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# Accepted Manuscripts

BASIC VIEW  |  EXPANDED VIEW
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037318
This work presents a numerical investigation of the undesired lateral vibrations (whirls) occurring in drill-strings, which is one of the main source of losses in drilling applications. The numerical studies are conducted using a non-smooth 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 behaviour. As a result, the parameter space is divided into different regions showing dynamically relevant responses of the model, with special interest in the influence of the mass and angular velocity of the drill-string system. In particular, the study reveals the co-existence 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, Drilling rigs, Drilling, Lumped parameter models, Vibration
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037225
The Kidder equation, $y''(x)+2xy'(x)/\sqrt{1-\beta y(x)}=0$, $x\in [0,\infty)$, $\beta \in [0,1]$ with $y(0)=1$, and $y(\infty)=0$, is a second order non-linear two-point boundary value ordinary differential equation on the semi-infinite domain, with a boundary condition in the infinite that describes the unsteady isothermal flow of a gas through a semi-infinite micro-nano porous medium and has widely used in the chemical industries. In this paper, a hybrid numerical method is introduced for solving this equation. First, using the quasilinearization method, the equation is converted into a sequence of linear ordinary differential equations (LDEs), and then these LDEs are solved using the rational Legendre functions collocation method. Using $200$ collocation points, we have obtained a very good approximation solution and the value of the initial slope $y'(0)=-1.19179064971942173412282860380015936403$ for $\beta=0.50$, highly accurate to 38 decimal places. The convergence of numerical results is shown.
TOPICS: Numerical analysis, Flow (Dynamics), Porous materials, Differential equations, Approximation, Boundary-value problems, Chemical industry
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037226
While several curvature expressions have been used in the literature, some of these expressions are not consistent with basic geometry definitions and are not consistent with the fact that the bending and shear deformations are independent. These inconsistencies are attributed to the fact that when low order of interpolation is used for the finite element (FE) position field, non-zero curvature cannot be defined, thereby leaving only the option of using expressions that are inconsistent with differential geometry and basic mechanics principles. This paper uses three different elastic force formulations in order to examine the effect of the curvature definition in the large displacement analysis of beams. In the first elastic force formulation, a general continuum mechanics approach (Method 1) based on nonlinear strain-displacement relationship is used. The second approach (Method 2) is based on a classical nonlinear beam theory, in which a curvature expression consistent with differential geometry and independent of the shear deformation is used. The third elastic force formulation (Method 3) employs a curvature expression obtained from the shear angle. In the literature, one resorted to using this latter inconsistent curvature definition because of the use of low order finite elements in which non-zero curvature cannot be defined within the element.
TOPICS: Displacement, Geometry, Shear (Mechanics), Finite element analysis, Deformation, Continuum mechanics, Interpolation, Shear deformation, Euler-Bernoulli beam theory
research-article
J. Comput. Nonlinear Dynam   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 systems with impacts. 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 with impacts 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.
TOPICS: Phase transitions, Equations of motion, Differential equations, Dynamic systems, Cycles, Errors, Matlab, Stiffness, Flight
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037032
A non-linear mechanical study of a vibro-impact system influenced by a double non-smooth mechanical factor that combined elastic and rigid impact was studied. The theoretical solutions to judge the periodic motion stability of the system are described. Three different “gazing” motions and the corresponding conditions for each “gazing” motion are described. The transition and coupling regularities of periodic motion distribution under these combined effects were demonstrated. The formation mechanism of sticking motion, chattering motion, and formation of the periodic cavity under the induction of gazing bifurcation were analyzed. The extreme sensitivity of the initial value when the system is within the high frequency region was studied. The distribution of attractors and the relevant regions of attraction in the state space of different periodic motions in the periodic coexistence region were also studied.
TOPICS: Stability, Electromagnetic induction, Bifurcation, Cavities, Attractors
research-article
J. Comput. Nonlinear Dynam   doi: 10.1115/1.4037033
In this article we will introduce the phenomenon known as the Painlev\'e paradox, and further discuss the associated coupled phenomena, jam and lift-off. We analyse under what conditions the Painlev\'e 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 visualise jam and lift-off in a numerical simulation, we choose to use the 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 Painlev\'e 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 Painlev\'e paradox for relatively low coefficient of friction values, and second we will use the energetic impact law to numerically show the occurrence of the Painlev\'e paradox in a double pendulum system.
TOPICS: Pendulums, Friction, Computer simulation, Collisions (Physics)
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: 10.1115/1.4036816
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
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
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