Forest UCM Osc 2-DOsc - Revision history

Foretony: /* Isotropic Oscillator */

2021-10-14T23:10:44Z

Isotropic Oscillator

2014-10-08T17:33:09Z

Lissajous figure

2014-10-06T17:49:18Z

Lissajous figure

2014-10-06T17:47:36Z

Lissajous figure

2014-10-06T17:46:41Z

Lissajous figure

2014-10-06T17:43:36Z

Lissajous figure

2014-10-06T17:35:04Z

Lissajous figure

2014-10-05T12:27:11Z

Lissajous figure

2014-10-05T12:26:29Z

Lissajous figure

2014-10-05T12:16:00Z

Anisotropic Oscillator

@@ Line 42: / Line 42: @@
 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)</math>

@@ Line 113: / Line 113: @@
 ;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
@@ Line 118: / Line 120: @@
 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>\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>

@@ Line 133: / Line 133: @@
 : <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 a.) show motion is not periodic (no repetition) if the ratio of frequencies is a IR-rational number
+;problem 5.17 b.) show motion is not periodic (no repetition) if the ratio of frequencies is a IR-rational number
 [[Forest_UCM_Osc#2-D_Oscillators]]

@@ Line 114: / Line 114: @@
 ;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
+:<math>\frac{\omega_x}{\omega_y} =\frac{p}{q} </math> where p & q are integers
 The period of the oscillation will be given by

@@ Line 112: / Line 112: @@
-problem 5.17
+;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
@@ Line 134: / Line 132: @@
 : <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
 [[Forest_UCM_Osc#2-D_Oscillators]]

← Older revision		Revision as of 17:43, 6 October 2014
Line 122:		Line 122:
	:<math>\tau = \frac{2 \pi p}{\omega_x} = \frac{2 \pi q}{\omega_y}</math> Definition of period		:<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



	[[Forest_UCM_Osc#2-D_Oscillators]]		[[Forest_UCM_Osc#2-D_Oscillators]]

← Older revision		Revision as of 17:35, 6 October 2014
Line 113:		Line 113:

	problem 5.17		problem 5.17
		+
		+	Assume that 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
		+
		+
		+


	[[Forest_UCM_Osc#2-D_Oscillators]]		[[Forest_UCM_Osc#2-D_Oscillators]]

@@ Line 105: / Line 105: @@
 if
 :<math>\frac{\omega_x}{\omega_y} =</math> rational number (a fraction formed two integers, denominator is non-zero)
 problem 5.17

← Older revision		Revision as of 12:16, 5 October 2014
Line 100:		Line 100:

	[[File:TF_2-D_AnIsoOscilator.png \| 600 px]]		[[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)

	[[Forest_UCM_Osc#2-D_Oscillators]]		[[Forest_UCM_Osc#2-D_Oscillators]]