While some attempts have been made to model the behaviour of the grand piano action (the mechanism that translates a key press into a hammer striking a string), most researchers have reduced the system to a simple model with little relation to the components of a real action. While such models are useful for certain applications, they are not appropriate as design tools for piano makers, since the model parameters have little physical meaning and must be calibrated from the behaviour of a real action. A new model for a piano action is proposed in this thesis. The model treats each of the five main action components (key, whippen, jack, repetition lever, and hammer) as a rigid body. The action model also incorporates a contact model to determine the normal and friction forces at 13 locations between each of the contacting bodies. All parameters in the model are directly measured from the physical properties of individual action components, allowing the model to be used as a prototyping tool for actions that have not yet been built. To test whether the model can accurately predict the behaviour of a piano action, an experimental apparatus was built. Based around a keyboard from a Boston grand piano, the apparatus uses an electric motor to actuate the key, a load cell to measure applied force, and optical encoders and a high speed video camera to measure the positions of the bodies. The apparatus was found to produce highly repeatable, reliable measurements of the action. The behaviour of the action model was compared to the measurements from the experimental apparatus for several types of key blows from a pianist. A qualitative comparison showed that the model could very accurately reproduce the behaviour of a real action for high force blows. When the forces were lower, the behaviour of the action model was still reasonable, but some discrepancy from the experimental results could be seen. In order to reduce the discrepancy, it was recommended that certain improvements could be made to the action model. Rigid bodies, most importantly the key and hammer, should be replaced with flexible bodies. The normal contact model should be modified to account for the speed-independent behaviour of felt compression. Felt bushings that are modelled as perfect revolute joints should instead be modelled as flexible contact surfaces.

The piano action is the mechanism that transforms the finger force applied to a key into the motion of a hammer that strikes a piano string. This thesis focuses on improving the fidelity of the dynamic model of a grand piano action which has been already developed by Hirschkorn et al. at the University of Waterloo. This model is the state-of-the-art dynamic model of the piano in the literature and is based on the real components of the piano action mechanism (key, whippen, jack, repetition lever, and hammer). Two main areas for improving the fidelity of the dynamic model are the hammer shank and the connection point between the key and the ground. The hammer shank is a long narrow wooden rod and, by observation with a high-speed video camera, the flexibility of this part has been confirmed. In previous work, the piano hammer had been modelled as a rigid body. In this work, a Rayleigh beam model is used to model the flexible behaviour of the hammer shank. By comparing the experimental and analytical results, it turns out that the flexibility of the hammer shank does not significantly affect the rotation of the other parts of the piano mechanism, compared with the case that the hammer shank has been modelled as a rigid part. However, the flexibility of the hammer shank changes the impact velocity of the hammer head, and also causes a greater scuffing motion for the hammer head during the contact with the string. The connection of the piano key to the ground had been simply modelled with a revolute joint, but the physical form of the connection at that point suggests that a revoluteprismatic joint with a contact force underneath better represents this connection. By comparing the experimental and analytical results, it is concluded that incorporating this new model significantly increases the fidelity of the model for the blows. In order to test the accuracy of the dynamic model, an experimental setup, including a servo motor, a load cell, a strain gauge, and three optical encoders, is built. The servo motor is used to actuate the piano key. Since the purpose of the motor is to consistently mimic the finger force of the pianist, the output torque of the motor is controlled. To overcome the problem associated with the motor torque control method used in previous work, a new torque control method is implemented on a real-time PC and a better control of the motor torque output is established. Adding a more realistic model of the piano string to the current piano action model and finding a better contact model for the contacts that happen between the surfaces that are made of felt (or leather), are two main areas that can be worked on in the future research. These two areas will help to further increase the fidelity of the present piano action model.

While some attempts have been made to model the behaviour of the grand piano action (the mechanism that translates a key press into a hammer striking a string), most researchers have reduced the system to a simple model with little relation to the components of a real action. While such models are useful for certain applications, they are not appropriate as design tools for piano makers, since the model parameters have little physical meaning and must be calibrated from the behaviour of a real action. A new model for a piano action is proposed in this thesis. The model treats each of the five main action components (key, whippen, jack, repetition lever, and hammer) as a rigid body. The action model also incorporates a contact model to determine the normal and friction forces at 13 locations between each of the contacting bodies. All parameters in the model are directly measured from the physical properties of individual action components, allowing the model to be used as a prototyping tool for actions that have not yet been built. To test whether the model can accurately predict the behaviour of a piano action, an experimental apparatus was built. Based around a keyboard from a Boston grand piano, the apparatus uses an electric motor to actuate the key, a load cell to measure applied force, and optical encoders and a high speed video camera to measure the positions of the bodies. The apparatus was found to produce highly repeatable, reliable measurements of the action. The behaviour of the action model was compared to the measurements from the experimental apparatus for several types of key blows from a pianist. A qualitative comparison showed that the model could very accurately reproduce the behaviour of a real action for high force blows. When the forces were lower, the behaviour of the action model was still reasonable, but some discrepancy from the experimental results could be seen. In order to reduce the discrepancy, it was recommended that certain improvements could be made to the action model. Rigid bodies, most importantly the key and hammer, should be replaced with flexible bodies. The normal contact model should be modified to account for the speed-independent behaviour of felt compression. Felt bushings that are modelled as perfect revolute joints should instead be modelled as flexible contact surfaces.

