Modelling the dynamics of the piano action: is apparent success real? 

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MODELLING THE DYNAMICS OF THE PIANO ACTION: IS APPARENT SUCCESS REAL?

Abstract

The kinematics and the dynamics of the piano action mechanism have been much studied in the last 50 years and fairly sophisticated models have been proposed in the last decade. Surprisingly, simple as well as sophisticated models seem to yield very valuable simulations, when compared to measurements. We propose here a too simple model, with only 1-degree of freedom, and compare its outcome with force and motion measurements obtained by playing a real piano mechanism. The model appears either as very good or as very bad, depending on which physical quantities are used as the input and output. We discuss the sensitivity of the simulation results to the initial conditions and to noise and the sensitivity of the experimental/simulation comparisons to the chosen dynamical model. It is shown that only motion-driven simulations should be used for validating a dynamical model of the piano action, contrary to what has been proposed in the literature.

Introduction

The mechanical function of the piano action is to throw the hammer towards the strings. As a human-machine interface, its role is to provide the pianist with a means to perform the following musical task: obtain a given impact velocity of the hammer on the strings at a given instant, with as much as precision as possible. We focus here on the grand piano action but all what is proposed here would apply to the upright piano mechanism.
Piano actions are complex systems mostly resulting from engineering during the 18th and 19th centuries, mostly by trial-and-error. The assembly of dozens of pieces is the fruit of a few major inventions (particularly by Cristofori and Érard, for the mechanisms that remained in the 20th century) and many minor refinements. In the resulting sophisticated design, it is not any more obvious to distinguish what are the features due to engineering – economy, ease of manufacturing in given historical conditions, necessity of a silent motion, ease of repair and adjustment, etc.– and those imposed by piano playing requirements: ease and precision of control, compliance with the playing tradition. Actions that are built for digital pianos (sound synthesisers) can be seen as tentatively complying with the latter group of requirements by means of markedly different engineering solutions. Although constantly improving over years, it is interesting to notice that the results are not yet judged as entirely convincing.
We focus here on the dynamics of the piano action – the force-motion relationship – as seen from (or felt at) the finger-end of the key. In this paper, the piano action is considered either subject to a given force or to a given motion, which would be imposed by an operator. As of today, the physical quantity controlled by the pianist during the keystroke (or before) in order to perform the musical task has not been identified. In reality, the dynamics of the piano action is coupled to that of the finger/hand/arm/. . .
musculoskeletal system which is coupled itself to a neurological system of efferent and afferent nerves. A vast literature is available on various questions pertaining to the pianist control, involving sensory-motor questions as well as the dynamics of the pianist limbs and fingers. This complex question is not analysed here.
Since the 60’s, many dynamical or mechanical models of the piano action have been proposed, each of their authors more or less claiming that it emulates successfully the kinematics (usually the angular positions of the key and hammer) or the dynamics of the mechanism as seen at the end of the key. These claims are usually supported by comparisons between experimental measurements and numerical results issued by the model. The experimental results are generally obtained by imposing a force (constant or varying in time) on the key, by measuring this force, the resulting motion of the key, that of the hammer and, sometimes, of other pieces. Since it is not known whether the control by the pianist is more of a force- or a motion-nature, the choice of given force-profiles or motion-profiles for controlling the dynamics in experiments and simulations may appear as more or less arbitrary, and irrelevant for validating a given model. This paper aims at demonstrating that this is not so. To this end, we analyse the predictions of very simple models of the mechanism, with only one degree-of-freedom.
A few elementary models of the piano action or of some of its parts have been proposed in the first half of the 20th century. In 1965, a frictionless model with superimposed masses is proposed by Dijksterhuis Dijksterhuis [1965]. Oledzki Oledzki [1973] studied a model where two masses (one for the hammer and one for all the other parts) were connected by a spring, representing the internal flexibility of the action. Gillespie and Cutkosky Gillespie and Cutkosky [1992] presented a model with four bodies (key, whippen, jack and hammer) where damping, compliances and friction were neglected. In Gillespie [1994], one model is considered for each set of kinematic constraints. A unidimensional model was exposed by Mori Mori [1997], who applied forces to the key with calibrated weights. Another model was proposed by Hayashi et al. Hayashi et al. [1999], consisting in a 2-DOF model with a free mass representing the hammer. Contrary to all the other simulations in the literature which are driven by forces, Hayashi’s are driven either by a constant velocity, or by a constant acceleration. However, forces are not considered in this paper. A 2-DOF model is also proposed by Oboe Oboe [2006]. The key and the hammer are modelled, neglecting friction, but the escapement is not considered. In 1995, Van den Berghe et al. Van den Berghe et al. [1995] considered a 3-DOF model where the whippen-lever-jack assembly is rigid. The escapement is therefore not modelled either. The kinematics in response to a force input is discussed.
More complex models appear in the late 90s. The repetition lever is taken into account in Gillespie [1996]. A complete model (5-DOF, the damper is ignored), with measured parameters, is proposed by Hirschkorn Hirschkorn [2004]. Links presents a similar model Links [2011]. Lozada Lozada [2007] gives a different model with all the values of its parameters. It also includes the first attempt of driving the simulations with a position, without success. Recently, Bokiau et al. Bokiau et al. [2012] have also proposed a rather sophisticated model. Force-driven simulations yield the motion of various pieces.
Except those of Hayashi et al. [1999] and Lozada [2007], all the simulations were driven with a force input (sometimes, the applied force is constant or idealised), and the resulting kinematics was observed.
In this paper, the experiments consist in playing a real key mechanism almost like a pianist, at three different dynamical levels, and in recording the motion of the key and the force acting on it (Section 2.2). We then consider very simple models, so simple that they can hardly be considered as valid (Section 2.3). Their parameters are derived from static measurements on the real mechanism and from measurements on separate pieces that have been taken apart. The models predict the resulting motion for a given force exerted at the end of the key. When driven by an imposed motion, they can predict the reacting force as well. Comparisons are made between the measured and the predicted motions in the first case, and between the measured force and the predicted force in the other one (Section 2.4). The matching between the former appears to be much better than between the latter, motivating the discussion in Section 2.5.

