Complex Multibody Systems Research Articles

A regularization method for solving the redundant problems in multibody dynamic system

Redundant constraints may cause great difficulties during the numerical simulation, when solving the differential algebraic equations (DAEs). The article presents a regularization method based on optimization form of the Gauss principle of least constraint and the Udwadia and Kalaba (UK) method, which expresses the solution of the dynamic motion in an explicit expression. Compared with Lagrange method, the proposed method does not need to solve the Lagrange multipliers any more for the complex multibody mechanical systems. In the end, a numerical example is given to illustrate that the proposed method not only has the ability to handle the problems with redundant constraints, but also has higher accuracy and keep a stable constraint violation level than the traditional methods.

Harmonic wavelets based response evolutionary power spectrum determination of linear and nonlinear structural systems with singular matrices

A new approximate analytical technique is proposed for determining the response evolutionary power spectrum (EPS) of stochastically excited structural multi-degree-of-freedom (MDOF) linear and nonlinear systems with singular matrices. Such systems can appear, indicatively, when a redundant coordinates modeling is adopted for forming the equations of motion of complex multi-body systems. For this case, it can be argued that this modeling approach facilitates the system’s stochastic response analysis, since employment of redundant DOFs is associated with computational cost efficient solution frameworks, and potentially provides with enhanced modeling flexibility. In this context, aiming at the joint time–frequency response analysis of MDOF systems, recently developed wavelet-based solution frameworks, which generalize classic input–output relationships of random vibration, are adopted and further generalized in this paper to account for systems with singular matrices. Specifically, resorting to the theory of generalized inverses of singular matrices, as well as to the theory of harmonic wavelets, a Moore–Penrose generalized matrix inverse excitation-response relationship is derived herein for determining the response EPS of linear MDOF systems. Further, a recently developed harmonic-wavelet-based statistical linearization technique is also generalized to account for the case of nonlinear MDOF systems. The validity of the proposed technique is demonstrated by pertinent numerical examples.

Extended state observer based smooth switching control for tilt-rotor aircraft

A tilt-rotor aircraft has three flight modes: helicopter mode, airplane mode and conversion mode. Unlike the traditional aircraft, the tilt-rotor aircraft, which combines the characteristics of helicopters and fixed-wing aircraft, is a complex multi-body system with the violent variation of the aerodynamic parameters. For these characteristics, a new smooth switching control scheme is provided for the tilt-rotor aircraft. First, the reference commands for airspeed and nacelle angles are calculated by analyzing the conversion corridor and the conversion path. Subsequently, based on the finite-time switching theorem, an average dwell time condition is designed to guarantee the stability in the switching process. Besides, considering the state vibrations and bumps may appear in switching points, the fuzzy weighted logic is employed to improve the system transient performance. For disturbance rejection, three extended state observers are designed separately to estimate the disturbances in the switched systems. Compared with the traditional auto disturbance rejection control and proportion integration differentiation control, this method overcomes the conservatism of wasting the whole model information. The control performances of robustness and smoothness are verified with simulation, which shows that the new smooth switching control scheme is more targeted and superior than the traditional design method.

A new version of the Riccati transfer matrix method for multibody systems consisting of chain and branch bodies

Computational speed and stability are two important aspects in the dynamics analysis of large-scale complex multibody systems. In order to improve both in the context of the multibody system transfer matrix method, a new version of the Riccati transfer matrix method is presented. Based on the new version of the general transfer matrix method for multibody system simulation, recursive formulae are developed which not only retain all advantages of the transfer matrix method, but also reduce the truncation error. As a result, the computational speed, accuracy and efficiency are improved. Numerical computation results obtained by the proposed method and an ordinary multibody system formulation show good agreement. The successful computation for a spatial branch system with more than 100000 degrees of freedom validates that the proposed method is also working for huge systems.

On the Projection of a Flexible Bodies Modal Coordinates Onto Another Finite Element Model With Local Modifications

The time integration of a complex multibody system is a time consuming part of the entire evaluation process of a flexible component. A multibody simulation of a flexible crankshaft, for instance, interacting with pistons, con rods, fly wheel, hydrodynamic bearings and further takes several hours of central processing unit (CPU) time and may dominate the entire simulation chain. Small, local changes in the involved finite element (FE) models, for example, another notch radius, normally require a new time integration of the entire multibody system. In this publication, a remarkably simple method is presented, so that the multibody simulation of such a variant can be skipped entirely. Instead, a simple and cheap projection of the original results to the modified FE model is proposed. One simple and one elaborate example demonstrate the extraordinary resulting quality for minor design changes like notch radius variations.

