4-DOF Acoustic Guitar Simulator

This tool simulates the acoustic frequency response of a guitar body, providing a window into the core physics that shape its primary resonances. It is designed for luthers, researchers, and acoustics enthusiasts who want to explore the complex relationships between physical parameters and the resulting sound.

The simulation is based on the 4-Degree-of-Freedom (4-DOF) lumped-parameter model as described by Trevor Gore. This approach simplifies the complex, vibrating guitar body into four key components that can move and interact:

• The Top Plate
• The Back Plate
• The Soundhole Air
• The Sides

The model treats the top, back, sides, and soundhole air as interacting “pistons” (monopoles).

How to Use the Simulator

1. Main Tab (Physical Parameters)

This is where you input the physical properties of the guitar.

• Mass (g): The effective moving mass for each of the four components.
• Stiffness (N/mm): This is the purely mechanical stiffness of the top, back, and sides. This value is what you would measure with a static displacement test when the component is isolated.
• Area (cm²): The effective radiation surface area of each component.
• Volume (L): The internal net air volume of the guitar box.

2. Q Factor Tab (Damping)

This tab controls the damping for each component.

• Q Factor: A number representing how resonant a component is. A high Q means low damping, resulting in a sharp, ringing resonance peak. A low Q means high damping and a broad, gentle peak. This is a key simplification, as real-world guitar damping is far more complex and frequency-dependent.
• Note on Air Damping (Qa): In this specific implementation, unlike the standard Gore model where all Q factors are independent inputs, the air damping (Q_a) is calculated automatically based on the Side Stiffness (K_s). This reflects the physical intuition that stiffer sides absorb less energy from the internal air, leading to lower air damping (higher Q_a). This is a small adaptation intended to capture some of the interaction between the structure and the air mode’s damping.

3. The Graph & Outputs

This is where you see the results of your simulation.

• Response Curves: The graph shows the simulated Sound Pressure Level (dB). You can toggle the individual contributions:
  • Total Sum (Orange): The final, combined sound radiated from all components. This is what you would “hear.”
  • Top (Blue): Sound radiated by the top plate only.
  • Back (Green): Sound radiated by the back plate only.
  • Air (Purple): Sound radiated by the soundhole air.
• Uncoupled Modes: This box shows the true resonance frequency of the main components in a vacuum (i.e., with no air spring). 
• Coupled Modes: This box identifies the main peaks of the Total Sum curve. These are the new frequencies of the entire system vibrating together.

4. Management Tab

This allows you to save and compare your work:

• Set as Reference: Stores the current “Total Sum” curve as a red dashed line for easy comparison.
• Curve Management: Save, load, or delete specific response curves.
• Configuration Management: Save, load, or delete entire sets of parameters.

The Core Physics: Dual Coupling Mechanisms

This model incorporates the two primary coupling mechanisms that link the components:

1. Acoustic Coupling (Air Spring): The internal air volume (V) acts as a shared spring. As the top moves, it compresses or expands the internal air, creating pressure changes that simultaneously act on the back, sides, and soundhole air.
2. Mechanical Coupling (Through Sides): The model also mathematically accounts for the physical connection between the top, sides, and back. Vibrations from the top can be transmitted mechanically through the sides to the back, and vice-versa.

The simulation’s logic follows this path:

1. Force Application: A constant force (F), representing the string’s vibration, is applied to the top plate.
2. Coupled Motion: This force sets the entire system in motion. Both the internal air pressure changes and the mechanical flex through the sides transfer energy between all four components.
3. Complex Interaction: The motion of each component feeds back through both coupling paths, influencing the motion of all other components simultaneously.
4. Solving the System: For every frequency on the graph, the code solves the system of four coupled differential equations (using the complex algebra frequency-domain solution presented by Gore) to find the final displacement (both amplitude and phase) for all four components.
5. Acoustic Output: The final sound is the sum of the air “pumped” by each moving part. The code calculates the sound contribution (volume velocity) from each (Displacement × Area) and adds them together, accounting for phase, to create the final “Total Sum” curve.

Model Origins and Acknowledgements

The 4-DOF lumped-parameter model is a well-established method for understanding guitar acoustics, with foundational work developed and popularized by Trevor Gore. This simulator is an implementation based on those principles, further informed by valuable discussions within communities like the New Zealand ANZLF Forum.