Hammer shank vibration has an outstanding effect on piano sound. Hart,Lusby,Fuller thought that shank vibration energies were negligible compared to the hammer striking velocities.
They could neither observe shank vibrations nor their effect because a strong piece of cardboard was glued to the shank for the experiments.
Later Askenfelt&Jansson launched the idea of the importance of shank vibrations. Following the results of Ask&J we develop the theory of the vibrating hammer shanks.
A) The vibrations of the hammer : the experiments of Ask&J.
Slow vibrations of flexion.
Fast vibrations ``flip-flop`` - at the touch, - after escapement.
Super fast bending modes.
B) The vibrations of the shank : the bending forms.
B/I The starting point : statics, the absolute values of bending.
Elasticity constant of the shank at the roller.
B/II Equations of slow and fast vibrations.
B/III Vibration of the shank pushed by the jack.
Movement till the ``let-off``.
B/IV Equations of the free vibration after the ``let-off``.
C) The movement of the hammer : the time functions.
(See: Suite)
C/1 The electrical model of the hammer ``flip-flop``, generation of the ?flip-flop? motion.
C/2 The motion of the hammer after ``let-off``: the analogous web with braking and the free way.
C/3 The flip-flop motion and braking together, the phase relations.
To avoid all confusions we will mostly substitute jack-dolly collision to the term of jack-dolly contact, as in scientific literature jack-dolly contact means often the whole period between collision and impact on the string.
We will have for ``let off`` the restricted definition of loosing contact between roller and jack. ?Let off? distance will have also a restricted definition as the distance between hammer and strings when the hammer reaches his highest point when we push on the key very slowly.
Our discussions are somewhat idealized. We consider the following simplifications :
The hammer touches the strings only on one point, the dispersion of the spectrum because the finite hammer head is ignored.
The strings are completely flexible.
The hammer head top and bottom are moving together.
A. The vibrations of the hammer : the experiments of Askenfelt&Jansson
1. The slow vibrations 50 Hz
We found in a preceding work that the slow bending vibrations of the shank are possible only with a clamped shank. If this clamping is substituted by the roller-jack contact, the slow vibrations result from the whole system, key and hammer. Then the slow bending mode cannot be separated from the piano action, and will not be the target of this chapter. The slow vibrations will furnish some data to the other ``fast`` flip-flop mode.
2. Flip-flop vibration after the jack-dolly collision.
Horizontal components : this was the only component investigated by Ask&J, who suggested his role of generating longitudinal string vibrations, which are substantial for a good authentic piano sound.
We have found that vertical flip-flop motions exist too. Although the vertical deflection of the head is only 12 - 20% of the maximal shank flexing, it has a major influence on the shape of the impact velocity pulse, and enhances considerably the fundamental or lower spectral frequencies. This effect is very important at mezzoforte and piano levels.
3. Flip-flop movement at the stroke.
Only a staccato touch can produce flip-flop move at the beginning. On some pianos they are strong but are also strongly damped. On Steinway?s they are smaller but lasting till the jack-dolly collision. Utility of the initial flip-flop is the larger contact zone between jack and roller, which facilitates calculating and authorizes larger negative compressions (about twofold), than they would be if only due to gravitation.
4) Super fast bending modes 650 Hz.
Probably due to the two zones of the shank, with a strong discontinuity at the roller. We did not find it with our homogeneous shank model.
B. The vibrations of the shank : the bending forms
I. The starting point : statics
The absolute values of bending.
Elasticity constant of the shank at the roller.
At the beginning of the stroke the hammer remains immobile, the inertia of the head is in equilibrium with the striking force. The jack force is divided : one part is pushing the hinge upwards, the other is setting the head in motion.
In this early static state it is easy to calculate the motion of the jack and the corresponding Young modulus.
II. The equations of slow and fast vibrations
As we have mentioned, the study of shank vibration is limited to the few milliseconds after jack-dolly collision. The flip-flop motion beginning at this moment is responsible for the spectral components, which can be modified by the pianist. Shortly, it is important that at the moment of jack-dolly collision, the compression of the roller and the flexion of the shank should be of high level.
As from the jack-dolly collision on, the behaviour of the hammer is governed by the sum of four motions :
- the hammer-head continues its motion to the string because of its inertia,
- the rubbing between the jack and the roller are braking the motion,
- the bended shank straightens (low bending motion),
- the fast flip-flop motion.
Our scope is to certify theoretically the works of Ask.&J. and to clarify the origin of the slow and fast shank vibrations. Only the fast vibrations will be matter of detailed discussions. The slow bending modes will be mentioned only when they can be helpful to find some material constants (e.g. Young modulus of the roller). The coherent results of the slow vibrations will contribute to confirm our calculations.
The simplified hammer model is represented in the Figure 1, the notations and constants are below :
To calculate the dynamic motion we cannot consider the hammer shank divided in two parts : the higher rigidity of the short end was taken in account by the static calculations. The extra masses of the roller and of the larger short end can be introduced as a supplementary mass of the whole shank.
Instead of the jack force and global Young modulus we will suppose a torque and a spiral spring constant.
Notations and material constants :
Our sources : the numerical values and diagrams in the publications (Ask.&J. ).
Own measurements. Hypothetical calculations to eliminate the incoherencies.
