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# Accepted Manuscripts

BASIC VIEW  |  EXPANDED VIEW
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036830
We study a ball-beam impact in detail; and in particular, we study the interplay between dissipation and modal truncation. With Hertzian contact between a solid ball and an Euler-Bernoulli beam model, we find using detailed numerical simulations that many (well above 60) modes are needed before convergence occurs; that contact dissipation (either viscous or hysteretic) has only a slight effect; and that contact location plays a significant role. However, and more interestingly, we find that as little as two percent modal damping speeds up convergence of the net interaction so that only about 25 modes are needed. We offer a qualitative explanation for this effect in terms of the many sub-impacts that occur in the overall single macroscopic impact. In particular, we find that in cases where the overall interaction time is long enough to damp out high modes yet short enough to leave lower modes undissipated, modal truncation at about 25 modes gives good results. In contrast, if modal damping is absent so that higher-mode vibrations persist throughout the interaction, final outcomes are less regular and many more modes are needed. The regime of impact interactions studied here occur for reasonable parameter ranges, e.g., for a 3-4 cm steel ball dropped at speeds of 0.1-1.0 m/s on a meter-long steel beam of net mass 1 kg. We are unaware of any prior similarly detailed numerical study which clearly offers the one summarizing idea that we obtain here.
TOPICS: Energy dissipation, Damping, Steel, Computer simulation, Performance, Vibration
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036831
The paper proposes a time-delayed hyper-chaotic system composed of multi-scroll attractors with multiple positive Lyapunov exponents (LEs), which is described by a three-order nonlinear retarded type delay differential equation. The dynamical characteristics of the time-delayed system are far more complicated than those of the original system without time delay. The three-order time-delayed system not only generates hyper-chaotic attractors with multi-scroll but also has multiple positive LEs. We observe that the number of positive LEs increases with increasing time delay. Through numerical simulations, the time-delayed system exhibits a larger number of scrolls than the original system without time delay. Moreover, different numbers of scrolls with variable delay and coexistence of multiple attractors with a variable number of scrolls are also observed in the time-delayed system. Finally, we set up electronic circuit of the proposed system, and make Pspice simulations to it. The Pspice simulation results agree well with the numerical results.
TOPICS: Attractors, Delays, Time delay systems, Large eddy simulation, Simulation results, Delay differential equations, Computer simulation, Simulation, Engineering simulation, Circuits
research-article
Marwan Abukhaled
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036813
In this paper, a Green’s function based iterative algorithm is proposed to solve strong nonlinear oscillators. The method’s essential part is based on finding an appropriate Green’s function that will be incorporated into a linear integral operator. An application of fixed point iteration schemes such as Picard’s or Mann’s will generate an iterative formula that gives reliable approximations to the true periodic solutions that characterize this kind of equations. The applicability and stability of the method will be tested through numerical examples. Since exact solutions to these equations usually do not exist, the proposed method will be tested against other popular numerical methods such as the modified homotopy perturbation, the modified differential transformation, and the fourth order Runge-Kutta methods.
TOPICS: Stability, Algorithms, Numerical analysis, Approximation, Runge-Kutta methods
research-article
David Chelidze
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036814
False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay-coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Here, nearest-neighbor statistics are studied for uncorrelated random time series and contrasted with the corresponding deterministic and stochastic statistics. New composite FNN metrics are constructed and their performance is evaluated for deterministic, stochastic, and random time series. In addition, noise-contaminated deterministic data analysis shows that these composite FNN metrics are robust to noise. All FNN results are also contrasted with surrogate data analysis to show their robustness. The new metrics clearly identify random time series as not having a finite embedding dimension and provide information about the deterministic part of stochastic processes. These metrics can also be used to differentiate between chaotic and random time series.
