Multibody Dynamics Equations Research Articles

Redundant Serial Manipulator Inverse Position Kinematics and Dynamics

AbstractA redundant serial manipulator inverse position kinematic mapping is employed to define a new manipulator operational space differentiable manifold and an associated system of well posed operational space differential equations of manipulator dynamics. A review of deficiencies in the conventional generalized inverse velocity approach to manipulator redundancy resolution and a numerical example show that the conventional approach is incompatible with kinematics of redundant serial manipulators. The inverse position kinematic mapping presented is shown to define a differentiable manifold that is parameterized by either input or operational space coordinates. Differentiation of the inverse position mapping yields an inverse velocity mapping that is a total differential, in contrast with generalized inverse velocity mappings, hence avoiding the deficiencies identified. A second differentiation yields an inverse acceleration mapping that is used, without ad-hoc derivation, to obtain well posed operational space ordinary differential equations of redundant manipulator dynamics that are equivalent to the equations of multibody dynamics.

Transport vibration analysis of mobile liquid Pb-Bi micro-reactors with non-uniform mass distribution

The mobile micro-reactors, as advanced nuclear systems, had the capability to provide energy to remote areas and islands. However, the non-uniform mass distribution characteristic of Liquid Lead-Bismuth Eutectic (LBE), which possessed a high density, substantial mass, and compact volume, mobile micro-reactors required special attention during transportation when mounted on trucks and passing through rough roads. This paper aimed to investigate the vibration responses of the mobile micro-reactor with non-uniform mass distribution during transportation. To achieve this, a separated model was developed using the Lagrange multi-body dynamic equations, the recorded responses data were evaluated using MATLAB, a numerical simulation software. Furthermore, a parametric study was conducted to examine the influence of the mass of LBE mass and the positions of the micro-reactor on the vibration responses during transportation. The results indicated that the mass of the micro-reactorwas directlyproportionaltothe vibration responses, and the responses could be different with the moveable situation of the reactor. This study demonstrated that the separated model could evaluate the transportation vibration responses of both the mobile micro-reactors and the truck more accurately, thereby facilitating the safe transportation of mobile micro-reactors with non-uniform mass distribution.

Dynamic analysis and validation of a multi-body floating wind turbine using the moving frame method

This research applies the moving frame method (MFM) to the multi-body dynamic analysis of an OC3 phase IV spar buoy with the NREL 5MW turbine. Further, it verifies previous results obtained through numerical comparisons with commercial software. The long-term goal is to lay the foundation for leveraging the MFM to create a self-contained software system for future analyses that can incorporate effects that are more sophisticated, when commercial codes fall short. In this first evidentiary phase, this project treats the floating turbine as a three-bodied system consisting of the platform (platform + tower), nacelle and rotor (hub + blades). Then the paper presents the MFM in a tutorial style—in the context of this problem’s resolution. The paper supplements the multi-body dynamic equations of motion obtained through the MFM with simplified and reduced hydrodynamic, aerodynamic and mooring loads to simulate the translational and rotational response of the floating turbine under various load conditions. The results closely approximate those found in previous work and, in the process, demonstrates MFM’s analytical advantage. Current results capture the coupled responses in all degrees of freedom and gyroscopic effects occurring when the platform pitches with the spinning rotor. The project thus provides an accurate model for the dynamics of the turbine and opens the door to inserting correct advanced hydrodynamics to validate the model further. The work presents simulations for the different load cases through a 3D web page using WebGL and the ThreeJS library. Users may download all software to verify the results. An undergraduate student conducted the work alone, demonstrating the ease of implementation of the MFM.

