Computational structural analysis of spatial multibody systems based on mobility criteria

<div class="htmlview paragraph">In this paper methods for the nonlinear dynamic analysis of multibody vehicle systems are presented. A total Lagrangian finite element formulation for deformable bodies of the vehicle systems that undergo large translational and rotational displacements is developed. A set of nonlinear dynamic differential equations are developed in a closed form for each finite element in terms of the reference as well as the nodal elstic coordinates and their time derivatives. These equations are expressed in terms of a set of element invariants that depend on the assumed displacement field. The invariants of the deformable body that undergoes large reference translational and rotational displacements can be obtained by assembling the invariants of its elements using a standard finite element procedure. The nonlinear dynamic equations of the deformable body can then be obtained by assembling the nonlinear element equations. Both the consistent and lumped mass finite element formulations are discussed. The dynamic formulations presented in this paper are exemplified using multibody vehicle systems that consist of interconnected rigid and deformable bodies. Numerical results are obtained using the general purpose computer program <b>DAMS</b> (Dynamic Analysis of Multibody Systems).</div>

Methods and identities for computation of kinematic and kinetic derivatives required for a broad spectrum of multibody system analyses are presented. Analyses such as implicit numerical integration of the differential–algebraic equations of multibody dynamics, dynamic sensitivity analysis, and workspace analysis are shown to require computation of three derivatives of algebraic constraint functions and first derivatives of inertia and force expressions. Computationally efficient derivative calculation methods and associated identities are presented for Cartesian generalized coordinates, with Euler parameters for orientation. Results presented enable practical and efficient computation of all derivatives required in multibody mechanical system analysis.

Analytical formulas for kinematic and kinetic derivatives needed in multibody system analyses are derived. A broad spectrum of problems, including implicit numerical integration, dynamic sensitivity analysis, and kinematic workspace analysis, require evaluation of first derivatives of generalized inertia and force expressions and at least three derivatives of algebraic constraint functions. In the setting of a formulation based on Cartesian generalized coordinates with Euler parameters for orientation, basic identities are developed that enable practical and efficient computation of all derivatives required for a large number of multibody mechanical system analyses. The formulation is verified through application to a spatial slider crank mechanism and a 14 body vehicle model. Efficiency of computation using the expressions derived is compared with results obtained employing finite differences, showing significant computational advantage using the analytically derived expressions.

This paper presents the validation of a novel code for the dynamic analysis of flexible multibody systems, which leverages projective geometric algebra for the kinematic description of the system. The theoretical foundations of geometric algebra are introduced briefly. A closed-form solution for the interpolation of motion is derived within the proposed framework and a finite element formulation of geometrically exact beams is developed. The accuracy of this code is assessed by comparing its prediction with those of eight multibody dynamics codes developed by independent researchers. Results are presented for three benchmark problems: the flexible four-bar mechanism, the lateral buckling of a thin beam, and the stability of a rotating shaft. Excellent agreement is found between the predictions of the proposed code and the previously published results, confirming the accuracy of the PGA-based approach. These findings demonstrate the potential of geometric algebra as a robust and elegant basis for the analysis of complex multibody systems.

This paper presents a generalization of the divide and conquer algorithm for sensitivity analysis of dynamic multibody systems based on direct differentiation. While similar sensitivity analysis approach has been demonstrated for multi-rigid and multi-flexible systems in tree topologies and a limited set of kinematically closed loop topologies, this paper presents the generalization of these approaches to systems in generalized topologies including many coupled kinematically closed loops. This generalization retains the efficient complexity of the underlying formulations i.e. linear and logarithmic complexity in serial and parallel implementation. Other than the computational efficiency, the advantages of this method include concurrent sensitivity analysis with forward dynamics, no numerical artifacts arising from parametric perturbation and significantly reduced data storage compared to traditional methods. An interesting application of this work in control of multibody systems is discussed.

Clearance is inevitable for manufacture and assembly in the revolute joints of multibody systems. Excessive value of joint clearance will lead to the poor dynamic performance of mechanism, namely noise, vibration and fatigue failure. This paper presents the development of a dynamic model for planar multibody systems with clearance joints. Based on the continuous contact law, the contact force model proposed by Lankarani and Nikravesh is employed to describe the contact–impact phenomenon between pin and hole. And, the incorporation of the friction influence on revolute joint clearance is conducted by a modified friction force model. According to the geometric relationship between contact elements, the kinematics of the multibody systems is mapped into the global coordinate systems. Additionally, an experimental test platform is set up whereby a slider–crank mechanism with two clearance joints is used as an example to demonstrate the correctness and effectiveness of the proposed approach. Finally, the effects of joint clearance on the dynamic characteristics of planar multibody systems are investigated.

