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.

References

References
1.
Pfeiffer
,
W.
,
1921
,
The Piano Key and Whippen: An Analysis of Their Relationship in Direct Blow Actions
,
Verlag Das Musikinstrument, Frankfurt am Main
,
Germany
(English transl. by J Engelhardt, 1965, incorporating material from second (1931) and third (1955) German editions).
2.
Matveev
,
P.
, and
Rymskij-Korsakov
,
A.
,
1937
, “
Sbornik
,”
NIMP
,
Moskva
,
1
.
3.
Oledzki
,
A.
,
1972
, “
Dynamics of Piano Mechanisms
,”
Mechanism and Machine Theory
,
7
, pp.
373
385
.10.1016/0094-114X(72)90047-X
4.
Topper
,
T.
, and
Wills
,
B.
,
1987
, “
The Computer Simulation of Piano Mechanisms
,”
Int. J. Model. Sim.
,
7
, pp.
135
139
.
5.
Hayashi
,
E.
,
Yamane
,
M.
, and
Mori
,
H.
,
1999
, “
Behavior of Piano-Action in a Grand Piano. I. Analysis of the Motion of the Hammer Prior to String Contact
,”
J. Acoust. Soc. Am.
,
105
, pp.
3534
3544
.10.1121/1.424678
6.
Gillespie
,
B.
,
1992
, “
Dynamical Modeling of the Grand Piano Action
,”
Proc. of the 1992 International Computer Music Conference
, pp.
77
80
.
7.
Gillespie
,
B.
,
1996
, “
Haptic Display of Systems With Changing Kinematic Constraints: The Virtual Piano Action
,” Ph.D. thesis, Stanford University, Stanford, CA.
8.
Van den Berghe
,
G.
,
De Moor
,
B.
, and
Minten
,
W.
,
1995
, “
Modeling a Grand Piano Key Action
,”
Comp. Music J.
,
19
, pp.
15
22
.10.2307/3680597
9.
Hirschkorn
,
M.
,
McPhee
,
J.
, and
Birkett
,
S. H.
,
2006
, “
Dynamic Modeling and Experimental Testing of a Piano Action Mechanism
,”
ASME J. Comp. Nonlin. Dyn.
,
1
(1), pp.
47
55
.10.1115/1.1951782
10.
DynaFlexPro
,
2006
,
DynaFlexPro User's Manual
,
MotionPro Inc.
,
Waterloo, ON, Canada
.
11.
McPhee
,
J.
,
Schmitke
,
C.
, and
Redmond
,
S.
,
2004
, “
Dynamic Modelling of Mechatronic Multibody Systems With Symbolic Computing and Linear Graph Theory
,”
Math. Comp. Model. Dyn. Syst.
,
10
, pp.
1
23
.10.1080/13873950412331318044
12.
Maplesoft
,
2007
,
Maple 11 User Manual
,
Maplesoft
,
Waterloo, ON, Canada
.
13.
Hunt
,
K. H.
, and
Crossley
,
F. R. E.
,
1975
, “
Coefficient of Restitution Interpreted as Damping in Vibroimpact
,”
ASME J. Appl. Mech.
,
42
(2), pp.
440
445
.10.1115/1.3423596
14.
Izadbakhsh
,
A.
,
McPhee
,
J.
, and
Birkett
,
S. H.
,
2008
, “
Dynamic Modeling and Experimental Testing of a Piano Action Mechanism With a Flexible Hammer Shank
,”
ASME J. Comp. Nonlin. Dyn.
,
3
(3), p. 031004.10.1115/1.2908180
15.
Izadbakhsh
,
A.
,
2006
, “
Dynamics and Control of a Piano Action Mechanism
,” M.Sc. thesis, University of Waterloo, Waterloo, ON, Canada.
16.
Vyasarayani
,
C. P.
,
Birkett
,
S. H.
, and
McPhee
,
J.
,
2009
, “
Modelling the Dynamics of a Compliant Piano Action Mechanism Impacting an Elastic Stiff String
,”
J. Acoust. Soc. Am.
,
125
, pp.
4034
4042
.10.1121/1.3125343
17.
Masoudi
,
R.
, and
Birkett
,
S. H.
,
2013
, “
Experimental Validation and Analysis
,”
ASME J. Comp. Nonlin. Dyn.
(submitted).
18.
Kane
,
T. R.
, and
Levinson
,
D. A.
,
1985
,
Dynamics: Theory and Applications
,
McGraw-Hill
,
New York
.
19.
McPhee
,
J.
,
2005
, “
Unified Modeling Theories for the Dynamics of Multibody Systems
,”
Advances in Computational Multibody Systems
,
Springer
, Netherlands, pp.
129
158
.
20.
Masoudi
,
R.
,
Birkett
,
S. H.
, and
McPhee
,
J.
,
2009
, “
Dynamic Model of a Vertical Piano Action Mechanism
,” ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference,
Proc. IDETC/CIE 2009
, Aug. 30–Sep. 2, San Diego CA, Paper No.DETC2009-87680.10.1115/DETC2009-87680
21.
van Wyk
,
C. M.
,
1946
, “
Note on the Compressibility of Wool
,”
J. Textile Inst.
,
37
, pp.
285
292
.10.1080/19447024608659279
22.
Carnaby
,
G. A.
, and
Pan
,
N.
,
1989
, “
Theory of Compression Hysteresis of Fibrous Assemblies
,”
Textile Res. J.
,
59
, pp.
275
284
.10.1177/004051758905900505
23.
Komori
,
T.
, and
Itoh
,
M.
,
1991
, “
A New Approach to the Theory of the Compression of Fiber Assemblies
,”
Textile Res. J.
,
61
, pp.
420
428
.10.1177/004051759106100709
24.
Alkhagen
,
M.
, and
Toll
,
S.
,
2007
, “
Micromechanics of a Compressed Fiber Mass
,”
ASME J. Appl. Mech.
,
74
(4), pp.
723
731
.10.1115/1.2711223
25.
Dunlop
,
K. H.
,
1983
, “
On the Compression Characteristics of Fibre Masses
,”
J. Textile Inst.
,
74
, pp.
92
97
.10.1080/00405008308631770
26.
Coleman
,
T. F.
, and
Li
,
Y.
,
1996
, “
An Interior, Trust Region Approach for Nonlinear Minimization Subject to Bounds
,”
SIAM J. Optimization
,
6
, pp.
418
445
.10.1137/0806023
27.
matlab®
,
2010
,
Version 7.10.0 (R2010a)
,
The MathWorks Inc.
,
Natick, MA
.
28.
Cull
,
S.
, and
Tucker
,
R.
,
1999
, “
On the Modeling of Coulomb Friction
,”
J. Phys. A Math. Gen.
,
32
, pp.
2103
2113
.10.1088/0305-4470/32/11/006
29.
Meirovitch
,
L.
,
2001
,
Fundamentals of Vibrations
,
McGraw-Hill
,
New York
.
30.
Fletcher
,
H.
,
1964
, “
Normal Vibration Frequencies of a Stiff Piano String
,”
J. Acoust. Soc. Am.
,
36
, pp.
203
209
.10.1121/1.1918933
31.
SolidWorks® Office Premium
,
2007
,
SolidWorks 2007 Online User's Guide—SP0.0
,
SolidWorks Corporation
,
Concord, MA
.
32.
Steinway & Sons
,
2007
,
World-Wide Technical Reference Guide
,
Steinway & Sons
,
New York
.
You do not currently have access to this content.