\frac{a}{k-m\omega^2} e^{i\omega t}$ .

In general if $k\in\mathbb{C}$ then:

$\displaystyle{ e^{i\sqrt{\frac{k}{m}}t}=\text{Exp}\left[\pm it(r^{1/2}e^{i/2\Theta})\right],\quad \Theta=Arg(k/m) },\quad r=|k/m|$

however, the solution above already contains both roots of the square root so that in the solution, $\sqrt{\frac{k}{m}}=r^{1/2}e^{\frac{i}{2}\theta}$ , the principal value.

Also, if you're into Mathematica, here's the code to study a particular IVP numerically:

$u''+(1-\frac{i}{2})u=-\frac{1}{2}e^{it},\quad u(0)=1+i,\quad u'(0)=2-i$

Code:

eqn = Derivative[2][u][t] + (k/m)*u[t] == 
    (a/m)*Exp[I*w*t] //. {k -> 2 - I, 
    m -> 2, w -> 1, a -> -1}
sol = NDSolve[{eqn, u[0] == 1 + I, 
    Derivative[1][u][0] == 2 - I}, u, 
   {t, 0, 5}]
p1 = Plot[{Re[Evaluate[u[t] /. sol]], 
    Im[Evaluate[u[t] /. sol]]}, 
   {t, 0, 5}, PlotStyle -> {Red, Blue}]

And then the code to solve for the two constants and plot the analytic solution:

Code:

clist = First[{c1, c2} /. FullSimplify[
     Solve[{c1 + c2 + a/(k - m*w^2) == u, 
        c1*I*Sqrt[k/m] - c2*I*Sqrt[k/m] + 
          (I*w*a)/(k - m*w^2) == v}, 
       {c1, c2}] //. {k -> 2 - I, m -> 2, 
       w -> 1, a -> -1, u -> 1 + I, 
       v -> 2 - I}]]
solution = k1*Exp[I*Sqrt[k/m]*t] + 
    k2*Exp[(-I)*Sqrt[k/m]*t] + 
    (a/(k - m*w^2))*Exp[I*w*t] /. 
   {k -> 2 - I, m -> 2, w -> 1, a -> -1, 
    k1 -> clist[[1]], k2 -> clist[[2]]}
p2 = Plot[{Re[solution], Im[solution]}, 
   {t, 0, 5}, PlotRange -> 
    {{0, 5}, {-3, 3}}, PlotStyle -> 
    {Red, Blue}]
Show[{p1, p2}]

and then plots of the real component of the solution in red and complex component in blue (pretty neat I think-- I never worked one like this before

):

jfortiv · #3 $Old$ November 5th, 2009, 07:16 AM

Thanks for helping with this. I worked through your math. I never would have thought to apply an operator like that to both sides.

As it turns out, it doesn't make sense to use an imaginary stiffness in a time-domain solution. It's only applicable to frequency-domain. It was an interesting mathematical exercise anyway.

Thanks!