Differences between revisions 4 and 16 (spanning 12 versions)

Driven Mass on a Spring, 3A60.39

Topic and Concept:

Oscillations, 3A60. Driven Mechanical Resonance

Location:

Cabinet: Waves and Sound (WS)
Bay: (A4 Right)
Shelf: #1

Abstract:

A function generator drives a spring with a suspended mass. The frequency is adjusted until resonance is reached.

Equipment	Location	ID Number

Function Generator	EM, Bay B1 Left, Shelf #3
Pasco Driver	WS, Bay A4, Shelf #2
Coaxial Cable	Rod and tackle cabinet located near main lecture halls
Springs	WS, Bay A4, Shelf #2
100g Mass	ME, Bay A1, Shelf #5
Rod and Clamps	Rod and tackle cabinet located near main lecture halls

Important Setup Notes:

Demonstration may require practice.

Setup and Procedure:

Vertically mount 3/4" rod to lecture bench using blue table clamp.
Horizontally mount 1/4" rod to the vertical rod using a right angle clamp.
Mount the Pasco driver to the 1/4" rod so that the driving component will move in a vertical direction.
Place the function generator on the table near the rod, and plug it in.
Connect the driver to the function generator with the two wires. Positive to positive. Ground to ground.
Find the shorter of the two springs near the Pasco Drivers that has a banana connector soldered onto one end (see picture below). Hang the spring from the driver.
Hang the 100g mass from the spring.
When ready, turn on the function generator by pressing the "On" button.
The mass will begin to oscillate in a vertical direction. Adjust the frequency of oscillation by changing the driving frequency set on the function generator so that the system approaches resonance (some practice beforehand is recommended). On 12/12/19, resonance frequency was found to be about
0.797 Hz, and resonance amplitude was about 30 cm (ie 60 cm peak to peak).

Cautions, Warnings, or Safety Concerns:

Discussion:

The dynamics of this system are described by a linear, inhomogeneous, second order, 2D (one spatial dimension, one time dimension) differential equation (see Hyperphysics- Driven Oscillator). The inhomogeneous part comes from the driving force which is sinusoidal having a general form of

F(t) = F₀ * cos(ω*t - φ_d)

where F₀ is the driving amplitude, ω is the driving frequency, and φ_d is the phase angle. F₀ and ω are set on the function generator. The steady-state solution (letting t go to infinity) has the form

x(t) = A*cos(ω*t - φ) where A, and φ depend on the specifics of the driving force, the spring constant, damping, and the mass used. After more math, we find that A has the form

A = (F₀/m) / Sqrt((ω²-ω₀²)² + 4*γ²*ω²)

where m is the mass, γ encodes the damping information, and ω₀ = Sqrt(k/m), k being the spring constant. Since the only variable in our expression for A is ω (we're only varying the driving frequency ω), A will be at a maximum when we minimize the denominator. This gives us ω_resonance = Sqrt(ω₀² - 2*γ²). We call this value of ω the resonant frequency of the system.

Videos:

Lecture Demonstration's Youtube Channel

References:

Home

-  ⇤ ← Revision 4 as of 2013-06-19 17:30:32 → 
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+   ← Revision 16 as of 2019-12-11 22:12:56 → ⇥
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-||<:30%>[:PiraScheme#Mechanics: Table of Mechanics Demonstration]||<:30%>[:MEEquipmentList: List of Mechanics Equipment & Supplies]||<:30%>[:Demonstrations:Lecture Demonstrations]||
+||<30%  style="&quot; &amp; quot;text-align:center&amp; quot;  &quot; ">[[PiraScheme#WavesSound|Table of Waves and Sound Demonstration]] ||<30%  style="&quot; &amp; quot;text-align:center&amp; quot;  &quot; ">[[MEEquipmentList|List of Mechanics Equipment & Supplies]] ||<30%  style="&quot; &amp; quot;text-align:center&amp; quot;  &quot; ">[[Demonstrations|Lecture Demonstrations]] ||
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+'''Topic and Concept:'''
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-'''Topic and Concept:'''
  Oscillations, [:Oscillations#DrivenRes: 3A60. Driven Mechanical Resonance]
+ . Oscillations, [[Oscillations#DrivenRes|3A60. Driven Mechanical Resonance]]
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- * '''Cabinet:''' [:MechanicsCabinet:Mechanic (ME)]
 * '''Bay:''' [:MechanicsCabinetBayA1:(A1)]
 * '''Shelf:''' #1,2,3..
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+Line 12:
-attachment:DrivenMassSpring18-400.jpg
+ * '''Cabinet:''' [[WavesSoundCabinet|Waves and Sound (WS)]]
 * '''Bay:''' [[WSCabinetRtBayA4|(A4 Right)]]
 * '''Shelf:''' #1

