Forest UCM Osc 2-DOsc

Two Dimensional Oscillators

[math] \vec F = -k_x x \hat i + k_y y \hat j)[/math]

Potential

[math]U = - \int \vec F \cdot d \vec r = k_x \int x dx + k_y \int y dy = \frac{1}{2} \left ( k_x x^2 + k_y y^2 \right )[/math]

Isotropic Oscillator

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

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

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

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

[math]\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]\delta = \delta_y - \delta_x[/math]

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

let

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

then

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

let

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

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

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

similarly

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

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

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

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

Interpretation

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

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

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

[math] \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]

The above is an equation of a circle if [math]A_x=A_y[/math] otherwise it is an ellipse with semi-major and semi-minor axes [math]A_x[/math] and [math]A_y[/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

Consider the case of having two different spring constants in each direction

[math] \vec F = -k_x x \hat i - k_y y \hat j[/math]

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

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

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

where

[math]\delta = \delta_y - \delta_x[/math]

Lissajous figure

if

[math]\frac{\omega_x}{\omega_y} =[/math] rational number (a fraction formed two integers, denominator is non-zero)

then the motion repeats itself.

If the ratio is irrational the motion does not repreat.

problem 5.17 a.) show motion is periodic if the ratio of frequencies is a rational number ( p/q)

[math]\frac{\omega_x}{\omega_y} =\frac{p}{q} [/math] where p & q are integers

The period of the oscillation will be given by

[math]\tau = \frac{2 \pi p}{\omega_x} = \frac{2 \pi q}{\omega_y}[/math] Definition of period

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

The solution for the next period is

[math]x(t + \tau) = A_x \cos(\omega_x (t+\tau) )= A_x \cos(\omega_x t+ 2\pi p) = A_x \cos(\omega_x t) \Rightarrow[/math] Motion is periodic

[math]\cos(\theta + 2 \pi p) = \cos(\theta)[/math]if p is an integer

Similarly for the y-direction

[math]y(t+\tau) = A_y \cos(\omega_y (t+\tau) ) - \delta_y)= A_y \cos(\omega_y t+ 2\pi q- \delta) \Rightarrow[/math] Motion is periodic

problem 5.17 b.) show motion is not periodic (no repetition) if the ratio of frequencies is a IR-rational number

For this solution you just assume the motion is periodic and start the solution to part 5.17 a.) from the end and work backwards.

Forest_UCM_Osc#2-D_Oscillators