Comprehensive Notes on Oscillations and Resonance

Simple Phenomena in Unrelated Situations

• Dynamics manifest in:
  • Spring-mass systems
  • Pendulums
  • Oscillatory electromagnetic circuits
  • Bio rhythms
  • Share market fluctuations
  • Electrocardiography (ECG)
  • Radiation oscillators
  • Molecular vibrations
  • Atomic, molecular, solid-state, and nuclear physics
  • Electrical, Mechanical, and Chemical engineering

Small Oscillations

• Galileo Galilei (1581): Observed swaying chandeliers at the Pisa cathedral.
  • Recognized the constancy of the periodic time for small oscillations at age 17.
  • This was the first theory of dynamical systems.

Taylor Series Expansion

• Taylor Series Expansion:
  $U(x) = U(x<em>0) + \frac{\partial U}{\partial x}(x - x</em>0)|<em>{x</em>0} + \frac{1}{2!} \frac{\partial^2 U}{\partial x^2}(x - x<em>0)^2|</em>{x<em>0} + \frac{1}{3!} \frac{\partial^3 U}{\partial x^3}(x - x</em>0)^3|<em>{x</em>0} + …$
• Small oscillations imply neglecting higher-order terms in the Taylor series.
• For a Linear Harmonic Oscillator:
  • The constant term in the potential has no physical significance; it only adds a constant value and does not contribute to the force.
  • 'Small' oscillations: We take the origin at $x = x<em>0$, setting $x</em>0 = 0$.

Periodic Motion

• Cause: A body attached to a spring, when displaced from its equilibrium position, experiences a restoring force.
  • This force tends to bring the object back to its equilibrium position, causing oscillation or periodic motion.

Glider-Spring System

• x > 0:
  • Glider displaced to the right from equilibrium.
  • Restoring force $F<em>x < 0$ and acceleration ax < 0.
  • The stretched spring pulls the glider toward equilibrium.
• x = 0:
  • The relaxed spring exerts no force.
  • The glider has zero acceleration.
• x < 0:
  • Glider displaced to the left from equilibrium.
  • Restoring force $F<em>x > 0$ and acceleration ax > 0.
  • The compressed spring pushes the glider toward equilibrium.

Characteristics of Periodic Motion

• Amplitude (A): Maximum magnitude of displacement from equilibrium.
• Period (T): Time for one complete cycle.
• Frequency (f): Number of cycles per unit time.
• Angular Frequency (ω):
  • $ω = 2πf$
• Relationship between Frequency and Period:
  • $f = \frac{1}{T}$
  • $T = \frac{1}{f}$

Simple Harmonic Motion (SHM)

• Occurs when the restoring force is directly proportional to the displacement from equilibrium.
• In many systems, the restoring force is approximately proportional to displacement for sufficiently small displacements.
  • If the amplitude is small enough, oscillations are approximately simple harmonic.

SHM as a Projection

• SHM can be viewed as a projection of uniform circular motion.
• Reference Circle: The circle in which a point moves such that its projection matches the motion of the oscillating body.
• Phasor: A rotating vector (like OQ) that rotates with constant angular speed $ω$ around the reference circle.

Conditions for SHM

• (i) Stable Equilibrium: A position of stable equilibrium must exist.
• (ii) No Energy Dissipation: There should be no dissipation of energy.
• (iii) Acceleration Proportional to Displacement: Acceleration should be proportional to the displacement and opposite in direction.

Equations for SHM

• Restoring Force: $F = -kx$
• Newton's Second Law: $ma = -kx$
• Acceleration: $a = -\frac{k}{m}x$
• Differential Equation: $\frac{d^2x}{dt^2} = -\frac{k}{m}x$
• Solution: $x = x<em>0 \sin(ω</em>0t)$
• Second Derivative: $\frac{d^2x}{dt^2} = -ω_0^2x$
• Angular Frequency: $ω_0^2 = \frac{k}{m}$
• Initial Conditions:
  • At $t = 0$, $x = 0$
  • At $t = \frac{T}{4}$, $x = x_0$
• Thus, $ω<em>0t = \frac{π}{2}$, so $ω</em>0\frac{T}{4} = \frac{π}{2}$
• Period: $T = \frac{2π}{ω_0} = 2π\sqrt{\frac{m}{k}}$

Characteristics of SHM

• For a mass $m$ vibrating on an ideal spring with force constant $k$.
• The greater the mass in a tuning fork's tines, the lower the frequency of oscillation and the lower the pitch of the sound.

