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### Research Papers

J. Comput. Nonlinear Dynam. 2008;3(4):041001-041001-12. doi:10.1115/1.2960486.

In this paper, a methodology for designing efficient energy scavengers is proposed. The scavenger consists of a cantilever beam on which piezoelectric films and a mass are mounted. The mass at the tip of the beam is known as the proof mass and the device is called either an energy scavenger or a beam-mass system. The proof mass is a permanent magnet, where in its vicinity attracting permanent magnets are placed. When a scavenger is mounted on a vibration source, the cantilever beam would vibrate. Due to the vibration of the beam, the piezoelectric films generate electric charge. The generated charge is proportional to the amplitude of vibration of the tip of the beam. It is shown that when the magnets have appropriate strengths and are placed appropriately, the vibration of the tip of the beam can be amplified, thereby the scavenger efficiency is increased. Examples are given throughout the paper.

Topics: Magnets , Force , Vibration , Springs
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041002-041002-8. doi:10.1115/1.2960482.

In this paper, a simple and efficient numerical method to solve for the dynamic interaction of a high-speed train and railway structure during an earthquake is given. The motion of the train is modeled in multibody dynamics with nonlinear springs and dampers used to connect components. An efficient mechanical model for contact dynamics between the wheel and rail during an earthquake is presented. The railway structure is modeled with various finite elements. A nonlinear spring element based on a trilinear elastic-plastic material model is given for the concrete railway structure during an earthquake. A substructure model where a train runs repeatedly has been devised to obtain an approximated combined motion of the long train with many cars connected and the railway structure during an earthquake. A modal method has been developed to solve large-scale nonlinear equations of motion of the train and railway structure effectively. Based on the present method, a computer program DIASTARS for the dynamic interaction analysis of a Shinkansen train (high-speed train in Japan) and the railway structure during an earthquake has been developed. Numerical examples are demonstrated.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041003-041003-10. doi:10.1115/1.2960481.

This paper proposes a new design approach for control of periodically time-varying systems. The approach is based on the point-mapping technique to obtain an equivalent linear time-invariant sampled-data system for the linear periodically time-varying system with a piecewise parametrization of the control vector. This allows the known control design techniques for sampled-data systems to be applied. The proposed approach is then extended for analysis of robustness of the control design with respect to plant parametric uncertainties. This is achieved by computation of approximate discrete-time dynamics of the perturbed system by truncated point-mappings. By computing an upper norm bound on the error due to the truncated approximations, the robustness analysis of the system with respect to the parametric uncertainties is then formulated as a discrete-time structured singular value problem. Two numerical examples are considered to illustrate the approach and the extension of the approach for robust stability analysis.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041004-041004-7. doi:10.1115/1.2960467.

Following an electrical stimulus, the transmembrane voltage of cardiac tissue rises rapidly and remains at a constant value before returning to the resting value, a phenomenon known as an action potential. When the pacing rate of a periodic train of stimuli is increased above a critical value, the action potential undergoes a period-doubling bifurcation, where the resulting alternation of the action potential duration is known as alternans in medical literature. Existing cardiac models treat alternans either as a smooth or as a border-collision bifurcation. However, recent experiments in paced cardiac tissue reveal that the bifurcation to alternans exhibits hybrid smooth∕nonsmooth behaviors, which can be qualitatively described by a model of so-called unfolded border-collision bifurcation. In this paper, we obtain analytical solutions of the unfolded border-collision model and use it to explore the crossover between smooth and nonsmooth behaviors. Our analysis shows that the hybrid smooth∕nonsmooth behavior is due to large variations in the system’s properties over a small interval of the bifurcation parameter, providing guidance for the development of future models.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041005-041005-17. doi:10.1115/1.2908134.

In this work, chaotic vibrations of shallow sector-type spherical shells are studied. A sector-type shallow shell is understood as a shell defined by a sector with associated boundary conditions and obtained by cutting a spherical shell for a given angle $θk$, or it is a sector of a shallow spherical cap associated with the mentioned angle. Both static stability and complex nonlinear dynamics of the mentioned mechanical objects subjected to transversal uniformly distributed sign-changeable load are analyzed, and the so-called vibration charts and scales regarding the chosen control parameters are reported. In particular, scenarios of transition from regular to chaotic dynamics of the mentioned shells are investigated. A novel method to control chaotic dynamics of the studied flexible spherical shells driven by transversal sign-changeable load via synchronized action of the sign-changeable antitorque is proposed and applied. All investigations are carried out within the fields of qualitative theory of differential equations and nonlinear dynamics.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041006-041006-6. doi:10.1115/1.2960470.

This paper presents an efficient treatment of gyroscopic bodies in the recursive solution of the dynamics of an $N$-body system. The bodies of interest include the reaction wheels in satellites, wheels on a car, and flywheels in machines. More specifically, these bodies have diagonal inertia tensors. They spin about one of its principal axes, with the moment of inertia along the transverse axes identical. Their center of mass lies on the spin axis. Current recursive solution methods treat these bodies identically as any other body in the system. The proposition here is that a body with gyroscopic children can be collectively treated as a composite body in the recursive solution process. It will be shown that this proposition improves the recursive solution speed to the $order(N−m)$ where $m$ is the number of gyroscopic bodies in the system. A satellite with three reaction wheels is used to illustrate the proposition.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041007-041007-10. doi:10.1115/1.2960475.

