From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- source/know/concept/harmonic-oscillator/index.md | 287 +++++++++++++++++++++++ 1 file changed, 287 insertions(+) create mode 100644 source/know/concept/harmonic-oscillator/index.md (limited to 'source/know/concept/harmonic-oscillator') diff --git a/source/know/concept/harmonic-oscillator/index.md b/source/know/concept/harmonic-oscillator/index.md new file mode 100644 index 0000000..f17203b --- /dev/null +++ b/source/know/concept/harmonic-oscillator/index.md @@ -0,0 +1,287 @@ +--- +title: "Harmonic oscillator" +date: 2021-06-02 +categories: +- Physics +- Mathematics +layout: "concept" +--- + +A **harmonic oscillator** obeys +the simple 1D version of [Hooke's law](/know/concept/hookes-law/): +to displace the system away from its equilibrium, +the needed force $F_d(x)$ scales linearly with the displacement $x(t)$: + +$$\begin{aligned} + F_d(x) = k x +\end{aligned}$$ + +Where $k$ is a system-specific proportionality constant, +called the **spring constant**, +since a spring is a good example of a harmonic oscillator, +at least for small displacements. +Hooke's law is also often stated for +the restoring force $F_r(x)$ instead: + +$$\begin{aligned} + F_r(x) = - k x +\end{aligned}$$ + +Let a mass $m$ be attached to the end of the spring. +After displacing it, we let it go $F_d = 0$, +so Newton's second law for the restoring force $F_r$ demands that: + +$$\begin{aligned} + F_r = m x'' +\end{aligned}$$ + +But $F_r = - k x$, +meaning $m x'' = - k x$, +leading to the following equation for $x(t)$: + +$$\begin{aligned} + \boxed{ + x'' + \omega_0^2 x = 0 + } +\end{aligned}$$ + +Where $\omega_0 \equiv \sqrt{k / m}$ is the **natural frequency** of the system. +This differential equation has the following general solution: + +$$\begin{aligned} + \boxed{ + x(t) + = C_1 \sin(\omega_0 t) + C_2 \cos(\omega_0 t) + } +\end{aligned}$$ + +Where $C_1$ and $C_2$ are constants determined by the initial conditions. +For example, for $x(0) = 1$ and $x'(0) = 0$, the solution becomes: + +$$\begin{aligned} + x(t) = \cos(\omega_0 t) +\end{aligned}$$ + +When using [Lagrangian](/know/concept/lagrangian-mechanics/) +or Hamiltonian mechanics, +we need to know the potential energy $V(x)$ +added to the system by a displacement to $x$. +This equals the work done by the displacement, +and is therefore given by: + +$$\begin{aligned} + V(x) = \int_0^x F_d(x) \:dx = \frac{1}{2} k x^2 = \frac{1}{2} m \omega_0^2 x^2 +\end{aligned}$$ + + +## Damped oscillation + +If there is a **friction force** $F_f$ affecting the system, +then the oscillation amplitude will decrease, +or it might not oscillate at all. +We define $F_f$ using a **viscous damping coefficient** $c$: + +$$\begin{aligned} + F_f = - c x' +\end{aligned}$$ + +Both $F_r$ and $F_f$ are acting on the system, +so Newton's second law states that: + +$$\begin{aligned} + m x'' = - c x' - k x +\end{aligned}$$ + +This can be rewritten in the following conventional form +by defining the **damping coefficient** $\zeta \equiv c / (2 \sqrt{m k})$, +which determines the expected behaviour of the system: + +$$\begin{aligned} + \boxed{ + x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = 0 + } +\end{aligned}$$ + +The general solution is found from the roots $u$ of the auxiliary quadratic equation: + +$$\begin{aligned} + u^2 + 2 \zeta \omega_0 u + \omega_0^2 = 0 +\end{aligned}$$ + +The discriminant $D = 4 \zeta^2 \omega_0^2 - 4 \omega_0^2$ +tells us that the behaviour changes substantially +depending on the damping coefficient $\zeta$, +with three possibilities: $\zeta < 1$ or $\zeta = 1$ or $\zeta > 1$. + +If $\zeta < 1$, there is **underdamping**: +the system oscillates with exponentially decaying +amplitude and reduced frequency $\omega_1 \equiv \omega_0 \sqrt{1 - \zeta^2}$. +The general solution is: + +$$\begin{aligned} + \boxed{ + x(t) + = \big( C_1 \sin(\omega_1 t) + C_2 \cos(\omega_1 t) \big) \exp(- \zeta \omega_0 t) + } +\end{aligned}$$ + +If $\zeta = 1$, there is **critical damping**: +the system returns to its equilibrium point in minimum time. +The general solution is given by: + +$$\begin{aligned} + \boxed{ + x(t) + = \big( C_1 + C_2 t \big) \exp(- \zeta \omega_0 t) + } +\end{aligned}$$ + +If $\zeta > 1$, there is **overdamping**: +the system returns to equilibrium slowly. +The general solution is as follows, +where $\omega_1 \equiv \omega_0 \sqrt{\zeta^2 - 1}$: + +$$\begin{aligned} + \boxed{ + x(t) + = \big( C_1 \exp(\omega_1 t) + C_2 \exp(- \omega_1 