Pendulum Motion

• Proposed Solution: $θ = θ<em>0 \sin(ω</em>0t)$
• Initial Conditions:
  • $θ = 0$ at $t = 0$
  • $θ = θ_0$ at $t = \frac{T}{4}$
• Angular Frequency: $ω_0 = \sqrt{\frac{g}{l}}$
• Calculating Period:
  • We know $θ = θ<em>0$ at $t = \frac{T}{4}$, but $θ = θ</em>0$ at $ω_0t = \frac{π}{2}$
  • $ω \frac{T}{4} = \frac{π}{2} \implies T = \frac{2π}{ω_0}$
• Thus, $T = 2π\sqrt{\frac{l}{g}}$
• We have $θ = θ<em>0 \sin(ω</em>0t)$
• Frequency: $f = \frac{1}{2π} \sqrt{\frac{g}{l}}$

Displacement in SHM

• Displacement as a function of time:
  • $x = A \cos(ωt + Φ)$

Graphs of Displacement and Velocity

• Displacement: $x = A \cos(ωt + Φ)$
• Maximum Displacement: $x_{max} = A$
• Velocity: $v_x = -ωA \sin(ωt + Φ)$
• Maximum Velocity: $v_{max} = ωA$
• The $v_x-t$ graph is shifted by $\frac{1}{4}$ cycle from the $x-t$ graph.

Graphs of Displacement and Acceleration

• Displacement: $x = A \cos(ωt + Φ)$
• Maximum Displacement: $x_{max} = A$
• Acceleration: $a_x = -ω^2 A \cos(ωt + Φ)$
• Maximum Acceleration: $a_{max} = ω^2A$
• The $a-t$ graph is shifted by $\frac{1}{4}$ cycle from the $v-t$ graph and $\frac{1}{2}$ cycle from the $x-t$ graph.

Simple Harmonic Oscillator

• A body executing simple harmonic oscillations.
• It possesses both potential and kinetic energy.
• Elastic property (spring) stores potential energy by virtue of displacement from equilibrium.
• Inertia (mass) stores kinetic energy due to its velocity.
• Kinetic Energy: $E_k = \frac{1}{2}mv^2 = \frac{1}{2}mω^2A^2\cos^2(ωt + φ) = \frac{1}{2}mω^2A^2(1 - \sin^2(ωt + φ))$
• $E_k = \frac{1}{2}mω^2(A^2 - x^2) = \frac{1}{2}k(A^2 - x^2)$

Potential Energy

• Potential energy, $U$:
  • $U = -\int<em>0^x F dx = -\int</em>0^x -kx dx = \frac{1}{2}kx^2$
• Potential energy is proportional to the square of displacement.
• The mass is in a potential well created by the spring, characterized by a parabolic potential well.
• Total Energy:
  • $E = K.E + P.E = \frac{1}{2}k(A^2 - x^2) + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$
• Total energy is constant and does not depend on time.
• Total energy is proportional to the square of the amplitude.

Energy Diagrams for SHM

• Potential energy $U$ and total mechanical energy $E$ as a function of displacement $x$.
• Potential energy $U$, kinetic energy $K$, and total mechanical energy $E$ as a function of displacement $x$.

SHM, Reference Circle, and Wave Motion

• Relationship between simple harmonic motion, circle of reference, and wave motion.

Electromagnetic Oscillations

• Illustrative diagrams of electromagnetic oscillations involving capacitors and inductors.