Optimal reconfiguration with collision avoidance for a granular spacecraft using laser pressure

Granular spacecraft, as a new type of distributed spacecraft system, “are complex multibody systems composed of a spatially disordered distribution of a large number of elements” (Quadrelli et al., 2013), designed and controlled to perform certain desired functions. To achieve the collision-free reconfiguration of granular spacecraft, an optimal reconfiguration algorithm with collision avoidance for a granular spacecraft using laser pressure is presented in this paper. The three-dimensional model of granular spacecraft using laser pressure is established first, and the reconfiguration problem statement is presented. The optimal reconfiguration plan is designed based on optimal transport, and the collision-free trajectories for all the particles are generated based on Voronoi partitioning, with the motion constraints under the control of laser pressure. Finally, the numerical simulations are provided to demonstrate the effectiveness of the proposed reconfiguration algorithm and the feasibility of using laser actuators for the reconfiguration of a granular spacecraft.

Comprehensive analyses of the elasto-plastic oblique contact-impact with vibration response

Contact-impact between different surfaces is a ubiquitous phenomenon especially in the mechanical systems. Previous work of authors indicated that the effect of longitudinal and/or transverse vibration response during the motion and dynamic modeling cannot be neglected for the elasto-plastic contact-impact events. In this study, further analyses were performed to characterize the contribution of longitudinal and transverse vibration responses during the elasto-plastic oblique contact-impact, and the formula to calculate the influence factor of vibration, ξ, was proposed based on the contact force with different vibrations during the impact. The momentum theorem and assumed mode method were used to develop the equations of motion of a flexible bar with a solid flat surface. Simulation results were compared with experimental results reported in the literature to verify the accuracy of the established model. The tangential and normal velocities of the contact point after the impact were compared with the simulations for different vibration cases, and the comparison between the simulations and the experimental results had yield encourage results. For different elasto-plastic materials, three critical initial impact angles had been found from the simulation to determine whether the flexible bar slides or sticks the flat surface. Although considering the effect of vibration response during the motion and dynamic modeling is very significant, increasing the number of shape functions did not effect the result significantly. The longitudinal and transverse vibration responses were found to effect the normal and tangential velocities of the contact point after the impact, respectively. Moreover, it had been shown that for the oblique contact-impact with sliding, considering the response of longitudinal and transverse vibration at the same time is more reasonable than other cases, which also reveals that, in this case, there is a good agreement between the simulation and experimental results. Besides, the mathematical expression on the influence factor of vibration depending on the initial velocity also had been obtained by a numerical analysis. This work can provide useful insights for dynamic modeling of complex multi-body systems during the contact-impact.

Study on automatic deduction method of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems

The transfer matrix method for multibody systems has been developed for 20 years and improved constantly. The new version of transfer matrix method for multibody system and the automatic deduction method of overall transfer equation presented in recent years make it more convenient of the method for engineering application. In this article, by first defining branch subsystem, any complex multibody system may be regarded as the assembling of branch subsystems and simple chain subsystems. If there are closed loops in the system, the loops should be “cut off,” thus a pair of “new boundaries” are generated at each “cutting-off” point. The relationship between the state vectors of the pair of “new boundaries” may be described by a supplement equation. Based on above work, the automatic deduction method of overall transfer equation for tree systems as well as closed-loop-and-branch-mixed systems is formed. The results of numerical examples obtained by the automatic deduction method and ADAMS software for tree system dynamics as well as mixed system dynamics have good agreements, which validate the features of proposed method such as high computational speed, more effective for complex systems, no need of the system global dynamics equation, highly programmable, as well as convenient popularization and application in engineering.

A new perspective towards decomposition of the generalized inertia matrix of multibody systems

This paper presents a new perspective into the decomposition of the Generalized Inertia Matrix (GIM) of multibody systems with open kinematic architecture, serial or tree-type. Links and kinematic pairs are the two constituting elements of multibody systems. In this work, we propose to decompose a multi-branch multibody system into several kinematic modules. Each module is a set of serially connected links like a serial-chain system. Such a description allows one to obtain a block decomposition $\bar{\mathbf{U}}\bar{\mathbf{D}}\bar{\mathbf{U}}^{T}$ of the GIM where $\bar{\mathbf{U}}$ and $\bar{\mathbf{D}}$ are the block upper-triangular and diagonal matrices, respectively. The results provide a recursive inverse of the GIM on module-level. Many new perspectives leading to macroscopic purview of the complex multibody systems are provided. Empowered with the proposed decomposition, an inter- and intra-modular efficient and numerically stable recursive dynamics algorithm for forward dynamics and simulation was possible. While recursive expressions are derived for a four degree-of-freedom gripper, numerical results are shown for a spatial biped.

