Normal Mode and Musical Theory

​Normal Mode

By Judy Hutton

What is a Normal Mode?

​Normal modes are fundamental, independent patterns of vibration in a physical system (like molecules, bridges, Musical instruments or oscillators) where all parts move sinusoidally at the same natural frequency. They represent the basic, characteristic ways a system oscillates, with complex motions often being a combination of these independent modes.

​Key Concepts

Natural Frequency: Every object has one or more natural frequencies at which it "wants" to vibrate after an initial push.
Resonance: Occurs when an external driving force's frequency matches an object's natural frequency, causing the amplitude of vibration to build up. Amplitude: At resonance, the vibration's intensity (amplitude) becomes much larger, even with a small driving force.
Damping: The force that opposes vibration (like friction); minimal damping allows for greater resonance effects.

Match

Match the following Key Concepts

• Natural Frequency

• Resonance

• Amplitude

• Dampening

• Normal Mode

Every object has one

when an external driving frequency match

The maximum extent of a vibration

The force that opposes vibration

independentpatterns of vibration

• Every object has one

• when an external driving frequency match

• The maximum extent of a vibration

• The force that opposes vibration

• independentpatterns of vibration

​ Key Characteristics: In a normal mode, all components of a system move with a fixed phase relation and a single, characteristic frequency known as the resonant frequency.

​Resonance frequency is the specific frequency at which a system naturally vibrates with maximum amplitude when disturbed, determined by its physical properties (mass, shape, material). When an external force matches this frequency, energy efficiently transfers, causing vibrations to amplify dramatically, a phenomenon seen in everything from shattering wine glasses to tuning radios and bridges swaying. It's the frequency where a system stores vibrational energy best and oscillates most easily, leading to large responses from small pushes.

​Resonance frequency

​Physical Systems: Examples include a stretched string vibrating in harmonics, or atoms in a molecule vibrating at specific infrared frequencies.

Linear Systems: For linear systems, any complex movement can be described as a linear superposition (combination) of these normal modes.

​The Basic Principle of Oscillator Circuits
The core principle behind oscillator operation is positive feedback combined with an amplification process.
Energy needs to move back and forthbetween the two forms this is what causes the oscillation. Eventually, any physical oscillator stops moving because of friction. To keep it going, you have to add a little bit of energy on each cycle.

The Pendulum Example

One of the most commonly used oscillators is the pendulum of a clock. If you push on a pendulum to start it swinging, it will oscillate at a desired frequency — it will swing back and forth a certain number of times per second. The length of the pendulum is the main thing that controls the frequency.

Drag and Drop

The core principle behind oscillator operation is positive ​

Placeholder

combined with an ​

Placeholder

process. Energy needs to move back and forth between the two forms this is what causes the ​

Placeholder

. Eventually, any physical oscillator stops moving because of ​

Placeholder

. To keep it going, you have to add a little bit of ​

Placeholder

on each cycle.

Drag these tiles and drop them in the correct blank above

amplification

energy

friction

oscillation

feedback

Multiple Choice

What is an oscillator?

• a perpetual motion machine

• a system that exchanges heat energy

• a system with motion that repeats in cycles

Multiple Choice

What is the symbol that represents period?

• T

• p

• P

• .

Multiple Choice

The time it takes for one complete cycle.

• frequency

• period

Multiple Choice

What does this formula help you calculate?

• frequency

• period

Multiple Choice

What units measure frequency?

• meters per second

• Hertz

• seconds

• joules

Multiple Choice

An object has a frequency of 50 Hz. What is the period?

• 50 s

• 0.02 s

• 25 s

Multiple Choice

What is the period of the cycle indicated on this graph?

• T = 4 s

• T = 8 s

• T = 20 s

Multiple Choice

What is the frequency of the cycle in this graph?

• 0.12 Hz

• 20 Hz

• 0.04 Hz

• 4 Hz

Coupled oscillators are systems of two or more, usually harmonic, oscillators connected such that they exchange energy, with the motion of one influencing the others.
Common examples include pendulums linked by a spring, electrical circuits with coupled branches, or interconnected atomic lattices. These systems exhibit complex motion often analyzed through normal modes, where all parts oscillate at a single, specific frequency. 

