Forest UCM Osc Driven - Revision history

Foretony: /* What is delta? */

2021-10-14T23:23:08Z

What is delta?

2021-10-14T23:13:51Z

Case 1 : f(t) is sinusoidal

2021-10-14T23:13:32Z

Solving the Inhomogeneous Diff. Eq.

2021-10-14T23:13:16Z

Differential equations in Operator form

2014-10-08T12:14:06Z

Complete Solution for the Driven Damped oscilaltor=

2014-10-08T12:13:35Z

Complete Solution for the Driven Damped oscilaltor=

2014-10-08T12:13:09Z

Complete Solution for the Driven Damped oscilaltor=

2014-10-08T12:11:16Z

Complete Solution for the Driven Damped oscilaltor=

2014-10-08T12:10:54Z

What is delta?

2014-10-08T12:09:17Z

What is delta?

@@ Line 138: / Line 138: @@
 :<math>  {f_0} e^{i\delta} = A \left ( (\omega_0^2- \omega^2) + 2 i \beta \omega  \right )   </math>
-Since <math>f_0</math> and <math>A</math> are not imaginary variable
+Since <math>f_0</math> and <math>A</math> are not imaginary variables
-The phase and <math>\delta</math> must be due to the remaining complex term <math> \left ( (\omega_0^2- \omega^2) + 2 i \beta \omega  \right )</math>
+The phase (<math>\delta</math>) must be due to the remaining complex term <math> \left ( (\omega_0^2- \omega^2) + 2 i \beta \omega  \right )</math>
-To detmine the angle <math>\delta</math> we construct a right trianble in the complex plane where the leg along the real axis is given by <math>(\omega_0^2- \omega^2)</math>  and the other leg of the triangle is along the imaginary axis with a length of <math> 2 \beta \omega</math>
+To determine the angle <math>\delta</math> we construct a right triangle in the complex plane where the leg along the real axis is given by <math>(\omega_0^2- \omega^2)</math>  and the other leg of the triangle is along the imaginary axis with a length of <math> 2 \beta \omega</math>
 :<math>\tan \delta = \frac{2 \beta \omega}{(\omega_0^2- \omega^2)}</math>

@@ Line 102: / Line 102: @@
 ;Trick: If a differential equation has the cosine function as a solution then the sine function may also be a solution since the difference between the two is only a phase shift.
-It must also be try that
+It must also be true that
 :<math>D(y_p) = f_0 \sin ( \omega t)</math>

@@ Line 57: / Line 57: @@
-Since the solution of the 2nd order differential equation requires exactly two arbitrary constant,
+Since the solution of the 2nd order differential equation requires exactly two arbitrary constants,
 We can formulate the solution by tacking a particular solution onto the homogeneous solution that already has two arbitrary constants.
@@ Line 154: / Line 154: @@
 with
 :<math>x_p = \mbox{Real} \left |z_p \right | = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4  \beta^2 \omega^2  }} \cos(\omega t-\delta)</math>
 ==Complete Solution for the Driven Damped oscilaltor==

@@ Line 36: / Line 36: @@
 :<math>  D x  = f(t)</math>
-;Linear differential equations have coefficient that can constant or variable coefficients that can be transformed into constant coefficients.
+;Linear differential equations have coefficients that can be constant or variable coefficients that can be transformed into constant coefficients.
@@ Line 49: / Line 49: @@
 :<math>D(ax) = aD(x)</math> and <math>D(x_1 + x_2) = D(x_1) + D(x_2)</math>
 ==Solving the Inhomogeneous Diff. Eq. ==

@@ Line 157: / Line 157: @@
 :<math>x_p = \mbox{Real} \left |z_p \right | = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4  \beta^2 \omega^2  }} \cos(\omega t-\delta)</math>
-==Complete Solution for the Driven Damped oscilaltor===
+==Complete Solution for the Driven Damped oscilaltor==
 :<math>x(t) =x_h + x_p = C_1 e^{r_1 t} + C_2 e^{r_2 t} + A \cos(\omega t-\delta)</math>

@@ Line 166: / Line 166: @@
 :<math>r_2 = - \beta + \sqrt{\beta^2 + \omega_0^2}</math>
 :<math>A=\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4  \beta^2 \omega^2  }}</math>
 [[Forest_UCM_Osc#Damped_Oscillations_with_driving_source]]

@@ Line 159: / Line 159: @@
 ==Complete Solution for the Driven Damped oscilaltor===
-:x=x_h + x_p =
+:x=x_h + x_p = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4  \beta^2 \omega^2  }} \cos(\omega t-\delta)
 [[Forest_UCM_Osc#Damped_Oscillations_with_driving_source]]

← Older revision		Revision as of 12:10, 8 October 2014
Line 156:		Line 156:

	:<math>x_p = \mbox{Real} \left \|z_p \right \| = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }} \cos(\omega t-\delta)</math>		:<math>x_p = \mbox{Real} \left \|z_p \right \| = \frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4 \beta^2 \omega^2 }} \cos(\omega t-\delta)</math>
		+
		+	==Complete Solution for the Driven Damped oscilaltor===
		+
		+	:x=x_h + x_p =

	[[Forest_UCM_Osc#Damped_Oscillations_with_driving_source]]		[[Forest_UCM_Osc#Damped_Oscillations_with_driving_source]]

@@ Line 146: / Line 146: @@
 :<math>\tan \delta = \frac{2 \beta \omega}{(\omega_0^2- \omega^2)}</math>
-:< math>\delta = \tan^{-1}\left ( \frac{2 \beta \omega}{(\omega_0^2- \omega^2)} \right )</math>
+:<math>\delta = \tan^{-1}\left ( \frac{2 \beta \omega}{(\omega_0^2- \omega^2)} \right )</math>
 ;The final complex solution si :
@@ Line 152: / Line 152: @@
 :<math>z_p =\frac{f_0} { \sqrt{(\omega_0^2 - \omega^2)^2 + 4  \beta^2 \omega^2  }}e^{i (\omega t-\delta)}</math>
 [[Forest_UCM_Osc#Damped_Oscillations_with_driving_source]]