Four cases of a driven oscillator

<h3 style="text-align:center">Pure resonance $b=0,\quad\omega=\sqrt{k/m}$</h3>

<p>If an undamped oscillator is driven at its resonance frequency, $\Omega = \omega = \sqrt{k/m}$, the solution grows continuously with time. This is called a pure resonance. The differential equation that describes a pure resonance is,</p>

$$m\frac{d^2x}{dt^2} + kx = F_0\cos(\Omega t).\qquad (1)$$

<p>The solution is the sum of the homogeneous solution and the particular solution. The particular solution is,</p>

$$x_p(t) = \frac{F_0}{2\Omega m}t\sin (\Omega t).$$

<p>We can show that this is a solution by differentiating twice and substituting it into the differential equation (1).</p>

$$\frac{dx_p}{dt} = \frac{F_0}{2\Omega m}\sin (\Omega t)+\frac{F_0}{2 m}t\cos (\Omega t).$$

$$\frac{d^2x_p}{dt^2} = \frac{F_0}{2 m}\cos (\Omega t)+\frac{F_0}{2 m}\cos (\Omega t)-\frac{\Omega F_0}{2 m}t\sin (\Omega t).$$

<p>Because $k/m = \Omega^2$, $x_p$ solves Eq. (1).</p>

<p>The homogeneous solutions solve the differential equation,</p>

$$m\frac{d^2x}{dt^2} + kx = 0.\qquad (2)$$

<p>The homogeneous solutions have the form,</p>

$$x_h(t) = C_1\cos (\omega t) +C_2\sin (\omega t),$$

<p>where $\omega = \sqrt{k/m}$. The homogeneous solutions solve Eq. (2) for any value of $C_1$ and $C_2$ as can be demonstrated by substituting the homogeneous solution into the differential equation (2). The total solution is,</p>

$$x(t) = C_1\cos (\omega t) +C_2\sin (\omega t)+\frac{F_0}{2\Omega m}t\sin (\Omega t).$$

<p>The constants $C_1$ and $C_2$ can be determined using the initial conditions: $x_0$ is the position at $t=0$ and $v_{x0}$ is the velocity at $t=0$. Conveniently, $x_p(t=0) = 0$ and $\frac{dx_p}{dt}(t=0) = 0$ so the $C_1$ and $C_2$ are easily evaluated and the solution is, </p>

$$\bbox[10px, border: 1px solid black]{x(t) = x_0\cos (\omega t) +\frac{v_{x0}}{\omega}\sin (\omega t)+\frac{F_0}{2\Omega m}t\sin (\Omega t).}$$

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<h3 style="text-align:center">Underdamped $b^2\lt 4km$</h3>

<p>A damped, driven oscillator is described by the equation,</p>

$$m\frac{d^2x}{dt^2}+b\frac{dx}{dt} + kx = F_0\cos(\Omega t).\qquad (1)$$

$$x_p(t) = \frac{F_0}{\rho}\cos (\Omega t -\theta),$$

<p>where</p>

$$\rho = \sqrt{(k-m\Omega^2)^2+\Omega^2b^2}\hspace{1cm}\text{and}\hspace{1cm} \tan\theta = \frac{\Omega b}{k-m\Omega^2}.$$

<p>This particular solution was derived in the discussion of the <a href="http://lampz.tugraz.at/~hadley/physikm/script/oscillations/resonance.en.php">Resonance of a damped driven harmonic oscillator</a>. It can be shown that this is a solution by differentiating twice and substituting it into the differential equation (1). The homogeneous solutions solve the differential equation,</p>

$$m\frac{d^2x}{dt^2} +b\frac{dx}{dt} + kx = 0.\qquad (2)$$

<p>The homogeneous solutions have the form,</p>

$$x_h(t) = \exp \left(-t/\tau\right)\left(C_1\cos (\omega t) + C_2\sin(\omega t)\right),$$

<p>where

$$\omega=\frac{\sqrt{4mk-b^2}}{2m}\qquad\text{and}\qquad\tau=\frac{2m}{b}.$$

<p>The homogeneous solutions solve Eq. (2) for any value of $C_1$ and $C_2$ as can be demonstrated by substituting the homogeneous solution into the differential equation (2). The total solution is,</p>

$$x(t) = \exp \left(-t/\tau\right)\left(C_1\cos (\omega t) + C_2\sin(\omega t)\right) + \frac{F_0}{\rho}\cos (\Omega t -\theta),\\ v(t)= -\frac{\exp \left(-t/\tau\right)}{\tau}\left(C_1\cos (\omega t) + C_2\sin(\omega t)\right)+\exp \left(-t/\tau\right)\left(- C_1\omega\sin (\omega t) + C_2\omega\cos(\omega t)\right) - \frac{\Omega F_0}{\rho}\sin (\Omega t -\theta).$$

