Resonator Ring-Down Simulator

Simulate the ring-down of tuning forks and beam resonators using geometry-based frequency estimates, damping ratios, and time-domain decay plots.

Tool Purpose & README

About This Tool

This tool models a struck resonator as a damped, single-mode oscillator. It estimates the fundamental frequency from beam geometry and material properties, then simulates the decay response.

Geometry-driven frequency using Euler-Bernoulli beam theory
Damping and decay metrics including Q, zeta, and t60
Time-domain waveform with an exponential envelope
Decay plot showing amplitude in dB over time

Use the Advanced options to enter custom material properties and simulation settings.

Inputs

Resonator

Resonator Type ?

Support Condition ?

Material ?

Quality Factor Q ?

Initial Displacement (mm) ?

Geometry

Cross-Section ?

Active Length (mm) ?

Width (mm) ?

Thickness (mm) ?

Custom Material

Simulation

Duration (s) ?

Sample Rate (Hz) ?

Noise RMS (mm) ?

Key Results

Enter inputs and press Calculate to see results.

Fundamental Frequency

1. Beam Frequency Equation

$$ f_n = \frac{\beta_1^2}{2\pi} \sqrt{\frac{E I}{\rho A L^4}} $$

2. Substituted Values

Damped Frequency

1. Damped Frequency

$$ f_d = f_n \sqrt{1 - \zeta^2} $$

2. Substituted Values

Damping Ratio

1. Q to Damping Ratio

$$ \zeta = \frac{1}{2Q} $$

2. Substituted Values

Decay Time Constant

1. Time Constant

$$ \tau = \frac{1}{\zeta \omega_n} $$

2. Substituted Values

Time to -20 dB

1. Amplitude Decay

$$ t_{20} = \frac{\ln(10)}{\zeta \omega_n} $$

2. Substituted Values

Time to -60 dB

1. Amplitude Decay

$$ t_{60} = \frac{\ln(1000)}{\zeta \omega_n} $$

2. Substituted Values

Ring-Down Waveform

What this shows: Tip displacement versus time with an exponential envelope. Key insight: Higher Q values reduce damping and extend the ring-down.

Decay Envelope (dB)

What this shows: Envelope amplitude in dB relative to the initial displacement. Key insight: t20 and t60 are derived from the slope of this line.

Contents

Overview
Geometry and Section Properties
Mode Shape and Natural Frequency
Damping and Ring-Down Response
Decay Metrics
Sampling and Aliasing
Interpreting Results
Limitations
References

1. Overview

This simulator models the ring-down of a struck resonator using a single dominant bending mode. Geometry and material properties set the natural frequency. The quality factor Q controls damping and therefore how quickly the amplitude decays.

The workflow is: compute section properties (area and second moment), select the mode constant for the support condition, calculate the natural frequency, then apply damping to simulate the time response and decay metrics.

Tuning fork vs rod: A tuning fork keeps energy in the tines by canceling motion at the handle, reducing losses to your hand or a table. A rod can ring too, but support conditions and mounting losses usually shorten the decay unless carefully suspended.

2. Geometry and Section Properties

Section properties define bending stiffness and mass per length. The bending axis is assumed to align with the thickness dimension for rectangular sections.

Equation (1): Rectangular area $$ A = b h $$

A: cross-section area
b: width
h: thickness (bending direction)

Equation (2): Rectangular second moment $$ I = \frac{b h^3}{12} $$

I: second moment of area
b: width
h: thickness (bending direction)

Equation (3): Circular area $$ A = \frac{\pi d^2}{4} $$

A: cross-section area
d: diameter
pi: 3.14159...

Equation (4): Circular second moment $$ I = \frac{\pi d^4}{64} $$

I: second moment of area
d: diameter
pi: 3.14159...

3. Mode Shape and Natural Frequency

The first bending mode constant beta_1 depends on boundary conditions. The tool uses the standard values: cantilever 1.8751, simply supported pi, clamped-clamped 4.7300, free-free 4.7300.