An equivalent stiffness approach for modelling the behaviour of compression members according to Eurocode 3

The kinematic and dynamic analysis of the high-degree of freedom (dof) hybrid systems that using serial and parallel robots together on fixed or mobile platform such as legged robots and service robots, is a rather difficult and complex, time consuming problem and a topic of great interest for researchers. There are many educational and commercial toolboxes which analyze serial and parallel robots separately. With the toolbox developed here, high-dof serial, parallel and hybrid robotics systems used on both fixed and mobile platforms can be designed and studied easily. Forward and inverse kinematics, Jacobian matrix, singularities, trajectory planning, torques and forces in joint and operational space can be examined with different robot designs. Using the toolbox that proposed in this paper, robot designer, researchers, students and educators can easily develop their own robotics systems and visualize them in the virtual reality environment provided by MATLAB. Also, it helps understanding fundamentals of high-dof multi-body hybrid robotics systems through user friendly interactive simulation. The most of existing toolboxes in the literature allow only user to analyze the predefined robots in their library with limited interactive capabilities. In some of them, the user can define a new robot with limited dof. But, the user must already have a prior knowledge about designing process of robots. In the proposed toolbox, designing and defining any new robotics systems become very easy. What makes this toolbox unique compared with the existing toolboxes is that any type of robot structures with high dof can be developed, analyzed and studied in detail.

Optimal design of multibody systems is of primary importance to researchers working in various fields. The goal of the paper is to present a novel Hamiltonian based approach for finding design sensitivities of multibody systems (MBS) through the use of direct differentiation method. The total derivatives of Hamiltonian equations of motion (EOM) based on the augmented Lagrangian in terms of the design are determined in the text. An algorithm for the evaluation of a gradient of an objective function is also delivered. Sample test case is presented that demonstrates the validity of the proposed approach.

In this work, a methodology for dynamic analysis of rigid-flexible multibody systems with elastohydrodynamic (EHD) lubricated joints is presented. The EHD lubricated cylindrical joint is formulated by the Natural Coordinate Formulation (NCF) and the twenty-node hexahedral element of Absolute Nodal Coordinate Formulation (ANCF), being the lubricant pressure determined through the resolution of the Reynolds’ equation employing the finite difference method. The outcomes are validated with those obtained by using the commercial software ADINA. It is demonstrated that the bearing flexibility plays a significant role in the system responses, extends the lubricant distribution space and reduces the lubricant pressure.

This chapter provides a brief and very selective survey of some software useful for solving problems in analytical dynamics. I have chosen to include only software intended for academic and industrial use that has been validated for modeling accuracy. Consequently, I have omitted the many dynamics engines used for video games and entertainment purposes. General Purpose Mathematical Software A number of powerful tools for mathematical analysis exist. Two of the most popular are Mathematica by Wolfram Research Inc. and MATLAB by The MathWorks Inc. They provide capabilities for both symbolic manipulation of expressions (i.e., computer algebra systems) and numerical computation and evaluation of expressions. While the basic Mathematica and MATLAB applications do not provide high-level programming abstractions specific to multibody system modeling and analysis, they are extremely useful for the low-level symbolic and numerical processing required in solving problems in analytical dynamics (see Figure 10.1 for a screenshot of Mathematica). This has a benefit in the learning process, as they require the student to understand the central concepts of analytical dynamics and to set up the solution procedure in a detailed manner, while alleviating much of the tedium of the algebraic and numerical work. Packages and Extensions for Multibody Dynamics Wolfram Research and MathWorks both provide domain-specific extensions for analyzing multibody systems. This gives the user access to high-level abstractions for modeling, simulating, and analyzing multibody systems. These high-level abstractions can be accessed through a programming language or through a diagrammatic interface supporting both acausal and causal modeling. Acausal modeling involves component-based schematic representations of the physical system, whereas causal modeling involves block diagram representations of the underlying mathematical description of the system. Simscape Multibody (formerly SimMechanics) by The MathWorks Inc. allows users to model multibody systems using a library of blocks associated with bodies, joints, constraints, spatial transformations, force elements, and sensors (see Figure 10.2 for a screenshot of Simscape Multibody). Entire CAD assemblies can be imported along with associated geometry, inertial properties, joint parameters, and constraints. Simscape Multibody then formulates and integrates the equations of motion for the multibody system. Animations are generated during simulation to allow visualization of the motion of the system.

This work presents a novel motion planning framework, rooted in nonlinear programming theory, that treats uncertain fully and underactuated dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. Treatment of uncertainty in design is of paramount practical importance because all real-life systems are affected by it, and poor robustness and suboptimal performance result if it is not accounted for in a given design. In this work uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The computational efficiency of this approach enables the inclusion of uncertainty statistics in the nonlinear programming optimization process. As such, the proposed framework allows the user to pose, and answer, new design questions related to uncertain dynamical systems. Specifically, the new framework is explained in the context of forward, inverse, and hybrid dynamics formulations. The forward dynamics formulation, applicable to both fully and underactuated systems, prescribes deterministic actuator inputs that yield uncertain state trajectories. The inverse dynamics formulation is the dual to that of forward dynamics, and is only applicable to fully actuated systems; deterministic state trajectories are prescribed and yield uncertain actuator inputs. The inverse dynamics formulation is more computationally efficient as it requires only algebraic evaluations and completely avoids numerical integration. Finally, the hybrid dynamics formulation is applicable to underactuated systems where it leverages the benefits of inverse dynamics for actuated joints and forward dynamics for unactuated joints; it prescribes actuated state and unactuated input trajectories that yield uncertain unactuated states and uncertain actuated inputs. The benefits of the ability to quantify uncertainty when planning the motion of multibody dynamic systems are illustrated through several case studies. The resulting designs determine optimal motion plans—subject to deterministic and statistical constraints—for all possible systems within the probability space.