Electromagnetic Waves

• Observed EM waves (Maxwell).
• Unit of frequency: Hertz (Hz).
• Observed Photoelectric Effect.
• Experiments by Hertz and Lenard in 1887.
• Heinrich Hertz (1857-1894): Electronics Engineer, Physicist.

Voltage and Current in Capacitors and Inductors

• Voltage lags the current in a capacitor by 90 degrees but leads the current in an inductor by the same amount.
• $V<em>{max} \propto Q</em>{max}$, not to $V$, as in a resistor.
• $V_L = L\frac{dI}{dt} = L\frac{d^2Q}{dt^2}$
• $V_C = \frac{Q}{C}$
• $L\frac{dI}{dt} = -\frac{Q}{C}$
• $I = \frac{dQ}{dt}$
• $\frac{dV}{dt} = \frac{I}{C}$

Capacitor Discharge

• Diagram illustrating capacitor discharge.
• Current is the rate at which the capacitor is discharging: $\frac{dq}{dt}$
• Voltage drop across the inductor depends upon the rate at which the current changes and the inductance $L$.

Energy in Capacitor and Inductor

• At an instant, charge on capacitor $q = q_0 \cos(ωt)$
• $I = \frac{dq}{dt} = -q_0ω \sin(ωt)$
• Energy in Capacitor: $U<em>C = \frac{q^2}{2C} = \frac{(q</em>0 \cos(ωt))^2}{2C} = \frac{q_0^2}{2C} \cos^2(ωt)$
• Energy in Inductor: $U<em>L = \frac{1}{2}LI^2 = \frac{1}{2}L(-q</em>0ω \sin(ωt))^2$
• But $ω = \frac{1}{\sqrt{LC}} \implies Lω^2 = \frac{1}{C}$
• $U<em>L = \frac{q</em>0^2}{2C} \sin^2(ωt)$
• $U<em>C + U</em>L = \frac{q<em>0^2}{2C} \cos^2(ωt) + \frac{q</em>0^2}{2C} \sin^2(ωt) = \frac{q_0^2}{2C} = Constant$

Energy Transfer

• Diagram depicting energy transfer between capacitor and inductor during oscillations.
• $i = 0, q = q_0$
• $i = i_{max}, q = 0$
• Energy stored in electric field of the capacitor.
• Energy stored in the magnetic field of the inductor.

Mechanical vs. Electrical Oscillations

• Mass/inertia corresponds to inductance.
• Capacitance corresponds to compliance $\frac{1}{k}$.
• Resistance corresponds to friction.
• Capacitor stores energy in electrical form.
• Inductor stores energy in magnetic form.

Simple Pendulum and Spring-Mass System

• Spring-Mass System:
  • $U(x) = \frac{1}{2}Kx^2$
  • $F(x) = -Kx$
  • $m\ddot{x} = -Kx$
  • $\ddot{x} = -\frac{K}{m}x$
  • Natural frequency: $ω_0 = \sqrt{\frac{K}{m}}$
• Simple Pendulum:
  • $ml\ddot{θ} = -mg \sin(θ)$
  • $\ddot{θ} = -\frac{g}{l} \sin(θ)$
  • For small angles: $\ddot{θ} \approx -\frac{g}{l}θ$
  • Natural frequency: $ω_0 = \sqrt{\frac{g}{l}}$

Generalized Coordinates

• In general, $\ddot{q} = -ω_0^2 q$
• $q$ is the generalized coordinate.
• $ω_0^2$ is a constant.
• There is a restoring force, equal and opposite, proportional to the displacement $q$.

Solutions

• The solution is $q(t) = Ae^{iω<em>0t} + Be^{-iω</em>0t}$
• $\omega_0 = \sqrt{\frac{K}{m}}$, (for a spring-mass system)
• $\omega_0 = \sqrt{\frac{1}{LC}}$, (for LC oscillator circuit)
• $\omega_0 = \sqrt{\frac{g}{l}}$, (for a simple pendulum)

Damped Oscillator

• Real situations involve energy dissipation (damping).
• Introduce velocity-dependent damping.
• $F_{damping} = -bv = -b\dot{x}$, where $b$ is the damping coefficient.