Drag and Drop

​

Placeholder

oscillators are systems of two or more, usually ​

Placeholder

, oscillators connected such that they exchange ​

Placeholder

, with the motion of one influencing the others. These systems exhibit complex motion often analyzed through ​

Placeholder

, where all parts oscillate at a single, specific ​

Placeholder

.

Drag these tiles and drop them in the correct blank above

normal modes

frequency

Coupled

harmonic

energy

​Independence:
Normal modes are orthogonal, meaning they do not exchange energy, allowing each to be excited independently. Orthogonal normal modes are independent, unique vibration patterns (eigenvectors) of a linear system that are perpendicular to each other, meaning they do not exchange energy.
We know that the word orthogonal is kind of like the word perpendicular. It implies that two vectors have an angle of ninety degrees or half pi radians between them. But this term means much more than this, as we can have orthogonal matrices, or entire subspaces that are orthogonal to one another.

​It means that the different modes of motion of the system that you are studying are decoupled. In other words, they can be 'told apart', as they are physically distinct ways for the system to evolve in time. The catch is that each mode has a distinct energy, so that the total energy of a linear combination made up with different modes of vibration may be written as a sum over their energies

​Mathematics: Finding normal modes involves solving an eigenvalue problem, where eigenvectors represent mode shapes and eigenvalues represent frequencies.

Simple Harmonic Motion

By Stacy King

Equilibrium ​position

• ​resting (non moving)

• ​balanced forces

​

​

Oscillating Motion

​

External force that alters the equilibrium ​position

​

​

​Forced Vibration

​

The gradual decrease in amplitude and energy of a wave due to friction or other energy-loss mechanisms.

Damping

​

​Damping

•All vibrational systems are subject to some dampening.
•Underdamped oscillation is when the damping force is less than the critical damping force. This results in the oscillation decaying slowly. 

•Over-damping occurs when oscillations come to a halt after a significant period of time has passed since the resistive force was applied.

•It moves towards the equilibrium point more slowly than a critically damped object.

•There are no oscillations.

•Critical damping is defined as the threshold between overdamping and underdamping.

•In the case of critical damping, the oscillator returns to the equilibrium position as quickly as possible, without oscillating, and passes it once at most.

Drag and Drop

The gradual decrease in ​

Placeholder

and energy of a wave due to ​

Placeholder

or other ​

Placeholder

-loss mechanisms. All ​

Placeholder

systems are subject to some ​

Placeholder

Drag these tiles and drop them in the correct blank above

amplitude

friction

energy

vibrational

dampening

A force that acts on the oscillating object to restore ​it back to equilibrium

​

​

​​Restoring Force

​

​

Periodic motion

• Is motion around a central equilibrium point. The motion repeats itself over and over again in cycles.

​

What does the graph of position versus time look like?

​

​

​Oscillation

Drag and Drop

An ​

Placeholder

is Periodic motion around a central ​

Placeholder

point that repeats itself over and over again in ​

Placeholder

. It loses ​

Placeholder

over time due to ​

Placeholder

and other forces.

Drag these tiles and drop them in the correct blank above

Oscillation

equilibrium

cycles

energy

friction

Multiple Choice

Which graph do you think would represent oscillation?

The graph indicates the repeating cycles

​

​Graphic Analysis of Oscillation

Multiple Select

MORE THAN 1 is correct.

Which of these would be considered oscillators?

A full cycle is one complete back and forth motion.

​

The period is the time it takes to complete one full cycle.

​

Period T is measured in seconds.

​

​Cycles and Periods

Multiple Choice

What is the period of the Earths rotation?

• 24 hrs

• 48 days

• 24 days

• 48 hrs

Frequency is how many cycles are completed each second.

​

Frequency f is measured in hertz, or Hz.​

​Frequency

​Write This Down

Multiple Choice

Javier is on a swing. His feet brush the ground every 3.0 seconds.

