A differential approach for modeling revolute clearance joints in planar rigid multibody systems

A non-penetration approach of frictional contact analysis is presented for modeling revolute clearance joints of planar rigid multibody systems. In the revolute clearance joint, the motion modes of the journal are divided into three categories, namely, the free motion, collision, and permanent contact modes. The switch between different contact modes is identified by the state of the journal and bearing, including the gap and the normal relative velocity. When impact in the revolute clearance joint is detected, the collision process is simulated by the impulse-based differential approach, where Stronge’s improved model for restitution is employed to determine the relative velocity after impact. Instead of algebraic equations, the impact process is described by a set of ordinary differential equations (ODEs), which avoids solving complementarity problems. Moreover, in the permanent contact mode, the constraint-based approach and modified Coulomb’s friction law are adopted. The permanent contact mode maintains for most of the time and the governing ODEs are non-stiff. There is general agreement that the constraint-based approach is more efficient than the force-based method. A slider–crank mechanism with a revolute clearance joint is considered as a demonstrative application example where the comparison with the continuous contact force model is investigated.     

A modified HHT method for the numerical simulation of rigid body rotations with Euler parameters

Multibody System Dynamics - In multibody dynamics, the Euler parameters are often used for the numerical simulation of rigid body rotations because they lead to a relatively simple form of the...     

Modeling and simulation of the nonsmooth planar rigid multibody systems with frictional translational joints

The main purpose of this paper is to present a modeling and simulation method for the rigid multibody system with frictional translational joints. The small clearance between a slider and guide is considered. The geometric constraints of the translational joints are treated as bilateral constraints and the impacts between sliders and guides are neglected when the clearance sizes of the translational joints are very small. The contact situations of the normal forces acting on the sliders are described by inequalities and complementarity conditions, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The dynamic equations of the multibody systems with normal and tangential contact forces are written on the acceleration-force level using the Lagrange multiplier technique. The problem of the transitions of the contact situation of the normal forces acting on sliders and the transitions of the stick-slip of the sliders in the system is formulated as a horizontal linear complementarity problem (HLCP), which is solved by event-driven method. Baumgarte’s stabilization method is used to decrease the constraint drift. Finally, two typical mechanisms are considered as demonstrative application examples. The numerical results obtained show some dynamical behaviors of the systems with frictional translational joints and constraint stabilization effect.     

Performance of automatic differentiation tools in the dynamic simulation of multibody systems

Within the multibody systems literature, few attempts have been made to use automatic differentiation for solving forward multibody dynamics and evaluating its computational efficiency. The most relevant implementations are found in the sensitivity analysis field, but they rarely address automatic differentiation issues in depth. This paper presents a thorough analysis of automatic differentiation tools in the time integration of multibody systems. To that end, a penalty formulation is implemented. First, open-chain generalized positions and velocities are computed recursively, while using Cartesian coordinates to define local geometry. Second, the equations of motion are implicitly integrated by using the trapezoidal rule and a Newton–Raphson iteration. Third, velocity and acceleration projections are carried out to enforce kinematic constraints. For the computation of Newton–Raphson’s tangent matrix, instead of using numerical or analytical differentiation, automatic differentiation is implemented here. Specifically, the source-to-source transformation tool ADIC2 and the operator overloading tool ADOL-C are employed, in both dense and sparse modes. The theoretical approach is backed with the numerical analysis of a 1-DOF spatial four-bar mechanism, three different configurations of a 15-DOF multiple four-bar linkage, and a 16-DOF coach maneuver. Numerical and automatic differentiation are compared in terms of their computational efficiency and accuracy. Overall, we provide a global perspective of the efficiency of automatic differentiation in the field of multibody system dynamics.     

