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.

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. 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.
7. Hang the 100g 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). On 12/12/19, resonance frequency was found to be about
10. 0.797 Hz, and resonance amplitude was about 30 cm (ie 60 cm peak to peak).

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 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:

• Wikipedia - Harmonic Oscillator

• Wikipedia - Resonance

• Hyperphysics- Driven Oscillator

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