HARMONIC OSCILLATORS

By John Meth

"A dashpot is a mechanical device, a damper which resists motion via viscous friction.[1] The resulting force is proportional to the velocity, but acts in the opposite direction,[2] slowing the motion and absorbing energy. It is commonly used in conjunction with a spring (which acts to resist displacement). The process and instrumentation diagram (P&ID) symbol for a dashpot is ." https://en.wikipedia.org/wiki/Dashpot

1. Simple Harmonic Oscillator: This is a mass spring system where there is no damping, $c = 0.$ Braun calls these "free vibrations".
2. Damped Harmonic Oscillator: This is a mass spring system where there is damping, $c > 0.$ Braun calls these "damped free vibrations".
3. Driven Harmonic Oscillator: This is a mass spring system with an external force, $g(t),$ acting on the mass: $$ m \cdot y^{\ \prime \prime}(t) = - c \cdot y^{\ \prime}(t) - k \cdot y(t) + g(t) \quad \Rightarrow \quad m \cdot y^{\ \prime \prime}(t) + c \cdot y^{\ \prime}(t) + k \cdot y(t) = g(t). $$ These produce nonhomogeneous second order O.D.E.'s with constant coefficients. An example is the vehicle suspension figure, above, with $g(t)$ given by the height of the road. (This term, $g(t),$ is $y(t)$ in their figure, and their solution is $x(t).)$

• Case 1: Simple Harmonic Oscillator
  When $c = 0,$ our equation becomes $$ m y^{\prime \prime} + k y = 0 $$ which has characteristic equation $$ mr^2 + k = 0 $$ with complex solutions $$ r = \pm \sqrt{-\frac{k}{m}} = \pm \, \, i \cdot \sqrt{\frac{k}{m}}. $$ These give us the following general solution to the homogeneous equation: $$ y_c(t) = c_1 \cos \left( \sqrt{\frac{k}{m}} \cdot t \right) + c_2 \sin \left( \sqrt{\frac{k}{m}} \cdot t \right). $$ Initial conditions will turn this O.D.E. into an I.V.P. Usually these are given as
  • An initial position, $y(0) = y_0.$ Imagine holding the mass, lifting it (compressing the spring) or pulling it (stretching the spring), and then releasing the mass.
  • An initial velocity, $y^{\prime}(0) = v_0.$ Imagine tapping the mass, giving it an upward or downward velocity at time $t = 0.$
  Using these initial conditions to solve for our constants $c_1$ and $c_2$ gives $$ y_c(t) = y_0 \cdot \cos \left( \sqrt{\frac{k}{m}} \cdot t \right) + \frac{v_0}{\sqrt{\frac{k}{m}}} \cdot \sin \left( \sqrt{\frac{k}{m}} \cdot t \right) $$ Lemma 1 in section 2.6 of Braun uses simple trigonometric identities to represent such a function as a single shifted cosine function: $$ y(t) = R \cdot \cos \left( \omega \cdot t - \delta \right) $$ $$ R = \sqrt{y_0^2 + \frac{m}{k} v_0^2} \quad \quad \quad \quad \omega = \sqrt{\frac{k}{m}} \quad \quad \quad \quad \delta = \tan^{-1}\left( \frac{v_0}{y_0} \sqrt{\frac{m}{k}} \right) $$ The relevant piece is that every one of these graphs looks like a cosine graph that's been shifted and stretched.


• Case 2: Damped Harmonic Oscillator
  When $c > 0,$ our equation becomes $$ m y^{\prime \prime} + c y^{\prime} + k y = 0 $$ which has characteristic equation $$ mr^2 + cr + k = 0 $$ with solutions $$ r = \frac{-c \pm \sqrt{c^2 - 4mk}}{2m}. $$ Let $\Delta = c^2 - 4mk,$ the discriminant of the characteristic equation. This breaks Case 2 into subcases:
  1. $\Delta < 0,$ underdamped
  2. $\Delta = 0,$ critically damped
  3. $\Delta > 0,$ overdamped


• Case 2 (a): Underdamped Harmonic Oscillator; $\Delta < 0$
  With $\Delta < 0$ we get two complex solutions to our characteristic equation and our general solution to the homogeneous O.D.E. is: $$ y_c(t) = e^{- \frac{c}{2m} t} \left[ c_1 \cos \left( \frac{\sqrt{4mk-c^2}}{2m} \cdot t \right) + c_2 \sin \left( \frac{\sqrt{4mk-c^2}}{2m} \cdot t \right) \right] $$ Again, initial conditions will turn this O.D.E. into an I.V.P. Using these initial conditions to solve for our constants $c_1$ and $c_2$ gives $$ y_c(t) = e^{- \frac{c}{2m} t} \left[ y_0 \cos \left( \frac{\sqrt{4mk-c^2}}{2m} \cdot t \right) + \left( \frac{2mv_0 + cy_0}{\sqrt{4mk-c^2}} \right) \sin \left( \frac{\sqrt{4mk-c^2}}{2m} \cdot t \right) \right] $$