Application of a parallel physics-informed neural network to solve the multi-body dynamic equations for full-scale train collisions

The prohibitive cost of acquiring data from full-scale train collision experiments limits the applicability of data-driven machine learning methods in train collision simulation. Physics-informed neural networks (PINN) attempt to address this challenge by incorporating physics equations as part of the loss function construction. However, the PINN approach is relatively time-consuming when it comes to solving a large number of physical equations. In this paper, a parallel physics-informed neural network (PPINN) methodology is developed for the solution of multibody dynamics equations to further reduce the computational cost. As well, a PPINN-based framework for engineering applications is proposed, investigating the dynamic responses, absorbed energy, collision forces, and wheel vertical rises of the full-scale train collision. Automatic differentiation and parallelization algorithms are applied to the multi-body dynamic equations including mass, damping, and stiffness matrices, as well as the. The residuals and the initial conditions are included in the loss function. The dynamic responses, absorbed energy, collision forces, and wheel vertical rise of the full-scale train collision are simulated and studied in detail. The results obtained from the PPINN method are in excellent agreement with those obtained from the finite element method (FEM), Newmark-β, and fourth-order Runge–Kutta methods for all four train collision scenarios. Additionally, the PPINN methodology keeps better stability even under large time steps which implies a big potential for computational cost reduction.

Internal Ballistic Modeling and Simulation Analysis of High-low Pressure Low-overload Launch of Unmanned Aircraft

The low-overload launch is an important research direction in weapon launch. In this paper, a new low-overload launch method is designed by using the piston-type high-low pressure launch principle combined with the secondary launch principle. Based on the classical internal ballistic equations and multi-body dynamics equations, the internal ballistic model of this low overload launch system is established. A joint simulation model based on MATLAB and RecurDyn is constructed to simulate the internal ballistic state during the launch process. The sensitivity of each parameter change of the high-low pressure system to the launch overload is also studied. The results show that the maximum low-pressure chamber pressure of the low overload launcher in this paper is reduced by 62% compared to the conventional internal ballistic pressure, the saturation of the pressure curve is also improved, and the muzzle velocity is accelerated; the maximum overload is reduced by 65%, which can meet the requirements of muzzle initial velocity and low overload when launching unmanned aircraft.

Attitude Error and Contact Influencing Characteristic Analysis for a Composite Docking Test Platform

The design and analysis of a new type of three-jaw docking mechanism capable of space rendezvous and docking are presented. In addition, a composite docking test platform capable of both vertical and horizontal docking is created. On the basis of kinematics theory, a global coordinate system is built, and the attitude error is assessed based on the error angle. On the basis of dynamic theory, the multi-body dynamic differential equation of the composite docking platform is derived, and the impact-induced interaction state of the locking pawls is studied. The simulation software is then used to jointly simulate the test platform and the docking mechanism under the two conditions of frontal and oblique docking, and to analyze the attitude law caused by the change of docking impact force. This provides a solid foundation for future research into the application of space rendezvous theory to small spacecraft.

On the Use of Half-Implicit Numerical Integration in Multibody Dynamics

Abstract This work highlights the use of half-implicit numerical integration in the context of the index three differential algebraic equations (DAEs) of multibody dynamics. Although half-implicit numerical integration is well established for ordinary differential equations problems, to the best of our knowledge, no formal discussion covers its use in the context of index three DAEs of multibody dynamics. We wish to address this since when compared to fully implicit methods, half-implicit integration has two attractive features: (i) the solution method does not require the computation of the Jacobian associated with the constraint, friction, contact, or user-defined applied forces; and (ii) the solution is simpler to implement. Moreover, for nonstiff problems, half-implicit numerical integration yields a faster solution. Herein, we outline the numerical method and demonstrate it in conjunction with three mechanisms. We report on convergence order behavior and solution speed. The Python software developed to generate the results reported is available as open in a public repository for reproducibility studies.

