# Chapter 3 - Ordinary Differential Equations¶

### (3.1) Consider the motion of a damped, driven harmonic oscillator (such as a mass on a spring, a ball in a well, or a pendulum making small motions): $m\ddot{x}+\gamma\dot{x}+kx=e^{i\omega t}.$¶

#### (a) Under what conditions will the governing equations for small displacements of a particle around an arbitrary 1D potential minimum be simple undamped harmonic motion?¶

From Newton's second law: $F=m\ddot{x}$.

For an 1D potential $V(x)$ we can calculate the force acting on a particle as $F=-\frac{dV(x)}{dx}$.

Assuming small perturbations ($x\rightarrow 0$), we can expand the potential as a Taylor series and keep the terms up to second order. Expanding around the minimum at $x^*$, $$V(x) = V(x)\big|_{x=x^*} + \frac{dV}{dx}x\Big|_{x=x^*} + \frac{1}{2}\frac{d^2V}{dx^2}x^2\Big|_{x=x^*} + O(x^3) = V(x)\big|_{x=x^*} + \frac{1}{2}\frac{d^2V}{dx^2}x^2\Big|_{x=x^*}$$, as $\frac{dV}{dx}x\Big|_{x=x^*} = 0$ since $x^*$ is a minimum point.

Combining the above, we have $$m\ddot{x} = -\frac{d^2V}{dx^2}x\Big|_{x=x^*} \Rightarrow m\ddot{x} + kx = 0$$ , with $k = -\frac{d^2V}{dx^2}\Big|_{x=x^*}$.

Which is the equation for a simple, undamped harmonic oscillator under the condition that $k\neq0$, i.e. the second derivative at the minimum should be non-zero.

#### (b) Find the solution to the homogeneous equation, and comment on the possible cases. How does the amplitude depend on the frequency?¶

Deviding by the mass, we have $$\ddot{x} + \alpha\dot{x} + \omega^2 = 0$$, where $\alpha=\frac{\gamma}{m}$ and $\omega=\sqrt{\frac{k}{m}}$.

Assuming the solution has the form $e^{\lambda t}$, and substituting above we get $$\lambda^2 + \alpha\lambda + \omega^2 = 0$$, which has the solution $$\lambda = \frac{-\alpha \pm \sqrt{\alpha^2-4\omega^2}}{2}$$

Depending on whether $\alpha^2 >,<,= 4\omega^2$ we have the well-known three cases of the harmonic oscillator (overdamped, underdamped, critically damped). A nice treatment can be found here: http://www.entropy.energy/scholar/node/damped-harmonic-oscillator

In all cases, the amplitude does not depend on the frequency.

#### (c) Find a particular solution to the inhomogeneous problem by assuming a response at the driving frequency, and plot its magnitude and phase as a function of the driving frequency for $m = k = 1, γ = 0.1$.¶

For an input $x=Ae^{i\omega t}$ we have $$-m\omega^2A+i\gamma\omega A + kA = 1$$ or $$A = \frac{1}{-m\omega^2 + i\gamma\omega + k}$$ where $$|A| = \frac{1}{\sqrt{(k-m\omega^2)^2 + \gamma^2\omega^2}}\text{ and } \phi_A = \tan^{-1}{\frac{\gamma\omega}{k-m\omega^2}}$$

Plotting for $m=k=1, \gamma=0.1$ below.

In [28]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
m=1
k=1
gamma=.1

w= np.logspace(-1,1, num=1000)
A = 1/(-m*w**2 + 1j*gamma*w + k)
mag = 1.0/(np.sqrt((k-m*w**2)**2 + gamma**2*w**2))
# or
# mag = np.absolute(A)
phase = np.arctan2(-gamma*w,(k-m*w**2))
# or
# phase = np.angle(A)

plt.figure()
plt.semilogx (w, mag, color="blue")
plt.xlabel ("Frequency")
plt.ylabel ("Amplitude")

plt.figure()
plt.semilogx (w, phase, color="blue")
plt.xlabel ("Frequency")
plt.ylabel ("Phase")

plt.show()


#### (d) For a driven oscillator the $Q$ or Quality factor is defined as the ratio of the center frequency to the width of the curve of the average energy (kinetic + potential) in the oscillator versus the driving frequency (the width is defined by the places where the curve falls to half its maximum value). For an undriven oscillator the Q is defined to be the ratio of the energy in the oscillator to the energy lost per radian (one cycle is $2\pi$ radians). Show that these two definitions are equal, assuming that the damping is small. How long does it take the amplitude of a $100Hz$ oscillator with a $Q$ of $109$ to decay by $1/e$?¶

The average energy is the sum of the average kinetic ($T$) and potential ($U$) energies, $$\langle E\rangle = \langle T\rangle + \langle U\rangle = \frac{1}{2}m\langle\dot{x}^2\rangle + \frac{1}{2}k\langle x^2\rangle$$

Assuming a solution of the form $x = A*e^{i\omega t}=A(\cos{\omega t}+i\sin{\omega t})$, then we have (easily proven that the average of $cos^2$ and $sin^2$ in a period is $1/2$) $$\langle x^2\rangle=\frac{1}{2} \text{ and } \langle \dot{x}^2\rangle=\frac{1}{2}.$$

Thus, \begin{align} \langle E\rangle &= \frac{1}{4}(m\omega^2 + k)A^2 \\ &= \frac{1}{4}\frac{m\omega^2 + k}{(k-m\omega^2) + \gamma^2\omega^2} \end{align}

At the center (resonant frequency) $\omega_0 = \frac{k}{m} \Rightarrow k=\omega_0m$ we have $$\langle E\rangle = \frac{1}{4m}\frac{\omega_0^2 + \omega^2}{(\omega_0^2 - \omega^2)^2 + \omega^2\gamma^2/m^2}$$

### (3.3) A common simple digital filter used for smoothing a signal is $y(k)=\alpha y(k-1)+(1-\alpha)x(k)$, where $\alpha$ is a parameter that determines the response of the filter. Use $z$-transforms to solve for $y(k)$ as a function of $x(k)$ (assume $y(k<0)=0$). What is the amplitute of the frequency response?

In [33]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

a=.01
n=1

w = np.logspace(-1,4, num=1000)
H = (1-a)*np.exp(1j*w*n)/(np.exp(1j*w*n)-a)
mag = np.absolute(H)

plt.figure()
plt.semilogx (w, mag, color="blue")
plt.xlabel ("Frequency")
plt.ylabel ("Amplitude")
plt.show()