Waggling conical pendulums and visualization in mechanics

Students often find mechanics a difficult area to grasp. This paper describes an equation of motion for a waggling conical pendulum. A wide range of pendulum dynamics can be simulated with this model. The equation of motion is embedded in a graphical user interface (GUI) for its numerical solution in MATLAB. This allows a student's focus to be on the influence of different parameters on the pendulums dynamics. The simulation tool can be used as a dynamics demonstrator in a lecture or as an educational tool driven by the imagination of the student. By way of demonstration, the simulation tool has been applied to two damped pendulums and an inverted damped pendulum. The model has also been used to simulate resonance and has shown that there is a wide range of behaviour possible depending on the type of forcing applied. Finally, a forced conical pendulum as a system for harnessing wave energy is considered.     

An integrated simulation-optimization framework for the operational planning of seaport container terminals

In this article the operational planning of seaport container terminals is considered by defining a suitable integrated framework in which simulation and optimization interact. The proposed tool is a simulation environment (implemented by using the Arena software) representing the dynamics of a container terminal. When the system faces some particular conditions (critical events), an optimization procedure integrated in the simulation tool is called. This means that the simulation is paused, an optimization problem is solved and the relative solution is an input for the simulation environment where some system parameters are modified (generally, the handling rates of some resources are changed). For this reason, in the present article we consider two modelling and planning levels about container terminals. The simulation framework, based on an appropriate discrete-event model, represents the dynamic behaviour of the terminal, thus it needs to be quite detailed and it is used as an operational planning tool. On the other hand, the optimization approach is devised in order to define some system parameters such as the resource handling rates; in this sense, it can be considered as a tool for tactical planning. The optimization procedure is based on an aggregate representation of the terminal where the dynamics is modelled by means of discrete-time equations.     

Rigid–flexible interactive dynamics modelling approach

Dynamics modelling of multi-body systems composed of rigid and flexible elements is elaborated in this article. The control of such systems is highly complicated due to severe underactuated conditions caused by flexible elements and an inherent uneven non-linear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, simulation studies for design improvement and also practical implementations. In this article, the rigid–flexible interactive dynamics modelling (RFIM) approach is proposed as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than a common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. The proposed RFIM approach is first detailed for multi-body systems with flexible joints, and then with flexible members. Then, to reveal the merits of this new approach, few case studies are presented. A flexible inverted pendulum is studied first as a simple template for lucid comparisons, and next a space free-flying robotic system that contains a rigid main body equipped with two manipulating arms and two flexible solar panels is considered. Modelling verification of this complicated system is vigorously performed using ANSYS and ADAMS programs. The obtained results reveal the outcome accuracy of the new proposed approach for explicit dynamics modelling of rigid–flexible multi-body systems such as mobile robotic systems, while its limited computations provide an efficient tool for controller design, simulation studies and also practical implementations of model-based algorithms.     

Dynamics modelling and Hybrid Suppression Control of space robots performing cooperative object manipulation

Dynamics modelling and control of multi-body space robotic systems composed of rigid and flexible elements is elaborated here. Control of such systems is highly complicated due to severe under-actuated condition caused by flexible elements, and an inherent uneven nonlinear dynamics. Therefore, developing a compact dynamics model with the requirement of limited computations is extremely useful for controller design, also to develop simulation studies in support of design improvement, and finally for practical implementations. In this paper, the Rigid–Flexible Interactive dynamics Modelling (RFIM) approach is introduced as a combination of Lagrange and Newton–Euler methods, in which the motion equations of rigid and flexible members are separately developed in an explicit closed form. These equations are then assembled and solved simultaneously at each time step by considering the mutual interaction and constraint forces. The proposed approach yields a compact model rather than common accumulation approach that leads to a massive set of equations in which the dynamics of flexible elements is united with the dynamics equations of rigid members. To reveal such merits of this new approach, a Hybrid Suppression Control (HSC) for a cooperative object manipulation task will be proposed, and applied to usual space systems. A Wheeled Mobile Robotic (WMR) system with flexible appendages as a typical space rover is considered which contains a rigid main body equipped with two manipulating arms and two flexible solar panels, and next a Space Free Flying Robotic system (SFFR) with flexible members is studied. Modelling verification of these complicated systems is vigorously performed using ANSYS and ADAMS programs, while the limited computations of RFIM approach provides an efficient tool for the proposed controller design. Furthermore, it will be shown that the vibrations of the flexible solar panels results in disturbing forces on the base which may produce undesirable errors and perturb the object manipulation task. So, it is shown that these effects can be significantly eliminated by the proposed Hybrid Suppression Control algorithm.     

Computational fluid dynamics based dynamic modeling of parafoil system

The calculation of aerodynamic coefficients has been one of the key issues when modeling parafoil systems, that directly affects model precision. This study relates to investigate limitations of traditional calculation methods. As a result, we achieve aerodynamic parameters of a parafoil using computational fluid dynamics simulations. Also we employ the least square method as a tool for the rapid identification of deflection factors of aerodynamic coefficients. The estimated aerodynamic coefficients of the system were incorporated into the dynamic equations of the parafoil to implement a six degree of freedom model of a parafoil system according to the Kirchhoff equations. Numerical results generated by simulation and airdrop testing demonstrate that the established model can accurately describe the flight characteristics of the parafoil system.     