Experiments

Figure 2.1 – Experimental set-up. The black-and-white patterns were used for experiments which are not reported here.
The experiments are performed on a single piano key mechanism (Figure 2.1) manufac-tured by the Renner factory for demonstration purposes but similar to the mechanisms in use in grand pianos, particularly with respect to its regulation possibilities. The key action has been carefully adjusted by a professional piano technician in line with the standards observed in a piano keyboard.
Compared to normal playing, a few modifications have been introduced. The damper has been removed (which may happen in « normal » playing). It appeared that some experiment-simulation comparisons are sensitive to the precise initial position of the key. Since investigating this question is not important for the object of this paper, the felt supporting the key at rest (left end of the key in Figure 2.1) has been replaced by a rigid support.
We consider four phases during a keystroke: the first phase of the motion ends when the hammer escapes, the second phase when it is checked, the third phase lasts until the key is released and the last phase when the key comes back to rest. For a detailed description of the timing of the piano action, see Askenfelt and Jansson [1990].
The position of the key is measured by laser-sensors (Keyence LB12, with LB72 condition-ing amplifier) at the end of the key and approximately mid-way between the finger-end and its rotation centre. Two particular angular positions of the key (θ = θe and θ = θp) and the corresponding times at which they are measured are reported in the figures of this article by gray dashed-lines and gray continuous lines, respectively. The angular position θ = θe ≈ 0.035 rad has been evaluated in a quasi-static test as the angular position of the key when the jack meets the let-off button. When playing, this position corresponds closely to escapement but not exactly since the felts are compressed, depending on how the key has been played. In fact, escapement occurs slightly after (by a variable margin) θ (t) reaches θe. For the sake of brevity in formulation, this slight difference is ignored in the rest of the article. The angular position θ = θp ≈ 0.040 rad corresponds to the key meeting the front rail punching and has also been evaluated in a quasi-static test.
The minimum force initiating down-motion and the maximum force preventing up-motion have been estimated with the standard procedure (adding and removing small masses at the end of the key). They are respectively Fdown ≈ 0.70 N and Fup ≈ 0.38 N, both exceeding by about 0.15 N the values normally adjusted by technicians.
The key acceleration is measured by a light (0.4 g) accelerometer (Endevco 2250A-10, with B&K Nexus measurement amplifier) glued approximately mid-way between the end and the rotation centre of the key. The force exerted on the end of the key is measured with a light-weight (1.2 g) piezoelectric sensor (Kistler 9211, with charge amplifier 5015). The data are sampled at 50 kHz (ADC USB-6211 by National Instruments).
In what follows, the motion of the key is reported at the end of the key (with measured signals multiplied by the appropriate factor) and shifted so that the zero-values correspond to the rest position. The force signal is also shifted so that its value is zero as long as the user has not touched the key.
Since the models include viscosity, the key velocity must be estimated. The velocity is obtained numerically by two independent algorithms: integration of the acceleration signal (after removal of the average value of the signal at rest) and differentiation of the position signal, using a total-variation regularisation Chartrand [2011] (here: 30 iterations, 200 subiterations, a regularisation parameter of 5.10−5 and ǫ = 10−9). In practice, choosing one or the other estimation of the velocity has very small influence on the simulation results.
Typical results for mezzo forte playing are displayed in Figure 2.2. The position of the key at the finger’s location is positive when the key is pushed down. The same convention applies to the force F(t) on the key.

