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Research Papers

Experimental Validation of a Mechanistic Multibody Model of a Vertical Piano Action

[+] Author and Article Information
Ramin Masoudi

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

Stephen Birkett

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

matlab 7.10.0.499 (R2010a) ‘ode15s' method. Mathworks, 3 Apple Hill Drive, Natick, MA 01760-2098 USA.

Photron USA, Inc., 9520 Padgett Street, Suite 110 San Diego, CA 92126-4446.

MaxTraq 2D, Innovision Systems Inc., 3717 Peters Rd., Columbiaville, MI 48421-9304.

Endevco Model 2311-10, mass 28 g, resonant frequency 75 kHz.

National Instruments NI PXIe-6356 with Labview 2010 version 10.0 interface.

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 4, 2014; final manuscript received July 28, 2014; published online April 9, 2015. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 10(6), 061004 (Nov 01, 2015) (11 pages) Paper No: CND-14-1088; doi: 10.1115/1.4028194 History: Received April 04, 2014; Revised July 28, 2014; Online April 09, 2015

The validity and accuracy of a high-fidelity mechanistic multibody model of a vertical piano action mechanism is examined experimentally and through simulation. An overview of the theoretical and computational framework of this previously presented model is given first. A dynamically realistic benchtop prototype mechanism was constructed and driven by a mechanical actuator pressing the key. For simulations, a parameterization based on geometric and dynamic component properties and configuration is used; initial conditions are established by a virtual regulation that mimics a piano technician's procedure. The motion of each component is obtained experimentally by high-speed imaging and automated tracking. Simulated response is shown to accurately represent that of the real action for two different (pressed) key inputs using a single fixed parameterization. Various specialized model features are separately evaluated. Proper simulated dynamic behavior supports the accuracy of the friction representation; this is especially so for softer key inputs which demand a more actively controlled playing technique. The accuracy of the contact model is confirmed by the proper timing and function of the mechanism, as the relationship between components is strongly dependent on the state of compression of the interface between them. The value of including three flexible components is weighed against their significant computational cost. Compared to a rigid fixed ground point target, hammer impact on a compliant string reduces impact force, contact duration, and postimpact hammer velocity to improve accuracy. Flexibility of the backcheck wire and hammer shank also strongly affects postimpact behavior of the mechanism. The sophisticated balance pivot model is seen to be valuable in creating a realistic key response, with compression of felt balance punching and lift-off of the key, very important for achieving the proper key–hammer relationship. Finally, two components unique to the vertical mechanism—the bridle strap and butt spring—are shown to be essential in controlling the hammer for detached key inputs, where the key is released before it has reached the front punching. Accurate postimpact response is important for proper simulation of repeated notes, as well as the “feel” of the action. In general, the results reported can be considered as a validation of the method for constructing and parameterizing a dynamically accurate multibody model of a specific prototype mechanism or system including compliant contacts and flexibility of some components, as well as ad hoc components to cover unusual dynamic behavior.

Copyright © 2015 by ASME
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References

Figures

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Fig. 1

Components and configuration of a typical vertical piano action mechanism

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Fig. 2

Graph–theoretic model of a vertical piano action mechanism (Steinway Model 45), showing the five bodies and the graph edges representing their physical behavior. Generalized coordinates for the rigid-body motions are also indicated.

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Fig. 3

Experimental apparatus showing locations of tracking points on the mechanism components used for model validation comparisons.

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Fig. 4

Piano-pressed input force profile (top) and simulated and experimental vertical displacement of key front and whippen (middle) and hammer tip (bottom) tracking points. Time zero corresponds to the moment of maximum hammer–string contact force. Inset detail shows moment of first hammer–string contact (point B).

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Fig. 5

Forte-pressed input force profile (top) and simulated and experimental vertical displacement of tracking points on key front and whippen (middle) and hammer (bottom). Time zero corresponds to the moment of maximum hammer–string contact force.

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Fig. 6

Sensitivity of hammer dynamic response to key pivot revolute joint friction. Simulated hammer tip horizontal displacement for piano-pressed (top) and forte-pressed (bottom) inputs with original friction parameters and increased by factor of three.

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Fig. 7

Sensitivity of hammer dynamic response to hammer–backcheck sliding friction. Simulated hammer tip horizontal displacement for piano-pressed (top) and forte-pressed (bottom) inputs with original friction parameters and reduced by 50%.

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Fig. 8

Comparison between simulated horizontal hammer displacement with hysteretic and nonhysteretic contact models in case of piano-pressed (top) and forte-pressed (bottom) inputs.

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Fig. 9

Simulated and experimental key vertical displacement at the balance point (balance rail punching compression) and difference between them (absolute simulation “error”) in the insets, for piano- and forte-pressed inputs.

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Fig. 10

Simulated hammer versus key front vertical displacement for piano- and forte-pressed inputs, showing effect due to a 50% softer prismatic contact.

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Fig. 11

Simulated horizontal displacement of hammer tracking point with flexible and rigid backcheck wire for piano- and forte-pressed inputs.

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Fig. 12

Simulated horizontal displacement of hammer tip with flexible and rigid hammer shank for forte- and piano-pressed inputs.

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Fig. 13

Horizontal displacement of the hammer tip for forte- and piano-pressed input forces with elastic string or rigid stop ground contact.

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Fig. 14

Staccato force input (top) and simulated response as hammer horizontal displacement (bottom) for different activation scenarios for the butt spring and bridle strap.

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Fig. 15

Simulated bridle strap force (top) and simulated and experimental horizontal hammer displacement (bottom) for staccato key input force with operative bridle strap and normal butt spring.

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Fig. 16

Simulated and experimental hammer horizontal displacement (top) and simulated bridle strap force (bottom) for staccato key input force with disconnected butt spring and normal bridle strap.

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Fig. 17

Simulated and experimental hammer horizontal displacement for staccato key input force with inoperative bridle strap and normal butt spring.

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