Takafumi Horiuchi

Motion Analysis of the Piano Action

January 2020

Overview

A grand piano’s action is the system of levers that transfers a keystroke to a hammer, which rises to strike a string and set it vibrating. This project recreated the action as a 3D model in SolidWorks* and used motion analysis to examine how input at the key translates into hammer speed and reaction force at the string.

The mechanism was modeled from scratch using reference images, then examined in two analyses: the relationship between the downward speed of the key and the upward speed of the hammer, and the relationship between the force applied to the key and the reaction force at the string. The simulation results were compared with theoretical values calculated from the model’s dimensions and pivot locations to check how it transmitted motion and force.

* SolidWorks is a 3D CAD system with tools for structural analysis of parts and motion analysis of mechanisms.

  1. Title slide: “Motion Analysis of the Piano Action”.
  2. A photograph of the inside of a grand piano, introducing the row of action mechanisms behind the keys.
  3. A 3D model showing how the parts of the action move together when a key is pressed.
  4. A sectional model of the piano action, showing that pressing a key raises the hammer to strike the string.
  5. Tracing a cross-sectional reference drawing to model each part from scratch in SolidWorks.
  6. The completed 3D assembly of the piano action in SolidWorks.
  7. Part names and the degrees-of-freedom calculation for the model, showing 7 degrees of freedom after constraints are applied.
  8. The two analysis topics: the linear speeds of key and hammer, and the relationship between key input and string reaction force.
  9. The observation points for analysis 1: the downward speed at the end of the key and the upward speed of the hammer.
  10. A diagram simplifying the piano mechanism into three rotating elements.
  11. Derivation of the key-to-hammer speed relationship from the output-to-input radius ratios of three rotating elements.
  12. The analysis conditions: 5 newtons applied downward at the end of the key from 1 to 2 seconds.
  13. Graphs of key and hammer speeds, with peaks of 1,027 and 4,284 millimeters per second, respectively.
  14. Comparison of the theoretical hammer speed of 4,614 millimeters per second, calculated from the measured radii, with the simulated speed of 4,284 millimeters per second.
  15. The conditions for analysis 2: 15 newtons applied downward to the key, with the reaction force on the string side observed.
  16. A comparison between the maximum string reaction force of 31 newtons and the theoretical value of 31.1 newtons from the lever principle.
  17. A list of references on mechanical models of the piano action.

Building a piano-action model for analysis

How a keystroke makes the hammer strike

The piano action converts the downward motion of a key into the upward motion of a hammer. Pressing a key moves the capstan, wippen, jack, and repetition lever together, raising the hammer to strike the string. The player makes a single keystroke, but inside the mechanism, several parts rotate around different pivots and transfer motion through contact.

The 3D model includes the key, hammer, jack, wippen, repetition lever, capstan, damper, damper lever, and wire, along with their supporting frame. To model the motion, each part’s pivot and its points of contact with other parts had to be defined.

Building the 3D model from a cross-sectional drawing

Each part was modeled from scratch in 3D by tracing its outline over a cross-sectional drawing of the piano action. The parts were then assembled, with fixed connections and rotational joints corresponding to the physical mechanism. Defining these relationships turned the geometry into an analysis model that could move under applied forces.

SolidWorks screenshot showing a semitransparent cross-sectional drawing of a piano action, with the 3D parts modeled over its outlines. The Windows taskbar and component tree appear on the left.
A cross-sectional reference drawing was used as a semitransparent underlay for tracing and modeling the parts in SolidWorks.
SolidWorks on Windows, with the taskbar and component tree on the left and the piano-action 3D model in the center.
The completed 3D model of the piano action in SolidWorks.

Verifying the assembly’s degrees of freedom

The assembly contains nine moving parts, excluding the fixed frame, for a total of 54 unconstrained degrees of freedom. Fixing the capstan and wire to other parts removes 12 degrees of freedom. Rotational joints on the remaining seven parts remove another 35, leaving the CAD assembly with 7 degrees of freedom:

54 − 12 − 35 = 7

SolidWorks’ constraint diagnostics also reported 7 estimated and 7 actual degrees of freedom, with 0 redundant constraints. This confirmed that the parts and constraints defined for this project had been assembled as intended; it does not establish the degrees of freedom of every physical piano action.

The labeled piano-action model and its degrees-of-freedom calculation: 54 minus 12 removed by fixed connections and 35 removed by rotational joints, leaving 7.
Part names and assembly constraints. SolidWorks’ diagnostics reported 7 degrees of freedom and 0 redundant constraints.

Two analyses of key input

The model was examined in two separate analyses. Analysis 1 compared key speed with hammer speed. Analysis 2 compared the force applied to the key with the reaction force at the string. Separating these input–output relationships made the complex linkage easier to analyze using geometric speed ratios and lever balance.

For comparison, the mechanism was simplified into rotating elements and analyzed theoretically. Agreement between the theoretical and simulated values would indicate that the model’s motion was consistent with its specified dimensions and constraints.

Two diagrams pairing the key and hammer speeds in analysis 1, and the force applied to the key and the reaction force at the string in analysis 2.
The two input–output comparisons: key speed versus hammer speed in analysis 1, and force applied to the key versus reaction force at the string in analysis 2.