A nonsmooth method for spatial frictional contact dynamics of flexible multibody systems with large deformation

AbstractA nonsmooth contact dynamics computation methodology for spatial frictional contact dynamics of flexible multibody systems with large deformation is presented. The contact and friction conditions between contact bodies are formulated as a cone complementary problem (CCP). Based on the mortar discretization method, nonsmooth frictional contact formulations for surface‐to‐surface and beam‐to‐beam contact problems are derived. These formulations can be used to study a wide range of contact problems between three‐dimensional bodies, thin plates and beams. The nonsmooth multibody dynamics equations are solved by using the generalized‐ method. The CCP is solved by using the accelerated projected gradient descend (APGD) algorithm in each Newton's iteration. Finally, six numerical examples are provided to validate the proposed computation methodology.

Feedback tracking control of optimal reference trajectories for spacecraft relative motion

In this paper, first the optimal trajectories are defined by a fixed end-time, continuous-thrust trajectory which minimizes a terminal cost at the end of an inter-planetary trajectory by appending the multi-body dynamic equations defining the motion by a set of Lagrange multipliers or co-states. The solution of these co-states and the optimal trajectories for a fixed end-time are obtained by solving a two-point boundary value problem along with a steering control law, which is used to provide a reference trajectory. Inspired by the general expression for the gravitation potential and the Q-law, a tracking control law is designed to track the reference trajectory. As the steering controller is not a feedback controller, a feedback tracking control law is proposed to continuously correct the spacecraft's actual relative motion trajectory so it tracks the reference relative motion trajectory. The nonlinear relative motion tracking control law is reduced to a linear feedback control law, when the spacecraft is sufficiently close to the reference relative motion trajectory. The feedback laws demonstrate the feasibility of developing practical tracking laws for continuous trajectory correction manoeuvre control of spacecraft relative motion, for implementing low-thrust inter-planetary trajectory following control in electrically propelled spacecraft, in practice.

Multi-body modelling and adaptive control of a re-configurable uninhabited airship

This paper presents pitch control of an uninhabited airship using changes in center of gravity. Changes in center of gravity are achieved by controlling the position of a movable gondola along a rigid keel fixed to the bow of the helium envelope. The longitudinal multi-body dynamic equations of motion were developed using the measured and estimated physical properties and an adaptive PID controller was designed to control the pitch angle. Flight tests on a 4 m long uninhabited airship prototype were conducted to demonstrate the effectiveness of the method.

Kinematic characteristics of longitudinal double folding wings

ABSTRACTA folding wing is a tactical missile launching device that needs to be miniaturised to facilitate storage, transportation, and launching; save missile and transportation space; and improve the combat capability of weapon systems. This study investigates the aeroelastic characteristics of the secondary longitudinal folding wing during the unfolding process. First, the Lagrange equation is used to establish the structural dynamics model of the folding wing, the kinematics characteristics during the deformation process are analysed, and the unfolding movement of the folding wing is obtained using the dynamic equations in the process. Then, the generalised unsteady aerodynamic force is calculated using the dipole grid method, and the multi-body dynamics equation of the folding wing is obtained. The initial angular velocity required for the deployment of the folding wing is analysed through structural model simulation, and the influence of the initial angular velocity on the opening process is studied. Finally, aeroelastic flutter analysis is performed on the folding wing, and the physical model of the folding wing verified experimentally. Results show that the type of aeroelastic response is sensitive to the initial conditions and the way the folding wing opens.

On the use of multibody dynamics techniques to simulate fluid dynamics and fluid–solid interaction problems