What is Javier’s frequency?

• .33

• .17

• 1

• 6

Multiple Choice

Marie has a spring-mass system with a frequency of 4 Hz.  What is the system’s period?

• .25

• 4

• 1

• .17

Oscillations occur in systems with stable equilibrium.

​

Stable systems have restoring forces that act to return them to the equilibrium position if they are displaced.

​

​Restoring Force

Multiple Choice

What provides the restoring force for a simple pendulum?

• Gravity

• Centripetal

• Friction

​The maximum displacement of the object from equilibrium

​

X signifies the displacement variable​ and is measured in meters....

​

​

​

​Amplitude

Multiple Choice

What is the amplitude of the graph? 

• 1
• 3
• 6
• 9

Equilibrium

• restoring force is zero

​Displacement

• ​restoring force is toward the line of equilibrium

​

​

​Force and Position

​

Multiple Choice

.......... is the diatance of the diatance of the oscillating object from yhe equilibrium position at a given instant

• frequency

• time

• period

• displacement

​Kinetic- energy due to motion

Potential-​ energy that is stored

Total mechanical energy​- energy acquired by the objects upon which work is done

​

​

​

​

​Some text here about the topic of discussion.

​

​Energy and Oscillation Pendulum

​

KE = ​1/2 mv²

​

KE increases as the object moves toward equilibrium ​

​

​

​Some text here about the topic of discussion.

Calculating Kinetic Energy Pendulum

​

Multiple Choice

A body executes simple harmonic motion. The potential energy, the kinetic energy and total energy are measured as a function of displacement x. Which of the following statements is true?

• Kinetic energy is maximum when x = 0

• Total energy is zero, when x = 0

• Kinetic energy is maximum when x is maximum

• Potential energy is maximum when x = 0

PE= mgh​

​

​Potential energy is greatest at maximum displacement

​

​

​Some text here about the topic of discussion.

Calculting Potential Energy​ Pendulum

Multiple Choice

Where does a pendulum have maximum potential energy?

• at the bottom of its arc

• at the end of its swing

• as it rises up

• as it falls

Multiple Choice

A pendulum is 0.75 meters long and has a period of 4.17 seconds. The Pendulum is on an unknown planet.  What is the gravity of the Unknown Planet?

• 9.8
• 3.4
• 1.7
• Greater than 9.8

F_spring = -k•x​

​

restorative Force is dependent on displacement and the springs ability to stretch (k= the springs constant)

​Motion of Mass on a Spring Hooke's Law

​

Multiple Choice

A spring has a spring constant of 450 N/m. How far is the spring compressed if 150 N of force are used?

• 2.2 m

• 3m

• 5.0 m

• 0.3 m

KE = ½•m•v²

​Kinetic Energy

​

​PEspring = ½ • k•x²

​elastic potential energy

Kinetic and Potential Energy of a Mass on a Spring

​

Multiple Choice

A force of 16 N is required to stretch a spring a distance of 40 cm from its rest position. What force (in Newtons) is required to stretch the same spring …

• 32 N

• not enough info given

• 8 N

• 16 N

​Music is the art of arranging vibrations through elements of: timing, melody, harmony, rhythm, and timbre It serving as a universal form of expression and communication,

​How is Music Created?

​Sound is created through vibrations.

If something vibrates, sound is created. ​

Sound​

Multiple Select

Music is the art of arranging vibrations through elements of:

• timing

• melody

• harmony

• rhythm

• timbre

Music is based on vibrating oscillations

Key Aspects of Musical Note Vibrations 

• Pitch and Frequency: The faster an object vibrates, the higher the pitch. A

  256 Hz256 Hz vibration corresponds to Middle C.

• Octaves: Doubling the frequency (e.g., 440 Hz to 880) results in the same note one octave higher.

• Measurement: Frequencies are measured in Hertz (Hz), representing cycles per second.

• Structure: Musical notes are often composed of a fundamental frequency combined with harmonics (multiples of that frequency), which gives instruments their unique timbre. 

Frequency is the amount of times an object vibrates in a given amount of time.​

​

​

​What is frequency?

