Research Papers

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

[+] Author and Article Information
Ramin Masoudi

Research Fellow
Department of Systems Design Engineering,
University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada
e-mail: rmasoudi@uwaterloo.ca

Stephen Birkett

Associate Professor
Department of Systems Design Engineering,
University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada
e-mail: sbirkett@uwaterloo.ca

John McPhee

Department of Systems Design Engineering,
University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada
e-mail: mcphee@real.uwaterloo.ca

The diminishing key holding force evident in Fig. 7 is due to relaxation of the dashpot component in the mechanical actuator.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 19, 2013; final manuscript received November 29, 2013; published online February 13, 2014. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 9(3), 031014 (Feb 13, 2014) (10 pages) Paper No: CND-13-1202; doi: 10.1115/1.4026157 History: Received August 19, 2013; Revised November 29, 2013

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.

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Fig. 1

Components and configuration of a typical vertical piano action mechanism (Essex EUP-123)

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Fig. 5

Loading and unloading fit curves obtained from experiment for a contact interface in the vertical action and simulated partial unloading curves calculated using the technique described in the text

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Fig. 4

Loading and unloading curves for dynamic force-compression of a piano hammer impacting a rigid stop at various velocities

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Fig. 3

Contact locations in the vertical action mechanism

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Fig. 2

Schematic of the vertical piano action (Steinway Model 45) components and rigid-body generalized coordinates. Flexible shank and wire in exploded hammer and backcheck each introduce three additional generalized coordinates (see text).

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Fig. 6

Generalized coordinate system for the hammer-string interaction

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Fig. 7

Piano pressed and forte pressed key stroke force profile model inputs

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Fig. 8

Regulation points for the vertical piano action mechanism

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Fig. 9

Simulated and experimental horizontal velocity of hammer tip for the piano pressed input

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Fig. 10

Simulated and experimental horizontal velocity of hammer tip for the forte pressed input

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Fig. 11

Simulated time history of angular position of hammer θh(t), whippen θw(t), and key 6θk(t) for piano pressed key input

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Fig. 12

Simulated time history of angular position of hammer θh(t), whippen θw(t), and key 4θk(t) for forte pressed key input



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