TOPICS: Dimensions, Statistical analysis, Time series, Fuzzy neural nets, Noise (Sound), Statistics as topic, Composite materials, Stochastic processes, Delays, Robustness
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036815
The nonlinear modal coupling between the vibration modes of an arch shaped microstructure is an interesting phenomenon, which may have desirable features for numerous applications, such as vibration-based energy harvesters. This works presents an investigation into the potential nonlinear internal resonances of a Micro electro mechanical systems MEMS arch when excited by static (DC) and dynamic (AC) electric forces. The influences of initial rise and mid-plane stretching are considered in the governing equation. The cases of one-to-one and three-to-one internal resonances between the first and second modes and between the first and third modes are studied using the method of multiple scales and the direct attack of the partial differential equation of motion. It is interestingly shown that for distinct domain of actuation voltages, there exist three-to-one internal resonance between the first and third symmetric modes and one-to-one internal resonance case between the first symmetric and the second antisymmetric mode. These results can shed light on such interactions that are commonly found on micro and nano structures, such as carbon nano tubes.
TOPICS: Resonance, Arches, Microelectromechanical systems, Vibration, Energy harvesting, Nanotubes, Partial differential equations, Carbon, Nanostructures
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036816
Seismic isolation can protect delicate equipment housed in structures under earthquake attacks. One of the common approaches to isolate equipments is by using various base isolation systems on which the equipments are mounted. Base isolation requires a gap between the base-isolated equipment and its surroundings to provide space for the deformation of isolation system. During strong earthquakes, structural poundings may occur between the equipment and the surrounding moat wall because of the limited separation distance and the deformations of the isolator. Bumping against the surroundings may change the performance of the base-isolated equipment. A potential mitigation measure for this problem is the incorporation of layers of soft material, which can act as collision bumpers, in order to prevent the sudden impact pulses. Thus, shaking table tests have been carried out to investigate the pounding phenomenon between a mass and two-sided shock absorbers, subject to sinusoidal excitations. To investigate the effectiveness of such an impact mitigation measure, preliminary tests were carried out: first, the dynamic response was recorded without pounding, and secondly the test structure was placed with gap separation and pounding was induced. Absolute acceleration, relative excursion, mean contact force, coefficient of restitution and dissipated energy were recorded at steady state and the excitation frequency range for pounding occurrences were determined. Numerical predictions were made by using a contact model for the simulation of impacts, able to appropriately describe the behavior of rubber under impact loading. Good agreement between the experimental and the numerical results was achieved.
TOPICS: Excitation, Earthquakes, Deformation, Separation (Technology), Rubber, Simulation, Collisions (Physics), Dynamic response, Shock absorbers, Steady state
research-article
Guang-Da Hu
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036761
In this paper, explicit Runge-Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, $\varepsilon-$preserving is introduced to describe the conserved quantities being approximately retained. Then a modified version of explicit Runge-Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge-Kutta method is superior to implicit numerical methods in literature. The order of the modified Runge-Kutta method is the same as the standard Runge-Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge-Kutta method.
TOPICS: Nonlinear dynamical systems, Runge-Kutta methods, Numerical analysis, Optimization
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036760
The results of two sets of impact experiments are reported within. To assist with model development using the impact data reported, the materials are mechanically characterized using a series of standard experiments. The first set of impact data comes from a series of coefficient of restitution experiments, in which a 2 meter long pendulum is used to study "in context" measurements of the coefficient of restitution for eight different materials (6061-T6 Aluminum, Phosphor Bronze alloy 510, Hiperco, Nitronic 60A, Stainless Steel 304, Titanium, Copper, and Annealed Copper). The coefficient of restitution is measured via two different techniques: digital image correlation and laser Doppler vibrometry. Due to the strong agreement of the two different methods, only results from the digital image correlation are reported. The coefficient of restitution experiments are "in context" as the scales of the geometry and impact velocities are representative of common features in the motivating application for this research. Finally, a series of compliance measurements are detailed for the same set of materials. The compliance measurements are conducted using both nano-indentation and micro-indentation machines, providing sub-nm displacement resolution and uN force resolution. Good agreement is seen for load levels spanned by both machines. As the transition from elastic to plastic behavior occurs at contact displacements on the order of 30 nm, this data set provides a unique insight into the transitionary region.
