Grand-Piano

Vittorio Sartor – vittorio.sartor@studenti.unipd.it
updated on July, 2020
 

Introduction

The piano is a very complex tool constructively, and many solutions have been found to allow an effective transmission of forces from our fingers to the strings. One of the main mechanism for transmitting motion from the key to the string is the piano action, which includes many components, and each of them has an important role in guaranteeing percussion reliability, speed, repeatability and allowing that sensitivity that a pianist must know in order to strike the string appropriately. Transmission is so important because it is the means of communication between the pianist’s thought and his instrument, and thus many variables have to be taken into account. The mechanism studied in this project was a grand-piano one, which differs in some aspects from the vertical one.

 

Immagine_piano_action

 

 

Note that each component, including material, has its own evolution within each manufacturer, and has been specifically designed to maximize performance. It is therefore not surprising that such a high cost for high-quality instruments.

 

Objectives

The aim of this project is certainly not that of an effective design of the mechanism, because it would require enormous experience and advanced experimental study on the parameters involved

The goal of this project was instead to understand the mechanism of the piano, and to study the dynamics of the hammer and the forces between hammer and wire, and between hammer and jack, as functions of the jack_toe, which is an extremely important component for the response to the percussion of the string.

An attempt was also made to evaluate the impulse given to the piano string, with the future goal of studying the vibrations induced on the string as the pulse changes, which is in turn a function of the parameters of the whole mechanism.

 

The modelling problem

The piano_action model was designed by the modification of the downloaded geometry from free3D. In order to transfer it to Adams, the model was first exported to Solidworks, and, as some changes had been made, saved to Parasolid format.

Immagine_piano_action_da_solidworks

It can be observed the great approximation made in the reconstructive model, in which the felt pads have been eliminated, together with a geometric simplification of all its components. For a realistic study of the rope, moreover, it would be advisable to provide the triple rope double wedged at the ends, and not simply a rope modeled by a cantilever beam.

Kinematic chain of the system

Immagine_piano_action_da_adams

For illustrative purposes only, colors have been introduced for each component of the system.

The key is in orange, and this is where the force that sets the whole mechanism in motion comes from. Force, which, of course, comes from pressing the finger on the button. Through a pin, called capstain, the wippen is rotated, which is the component in light brown. The jack is the purple component, which pushes the hammer, dark brown component, as long as, thank to the acquired inertia, it proceeds to impact on the string, located horizontally at the top.

On the left you can see the damping mechanism, which is raised by the button itself, during its lifting. This allows you to release the piano string to allow it to vibrate.

The importance of the jack can be understood if one thinks of the movement of a hypoteric hammer pressed on the string. There would be no sound, because the swing would be immediately dampened by the hammer itself attached to the string. In the real mechanism instead, the jack, pivoting on the jacktoe, which is the little red body on the right, rotates and detaches itself from the hammer, interrupting the communication of the push. In this way the hammer will be free to continue the motion, impact the rope and immediately detach. It will precipitate, by gravity, on the support obtained on the key in orange, which is used for repeating the note, so as to avoid having, for each consecutive consecutive note, to rotate the hammer again from the beginning.

Here it is shown the role of the jack in this mechanism.

Bodies:

  • frame
  • key
  • capstan
  • wippen
  • repetition lever
  • jack
  • hammer
  • damper_lever
  • damper
  • wire
  • jack_toe
  • vertical damper support

note: the grey bodies are parts of the structure of the piano, so they are not listed among the other bodies and connectors.

Connectors:

  • cylindric_damper_damper_lever
  • fix_jack_toe
  • cylindric_damper_support
  • fix_wire_frame
  • fix_campstain_key
  • rvl_key_keybase
  • revolute_damper_lever
  • fix_all_frame_to_ground
  • rvl_repetition_lever
  • sfere_jack_wippen
  • rvl_wippen_frame
  • cylindric_damper_vertical_support_ground
  • revolute_hammer

Motion:

  • motor_key

 

Some of the mass of the components were realistic, taken from a Gavou Paris(1927), which is a baby grand piano. The others were calculated by Adams

ID Rigid bodies Mass (kg)
8 hammer 8e-3
10 damper 11e-3

 

The model has 5 degrees of freedom, which consider the rotation of the button, the rotation of the wippen, the hammer, the repetition lever, the jack and the lifting of the damper, but the degree of freedom of the key locked by the motor must be removed.

5 Gruebler Count (approximate degrees of freedom)

12 Moving Parts (not including ground)

3 Cylindrical Joints

6 Revolute Joints

4 Fixed Joints

1 Motions

To improve the performance of the simulation, three analytical contacts (sphere-sphere and cylinder-cylinder) have been added between the wippen and the capstain, and between the jack and the hammer, and between the hammer and the piano string.

Furthermore, to further lighten the simulation, the kinematic motion of the damper was created by eliminating the contacts and replacing them with ideal joints (there are four bodies and 5 ideal joints, e.g. a fix joint, a revolute joint and three cylindrical joints).

