Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Oscillations and waves ↓


Damped and driven oscillations


Oscillations are a fundamental concept in physics that describes how systems evolve over time. From simple pendulums to complex electronic circuits, oscillations are everywhere. A particularly essential type of oscillation to understand is damped and driven oscillations. In this exploration, we will delve deeper into these concepts, using both textual and visual examples to aid understanding.

Basic concepts

Let's start by reviewing some basic concepts. Oscillations typically occur in systems where there is a restoring force that tries to return the system to a state of equilibrium. A classic example is a spring-mass system, where a mass attached to a spring oscillates back and forth when displaced.

Simple Harmonic Motion (SHM)

Simple harmonic motion (SHM) refers to a type of oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position. Mathematically, it is represented by:

F = -kx

Where F is the force, k is the spring constant, and x is the displacement.

balance maximum displacement maximum displacement

The motion is periodic and the position x(t) as a function of time t can be described as:

x(t) = a cos(ωt + φ)

Here, A is the amplitude, ω is the angular frequency, and φ is the phase constant.

Damped oscillation

In a real-world system, oscillations are often not ideal. They lose energy over time due to resistive forces, such as friction or air resistance. This energy loss results in an effect called damping.

The damping force is usually proportional to the velocity of the moving object and can be expressed as:

F_d = -bv

Where F_d is the damping force, b is the damping coefficient, and v is the velocity.

Types of damping

  • Underdamped: Oscillations occur with gradually decreasing amplitude. The system eventually stops.
  • Critically damped: The system returns to equilibrium as quickly as possible without oscillation.
  • Overdamped: The system returns to equilibrium slowly, without oscillating.
Underdamped severely damped overdamped

The equation of damped oscillation is given as:

m*d^2x/dt^2 + b*dx/dt + kx = 0

Where m is the mass, b is the damping coefficient, and k is the spring constant.

Driven oscillation

Driven oscillations occur when an external force is applied to the system, providing it with a continuous supply of energy. This external force is usually periodic, leading to the formation of a driven harmonic oscillator.

The equation governing the driven oscillations can be expressed as:

m*d^2x/dt^2 + b*dx/dt + kx = F_0 cos(ω_d t)

Where F_0 is the amplitude of the external force and ω_d is its angular frequency.

Echo

An important phenomenon related to driven oscillations is resonance. Resonance occurs when the frequency of the driving force matches the natural frequency of the system. At resonance, the amplitude of the system can increase dramatically.

An everyday example of resonance is pushing a swing. When you match the swing's natural frequency with your pushes, you can make it sound even louder.

Echo

Combination of damped and driven oscillations

In reality, most systems have both damping and driving forces. Mathematically, this is expressed as:

m*d^2x/dt^2 + b*dx/dt + kx = F_0 cos(ω_d t)

The presence of both damping and driving elements leads to a complex interplay of forces. The system will come to a steady state where the energy supplied by the external force balances the energy lost due to damping. Thus this steady-state response of the system depends critically on the driving frequency, damping, and natural frequency.

Conclusion

Understanding damped and driven oscillations in classical mechanics provides insight into many physical systems and phenomena. Whether it is a pendulum, an electrical circuit, or even celestial mechanics, the theory of oscillations provides invaluable tools to predict and explain system behavior. By mastering these concepts through examples and mathematical expressions, you empower yourself to tackle a wide range of problems in physics and engineering.


Undergraduate → 1.8.2


U
username
0%
completed in Undergraduate

← Prev (1.8.1)
Simple Harmonic Motion
Next (1.8.3) →
Coupled oscillations and common modes

Comments 



Damped and driven oscillations

https://www.physicsflow.com/ug/1.8.2?pfal=en