Stephen Birkett

Modeling the dynamics of a vibrating string with a finite distributed unilateral constraint: Application to the sitar

The free vibration response of an ideal string impacting a distributed parabolic obstacle located at its boundary has been analyzed, the goal being to understand and simulate a sitar string. The portion of the string in contact with the obstacle is governed by a different partial differential equation (PDE) from the free portion represented by the classical string equation. These two PDEs and corresponding boundary conditions, along with the transversality condition that governs the dynamics of the moving boundary, are obtained using Hamiltons principle. A Galerkin approximation is used to convert them into a system of nonlinear ordinary differential equations, with lower mode-shapes parametrized with respect to the location of the moving boundary as basis functions. This system is solved numerically and the behavior of the string studied from simulations. The advantages and disadvantages of the proposed method are discussed in comparison to the penalty approach for simulating wrapping contacts. Simulations with bridge-string parameters consistent with the configuration of a real sitar show that any degree of obstacle wrapping may occur during normal playing. Finally, the model is used to investigate the mechanism behind the generation of the buzzing tone in a sitar.

Dynamic Modeling and Experimental Testing of a Piano Action Mechanism With a Flexible Hammer Shank

The piano action is the mechanism that transforms the finger force applied to a key into a motion of a hammer that strikes a piano string. This paper presents a state-of-the-art model of a grand piano action, which is based on the five main components of the action mechanism (key, whippen, jack, repetition lever, and hammer). Even though some piano action researchers (e.g., Askenfelt and Jansson) detected some flexibility for the hammer shank in their experiments, all previous piano models have assumed the hammers to be rigid bodies. In this paper, we have accounted for the hammer shank flexibility using a Rayleigh beam model. It turns out that the flexibility of the hammer shank does not significantly affect the rotation of the other parts of the piano mechanism and the impact velocity of the hammer head, compared to the case that the hammer shank has been modeled as a rigid part. However, the flexibility of the hammer shank causes a greater scuffing motion for the hammer head during the contact with the string. To validate the theoretical results, experimental measurements were taken by two strain gauges mounted on the hammer shank, and by optical encoders at three of the joints.

Modeling MEMS Resonators Past Pull-In

In this paper, we develop a mathematical model of an electrostatic MEMS (Micro-Electro-Mechanical systems) beam undergoing impact with a stationary electrode subsequent to pull-in. We model the contact between the beam and the substrate using a nonlinear foundation of springs and dampers. The system partial differential equation is converted into coupled nonlinear ordinary differential equations using the Galerkin method. A numerical solution is obtained by treating all nonlinear terms as external forces. We use the model to predict the contact length, natural frequencies, and mode shapes of the beam past pull-in voltage as well as the dynamic response of a shunt switch in a closing and opening sequence.

Modeling Impacts Between a Continuous System and a Rigid Obstacle Using Coefficient of Restitution

In this work, we discuss the limitations of the existing collocation-based coefficient of restitution method for simulating impacts in continuous systems. We propose a new method for modeling the impact dynamics of continuous systems based on the unit impulse response. The developed method allows one to relate modal velocity initial conditions before and after impact without requiring the integration of the system equations of motion during impact. The proposed method has been used to model the impact of a pinned-pinned beam with a rigid obstacle. Numerical simulations are presented to illustrate the inability of the collocation-based coefficient of restitution method to predict an accurate and energy-consistent response. We also compare the results obtained by unit impulse-based coefficient of restitution method with a penalty approach.

Modeling the dynamics of a compliant piano action mechanism impacting an elastic stiff string

A realistic model of the piano hammer-string interaction must treat the action mechanism and string as a single system. In this paper an elastic stiff string model is integrated with a dynamic model of a compliant action mechanism with flexible hammer shank. Action components represented as rotating bodies interact through felt-lined interfaces for which a specialized contact model with hysteretic damping and tangential friction was developed. The motion of the hammer during string contact is governed by the dynamics of the action mechanism, thereby providing a more sophisticated hammer-string interaction than a simple transverse impact hammer model with fixed contact location. Simulations have been used to compare mechanism response for impact on the elastic string as compared to a rigid stop. Hammer head scuffing along the string and time in contact were predicted to increase, while hammer shank vibration amplitude and peak contact force were decreased. Introducing hammer-string friction decreases the duration of contact and reduces the extent of scuffing. Finally, significant differences in hammer and string motion were predicted for a highly flexible hammer shank. Initial contact time and location, length of contact period and peak force, hammer vibration amplitude, scuffing extent, and string spectral content were all influenced.

