Lie symmetries and conserved quantities of constrained mechanical systems

Summary The Lie symmetries and conserved quantities of constrained mechanical systems are studied. Using the invariance of the ordinary differential equations under the infinitesimal transformations, the determining equations and the restriction equations of the Lie symmetries of the systems are established. The structure equation and the form of conserved quantities are obtained. We find the corresponding conserved quantity from a known Lie symmetry, that is a direct problem of the Lie symmetries. And then, the inverse problem of the Lie symmetries-finding the corresponding Lie symmetry from a known conserved quantity-is studied. Finally, the relation between the Lie symmetry and the Noether symmetry is given.     

Equations of motion for general constrained systems in Lagrangian mechanics

This paper develops a new, simple, explicit equation of motion for general constrained mechanical systems that may have positive semi-definite mass matrices. This is done through the creation of an auxiliary mechanical system (derived from the actual system) that has a positive definite mass matrix and is subjected to the same set of constraints as the actual system. The acceleration of the actual system and the constraint force acting on it are then directly provided in closed form by the acceleration and the constraint force acting on the auxiliary system, which thus gives the equation of motion of the actual system. The results provide deeper insights into the fundamental character of constrained motion in general mechanical systems. The use of this new equation is illustrated through its application to the important and practical problem of finding the equation of motion for the rotational dynamics of a rigid body in terms of quaternions. This leads to a form for the equation describing rotational dynamics that has hereto been unavailable.     

An accelerated iterative method for the dynamics of constrained multibody systems

An accelerated iterative method is suggested for the dynamic analysis of multibody systems consisting of interconnected rigid bodies. The Lagrange multipliers associated with the kinematic constraints are iteratively computed by the monotone reduction of the constraint error vector, and the resulting equations of motion are easily time-integrated by a well established ODE technique. The velocity and acceleration constraints as well as the position constraints are made to be satisfied at the joints at each time step. Exact solution is obtained without the time demanding procedures such as selection of the independent coordinates, decomposition of the constraint Jacobian matrix, and Newton Raphson iterations. An acceleration technique is employed for the faster convergence of the iterative scheme and the convergence analysis of the proposed iterative method is presented. Numerical solutions for the verification problems are presented to demonstrate the efficiency and accuracy of the suggested technique.     

Geometric non-linear substructuring for dynamics of flexible mechanical systems

A substructure synthesis formulation is presented that permits use of established flexible multibody dynamic analysis computer codes to account for structural geometric non-linear effects. Large relative displacement is permitted between points within bodies that undergo small strain elastic deformation. Components are divided into substructures, on each of which the theory of linear elasticity relative to a body reference frame is adequate to describe deformation and its coupling with system motion. Normal vibration and static correction deformation modes are used to account for elastic deformation within each substructure. Compatibility conditions are derived and imposed as constraint equations at boundary points between substructures. System equations of motion that include geometric non-linear effects of large rotation, in terms of generalized co-ordinates of a reference frame for each substructure and a set of deformation modes that are defined within the substructure, are assembled. The method is implemented in an industry standard flexible multibody dynamics code, with minimal modification. Use of the formulation is illustrated on the classical problem of a spinning beam with geometric stiffening and on a space structure that experiences large deformation.     

A recursive method for the dynamic analysis of mechanical systems in spatial motion

Summary. In the present study, a recursive method for generating the equations of motion of mechanical systems that undergo spatial motion is presented. The method uses the force and moment equations to generate the rigid body equations of motion in terms of the Cartesian coordinates of a dynamically equivalent constrained system of particles, without introducing any rotational coordinates and the corresponding rotation matrices. For the open loop case, the equations of motion are generated recursively along the serial chains. Closed loop systems are transformed to open loop systems by cutting suitable kinematic joints and introducing cut-joint constraints. The method is simple and suitable for computer implementation. An example is chosen to demonstrate the generality and simplicity of the developed formulation.     

Stability analysis of constrained multibody systems

Automated algorithms for the dynamic analysis and simulation of constrained multibody systems usually assume the rows of the constraint Jacobian matrix to be linearly independent. But during the motion, at instantaneous configurations, the Jacobian matrix may become less than full rank resulting in singularities. This occurs when the closed-loop goes from 3D to 2D type of configuration. In this paper the linearly dependent rows are identified by an uptriangular decomposition process. The corresponding constraint equations are modified so that the singularities in the numerical procedure are avoided. The conditions for the validity of the modified equations are described. Furthermore, the constraint equations expressed in accelerations are modified by Baumgarte's approach to stabilize the accumulation of the numerical errors during integration. A computational procedure based on Kane's equations is presented. Two and three-link robotic manipulators will be simulated at singular configurations to illustrate the use of the modified constraints.     