The piano action is the mechanism that transforms the finger force applied to a key into the motion of a hammer that strikes a piano string. This thesis focuses on improving the fidelity of the dynamic model of a grand piano action which has been already developed by Hirschkorn et al. at the University of Waterloo. This model is the state-of-the-art dynamic model of the piano in the literature and is based on the real components of the piano action mechanism (key, whippen, jack, repetition lever, and hammer). Two main areas for improving the fidelity of the dynamic model are the hammer shank and the connection point between the key and the ground. The hammer shank is a long narrow wooden rod and, by observation with a high-speed video camera, the flexibility of this part has been confirmed. In previous work, the piano hammer had been modelled as a rigid body. In this work, a Rayleigh beam model is used to model the flexible behaviour of the hammer shank. By comparing the experimental and analytical results, it turns out that the flexibility of the hammer shank does not significantly affect the rotation of the other parts of the piano mechanism, compared with the case that the hammer shank has been modelled as a rigid part. However, the flexibility of the hammer shank changes the impact velocity of the hammer head, and also causes a greater scuffing motion for the hammer head during the contact with the string. The connection of the piano key to the ground had been simply modelled with a revolute joint, but the physical form of the connection at that point suggests that a revoluteprismatic joint with a contact force underneath better represents this connection. By comparing the experimental and analytical results, it is concluded that incorporating this new model significantly increases the fidelity of the model for the blows. In order to test the accuracy of the dynamic model, an experimental setup, including a servo motor, a load cell, a strain gauge, and three optical encoders, is built. The servo motor is used to actuate the piano key. Since the purpose of the motor is to consistently mimic the finger force of the pianist, the output torque of the motor is controlled. To overcome the problem associated with the motor torque control method used in previous work, a new torque control method is implemented on a real-time PC and a better control of the motor torque output is established. Adding a more realistic model of the piano string to the current piano action model and finding a better contact model for the contacts that happen between the surfaces that are made of felt (or leather), are two main areas that can be worked on in the future research. These two areas will help to further increase the fidelity of the present piano action model.

Many engineering applications make use of fiber assemblies under compression. Unfortunately, this compression behavior is difficult to predict, due to nonlinear compliance, hysteresis, and anelasticity. The main objective of this research is to develop an algorithm which is capable of incorporating the microscale features of the fiber network into macroscopic scale applications, particularly the modeling of contact mechanics in multibody systems. In micromechanical approaches, the response of a fiber assembly to an external force is related to the response of basic fiber units as well as the interactions between these units, i.e. the mechanical properties of the constituent fibers and the architecture of the assembly will both have a significant influence on the overall response of the assembly to compressive load schemes. Probabilistic and statistical principles are used to construct the structure of the uniformly-distributed random network. Different micromechanical approaches in modeling felt, as a nonwoven fiber assembly with unique mechanical properties, are explored to gain insight into the key mechanisms that influence its compressive response. Based on the deformation processes and techniques in estimating the number of fiber contacts, three micromechanical models are introduced: (1) constitutive equations for micromechanics of three-dimensional fiberwebs under small strains, in which elongation of the fibers is the key deformation mechanism, adapted for large deformation ranges; (2) micromechanical model based on the rate theory of granular media, in which bending and torsion of fibers are the predominant elemental deformations used to calculate compliances of a particular contact; and (3) a mechanistic model developed using the general deformation theory of the fiber networks with fiber bending at the micro level and a binomial distribution of fiber contacts. A well-established mechanistic model, based on fiber-to-fiber friction at the micro level, is presented for predicting the hysteresis in compression behavior of wool fiberwebs. A novel algorithm is introduced to incorporate a hysteretic micromechanical model - a combination of the mechanistic model with microstructural fiber bending, which uses a binomial distribution of the number of fiber-to-fiber contacts, and the friction-based hysteresis idea - into the contact mechanics of multibody simulations with felt-lined interacting bodies. Considering the realistic case in which a portion of fibers slides, the fiber network can be treated as two subnetworks: one from the fibers with non-sliding contact points, responsible for the elastic response of the network, and the other consisting of fibers that slide, generating irreversible hysteresis phenomenon in the fiberweb compression. A parameter identification is performed to minimize the error between the micromechanical model and the elastic part of the loading-unloading experimental data for felt, then contribution of friction was added to the obtained mechanistic compression-recovery curves. The theoretical framework for constructing a mechanistic multibody dynamic model of a vertical piano action is developed, and its general validity is established using a prototype model. Dynamic equations of motion are derived symbolically for the piano action using a graph-theoretic formulation. The model fidelity is increased by including hammer-string interaction, backcheck wire and hammer shank flexibility, a sophisticated key pivot model, nonlinear models of bridle strap and butt spring, and a novel mathematical contact model. The developed nonlinear hysteretic micromechanical model is used for the hammer-string interaction to affirm the reliability and applicability of the model in general multibody dynamic simulations. In addition, dynamic modeling of a flexible hub-beam system with an eccentric tip mass including nonlinear hysteretic contact is studied. The model represents the mechanical finger of an actuator for a piano key. Achieving a desired finger-key contact force profile that replicates that of a real pianist's finger requires dynamic and vibration analysis of the actuator device. The governing differential equations for the dynamic behavior of the system are derived using Euler-Bernoulli beam theory along with Lagrange's method. To discretize the distributed parameter flexible beam in the model, the finite element method is utilized. Excessive vibration due to the arm flexibility and also the rigid-body oscillations of the arm, especially during the period of key-felt contact, is eliminated utilizing a simple grounded rotational dashpot and a grounded rotational dashpot with a one-sided relation. The effect on vibration behavior attributed to these additional components is demonstrated using the simulated model.

