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C. The motion of the hammer : the time function


The differential equation of the vibrating hammer can be represented as the product of the differential equation of the form function and that of the time function. As we have seen, the time function is a pure sinusoidal enabling to substitute a simple oscillating circuit for the shank motion.



C/1 Electrical model of a flip-flopping hammer and the excitation of the vibrations


The excitation of the vibration :
- at the beginning of the touch : Only a staccato touch can create a flip-flop motion, which will be already damped when the jack-dolly collision occurs.
- excitation at the jack-dolly collision and at the impact on the strings :
We arrive at the crucial point of our study. The jack-dolly collision moment is decisive to discuss the origin of the flip-flop motion and the corollary braking process of the hammer velocity. Both phenomena need a force : the necessary forces are five to ten times stronger then the gravitation. (The gravitation is negligible in the middle register)
We anticipate the very important information of Ask.&J. (6) reproduced in the Fig. 15, 16 & 17 :
1) the roller remains in contact with the jack in many cases at mezzoforte and forte levels
2) The hammer continues its motion with very inequal velocity up to the strings.
Both of these observations are of crucial interest : As Suzuki&Nakamura(9) resume : two basic principles become obsolete.
These two dogmas: the uniform velocity of the hammer after the jack-dolly collision and the hammer completely liberated from the action, dominated minds so strongly, that nobody had the idea till the second half of the 1980s, to see what happens in reality.
3) In two figures we see that two very different kinds of touch produce the same shank vibration, the same final velocity of the hammer and the same sound.
This can be explained in two manners : one (wrong) is that the touch of the pianist has no importance, the other (right) that there are several different kinds of staccato touch (as Ask&J mentioned: there are other “modes” of staccato).
Our previous work permits to assert that an essential condition is the high degree of compression of the action parts and of the roller and shank at the moment of the jack-dolly collision. The pianist has to dose the key force with care to produce this compression, and as we have discussed in the previous work, the touch cannot be named by the narrow categories of simple stacato’s or legato.

The oscillating circuit formed by the flip-flopping shank
Every point of the shank, has a sinusoidal motion (the flip-flop in our denomination does not mean a relaxation movement) it is then possible to substitute an electrical oscillating circuit for the shank composed of the head mass “M” with a parallel spring constant “C”, which form an electrical analogic series circuit.

First we have to calculate the spring constant of the shank at the head end :



The two methods yield very similar results, the small discrepancies (-5%) come probably from the neglected terms, such as the shank mass.
The electrical analogic model is fed by the jack force “F” which decreases after the jack-dolly collision point because a great part of the jack force will be absorbed by the “dry” friction between roller and jack. This negative force pulse is the excitation source of the shank vibrations. But this vanishing cannot be instantaneous. As the frictional force reaches progressively its final value, so the force “F” will diminish with some delay.
We admit, for practical reasons of calculation, that the above diminution obeys to an exponential law with a tendency to zero. This exponential law supposes that the felt or leather layer on the dolly is an absorbing resistive material without elasticity. This is only a working hypothesis, but we can be sure, that each decay law, exponential or sinusoidal or other, has at the beginning the same linear decreasing feature in the vicinity of the excitation.
The shank vibration exists only if the global spring of the whole structure is highly compressed, which means the good bending of the shank upward. If the pushing force disappears, the strain of the shank
pushes the whole hammer downwards and the oscillations begin. The damping is slow, which means that the oscillations continue without external force. The amplitude of the oscillations in mezzoforte with our typical 10 N jack force is of 0,18 mm at the center and of 0,04 mm at the head when "F" is concentrated and its vanishing of is infinitely short.
The study of the shank can be done independently of the interaction with the keyboard and jack action. The rapid oscillations of the shank cannot much perturb the keyboard motion, which can be considered as immobile. The two motions will be simply additioned when necessary, without any mutual influence.

After this introduction let us examine the calculations and the resulting diagrams of the shank vibrations whith different time constants of the decreasing pushing force :

With a jack force of 10 N, we have a head deflection of 0,041 mm. So, with a spring constant of 0,000028 m/N the force at the head F_h will be of 1,47 N and the product F_h x c, evidently, of 0,041 mm.
The shank deflection at the head will be “y” :


In the following figures we represent the shank excitation at various collision speeds between roller and dolly. The vibration amplitude begins to decrease from 0,2 msec on.


The exponential time constants are chosen only as an example. The determination of the exact values is the task of the future researches.



