Forest UCM Osc 2-DOsc

Two Dimensional Oscillators

Isotropic Oscillator

The simplest 2-D oscillator that is composed of identical springs (same spring constant).

[math]\displaystyle{ \vec F = -k( x \hat i + y \hat j) }[/math]

The equations of motion are separable two equations, one for each direction

[math]\displaystyle{ \ddot x = - \omega^2x \Rightarrow x = A_x \cos(\omega t - \delta_x) }[/math]

[math]\displaystyle{ \ddot y = - \omega^2 y \Rightarrow y = A_y \cos(\omega t - \delta_y) }[/math]

you could define a relative phase between the two oscillators as

[math]\displaystyle{ \delta = \delta_y - \delta_x }[/math]

it can be substituted into the above equations by shifting the time origin (problem 5.15)

let

[math]\displaystyle{ t^{\prime} = t + t_0 }[/math]

then

[math]\displaystyle{ x = A_x \cos(\omega t^{\prime} - \omega t_0 - \delta_x) }[/math]

let

[math]\displaystyle{ t_0 = \frac{-\delta_x}{\omega} }[/math]

[math]\displaystyle{ x=A_x \cos(\omega t^{\prime} - \omega\frac{-\delta_x}{\omega} - \delta_x) }[/math]

[math]\displaystyle{ =A_x \cos(\omega t^{\prime}) }[/math]

similarly

[math]\displaystyle{ y = A_y \cos(\omega t^{\prime} - \omega t_0 - \delta_x) }[/math]

[math]\displaystyle{ =A_y \cos(\omega t^{\prime} - \omega \frac{-\delta_x}{\omega} - \delta_x) }[/math]

[math]\displaystyle{ = A_y \cos(\omega t^{\prime} +\delta_x - \delta_y) }[/math]

[math]\displaystyle{ = A_y \cos(\omega t^{\prime} -\delta) }[/math]

Interpretation

[math]\displaystyle{ =A_x \cos(\omega t^{\prime}) }[/math]

[math]\displaystyle{ = A_y \cos(\omega t^{\prime} -\delta) }[/math]

There are several modes of motion for the above system of equation

No phase difference

If there is no phase difference then (\delta =0 ) and you have oscillating motion along a line.

90 degree phase difference

If the x and y motions are completely out of phase ( phase difference of 90 degrees or [math]\displaystyle{ \frac{\pi}{2} }[/math])

then the motion is an ellipse (unless their amplitudes are equal in which case the mass moves in a circle).

[math]\displaystyle{ =A_x \cos(\omega t^{\prime}) }[/math]

[math]\displaystyle{ = A_y \cos(\omega t^{\prime} +\frac{\pi}{2})= A_y \sin(\omega t^{\prime}) }[/math]

[math]\displaystyle{ \left (\frac{x}{A_x} \right )^2 + \left (\frac{y}{A_y} \right )^2 = \cos^2(\omega t^{\prime}) + \sin^2(\omega t^{\prime})= 1 }[/math]

If the phase shift is less than 90 then the ellipse is slanted towards the x-axis otherwise it is toward the y-axis.

Anisotropic Oscillator

Forest_UCM_Osc#2-D_Oscillators