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Simple models

The mechanism (Figure 2.3) consists in several quasi-rigid bodies – key, whippen, jack, lever, hammer (the damper has been excluded) – which are coupled together by felts and pivots. A first simplification consists in considering three blocks in the mechanism: the key, the whippen–lever–jack assembly, the hammer (Figure 2.4). The angular positions of the three blocks shown in Figure 2.4 – {Key}, {Whippen-Jack-Lever}, {Hammer} – are denoted by θ1, θ2, θ3 and their inertia with respect to their rotation axes by J1, J2, J3. The sign convention is counterclockwise for angles and torques. However, for the sake of simplicity in the representation of force and motion, the force F exerted at the end of the key and the motion y of the end of the key are counted positively when the key is pushed down: F = −C/L and y = −Lθ1 = −Lθ.
This model does not take into account the compliance between the bodies (key, whippen, etc.) and the contacts with the support (let-off button, drop screw). In line with the spirit of a simple model, it is considered that the variations of the θi are small (see Figure 2.4): the geometrical non-linearities are ignored so that coupling between the parts of the real mechanism does not alter significantly the parameters of the model.
For very small key displacements from its rest position, dry friction in the hammer’s and the whippen’s axes prevents their motion. Experimentally, we also observed that the compression of the small felt below the centre of rotation of the key (in Figure 2.1 see the small red felt between the middle of the key and the piece of wood supporting it) cannot be neglected any more. This lasts at least as long as the force applied to the key is less than Fdown. By various inspections of the motions of the different pieces (position tracking, not reported here), it was found that the dynamics was distinctively different whether θ L was less or more than ≈ 0.8 mm. Before the force reaches that threshold, a different model must be used, which is proposed further.
The momentum of the hammer is several times that of the rest of the mechanism. It follows that the inertia of the whole mechanism differs strongly from that of the simplistic key model when the hammer is dissociated from the rest of the mechanism (a few milliseconds between the escapement and the check of the hammer). Continuation of the model is discussed further in Section 2.4.
The actions of the torques exerted on the real mechanism are transposed on the simplistic key as follows. Non-permanent torques imposed by the stops limiting the motion of the key are considered and denoted by Cs. This torque includes the reaction of the rest support which disappears as soon as θ(t) > 0 and the reaction of the front rail punching LF which appears when θ(t) > θp. The compression law F(Lθ) of the front rail punching is that of a felt.
After the hammer check, an additional source of dissipation is located in the back-check: the key (with the whippen block resting on it) becomes also coupled to the support through the hammer and the back-check felt. Hypothetically, the corresponding friction dissipates significantly more than the internal compression of the felt of the front rail punching. Therefore, the value of b has been arbitrarily taken 100 times more than that measured by Brenon. The values of k, b and r are given in Table 2.2.

Simulations

This part presents simulations of the position of the key in response to given forces (force-driven simulations) and conversely, the reaction of the key to a prescribed motion at its end (motion-driven simulations).
As mentioned in Section 2.3, the inertial aspect of the model is invalid between the escapement and the check of the hammer. However, we chose to continue the simulation all along.
For a given force profile (here: Fmeas(t)), the angular position of the key θsimul(t) (and the displacement ysimul = L θ of the end) has been obtained by solving numerically Eq. (2.5) followed by Eq. (2.6) with C(t) = F(t)meas/L. The link between the two models was done by linear interpolation of the momentum of inertia ˜(θ) and the momentum of weights ˜ (θ) from their values in the first phase to their values in the second phase. The Cw numerical integration has been done by the NDSolve function of Mathematica R , using an Adams method with a maximum step limit of 30000. The results of these force-driven simulations are presented in Figure 2.6 for three different strengths of the keystroke: piano, mezzo forte and forte.
We also present the result of the simulation of Eq. (2.6) alone, with initial conditions given by the observation of θ and θ˙ at t corresponding to F(t) = Fdown. The drift that can be observed in these simulations is discussed in Section 2.5.
Conversely, motion-driven simulations yield Fsimul(t) = C(t)/L, the opposite of the reac-tion force exerted by the key for a prescribed motion θ(t) (here, θ(t) = xmeas/L). Ac-cording to Eq. (2.5) and Eq. (2.6), such simulations are straightforward, once the position and the acceleration of the key have been measured and the velocity has been estimated (see Section 2.2). The results are presented in Figure 2.7 for the same keystrokes as in Figure 2.6.