A multibody dynamics-based solution to the fluid dynamics problem is compared herein to two established Lagrangian-based techniques used by the computational fluid dynamics (CFD) community. The multibody dynamics-based solution has two salient attributes: it enforces the incompressibility condition through bilateral kinematic constraints, and it treats the coupling with the solid phase via unilateral kinematic constraints. The multibody dynamics-based solution, called herein the Kinematically Constrained Smoothed Particle Hydrodynamics (KCSPH) method, is a Lagrangian approach to solving the CFD problem. It relies on the Smoothed Particle Hydrodynamics (SPH) method to discretize the spatial differential operators in the Navier–Stokes equations, and on the Newton–Euler equations of multibody dynamics to convect the SPH particles forward in time. We show that the multibody dynamics-based approach is efficient and accurate by comparing its performance with the two most commonly used SPH algorithms in the CFD community: the weakly compressible SPH (WCSPH), and the implicit SPH (ISPH) methods. The comparison is carried out in conjunction with four tests: an incompressibility benchmark test, dam break, floating cylinder, and sloshing tank. We conclude that KCSPH is a robust alternative to conventional CFD approaches for fluid–solid interaction (FSI) problems with complex/moving boundaries. The solvers and models used herein are publicly available in an open-source software called Chrono; the implementations use GPU (for WCSPH and ISPH), and multicore CPU (for KCSPH) parallel computing.

Multibody Dynamics on Differentiable Manifolds

Abstract Topological and vector space attributes of Euclidean space are consolidated from the mathematical literature and employed to create a differentiable manifold structure for holonomic multibody kinematics and dynamics. Using vector space properties of Euclidean space and multivariable calculus, a local kinematic parameterization is presented that establishes the regular configuration space of a multibody system as a differentiable manifold. Topological properties of Euclidean space show that this manifold is naturally partitioned into disjoint, maximal, path connected, singularity free domains of kinematic and dynamic functionality. Using the manifold parameterization, the d'Alembert variational equations of multibody dynamics yield well-posed ordinary differential equations of motion on these domains, without introducing Lagrange multipliers. Solutions of the differential equations satisfy configuration, velocity, and acceleration constraint equations and the variational equations of dynamics, i.e., multibody kinematics and dynamics are embedded in these ordinary differential equations. Two examples, one planar and one spatial, are treated using the formulation presented. Solutions obtained are shown to satisfy all three forms of kinematic constraint to within specified error tolerances, using fourth-order Runge–Kutta numerical integration methods.

The Road Spectrum of Combine Harvester Based on Virtual Iteration

The combine harvester has a complex work environment, and the reliability of the rack is critical to the functionality of the machine. In order to shorten the test time, reduce the cost of research and development, and make the road spectrum can be reproduced on the test-bed, this paper takes the combine harvester as the research object, the acceleration spectrum was measured on asphalt road, hard road in field and wheat field, and the multi-body dynamic equation was established according to the actual parameters of the vehicle. The transfer function between the tire displacement signal and the target signal is established, and the tire displacement spectrum is calculated. The road spectrum is taken as the input load of the four-column test-bed, and the actual load condition is reproduced on the test-bed, which provides the input condition for the fatigue analysis of the combine harvester rack in the later stage, and greatly enhances the efficiency of durability design.

Multi Body Dynamic Equations of Belt Conveyor and the Reasonable Starting Mode

Based on the rigid finite element method and multibody dynamics, a discrete model of a flexible conveyor belt considering the material viscoelasticity is established. RFE (rigid finite element) and SDE (spring damping element) are used to describe the rigidity and flexibility of a conveyor belt. The dynamic differential equations of the RFE are derived by using Lagrange’s equation of the second kind of the non-conservative system. The generalized elastic potential capacity and generalized dissipation force of the SDE are considered. The forward recursive formula is used to construct the conveyor belt model. The validity of dynamic equations of conveyor belt is verified by field test. The starting mode of the conveyor is simulated by the model.

Integral dynamics modelling of chain-like space robot based on n-order dual number

In this study, n-order dual numbers were introduced to derive novel dynamics equations of chain-like multi-body deployed on spacecraft. A multi-body system dynamics equation was proposed in the Newton-Euler form, after dual motor, dual mass and dual momentum were provided by establishing operator σ and operation properties. Translation and rotation motion around base are described in the presented integral models based on n-order dual number. Finally, a chain-like space robot manipulator was designed as a verifying example of the n-order dual number scheme applied to multi-body system dynamics. The simulation results showed that the dynamics of each joint was simplified and the coupling effect between the links was reduced.