C/2. The hammer motion after the jack-dolly collision. The analogic circuitry with braking.



The generation of flexional “slow” vibrations :
- at the beginning of the touch, the shank vibration is inseparable from the vibration of the piano action structure,
- at the jack-dolly collision : the vibration is connected with the very delicate braking problem of the hammer and will be the theme of the present chapter.

The braking of the hammer before reaching the strings represents a very crucial and mysterious problem. The decreasing speed of the hammer after the jack-dolly collision is not comprehensible without a negative force. Or to pull on the roller is hard to conceive. A diminishing push on the roller can act as a negative force within the limit of the existing contact between roller and jack.
We consider the piano action and the hammer as a whole, moving upwards, pushed by the key force with a high level of compression of all parts, included the roller felt. If a braking force appears, the roller compression diminishes but does not disappear immediately. During the decrease of the roller compression the weakening of the jack force appears as a braking force on the hammer.
The jack force begins to decrease when the jack tail reaches the dolly. At this moment the pushing force of the key is divided between a small pushing force on the jack and a major frictional force of the roller leather on the jack top which resists the jack rotation. Much care must be taken of the motion and compression senses.The following figure facilitates the understanding of this process :


The mechanical model of the above structure is represented schematically in the Figure 7 :


In this case the initial conditions are not zero and their determination is important. The initial conditions concern the forces and positions.
The starting point of the braking process is a delicate problem : namely the braking force is null in quiet state, it is stronger at the start but later does not depend on velocity and his sense changes if the roller moves downwards.
At zero time the springs are compressed independently from the braking force which is null without motion.
At time 0+DELTA(t) the motion begins and the equations of motion become valid depending on the initial conditions at time 0-DELTA(t) (before or at the jack-dolly collision moment).
We shall represent the above system in the form of mathematical equations :

The treatise will be easier if we substitute an electrical analogic circuit for the mechanical model :



We write the equations in matrix form with positions instead of velocities.
On the left side are the loops of “paths”, on the right side are the sources of force. On the top line are ranged the variables of positions y, z, x with the initial values y_o, z_o and x_{o :}

As we see the braking force F_BR decreases with the variables “y” and “x”. This is normal because F_BR decreases when the compression, that is the y - x difference, falls as the hammer reaches progressively the position prescribed by the key. The reorganized matrix is shown on the right side.
The notations and the adopted numerical constants are as follows :


c = 12.10^-5 m/N, the “global” spring constant, c = a + b
b = 3.10^-5m/N, the spring constant of the roller and the shank,
a = 9.10^-5m/N the spring constant of the key and lever system
k = coefficient of friction between the leather layer of the roller and the jack top multiplied by the lever arms ratio of the jack, k = 0,3 x 3 = 0,9
V, v = hammer and key velocities converted to the roller position before the jack-dolly collision
at (roughly) mezzoforte level : V = 2/6,5 m/sec and v = V/5 m/sec.
v = key velocity reduced to the jack level : v = V/5
y_o-x_o = path difference of key and hammer, both converted to the roller position. This serves as compression indicator at the moment of jack-dolly collision. To simplify we write (y_o-x_o)/c = Q

The converted hammer path : “x”
The determinant of the matrix “DET_o”, yields the frequency of the whole structure between jack-dolly collision and the strings and the sub-determinant “DET_x” gives the position of the hammer :


The examination of the hammer path shows that the last term is negligible. That means that the pianist does not influence the hammer motion after the jack-dolly collision, that is during the last 5 - 10 msec before the impact on the strings. We must not be misled by this result, the touch of the pianist remains decisive at the beginning of the whole motion during the whole period before string impact.

The path of the jack top : “z”



In the jack path equation the force “F” in the third term, depending on touch, influences the jack motion and the condition to maintain the roller jack contact. Although the hammer motion is mostly independent from the jack force or touch, the jack-roller contact before impact is important and may have an influence on the form of the string impact velocity.
The examples of hammer motion published by Ask.&J. and our own earlier results concerning mezzoforte levels and legato touch show that this situation requires a touch with progressively increasing force. As in the case of mezzoforte, the key bottom is not reached, there is no reason why the progressively increasing force should stop and should not continue after the jack-dolly collision. We will complete the force “F” and multiply it by (1+1/ßp), where ß represents the acceleration rate of the key motion before the jack-dolly collision.