Discussion

After the escapement, the key hits the front rail punching. This piece does not differ between traditional keyboards and numerical keyboards. Since the latter are not judged as of equivalent quality by pianists, one should infer from this observation as well as from how pianists test and feel a keyboard, that the haptic feedback before escapement is of prime interest for them. Therefore, the discussion is primarily focused on this phase of the motion.
A first and basic finding can be deduced from the experimental observations reported in Figure 2.2. As can be seen in the bottom frame of this figure, the dynamics of the mechanism is dominated before escapement by the inertia of its pieces, taken as a whole. The other-than-inertial dynamical effects due the internal degrees of freedom, the various stops that are met or left by the pieces, etc. appear as time variations of the difference F(t) − ¨y(t)J /L2. The corresponding wiggles can easily be distinguished in the bottom frame of Figure 2.2, even though the motion (top frame) is quite smooth. Although not surprising, this elementary observation has important implications with regard to the main point raised in the introduction: in order to validate a dynamical model, should the dynamics be examined as producing a force in response to an imposed displacement or vice-versa?
From a purely experimental point of view, it is generally difficult to drive a mechanism with a rapidly changing force or acceleration. In this particular case, it would not be advisable either, since it would generate vibrations in the key that would mask the time-variation of the key displacement that are expected to be characteristic of the dynamics mechanism, considered as an assembly of rigid bodies. Altogether, realistic experimentations would consist in pushing this mechanism with a smooth force-profile, or with a smooth motion-profile similar to the ones reported here. Since the dynamics of this particular mechanism is dominated by inertia until escapement, it follows that the acceleration is generally smooth, possibly displaying some wiggles. When looking at the angular position of the key or displacement of the end of the key (imposed force), these potential wiggles in the acceleration are heavily filtered by the double time-integration: the differences between inertia and the complete dynamics of the system becomes hard, if not impossible, to distinguish. In other words, any model, provided that it is inertia-dominated, is likely to appear as very good when checking its validity by means of comparisons of motion-results obtained in force-driven simulations and tests, before escapement. This lack of sensitivity of the results to the model is represented by a « 0 » in upper left cell in Table 2.3.

Table of contents :

General introduction 
1. Introduction 
1.1. Presentation of the grand piano
1.2. Description of the grand piano’s action
1.3. History
1.4. Motivations and objectives
1.5. Literature review
1.6. Studied action
2. Modelling the dynamics of the piano action: is apparent success real? 
2.1. Introduction
2.2. Experiments
2.3. Simple models
2.4. Simulations
2.5. Discussion
2.6. Conclusion
3. Experimental set-up 
3.1. Kinematics
3.2. Force
3.3. Actuation of the key
3.4. Trigger and acquisition
3.5. Summary of measurement errors and uncertainties
4. Model 
4.1. Characteristics of the piano action
4.2. Generalities
4.3. Physical elements
4.4. Classification of parameters
4.5. Dynamics: description and values
4.6. Matrix formulation of the dynamics
5. Simulation methods 
5.1. Limitations of the regularising approach
5.2. Formulation of the piano action as a non-smooth dynamical multibody system
5.3. eXtended Dynamic Engine (XDE)
5.4. Adjustments of the model for its implementation
5.5. Adjustments of XDE for the simulation
5.6. Simulating in practice
6. Results and discussions
6.1. Regulation of the virtual piano action
6.2. Position-driven simulations
6.3. Force-driven simulations
6.4. Discussion
6.5. A sensitivity analysis
Conclusion 
A. Adjustment procedure of the grand piano action I
B. Application of the Lagrangian to a double pendulum V
C. Example of the calculation of ∂xδKH

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