The differential equation for forced oscillation is:
$$m \ddot{x} + b\dot{x}+kx = F_{o}\sin(\omega''t)$$
I don't find this equation intuitively satisfying. My mind tends to think that as $F_{o}\sin(\omega''t)$ is the restoring force, it should be equal to the damping force $b\dot{x}$. However, that's not true (and I know that, somewhat mechanically).
If $m\ddot{x}$ is the total force on the oscillating body, then why is in the equation that equates to only the restoring force. Why is the $kx$ term there? How does the sign of those terms make sense?
I know I sound lost, and perhaps silly, but I really don't feel intuitively good about this equation.
Best Answer
It's better to write the differential equation for forced oscillations in a different way that makes it more relatable to Newton's second law:
$$m\ddot{x}=F_0\sin{(\omega''t)}-b\dot{x}-kx$$
Now we can break this equation down term by term.
$kx$ refers to the typical restoring force of simple harmonic motion given by Hooke's law, where $k$ is a positive force constant representing the magnitude of the restoring force. The sign of the term is negative because the direction of the force is oriented towards $x=0$, the equilibrium position of the oscillating body.
$b\dot{x}$ refers to the damping force of the oscillating system where $b$ is a positive constant representing the magnitude of the damping force. This typically represents friction of some other internal force that siphons kinetic energy away from the system to be converted into heat. The sign of this term is negative because the direction of the force opposes the direction of oscillation, similarly to how the friction tends slow something down by being directed opposite to its velocity.
$F_o\sin{(\omega''t)}$ refers to the driving force of the system. This typically represents some kind of external apparatus attached to our oscillating system (like a piston or an AC power source) which forces the system to keep oscillating rather than lose all its energy to the damping force described above. The sign on this term doesn't really matter since the driving force oscillates between $F_o$ and $-F_o$ due to the sine function.
Hopefully that all makes sense, does that help clarify things?