TOPICS: Metals, Copper, Aluminum, Lasers, Machinery, Bronze, Stress, Resolution (Optics), Aerospace industry, Displacement, Geometry, Model development, Nanoindentation, Pendulums, Phosphors, Stainless steel, Titanium
research-article
Shahrokh Esmaeili
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036710
Since the solutions of the fractional differential equations have, in general, unbounded derivatives at zero, their numerical solutions by piecewise polynomial collocation method on uniform meshes will lead to poor convergence rates. This paper presents a piecewise nonpolynomial collocation method for solving such equations which reflects the singularity of the exact solution. The entire domain is divided into several small subdomains, and the nonpolynomial pieces are constructed using a block-by-block scheme on each subdomain. The method is applied to solve linear and nonlinear fractional differential equations. Numerical examples are given and discussed to illustrate the effectiveness of the proposed approach.
TOPICS: Differential equations, Polynomials
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036712
Impact oscillators exhibit an abrupt onset of chaos close to grazing due to the square root singularity in their discrete-time maps. In practical applications, this large-amplitude chaotic vibration needs to be avoided. It has been shown that this can be achieved if the ratio of the natural frequency of the oscillator and the forcing frequency is an even integer. But in practice it is difficult to set a parameter at such a precise value. We show that in systems with square root singularity (pre-stressed impacting surface), there exists a range of $\omega_0$ around the theoretical value over which the chaotic orbit does not occur, and that this is due to an interplay between the main attractor and coexisting orbits. We show that this range of forcing frequency has exponential dependence on the amount of pre-stress as well as on the stiffness ratio of the springs.
TOPICS: Stress, Vibration, Chaos, Springs, Stiffness, Attractors
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036614
An analysis of the shape memory prosthesis of a middle ear is presented in the paper. A shape memory prosthesis enables adjusting its length to individual patient’s needs but sometimes the prosthesis can not be fixed properly to the stapes. In this case an impact phenomenon between the prosthesis and the stapes can be meaningful. Therefore, a reconstructed middle ear is modelled as a two degree of freedom system with a nonlinear shape memory element and soft impact to explain its behaviour when the prosthesis is not placed or fixed properly. The properties of the shape memory prosthesis, in the form of helical spring, are described here by a polynomial function. The system demonstrates advisable periodic and undesirable aperiodic and irregular behaviour depending on excitation amplitude, frequency and a prosthesis length. The prosthesis length can change causing a modification of a distance between the prosthesis and the stapes. The results of the study give an answer how the prosthesis length, which produces the ossicular chain tension, influences system dynamics and what it means in medical practice.
TOPICS: Biomechanics, Shapes, Prostheses, Ear, System dynamics, Degrees of freedom, Chain, Springs, Tension, Biomedicine, Excitation, Polynomials
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036548
A vibrating system with impacts, which can be applied to model the cantilever beam with a mass at its end and two-sided impacts against a harmonically moving frame, is investigated. The objective of this study is to determine in which regions of parameters characterizing the system, the motion of the oscillator is periodic and stable. An analytical method to obtain stable periodic solutions to the equations of motion on the basis of Peterka’s approach is presented. The results of analytical investigations have been compared to the results of numerical simulations. The ranges of stable periodic solutions determined analytically and numerically with bifurcation diagrams of spectra of Lyapunov exponents show a very good conformity. The locations of stable periodic solution regions of the system with a movable frame and two-sided impacts differ substantially from the locations of stable periodic solution regions for the system: (i) with a movable frame and one-sided impacts, (ii) with an immovable frame and two-sided impacts.
TOPICS: Spectra (Spectroscopy), Cantilever beams, Computer simulation, Equations of motion, Bifurcation
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036518
In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcation in discontinuous dynamical systems. Multi-valued linear vector fields are employed in the discontinuous dynamical system, and generic mappings are defined among the boundaries and corners. From mapping structures, periodic motions switching on the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory in discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map are developed for periodic motions in such multi-valued dynamical systems in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in the discontinuous dynamical system.
