ENGINEERING-MATHVerified Reference Parity

Damped Sine Wave (Harmonic Oscillator)

Exponentially decaying sinusoidal oscillation model for transient mechanical vibration and RLC circuit responses.

Primary Disciplines:Mechanical EngineeringElectrical EngineeringAcousticsGeophysics

Mathematical Formulation

$$y(t) = y_0 + A \exp(-\gamma t) \sin(\omega t + \phi)$$
Plaintext:y = y0 + A * exp(-gamma * t) * sin(omega * t + phi)

Parameters & Physical Meanings

Detailed parameter breakdown, standard units, valid mathematical constraints, and initial guess heuristic algorithms.

SymbolNameUnitConstraintsPhysical InterpretationInitial Guess Heuristic
$y_0$Equilibrium OffsetDisplacement / VoltageunconstrainedSteady-state resting position or DC offsetmean(y)
$A$Initial Oscillation AmplitudeDisplacement / VoltageA > 0Peak displacement amplitude at t = 0max(y) - mean(y)
$\gamma$Damping Factor (Decay Rate)1 / time (s⁻¹)gamma >= 0Exponential decay rate of oscillation envelope (gamma = zeta * omega0)Envelope logarithmic decrement estimation
$\omega$Damped Angular Frequencyrad / somega > 0Natural oscillation frequency under damping (omega = 2 * pi * f_d)FFT dominant peak frequency: 2 * pi * f_peak
$\phi$Phase AngleRadians-pi <= phi <= piInitial oscillation phase angle at t = 00.0

When to Choose This Model

  • Analyzing transient vibration decay from accelerometer impulse response testing
  • Extracting damping ratios (Q-factors) in acoustic resonators and MEMS cantilevers
  • Fitting ring-down decay transients in laser cavity ring-down spectroscopy (CRDS)

Typical Scientific Applications

  • Mechanical vibration testing
  • Cavity Ring-Down Spectroscopy (CRDS)
  • RLC circuit transient analysis
  • Seismic wave damping estimation

Expected Fit Profile & Curve Morphology

Shape: oscillatory
Fitted ModelRaw Data

Example Dataset & Expected Fit Output

Synthetic 30-point cantilever impulse vibration decay with damping factor gamma = 0.45 s⁻¹ and frequency f = 2.0 Hz

R²:0.9992RMSE:0.2800

Expected Converged Parameters

  • y00.010
  • A9.850
  • gamma0.450
  • omega12.570
  • phi0.020

Sample Experimental Vectors (21 points)

#x (Independent)y (Observed)
10.0000.020
20.0504.820
30.1008.410
40.1507.950
50.2003.820
60.250-2.140
70.300-6.840
80.350-7.820
+ 13 more points available in AltaiPlot preset

Python / SciPy Reference Implementation

Reference library: scipy.optimize (Levenberg-Marquardt).Exact parameterization parity verified.

import numpy as np
from scipy.optimize import curve_fit

def damped_sine_model(t, y0, A, gamma, omega, phi):
    return y0 + A * np.exp(-gamma * t) * np.sin(omega * t + phi)

p0 = [np.mean(y_data), np.max(y_data)-np.mean(y_data), 0.5, 2.0*np.pi*2.0, 0.0]
popt, pcov = curve_fit(damped_sine_model, t_data, y_data, p0=p0)
Numerical Verification Tolerance: Max Relative Error < 4.1e-6Verified: 2026-08-16

Comparative & Alternative Models

Key decision trade-offs between this model and related functional alternatives:

Alternative Model

Bi-Exponential Decay

Two-phase exponential decay model resolving fast and slow relaxation lifetimes.

$y(t) = y_0 + A_1 \exp\left(-\frac{t}{\tau_1}\right) + A_2 \exp\left(-\frac{t}{\tau_2}\right)$

Related Mathematical Models

Models sharing mathematical formulations or physical domains in AltaiPlot:

kinetics

Bi-Exponential Decay

Two-phase exponential decay model resolving fast and slow relaxation lifetimes.

$y(t) = y_0 + A_1 \exp\left(-\frac{t}{\tau_1}\right) + A_2 \exp\left(-\frac{t}{\tau_2}\right)$

Frequently Asked Questions (Damped Sine Wave (Harmonic Oscillator))

How can I obtain a reliable initial guess for the angular frequency omega?

Perform an FFT on the dataset to extract the dominant peak frequency f_0, then initialize omega = 2 * pi * f_0. AltaiPlot performs this automatic spectral peak initialization internally.

Scientific References & Citations

  • Rao, S. S. (2018). Mechanical Vibrations (6th ed.). Pearson.[Source / DOI]