Sound​

​What is pitch?

Pitch is the highness or lowness of a sound. The higher the frequency of the sound, the higher the pitch. The lower the frequency of the sound, the lower the pitch.

​

Higher Frequency = High Pitch

Lower Frequency = Low Pitch​

Science​

Sound​

​- Name of Author

​- Name of Author

" Replace this with a quote, words full of wisdom that someone important said and can inspire the reader. " 

1. ​Choose some different size rubber bands.

   2. ​Wrap the rubber bands around objects. Create sound by vibrating the rubber bands.

   3. Using your rubber bands, create sounds that are low pitch. What did you do to get low pitch sounds?

   4. Using your rubber bands, create sounds that are high pitch.​ What did you do to get high pitch sounds?

​Thick rubber bands usually create lower pitch sound because the frequency is lower.

​​Thick Rubber Bands

​Thin rubber bands usually create higher pitch sound because the frequency is higher.

​​Thin Rubber Bands

​

​Looser rubber bands usually create lower pitch sound because the frequency is lower.

​​Looser Rubber Bands

​

​Tighter rubber bands usually create higher pitch sound because the frequency is higher.

​​Tighter Rubber Bands

​

​Thicker, Looser = Low Pitch

Thinner, Tighter = High Pitch​

Multiple Choice

How is sound created?

• Through light

• Through pitch

• Through vibrations

• Through frequencies

Multiple Choice

What is frequency?

• The amount of times an object vibrates in a given time.

• High pitch

• Low pitch

• The highness or lowness of a sound

Multiple Choice

True or false? High frequency vibrations create high pitch sounds.

• True

• False

Notes

• Sounds are caused by vibrations

• Sound travels in waves

• Frequency is the number of times a sound wave repeats in a second

• Pitch is the highness or lowness of sounds

• Volume is the loudness or softness of sounds.

Vocabulary

• Vibration- something that moves back and forth quickly.

• Frequency- the number of wave cycles per second.

• Decibels- a unit for measuring the volume of sounds. The quietest sound a human ear can hear is 0 decibels.

• Pitch- the highness or lowness of a musical sound.

• Hertz- A unit for measuring the frequency of vibrations and waves, equal to one cycle per second. The abbreviation for hertz is Hz.

• Echo- when a sound repeats because the sound wave hit a large object and bounced back.

Multiple Choice

What role does energy play in sound?

• Sound waves have nothing to do with energy.

• Sounds waves carry energy.

• Energy is present in loud sounds only.

• Energy is present in high-pitched sounds but not in low-pitched sounds.

Multiple Choice

What sort of frequency do high-pitched sounds have?

• high

• low

• medium

• no frequency

Multiple Choice

What does volume measure?

• the speed of a sound

• the frequency of a sound

• the highness or lowness of a sound

• the loudness or softness of a sound (amplitude)

Multiple Choice

Which two aspects of sound go together?

• pitch and frequency

• decibels and pitch

• pitch and volume

• amplitude and pitch

Multiple Choice

What does amplitude measure?

• frequency

• volume

• pitch

• echoes

Multiple Choice

What happens to the vibrations as the volume goes up?

• The vibrations get stronger

• The vibrations get weaker

• The vibrations stay the same.

Match

Match the following

• The faster an object vibrates, the higher the __.

• Doubling the frequency (e.g.,

  440 Hz to

  880 Hz results in the same note one __ higher

• Frequencies are measured in __

• __ a fundamental frequency combined with multiples of the same frequency

•   A 256 Hz vibration

pitch

octave

Hertz

Harmonics

Middle C

• pitch

• octave

• Hertz

• Harmonics

• Middle C

​The Seven Modes (Light to Dark):
1. Lydian: Bright, dreamy, or anxious.
2. Ionian: Happy, stable.
3. Mixolydian: Bluesy, rock-oriented,, relaxed.
4. Dorian: Jazzy, minor, or "cool".
5. Aeolian: Sad, serious.
6. Phrygian: Exotic, intense, dark.
7. Locrian: Tense, unstable.