t) \big) \exp(- \zeta \omega_0 t) + } +\end{aligned}$$ + + +## Forced oscillation + +In the differential equations given above, +the right-hand side has always been zero, +meaning that the oscillator is not affected by any external forces. +What if we put a function there? + +$$\begin{aligned} + x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = f(t) +\end{aligned}$$ + +Obviously, there exist infinitely many $f(t)$ to choose from, +and each needs a separate analysis. +However, there is one type of $f(t)$ that deserves special mention, +namely sinusoids: + +$$\begin{aligned} + \boxed{ + x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = \frac{F}{m} \cos(\omega t + \chi) + } +\end{aligned}$$ + +Where $F$ is a constant force, $\chi$ is an arbitrary phase, +and the frequency $\omega$ is not necessarily $\omega_0$. +We solve this case for $x(t)$ in detail. +Consider the complex version of the equation: + +$$\begin{aligned} + X'' + 2 \zeta \omega_0 X' + \omega_0^2 X = \frac{F}{m} \exp\!\big(i (\omega t + \chi)\big) +\end{aligned}$$ + +Then $x(t) = \Real\{X(t)\}$. +Inserting the ansatz $X(t) = C \exp(i \omega t)$, +for some constant $C$: + +$$\begin{aligned} + - C \omega^2 + C 2 i \zeta \omega_0 \omega + C \omega_0^2 = \frac{F}{m} \exp(i \chi) +\end{aligned}$$ + +Where $\exp(i \omega t)$ has already been divided out. +We isolate this equation for $C$: + +$$\begin{aligned} + C + = \frac{F}{m \big((\omega_0^2 - \omega^2) + 2 i \zeta \omega_0 \omega\big)} \exp(i \chi) + = \frac{F \big((\omega_0^2 - \omega^2) - 2 i \zeta \omega_0 \omega\big)} + {m \big((\omega_0^2 - \omega^2)^2 + 4 \zeta^2 \omega_0^2 \omega^2\big)} + \exp(i \chi) +\end{aligned}$$ + +We would like to rewrite this in polar form $C = r \exp(i \theta)$, +which turns out to be as follows: + +$$\begin{aligned} + C + &= \frac{F}{m \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \zeta^2 \omega_0^2 \omega^2}} + \exp\!\bigg(i \chi - i \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big)\bigg) +\end{aligned}$$ + +For brevity, let us define the **impedance** $Z$ +and the **phase shift** $\phi$ +in the following way: + +$$\begin{aligned} + Z + \equiv \sqrt{(\omega_0^2 - \omega^2)^2 / \omega^2 + 4 \zeta^2 \omega_0^2} + \qquad \quad + \phi + \equiv \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big) +\end{aligned}$$ + +Returning to the original ansatz $X(t) = C \exp(i \omega t)$, +we take its real part to find $x(t)$: + +$$\begin{aligned} + \boxed{ + x(t) + = \frac{F}{m \omega Z} \sin(\omega t + \chi - \phi) + } +\end{aligned}$$ + +Two things are noteworthy here. +Firstly, $f(t)$ and $x(t)$ are out of phase by $\phi$; there is some lag. +This is caused by damping, because if $\zeta = 0$, it disappears $\phi = 0$. + +Secondly, the amplitude of $x(t)$ depends on $\omega$ and $\omega_0$. +This brings us to **resonance**, +where the amplitude can become extremely large. +Actually, resonance has two subtly different definitions, +depending on which one of $\omega$ and $\omega_0$ is a free parameter, +and which one is fixed. + +If the natural $\omega_0$ is fixed and the driving $\omega$ is variable, +we find for which $\omega$ resonance occurs by minimizing the amplitude denominator $\omega Z$. +We thus find: + +$$\begin{aligned} + 0 + = \dv{(\omega Z)}{\omega} + = \frac{- 4 \omega_0^2 \omega + 4 \omega^3 + 8 \zeta^2 \omega_0^2 \omega}{2 \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \zeta^2 \omega_0^2 \omega^2}} + \quad \implies \quad + \boxed{ + \omega = \omega_0 \sqrt{1 - 2 \zeta^2} + } +\end{aligned}$$ + +Meaning the resonant $\omega$ is lower than $\omega_0$, +and resonance can only occur if $\zeta < 1 / \sqrt{2}$. + +However, if the driving $\omega$ is fixed and the natural is $\omega_0$ is variable, +the problem is bit more subtle: +the damping coefficient $\zeta = c / (2 m \omega_0)$ +depends on $\omega_0$. +This leads us to: + +$$\begin{aligned} + 0 + = \dv{(\omega Z)}{\omega_0} + = \frac{4 \omega_0^3 - 4 \omega^2 \omega_0}{2 \sqrt{(\omega_0^2 - \omega^2)^2 + c^2 \omega^2 / m^2}} + \quad \implies \quad + \boxed{ + \omega_0 = \omega + } +\end{aligned}$$ + +Surprisingly, the damping does not affect $\omega_0$, if $\omega$ is given. +However, in both cases, the damping *does* matter for the eventual amplitude: +$c \to 0$ leads to $x \to \infty$, +and resonance disappears or becomes negligible for $c \to \infty$. + + + +## References +1. M.L. Boas, + *Mathematical methods in the physical sciences*, 2nd edition, + Wiley. -- cgit v1.2.3