Grubler Count = 6×4-6×1-5×1-4×3 = 1

In the case of the model with only the use of contacts (without kinematic motion of the damper with ideal constraints), the number of degrees of freedom rises to six, because the damper can also rotate relative to the damper lever

 

Contacts:

  • CONTACT jack_jacktoe, which allows the jack to rotate e free the hammer from its thrust
  • CONTACT damper_sostegno_verticale_damper
  • CONTACT wippen_hammer
  • CONTACT key_frame_uplimit, which eventually stops the key if it’s pushed too hard
  • CONTACT wippen_repetition_lever
  • CONTACT damper_lever_frame
  • CONTACT dumper_wire, which holds the damper right on the string
  • CONTACT key_hammer, which supports the hammer for a possible second hit on the string
  • CONTACT hammer wire, which allows the hammer to hit the string
  • CONTACT key_damper_lever, that allows the push of the button on the damper to lift it
  • CONTACT jack_hammer, which is responsible for the final transmission of force from the key to the hammer
  • CONTACT key_key_base, which allows the key to stay in place on the carcass of the piano
  • CONTACT wippen_capstain, taht is responsible for the trasmission of the force between che key and the wippen, which pushes the jack toe, which pushes the hammer straight upward towards the wire

Design_variable:

There is a design variable (DV_jack_toe) that allows to change the high of the jack_toe for the different simulations, in order to rotate the jack in different ways

 

In this figure it is possible the see the joints and the contacts between the bodies.
Immagine_con_giunti_visibili

Simulations and analysis of results

At first dynamicsimulation has been performed to check if the motion of the hammer was correct, at least at first sight. Since it is a rather simple measure (the pure angular velocity around a marker), convergence was immediately found, regardless of the integrator error and the size step.

 


Convergence of hammer angular velocity_jacktoe_1_5The convergence of the force established between the jack and the hammer was subsequently found:convergence_hammer_jack_force

 

Here follows an image relating to the variation of the force established between jack and hammer when damping changes. The model of the interaction is however not valid, because in reality the jack continues and crawls on the roll of the hammer, and not, as happens in the simulation, it comes off as soon as it has transferred the force.

Jack hammer force with jacktoe changes

 

Subsequently, the interaction force between the hammer and the string was assessed as the simulation parameters changed.

convergenza_hammer_force_with_integrator_stepsize_and_k

The problem is that the measured force (magnitude of the hammer-wire contact between the hammer and the string) continues to change value (and changes a lot if I decrease the size step), without ever giving the appearance of convergence. By increasing the penetration between the hammer and the string, the results of the impulsive force during contact approached between 20 and 40 N, maintaining a size step of 1exp-4, thanks to the increase of the penetration between the two bodies. As sizestep decreases though , the force during the contact between the hammer and the wire increased too rapidly that the results are no more meaningful.

The impulse seems to occur so quickly that sometimes, if the size step is not small enough, the calculator provides a flat graph of the force, as if the impulse had not occurred. By increasing the size step, however, the estimated force increases in an unrealistic way (800-900N of force exchanged between the hammer and the rope for a size step equal to 1exp-6). By increasing the penetration, the calculator manages to highlight the peak.

unreached_convergence_of_hammer_force

 

Lastly, the graph of the pulse transferred to the hammer from the jack was compared with the impulse transferred to the string from the hammer.

A loss of energy can be observed due to the fact that the hammer continues, for a certain angle of rotation, its motion opposed by the force of gravity:

comparison_hammer_wire_forces

 

 

Conclusions

A kinematic analysis of the action plan has been carried out and we have faced the problems that today concern the computer design of the new piano models, which see the cohesion of experimental data and computer simulations.
Dynamic analysis has instead touched on a few aspects taken individually, but the mechanics of the piano is much more complex and requires more indepth studies.
Graphs have been obtained that show the trend of the force according to the damping between the contacts, and the speeds that can be obtained on the hammer when the jack-toe changes.

The elements that make up the simplified model were not able to simulate the contact between jack and hammer, since it should only be separated at the end of the movement, but a certain connection could be seen between the jacktoe and the thrust carried on the hammer.
The convergence of the contact force between the string and the hammer has not been obtained, because it is too sensitive to parameters set for each simulation.
Once the appropriate parameters have been established to obtain good control over the strength of the hammer, it would be interesting to model the string with a flexible body, and to observe the vibrations induced on the piano string as the impulse given by the hammer changes.

References

[1]  Il Martello/Tasto e Cavalletto del Pianoforte/Smorzi, Molle e Tocco, Walter pfeiffer, Rugginenti

[2] Il Pianoforte, Alfonso Alberti

[2] https://it.wikipedia.org/wiki/Pianoforte

[3] https://free3d.com/3d-model/piano-action-227253.html

[4] https://www.youtube.com/watch?v=XthnCDTnAGw

[5] https://www.youtube.com/watch?v=qCgvbtDUm6Y

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