Plucked String Dynamics Simulator

How the Plucked String Dynamics Simulator Works

This tool visualizes the vibration of a plucked guitar string and how its energy is transferred to the guitar body through the bridge.

The simulation starts from an ideal triangular pluck and separates the string motion into its harmonic components. It then calculates the forces acting at the saddle and uses a simplified 4-DOF model of the guitar body to show how bridge mobility affects energy transfer and string decay.

How to Use the Simulator

String Fundamental selects the played note using the equal-tempered scale. Changing the note moves all the string harmonics while the resonances of the guitar body remain in place.

Pluck Position changes where the string is displaced before release. This changes the relative strength of the harmonics and therefore the initial spectrum of the pluck.

Displayed Harmonics controls how many individual harmonic components are shown. The black curve represents the complete string motion using a higher-resolution internal model, while the colored curves show the selected harmonics separately.

Bridge Mechanical Impedance changes the boundary condition at the saddle:

• Fixed Bridge represents an ideally rigid termination. No energy is transferred to the guitar body.
• Real Guitar introduces the frequency-dependent bridge mobility calculated from the 4-DOF guitar-body model.
• Intermediate positions allow you to explore the transition between these two conditions.

Coupled Body Modes

Activate Edit Coupled Body Modes to change the main resonances of the guitar body:

• Air-dominant Mode
• Top-dominant Mode
• Back-dominant Mode

These are final coupled resonances of the complete guitar system, not isolated plate frequencies. The names indicate which component contributes most strongly to each mode.

Moving these resonances changes the mechanical impedance seen by the string and therefore changes which harmonics transfer energy most efficiently to the body.

Press Pluck to start the slow-motion simulation. The vibration is deliberately slowed down so that the motion of the string and the evolution of the forces can be observed over complete cycles.

What the Graphs Show

1. String Motion

The first graph shows the instantaneous shape of the string between the nut and the saddle.

The black line is the complete string motion. The colored lines show its individual harmonic components.

As the vibration decays, different harmonics lose energy at different rates. The shape of the string therefore changes progressively with time.

2. Spectrum or Bridge Forces

The second graph can show either the Spectrum or the Bridge Forces.

Spectrum

The Spectrum view offers two different quantities:

String Displacement shows the harmonic content of the motion of the string itself: how much each harmonic contributes to the string displacement.

Bridge F⊥ shows the harmonic content of the transverse force applied by the string to the saddle. Higher harmonics can contribute more strongly to bridge force than their displacement alone would suggest.

In simple terms:

String Displacement shows what is vibrating in the string.
Bridge F⊥ shows what the string is sending into the guitar.

The spectrum is referenced to the initial pluck, so you can watch each harmonic decrease as the string decays.

Optional references allow you to compare:

• Initial Spectrum: the harmonic content at the instant of release.
• Fixed-Bridge Reference: how the same string would decay without transferring energy to the body.
• 4-DOF Bridge Mobility: where the guitar body is most easily driven.

This makes it possible to see directly why some harmonics decay much faster than others.

Bridge Forces

The Bridge Force view shows the forces generated at the saddle:

• Transverse force F⊥: the main oscillating force driving the soundboard.
• Longitudinal force $\Delta T$: the smaller variation in string tension produced by the changing geometry of the vibrating string.

Both are calculated at the saddle, corresponding to the right-hand endpoint of the first graph.

3. Energy Transfer to the Guitar

The third graph shows the instantaneous mechanical power exchanged between the string and the guitar body.

It is calculated from the transverse force applied by the string and the resulting velocity of the bridge:

P(t)=F⊥(t)vbridge(t)

The power curve does not necessarily have the same shape as the force curve. A plucked string contains several harmonics, and the guitar body responds differently to each frequency. Their different amplitudes and phases can therefore produce several power peaks within one string cycle.

• Positive values: energy is transferred from the string to the guitar body.
• Negative values: some energy is temporarily returned from the body to the string.
• Mean absorbed power: represents the net energy progressively removed from the string by the guitar body.

This back-and-forth exchange is normal in a coupled vibrating system. What matters for string decay is the net energy transferred over time, not the number of individual peaks in the instantaneous power curve.

String and Body Damping

Even with a fixed bridge, the simulated string has a small intrinsic damping. This loss is frequency-dependent, so higher harmonics naturally decay faster than the fundamental.

Connecting the string to the guitar adds another loss mechanism.

The mechanical impedance of the bridge changes with frequency, so every harmonic sees a different boundary condition. Harmonics close to regions of high bridge mobility transfer energy more efficiently to the body and therefore decay more rapidly.

Changing the String Fundamental moves the harmonic frequencies relative to the guitar resonances. Changing the Coupled Body Modes moves the guitar resonances relative to the string harmonics.

This makes it possible to explore one of the central ideas behind string–body interaction:

The guitar body is not simply driven by the strings. Its mechanical behavior also changes the way the strings vibrate, evolve, and decay.