Classical Rigid Body Dynamics Research Articles

Brachistochrone of off-centered cylinders

We consider the problem of finding paths of shortest transit time between two points (popularly known as brachistochrone) for cylinders with off-centered center of mass, rolling down without slip, subject solely to the force of gravity. This problem is set up using principles of classical rigid body dynamics and the desired path function is solved for numerically using the method of discrete calculus of variations. We discover a distinct array of brachistochrone trajectories for off-centered cylinders, demonstrate a critical dependence of such paths on the initial location and orientation of cylinders’ centers of mass and bring new insights into the family of brachistochrone problems and solutions.

Modelling and attitude control of novel multi-ducted-fan aerial vehicle in forward flight

Multi-ducted-fan aerial vehicle (MDFAV) is a novel platform for future transportation with promising prospects, which presents brand new challenges to the modelling and control approaches. In this paper, a comprehensive flight dynamical model of MDFAV is established utilising the aerodynamics principles and classical rigid body dynamics theory aiming to balance the complexity and covering necessary important dynamical behaviour of MDFAV, which is converted into state space by linearisation at the specific speed in forward flight mode in the following step. Subsequently a dual closed loop PID controller is put forward to keep flight fast and stable with the key coefficients tuned by non-smooth parameters optimisation method. The software and hardware of flight controller are designed for flight test of MDFAV systematically. Experiment results demonstrate excellent tracking performance and robustness, and at the same time, environment disturbances are suppressed effectively.

Modelling and attitude control of novel multi-ducted-fan aerial vehicle in forward flight

Multi-ducted-fan aerial vehicle (MDFAV) is a novel platform for future transportation with promising prospects, which presents brand new challenges to the modelling and control approaches. In this paper, a comprehensive flight dynamical model of MDFAV is established utilising the aerodynamics principles and classical rigid body dynamics theory aiming to balance the complexity and covering necessary important dynamical behaviour of MDFAV, which is converted into state space by linearisation at the specific speed in forward flight mode in the following step. Subsequently a dual closed loop PID controller is put forward to keep flight fast and stable with the key coefficients tuned by non-smooth parameters optimisation method. The software and hardware of flight controller are designed for flight test of MDFAV systematically. Experiment results demonstrate excellent tracking performance and robustness, and at the same time, environment disturbances are suppressed effectively.

Effect of breakup and transfer on complete and incomplete fusion in6Li+209Bi reaction in multi-body classical molecular dynamics calculation

The effect of breakup and transfer in 6 Li+209 Bi reaction is studied in a multi-body classical molecular dynamics approach in which the weakly-bound projectile 6 Li is constructed as a 2-body cluster of 4 He and 2 H in a configuration corresponding to the observed breakup energy. This 3-body system with their individual nucleon configuration in their ground state is dynamically evolved with given initial conditions using Classical Rigid Body Dynamics (CRBD) approach up to distances close to the barrier when the rigid-body constraint on the target, inter-fragment distance, and 2 H itself are relaxed, allowing for possible breakup of 2 H which may result in incomplete fusion following the transfer of the n or p . Relative probabilities of the possible events such as scattering with and without breakup, DCF, SCF, ICF(x ) where x may be 4 He, 2 H, 4 He+n , 4 He+p , n , p are calculated. Comparison of the calculated event-probabilities, complete, and incomplete fusion cross sections with the calculation in which 2 H is kept rigid demonstrates the effect of the transfer reactions on complete and incomplete fusion in the 4-body reaction. Events ICF(4 He+n ) corresponding to nstripping followed by breakup of the resultant 5 Li to 4 He+p are found to contribute significantly in the fusion process in agreement with a recent experimental observation of direct reaction processes in breakup of weakly-bound projectiles.

Three-stage classical molecular dynamics model for simulation of heavy-ion fusion

A three-stage Classical Molecular Dynamics (3S-CMD) approach for heavy-ion fusion is developed. In this approach the Classical Rigid-Body Dynamics simulation for heavy-ion collision involving light deformed nucleus is initiated on their Rutherford trajectories at very large initial separation. Collision simulation is then followed by relaxation of the rigid-body constrains for one or both the colliding nuclei at distances close to the barrier when the trajectories of all the nucleons are obtained in a Classical Molecular Dynamics approach. This 3S-CMD approach explicitly takes into account not only the long range Coulomb reorientation of the deformed collision partner but also the internal vibrational excitations of one or both the nuclei at distances close to the barrier. The results of the dynamical simulation for 24 Mg+208 Pb collision show significant modification of the fusion barrier and calculated fusion cross sections due to internal excitations.