{{attachment:DrivenMassSpring18-400.jpg}}
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-||<:style="width: 60%" :40%>'''Equipment'''||<:30%>'''Location'''||<:25%>'''ID Number'''||
+||<40%  style="&quot; &amp; quot;  &amp; amp;  quot;   ;text-align:center&amp; quot;  &quot; ">'''Equipment''' ||<30%  style="&quot; &amp; quot;text-align:center&amp; quot;  &quot; ">'''Location''' ||<25%  style="&quot; &amp; quot;text-align:center&amp; quot;  &quot; ">'''ID Number''' ||
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-||Function Generator||[:MechanicsCabinetBayB1: ME, Bay B1, Shelf #2]|| ||
||Pasco Driver||[:MechanicsCabinetBayB1: ME, Bay B1, Shelf #2]|| ||
||BNC Wire||[:MechanicsCabinetBayA5: ME, Bay A5, Shelf #2]|| ||
||Springs||[:MechanicsCabinetBayA5: ME, Bay A5, Shelf #2]|| ||
||Mass||[:MechanicsCabinetBayA5: ME, Bay A5, Shelf #2]|| ||
||Rod and Clamps||Rod and tackle cabinet located near main lecture halls.|| ||
+||Function Generator ||[[EMCabinetLtBayB1|EM, Bay B1 Left, Shelf #3]] || ||
||Pasco Driver ||[[WSCabinetRtBayA4|WS, Bay A4, Shelf #2]] || ||
||Coaxial Cable ||Rod and tackle cabinet located near main lecture halls || ||
||Springs ||[[WSCabinetRtBayA4|WS, Bay A4, Shelf #2]] || ||
||100g Mass ||[[MechanicsCabinetBayA1|ME, Bay A1, Shelf #5]] || ||
||Rod and Clamps ||Rod and tackle cabinet located near main lecture halls || ||
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-'''''Important Setup Notes:''''' 
 *  Demonstration may require practice.
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+Line 33:
-'''Setup and Procedure:'''
+'''''Important Setup Notes:'''''

 * Demonstration may require practice.

'''Setup and Procedure:'''
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-. Hang the spring from the driver.
 1. Hang the mass from the spring.
+. Find the shorter of the two springs near the Pasco Drivers that has a banana connector soldered onto one end (see picture below).  Hang the spring from the driver.
 1. Hang the 100g mass from the spring.
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+Line 47:
-. The mass will begin to oscillate in a vertical direction. Adjust the frequency of oscillation by changing the driving frequency set on the function generator so that the system approaches resonance (some practice beforehand is recommended).
+. The mass will begin to oscillate in a vertical direction. Adjust the frequency of oscillation by changing the driving frequency set on the function generator so that the system approaches resonance (some practice beforehand is recommended).  On 12/12/19, resonance frequency was found to be about
 1. 0.797 Hz, and resonance amplitude was about 30 cm (ie 60 cm peak to peak).
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+'''Discussion:'''
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-'''Discussion:'''
+The dynamics of this system are described by a linear, inhomogeneous, second order, 2D (one spatial dimension, one time dimension) differential equation (see [[http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html|Hyperphysics- Driven Oscillator]]). The inhomogeneous part comes from the driving force which is sinusoidal having a general form of
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+Line 58:
-Discuss the physics behind the demonstration, explaining some of the various steps of the demonstration when appropriate.
+F(t) = F,,0,, * cos(ω*t - φ,,d,,)
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-||attachment:DrivenMassSpring13-250.jpg||attachment:DrivenMassSpring14-250.jpg||attachment:DrivenMassSpring15-250.jpg||attachment:DrivenMassSpring16-250.jpg||
||attachment:DrivenMassSpring17-250.jpg||attachment:DrivenMassSpring19-250.jpg||attachment:DrivenMassSpring20-250.jpg||attachment:DrivenMassSpring21-250.jpg||
||attachment:DrivenMassSpring22-250.jpg||attachment:DrivenMassSpring23-250.jpg||attachment:DrivenMassSpring24-250.jpg||attachment:DrivenMassSpring25-250.jpg||
||attachment:DrivenMassSpring26-250.jpg||attachment:DrivenMassSpring27-250.jpg||attachment:DrivenMassSpring28-250.jpg||attachment:DrivenMassSpring29-250.jpg||
||attachment:DrivenMassSpring30-250.jpg||
+where F,,0,, is the driving amplitude, ω is the driving frequency, and φ,,d,, is the phase angle. F,,0,, and ω are set on the function generator. The steady-state solution (letting t go to infinity) has the form