Frictional impact analysis of an elastoplastic multi-link robotic system using a multi-timescale modelling approach

Considering tangential contact compliance, material nonlinearity and contact nonlinearity, an efficient multi-timescale computational approach (MCA) is presented to analyse the contact forces and transient wave propagation generated by the frictional impact of an elastoplastic multi-link robotic system. In the formulation of the MCA, a rigid multibody dynamic equation is derived to calculate the large overall motion without a unilateral contact constraint (large timescale) and the nonlinear finite element dynamic equation, including the penalty function algorithm, is given to calculate the contact force and stress wave propagation (small timescale). An experimental test for a manipulator’s oblique impact against a rough moving plate is also introduced. The accuracy, convergence and efficiency of the MCA are validated by a comparison with the pure finite element method (FEM) solution and experimental data. The error between the MCA and pure FEM solutions for the first peak value of the normal contact force $$F_{\mathrm{n}}$$ is less than 1.5%. The total time cost of the MCA is only 0.705% of the pure FEM. The numerical results also show that the presented MCA can effectively depict the transmission and reflection of the impact-induced waves at the middle hinge. In addition, the influences of the structural compliance, the velocity of plate $$v_{\mathrm{plate}}$$ and the coefficient of friction $$\mu $$ on the transient responses are investigated. The peak value of $$F_{\mathrm{n}}$$ will increase as $$v_{\mathrm{plate}}$$ increases, which has a strong relationship with the so-called dynamic self-locking phenomenon. Due to the large structural compliance of the rod, the “succession collisions” phenomenon (i.e. multiple contacts in a very short time) can be captured during an oblique impact event, and the normal relative motion at the local contact zone may experience two transitions between compression and restitution during a single contact process. The tangential stick-slip motion occurs as a reverse phenomenon. All investigations show that the MCA has sufficient accuracy to analyse the frictional impact of flexible robotic manipulators.

L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics

An L-stable method over time intervals for differential-algebraic equations (DAEs) of multibody system dynamics is presented in this paper. The solution format is established based on equidistant nodes and nonequidistant nodes such as Chebyshev nodes and Legendre nodes. Based on Ehle’s theorem and conjecture, the unknown matrix and vector in the L-stable solution formula are obtained by comparison with Pade approximation. Newton iteration method is used during the solution process. Taking the planar two-link manipulator system as an example, the results of L-stable method presented are compared for different number of nodes in the time interval and the step size in the simulation, and also compared with the classic Runge-Kutta method, A-stable method, Radau IA, Radau IIA, and Lobatto IIIC methods. The results show that the method has the advantages of good stability and high precision and is suitable for multibody system dynamics simulation under long-term conditions.

The constraint-stabilized implicit methods on Lie group for differential-algebraic equations of multibody system dynamics

The implicit methods on Lie group are developed for multibody system dynamics. To avoid the violation of the displacement, velocity and acceleration constraints of the nonlinear differential-algebr...

Dynamic Analysis of a 5-DOF Flexure-Based Nanopositioning Stage

A multibody dynamic model is developed for dynamic analysis of a 5-DOF flexure-based nanopositioning stage in the projection optical system of the semiconductor lithography in this paper. The 5-DOF stage is considered as an assembly of rigid bodies interconnected by elastic flexure hinges. Considering the length effects of flexure hinges, multibody dynamic equations are established according to spatial motions of rigid bodies by using Lagrangian method. The shear effects and the torsional compliances of the commonly used circular flexure hinges are considered to enhance the modeling accuracy. The accuracies of various out-of-plane compliance formulas are also discussed. To verify the developed dynamic model, the finite element analyses (FEA) by using ANSYS and modal hammer experimental tests of the primary flexure-based composition structures and the integral 5-DOF stage are performed. The analytical modal frequencies are well in agreement with FEA and experimental test. The results are significant to analyze and optimize the 5-DOF flexure-based nanopositioning stage.