Harmonic Load in the Time and Frequency Domains

[CivilSigma]

Homework Statement

For any harmonic load:

$$F(t)=F_0\cdot \sin(\omega t)$$

What is the corresponding Frequency domain equivalent?

My lecture notes is suggesting:

$$ F(t)=F_0 \cdot e^{i \omega t} $$

But I am failing to see how they are equal?

The lesson is about Stochastic Response of single degree of freedom structures and how to obtain the Transfer function from solving the dynamic equation of motion ( I could provide more detail if it is necessary, but it is the above giving me a hard time)

Thank you

 

[scottdave]

I'm on my phone. If you search for Euler's relationship and look how sine cosine and e are related.

 

[CivilSigma]

Well,

$$\sin x = \frac{e^{ix}-e^{-ix}}{2}$$

But my lecture notes is only using one exponential function to replace the sin function.

 

[scottdave]

Actually there is an $i$ in the denominator, like this:
$$\sin x = \frac{e^{ix}-e^{-ix}}{2i}$$
It is a possibility that they wanted you to take the imaginary portion of $e^{i\omega t}$, since $e^{ix} = cos (x) + i sin (x)$

 

[rude man]

Homework Statement

For any harmonic load:
$$F(t)=F_0\cdot \sin(\omega t)$$
What is the corresponding Frequency domain equivalent?
My lecture notes is suggesting:
$$ F(t)=F_0 \cdot e^{i \omega t} $$
But I am failing to see how they are equal?
The lesson is about Stochastic Response of single degree of freedom structures and how to obtain the Transfer function from solving the dynamic equation of motion ( I could provide more detail if it is necessary, but it is the above giving me a hard time)
Thank you

Both are time domain.
A frequency domain expression of sin(wt) would be a Laplace, Fourier, or other frequency transform (the simplest is for steady-state situations for which that transform would be just F_0., known as a phasor. Or more commonly as F₀/√2.