Parameter identification of damping models in multibody dynamic simulation of mechanical systems

To reduce vibration and noise, a damping mechanism is often required in mechanical systems. Many types of dampers are currently used. In this paper, several typical damping models, i.e., structural damping, frictional damping, and viscoelastic damping, are illustrated, and their parameters are identified for multibody dynamic simulation. Linear damping, widely adopted for structural damping, is applied to beam deflection. Quadratic damping including air resistance is applied to plate deflection. To model stick phenomenon in mechanical dampers, a STV (stick-transition velocity) model was first introduced. To identify parameters, an optimization process is applied to the damping parameters. A new MSTV (modified stick-transition velocity) model is proposed for a friction damper. A modified Kelvin–Voight model is suggested for a rubber bushing model used in vehicle dynamics, and its parameters are identified. A modified Bouc–Wen model is also proposed; it includes the hysteretic behavior of an elastomer, and optimized results with parameter identification are compared to test results.     

Use of penalty formulations in dynamic simulation and analysis of redundantly constrained multibody systems

The determination of particular reaction forces in the analysis of redundantly constrained multibody systems requires the consideration of the stiffness distribution in the system. This can be achieved by modeling the components of the mechanical system as flexible bodies. An alternative to this, which we will discuss in this paper, is the use of penalty factors already present in augmented Lagrangian formulations as a way of introducing the structural properties of the physical system into the model. Natural coordinates and the kinematic constraints required to ensure rigid body behavior are particularly convenient for this. In this paper, scaled penalty factors in an index-3 augmented Lagrangian formulation are employed, together with modeling in natural coordinates, to represent the structural properties of redundantly constrained multibody systems. Forward dynamic simulations for two examples are used to illustrate the material. Results showed that scaled penalty factors can be used as a simple and efficient way to accurately determine the constraint forces in the presence of redundant constraints.     

Selection of modeling coordinates for forward dynamic multibody simulations

Modeling mechanical systems in a manner that allows the models to be simulated quickly is vital in many fields, such as real-time simulation and control. Modeling these systems using their symbolic equations, rather than the more widely-used numerical methods, generally produces faster solution times. However, the number, complexity, and computational efficiency of these equations is highly dependent upon which coordinate set was used to model the system. Most coordinate selection methods established thus far are based on the assumption that minimizing the number of modeling coordinates will produce models with faster simulation times. This paper will show that this technique is not always valid and proposes a new technique of selecting a system’s coordinates based on a series of heuristics. A large part of these heuristics will be established by closely analyzing a specific technique used to formulate a system’s equations, and the effect each step of this formulation process will have on the complexity of the final system equations.     

Impacts in multibody systems: modeling and experiments

This paper outlines a novel approach to the modeling and analysis of impact involving multibody systems. This approach is based on an analysis of energy absorption and restitution during impact, using a decomposition of the kinetic energy, which decouples the parts associated with the spaces of admissible and constrained motions of the underlying unilateral constraints. Such a decomposition turns out to be useful in the analysis of energy dissipation during impact, and leads to a generalized definition of the energetic coefficient of restitution, which targets particularly collisions in multibody systems. The applicability of the approach reported is investigated by conducting an experimental study on a robotic testbed. It is shown that impact between multibody systems is considerably affected not only by the local dynamics characteristics of the interacting bodies, but also the configuration of the whole multibody system. The results reported here show that our decomposition can offer a sound characterization of impact in several problems of multibody systems.     

A new version of transfer matrix method for multibody systems

In order to avoid the global dynamics equations and increase the computational efficiency for multibody system dynamics (MSD), the transfer matrix method of multibody system (MSTMM) has been developed and applied very widely in research and engineering in recent 20 years. It differs from ordinary methods in multibody system dynamics with respect to the feature that there is no need for a global dynamics equation, and it uses low-order matrices for high computational efficiency. For linear systems, MSTMM is exact even if continuous elements like beams are involved. The discrete time MSTMM, however, has to use local linearization. In order to release the method from such approximations, a new version of MSTMM is presented in this paper where translational and angular accelerations, on the one hand, and internal forces and moments, on the other hand, are used as state variables. Already linear relationships among these quantities are utilized, which results in new element transfer matrices and algorithms making the study of multibody systems as simple as the study of single bodies. The proposed approach also allows combining MSTMM with any general numerical integration procedure. Some numerical examples of MSD are given to demonstrate the proposed method.     