x(t) = A*cos(ω*t - φ) where A, and φ depend on the specifics of the driving force, the spring constant, damping, and the mass used. After more math, we find that A has the form

A = (F,,0,,/m) / Sqrt((ω^2^-ω,,0,,^2^)^2^ + 4*γ^2^*ω^2^)

where m is the mass, γ encodes the damping information, and ω,,0,, = Sqrt(k/m), k being the spring constant. Since the only variable in our expression for A is ω (we're only varying the driving frequency ω), A will be at a maximum when we minimize the denominator. This gives us ω,,resonance,, = Sqrt(ω,,0,,^2^ - 2*γ^2^). We call this value of ω the resonant frequency of the system.
|| {{attachment:DrivenMassSpring13-250.jpg}} || {{attachment:DrivenMassSpring14-250.jpg}} || {{attachment:DrivenMassSpring15-250.jpg}} || {{attachment:DrivenMassSpring16-250.jpg}} ||
|| {{attachment:DrivenMassSpring17-250.jpg}} || {{attachment:DrivenMassSpring19-250.jpg}} || {{attachment:DrivenMassSpring20-250.jpg}} || {{attachment:DrivenMassSpring21-250.jpg}} ||
|| {{attachment:DrivenMassSpring22-250.jpg}} || {{attachment:DrivenMassSpring23-250.jpg}} || {{attachment:DrivenMassSpring24-250.jpg}} || {{attachment:DrivenMassSpring25-250.jpg}} ||
|| {{attachment:DrivenMassSpring26-250.jpg}} || {{attachment:DrivenMassSpring27-250.jpg}} || {{attachment:DrivenMassSpring28-250.jpg}} || {{attachment:DrivenMassSpring29-250.jpg}} ||
|| {{attachment:DrivenMassSpring30-250.jpg}} ||
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- * [https://www.youtube.com/user/LectureDemostrations/videos?view=1 Lecture Demonstration's Youtube Channel]
+ * [[https://www.youtube.com/user/LectureDemostrations/videos?view=1|Lecture Demonstration's Youtube Channel]]
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+Line 82:
- * [https://en.wikipedia.org/wiki/Harmonic_oscillator Wikipedia - Harmonic Oscillator]
 * [https://en.wikipedia.org/wiki/Resonance Wikipedia - Resonance]
+ * [[https://en.wikipedia.org/wiki/Harmonic_oscillator|Wikipedia - Harmonic Oscillator]]
 * [[https://en.wikipedia.org/wiki/Resonance|Wikipedia - Resonance]]
 * [[http://hyperphysics.phy-astr.gsu.edu/hbase/oscdr.html|Hyperphysics- Driven Oscillator]]
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-[:Instructional:Home]
+[[Instructional|Home]]

fw: Diff for "DrivenMassSpring"

Driven Mass on a Spring, 3A60.39