This paper presents a consistent formulation of the generalized-$α$ time integration scheme for mechanical and mechatronic systems. The algorithm can deal with a nonconstant mass matrix, controller dynamics, and kinematic constraints. The theoretical background relies on the analogy with linear multistep formulas, which leads to elegant results related to consistency, order conditions for constant and variable step-size methods, as well as global convergence. Those results are illustrated for a controlled spring-mass system, and the method is also applied for the simulation of a vehicle semi-active suspension.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041008-041008-14. doi:10.1115/1.2960477.

This paper introduces a new interpretation of the energetic coefficient of restitution, especially applicable to contact involving multibody systems. This interpretation generalizes the concept of the energetic coefficient of restitution and allows for consideration of simultaneous multiple-point contact scenarios. Such a generalization is obtained by an analysis of energy absorption and restitution during impact, using a decomposition technique, which exactly decouples the kinetic energy associated with the normal and tangential directions of the contact pairs. The main advantages of the new definition and its potential applications are highlighted.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041009-041009-6. doi:10.1115/1.2960463.

Switch-mode power supplies usually emit electromagnetic interferences at the switching frequency and its harmonics. Inducing chaos in these systems has recently been suggested as a means of reducing these spectral emissions, yet at the expense of aggravating the overall magnitude of the ripple in the output voltage. We propose here a new nonlinear feedback, which induces chaos and which is able at the same time to achieve a low spectral emission and to maintain a small ripple in the output. The design of this new and simple controller is based on the propriety that chaotified nonlinear systems present many independent chaotic attractors of small dimensions.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041010-041010-9. doi:10.1115/1.2960468.

The competing solutions of a planar pendulum parametrically excited by the vertical motion of the pivot are investigated in terms of both attractor robustness and basin integrity. Two different measures are considered to highlight how the integrity of the system is modified by changing the amplitude of the excitation. Various attractors, both in-well and out-of-well, are considered, and the integrity profiles of each of them are determined. A detailed discussion of the interaction and mutual erosion of the various attractors is performed by clarifying the role of the two complementary measures, and the most relevant characteristics of the erosion profiles are interpreted in terms of the underlying topological mechanisms involving local or global bifurcations.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041011-041011-8. doi:10.1115/1.2960472.

Chimneys in operating power plants cannot be demolished in the classical way by dynamite. A new technology has been invented where they are cut off starting from the top and where the pieces of the reinforced concrete chimney are dropped through the inner pipe of the chimney. In order to avoid the falling down of pieces on the outer side, a ring platform is attached to the chimney by pure force contact, which allows semi-automatic climbing up and down. The paper addresses two crucial design decisions concerning the ring platform: the design of the contact geometry and the shape of the ring structure. Both design problems are formulated as multicriterion optimization problems where the first one can be solved partially in an analytical way whereas the second one has to be solved numerically. It will be shown how the problems can be formulated in various ways depending on the technical requirements.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041012-041012-8. doi:10.1115/1.2960479.

In this paper, two triangular plate elements based on the absolute nodal coordinate formulation (ANCF) are introduced. Triangular elements employ the Kirchhoff plate theory and can, accordingly, be used in thin plate problems. As usual in ANCF, the introduced elements can exactly describe arbitrary rigid body motion when their mass matrices are constant. Previous plate developments in the absolute nodal coordinate formulation have focused on rectangular elements that are difficult to use when arbitrary meshes need to be described. The elements introduced in this study have overcome this problem and represent an important addition to the absolute nodal coordinate formulation. The two elements introduced are based on Specht’s and Morley’s shape functions, previously used in conventional finite element formulations. The numerical solutions of these elements are compared with previously proposed rectangular finite element and analytical results.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2008;3(4):041013-041013-8. doi:10.1115/1.2960483.

The biped walking robot demonstrates a stable limit cycle on shallow slopes. In previous researches, this passive gait was shown to be sensitive to ground slope and initial conditions. In this paper, we discuss the feedback stabilization of a biped robot by the “energy shaping” technique. Two designs are proposed to reduce the sensitivity of the biped walking robot to slope and initial conditions. In the first design, a moving mass actuator is located on each link of the robot. The actuators are used to shape the potential energy of the biped robot so that it tracks the potential energy of a known passive gait of a similar biped robot on a different slope. Although the method is applied to a simple kneeless planar biped, our results are completely generalizable and may be applied to general $n$-link bipeds. The second design uses a momentum wheel, which is placed on the hip of the robot to shape the energy of the biped. We use the controlled Lagrangian method to design the controller, and the simulation is carried out to show its performance. In the controlled Lagrangian method, either the total energy or the Lagrangian of the uncontrolled system is modified so that the Euler–Lagrange equations derived from this modified expression, called the controlled Lagrangian function, describe the closed loop equations of the system.

Commentary by Dr. Valentin Fuster