Equation (5): Beam natural frequency $$ f_n = \frac{\beta_1^2}{2\pi} \sqrt{\frac{E I}{\rho A L^4}} $$

f_n: natural frequency (Hz)
beta_1: mode constant
E: elastic modulus
I: second moment of area
rho: density
A: area
L: active length

Equation (6): Angular frequency $$ \omega_n = 2\pi f_n $$

omega_n: undamped angular frequency (rad/s)
f_n: natural frequency (Hz)
pi: 3.14159...

4. Damping and Ring-Down Response

Damping is represented with a single viscous damping ratio zeta derived from Q. The response is underdamped when zeta is less than 1.0.

Equation (7): Damping ratio from Q $$ \zeta = \frac{1}{2Q} $$

zeta: damping ratio
Q: quality factor

Equation (8): Damped angular frequency $$ \omega_d = \omega_n \sqrt{1 - \zeta^2} $$

omega_d: damped angular frequency (rad/s)
omega_n: undamped angular frequency (rad/s)
zeta: damping ratio

Equation (9): Ring-down response $$ x(t) = x_0 e^{-\zeta \omega_n t} \cos(\omega_d t) $$

x(t): displacement at time t
x_0: initial displacement
omega_n: undamped angular frequency (rad/s)
omega_d: damped angular frequency (rad/s)
zeta: damping ratio
t: time (s)

5. Decay Metrics

The exponential envelope defines decay time constants and dB drop times. These are derived directly from the damping ratio and natural frequency.

Equation (10): Time constant $$ \tau = \frac{1}{\zeta \omega_n} $$

tau: amplitude time constant (s)
zeta: damping ratio
omega_n: undamped angular frequency (rad/s)

Equation (11): Time to -20 dB $$ t_{20} = \frac{\ln(10)}{\zeta \omega_n} $$

t_20: time for 20 dB amplitude drop (s)
zeta: damping ratio
omega_n: undamped angular frequency (rad/s)

Equation (12): Time to -60 dB $$ t_{60} = \frac{\ln(1000)}{\zeta \omega_n} $$

t_60: time for 60 dB amplitude drop (s)
zeta: damping ratio
omega_n: undamped angular frequency (rad/s)

Equation (13): Logarithmic decrement $$ \delta = \frac{2\pi \zeta}{\sqrt{1-\zeta^2}} $$

delta: log decrement per cycle
zeta: damping ratio
pi: 3.14159...

6. Sampling and Aliasing

The waveform is sampled for plotting. If the sample rate is too low relative to the damped frequency, the plotted waveform can appear to change frequency due to aliasing. This tool uses a minimum samples-per-cycle rule to keep the plot stable.

Equation (14): Samples per cycle $$ N_{spc} = \frac{f_s}{f_d} $$

N_spc: samples per cycle
f_s: sample rate (Hz)
f_d: damped frequency (Hz)

Rule of thumb: Keep N_spc above 2.5 to avoid aliasing in the time plot. The simulator will raise the effective sample rate if needed.

7. Interpreting Results

A larger quality factor means lower damping, a longer decay time, and a narrower resonance peak. The decay time constant and the t20/t60 values show how quickly the amplitude drops.

Higher Q: Longer ringing, smaller damping ratio, slower decay.

Support condition: Changing from cantilever to free-free or clamped-clamped can shift frequency by a factor of 2 to 6 due to different mode constants.

Geometry trends are strong: frequency increases with thickness and decreases sharply with length. For a tuning fork, the two tines are treated as identical cantilevers, so the frequency follows the same beam relation while the moving mass accounts for both tines.

8. Limitations

Single-mode Euler-Bernoulli beam model (no shear or rotary inertia effects)
Linear viscous damping with a constant Q factor
Tuning fork modeled as two identical cantilever tines without base compliance
No acoustic radiation or mounting loss model beyond Q

9. References

Inman, D. J., Engineering Vibration, 4th ed.
Blevins, R. D., Formulas for Natural Frequency and Mode Shape
Thomson, W. T., Theory of Vibration with Applications