  Simple Harmonic Oscillator (No Damping)	Underdamped Harmonic Oscillator (Some Damping)



• Case 2 (b): Critically Damped Harmonic Oscillator; $\Delta = 0$
  With $\Delta = 0$ the characteristic equation has root $$ r = -\frac{c}{2m} = -\sqrt{\frac{k}{m}} \quad \quad \textrm{ (using } \Delta = c^2 - 4mk = 0 \textrm{.)} $$ with multiplicity $2$ (a double root.) In this case, our solution is $$ y(t) = \left( c_1 + c_2 t \right) e^{-\sqrt{\frac{k}{m}} \cdot \, t} $$ and solving for $c_1$ and $c_2$ using initial conditions yields $$ y(t) = \left[ y_0 + \left( v_0 + \sqrt{\frac{k}{m}} y_0 \right) t \right] e^{-\sqrt{\frac{k}{m}} \cdot \, t} $$


• Case 2 (c): Overdamped Harmonic Oscillator; $\Delta > 0$
  Let $$ r_+ = \frac{-c + \sqrt{c^2 - 4mk}}{2m} \quad \quad \textrm{ and } \quad \quad r_- = \frac{-c - \sqrt{c^2 - 4mk}}{2m} $$ then $\Delta > 0$ implies that both roots are real and negative. So we get the following general solution to the homogeneous equation: $$ y_c(t) = c_1 e^{(r_+) t} + c_2 e^{(r_-) t} $$ Again, initial conditions will turn this O.D.E. into an I.V.P. Using these initial conditions to solve for our constants $c_1$ and $c_2$ gives $$ y_c(t) = \left( \frac{y_0 r_- - v_0}{r_--r_+} \right) e^{(r_+) t} + \left( \frac{v_0 - y_0 r_+}{r_--r_+} \right) e^{(r_-) t} $$
  Note that we can also consider the solution when the damping becomes infinite: $$ \lim_{c \rightarrow \infty} r_+ = 0 \quad \quad \textrm{ and } \quad \quad \lim_{c \rightarrow \infty} r_- = -\infty \textrm{.} $$ Thus $$ \lim_{c \rightarrow \infty} y_1(t) = \lim_{c \rightarrow \infty} e^{(r_+) t} = e^{0 t} = 1 $$ $$ \lim_{c \rightarrow \infty} y_2(t) = \lim_{c \rightarrow \infty} e^{(r_-) t} = e^{- \infty t} = 0 $$ $$ \lim_{c \rightarrow \infty} c_2 = \lim_{c \rightarrow \infty} \left( \frac{v_0 - y_0 r_+}{r_--r_+} \right) = 0 $$ and since $c_1 = y_0 - c_2$ we get $$ \lim_{c \rightarrow \infty} c_1 = \lim_{c \rightarrow \infty} y_0 - c_2 = y_0 $$ Finally, we see $$ \lim_{c \rightarrow \infty} y(t) = \lim_{c \rightarrow \infty} c_1 y_1 + c_2 y_2 = y_0 \cdot 1 + 0 \cdot 0 = y_0 $$

$\Delta, c$	Solutions:
$c = 0 , \quad \Delta < 0$	$$y(t) = c_1 \cos \left( \sqrt{\frac{k}{m}} \cdot t \right) + c_2 \sin \left( \sqrt{\frac{k}{m}} \cdot t \right)$$
Name:	Graph:
Simple Harmonic Oscillator
Behavoir:
Mass oscillates forever without changing amplitude or frequency.

$\Delta, c$	Solutions:
$c > 0, \quad \Delta < 0$	$$y(t) = e^{- \alpha t} \Big[ c_1 \cos \left( \beta t \right) + c_2 \sin \left( \beta t \right) \Big] $$
Name:	Graph:
Underdamped Harmonic Oscillator
Behavoir:
Mass oscillates with constant frequency and diminishing amplitude.

$\Delta, c$	Solutions:
$c = 2 \sqrt{mk}, \quad \Delta = 0$	$$y(t) = \Big( c_1 + c_2 t \Big) e^{-\sqrt{\frac{k}{m}} \, t}$$
Name:	Graph:
Critically Damped Harmonic Oscillator
Behavoir:
Unique, fastest solution returning mass to equilibrium without oscillation.

$\Delta, c$	Solutions:
$c > 0, \quad \Delta > 0$	$$y(t) = c_1 e^{- r_1 t} + c_2 e^{-r_2 t}$$
Name:	Graph:
Overdamped Harmonic Oscillator
Behavoir:
Mass takes longer and longer to return to equilibrium as $c \rightarrow \infty.$

$\Delta, c$	Solutions:
$c \rightarrow \infty, \quad \Delta \rightarrow \infty$	$$y(t) = y_0$$
Name:	Graph:
Infinitely Damped Harmonic Oscillator
Behavoir:
Mass never moves from initial position $y(t) = y_0.$