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diff --git a/source/know/concept/harmonic-oscillator/index.md b/source/know/concept/harmonic-oscillator/index.md index 6729dc2..6d9eac0 100644 --- a/source/know/concept/harmonic-oscillator/index.md +++ b/source/know/concept/harmonic-oscillator/index.md @@ -11,34 +11,34 @@ 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 $x(t)$: +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, +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: +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: +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 $x(t)$: +But $$F_r = - k x$$, +meaning $$m x'' = - k x$$, +leading to the following equation for $$x(t)$$: $$\begin{aligned} \boxed{ @@ -46,7 +46,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where $\omega_0 \equiv \sqrt{k / m}$ is the **natural frequency** of the system. +Where $$\omega_0 \equiv \sqrt{k / m}$$ is the **natural frequency** of the system. This differential equation has the following general solution: $$\begin{aligned} @@ -56,8 +56,8 @@ $$\begin{aligned} } \end{aligned}$$ -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: +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: $$\begin{aligned} x(t) = \cos(\omega_0 t) @@ -65,8 +65,8 @@ $$\begin{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 $x$. +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: @@ -77,16 +77,16 @@ $$\begin{aligned} ## Damped oscillation -If there is a **friction force** $F_f$ affecting the system, +If there is a **friction force** $$F_f$$ affecting the system, then the oscillation amplitude will decrease, or it might not oscillate at all. -We define $F_f$ using a **viscous damping coefficient** $c$: +We define $$F_f$$ using a **viscous damping coefficient** $$c$$: $$\begin{aligned} F_f = - c x' \end{aligned}$$ -Both $F_r$ and $F_f$ are acting on the system, +Both $$F_r$$ and $$F_f$$ are acting on the system, so Newton's second law states that: $$\begin{aligned} @@ -94,7 +94,7 @@ $$\begin{aligned} \end{aligned}$$ This can be rewritten in the following conventional form -by defining the **damping coefficient** $\zeta \equiv c / (2 \sqrt{m k})$, +by defining the **damping coefficient** $$\zeta \equiv c / (2 \sqrt{m k})$$, which determines the expected behaviour of the system: $$\begin{aligned} @@ -103,20 +103,20 @@ $$\begin{aligned} } \end{aligned}$$ -The general solution is found from the roots $u$ of the auxiliary quadratic equation: +The general solution is found from the roots $$u$$ of the auxiliary quadratic equation: $$\begin{aligned} u^2 + 2 \zeta \omega_0 u + \omega_0^2 = 0 \end{aligned}$$ -The discriminant $D = 4 \zeta^2 \omega_0^2 - 4 \omega_0^2$ +The discriminant $$D = 4 \zeta^2 \omega_0^2 - 4 \omega_0^2$$ tells us that the behaviour changes substantially -depending on the damping coefficient $\zeta$, -with three possibilities: $\zeta < 1$ or $\zeta = 1$ or $\zeta > 1$. +depending on the damping coefficient $$\zeta$$, +with three possibilities: $$\zeta < 1$$ or $$\zeta = 1$$ or $$\zeta > 1$$. -If $\zeta < 1$, there is **underdamping**: +If $$\zeta < 1$$, there is **underdamping**: the system oscillates with exponentially decaying -amplitude and reduced frequency $\omega_1 \equiv \omega_0 \sqrt{1 - \zeta^2}$. +amplitude and reduced frequency $$\omega_1 \equiv \omega_0 \sqrt{1 - \zeta^2}$$. The general solution is: $$\begin{aligned} @@ -126,7 +126,7 @@ $$\begin{aligned} } \end{aligned}$$ -If $\zeta = 1$, there is **critical damping**: +If $$\zeta = 1$$, there is **critical damping**: the system returns to its equilibrium point in minimum time. The general solution is given by: @@ -137,10 +137,10 @@ $$\begin{aligned} } \end{aligned}$$ -If $\zeta > 1$, there is **overdamping**: +If $$\zeta > 1$$, there is **overdamping**: the system returns to equilibrium slowly. The general solution is as follows, -where $\omega_1 \equiv \omega_0 \sqrt{\zeta^2 - 1}$: +where $$\omega_1 \equiv \omega_0 \sqrt{\zeta^2 - 1}$$: $$\begin{aligned} \boxed{ @@ -161,9 +161,9 @@ $$\begin{aligned} x'' + 2 \zeta \omega_0 x' + \omega_0^2 x = f(t) \end{aligned}$$ -Obviously, there exist infinitely many $f(t)$ to choose from, +Obviously, there exist infinitely many $$f(t)$$ to choose from, and each needs a separate analysis. -However, there is one type of $f(t)$ that deserves special mention, +However, there is one type of $$f(t)$$ that deserves special mention, namely sinusoids: $$\begin{aligned} @@ -172,25 +172,25 @@ $$\begin{aligned} } \end{aligned}$$ -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. +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 $C$: +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 $C$: +Where $$\exp(i \omega t)$$ has already been divided out. +We isolate this equation for $$C$$: $$\begin{aligned} C @@ -200,7 +200,7 @@ $$\begin{aligned} \exp(i \chi) \end{aligned}$$ -We would like to rewrite this in polar form $C = r \exp(i \theta)$, +We would like to rewrite this in polar form $$C = r \exp(i \theta)$$, which turns out to be as follows: $$\begin{aligned} @@ -209,8 +209,8 @@ $$\begin{aligned} \exp\!\bigg(i \chi - i \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big)\bigg) \end{aligned}$$ -For brevity, let us define the **impedance** $Z$ -and the **phase shift** $\phi$ +For brevity, let us define the **impedance** $$Z$$ +and the **phase shift** $$\phi$$ in the following way: $$\begin{aligned} @@ -221,8 +221,8 @@ $$\begin{aligned} \equiv \arctan\!\Big(\frac{2 \zeta \omega_0 \omega}{\omega_0^2 - \omega^2}\Big) \end{aligned}$$ -Returning to the original ansatz $X(t) = C \exp(i \omega t)$, -we take its real part to find $x(t)$: +Returning to the original ansatz $$X(t) = C \exp(i \omega t)$$, +we take its real part to find $$x(t)$$: $$\begin{aligned} \boxed{ @@ -232,18 +232,18 @@ $$\begin{aligned} \end{aligned}$$ 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 $\phi = 0$. +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 $\omega_0$. +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, +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 $\omega Z$. +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: $$\begin{aligned} @@ -256,13 +256,13 @@ $$\begin{aligned} } \end{aligned}$$ -Meaning the resonant $\omega$ is lower than $\omega_0$, -and resonance can only occur if $\zeta < 1 / \sqrt{2}$. +Meaning the resonant $$\omega$$ is lower than $$\omega_0$$, +and resonance can only occur if $$\zeta < 1 / \sqrt{2}$$. -However, if the driving $\omega$ is fixed and the natural is $\omega_0$ is variable, +However, if the driving $$\omega$$ is fixed and the natural is $$\omega_0$$ is variable, the problem is bit more subtle: -the damping coefficient $\zeta = c / (2 m \omega_0)$ -depends on $\omega_0$. +the damping coefficient $$\zeta = c / (2 m \omega_0)$$ +depends on $$\omega_0$$. This leads us to: $$\begin{aligned} @@ -275,10 +275,10 @@ $$\begin{aligned} } \end{aligned}$$ -Surprisingly, the damping does not affect $\omega_0$, if $\omega$ is given. +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 $c \to \infty$. +$$c \to 0$$ leads to $$x \to \infty$$, +and resonance disappears or becomes negligible for $$c \to \infty$$. |