Physics - Classical Mechanics - Exercises around Damped and Forced Oscillation
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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 Exercises around Damped and Forced Oscillation.
So, without further ado, let's get straight into it!
Example 1: Damped Oscillation
Let's consider a mass of 0.2 Kg is attached to one side of a spring with constant 320 N/m. The system starts oscillating, while a damping force of the form F = -bv acts upon it.
Calculate:
- The period of oscillation (T) for damping constant b = 8 Kg/s
- The value of b for critical damping
- The value of b for an initial amplitude of 0.5 m and decrease to 0.1 m in 3 sec
Solution
The period is related to the angular frequency (ω), which for the corresponding values can easily be calculated as follows:
And so, for b = 8 Kg/s the period is:
Now for the next question, as we know critical damping is achieved for a damping constant of:
The last question is a little more complicated. We will have to use the motion equation:
Everything is then known except for the damping constant. Of course, the initial phase is φ = 0, because we have maximum positive displacement. Because the angular frequency (ω) is unknown, we will also have to substitute with the square root equation, where again only the value of b is missing.
Well, what remains now is simply solving this "abomination" for b and we're done. Using wolfram alpha, the numerical result that was returned is:
Example 2: Forced Oscillation
Let's continue on from the previous example, by applying an external driving force Fd upon the system. Consider the damping constant b = 8 Kg/s for the calculations.
Calculate:
- The maximum value of the driving force (F0) if the amplitude of the system (after reaching steady state) is A = 1 m and the driving frequency is ωd = 20 rad /sec
- The maximum possible amplitude that the oscillation can reach for that same maximum driving force
Solution
Let's first calculate the natural frequency of the system:
The driving frequency is smaller and thus we expect the maximum possible amplitude to be larger for the same maximum driving force.
Substituting, the known values into the amplitude equation, the maximum driving force is:
The maximum amplitude is achieved for ωd = ω0, which turns the first term in the square root into zero, and yields:
As expected the amplitude is larger...
RESOURCES:
References
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Mathematical equations used in this article, where made using quicklatex.
Visualizations were made using draw.io.
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
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- Advanced Newton law examples -> advanced (more difficult) exercises
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- Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
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- Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis
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- Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
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- Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
- Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
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- Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
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- Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
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- Earth's Rotation and its Effect on Gravity -> Gravity on Earth, Apparent Weight
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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)
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- 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
- Forced Oscillation and Resonance -> Forced Oscillation (Differential Equation, Amplitude, Resonance)
Final words | Next up
And this is actually it for today's post!
Next time we will get into Chaotic Oscillation (or Chaos)...
See ya!
Keep on drifting!
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