On the motion of mechanical networks

This paper develops the differential equations governing the motion of spatial networks to which mechanical features such as masses, stiffness coefficients, tensions and bending moments have been associated. These networks generalize the concept of particle systems introduced for the simulation of flexible bodies and extend their application to elastic models. The network deformation is shown to be related to the internal tensions and moments by a set of vectors, the directors of the network. A numerical example describing a rotating flexible beam is presented.     

Designing an optimal water quality monitoring network for river systems using constrained discrete optimization via simulation

The problem of designing a water quality monitoring network for river systems is to find the optimal location of a finite number of monitoring devices that minimizes the expected detection time of a contaminant spill event while guaranteeing good detection reliability. When uncertainties in spill and rain events are considered, both the expected detection time and detection reliability need to be estimated by stochastic simulation. This problem is formulated as a stochastic discrete optimization via simulation (OvS) problem on the expected detection time with a stochastic constraint on detection reliability; and it is solved with an OvS algorithm combined with a recently proposed method called penalty function with memory (PFM). The performance of the algorithm is tested on the Altamaha River and compared with that of a genetic algorithm due to Telci, Nam, Guan and Aral (2009) Telci, I. T., K. Nam, J. Guan, and M.M. Aral, 2009. “Optimal Water Quality Monitoring Network Design for River Systems.” Journal of Environmental Management, 90 (3–4): 2987–2998. doi: 10.1016/j.jenvman.2009.04.011[Crossref], [PubMed], [Web of Science ®] , [Google Scholar].     

Molecular dynamics investigation of mechanical mixing in mechanical alloying

Molecular dynamic simulation is exploited to obtain a deep insight of atomic scale mixing and amorphization mechanisms happening during mechanical mixing. Impact–relaxation cycles are performed to simulate the mechanical alloying process. The results obtained by structural analysis shows that the final structure obtained through simulation of mechanical alloying is in an amorphous state. This analysis reveals that amorphization occurs concurrently with the attainment of a perfectly mixed alloy. The results indicate diffusion and deformation are two important mechanisms for mixing during mechanical alloying. The rate of diffusion is controlled by the temperature and by the density of defects in the structure. Deformation enhances mixing directly by sliding atomic layers on each other and increases the number of defects in the structure. The results agree with mechanical alloying experiments described in the literature.     

Molecular dynamics simulation of mechanical behaviour

A molecular dynamics (MD) simulation of strain and failure of a crystal, the effect of environment on these processes, the interaction between the adsorption-active atoms and the environment walls, the effect of stress on mobility of interstitial admixtures, the formation and the failure of a contact between two crystals has been performed using a two-dimensional system consisting of Lennard-Jones atoms. The basic features of strain observed by means of MD included generation and motion of dislocations, various mechanisms of shear and brittle-to-ductile transition at low temperature. Environmentsensitive mechanical behaviour has been studied for the first time on an atomic scale. It is shown that rapid local processes whose unit act takes about 10^–10 sec and involves several tens or a few hundred atoms may provide for environment-induced embrittlement. The features common to these microscopic processes are (1) pronounced interaction between the foreign atoms and the atoms of the solid, i.e. a sharp decrease in the surface energy of the solid in contact with the environment, and (2) direct participation of thermal fluctuations in the failure and rearrangement of interatomic bonds. By interacting with the crack walls, the environment atoms create a force compatible with the interatomic bond strength, which promotes crack propagation. Tensile stress causes appreciable acceleration of diffusion of interstitial admixtures in the direction normal to the strain axis and hinders diffusion along the axis. Under constant load the failure of interatomic bonds and sintering involve a thermal fluctuation mechanism.     

Brachistochronic motion of a nonholonomic rheonomic mechanical system

The brachistochrone problem of the rheonomic mechanical system whose motion is subject to nonholonomic constraints is solved with nonlinear differential equations of motion. Apart from control forces, the system is influenced by the action of other known potential and nonpotential forces as well. The problem of optimal control is solved by applying Pontryagin’s Maximum Principle and the singular optimal control theory. This procedure results in the two-point boundary value problem for the system of ordinary nonlinear differential equations of the first order, with a corresponding number of initial and end conditions. This paper determines the control forces that are realized by imposing on the system a corresponding number of independent ideal holonomic constraints, without the action of active control forces. These constraints must be in accordance with the previously determined brachistochronic motion. The method is illustrated with a single complex example that represents the first known concrete demonstration of brachistochronic motion of the nonholonomic rheonomic mechanical system.     