A tensor-based modelling and simulation method for multibody system

PurposeThe purpose of this paper is to provide a method with convenient modelling as well as precise computation to the research of complex multi-body system, such as robot arms and solar power satellite. Classical modelling method does not always fit these two requirements.Design/methodology/approachIn this paper, tensor coordinates (TC) and homogeneous tensor coordinates (HTC) method with gradient components are developed, which also have a convenient interface with classical theory.FindingsThe HTC proved its precision and effectiveness by two examples. In HTC model, equations have a more convenient form as matrix and the results coincide well with classical one.Research limitations/implicationsThere is no plenty detailed operations supported in mathematics yet, which may be developed in further research.Practical implicationsWith TC/HTC method, the research work can be separated more thoroughly: a simpler modelling work is left to scientists, when more computing work is handed to the computers. It may ease scientists’ brains in multibody modelling.Originality/valueThe HTC method has the advantages of absolute nodal coordinate formulations, tensor and homogeneous coordinate (HC) and it may be used in multibody mechanics, or other related engineerings.

Modeling of Two Link Flexible Manipulator using Bond Graph Technique

A mechanism involves a number of interconnected rigid bodies. To design and control a system, modeling of mechanism plays an important role .There are many methods used for modeling like Newton –Euler, Lagrange’s Euler and Hamiltonian method. These methods become very laborious and cumbersome for complex multibody systems. The above mention techniques however, do not expose the cause and effect relationship. In this paper, bond graph approach is used for modeling two link manipulator by considering the flexibility at the joints. Bond graph technique shows the pictorial representation of the dynamics of the system. Moreover it may also be used for modeling multi energy domain systems

Random vibration of linear and nonlinear structural systems with singular matrices: A frequency domain approach

A frequency domain methodology is developed for stochastic response determination of multi-degree-of-freedom (MDOF) linear and nonlinear structural systems with singular matrices. This system modeling can arise when a greater than the minimum number of coordinates/DOFs is utilized, and can be advantageous, for instance, in cases of complex multibody systems where the explicit formulation of the equations of motion can be a nontrivial task. In such cases, the introduction of additional/redundant DOFs can facilitate the formulation of the equations of motion in a less labor intensive manner. Specifically, relying on the generalized matrix inverse theory, a Moore-Penrose (M-P) based frequency response function (FRF) is determined for a linear structural system with singular matrices. Next, relying on the M-P FRF a spectral input-output (excitation-response) relationship is derived in the frequency domain for determining the linear system response power spectrum. Further, the above methodology is extended via statistical linearization to account for nonlinear systems. This leads to an iterative determination of the system response mean vector and covariance matrix. Furthermore, to account for singular matrices, the generalization of a widely utilized formula that facilitates the application of statistical linearization is proved as well. The formula relates to the expectation of the derivatives of the system nonlinear function and is based on a Gaussian response assumption. Several linear and nonlinear MDOF structural systems with singular matrices are considered as numerical examples for demonstrating the validity and applicability of the developed frequency domain methodology.

Active Vibration Control Design Method Based on Transfer Matrix Method for Multibody Systems

AbstractThe efficient vibration control of complex multibody systems is an important issue in engineering design and has drawn substantial attention in the field of multibody system dynamics and co...

Stabilization of Internal Dynamics of Underactuated Systems by Periodic Servo-Constraints

The model-based motion control of underactuated, multiple degree-of-freedom, complex multibody systems is in focus. Underactuated mechanical systems possess less number of independent control inputs than degrees-of-freedom. The main difficulty in their control is caused by the dynamics of the uncontrolled part of the system. The complexity of multibody systems makes the dynamical and control formulation difficult. The direct application of traditional control techniques available in the literature can lead to unstable dynamic behavior in many cases. In order to avoid instability, these general methods are usually adapted for specific problems in an intuitive way. Here, we present a direct, more algorithmic approach, and propose the use of periodic servo-constraints to overcome stability problems and enhance the dynamic behavior. An exact, stability analysis-based method is also proposed for tuning the control parameters. A stability analysis procedure is developed which is directly applicable for investigating the dynamics of mechanical systems described by dependent coordinates and mathematically formulated as a set of algebraic differential equations.

Exploiting Hybrid Parallelism in the Kinematic Analysis of Multibody Systems Based on Group Equations

Computational kinematics is a fundamental tool for the design, simulation, control, optimization and dynamic analysis of multibody systems. The analysis of complex multibody systems and the need for real time solutions requires the development of kinematic and dynamic formulations that reduces computational cost, the selection and efficient use of the most appropriated solvers and the exploiting of all the computer resources using parallel computing techniques. The topological approach based on group equations and natural coordinates reduces the computation time in comparison with well-known global formulations and enables the use of parallelism techniques which can be applied at different levels: simultaneous solution of equations, use of multithreading routines, or a combination of both. This paper studies and compares these topological formulation and parallel techniques to ascertain which combination performs better in two applications. The first application uses dedicated systems for the real time control of small multibody systems, defined by a few number of equations and small linear systems, so shared-memory parallelism in combination with linear algebra routines is analyzed in a small multicore and in Raspberry Pi. The control of a Stewart platform is used as a case study. The second application studies large multibody systems in which the kinematic analysis must be performed several times during the design of multibody systems. A simulator which allows us to control the formulation, the solver, the parallel techniques and size of the problem has been developed and tested in more powerful computational systems with larger multicores and GPU.

