A Mechanistic Multibody Model for Simulating the Dynamics of a Vertical Piano Action

Abstract

The theoretical framework for constructing a fully mechanistic multibody dynamic model of a vertical piano action is described, and its general validity is established. Equations of motion are derived symbolically using a graph-theoretic formulation. Model fidelity is increased by introducing several novel features: (i) a new contact model for representing the compression of the felt-lined interfaces between interacting parts, capable of capturing the intermittent loading and unloading of these contacts occurring through the key stroke, as well as providing smooth transitions between these states; (ii) models for two important components that are unique to the vertical action, the bridle strap and the butt spring; (iii) a sophisticated key pivot model that captures both the rotational motion and the vertical translation of the key as it can lift off the balance rail under some conditions; (iv) flexible beam models for backcheck wire and hammer shank so as to predict observed vibrations in the response accurately; and (v) coupling of the mechanism model to a flexible stiff string model for realistic hammer impact. For simulation, parameters were obtained by experimental testing and measurement of a physical prototype vertical action. Techniques are described for the virtual regulation of the model to ensure that initial conditions and pseudostatic response accurately represent the precise configuration and desired relationships between the parts during the key stroke. Two input force profiles were used for simulations, a forte pressed (hard) and piano pressed touch (soft), typical of those measured at the key surface when activated by a pianist. Simulated response to these quite different inputs is described, and compared to experimental observations obtained from a physical prototype.

A novel micromechanical model of nonlinear compression hysteresis in compliant interfaces of multibody systems

“…The lack of physical significance for contact modeling was evident in Ref. [16], which is rectified in the current paper.…”

“…A simple model of a vertical piano action mechanism was developed by Masoudi et al [15], in which all the main bodies were considered to be rigid and the string was replaced with a rigid stop. The model fidelity was increased in other works by the authors [3,16], considering hammer-string interaction, hammer shank flexibility, backcheck wire flexibility, and a sophisticated key pivot model. They thoroughly studied the effect of bridle strap and butt spring on the hammer motion.…”

“…Details of the contact locations and geometries can be found in Refs. [3,16]. To estimate contact forces, the proposed hysteretic micromechanical model discussed in Sec.…”

“…3 is applied to the action, and its performance in multibody system simulations will be analyzed. To include partial loading and unloading during contacts, a mathematical relation is utilized [3,16]:…”