Oscillation – Comprehensive Study Notes (Ch. 2: SHM, Damped & Forced Vibrations)

Oscillation – Comprehensive Study Notes

2.1 Introduction

Simple harmonic motion (SHM) is a special case of generalized periodic motion.
Two broad categories of motion encountered in daily life:
- Periodic motion: motion about a mean position (fixed point).
- Translatory motion: motion from one place to another with respect to time.
Examples:
- Translational: moving bus, airplane, football.
- Periodic: simple pendulum, spring–mass system, vibrations of a stretched string.
SHM is the periodic motion in which the concerned body moves along a straight line; the body is called a simple harmonic oscillator (SHO).
Free (undamped) vibrations:
- Vibration of a body completely free from external forces.
- Persistence of vibration without loss of amplitude is idealized; real systems show damping.
Real systems show damping due to friction, resistance, etc., causing energy dissipation and eventual rest.
Forced vibrations:
- When a system is acted on by an external periodic force, it undergoes forced vibration.
- Initially, the system tends to vibrate at its natural frequency, but under forcing it ultimately vibrates at the forcing frequency.
- If the forcing frequency matches the natural frequency, resonance occurs, producing large amplitudes.
Resonance: large-amplitude oscillations when forcing frequency equals natural frequency; energy transfer is efficient but can drain the forcing system if coupling is strong.
Coupled vs forced vibrations:
- If the forcing system is not affected by the forced system (energy of forcing system remains unchanged), the vibration is forced.
- If the forcing system is modified by the response (the forced system), a coupled system results.

2.2 Relation of Simple Harmonic Motion with Circular Motion

SHM is a projection of uniform circular motion onto one diameter of the reference circle.
Consider a circle with centre C and a point P moving on its circumference; a projection onto the diameter CY' yields y as the displacement.
Let P move uniformly with angular velocity
\,: the angle with CO is
\, = ω t.
Drop perpendicular PM to diameter YY'. Point M moves such that as P completes a revolution (angle 2π), M completes an oscillation along the straight line.
Relationship:
- y = a sin φ, where φ = ω t (or φ = ω t − δ for a phase shift).
- General SHM form: y = a sin(ω t − δ).
If we consider a phase shift, the generalized SHM equation is
y = a \, ext{sin}(ω t - δ)
In the case of a phase origin shift, we can write
y = a \, ext{sin}(ω t - 8)
which is equivalent to a shifted time origin, reinforcing the generalized SHO form
y = a \, ext{sin}(ω t + d)
By introducing a reference point P that moves uniformly on the circle with angular velocity ω, the motion of M is represented by
y = a \, ext{sin}(α t - δ)
If we set α = ω and replace t by t, the standard form is recovered.
Key: displacement, velocity, and acceleration are connected to the circular motion through the geometry of the reference circle.

2.3 Differential Equation of Simple Harmonic Motion

General displacement for SHM:
y = a \, ext{sin}(ω t + d)
Velocity (first derivative):
rac{dy}{dt} = a ω \, ext{cos}(ω t + d)
Acceleration (second derivative):
rac{d^2 y}{dt^2} = -a ω^2 \, ext{sin}(ω t + d) = -ω^2 y
Therefore, the differential equation of SHM is
rac{d^2 y}{dt^2} + ω^2 y = 0
This is the standard equation for SHM with angular frequency
ω = 2π ν
The time period is
T = rac{2π}{ω}

2.4 Various Characteristics of SHM

Characteristics of SHM:
- Periodic and oscillatory motion.
- Restoring force F restoring the mass is proportional to displacement and directed opposite to it: F ∝ −y.
- Acceleration a is always directed toward the mean position.
- Motion occurs along a straight line (one-dimensional).
Summary relation:
F = -k y \, ext{(restoring force)}, \, a = -ω^2 y, \, ext{where} \, ω^2 = rac{k}{m}.

2.5 Energy of a Particle Executing SHM and Law of Conservation of Energy

Kinetic energy (Ek) for SHM with velocity v is Ek = rac{1}{2} m v^2 = rac{1}{2} m [a ω \, ext{cos}(ω t + d)]^2 = rac{1}{2} m ω^2 (a^2 - y^2)
Potential energy (Ep) due to the restoring force is Ep = rac{1}{2} k y^2 \, ext{with} \, k = m ω^2 \
E_p = rac{1}{2} m ω^2 y^2
Total energy E is constant:
E = Ek + Ep = rac{1}{2} m ω^2 (a^2 - y^2) + rac{1}{2} m ω^2 y^2 = rac{1}{2} m ω^2 a^2.
Energy distribution (illustrative): at extreme positions y = ±a, Ek = 0, Ep = ½ k a^2; at mean position y = 0, Ep = 0, Ek = ½ m ω^2 a^2.
The energy of SHO is proportional to the square of the amplitude a.

2.6 Differential Equation of Free or Undamped Vibrations

For a free oscillator (no damping), the energy remains constant and the equation is
rac{d^2 y}{dt^2} + ω^2 y = 0 \,\text{with} \, ω^2 = rac{k}{m}.
This is the ideal equation for a free, undamped oscillator.