Damped Oscillations

• Force is proportional to the speed and acts opposite to the motion.
• Retarding force: $R = -bv$, where $b$ is the damping coefficient.
• Restoring force: $-kx$
• Newton's Second Law: $\sum F = ma \implies m \frac{d^2x}{dt^2} = -kx - b\frac{dx}{dt}$
• Solution for small damping:
  • $x(t) = Ae^{-\frac{b}{2m}t} \cos(ωt + φ)$
  • $x(t) = Ae^{-\frac{b}{2m}t} \sin(ωt + φ)$

Types of Damping

• Overdamped
• Critically damped
• Underdamped

Damped Oscillations in Real Systems

• Real-world systems have dissipative forces that decrease amplitude.
• The decrease in amplitude is called damping, and the motion is called damped oscillation.

Damping Coefficient

• As the value of "b" increases, the amplitude decreases more rapidly.
• Critical damping occurs when $b$ reaches a critical value $b_c$ where the system does not oscillate.
• Overdamped: b/2 > ω
• Critically damped: $b/2 = ω$

Forced Oscillations

• Applying a right push at the right time (appropriate frequency, amplitude, and phase).
• Inhomogeneous differential equation.
• General solution and particular integral.
• $\ddot{x} + 2 \gamma \dot{x} + ω_0^2 x = Fe^{iωt}$

Forced Oscillator

• A damped oscillator driven by an external periodic force.
• Periodic driving force: $F(t) = F_0 \sin(ωt)$
• Newton's Second Law: $\sum F = ma \implies F_0 \sin(ωt) - kx - b \frac{dx}{dt} = m \frac{d^2x}{dt^2}$
• Solution: $x = A \cos(ωt + φ)$
• $A = \frac{F<em>0/m}{\sqrt{(ω</em>0^2 - ω^2)^2 + (bω/m)^2}}$

Natural Frequency

• $ω_0 = \sqrt{\frac{k}{m}}$ is the natural frequency of the undamped oscillator $(b = 0)$.
• For small damping, amplitude is large when the driving force frequency is near the natural frequency, i.e., $ω ≈ ω_0$.
• Resonance occurs when the driving frequency is near the natural frequency.

Amplitude vs. Frequency

• Resonance occurs when the driving force equals the natural frequency.
• The curve's dependence changes as the value of the damping coefficient $b$ changes.

Forced Oscillations and Resonance

• A damped oscillator stops moving without an external force.
• Constant-amplitude oscillation can be maintained by applying a periodic driving force.
• Motion resulting from a periodic driving force applied to a damped harmonic oscillator is forced oscillation or driven oscillation.

Resonance

• The peaking of amplitude at driving frequencies close to the natural frequency.
• Examples:
  • Oscillations of a child on a swing.
  • Vibrating rattle in a car at a certain engine speed.
  • Tuned circuit in a radio receiver responding strongly to waves near its natural frequency.

Amplitude and Damping

• Curves of amplitude $A$ for an oscillator at various angular frequencies $ω_d$.
• Successive curves from blue to gold represent greater damping.
• A lightly damped oscillator exhibits a sharp resonance peak when $ω_d$ is close to $ω$, the natural angular frequency.
• Stronger damping reduces, broadens, and shifts the peak to lower frequencies.
• If b > \sqrt{2km}, the peak disappears completely.

Forced Oscillations in Nature

• Lady beetle flies by means of forced oscillation.
• Muscles apply a periodic driving force, deforming the exoskeleton rhythmically, causing the wings to beat up and down.
• The oscillation frequency of wings and exoskeleton is the same as the driving force.

Tacoma Bridge Collapse

• Example of when engineers neglected Physics (resonance).

Resonance Examples

• Rattling of loose pieces in vehicles.
• Bridge collapse while troops march.
• Airplane wings falling off.

Applications of Resonance

• Resonance is not always bad.
• NMR, resonance spectroscopy.
• MRI scanner.