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F(t) = F,,0,, * cos(ω*t - φ) F(t) = F,,0,, * cos(ω*t - φ,,d,,)
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where F,,0,, is the driving amplitude, ω is the driving frequency, and φ is the phase angle. F,,0,, and ω are set on the function generator.
The steady-state solution (letting t go to infinite) for no damping (we can assume that for this system) has the form		 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
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x(t) = A*cos(ω'*t - φ') where A, ω', and φ' depend on the specifics of the driving force, the spring constant, and the mass used. After more math, we find that A 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
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A = (F,,0,,/m)/(ω^2^-ω,,0,,^2^) A = (F,,0,,/m) / Sqrt((ω^2^-ω,,0,,^2^)^2^ + 4*γ^2^*ω^2^)
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where m is the mass and ω,,0,, = Sqrt(k/m), k being the spring constant. Since the only variable in our expression for A (we're only varying the driving frequency ω), A will be at a maximum when we set ω = ω,,0,,. If there is significant damping, our expression for A changes slightly changing what value of ω maximizes A. Both cases, we call this ω the resonant frequency of the system. 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.

[:PiraScheme#Mechanics: Table of Mechanics Demonstration]

[:MEEquipmentList: List of Mechanics Equipment & Supplies]

[:Demonstrations:Lecture Demonstrations]

Driven Mass on a Spring, 3A60.39

Topic and Concept:

  • Oscillations, [:Oscillations#DrivenRes: 3A60. Driven Mechanical Resonance]

Location:

attachment:DrivenMassSpring18-400.jpg

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

[: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.

Important Setup Notes:

  • Demonstration may require practice.

Setup and Procedure:

  1. Vertically mount 3/4" rod to lecture bench using blue table clamp.
  2. Horizontally mount 1/4" rod to the vertical rod using a right angle clamp.
  3. Mount the Pasco driver to the 1/4" rod so that the driving component will move in a vertical direction.
  4. Place the function generator on the table near the rod, and plug it in.
  5. Connect the driver to the function generator with the two wires. Positive to positive. Ground to ground.
  6. Hang the spring from the driver.
  7. Hang the mass from the spring.
  8. When ready, turn on the function generator by pressing the "On" button.
  9. 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).

Cautions, Warnings, or Safety Concerns:

  • N/A

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

F(t) = F0 * cos(ω*t - φd)

where F0 is the driving amplitude, ω is the driving frequency, and φd 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 ωresonance = Sqrt(ω02 - 2*γ2). We call this value of ω the resonant frequency of the system.

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Videos:

References:

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fw: DrivenMassSpring (last edited 2019-12-11 22:12:56 by srnarf)