Analysis 1: From key speed to hammer speed

For the speed calculation, the action was simplified into three rigid bodies: the key, an intermediate mechanism centered on the wippen, and the hammer. Treating motion as a sequence of contacts from ① to ③ made it possible to analyze the linkage as three stages of rotation.

The piano action is color-coded as three rigid bodies: ① the key, ② the intermediate mechanism centered on the wippen, and ③ the hammer.
The simplified model has three rigid bodies: ① the key, ② the intermediate mechanism, and ③ the hammer.

All points on a rigid body share the same angular velocity. If r is the distance from the pivot and v is the linear speed at that point, then v = ωr. Assuming that speed transfers between bodies without loss at each contact point, hammer speed can be calculated by multiplying the output-to-input radius ratios across the three stages, then multiplying that product by key speed.

vhammer = (r1′ r2′ r3′ / r1 r2 r3) vkey

Diagram showing the distances from each of the three pivots to its input and output points, labeled r1, r1 prime, r2, r2 prime, r3, and r3 prime.
The distance from each pivot to its input and output points defines the radii. Multiplying the three output-to-input radius ratios gives the overall speed ratio.

Input conditions and results

In SolidWorks Motion Analysis, a downward force of 5 N was applied at the end of the key. The linear speeds of the key end and hammer were recorded. Immediately after the force began, their peak speeds were 1,027 mm/s and 4,284 mm/s, respectively, giving a simulated hammer-to-key speed ratio of about 4.17.

Key and hammer speed plotted over time, with peaks just after the force is applied at 1,027 and 4,284 millimeters per second, respectively.
Key and hammer speeds both peak just after the force is applied. The hammer reaches the higher speed.
Peak speeds in analysis 1
Quantity and methodSpeed
Key (simulation)1,027 mm/s
Hammer (theory)4,614 mm/s
Hammer (simulation)4,284 mm/s

Comparison with theory

The measured radii were r1 = 200 mm, r1′ = 88 mm, r2 = 53 mm, r2′ = 78 mm, r3 = 16 mm, and r3′ = 111 mm. Substituting these values gives an idealized speed ratio of about 4.49. At a key speed of 1,027 mm/s, the predicted hammer speed is 4,614 mm/s.

The six radii used to calculate the speed ratio, marked on the SolidWorks 3D model.
The six radii measured from the 3D model, used to calculate the theoretical hammer speed of 4,614 mm/s.

The simulated hammer speed of 4,284 mm/s was about 7% below the theoretical value, while showing a similar degree of speed amplification. The theoretical calculation uses three idealized rigid bodies, whereas the simulation represents the individual parts. The numerical difference alone does not establish its cause.

Analysis 2: From key force to string reaction force

In the second analysis, a downward force of 15 N was applied to the key surface, and the reaction force at the string was recorded at impact. The key input, Fkey, began increasing at 1.0 seconds and reached 15 N. The reaction force, Vstring, peaked at 31 N when the hammer struck.

Illustrative graph of the key input, F key, over time. It begins rising at 1 second, holds at 15 newtons, then returns to zero.
The force applied to the key, Fkey, begins at 1.0 seconds and reaches 15 N.
Illustrative graph of the string reaction force, V string, over time, with a peak of 31 newtons on impact.
The reaction force at the string, Vstring, peaks at 31 N on impact.

For comparison, a simplified lever-balance calculation was made using the arm lengths on the key and hammer sides.

Vstring = (15 × 195) / 94 = 31.1 N

Using arm lengths of 195 mm on the input side and 94 mm on the string side gives a static lever-balance comparison value of 31.1 N. This is 0.1 N above the simulated peak of 31 N. The conditions under which this static calculation can be applied to the instantaneous reaction-force peak at impact still need confirmation.

Force-balance diagram showing the 195-millimeter arm on the key-input side and the 94-millimeter arm on the string side.
A static lever-balance comparison using a key input of 15 N and the arm lengths gives 31.1 N, close to the simulated peak of 31 N.

Comparing simulation with theory

Results of the two comparisons

In analysis 1, the theoretical hammer speed was 4,614 mm/s, compared with 4,284 mm/s in the simulation. In analysis 2, the static lever-balance comparison value was 31.1 N, compared with a simulated peak of 31 N. Both comparisons showed close agreement between the 3D model and the theoretical calculations.

Interpreting the differences

The theoretical calculation reduces the action to three rigid bodies and assumes that motion and force transfer through their contact points without loss. The simulation represents the individual components instead. These different representations are important when interpreting the results, but attributing the remaining gap to energy loss or material deformation requires confirmation of the simulation settings.

Conclusion

The 3D simulation closely matched the theoretical comparisons for hammer speed and string reaction force. Within these comparisons, the model built from a cross-sectional drawing reproduced the transfer of motion and force expected from the specified geometry and constraints.

The project connected several stages of modeling: interpreting a cross-sectional drawing, translating relationships between parts into CAD constraints, and simplifying the assembly into a mechanical model suited to each analysis.