I was reading Halliday's section on Damped Simple Harmonic Motion, which stated that this equation:
$$-b\dot{x} – kx = m \ddot{x}$$
Is the differential equation that dictates the displacement of the object, and $b$ is the damping constant of the system.
The author claims that the solution of the equation is:
$$
x(t)=x_m \mathrm e^{-bt/2m}\cos(\omega't+\phi)
$$
Suppose we want to set initial conditions $x(0) = x_0$ and $x'(0) = 0$. Giving the equations
$$
x_m \cos(\phi) = x_0
$$
$$
-\frac{x_m b}{2m} \cos(\phi) – x_m \omega' \sin(\phi) = 0 \implies \frac{b}{2m} \cos(\phi)+\omega' \sin(\phi) = 0
$$
Physically, I would love to say that the initial conditions imposed are, a priori, the maximum amplitude and the initial velocity. If that is correct, then the first equation would imply that $x_m = x_0$ and $\cos(\phi) = 1$ (and therefore $\sin(\phi) = 0$)
However, that does not makes sense, since if we substitute these values into the second equation, we obtain
$$
\frac{b}{2m} = 0
$$
But both $m$(mass) and $b$ (damping constant) are non-zero. What is wrong with my intuition?
Also, if we begin with the second equation and solve for $\phi$, we obtain $\phi = \tan^{-1}\left(\frac{-b}{2m\omega'}\right)$. And then we would be bound to accept that $x_m = \frac{x_0}{\cos(\phi)}$. If correct, then what does $x_m$ represent, if not just a mathematical constant?
Best Answer
Here $x_m$ represents the displacement of the undamped oscillator; with right initial conditions you can equate that to maximum displacement, i.e, amplitude. Where as the product $x_m e^{-bt/2m}$ is the decaying amplitude of the damped oscillator.
Given initial condition, $x(t=0) = x_0$, you get
$$ x_m = \frac{x_0}{\cos(\phi)} $$
and the initial condition $x'(t=0) = 0$ means the initial velocity of the oscillator is zero, during which the there is no damping takes place, because damping factor $f \propto -\dot{x}$, because of this initial condition you are getting
$$ \frac{b}{2m} = 0 \implies b = 0 $$
the above tells that damping takes place once the oscillator starts moving.