- The Hammer Model
- Traveling-Wave Decomposition of the Lumped System
- The Scattering Junction Connection
- Making the Felt Nonlinear
- The Lossless Digital Model

- The piano hammer, or a felt mallet, can be modeled as a non-linear spring and mass system.
- The spring represents the felt portion of the hammer or mallet and its compliance is considered to vary with felt compression. In this way, the felt is modeled as a non-linear spring with a stiffness ``constant'' that increases as it is compressed.
- First consider a linear mass and spring system as shown in Fig. 11. The mass is assumed to have an initial velocity at time zero.
- Since the mass and spring are in parallel, the driving force applied on the left side of the spring,
*f*, is equally applied to the spring and to the mass:

*f*=*f*_{k}=*f*_{m}.

- The driving point velocity,
*v*, is equal to the sum of the spring compression velocity and the mass velocity,

*v*=*v*_{k}+*v*_{m}.

- By Hooke's law, the restoring force of the spring is given by

where*x*(*t*) is the compression distance of the spring and*k*is the spring stiffness constant. - By Newton's law, the force of the mass is

where represents an initial momentum impulse setting the mass in motion with velocity*v*_{0}. - Both of these force equations can be expressed in terms of their Laplace transforms and written as a driving-point admittance:

The admittance function, (*s*^{2 }+*k*/*m*) /*k s*, is the steady-state response of the system with zeros at . The*v*_{0}/*s*term represents the transient effect of the momentum impulse applied at time zero.

- The lumped driving-point admittance of the mass-spring system can be defined in terms of wave variables in a manner analogous to distributed systems.
- An applied force can be written as
, where is viewed as a force wave traveling into the admittance and
is viewed as a force wave traveling out of the admittance.
- In a similar way, we can define
.
- In distributed systems, waves have a finite propagation speed and force and velocity wave components can be related by a wave impedance, which is defined by the parameters of the medium of travel.
- In lumped systems, however, traveling waves may only be understood to travel instantaneously and there is no particular medium. But an arbitrary reference impedance,
*R*_{h}, can still be defined such that:

- Making this change of variables, the velocity transfer function can be written

- Using the bilinear transform to convert from a continuous-time system to a discrete-time system, we obtain

where the continuous-time impulse function mapping has been bandlimited as and the coefficients are defined as

- Because the reference impedance
*R*_{h}is arbitrary, it is possible to choose a value such that the coefficient*b*_{0}= 0, as

- In this way, a unit of delay can be factored out of the second-order transfer function, such that

where

with*a*_{0}given by

- It is then possible to rewrite
*R*_{h}in terms of*a*_{0}, eliminating*k*, as

- A piano hammer strikes the string and then remains in contact with it for a certain time duration (before being thrown off by forces propagating in the string).
- During the time the hammer is in contact with the string, the string is ``loaded'' by the hammer impedance and wave scattering occurs.
- A lossless three-port scattering junction, as shown in Fig. 13, can be developed that defines the energy transfer between the hammer and the two sections of string.
- At the junction, the velocities of the string sections and the hammer spring must all be equal,

- In addition, the sum of the forces at the junction must be zero, since it is a massless point,

- Using these expressions and the previous wave variable definitions, the scattering equations become:

- As originally discussed, the felt is modeled as a nonlinear spring with a spring ``constant'' that increases as the compression of the spring increases. In this way, the spring stiffness constant
*k*(*x*_{k}) is dependent on*x*_{k}. - Because
*x*_{k}=*f*_{k}/*k*and because the driving-point force on the mass and spring system is equal to*f*_{k}, we can write

- The above expression can then be written in terms of
*a*_{0}as

- At any time in the model, the value of
*x*_{k}can be used to lookup or determine an appropriate value for the spring constant*k*.

- The resulting system of equations can be combined as shown in the block diagram of Fig. 14.
*S*is the three-port scattering junction.*H*is a time-varying allpass filter.*G*is the modified integrator that integrates the incoming initial hammer velocity impulse signal,*v*_{I}[*n*], taking into account the wave decomposition and the nonlinear effects of the hammer felt.*X*computes the actual felt compression, taking advantage of the fact that the force of compression on the felt is instantaneously proportional to the felt compression.*K*is a look-up table for the stiffness coefficient,*a*_{0}[*n*], indexed by felt compression.*R*computes the time-varying wave impedance of the hammer system from*a*_{0}.*R*_{h}[*n*] = 0 when the hammer is not touching the string and*k*(*x*_{k}) = 0 as well, so the string equations are then not affected by the hammer system.

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