<p>The constants $C_1$ and $C_2$ can be determined using the initial conditions: $x_0$ is the position at $t=0$ and $v_{x0}$ is the velocity at $t=0$.</p>

$$C_1 = x_0 - \frac{F_0}{\rho}\cos ( \theta), \qquad C_2 = \frac{v_{x0}}{\omega}+ \frac{x_0}{\omega \tau}  - \frac{F_0}{\rho\omega\tau}\cos ( \theta)-\frac{F_0\Omega}{\rho\omega}\sin ( \theta).$$

<tr><td><p style="text-align:right; margin-right:0cm;">$b/\sqrt{4km}=$ </b><span id="Btext">0.1</span> </p></td><td><input type="button" value="-" onclick="data.slide_B.value=data.slide_B.value-0.01; plot_under();"><input type='range' min='0' max='0.99' step='0.01' name='slide_B' value='0.01' oninput="plot_under();" /><input type="button" value="+" onclick="data.slide_B.value=eval(data.slide_B.value) + 0.1; plot_under();"></td><td> </td><td><p style="text-align:right; margin-right:0cm;">$\Omega=$ </b><span id="Etext">1</span> [rad/s]</p></td><td><input type="button" value="-" onclick="data.slide_E.value=data.slide_E.value-0.1; plot_under();"><input type='range' min='0' max='10' step='0.1' name='slide_E' value='1' oninput="plot_under();" /><input type="button" value="+" onclick="data.slide_E.value=eval(data.slide_E.value) + 0.1; plot_under();"></td></tr>

<tr><td colspan = "2">$\omega=$ <span id="omegatext"></span> rad/s, $\tau=$ <span id="tautext"></span> s, $\rho=$ <span id="rhotext"></span> N/m, $\theta=$ <span id="thetatext"></span> rad, $b=$ <span id="b_text"></span> N s/m.</td><td> </td><td><p style="text-align:right; margin-right:0cm;">$v_{x0}=$ </b><span id="Gtext">1</span> [m/s]</p></td><td><input type="button" value="-" onclick="data.slide_G.value=data.slide_G.value-0.1; plot_under();"><input type='range' min='-2' max='2' step='0.1' name='slide_G' value='1' oninput="plot_under();" /><input type="button" value="+" onclick="data.slide_G.value=eval(data.slide_G.value) + 0.1; plot_under();"></td></tr>

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<h3 style="text-align:center">Critically damped  $b^2= 4km$</h3>

<p>A damped, driven oscillator is described by the equation,</p>

$$m\frac{d^2x}{dt^2}+b\frac{dx}{dt} + kx = F_0\cos(\Omega t).\qquad (1)$$

<p>If $b^2-4km = 0$, the system is critically damped and the solution is the sum of the homogeneous solution and the particular solution. The particular solution is,</p>

$$x_p(t) = \frac{F_0}{\rho}\cos (\Omega t -\theta),$$

<p>where</p>

$$\rho = \sqrt{(k-m\Omega^2)^2+\Omega^2b^2}\hspace{1cm}\text{and}\hspace{1cm} \tan\theta = \frac{\Omega b}{k-m\Omega^2}.$$

$$m\frac{d^2x}{dt^2} +b\frac{dx}{dt} + kx = 0.\qquad (2)$$

<p>The homogeneous solutions have the form,</p>

$$x(t) = C_1 \exp \left(-t/\tau\right)+C_2t \exp \left(-t/\tau\right),$$

<p>where</p>

$$\tau=\frac{2m}{b}.$$

$$x(t) = C_1 \exp \left(-t/\tau\right)+C_2t \exp \left(-t/\tau\right) + \frac{F_0}{\rho}\cos (\Omega t -\theta).$$

<p>The constants $C_1$ and $C_2$ can be determined using the initial conditions: $x_0$ is the position at $t=0$ and $v_{x0}$ is the velocity at $t=0$.</p>

$$C_1 = x_0-\frac{F_0}{\rho}\cos (\theta),\qquad C_2 = v_{x0} +\frac{C_1}{\tau} -\frac{\Omega F_0}{\rho}\sin ( \theta).$$

<tr><td><p style="text-align:right; margin-right:0cm;">$b=\sqrt{4km}=$ </b><span id="Btext">0.1</span> </p></td><td></td><td> </td><td><p style="text-align:right; margin-right:0cm;">$\Omega=$ </b><span id="Etext">1</span> [rad/s]</p></td><td><input type="button" value="-" onclick="data.slide_E.value=data.slide_E.value-0.1; plot_critically();"><input type='range' min='0' max='10' step='0.1' name='slide_E' value='1' oninput="plot_critically();" /><input type="button" value="+" onclick="data.slide_E.value=eval(data.slide_E.value) + 0.1; plot_critically();"></td></tr>

