Difference between revisions of "Forest UCM Osc 2-DOsc"

From New IAC Wiki
Jump to navigation Jump to search
 
(22 intermediate revisions by the same user not shown)
Line 1: Line 1:
 
=Two Dimensional Oscillators=
 
=Two Dimensional Oscillators=
  
[[File:TF_2-D_oscilator.png | 200 px]]
+
[[File:TF_2-D_oscilator.png | 600 px]]
 +
 
 +
 
 +
:<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==
 
==Isotropic Oscillator==
Line 16: Line 23:
 
you could define a relative phase between the two oscillators as
 
you could define a relative phase between the two oscillators as
  
:\delta = \delta_y - \delta_x
+
:<math>\delta = \delta_y - \delta_x</math>
  
 
it can be substituted into the above equations by shifting the time origin (problem 5.15)
 
it can be substituted into the above equations by shifting the time origin (problem 5.15)
Line 35: Line 42:
 
similarly  
 
similarly  
  
:<math>y = A_y \cos(\omega t^{\prime} - \omega t_0 - \delta_x)</math>
+
:<math>y = A_y \cos(\omega t^{\prime} - \omega t_0 - \delta_y)</math>
::<math> =A_y \cos(\omega t^{\prime} - \omega \frac{-\delta_x}{\omega}  - \delta_x)</math>
+
::<math> =A_y \cos(\omega t^{\prime} - \omega \frac{-\delta_x}{\omega}  - \delta_y)</math>
 
::<math>= A_y \cos(\omega t^{\prime} +\delta_x - \delta_y)</math>
 
::<math>= A_y \cos(\omega t^{\prime} +\delta_x - \delta_y)</math>
::<math>= A_y \cos(\omega t^{\prime} +\delta)</math>
+
::<math>= A_y \cos(\omega t^{\prime} -\delta)</math>
  
 
==Interpretation==
 
==Interpretation==
  
 
:<math>  =A_x \cos(\omega t^{\prime})</math>
 
:<math>  =A_x \cos(\omega t^{\prime})</math>
:<math>= A_y \cos(\omega t^{\prime} +\delta)</math>
+
:<math>= A_y \cos(\omega t^{\prime} -\delta)</math>
  
 
There are several modes of motion for the above system of equation
 
There are several modes of motion for the above system of equation
Line 64: Line 71:
  
 
[[File:TF_2-D_Oscilator_delta90.png | 200 px]]
 
[[File:TF_2-D_Oscilator_delta90.png | 200 px]]
 +
 +
:<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.
 
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==
 
==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>
 +
 +
 +
 +
[[File:TF_2-D_AnIsoOscilator.png | 600 px]]
 +
 +
===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)
 +
 +
Assume that the ratio of oscillation frequencies is
 +
 +
:<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 = \left ( \frac{2 \pi }{\omega_x} \right ) p =  \left ( \frac{2 \pi }{\omega_y}\right ) q</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]]
 
[[Forest_UCM_Osc#2-D_Oscillators]]

Latest revision as of 23:10, 14 October 2021

Two Dimensional Oscillators

TF 2-D oscilator.png


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


TF 2-D Oscilator deltaZero.png

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).


TF 2-D Oscilator delta90.png

[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]


TF 2-D AnIsoOscilator.png

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)

Assume that the ratio of oscillation frequencies is

[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 = \left ( \frac{2 \pi }{\omega_x} \right ) p = \left ( \frac{2 \pi }{\omega_y}\right ) q[/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