Physics - Classical Mechanics - Forced Oscillation and Resonance

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Introduction

Hey it's a me again @drifter1!

Today we continue with Physics, and more specifically the branch of "Classical Mechanics" in order to get into Forced Oscillation and Resonance.

So, without further ado, let's get straight into it!

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Forced Oscillation

Continuing on from the damped oscillator discussed last time, it's possible to add a so called harmonic driving force, that causes the system to oscillate "by force". Such an oscillation is known as forced oscillation.

Differential Equation

The differential equation that describes the motion of a forced oscillator, is shown below.

where the driving (external) force F_d is of the form:

The ω_d in this equation is called the driving frequency (or frequency of the external force), and isn't necessarily that same as the natural frequency of oscillation (ω₀) nor the oscillation (ω) that the damping oscillator has without any external force applied (free system oscillation frequency).

Amplitude

At first, when the damped oscillator is forced through an external driving force, the motion is chaotic (transient state). After some time has passed it reaches a steady state, where the motion is then periodic:

Let's skip the proof, and directly state that the amplitude is then given by:

Thus, as the frequency of the driving force (ω_d) approaches the natural frequency of the system (ω₀), the amplitude of the oscillations increases. The amplitude is maximum when ω_d=ω₀.

Resonance

Continuing on from that last statement, when ω_d=ω₀, we say that the system is resonating. Resonance is an interesting phenomenon, and even though it's simple, it can still be critical in lots of situations.

Near resonance, the damping should not be ignored, because as the amplitude reaches infinity the damping constant b reaches zero, which is not physical. Of course, infinite amplitude never occurs in practice, but as the amplitude reaches it's peak there may occur non-linearities in the system (large oscillations) that can be quite catastrophic.

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RESOURCES:

References

1. https://math.libretexts.org/Bookshelves/Differential_Equations/Book%3A_Differential_Equations_for_Engineers_(Lebl)/2%3A_Higher_order_linear_ODEs/2.6%3A_Forced_Oscillations_and_Resonance
2. https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Book%3A_University_Physics_I_-Mechanics_Sound_Oscillations_and_Waves(OpenStax)/15%3A_Oscillations/15.07%3A_Forced_Oscillations
3. https://ocw.mit.edu/courses/physics/8-03sc-physics-iii-vibrations-and-waves-fall-2016/syllabus/MIT8_03SCF16_Text_Ch2.pdf

Images

1. https://commons.wikimedia.org/wiki/File:Damped_oscillation_function_plot.png

Mathematical equations used in this article, where made using quicklatex.

Visualizations were made using draw.io.

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Previous articles of the series

Rectlinear motion

• Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
• Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
• Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
• Rectlinear motion exercises -> examples and tasks in rectlinear motion

Plane motion

• Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
• Projectile motion as a plane motion -> missile/bullet motion as a plane motion
• Smooth Circular motion -> smooth circular motion theory
• Plane motion exercises -> examples and tasks in plane motions

Newton's laws and Applications

• Force and Newton's first law -> force, 1st law
• Mass and Newton's second law -> mass, 2nd law
• Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
• Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
• Contact forces and friction -> contact force, friction
• Dynamics of Circular motion -> circular motion dynamics, applications
• Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
• Contact force and friction examples -> exercises in force and friction
• Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
• Advanced Newton law examples -> advanced (more difficult) exercises

Work and Energy

• Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
• Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
• Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
• Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
• Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
• Internal Energy and Work -> Internal Energy, Internal Work

Momentum and Impulse

• Conservation of Momentum -> Momentum, Conservation of Momentum
• Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
• Collision Examples -> Various Elastic and Inelastic Collision Examples
• Impulse -> Impulse with Example
• Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
• Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis

Angular Motion

• Angular motion basics -> Angular position, velocity and acceleration
• Rotation with constant angular acceleration -> Constant angular acceleration, Example
• Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
• Parallel Axis Theorem -> Parallel axis theorem with example
• Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
• Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
• Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
• Angular Momentum -> Angular Momentum and its conservation
• Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
• Exercises around Angular motion -> Angular motion examples

Equilibrium and Elasticity

• Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
• Force Couple System -> Force Couple System, Example
• Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
• Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
• Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
• Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
• Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
• Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples

Gravity

• Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
• Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
• Gravitational Fields -> Gravitational Field Mathematics and Visualization
• Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
• Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
• Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
• Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
• Kepler's Laws of Planetary Motion -> Kepler's Story, Elliptical Orbits, Kepler's Laws
• Spherical Mass Distributions -> Spherical Mass Distribution, Gravity Outside and Within a Spherical Shell, Simple Examples
• Earth's Rotation and its Effect on Gravity -> Gravity on Earth, Apparent Weight
• Black Holes and Schwarzschild Radius -> Black Holes (Creation, Types, How To "See" Them), Schwarzschild Radius

Periodic Motion

• Periodic Motion Fundamentals -> Fundamentals (Period, Frequency, Angular Frequency, Return Force, Acceleration, Velocity, Amplitude), Simple Harmonic Motion, Example
• Energy in Simple Harmonic Motion -> Forms of Energy in SHM (Potential, Kinetic, Total and Maximum Energy, Maximum Velocity), Simple Example
• Simple Harmonic Motion Equations -> SHM Equations (Displacement, Velocity, Acceleration, Phase Angle, Amplitude)
• Simple Harmonic Motion and Reference Circle -> SHM and Smooth Circular Motion, Reference Circle
• Simple Harmonic Motion Exercises -> 2 Complete Examples on Simple Harmonic Motion
• Simple Pendulum -> Simple Pendulum (Return Force, Small Angle Approximations, More Accurate Period, Gravity Approximation)
• Physical Pendulum -> Physical Pendulum (Return Torque, Small Angle Approximations, Estimating Moment of Inertia)
• Exercises around Pendulums -> Complete Examples on the 2 types of Pendulums (Simple, Physical)
• Damped Oscillation -> Damping Force, Total Force and Differential Equation, Motion Equations, Special Cases

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Final words | Next up

And this is actually it for today's post!

Next time we will get into Exercises on the topics of Damped and Forced Oscillation...

See ya!

Keep on drifting!

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