The Drazin inverse in multibody system dynamics

Summary The numerical analysis of multibody system dynamics is based on the equations of motion as differential-algebraic systems. A thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches. The main tool is the Drazin inverse, a generalized matrix inverse, which preserves the eigenvalues. The results are illustrated by a realistic truck model. Finally, the approach is extended to the nonlinear index 2 formulation.     

Energy drift in molecular dynamics simulations

In molecular dynamics, Hamiltonian systems of differential equations are numerically integrated using the Störmer–Verlet method. One feature of these simulations is that there is an unphysical drift in the energy of the system over long integration periods. We study this energy drift, by considering a representative system in which it can be easily observed and studied. We show that if the system is started in a random initial configuration, the error in energy of the numerically computed solution is well modeled as a continuous-time stochastic process: geometric Brownian motion. We discuss what in our model is likely to remain the same or to change if our approach is applied to more realistic molecular dynamics simulations.     

Quasiperiodic motion induced by heterogeneous delays in a simplified internet congestion control model

In this paper, a simplified congestion control model is considered to study the quasiperiodic motion induced by heterogenous time delays. Analysis for the stability of the equilibrium shows that the Hopf bifurcation curves with diverse frequencies may intersect at the so-called non-resonant double Hopf bifurcation point. Choosing the delays as the bifurcation parameters and employing the method of multiple scales, the amplitude–frequency equations or normal form equations are obtained theoretically. Based on these equations, the dynamics near the bifurcation point is classified. The values of the delays for which the quasiperiodic motion exists can be predicted with an acceptable accuracy. This result provides a reference in designing and optimizing the network systems.     

Exact Solutions of Extended Boussinesq Equations

The problem addressed in this paper is the verification of numerical solutions of nonlinear dispersive wave equations such as Boussinesq-like system of equations. A practical verification tool for numerical results is to compare the numerical solution to an exact solution if available. In this work, we derive some exact solitary wave solutions and several invariants of motion for a wide range of Boussinesq-like equations using Maple software. The exact solitary wave solutions can be used to specify initial data for the incident waves in the Boussinesq numerical model and for the verification of the associated computed solution. The invariants of motions can be used as verification tools for the conservation properties of the numerical model.     

The effect of environmental parameters on product recovery

Economical and environmental issues are the main driving forces for the development of closed-loop supply chains. This paper examines the impact of environmental issues on long-term behaviour of a single product supply chain with product recovery. The environmental issues examined are the firm's `green image' effect on customer demand, the take back obligation imposed by legislation, and the state campaigns for proper disposal of used products. The behaviour of the system is analyzed through a dynamic simulation model based on the principles of the system dynamics (SD) methodology. This model includes all major inventories of new, used and recovered products and the flows among them. Inventory levels and flow rates are linked through differential equations. The dynamic model provides an experimental simulation tool, which can be used to evaluate the effect of environmental issues on long-term decision making in collection and remanufacturing activities and on product demand. Numerical analysis illustrates the potential uses of the methodology.     

Vibration analysis and bifurcations in the self-sustained electromechanical system with multiple functions

We consider in this paper the dynamics of the self-sustained electromechanical system with multiple functions, consisting of an electrical Rayleigh–Duffing oscillator, magnetically coupled with linear mechanical oscillators. The averaging and the harmonic balance method are used to find the amplitudes of the oscillatory states respectively in the autonomous and nonautonomous cases, and analyze the condition in which the quenching of self-sustained oscillations appears. The influence of system parameters as well as the number of linear mechanical oscillators on the bifurcations in the response of this electromechanical system is investigated. Various bifurcation structures, the stability chart and the variation of the Lyapunov exponent are obtained, using numerical simulations of the equations of motion.     

Multipulse Orbits in the Motion of Flexible Spinning Discs

Summary. The global dynamics of flexible spinning discs are studied. The discs studied are parametrically excited in their spin rate, and have imperfections that cause symmetry-breaking. After determining the equations of motion in a suitable form, the energy-phase method is employed to show the existence of chaotic dynamics by identifying multipulse jumping orbits in the perturbed phase space. We provide restrictions on the damping, forcing, and symmetry-breaking parameters in order for these complicated dynamics to occur. The dissipative version of the energy-phase method predicts a wider range of values for which chaotic dynamics occurs than the traditional Melnikov method. The results are then discussed in terms of the physical motion of the spinning disc system. The multipulse orbits are manifested in the physical system as a shifting between two different nodal configurations of the disc. When the motion is chaotic, an observer will see a random jumping between the two nodal configurations of the disc. Received February 7, 2000; accepted November 18, 2001     

Asymptotics and numerics for non-Newtonian jet dynamics

This work deals with the modeling and simulation of non-Newtonian jet dynamics. Proceeding from a 3d boundary value problem of upper-convected Maxwell equations, a 1d viscoelastic string model can be derived asymptotically. The resulting system of PDEs has a hyperbolic-elliptic character with an additional differential constraint. Its applicability regime is limited depending on physical parameters and boundary conditions. Numerical results are shown for gravitational in-/outflow set-ups. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion equations of cooperative multi flexible mobile manipulator via recursive Gibbs–Appell formulation