We have analyzed the computational dynamics of a complex one-dimensional structural system consisting of a number of alternating stiff and soft subsystems. In particular, the method of Proper Orthogonal Decomposition (POD) in the frequency domain has been applied to analyze the (single-frequency) steady state dynamics in terms of spectral amplitudes and spatial shapes of proper orthogonal modes. It is shown that the dominant POD modes of such a multi-body system are sensitive to imperfections. The processing of the computational dynamics by the spectral-POD method leads to a novel computation of the transfer function of the system.

This paper describes a programme of tyre tests using a purpose built tyre test rig designed specifically to investigate the behaviour of tyres that are used with Road-Rail Vehicles (RRVs). These vehicles are used extensively by the rail industry to support the maintenance of existing rail lines and new construction works. In many cases, using a vehicle that can only operate on the road, or a rail engine that can only operate on tracks will not provide a suitable or effective solution for the task in hand. RRVs are able to operate on both roads and rail, and as such provide a very flexible solution to the rail industry. When operating on the road, RRVs use pneumatic tyres to control their motion and act to all intent and purpose as a normal road vehicle. When operating on rail lines RRVs use a combination of the pneumatic tyres and standard steel rail wheels to control their motion. The rail wheels maintain the directional stability on the track and the pneumatic tyres provide the tractive force to drive and brake the vehicle. Unlike standard road or rail vehicles, there is to date no predictive engineering practice that allows the use of computer simulation to design and optimise the performance of RRVs when they are operating on rails. Computer tools, such as multi-body systems (MBS) analysis are used extensively to design both road and rail vehicles. For road vehicles a tyre model is needed to represent the behaviour in the contact patch between the tyre and the road. For rail vehicles a model is needed to represent the contact force between the train wheel and the track. In both these applications the behaviour is well understood and over the last half century mathematical models have been developed that allow accurate and useful simulation to support the design of new vehicles and trains. In contrast, RRVs have evolved essentially as modifications to standard road vehicles. While the base vehicle may be very well designed to perform on the road, the performance on rail is based on experience and some testing. There is no up-front science involved in designing a RRV to perform effectively during this very important phase of its operation. In order to develop an accurate model to predict RRV performance on rails it is clear that a model of the behaviour of the tyre when in contact with the rail is needed. To date, no such model exists and additionally the testing needed to generate data that could be used to develop a model has never been carried out. The work described here addresses this with the design and build of a unique test rig that can be used to test tyres on rails for a range of operating conditions, and produce a set of initial results that provide a framework for a future tyre/rail model. The paper concludes with a review of the behaviour measured and provides new insights into how well tyres perform on rails and also where the behaviour differs from the well understood behaviour of tyres operating on roads. The important behaviours that would provide the main parameters for a new empirical tyre model are also identified and discussed.

The Shrimp rover is highly suitable for planetary exploration missions because of its unconventional wheel order, in-built passive adaptability and good ability to climb obstacles. It is a spatial multi-body system and a multi-variable, multi-parameter coupled non-linear system. Thus, kinematic and dynamic analyses for such systems are complex and time consuming. Long-range robotic missions for Martian exploration imply a high degree of autonomy. The most advanced locomotion concepts are based on wheels or caterpillars (e.g. Sojourner, NASA or Nanokhod, ESA). These rovers have clear advantages regarding power efficiency and complexity if compared with walking robots. However, they still have quite limited climbing abilities. Typically they can only overcome obstacle smaller than their wheel size. In this paper we present Shrimp, an innovative long range rover architecture with 6 motorized wheels. Using a rhombus configuration, the rover has a steering wheel in both, the front and the rear, and two wheels arranged on a bogie on each side. The front wheel has a spring suspension to guarantee optimal ground contact of all wheels at any time. The steering of the rover is realized by synchronizing the steering of the front and rear wheel and the speed difference of the bogie wheels. This allows for high precision maneuvers and even turning on the spot with minimum slip. The use of parallel articulations for the front wheel and the bogies enables to set a virtual centre of rotation at the level of the wheel axis while maintaining a high ground clearance. This insures maximum stability and climbing abilities even for relatively low friction coefficients between the wheel and the ground. This rover is able to passively overcome unstructured obstacles of up to two times its wheel diameter. With this high mobility, this architecture is the perfect candidate for long range planetary missions.