The completed path equations of “x” and “z” without the negligible terms :
The path equations “X”, “x” and “z” are not valid if “x” is greater than “z”, that is if the roller leaves the jack. Therefore our next task is the presentation of both “z” and “x”, and their difference in a clear and completed form :




and the path difference “z-x” between the roller and the jack top :


With the hammer path and the path difference, which is the spring compression of the roller and the flexion of the shank, we can draw the hammer velocity and verify that the roller jack top contact is existing :



This relation tells us the condition of existing contact between roller and jack, and so, the limits of validity of the equations of motion. If this condition vanishes, the motion must be divided into sections :
First section : the contact exists, the equations are valid.
The following sections :
- the hammer becomes free and continues its motion independently, or
- the jack turns and the contact disappears between roller and jack.
Exceptional situation - the jack moves downward and the sense of the friction changes.

The detailed study of these cases is needless. We will show only two possibilities :
1) the contact remains till the impact, “free motion” is inexistent,
2) the turning of the jack precedes the impact, the path finishes with a free motion.

The two cases are illustrated by the following diagrams :
(We omitted the superposing shank vibration for give a clear picture)





Comparison of the two diagrams :
The impact happens after 7,5 msec in the first case, and after 6,2 msec in the second. This difference corresponds roughly to a half period of shank vibration (not represented here) as we will see it in the Figures 13 & 14.
In both cases the velocity at the jack-dolly collision is the same.

With a delayed and suddenly increasing key force (12 N) there is no free path, the braking lasts till the impact (that is the impact velocity and loudness are less). Related to the lower loudness, the shank vibration will have a growing importance. The shank vibration itself will be higher because the roller is leaning on the jack. The relations of phase make that the fundamental frequency, or low frequancy part of the spectrum will grow which results in a soft piano sound.

With a less brutal, not delayed touch and a free path, the hammer reaches earlier the impact level, with higher velocity because the braking time is shorter. The free shank vibration has a lower amplitude and the phase relation produces rather a decreased fundamental frequency. The sound is changing in the sense of more brilliance and of mezzoforte level. Although the hammer arrives to the string with a velocity valley, the relative increase of the high frequency components produces a higher loudness sensation.
The foregoing reflexions are valid chiefly for the C4 - C5 octave. One octave deeper it can be question of the octave enhancement or weakening.

In the free path discussion the major question is the determination of the moment when the roller leaves the jack. The only fixed point is the jack top position at the moment of the jack-dolly collision. The hammer position, converted to the roller line, depends on the shank flexion, converted also to the roller line, and on the roller compression. The other reference point is the end of the jack rotation. This is, in all cases, the path of the upward jack motion during the horizontal motion of the jack top relative to the roller. The upward path of the jack top is then the horizontal motion divided by the jack arms ratio.
Some experiments showed that the lateral path of the jack top is 4 mm during the rotation. With a jack arms ratio of 3,1, the corresponding upward path is 1,3 mm. ( The 4 mm for the lateral path is very approximative !)
Other estimations, a following calculation and other experiments suggest a lateral path of
3,7 mm with an upward motion of 1,2 mm :





C/3. The flip-flop motion and the braking together
The phase relations of the shank vibration


The cumulative effect of flip-flop and braking :
In the three following examples we show the velocity and acceleration of the hammer after the jack-dolly collision without free path on mezzoforte level.
The hammer reaches the string roughly in 6 to 8 msec





In reality we do not need these curves for the following discussion of the impact on the strings. Nevertheless, we publish them because they offer an excellent possibility to compare theoretical and experimental results namely those of Askenfelt & Jansson.
The comparisons show a very good concordance and confirm our calculations. Before a detailed discussion of both results, let us examine the phase and amplitude relations by means of the acceleration curves.

The phase relations. After the Figures 9 and 10 we have mentioned the phase of the shank vibrations. The phase does not depend only on the moment when the roller leaves the jack top, and the compression becomes null (negative on the curves), but also on the excitation, which determines the amplitude at the same time. With “fast” or “low” collision the phase differences may be of 0,8 msec which corresponds to 90°. This have a major influence on the impact pulse, and this effect is added to the phase contribution of the free path.
The amplitude relations. As the shank vibration amplitude with fast collision is about twice the amplitude of the slow one, the piano constructor must find an adequate material to cover the dolly with a high mechanical resistance without rebounding. But the pianist cannot change this element, thus it can be useful to him to understand the collision process and have somme ideas how to adapt his touch to different pianos.