2.7 Damped Vibrations

Real systems have damping forces (friction, resistance) leading to energy dissipation.
General damped equation (for mass m, damping coefficient B, stiffness k):
m \frac{d^2 y}{dt^2} + B \frac{dy}{dt} + k y = 0.
Introducing damping per unit mass: define
β = \frac{B}{m},\quad ω_0^2 = \frac{k}{m}.
Then the normalized equation becomes
\frac{d^2 y}{dt^2} + 2β \frac{dy}{dt} + ω_0^2 y = 0.

2.8 Solution of the Equation of a Damped Oscillator and Its Analysis

Trial solution: assume
y = A e^{α t}.
Substituting into the damped equation yields characteristic equation
α^2 + 2β α + ω_0^2 = 0.
Roots:
α = -β \pm \sqrt{β^2 - ω_0^2}.
Three damping regimes:
- Case 1: Under-damped (β^2 < ω0^2): y = e^{-β t} \left[ A \cos(\sqrt{ω0^2 - β^2}\; t) + B \sin(\sqrt{ω_0^2 - β^2}\; t) \right].
- Case 2: Over-damped (β^2 > ω0^2): non-oscillatory exponential decay y = A1 e^{(-β + \sqrt{β^2 - ω0^2}) t} + A2 e^{(-β - \sqrt{β^2 - ω_0^2}) t}.
- Case 3: Critically damped (β^2 = ω_0^2): fastest non-oscillatory return to equilibrium; solution of the form
  y = (A + Bt) e^{-β t}.
Logarithmic decrement (for under-damped case):
- If amplitudes on the same side are A1, A2 separated by one period T, then
  A1 = A2 e^{λ} \,\text{with} \, λ = \ln \left(\frac{A1}{A2}\right)
- More commonly written as A(t) = A e^{-β t}, so the decrement per cycle is
  λ = \frac{1}{N} \ln\left( \frac{A(t)}{A(t+NT)} \right) = β T.
The angular frequency of damped motion (underdamped) is
ωd = \sqrt{ω0^2 - β^2}.
Notes:
- In the aperiodic (over-damped) case, the system returns to equilibrium without oscillating.
- In the critically damped case, return is fastest without oscillation.

2.9 Electrical Analogy of SHM and DV

SHM and DV analogies in electrical circuits:
- Mass–spring–damper corresponds to an LC or LCR circuit depending on damping.

2.9.1 SHM in an LC Circuit

Consider a parallel LC circuit with inductor L and capacitor C; charge q on the capacitor satisfies
\frac{d^2 q}{dt^2} + \frac{1}{LC} q = 0.
This is identical in form to SHM with angular frequency
ω^2 = \frac{1}{LC},\quad T = 2π\sqrt{LC}.
Current and voltage relationships:
- Charge q(t) = q_0 \cos(ω t) (assuming appropriate initial conditions).
- Current i(t) = \frac{dq}{dt} = -q_0 ω \sin(ω t).

2.9.2 Damped Vibration (DV) in an LCR Circuit

LCR circuit with resistance R, inductance L, capacitance C: forcing by an initial charge and subsequent discharge with damping due to R leads to
\frac{d^2 q}{dt^2} + \frac{R}{L} \frac{dq}{dt} + \frac{1}{LC} q = 0.
This is the electrical analogue of the damped mechanical oscillator with
2β = \frac{R}{L},\quad ω_0^2 = \frac{1}{LC}.
Underdamped, overdamped, and critically damped regimes map directly onto the mechanical cases described above.

2.10 Analysis of Forced Vibration

External periodic force on a mass m:
- Forcing: F cos(ω' t).
- Restoring force: −k y; Damping: −c dy/dt (where c is a damping coefficient).
Equation of motion:
m \frac{d^2 y}{dt^2} + c \frac{dy}{dt} + k y = F \cos(ω' t).
Introducing natural frequency ω0 and damping ratio, the homogeneous part is m y'' + c y' + k y = 0 \,\Rightarrow \, y'' + 2β y' + ω0^2 y = 0,
with
ω_0^2 = \frac{k}{m}, \quad β = \frac{c}{2m}.
Assume steady-state solution of the form
y = A \cos(ω' t - α).
Differentiating and substituting gives relationships:
-A ω'^2 \cos(ω' t - α) - 2β A ω' \sin(ω' t - α) + ω0^2 A \cos(ω' t - α) = F \cos(ω' t), which, by matching coefficients of cos and sin, yields: A = \frac{F}{\sqrt{(k - m ω'^2)^2 + (2 β m ω')^2}}, \tan α = \frac{2 β m ω'}{k - m ω'^2} = \frac{2 β ω'}{ω0^2 - ω'^2}.
The forced vibration is SHM with frequency ω' (the forcing frequency) and phase lag α.