Multiphysics modeling and optimization of mechatronic multibody systems

Modeling mechatronic multibody systems requires the same type of methodology as for designing and prototyping mechatronic devices: a unified and integrated engineering approach. Various formulations are currently proposed to deal with multiphysics modeling, e.g., graph theories, equational approaches, co-simulation techniques. Recent works have pointed out their relative advantages and drawbacks, depending on the application to deal with: model size, model complexity, degree of coupling, frequency range, etc. This paper is the result of a close collaboration between three laboratories, and aims at showing that for “non-academic” mechatronic applications (i.e., issuing from real industrial issues), multibody dynamics formulations can be generalized to mechatronic systems, for the model generation as well as for the numerical analysis phases. Model portability being also an important aspect of the work, they must be easily interfaced with control design and optimization programs. A global “demonstrator”, based on an industrial case, is discussed: multiphysics modeling and mathematical optimization are carried out to illustrate the consistency and the efficiency of the proposed approaches.     

Solvability of reactions in rigid multibody systems with redundant nonholonomic constraints

The problem of calculating joint reaction forces in rigid body mechanisms with redundant constraints, both geometric and nonholonomic, is discussed. When constraint equations are dependent, some of the constraint reactions are unsolvable, i.e., cannot be uniquely determined using a rigid body model, whereas some others may be solvable. In this paper, analytic conditions, which must be fulfilled to obtain unique values of selected reaction forces in the presence of dependent nonholonomic constraints, are presented and proven. The concept of direct sum, known from linear algebra, is exploited. These purely mathematical conditions are followed by numerical methods that enable detection of constraints with uniquely solvable reactions. Similar conditions and methods were proposed earlier for holonomic systems. In this contribution, they are generalized to the case of linear nonholonomic constraints. An example of constraint reactions solvability analysis, for a mechanism subjected to redundant nonholonomic constraints, is presented.     

Rigid-elastic modeling of meshing gear wheels in multibody systems

In many applications in mechanical engineering, gear wheels are used to transmit power between rotating shafts and, therefore, the ability to incorporate them into multibody systems and to simulate contact between them has become an essential topic in multibody dynamics.However, in some applications gear wheels may not be considered as being perfectly rigid. Due to the effect of contact forces there occur relevant deformations in meshing teeth and it is required for a high quality of the analysis to introduce some elasticities in the model of meshing gear wheels. Therefore, in this work elastic elements between the teeth and the body of each gear wheel are considered. This approach is especially well suited for multibody systems since it is a compromise between a totally rigid model and a fully elastic model allowing the simulation of large motions with many revolutions while still important elasticities are considered. The teeth and the body of each gear wheel are still modelled as being rigid but they are connected to each other by elastic elements. In doing so, an efficient and physically motivated algorithm is described and implemented in order to find the effects of multi-tooth contact as well as backlash and left and right hand side contact of the meshing teeth. Some examples compare the simulation results of rigid, partially elastic and fully elastic models.     

Graph theoretic modeling and analysis of multibody planarmechanical systems

A methodology of modeling and analysis of planar mechanical systems is developed based on graph theoretic methods, with improvements in component models. The system model based on cutset and circuit topologies is used to derive a new hybrid cutset-circuit method of formulation of the equations of motion for planar systems. Computer-aided formulation is based on analysis of the substitution procedure mandated by the hybrid cutset-circuit formulation. A new graphical representation of the formulation process is introduced: substitution graphs. No special programming is needed for computer-aided formulation which can be achieved in a symbolic form using the off the shelf Maple symbolic mathematics system. Symbolic formulation requires only inputting the systems equations in an order and form as derived from the analysis of the hybrid formulation. An algorithm for symbolic formulation using Maple is given. A compact set of differential-algebraic equations results, which can be solved numerically. Some simple systems will result in closed-form solutions. A number of examples are given to illustrate the modeling and formulation. Numerical solutions are also given to demonstrate the effectiveness and correctness of the formulation procedure     