TOPICS: Dynamics (Mechanics), Corners (Structural elements), Dynamic systems, Bifurcation, Chatter, Eigenvalues, Stability, Flow (Dynamics), Computer simulation, Collisions (Physics)
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036483
Musculoskeletal simulations can be used to determine loads experienced by the ligaments and cartilage during athletic motions such as impact from a drop landing, hence investigating mechanisms for injury. An open-source discrete element knee model was used to perform a forward dynamic simulation of the impact phase of a drop landing. Since the cartilage contact loads are largely depending on the elastic moduli of the cartilage, the analysis was performed for varying moduli: nominal stiffness based on the literature, stiffness increased by 10%, and decreased by 10%. As the cartilage stiffness increased, the medial compartment contact load decreased. Conversely, the lateral compartment load and MCL force increased, causing a shift in the load distribution. However, these changes were insignificant compared to the overall magnitude of the contact forces (<4% change). The ACL, PCL, and LCL loads remain unchanged between varying cartilage stiffness values. The medial compartment bears a majority of the load (860 N in medial compartment versus 540 N in lateral) during the impact phase of a drop landing, which agrees with physiological data that the medial side of the knee is more affected by osteoarthritis than the lateral side. This is one of the few models to quantify this load distribution and show the results are invariant to changes in cartilage stiffness.
TOPICS: Stress, Stiffness, Cartilage, Knee, Simulation, Osteoarthritis, Anterior cruciate ligament, Musculoskeletal system, Wounds, Physiology, Elastic moduli
Technical Brief
Auni Aslah Mat Daud
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036418
A Galton board, also referred to as quincunx, is an instrument invented in 1873 by Francis Galton (1822-1911). It is a box with a glass front and many horizontal nails or pins embedded in the back, and a funnel. Galton and many modern statisticians claimed that a lead ball descending to the bottom of the Galton board would display random walk. In this study, a new mathematical model of Galton board is developed, to further improve three very recently proposed models. The novel contribution of this paper is the introduction of the velocity dependent coefficient of restitution. The developed model is then analyzed using symbolic dynamics. The results of the symbolic dynamics analysis prove that the developed Galton board model does not behave the way Galton envisaged. This study also confirms that the details of the of the deterministic models of Galton board are not essential for demonstrating deviations from the statistical models.
TOPICS: Dynamics (Mechanics), Glass, Pins (Engineering), Instrumentation
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036197
In this article, a recursive approach is used to dynamically model a tree-type robotic system with floating-base. Two solution procedures are developed to obtain the time responses of the mentioned system. A set of highly nonlinear differential equations is employed to obtain the dynamic behavior of the system when it has no contact with the ground or any object in its environment (flying phase); and a set of algebraic equations is exploited when this tree-type robotic system collides with the ground (impact phase). The Gibbs-Appell (G-A) formulation in recursive form and the Newton's impact law are applied to derive the governing equations of the aforementioned robotic system for the flying and impact phases, respectively. The main goal of this article is a systematic algorithm that is used to divide any tree-type robotic system into a specific number of open kinematic chains and derive the forward dynamic equations of each chain, including its inertia matrix and right hand side vector. Then, the inertia matrices and the right hand side vectors of all these chains are automatically integrated to construct the global inertia matrix and the global right-hand-side vector of the whole system. Finally, to show the effectiveness of the suggested algorithm in deriving the motion equations of multi-chain robotic systems, a ten-link tree-type robotic system with floating base is simulated.
TOPICS: Inertia (Mechanics), Collisions (Physics), Equations of motion, Algorithms, Chain, Robotics, Manipulators, Nonlinear differential equations, Algebra, Open kinematic chains
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036196
This paper proposes a method for process parameter optimization of a mobile robotic percussive riveting system with flexible joints to guarantee the rivet gun alignment during the operation. This development is motivated by the increasing interest in using industrial robots to replace human operators for percussive impact riveting in aerospace assembly. In percussive riveting, the rivet gun generates repetitive impacts acting on the rivet. These impacts not only deform the rivet but also induce forced vibration to the robot, and thus the robot must hold the gun firmly during riveting. The process parameters for the mobile robotic riveting system include those related to the impact force generation for planning the rivet gun input and those related to the robot pose with respect to the joined panels for planning the mobile platform motion. These parameters are incorporated into a structural dynamic model of the robot under a periodic impact force. Then an approximate analytical solution is formulated to calculate the displacement of the rivet gun mounted on the end-effector for its misalignment evaluation. It is found that both the force frequency and the mobile platform position have strong influence on the robotic riveting performance in terms of alignment during operation. Global optimization of these process parameters is carried out to demonstrate the practical application of the proposed method for the planning of the robotic percussive riveting system.