In this paper, the developed model of an N-flexible-link mobile manipulator with revolute-prismatic joints is presented for the cooperative flexible multi mobile manipulator. In this model, the deformation of flexible links is calculated by using the assumed modes method. In additions, non-holonomic constraints of the robots’ mobile platforms that bound its locomotion are considered. This limitation is alleviated through the concurrent motion of revolute and prismatic joints, although it results in computational complexity and changes the final motion equations to time-varying form. Not only is the proposed dynamic model implemented for the multi-mobile manipulators with arms having independent motion, but also for multi-mobile manipulators in cooperation after defining gripper's kinematic constraints. These constraints are imported to the dynamic equations by defining Lagrange multipliers. The recursive Gibbs–Appell formulation is preferred over other similar approaches owing to the capability of solving the equations without the need to use Lagrange multipliers for eliminating non-holonomic constraints in addition to the novel optimized process of obtaining system equations. Hence, cumbersome simultaneous computations for eliminating the constraints of platform and arms are circumvented. Therefore, this formulation is improved for the first time by importing Lagrange multipliers for solving kinematic constrained systems. In the simulation section, the results of forward dynamics solution for two flexible single-arm manipulators with revolute-prismatic joints while carrying a rigid object are presented. Inverse dynamics equations of the system are also presented to obtain the maximum dynamic load-carrying capacity of the two-rigid-link mobile manipulators on a predefined path. Two constraints, namely the capacity of joint motors torque and robot motion stability are considered as the limitation criteria. The concluded motion equations are used to accurately control the movement of sensitive bodies, which is not achievable through the use of one platform.     

Stable fuzzy logic design of point-to-point control for mechanical systems

An approach for the development of fuzzy point-to-point control laws for second-order mechanical systems is presented. Asymptotic stability of the resulting closed-loop system is proved using Lyapunov stability theory. Closed-loop performance and robustness are quantified in terms of the parameters of membership functions. As opposed to most existing fuzzy control laws, the closed-loop stability of the proposed controller does not depend on the knowledge of the entire dynamics. Moreover, the approach does not require the plant to be open-loop stable. The proposed approach is demonstrated on design and simulation study of a fuzzy controller for a two-link robotic arm.     

Hamiltonian modelling and buckling analysis of a nonlinear flexible beam with actuation at the bottom

ABSTRACT

The use of beams and similar structural elements is finding increasing application in many areas including micro and nanotechnology devices. For the purpose of buckling analysis and control, it is essential to account for nonlinear terms in the strains while modelling these flexible structures. Further, the Poisson’s effect can be accounted in modelling by the use of a two-dimensional stress–strain relationship. This paper studies the buckling effect for a slender, vertical beam (in the clamped-free configuration) with horizontal actuation at the fixed end and a tip-mass at the free end. Including also the inextensibility constraint of the beam, the equations of motion are derived. A preliminary modal analysis of the system has been carried out to describe candidate post-buckling configurations and study the stability properties of these equilibria. The vertical configuration of the beam under the action of gravity is without loss of generality, since the objective is to model a potential field that determines the equilibria. Neglecting the inextensibility constraint, the equations of motion are then casted in port-Hamiltonian form with appropriately defined flows and efforts as a basis for structure-preserving discretization and simulation. Finally, the finite-dimensional model is simulated to obtain the time response of the tip-mass for different loading conditions.     

Dirac reduction for nonholonomic mechanical systems and semidirect products

This paper develops the theory of Dirac reduction by symmetry for nonholonomic systems on Lie groups with broken symmetry. The reduction is carried out for the Dirac structures, as well as for the associated Lagrange–Dirac and Hamilton–Dirac dynamical systems. This reduction procedure is accompanied by reduction of the associated variational structures on both Lagrangian and Hamiltonian sides. The reduced dynamical systems obtained are called the implicit Euler–Poincaré–Suslov equations with advected parameters and the implicit Lie–Poisson–Suslov equations with advected parameters. The theory is illustrated with the help of finite and infinite dimensional examples. It is shown that equations of motion for second order Rivlin–Ericksen fluids can be formulated as an infinite dimensional nonholonomic system in the framework of the present paper.     

Critical damping in nonviscously damped linear systems

In structural dynamics, energy dissipative mechanisms with nonviscous damping are characterized by their dependence on the time-history of the response velocity, mathematically represented by convolution integrals involving hereditary functions. Combination of damping parameters in the dissipative model can lead the system to be overdamped in some (or all) modes. In the domain of the damping parameters, the thresholds between induced oscillatory and non-oscillatory motion are named critical damping surfaces (or critical manifolds, since several parameters can be involved). In this paper the theoretical foundations to determine critical damping surfaces in nonviscously damped systems are established. In addition, a numerical method to obtain critical curves is developed. The approach is based on the transformation of the algebraic equations, which define implicitly the critical curves, into a system of differential equations. The derivations are validated with three numerical methods covering single and multiple degree of freedom systems.