Guest Editorial

J. Comput. Nonlinear Dynam. 2017;12(6):060301-060301-3. doi:10.1115/1.4037433.

A system undergoing impacts can have strong nonlinear characteristics and the system response can experience unique qualitative changes. Impacts can occur in a wide range of systems, and the wide interest in them has spurred researchers to explore different tools for analyses and simulations and experiments with real, physical systems.

Commentary by Dr. Valentin Fuster

Technical Brief

J. Comput. Nonlinear Dynam. 2017;12(6):060901-060901-3. doi:10.1115/1.4036418.

A Galton board 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.

Commentary by Dr. Valentin Fuster

Research Papers

J. Comput. Nonlinear Dynam. 2017;12(6):061001-061001-7. doi:10.1115/1.4035824.

In this study, a cam and a flat-faced follower system with impacts and friction at the contact points are analyzed. The dynamic analysis has been done by simulating the follower displacement at a uniform cam angular velocity. Impact and friction are considered to determine the Lyapunov exponent based on different follower guides' clearances and cam rotational speeds. The simulation analysis has been carried out using solidworks. An experimental procedure is developed to capture the follower position through high-resolution optical markers mounted on the moving link. The experimental results are compared with the simulation results.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061002-061002-11. 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 electromechanical 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. The coupled electromechanical equations of motion then are solved numerically, and a comparison between experimental results and numerical predictions is presented.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061003-061003-14. 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. In fact, this work is an extension of Shafei and Shafei (2016, “A Systematic Method for the Hybrid Dynamic Modeling of Open Kinematic Chains Confined in a Closed Environment,” Multibody Syst. Dyn., 38(1), pp. 21–42.), which was restricted to a single open kinematic chain. So, to show the effectiveness of the suggested algorithm in deriving the motion equations of multichain robotic systems, a ten-link tree-type robotic system with floating base is simulated.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061004-061004-12. doi:10.1115/1.4037032.

A nonlinear mechanical model of a vibro-impact system influenced by double nonsmooth mechanical factors that combine elastic and rigid impact is described. The theoretical solutions to judge the periodic motion stability of the system are presented, and three different “gazing” motions and the corresponding conditions are described. The transition and coupling of periodic motions by the nonsmooth mechanical factors are demonstrated. The formation mechanism of sticking motion, chattering motion, and the periodic cavity by the influence of gazing bifurcation are analyzed. The coexistence of periodic motions and the extreme sensitivity of the initial value within the high frequency region are studied. The distribution of the attractor and the corresponding attracting domain corresponding to different periodic motions are also studied.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061005-061005-7. 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.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061006-061006-8. 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. 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 medial collateral ligament (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 anterior cruciate ligament (ACL), posterior cruciate ligament (PCL), and lateral collateral ligament (LCL) loads remain unchanged between varying cartilage stiffness values. The medial compartment bears a majority of the load (860 N in the medial compartment versus 540 N in the 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 (OA) than the lateral side. The model was then simplified using a linear Kelvin–Voight model for the cartilage and linear pretensioned springs representing the cumulative ligament bundles. This allowed for a validation of the system and the extrapolation of the results as the mass and cartilage stiffness varied. This is one of the few studies to quantify this load distribution and shows that the results are invariant to changes in cartilage stiffness. This effect is due to the precompression system created by the coordinated action of cartilage and ligaments.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061007-061007-8. doi:10.1115/1.4037033.

In this article, we will introduce the phenomenon known as the Painlevé paradox and further discuss the associated coupled phenomena, jam and lift-off. We analyze under what conditions the Painlevé paradox can occur for a general two-body collision using a framework that can be easily used with a variety of impact laws, however, in order to visualize jam and lift-off in a numerical simulation, we choose to use a recently developed energetic impact law as it is capable of achieving a unique forward solution in time. Further, we will use this framework to derive the criteria under which the Painlevé paradox can occur in a forced double-pendulum mechanical system. First, using a graphical technique, we will show that it is possible to achieve the Painlevé paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlevé paradox in the double-pendulum system.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061008-061008-11. 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 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 and (ii) with an immovable frame and two-sided impacts.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061009-061009-7. doi:10.1115/1.4037318.