Closed form solution for the equations of motion for constrained linear mechanical systems and generalizations: An algebraic approach

In this paper, a mathematical methodology is presented for the determination of the solution of motion for linear constrained mechanical systems applicable also to systems with singular coefficients. For mathematical completeness and also to incorporate some other interesting cases, the methodology is formulated for a general class of higher order matrix differential equations. Thus, describing the system in an autoregressive moving average (ARMA) form, the closed form solution is derived in terms of the finite and infinite Jordan pairs of the system׳s polynomial matrix. The notion of inconsistent initial conditions is considered and an explicit formula for the homogeneous system is given. In this respect, the methodology discussed in the present note provides an alternative view to the problem of computation of the response of complex multi-body systems. Two interesting examples are provided and applications of the equation to such systems are illustrated.

Distributed parallel computing of the recursive eigenvalue search in the context of transfer matrix method for multibody systems

The modeling and solving a transcendental eigenvalue problem are important issues in the transfer matrix method for linear multibody systems. Based on the recursive eigenvalue search algorithm for transfer matrix method for linear multibody system, the distributed parallel approach for assembling overall transfer matrix and searching eigenvalues is proposed. This is achieved based on Message Parallel Interface. The influence of the CPU core number as well as the distributed network environment on the final computational time is analyzed through numerical examples of both a non-uniform beam and a multiple launch rocket system. The results indicate that the computational time is significantly reduced by the proposed parallel computing method, so that the computational efficiency on optimization and design of complex multibody systems can be improved.

Unified Approach to Modeling and Control of Rigid Multibody Systems

A unified approach is developed to model complex multibody mechanical systems and design controls for them. The characterization of such complex systems often requires the use of more coordinates than the minimum number to describe their configurations and/or the use of modeling constraints to capture their proper physical descriptions. When required to satisfy prescribed control requirements, it becomes necessary that the generalized control forces they are subjected to exactly satisfy these modeling constraints so that their physical descriptions are correctly preserved. The control requirements imposed can always be interpreted as a set of additional control constraints, and they may or may not be consistent with the modeling constraints that describe the physical system. This paper considers both the cases when the control constraints are consistent with the modeling constraints and when they are inconsistent. Such inconsistencies can arise when dealing with underactuated systems. A user-prescribed control cost is minimized at each instant of time in both cases. No linearizations/approximations of the nonlinear mechanical systems are made throughout. Insights into the control methodology are afforded through its geometric interpretation. Numerical examples with full-state control and underactuated control are considered, demonstrating the simplicity of the approach, its ease of implementation, and its effectiveness.

Lifting and parallel lifting optimization by using sensitivity and fuzzy set for an earthmoving mechanism

Earthmoving equipment in motor graders, which can be considered to be complex multibody systems (MBSs), are critical components for earthwork, compaction and re-handling. They have not yet received much attention due to their unusual applications and complicated structures. In this paper, a comprehensive study of an earthmoving MBS, from the mechanism identification and sensitivity analysis to the multi-objective optimization, is presented. First, the earthmoving MBS is identified to be a six degrees-of-freedom spatial hybrid mechanism, where a three revolute-revolute-prismatic-spherical (RRPS) and one spherical subchain (so, RRPS-S) spatial parallel mechanism is the key subsystem, through the mechanism analysis and synthesis. An earthmoving virtual prototyping model is built according to the system topology and connectivity. The kinematic simulations are carried out by imposing corresponding driving functions. Afterwards, the sensitivity analysis is introduced to extract several most relevant design variables from the global ones. A multi-objective optimization process is carried out to improve working performance, where fuzzy sets are used to define different objectives. Results show that the optimal earthmoving mechanism provides better lifting and parallel lifting capabilities.

State estimation applied to non-explicit multibody models

A new state estimation methodology is presented and validated to be applied to complex multibody systems modeled in Adams $${^{\circledR }}$$ . Traditional state observers are developed to be applied to explicit mathematical models. Given the complexity of some special multibody systems, particularly in vehicle dynamics, simplifications are needed to be able to apply this control technique. Nevertheless, tridimensional multibody systems are usually completely modeled in multibody dynamics simulation software, such as Adams $${^{\circledR }}$$ from MSC Software, for simulation, analysis and optimization, including highly nonlinear dynamic elements, such as the tires and the silent blocks in vehicle dynamics. Therefore, applying state observers to these non-explicit models is a real need in order to continue improving control systems on these systems. The methodology being presented and validated is a first step in this direction.