Coarsening dynamics of phase-separating systems

When a system such as a binary liquid is cooled rapidly from a homogeneous phase into a two-phase region, domains of the two equilibrium phases form and grow ('coarsen') with time. In the absence of an external drive, such as gravity or an imposed shear flow, a dynamical-scaling regime emerges in which the domain morphology is statistically self-similar at different times, up to an overall length-scale (coarsening scale) that grows with time. In the first part of the paper, the scaling phenomenology will be reviewed and the time-dependence of the coarsening scale will be discussed in the context of a number of different physical systems and scaling regimes. In the second part, the influence of an external drive, in particular a shear flow, will be addressed and recent developments reviewed. Interesting open questions include the late-time behaviour under shear and whether the coarsening continues indefinitely or is ultimately arrested by the shear flow.     

Tolerance analysis and design of nesting forces for exactly constrained mechanical assemblies

In this research we investigate the analysis and design of nesting forces for exactly constrained mechanical assemblies. Exactly constrained assemblies have a number of important advantages over other assemblies including the ability to assemble over a wide range of conditions. Such designs often require nesting forces to keep the design properly seated. To date, little theory has been developed for the analysis and design of nesting forces. We show how the effects of tolerances on nesting forces, a key issue, can be analyzed and then apply the analysis to two example problems. Good agreement is obtained between the method and Monte Carlo simulation.     

Formation of magnesium nanocomposite via mechanical milling

Nanocomposite Mg5wt%Al has been synthesized via mechanochemical milling of elemental Mg, Al and Ti powders with polyethylene–glycol. TiH₂ phase was formed through the reaction between Ti and polyethylene–glycol during the milling as well as the sintering processes. Formation of ultra-fine particles of a few nanometer in size was observed within the matrix grains. The formation of TiH₂ particles is hypothesized to occur through two stages. In the first stage, Ti is supersaturated in the Mg matrix during high energy milling while polyethylene–glycol decomposes to react with Mg as well as Ti which is not solid solute in the Mg matrix. In the second stage during sintering, the hydrogen and Ti released from the Mg react with each other, thus forming nanoparticles. As a result of the nanoparticles, the nanocomposite manifests improved yield strength and ductility in comparison to it counterpart.     

An analytical solution for mechanical etching of glass by powder blasting

Masked erosion of glass by powder blasting is studied and a nonlinear partial differential equation of first order describing the displacement of the glass surface is proposed. This equation is solved by means of the characteristic-strip equations. If so-called transition regions are introduced near the edges of the mask, an analytical solution can be obtained which is in reasonable agreement with measurements.     

The use of analytical redundancy in navigational measuring systems

Algorithms for monitoring and for the diagnostics of navigation systems based on analytical redundancy, which provide high reliability of estimates of the navigational parameters of aircraft, are described. __________ Translated from Izmeritel’naya Tekhnika, No. 2, pp. 29–33, February, 2007.     

An analytical framework for reliability growth of one-shot systems

In this paper, we introduce a new reliability growth methodology for one-shot systems that is applicable to the case where all corrective actions are implemented at the end of the current test phase. The methodology consists of four model equations for assessing: expected reliability, the expected number of failure modes observed in testing, the expected probability of discovering new failure modes, and the expected portion of system unreliability associated with repeat failure modes. These model equations provide an analytical framework for which reliability practitioners can estimate reliability improvement, address goodness-of-fit concerns, quantify programmatic risk, and assess reliability maturity of one-shot systems. A numerical example is given to illustrate the value and utility of the presented approach. This methodology is useful to program managers and reliability practitioners interested in applying the techniques above in their reliability growth program.     

Second-order design sensitivity analysis of mechanical system dynamics

Dependence of dynamic response of nonlinear mechanical systems on design variables is analysed. An adjoint variable method is used to derive first- and second-order derivatives of measures of dynamic response with respect to design variables. A computational algorithm is presented for numerical calculation of first and second design derivatives. A simple oscillator example is solved analytically and by the adjoint variable method, with identical results. A burst fire automatic weapon mechanism with linear and nonlinear damping is treated numerically. It is shown that quadratic appriximations of dynamic response, using results of second-order design sensitivity analysis, can be substantially better than conventional linear approximations.     

Nonlinear semiclassical dynamics of open systems

A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian and 'open' Markovian Lindblad operators, is reviewed. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian is a quadratic function and the Lindblad operators are linear functions of positions and momenta. The semiclassical formulae are interpreted within a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term in the double Hamiltonian is generated by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by the definition of a propagator, here developed in both representations. Generalized asymptotic equilibrium solutions are thus presented for the first time.