Experimental investigation of the piano hammer-string interaction

Experimental techniques for investigating the piano hammer-string interaction are described. It is argued that the accuracy, consistency, and scope of conclusions of previous studies can be compromised by limitations of the conventional methods relating to key inputs; physical distortion; numerical distortion, particularly when differentiation or integration of measured signals is used to derive primary response variables; contact identification; and synchronization issues. These problems are discussed, and experimental methods that have been devised to avoid them are described and illustrated by detailed results from a study of the hammer-string interaction in a vertical piano. High resolution displacements are obtained directly by non-contact high-speed imaging and quantitative motion tracking. The attention focused on achieving very accurate and consistent temporal and spatial alignment, including the objective procedure used for contact identification, allows meaningful comparisons of responses from separate tests. String motion at the strike point and on each side of it, as well as hammer motion, is obtained for eight dynamic levels from 1.06 to 2.98 m/s impact velocity. Detailed observations of the force-compression behavior of the hammer interacting with real strings are presented. The direct effects of hammer shank deflection and agraffe string pulses on the interaction are also highlighted.

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

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 technicians 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.

Dynamic Model of a Vertical Piano Action Mechanism

The dynamic behavior of a vertical piano action mechanism is studied using a simulation model and compared qualitatively to observations obtained by high-speed imaging of a real action. The simulated response of all components is obtained for two different prescribed input force profiles applied at the key front. These inputs represent in simplified form the general shape of a typical force input by a pianist measured at the key surface for a strong (forte) strike, or two key strikes in rapid succession. The graph-theoretic multibody model constructed represents the components and their interactions. Explicit contact edges provide forces generated between two bodies as a function of their kinematic states, using a special contact model to represent the compression of felt lined interfaces that can separate during the key stroke. Masses and geometrical parameters of the action were measured by importing scanned images from a real action into CAD software. The highly nonlinear system of five ordinary differential equations of motion was derived symbolically and solved by a numerical stiff solver in Maple. The effects of two components not present in the horizontal grand piano action, the bridle strap and hammer butt spring, were examined using simulations. The butt spring is seen to serve an important function in assisting the return of the hammer to its rest position on key release. The model will be useful in future studies to compare vertical actions to horizontal grand piano actions, as these are known to exhibit quite different playing characteristics.Copyright

The slippery path from piano key to string

Everything that contributes to the excitation of a piano string, from key input to hammer–string interaction, is both deterministic and consistently repeatable. Sequences of identical experimental trials give results that are indistinguishable. The simplicity of this behavior contrasts with the elusive goal of predicting input–output response and the extreme difficulty of accurate physical characterization. The nature and complexity of the mechanisms and material properties involved, as well as the sensitivity of their parameterization, place serious obstacles in the way of the usual investigative tools. This paper discusses and illustrates the limitations of modeling and simulation as applied to this problem, and the special considerations required for meaningful experimentation.

Dynamic Simulation and Vibration Analysis of a Mechanical Piano Key Actuator

Dynamic modeling of a flexible hub-beam system with an eccentric tip mass including nonlinear hysteretic contact is studied in this paper. In reality, the model is intended to represent the mechanical finger of an actuator for a piano key. Developing a device to achieve a desired finger-key contact force profile that realistically replicates that of a real pianist’s finger is the main objective of this research. The proposed actuation system consists of a flexible arm which is attached to a DC brushless rotary motor thorough a hub. The compliant arm behaves as a cantilever beam to which an eccentric tip mass has been attached at its free end. During the piano key stroke, the contact force input from the tip causes the key to rotate and impact the ground through an interface lined with stiff felt to suppress vibrations and noise. Euler-Bernoulli beam theory in conjunction with Lagrange’s method is utilized to obtain the governing equations of motion for the system. The finite element method is used as the discretization procedure for the flexible cantilever beam, which can be considered a distributed parameter system. To include contact dynamics at the stop, the nonlinear hysteretic behavior of felt under compression is modeled in such a way that smooth transitions between loading and unloading stages are produced, thus ensuring accurate response under dynamic conditions, and particularly with partial loading and unloading states that occur during the contact period. Simulation results show excessive vibration is produced due to the arm flexibility and also the rigid-body oscillations of the arm, especially during the period of key-felt contact. To eliminate these vibrations, the system was supplemented with various dashpot models, including a simple grounded rotational dashpot, and a grounded rotational dashpot with a one-sided relation. The results of simulations are presented showing the effect on vibration behavior attributed to these additional components.Copyright