Equivalent hammer-length (distance of hammer-head to rotation axis) : 0,13 m,
hammer-head mass : m = 0,0085 kg (Ask.&Jan.II.p. 2389),
equivalent cross-section of the shank : S = 5,52.10-4m2
specific weight of maple : m = 650 kg/m3 (Lugosi)
Young modulus of the shank : E = 9,81.1,2.109 N/m2 (Ask&J and measurements)
inertia moment of the shank : I = 5,74.10-12/12
inertia moment of the hammerhead : m x 0,0312 kgm2 (equivalent radius, determined by planimetry : 3,1 cm
spring constant of the roller : c = 0,0002 m/Newton.
Differential equations of the vibrating shank
The form and motion of bending rods satisfy the equation (1) :
(1) 
, whose solutions are of the form Y(x,t) = y(x).y(t)
After separation of the variables, Eq.1 will take the form 
(2)
(3) 
The general solutions of the two equations are :
The coefficients will depend on the boundary and initial conditions either in the case of free vibrating shank or when the shank is leaning on the jack.
III. The vibration of the shank leaning on the jack
Boundary conditions at the rotation side x = 0 : y(0,t) = 0 that means A = -C
Bending torque near the rotating axis : 
(with + sign, as the spiral spring pushes the axis upwards).
We must convert the pushing of the jack into a torque in the rotation center. Because we want to treat the shank vibration separately from the piano action, we will consider two extreme cases, one with a heavy hand, the other with a light one. The difference will be rather slight.
We will prefer the first case because it seems more realistic and offers more common points with the experiments of Ask.&J. taking account also of the global spring constant. The slight difference prove that the separated treatise of the shank-hammer system is justified.
Boundary conditions on the head side x = l :
Equations of shearing force :
The two equations to determine B and D :
We enumerate the differentials to be placed in the above equations :
After the substitutions we write the system we want to solve in matrix form :
The right side of the matrix is ?0?, the only possibility to find solutions for the different bending modes are the zero?s of the determinant which yield the values of ?lambda?
Besides, to find the coefficients A and B we must suppose that D is known and constant, equal to Do.
The equation of the shank bending will be
As it is evident, Do is a common factor and it will be easy to determine it by comparison with the static bending forms which are exactly calculable.
The result is in the Figure 2.
The static curve represents the situation when the equations are calculated with a heavy mass (0,85 kg) theoretically fixed on a rotating axis in the vicinity of the hammer head. It is an obstacle for all motion in vertical direction, but does not perturb the flip-flop motion of the head and shank.
This arrangement allows to compare the dynamic and static curves with the possibility to fix the real value of the coefficient Do.
The hammer-head has a vertical motion of ± 0,043 mm for ?mezzoforte? at 1 m/s hammer impact velocity.
The shank velocity is ± 0,000043x2000 = 0,086 m/s or 17,2 %. This is to compare with the low frequency components of the impact velocity pulse (about 20 % of the whole pulse)
IV. Equations of the free vibrating shank.
With free vibration the effect is slighter but the low frequency components are reinforced through the higher value of the equivalent spring constant at the end of the shank.
The calculations are simpler because the coefficients A and C are both equal to 0, but they require higher precision.
The values of ?lambda? are given by the solution of the characteristic equation of the following matrix :
The upper curve shows a final deflection of about 0,029 mm. The free vibration is associated with a higher value of equivalent spring constant at the head side. This means a low frequency reinforcement although the deflection is smaller.
V. Correcting terms
We have already taken account of the higher rigidity of the short end in the static calculatios. We had to evaluate the error when comparing maximal deflection instead of whole potential energies of the bended shanks. The error is about -2,5 % with free vibration and much less (+0,5%) in the case of the shank leaning on the jack. The curves above are calculated with the correction.
Another step is the extra mass correction of the roller and of the larger section at the short end. If we transfer these masses at the center of the shank and compare with the mass of the whole shank we find about 2 % of increase. This has a negative effect on the deflection and velocity of the shank of about 1 %.
We may have a much higher error if we consider the variable specific mass of the different kinds of maple.
As it is well known, the shanks are classed before being glued in the head in function of their tap tones. These shanks are of standard fabrication, the differencies can be only of weight and/or rigidity, which may differ in the stem.
A short calculation shows that 20% of weight difference diminishes by 12 - 13 % the deflection and by 1,5 % the frequency. Both contribute to the decrease of the velocity amplitude of the vibrating shank.
(The weight difference between two species of maple can vary by 20 %)
VI. Final conclusion :
A jack force of 10 Newton (mezzoforte) gives a free velocity vibration of about
± 0,000029x6,28x262 = ± 0,048 m/s
and a velocity of ± 0,000043x6,28x320 = ± 0,086 m/s with the shank leaning on the jack.
At mezzoforte level, with an impact velocity of 1 m/s, this is a change of ± 8,2 % or ± 17,2 %
At piano level the change may be of 20 - 30 %. The importance of this effect is growing if we compare the spectra. The spectrum of the impact velocity pulse is large, the fundamental is only 20 % of the total pulse. The shank vibration transfers a nearly sinusoidal pulse to the strings in the low frequency range of the piano middle register.
It is important to mention, that the jack force at the jack-dolly collision is not in relation with the hammer velocity. The delicate task of the touch is the dosage of the key force so that the hammer should not have enough time to accelerate and the jack force should be maximal at the jack-dolly collision. We have illustrated a possiblity in the discussion of the ?delayed touch? in Piano Acoustics I.
C. The movement of the hammer : the time functions.
Piano-acoustics II. (next chapter)
Piano Acoustics II (suite)/
zhorusitzky@voila.fr