Simplifying Analysis Using Complex Numbers

We hold a clear perception that complex number is just a tool, use it only when it is useful, otherwise throw it away. One of the most common and vital applications is about TMD, which will be posted on other articles.

Here we focus on the solution of SDOF subjected separately to two harmonic loadings.🫡
$$
\ddot x+2\zeta w_1\dot x+w_1^2x=P_0coswt…(1)
\\ \ddot y+2\zeta w_1 \dot y+w_1^2y = P_0sinwt…(2)
$$
Instead of solving for \(x(t)\) and \(y(t)\) individually, let`s define the complex-valued function and solve it at once.
$$
z(t) = x(t) +iy(t)
$$
It turns out that the algebra involved is much easier, to such an extent that this approach almost becomes the standard procedure for steady state problems or more generally frequency domain approach.

Taking (1)+(2)×i gives:
$$
(\ddot x+i\ddot y) + 2\zeta w_1(\dot x+i\dot y)+w_1^2(x+iy)=p_0(coswt+isinwt)
\\ \Rightarrow\ddot z+2\zeta w_1\dot z +w_1^2z=p_0e^{iwt}…(3)
$$
Let the steady state response of z(t) be of the form:
$$
z(t)=Ze^{iwt}
$$
Where Z is a complex number to be determined. The important thing is this form implies:
$$
\dot z(t) = iwZe^{iwt};\ \ddot z(t) = (iw)^2Ze^{iwt} = -w^2Ze^{iwt}
$$
Subs. into (3) and the result must hold for all t:
\[
(w_1^2-w^2+2\zeta ww_1i)Ze^{iwt}=p_0e^{iwt}
\\ \Rightarrow Z=p_0 × \underbrace{\frac{1}{w_1^2-w^2+2\zeta ww_1i}} _ {h(w)}=p_0h(w)
\]
Where h(w) is named as frequency response function (FRF). And we are quite familiar with it, thus we can rewrite its modulus to link dynamic ampification \(D(w)\).
\[
h(w)=|h(w)|e^{-i\phi};\ \beta=\frac{w}{w_1}
\|h(w)| = \frac{1}{w_1^2}×\underbrace{\frac{1}{\sqrt{(1-\beta^2)^2+(2\zeta\beta)^2}}}_ {D(w)}= \frac{D(w)}{w_1^2}
\]
The phase of FRF is related to the phase lag of response from the harmonic load:
\[
∠ h(w) = -\phi\Rightarrow \phi = tan^{-1}\frac{2\zeta\beta}{1-\beta^2}
\]
Hence, subs. into \(z(t)=Ze^{iwt}\):
\[
z(t) = \underbrace{\frac{p_0}{w_1^2}} _ {static}×\underbrace{D(w)}_ {dyn.amp.}×\underbrace{e^{i(wt-\phi)}}_{phase}
\]
Taking real and imaginary part of z(t) gives the same expressions of x(t) and y(t) derived in the SDOF lecture notes, but now the algebra is much simpler:
\[
x(t) = Re\ z(t)=\frac{p_0}{w_1^2}D(w)cos(wt-\phi)
\\ y(t) = Im\ z(t)=\frac{p_0}{w_1^2}D(w)sin(wt-\phi)
\]

Euler`s identity: \(e^{i\theta}=cos\theta+isin\theta,\ |e^{i\theta}|=1\)

How we come up with it?

From laurent series: \(e^{i\theta}=1+\frac{i\theta}{1!}+\frac{(i\theta)^2}{2!}+\frac{(i\theta)^3}{3!}+… \)

We can separate it into 2 parts:

\(Re = 1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-…=cos\theta\)

\(Im = \frac{i\theta}{1!}-\frac{i\theta^3}{3!}+\frac{i\theta^5}{5!}-…=isin\theta\)

All above process is like magic, right? But don`t be too excited, as we mentioned at the beginning, complex number is just a tool so they don’t work in all cases, throw it away when it cannot help.😂