The theoretical acceleration curves compared to the curves published by Ask.&J.
To compare we must find the correspondance between dynamical levels and hammer velocities.
This is a head-splitting task because the authors did not take account of the slowing-down process before impact. We were informed of this effect only by the following curves of Ask&J but we had to find ourselves the explanation and origin. We found only one reference mentioning the velocity “just before the impact” (Giordano). The velocities, supposedly before slowing down, are concordant with the curves of Ask&J. The Giordano(12) velocity on forte seems to be coherent with the mezzoforte level of 1 m/s.
We could take as typical the following curves of Ask.&J. and consider the mezzoforte C4 velocities equal to 1,8 - 2 m/s at the jack-dolly collision and about 1 m/s at the impact.
The comparison of the measurement with the theory is limited. The piano model is not necessarily the same or rather it is surely not the same, that we considered as typical, and we ignore the exact behaviour of the pianist. Our model is based on mathematical and simplified laws.
The acceleration scales, not compatible with the velocity curves, were rectified in concordance with them.
The dynamical level is mezzoforte, the hammers are the B3 or C4.
It is regrettable that in the curves Fig.15 and 16 we miss the jack force and on the last diagram the velocity is missing. But even with these lacks the three figures permitted to understand the shank vibration and to validate our models and calculations.


The shank vibrates 2,5 periods during 6 msec separating the jack-dolly collision and string impact.
The mean flip-flop frequency is 250 Hz. The first three half-periods are shorter than the two last ones.
The damper has a braking effect at the half course. This effect was difficult to appreciate and we admit that it is compensated automatically by the pianist.
We did not care about key bottom contact, as we suppose that on “mezzoforte” this does not occur.


Here the key force was a pulse (see Piano Acoustics I. Fig. 15 & 16, “Double exponential touch”)
The compression at the beginning had negative portions. The negative portions are diminished by the negative half-periods of the initial shank vibration. The supposed hysteresis of the roller can integrate the half-periods of the downward shank movement. This integration acts as a multiplied gravitational force making the roller jack contact sure and so the linearity of the system even during the negative half-periods of compression.
This requires farther experimental works, but does not influence the results of the next chapter.
In Figure 16 we do not see the damper effect but it modifies surely the second negative portion of the velocity curve.
Both figures show the slowing-down of the hammer velocity after the jack-dolly collision. We can see a slight overshoot at the start of the slowing down. We dont see clearly the shank vibration effect on the velocity curve. Our calculations show also the slight overshoot and the relative slightness of the observable shank vibration effect on the velocity curves.


The "kick" of the jack force is comprehensible : the force that produced the positive (upward) bending of the shank is acting against the jack at the first downward motion.
The interesting question is why the second downward motion did not produce the same “kick”?

It is possible that the second downward pulse has a lower effect because the roller-jack contact becomes looser. Perhaps the motion is a semi-free vibration : the upper half period is free, the lower is under a loose contact, but for the apparatus the contact remains existing.
Another hypothesis is the supposed hysteresis of the roller which rests compressed after the first strong compression pulse.

Before to continue and pass to the chapter “Piano Acoustics III. Impact and Spectra” we resume an obsevation for the pianists :

The shank vibrations and the pianist :
The pianist can produce vibrations, either with legato or with staccato touch, he can dose the key force, delicately as a medecine, to obtain a maximal compression and shank flexion at the moment of the jack-dolly collision.
The problem is not so simple concerning the question of phase which will be examined in the next chapter.
We may suppose that the phase relations of the shank vibrations, combined with the reflected agraffe impulsions, will depend rather on the hazard and not of the will of the pianist. If it happens so, we cannot speak of beautiful or hollow, but of a monotonous or varying brilliant sound. The varying sound does not signify that every tone is brilliant and exciting, but that brilliant and insolite sounds are accompanied by less interesting ones. At the end, the ears of the pianist will decide how to create a pleasant sound effect with what kind of finger motion. The pianist will cultivate the force of his fingers till the heard result is satisfactory. In this sense we can say that the pianist plays with his ears.
But the success must be wanted and the pianist must be convinced that there is not any kind of physical laws, which could forbid the good result. On the contrary there are plenty of possibilities to create a world of sounds corresponding to his will and his sound perception.

Appendix and References

href="pianoac1.html">Piano Acoustics I Piano Acoustics III


Conclusion