<tr><td colspan="2">$\tau=$ <span id="tautext"></span> s, $\rho=$ <span id="rhotext"></span> N/m, $\theta=$ <span id="thetatext"></span> rad</td><td> </td><td><p style="text-align:right; margin-right:0cm;">$v_{x0}=$ </b><span id="Gtext">1</span> [m/s]</p></td><td><input type="button" value="-" onclick="data.slide_G.value=data.slide_G.value-0.1; plot_critically();"><input type='range' min='-2' max='2' step='0.1' name='slide_G' value='1' oninput="plot_critically();" /><input type="button" value="+" onclick="data.slide_G.value=eval(data.slide_G.value) + 0.1; plot_critically();"></td></tr>

</table>

<h3 style="text-align:center">Overdamped  $b^2 > 4mk$</h3>

<p>A damped, driven oscillator is described by the equation,</p>

$$m\frac{d^2x}{dt^2}+b\frac{dx}{dt} + kx = F_0\cos(\Omega t).\qquad (1)$$

<p>If $b^2-4km > 0$, the system is overdamped and the solution is the sum of the homogeneous solution and the particular solution. The particular solution is,</p>

$$x_p(t) = \frac{F_0}{\rho}\cos (\Omega t -\theta),$$

<p>where</p>

$$\rho = \sqrt{(k-m\Omega^2)^2+\Omega^2b^2}\hspace{1cm}\text{and}\hspace{1cm} \tan\theta = \frac{\Omega b}{k-m\Omega^2}.$$

$$m\frac{d^2x}{dt^2} +b\frac{dx}{dt} + kx = 0. \qquad (2)$$

<p>The homogeneous solutions have the form,</p>

$$x(t) = C_1 \exp \left(- t/\tau_1\right)+ C_2 \exp \left(- t/\tau_2\right),$$

<p>where</p>

$$ \tau_1 = \frac{2m}{b-\sqrt{b^2-4km}},\qquad\tau_2 = \frac{2m}{b+\sqrt{b^2-4km}}.$$

$$x(t) = C_1 \exp \left(- t/\tau_1\right)+ C_2 \exp \left(- t/\tau_2\right) + \frac{F_0}{\rho}\cos (\Omega t -\theta).$$

<p>The constants $C_1$ and $C_2$ can be determined using the initial conditions: $x_0$ is the position at $t=0$ and $v_{x0}$ is the velocity at $t=0$.</p>

$$C_1 = \frac{\tau_1}{\tau_1-\tau_2}\left(\tau_2v_{x0} +x_0 - \frac{\tau_2\Omega F_0}{\rho}\sin (\theta)-\frac{F_0}{\rho}\cos (\theta)\right),$$

$$C_2 =  \frac{\tau_2}{\tau_2-\tau_1}\left(\tau_1v_{x0} +x_0  - \frac{\tau_1\Omega F_0}{\rho}\sin (\theta)-\frac{F_0}{\rho}\cos (\theta)\right).$$

<tr><td><p style="text-align:right; margin-right:0cm;">$b/\sqrt{4km}=$ </b><span id="Btext">1</span> </p></td><td><input type="button" value="-" onclick="data.slide_B.value=data.slide_B.value-0.1; plot_over();"><input type='range' min='1.01' max='20' step='0.1' name='slide_B' value='1' oninput="plot_over();" /><input type="button" value="+" onclick="data.slide_B.value=eval(data.slide_B.value) + 0.1; plot_over();"></td><td> </td><td><p style="text-align:right; margin-right:0cm;">$\Omega=$ </b><span id="Etext">1</span> [rad/s]</p></td><td><input type="button" value="-" onclick="data.slide_E.value=data.slide_E.value-0.1; plot_over();"><input type='range' min='0' max='10' step='0.1' name='slide_E' value='1' oninput="plot_over();" /><input type="button" value="+" onclick="data.slide_E.value=eval(data.slide_E.value) + 0.1; plot_over();"></td></tr>

<tr><td colspan="2">$\tau_1=$ <span id="tau1text"></span> s, $\tau_2=$ <span id="tau2text"></span> s, $\rho=$ <span id="rhotext"></span> N/m, $\theta=$ <span id="thetatext"></span> rad $b=$ <span id="b_text"></span> N s/m</td><td> </td><td><p style="text-align:right; margin-right:0cm;">$v_{x0}=$ </b><span id="Gtext">1</span> [m/s]</p></td><td><input type="button" value="-" onclick="data.slide_G.value=data.slide_G.value-0.1; plot_over();"><input type='range' min='-2' max='2' step='0.1' name='slide_G' value='1' oninput="plot_over();" /><input type="button" value="+" onclick="data.slide_G.value=eval(data.slide_G.value) + 0.1; plot_over();"></td></tr>

</table>
<table align="center"><tr><td>
 $t$ [s] $x$ [m] $v_x$ [m/s]<br />
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	Physik für Geodäsie 511.018 / Physik M 513.805