Many engineering applications make use of fiber assemblies under compression. Unfortunately, this compression behavior is difficult to predict, due to nonlinear compliance, hysteresis, and anelasticity. The main objective of this research is to develop an algorithm which is capable of incorporating the microscale features of the fiber network into macroscopic scale applications, particularly the modeling of contact mechanics in multibody systems. In micromechanical approaches, the response of a fiber assembly to an external force is related to the response of basic fiber units as well as the interactions between these units, i.e. the mechanical properties of the constituent fibers and the architecture of the assembly will both have a significant influence on the overall response of the assembly to compressive load schemes. Probabilistic and statistical principles are used to construct the structure of the uniformly-distributed random network. Different micromechanical approaches in modeling felt, as a nonwoven fiber assembly with unique mechanical properties, are explored to gain insight into the key mechanisms that influence its compressive response. Based on the deformation processes and techniques in estimating the number of fiber contacts, three micromechanical models are introduced: (1) constitutive equations for micromechanics of three-dimensional fiberwebs under small strains, in which elongation of the fibers is the key deformation mechanism, adapted for large deformation ranges; (2) micromechanical model based on the rate theory of granular media, in which bending and torsion of fibers are the predominant elemental deformations used to calculate compliances of a particular contact; and (3) a mechanistic model developed using the general deformation theory of the fiber networks with fiber bending at the micro level and a binomial distribution of fiber contacts. A well-established mechanistic model, based on fiber-to-fiber friction at the micro level, is presented for predicting the hysteresis in compression behavior of wool fiberwebs. A novel algorithm is introduced to incorporate a hysteretic micromechanical model - a combination of the mechanistic model with microstructural fiber bending, which uses a binomial distribution of the number of fiber-to-fiber contacts, and the friction-based hysteresis idea - into the contact mechanics of multibody simulations with felt-lined interacting bodies. Considering the realistic case in which a portion of fibers slides, the fiber network can be treated as two subnetworks: one from the fibers with non-sliding contact points, responsible for the elastic response of the network, and the other consisting of fibers that slide, generating irreversible hysteresis phenomenon in the fiberweb compression. A parameter identification is performed to minimize the error between the micromechanical model and the elastic part of the loading-unloading experimental data for felt, then contribution of friction was added to the obtained mechanistic compression-recovery curves. The theoretical framework for constructing a mechanistic multibody dynamic model of a vertical piano action is developed, and its general validity is established using a prototype model. Dynamic equations of motion are derived symbolically for the piano action using a graph-theoretic formulation. The model fidelity is increased by including hammer-string interaction, backcheck wire and hammer shank flexibility, a sophisticated key pivot model, nonlinear models of bridle strap and butt spring, and a novel mathematical contact model. The developed nonlinear hysteretic micromechanical model is used for the hammer-string interaction to affirm the reliability and applicability of the model in general multibody dynamic simulations. In addition, dynamic modeling of a flexible hub-beam system with an eccentric tip mass including nonlinear hysteretic contact is studied. The model represents the mechanical finger of an actuator for a piano key. Achieving a desired finger-key contact force profile that replicates that of a real pianist's finger requires dynamic and vibration analysis of the actuator device. The governing differential equations for the dynamic behavior of the system are derived using Euler-Bernoulli beam theory along with Lagrange's method. To discretize the distributed parameter flexible beam in the model, the finite element method is utilized. Excessive vibration due to the arm flexibility and also the rigid-body oscillations of the arm, especially during the period of key-felt contact, is eliminated utilizing a simple grounded rotational dashpot and a grounded rotational dashpot with a one-sided relation. The effect on vibration behavior attributed to these additional components is demonstrated using the simulated model.