Dynamic investigation of a spatial multi-body mechanism considering joint clearance and friction based on coordinate partitioning method

The dynamic response of the model, which is the series connection of a planar four-bar mechanism and a spatial RSSR mechanism, is analyzed considering revolute joint clearance and friction. A non-holonomic constraint equation is proposed to solve the Euler angles. The dynamic equations are established by combining the Lagrange equation with the modified contact model and the LuGre friction model. A dynamic solution program based on the coordinate partitioning method is designed to solve the dynamic equations. The paper verifies the correctness and applicability of the solution program by comparing the numerical calculation results with Adams simulation. Compared with the results of eccentricity, it is found that the maximum penetration is very sensitive to the change of the slider speed rather than the clearance. The equivalent damping coefficient proposed by authors not only represents whether a collision occurs, but reflects the hysteresis caused by damping. The macroscopic manifestation of the up and down oscillation of eccentricity is the swing of the contact point. Besides, the system quickly changes from the collision into the stable state due to considering friction, and the peak value of each collision reduces greatly. Therefore, when the clearance is unavoidable, the clearance joint should be coated with a material with a large friction coefficient and not easy to wear.

A Generic and Modular Modeling Approach for Automotive Drivetrains Using a Coordinate Partitioning Method

Drivetrain models play an important role in state-of-the-art automotive drivetrain and control concept development. Based on a proposed set of elementary drivetrain components, this article contributes a generic straightforward approach to compute state-space models for various geared drivetrain layouts, including complex hybrid multi-mode transmissions. The modular approach follows Lagrange formalism: The free motion of rigid shafts is subsequently constrained, considering connecting elements like spur and planetary gear sets. The generalized coordinates are determined by a coordinate partitioning method, ensuring a physically reasonable coordinate system. The proposed approach features high potential for automation. This enables drivetrain modeling by non-experts in the field of mechanical engineering.

A Computational Method for Formulation and Solution of Dynamical Equations for Complex Mechanisms and Multibody Systems

This paper presents a computational method for formulating and solving the dynamical equations of motion for complex mechanisms and multibody systems. The equations of motion are formulated in a preconditioned form using kinematic substructuring with a heuristic application of Generalized Coordinate Partitioning (GCP). This results in an optimal split of dependent and independent variables during run time. It also allows reliable handling of end-of-stroke conditions and bifurcations in mechanisms, thereby facilitating dynamic simulation of paradoxical linkages such as Bricard’s mechanism that has been known to cause problems with some multibody dynamic codes. The new Preconditioned Equations of Motion are then solved using a recursive formulation of the Schur Complement Method combined with Sparse Matrix Techniques. In this fashion the Preconditioned Equations of Motion are recursively uncoupled and solved one kinematic substructure at a time. The results are demonstrated using examples.

Direct and Adjoint Sensitivity Analysis of Multibody Systems Using Maggi’s Equations

The importance of the sensitivity analysis of multibody systems for several applications is well known, concretely design optimization based on the dynamics of multibody systems usually requires the sensitivity analysis of the equations of motion. A broad range of methods for the dynamics of multibody systems include the state space formulations based on Maggis equations, nullspace methods or coordinate partitioning. Dynamic sensitivities, when needed, are often calculated by means of finite differences but, depending of the number of parameters involved, this procedure can be very demanding in terms of CPU time and the accuracy obtained can be very poor in many cases. In this paper, several ways to perform the sensitivity analysis are explored and analytical expressions for the direct and adjoint sensitivity analysis of multibody systems are presented, all of them based on Maggi’s formulations. Moreover, two different approaches to the adjoint sensitivity analysis of multibody systems are presented. Although particularized to one formulation, the general expressions provided in the paper, are intended to be easily generalized and applied to any other formulation that can be expressed as an ODE-like system of equations, including penalty formulations. Besides, to check the validity and correctness of the proposed equations, the solutions of all the methods proposed are compared: 1) between them, 2) with the third party code FATODE and 3) with the numerical solution using real and complex perturbations. Finally, all the techniques proposed are applied to the dynamical optimization of a multibody system.

An Alternative Formulation of Motion Equations in Redundant Coordinates for the Inverse Dynamics of Constrained Mechanical Systems

The basis for any model-based control of dynamical systems is an efficient formulation of the motion equations. These are preferably expressed in terms of independent coordinates. In other words the coordinates of a constrained system are split into a set of dependent and independent ones. It is well-known that such coordinate partitioning is not globally valid. A remedy is to switch between different possible sets of minimal coordinates. This drastically increases the numerical complexity and implementation effort, however. In this paper a formulation of the motion equations in redundant coordinates is presented for general non-holonomic systems. This gives rise to a redundant system of differential equations. The formulation is valid in any regular configuration. Because of the singular mass matrix it is not directly applicable for solving the forward dynamics but is tailored for solving the inverse dynamics. An inverse dynamics solution is presented for general full-actuated systems.

Optimization Problem in Biomolecular Simulations With DCA-Based Modeling of Transition From a Coarse to a Fine Fidelity

In multiscale modeling of highly complex biomolecular systems, it is desirable to switch the system model either to coarser, or higher fidelity models to achieve the appropriate accuracy and speed. These transitions are achieved by effectively imposing (or releasing) certain systems constraints from a fine scale model to a reduced order model (or vice versa). The transition from a coarse model to a fine one may not result in a unique solution. Therefore, a knowledge-based or physics-based optimization procedure may be used to arrive at the finite number of solutions. In this paper, it is shown that traditional approaches to address and solve the optimization problem such as Lagrange multipliers or changing the constrained optimization problem to an unconstrained one based on coordinate partitioning or basic linear algebra methods are computationally expensive for biomolecular systems. It is demonstrated that using a DCA based approach in modeling the transition can reduce dramatically the computational expense associated with the manipulations performed as part of optimization as well as the ones performed to derive the dynamics of the transition.

Analysis and Verification of a Watt I Six-Bar Furniture Hinge Mechanism

The analysis and verification of a Watt I six-bar furniture hinge mechanism were carried out in this article. Using the characteristics of the topological structure of existing hinge mechanisms and design requirement specification, six hinge designs were synthesized. The most suitable design was selected for kinematic analysis with the vector loop method. The equations of motion of the mechanism were then derived by the Hamilton principle and Lagrange undetermined multipliers method and simplified using coordinate partitioning method. The differential algebraic equation was subsequently solved by using the Runge-Kutta method. From numerical simulation, the dynamic response of the door under various loading conditions was obtained and was found to fulfil the design requirements and constraints. Results obtained from experiments and numerical simulation were comparable, which demonstrated the success of this study. It is anticipated that this research will be beneficial to the further development of hinge designs for use in furniture.

Recursive Coordinate Reduction: An Addendum

This paper presents an addendum to the Recursive Coordinate Reduction (RCR) algorithm for the efficient numerical analysis and simulation of modest to heavily constrained multi-rigid-body dynamic systems. The RCR algorithm can accommodate the spatial motion of broad categories of multi-rigid-body systems containing arbitrarily many closed loops in O(n + m) operations overall for systems containing n generalized coordinates, and m independent algebraic constraints. The presented approach does not suffer from the performance (speed) penalty encountered by most other of the so-called “(n)” state-space formulations, and does not require additional constraint violation stabilization procedures (e.g. Baumgartes method, coordinate partitioning, etc.). Due to these characteristics, the presented algorithm should offer superior computing performance relative to other methods in many situations involving both large n and m. This paper will specifically address an unpublished recursive step in the handling of “floating” loop base bodies, as well as present an extension to “spur” topologies.

On the Determination of Joint Reactions in Multibody Mechanisms

In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms—no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.