The previous section treated an ideal pointmass striking an ideal string. This can be considered a simplified pianohammer model. The model can be improved by adding a damped spring to the pointmass, as shown in Fig.9.22 (cf. Fig.9.12).

The impedance of this plucking system, as seen by the string, is the parallel combination of the mass impedance and the damped spring impedance . (The damper and spring are formally in seriessee §7.2, for a refresher on series versus parallel connection.) Denoting the drivingpoint impedance of the hammer at the string contactpoint by , we have
The impedance formulation of Eq. (9.19) assumes all elements are linear and timeinvariant (LTI), but in practice one can normally modulate element values as a function of time and/or statevariables and obtain realistic results for loworder elements. For this we must maintain filtercoefficient formulas that are explicit functions of physical state and/or time. For best results, state variables should be chosen so that any nonlinearities remain memoryless in the digitization [364,351,558,556].