TOPICS: Robotics, Optimization, Riveting, Rivets, Robots, Manufacturing, Structural dynamics, Aerospace industry, Vibration, Displacement, End effectors
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036115
A free-standing rack (FS rack) is a type of a spent nuclear fuel rack, which is just placed on a floor of a pool. For this characteristic, seismic loads can be reduced by fluid force and friction force, but a collision between a rack and another rack or a wall must be avoided. Therefore, it is necessary for designing an FS rack to figure out how it moves under seismic excitation. In this research, a dynamic model of an FS rack is developed considering seismic inertial force, friction force and fluid force. This model consists of two sub-models: a translation model, which simulates planar translational and rotational motion; and a rocking model, which simulates non-slide rocking motion. First, simulations with sinusoidal inertial force were conducted, changing values of a friction coefficient. Next, to validate this dynamic model, a miniature experiment was conducted. Finally, the model is applied to a real-size FS rack and actually observed seismic acceleration. It is found that translational movement of a rack varies depending on the value of friction coefficient in the simulation with sinusoidal and actual acceleration. Also, simulation results are similar to the experimental results in the aspects of translational and rocking motion provided friction coefficient is selected properly. Through this research, the knowledge is acquired that friction force plays a significant role in a motion of FS rack so that estimating and controlling a friction coefficient is important in designing an FS rack.
TOPICS: Fuels, Dynamic models, Excitation, Friction, Fluids, Design, Simulation, Stress, Collisions (Physics), Simulation results, Rotation, Spent nuclear fuels
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4036114
In this paper we perform a path following bifurcation analysis of church bell to gain an insight into the governing dynamics of the yoke-bell-clapper system. We use an experimentally validated hybrid dynamical model based on the detailed measurements of a real church bell. Numerical analysis is performed both by a direct numerical integration and a path-following methods using a new numerical toolbox ABESPOL (Chong, Numerical modelling and stability analysis of non-smooth dynamical systems via ABESPOL) based on Coco (Dankowicz et al., Recipes for continuation). We constructed one-parameter diagrams that allow to characterize the most common dynamical states and to investigate the mechanisms of their dynamic stability. A novel method allowing to locate the regions in the parameters space ensuring robustness of bells effective performance is presented.
TOPICS: Dynamic systems, Bifurcation, Dynamic stability, Robustness, Dynamics (Mechanics), Stability, Modeling, Non-smooth dynamics, Numerical analysis
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4035933
This paper presents a development in design, mathematical modeling and experimental study of a vibro-impact moling device which was invented by the author before. A vibratory unit deploying electro-mechanical interactions of a conductor with oscillating magnetic field has been realized and developed. The combination of resonance in an RLC circuit including a solenoid is found to create a relative oscillatory motion between the metal bar and the solenoid. This results in impacts of the solenoid on an obstacle block, which causes the forward motion of the system. Compared to the former model which employs impact from the metal bar, the improved rig can offer a higher progression rate of six times when using the same power supply. The novel geometrical arrangement allows for future optimization in terms of system parametric selection and adaptive control. This implies a very promising deployment of the mechanism in ground moling machines as well as other self-propelled mobile systems. In this paper, insight to the design development based on physical and mathematical models of the rig is presented. Then the obtained coupled electro-mechanical equations of motion are solved numerically, and a comparison between experimental results and numerical predictions is presented.
TOPICS: Design, Solenoids, Metals, Machinery, Magnetic fields, Adaptive control, Equations of motion, Modeling, Optimization, Circuits, Resonance