Forest UCM Osc Resonance - Revision history

Foretony: /* Resonance */

2021-10-14T23:23:33Z

Resonance

2014-10-12T13:50:50Z

Quality factor (Q)

2014-10-12T13:48:27Z

Quality factor (Q)

2014-10-12T13:40:10Z

Quality factor (Q)

2014-10-12T13:39:40Z

Quality factor (Q)

2014-10-12T13:34:35Z

Quality factor (Q)

2014-10-12T13:33:21Z

Quality factor (Q)

2014-10-12T13:32:07Z

Quality factor (Q)

2014-10-12T13:28:13Z

Quality factor (Q)

2014-10-12T13:27:07Z

Quality factor (Q)

@@ Line 47: / Line 47: @@
 :<math>\omega = \omega_0</math>
-then the denominator is minimazed thereby maximizing the amplitude of the forced oscillation.
+then the denominator is minimized thereby maximizing the amplitude of the forced oscillation.
 Resonance occurs when your applied sinusoidal force matches the natural frequency of the oscillaor.

@@ Line 142: / Line 142: @@
 The "sharpness" of the resonance is quantified in terms of a quality factor given by the ratio of the natural frequency to the dampening as
-:<math> Q \equiv \frac{\omega_0}{2 \beta}=\frac{\frac{1}{\beta}}{2 \frac{1}{\omega_0}}=\frac{2\tau}{2 \frac{1}{\omega_0}}</math>
+:<math> Q \equiv \frac{\omega_0}{2 \beta}=\frac{\frac{1}{\beta}}{2 \frac{1}{\omega_0}}=\frac{\tau_d}{2 \frac{1}{\omega_0}}</math>
 The above is proportions to the amount of energy stored in one cycle of the oscillation divided by the average energy dissipated in one cycle.

@@ Line 142: / Line 142: @@
 The "sharpness" of the resonance is quantified in terms of a quality factor given by the ratio of the natural frequency to the dampening as
-:<math> Q \equiv \frac{\omega_0}{2 \beta}</math>
+:<math> Q \equiv \frac{\omega_0}{2 \beta}=\frac{\frac{1}{\beta}}{2 \frac{1}{\omega_0}}=\frac{2\tau}{2 \frac{1}{\omega_0}}</math>
 The above is proportions to the amount of energy stored in one cycle of the oscillation divided by the average energy dissipated in one cycle.

@@ Line 160: / Line 160: @@
 |<math>10^7</math>|| excited atoms
 |-
-|<math>10^10</math>|| CEBAF's accelerator RF cryomodule
+|<math>10^{10}</math>|| CEBAF's accelerator RF cryomodule
 |-
 |<math>10^{12}</math>|| excited nuclei

← Older revision		Revision as of 13:39, 12 October 2014
Line 147:		Line 147:

	Some Q values for several oscilaltors		Some Q values for several oscilaltors
		+
		+	;Accelerator Settings
		+	{\| border="1"
		+	\| Q \|\| System
		+	\|-
		+	\| 5-10 \|\| rubberbands and loud speakers
		+	\|-
		+	\| <math>10^3</math> \|\| tuning forks and violin strings
		+	\|-
		+	\|<math>10^4</math>\|\| microwave cavity
		+	\|-
		+	\|<math>10^7</math>\|\| excited atoms
		+	\|-
		+	\|<math>10^10</math>\|\| CEBAF's accelerator RF cryomodule
		+	\|-
		+	\|<math>10^{12}</math>\|\| excited nuclei
		+	\|-
		+	\|<math>10^{14}</math>\|\| gas lasers
		+	\|-
		+	\|}


	[[Forest_UCM_Osc#Resonance]]		[[Forest_UCM_Osc#Resonance]]

← Older revision		Revision as of 13:34, 12 October 2014
Line 146:		Line 146:
	The above is proportions to the amount of energy stored in one cycle of the oscillation divided by the average energy dissipated in one cycle.		The above is proportions to the amount of energy stored in one cycle of the oscillation divided by the average energy dissipated in one cycle.

		+	Some Q values for several oscilaltors


	[[Forest_UCM_Osc#Resonance]]		[[Forest_UCM_Osc#Resonance]]

@@ Line 135: / Line 135: @@
 :<math>Q = \frac{\mbox{Energy stored in oscillator}}{\mbox{Energy dissipated}} = \frac{E}{\Delta E}</math>
 :: <math>= \frac{E}{2 \beta E  \frac{1}{\omega}}</math>
-:: <math>= \frac{\omega }{2 \beta E }</math>
+:: <math>= \frac{\omega }{2 \beta }</math>

← Older revision		Revision as of 13:32, 12 October 2014
Line 129:		Line 129:

	:<math>\Delta E = \left \| \frac{dE}{dt} \right \| \delta t = \left \| -2 \beta E_0 e^{- 2\beta t} \right \| \delta t = 2 \beta E \delta t</math>		:<math>\Delta E = \left \| \frac{dE}{dt} \right \| \delta t = \left \| -2 \beta E_0 e^{- 2\beta t} \right \| \delta t = 2 \beta E \delta t</math>
		+
		+	The energy lost from one oscillation is then
		+	:<math>\Delta t = \frac{1}{\omega}</math>
		+
		+	:<math>Q = \frac{\mbox{Energy stored in oscillator}}{\mbox{Energy dissipated}} = \frac{E}{\Delta E}</math>
		+	:: <math>= \frac{E}{2 \beta E \frac{1}{\omega}}</math>
		+	:: <math>= \frac{\omega }{2 \beta E }</math>
		+


−	~~As shown previously in [[Forest_UCM_Osc_Damped#Energy]]~~

−	~~The rate of energy loss is then given by~~
−	~~:<math> \frac{dE}{dt} = \frac{W_f}{dt} = -2\beta v^2</math>~~
	The "sharpness" of the resonance is quantified in terms of a quality factor given by the ratio of the natural frequency to the dampening as		The "sharpness" of the resonance is quantified in terms of a quality factor given by the ratio of the natural frequency to the dampening as

← Older revision		Revision as of 13:28, 12 October 2014
Line 128:		Line 128:
	The energy dissipated with time may be expressed as		The energy dissipated with time may be expressed as

−	:<math>\Delta E = \left \| \frac{dE}{dt} \right \| \delta t = 2 \beta E \delta t</math>	+	:<math>\Delta E = \left \| \frac{dE}{dt} \right \| \delta t = \left \| -2 \beta E_0 e^{- 2\beta t} \right \| \delta t = 2 \beta E \delta t</math>

← Older revision		Revision as of 13:27, 12 October 2014
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	: <math>E_0 =\frac{1}{2} m \omega^2 A</math>		: <math>E_0 =\frac{1}{2} m \omega^2 A</math>
		+
		+	The energy dissipated with time may be expressed as
		+
		+	:<math>\Delta E = \left \| \frac{dE}{dt} \right \| \delta t = 2 \beta E \delta t</math>
		+

	As shown previously in [[Forest_UCM_Osc_Damped#Energy]]		As shown previously in [[Forest_UCM_Osc_Damped#Energy]]