A new approach to fuzzy modeling and control for nonlinear dynamic systems: Neuro-fuzzy dynamic characteristic modeling and adaptive control mechanism

The study on nonlinear control system has received great interest from the international research field of automatic engineering. There are currently some alternative and complementary methods used to predict the behavior of nonlinear systems and design nonlinear control systems. Among them, characteristic modeling (CM) and fuzzy dynamic modeling are two effective methods. However, there are also some deficiencies in dealing with complex nonlinear system. In order to overcome the deficiencies, a novel intelligent modeling method is proposed by combining fuzzy dynamic modeling and characteristic modeling methods. Meanwhile, the proposed method also introduces the low-level learning power of neural network into the fuzzy logic system to implement parameters identification. This novel method is called neuro-fuzzy dynamic characteristic modeling (NFDCM). The neuro-fuzzy dynamic characteristic model based overall fuzzy control law is also discussed. Meanwhile the local adaptive controller is designed through the golden section adaptive control law and feedforward control law. In addition, the stability condition for the proposed closed-loop control system is briefly analyzed. The proposed approach has been shown to be effective via an example. Recommended by Editor Young-Hoon Joo. This work was jointly supported by National Natural Science Foundation of China under Grant 60604010, 90716021, and 90405017 and Foundation of National Laboratory of Space Intelligent Control of China under Grant SIC07010202. Xiong Luo received the Ph.D. degree from Central South University, Changsha, China, in 2004. From 2005 to 2006, he was a Postdoctoral Fellow in the Department of Computer Science and Technology at Tsinghua University. He currently works as an Associate Professor in the Department of Computer Science and Technology, University of Science and Technology Beijing. His research interests include intelligent control for spacecraft, intelligent optimization algorithms, and intelligent robot system. Zengqi Sun received the bachelor degree from Tsinghua University, Beijing, China, in 1966, and the Ph.D. degree from Chalmers University of the Technology, Gothenburg, Sweden, in 1981. He currently works as a Professor in the Department of Computer Science and Technology, Tsinghua University. His research interests include intelligent control of robotics, fuzzy neural networks, and intelligent flight control. Fuchun Sun received the Ph.D. degree from Tsinghua University, Beijing, China, in 1998. From 1998 to 2000, he was a Postdoctoral Fellow in the Department of Automation at Tsinghua University, where he is currently a Professor in the Department of Computer Science and Technology. His research interests include neural-fuzzy systems, variable structure control, networked control systems, and robotics.     

A hybrid switching control strategy for nonlinear and underactuated mechanical systems

This note presents a hybrid switching control strategy for nonlinear and underactuated mechanical systems. Sufficient conditions for constructing the hybrid switching control, stability proof, and experimental results for using the hybrid switching control are given.     

A unified dynamic algorithm for wheeled multibody systems with passive joints and nonholonomic constraints

This paper presents a systematic approach to develop a generalized symbolic/numerical dynamic algorithm for modeling and simulation of multibody systems with branches and wheels. The proposed dynamic algorithm includes the direct kinematic and inverse dynamic models of the wheeled systems with prismatic/revolute as well as actuated/passive degrees of freedom. Using the geometric configuration of the system through modified Denavit–Hartenberg convention, symbolic equations in general algorithmic form are developed for kinematic constraints associated with the wheel–ground contacts. The Newton–Euler equations are used to develop an algorithm for the inverse dynamic model of the multibody system. The complete algorithm is then used to solve the kinematics and dynamics of the system, and computes: (i) the kinematics of the external/internal passive degrees of freedom of the system, (ii) the Lagrange multipliers associated with the wheel–ground contacts, and (iii) the driving forces/torques of the actuated degrees of freedom. Some examples are solved with the help of the proposed algorithm, using MATLAB, to illustrate its implementation on different wheeled systems. These examples include a differential wheeled robot, a snake-like wheeled system, and a bicycle.     

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.