This work presents a numerical investigation of the undesired lateral vibrations (whirling) occurring in drill-strings, which is one of the main sources of losses in drilling applications. The numerical studies are conducted using a nonsmooth lumped parameter model, which has been calibrated based on a realistic experimental drilling rig. The numerical investigations are focused on identifying different types of whirling responses, including periodic and chaotic behavior, which have been previously observed experimentally. As a result, the parameter space is divided into different regions showing dynamically relevant responses of the model, with special interest on the influence of the mass and angular velocity of the drill-string system. In particular, the study reveals the coexistence of various types of whirling motion for a given set of parameters and their sensitivity to initial conditions. The obtained theoretical predictions confirm previous experimental studies carried out by the authors, which provides a solid basis for a better understanding of whirling phenomena in drill-string applications.

Topics: Whirls , Drill strings
Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061010-061010-10. doi:10.1115/1.4036816.

Shaking table tests have been carried out to investigate the pounding phenomenon between a mass and two-sided shock absorbers, subject to sinusoidal excitations. In an effort to investigate the effectiveness of such an impact mitigation measure, preliminary tests were carried out: first, the dynamic response was recorded without pounding, and second, 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 was 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.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061011-061011-23. 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 (COR) experiments, in which a 2 m 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 (DIC) and laser Doppler vibrometry (LDV). 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 μN 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.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061012-061012-9. doi:10.1115/1.4036115.

A free-standing (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 submodels: a translation model, which simulates planar translational and rotational motion, and a rocking model, which simulates nonslide 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.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061013-061013-9. doi:10.1115/1.4036614.

The analysis of the shape memory prosthesis (SMP) of the middle ear is presented in this paper. The shape memory prosthesis permits the adjustment of its length to individual patient needs, but sometimes the prosthesis cannot be properly fixed to the stapes. In this case, the impact between the prosthesis and stapes is important. Therefore, the reconstructed middle ear is modeled as a two degree-of-freedom system with a nonlinear shape memory element and soft impact to represent its behavior when the prosthesis is not properly placed or fixed. The properties of the shape memory prosthesis, in the form of a helical spring, are represented by a polynomial function. The system exhibits advisable periodic and undesirable aperiodic and irregular behavior depending on the excitation amplitude, the frequency, and the prosthesis length. The prosthesis length can change, resulting in a modification of the distance between the prosthesis and the stapes. The results of this study provide an answer in terms of how the prosthesis length, which produces the ossicular chain tension, influences the system dynamics and its implication in medical practice.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061014-061014-11. 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 bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at 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 of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061015-061015-4. 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 ω0 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 (prestressed impacting surface), there exists a range of ω0 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 prestress as well as on the stiffness ratio of the springs.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061016-061016-8. doi:10.1115/1.4037074.

In this paper, a method is developed that results in guidelines for selecting the best Ordinary Differential Equation (ODE) solver and its parameters, for a class of nonlinear hybrid system were impacts are present. A monopod interacting compliantly with the ground is introduced as a new benchmark problem, and is used to compare the various solvers available in the widely used matlab ode suite. To provide result generality, the mathematical description of the hybrid system is brought to a dimensionless form, and its dimensionless parameters are selected in a range taken from existing systems and corresponding to different levels of numerical stiffness. The effect of error tolerance and phase transition strategy is taken into account. The obtained system responses are evaluated using solution speed and accuracy criteria. It is shown that hybrid systems represent a class of problems that cycle between phases in which the system of the equations of motion (EoM) is stiff (interaction with the ground), and phases in which it is not (flight phases); for such systems, the appropriate type of solver was an open question. Based on this evaluation, both general and case-specific guidelines are provided for selecting the most appropriate ODE solver. Interestingly, the best solver for a realistic test case turned out to be a solver recommended for numerically nonstiff ODE problems.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061017-061017-8. 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, 2016, “Numerical Modeling and Stability Analysis of Non-Smooth Dynamical Systems Via ABESPOL,” Ph.D. thesis, University of Aberdeen, Aberdeen, UK) based on COCO (Dankowicz and Schilder, Recipes for Continuation (Computational Science and Engineering), Society for Industrial and Applied Mathematics, Philadelphia, PA). 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.

Commentary by Dr. Valentin Fuster
J. Comput. Nonlinear Dynam. 2017;12(6):061018-061018-8. 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 2% 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 subimpacts 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 occurs 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.

Commentary by Dr. Valentin Fuster

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