2.11 Resonance

Forced vibrations can lead to resonance when the forcing frequency matches the natural frequency (or near it in the presence of damping).
Amplitude resonance: amplitude A is maximal when the denominator of A is minimal, which leads to the condition for maximum A at a driving frequency ω'r given by: ω'r = \sqrt{ω_0^2 - 2β^2}.
The maximum amplitude depends on damping: smaller damping (smaller β) yields larger A at resonance; sharper resonance for smaller damping.
In the ideal undamped case (β → 0), resonance would yield infinite amplitude (physical systems always have some damping).

2.11.1 Amplitude Resonance (Summary)

Peak amplitude occurs at driving frequency ω'r approximately equal to ω0 for small damping, but corrected for damping by ω'r = \sqrt{ω0^2 - 2β^2} in the presented treatment.
Maximum amplitude expression (as given): a form involving ω0, β, and forcing parameters (exact constants given in the source, typically Amax ∝ 1/(2β ω'_r) under certain approximations).
Qualitative result: the smaller the damping parameter β, the higher and sharper the resonance peak.

2.12 Energy (or Velocity) Resonance

In forced vibration, when the forcing frequency ω' equals the natural frequency ω (or the relevant resonance condition), the energy transfer is maximized.
Energy resonance or velocity resonance refers to the situation where the energy pumped into the system is maximized for a given forcing amplitude and damping.
Expression for energy at resonance (as given in the source): E_max occurs when ω' = ω and relates to the forcing amplitude F, damping b, and mass m (specific form provided in the text as Eq. (2.47)); conceptually, the energy is maximized at resonance for a given drive and damping.

2.13 Sharpness of Resonance

The sharpness of resonance measures how quickly the energy falls off as you move away from the resonant frequency.
The resonance energy E(ω') can be expressed as a function of detuning from resonance; the fall-off is governed by damping.
A common quantitative measure is the bandwidth (Δω) between the two half-power points on the resonance curve:
Δω = ω2 - ω1 = 2β
The sharpness (S) can be defined as the reciprocal of the bandwidth with a standard relation
S = \frac{ω}{Δω} = \frac{ω}{2β} = Q,
where Q is the quality factor.
Visual: sharper resonance for smaller damping (larger Q); broader resonance for larger damping.
Application examples: tuning instruments, radio receivers, Helmholtz resonators, etc.

2.14 Quality Factor (Q)

Definitions and relationships:
- Work-energy perspective: Q is related to how underdamped the system is and is defined as
  Q = \frac{ω}{Δω},
  with Δω being the bandwidth at half-maximum power.
- In the damped oscillator with parameter β (per unit mass damping), the approximate relation is
  Q = \frac{ω_0}{2β}.
Higher Q implies sharper resonance and less energy loss per cycle; lower Q implies broader resonance and larger damping.

2.15 Forced Vibration in an LCR Circuit

LCR circuit can exhibit forced vibration when driven by an AC source and experiences damping due to resistance R.
Setup and forcing:
- Initially, a DC source charges the capacitor; when the circuit is closed with an AC source, the capacitor discharges and the circuit oscillates.
- The inductor voltage, capacitor voltage, and resistor drop interact with the source to produce oscillations.
Governing equation (in standard form):
\frac{d^2 q}{dt^2} + \frac{R}{L} \frac{dq}{dt} + \frac{1}{LC} q = \frac{E_0}{L} \cos(\omega' t).
Normalization and comparison with mechanical DV:
- This is analogous to mechanical damped forced oscillator
  \frac{d^2 y}{dt^2} + 2β \frac{dy}{dt} + ω0^2 y = \frac{F0}{m} \cos(ω' t).
Key outcomes:
- The circuit behaves as a forced oscillator with a damping term due to R.
- The resonance behavior, amplitude response, and phase lag have direct electrical analogues in the LCR circuit.
Particular results for the LC and LCR circuits emphasize the same mathematical structure across mechanical and electrical domains, illustrating the universality of SHM concepts.

Notes on notation used in the source (quick reference):

y(t): displacement (or charge displacement in circuits), a: amplitude, d: initial phase, ω: natural (undamped) angular frequency, ω' or ω_d: driving/damped frequency, β or b: damping constant per unit mass (or half the resistance per unit mass in mechanical form), k: restoring constant, m: mass.
Equations frequently appear in the following standard SHM forms:
- SHM: \frac{d^2 y}{dt^2} + ω^2 y = 0.
- Damped SHM: \frac{d^2 y}{dt^2} + 2β \frac{dy}{dt} + ω_0^2 y = 0.
- Forced SHM: m \frac{d^2 y}{dt^2} + c \frac{dy}{dt} + k y = F \cos(ω' t).
Key derived quantities:
- Amplitude in forced vibration: A = \frac{F}{\sqrt{(k - m ω'^2)^2 + (2 β m ω')^2}}.
- Phase lag: \tan α = \frac{2 β m ω'}{k - m ω'^2}.
- Damped natural frequency: ωd = \sqrt{ω0^2 - β^2} for the underdamped case.
- Resonant (peak) frequency: ω'r = \sqrt{ω0^2 - 2β^2}.
- Quality factor: Q = \frac{ω0}{2β} \, (\approx \frac{ω0}{Δω}).

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