Classical simulations of heavy-ion fusion reactions and weakly-bound projectile breakup reactions

Heavy-ion collision simulations in various classical models are discussed. Heavy-ion reactions with spherical and deformed nuclei are simulated in a classical rigid-body dynamics (CRBD) model which takes into account the reorientation of the deformed projectile. It is found that the barrier parameters not only on the initial orientations of the deformed nucleus, but also on the collision energy and the moment of inertia of the deformed nucleus. Maximum reorientation effect occurs at near- and below-barrier energies for light deformed nuclei. Calculated fusion cross-sections for 24Mg + 208Pb reaction are compared with a static-barrier-penetration model (SBPM) calculation to see the effect of reorientation. Heavy-ion reactions are also simulated in a 3-stage classical molecular dynamics (3S-CMD) model in which the rigid-body constraints are relaxed when the two nuclei are close to the barrier thus, taking into account all the rotational and vibrational degrees of freedom in the same calculation. This model is extended to simulate heavy-ion reactions such as6Li + 209Bi involving the weakly-bound projectile considered as a weakly-bound cluster of deuteron and 4He nuclei, thus, simulating a 3-body system in 3S-CMD model. All the essential features of breakup reactions, such as complete fusion, incomplete fusion, no-capture breakup and scattering are demonstrated.

Coulomb reorientation in near-barrier fusion of deformed+spherical systems in classical dynamical approach

A classical rigid-body dynamics model which takes into account all the translational and the rotational degrees of freedom is developed to study Coulomb reorientation of deformed nuclei in heavy-ion collisions. Various aspects of the collision dynamics in the case of near-barrier fusion of 24Mg + 208Pb system due to the Coulomb reorientation are studied; the dependence of the extent of reorientation of the symmetry axis of the deformed nucleus, isotropy of the initial orientations, barrier parameters, and rotational excitation energy are discussed in detail. It is found that the barrier parameters not only depend on the initial orientations of the deformed nucleus but also on the collision energy; with maximum reorientation effect at near- and below-barrier energies. Even small amount of the rotational excitation energy gained by the deformed nucleus at large separation distances is crucial in determining the conditions at the barrier. Study of 154Sm + 16O and 238U + 16O systems involving heavier deformed nuclei shows that the extent of reorientation also depends on the moment of inertia of the deformed nucleus.

Coulomb reorientation effect on near-barrier heavy-ion fusion reactions in rigid-body rotational dynamics calculation

A classical rigid-body dynamics model is used for study of Coulomb reorientation effect on the fusion dynamics. Model calculations show that barrier parameters depend on the initial orientations and also on the collision energy. Calculated fusion cross sections for 24Mg+ 208Pb and 16O+ 154Sm reactions show that reorientation effect is more pronounced in the case of 24Mg+ 208Pb reaction which involves a lighter deformed nucleus.

Analysis of Rigid-Body Dynamic Models for Simulation of Systems With Frictional Contacts

The use of Coulomb’s friction law with the principles of classical rigid-body dynamics introduces mathematical inconsistencies. Specifically, the forward dynamics problem can have no solutions or multiple solutions. In these situations, compliant contact models, while increasing the dimensionality of the state vector, can resolve these problems. The simplicity and efficiency of rigid-body models, however, provide strong motivation for their use during those portions of a simulation when the rigid-body solution is unique and stable. In this paper, we use singular perturbation analysis in conjunction with linear complementarity theory to establish conditions under which the solution predicted by the rigid-body dynamic model is stable. We employ a general model of contact compliance to derive stability criteria for planar mechanical systems. In particular, we show that for cases with one sliding contact, there is always at most one stable solution. Our approach is not directly applicable to transitions between rolling and sliding where the Coulomb friction law is discontinuous. To overcome this difficulty, we introduce a smooth nonlinear friction law, which approximates Coulomb friction. Such a friction model can also increase the efficiency of both rigid-body and compliant contact simulation. Numerical simulations for the different models and comparison with experimental results are also presented.

Exact energy-momentum conserving algorithms and symplectic schemes for nonlinear dynamics

It is shown that widely used implicit schemes, in particular the classical Newmark family of algorithms and its variants, generally fail to conserve total angular momentum for nonlinear Hamiltonian systems including classical rigid body dynamics, nonlinear elastodynamics, nonlinear rods and nonlinear shells. For linear Hamiltonian systems, it is well known that only the Crank-Nicholson scheme exactly preserves the total energy of the system. This conservation property is typically lost in the nonlinear regime. A general class of implicit time-stepping algorithms is presented which preserves exactly the conservation laws present in a general Hamiltonian system with symmetry, in particular the total angular momentum and the total energy. Remarkably, the actual implementation of this class of algorithms can be effectively accomplished by means of a simple two-step solution scheme which results in essentially no added computational cost relative to standard implicit methods. A complete analysis of these algorithms and a related class of schemes referred to as symplectic integrators is given. The good performance of the proposed methodology is demonstrated by means of three numerical examples which constitute representative model problems of nonlinear elastodynamics, nonlinear rods and nonlinear shells.