A general and comprehensive analysis on the continuous contact force models for soft materials in multibody dynamics is presented throughout this work. The force models are developed based on the foundation of the Hertz law together with a hysteresis damping parameter that accounts for the energy dissipation during the contact process. In a simple way, these contact force models are based on the analysis and development of three main issues: (i) the dissipated energy associated with the coefficient of restitution that includes the balance of kinetic energy and the conservation of the linear momentum between the initial and final instant of contact; (ii) the stored elastic energy, representing part of initial kinetic energy, which is evaluated as the work done by the contact force developed during the contact process; (iii) the dissipated energy due to internal damping, which is evaluated by modeling the contact process as a single degree-of- freedom system to obtain a hysteresis damping factor. This factor takes into account the geometrical and material properties, as well as the kinematic characteristics of the contacting bodies. This approach has the great merit that can be used for contact problems involving materials with low or moderate values of coefficient of restitution and, therefore, accommodate high amount of energy dissipation. In addition, the resulting contact force model is suitable to be included into the equations of motion of a multibody system and contributes to their stable numerical resolution. A demonstrative example of application is used to provide the results that support the analysis and discussion of procedures and methodologies described in this work.
One of the most important and complex parts of the simulation of multibody systems with contact-impact involves the detection of the precise instant of impact. In general, the periods of contact are very small and, therefore, the selection of the time step for the integration of the time derivatives of the state variables plays a crucial role in the dynamics of multibody systems. The conservative approach is to use very small time steps throughout the analysis. However, this solution is not efficient from the computational view point. When variable time-step integration algorithms are used and the preimpact dynamics does not involve high-frequencies, the integration algorithms may use larger time steps and the contact between two surfaces may start with initial penetrations that are artificially high. This fact leads either to a stall of the integration algorithm or to contact forces that are physically impossible which, in turn, lead to post-impact dynamics that is unrelated to the physical problem. The main purpose of this work is to present a general and comprehensive approach to automatically adjust the time step, in variable time-step integration algorithms, in the vicinity of contact of multibody systems. The proposed methodology ensures that for any impact in a multibody system the time step of the integration is such that any initial penetration is below any prescribed threshold. In the case of the start of contact, and after a time step is complete, the numerical error control of the selected integration algorithm is forced to handle the physical criteria to accept/reject time steps in equal terms with the numerical error control that it normally uses. The main features of this approach are the simplicity of its computational implementation, its good computational efficiency, and its ability to deal with the transitions between non-contact and contact situations in multibody dynamics. A demonstration case provides the results that support the discussion and show the validity of the proposed methodology.
A comprehensive computational methodology to study the coupling dynamics of a geared multibody system supported by ElastoHydroDynamic (EHD) lubricated cylindrical joints is proposed throughout this work. The geared multibody system is described by using the Absolute-Coordinate-Based (ACB) method that combines the Natural Coordinate Formulation (NCF) describing rigid bodies and the Absolute Nodal Coordinate Formulation (ANCF) characterizing the flexible bodies. Based on the finite-short bearing approach, the EHD lubrication condition for the cylindrical joints supporting the geared system is considered here. The lubrication forces developed at the cylindrical joints are obtained by solving the Reynolds’ equation via the finite difference method. For the evaluation of the normal contact forces of gear pair along the Line Of Action (LOA), the time-varying mesh stiffness, mesh damping and Static Transmission Error (STE) are utilized. The time-varying mesh stiffness is calculated by using the Chaari’s methodology. The forces of sliding friction along the Off-Line-Of-Action (OLOA) are computed by using the Coulomb friction models with a time-varying coefficient of friction under the EHD lubrication condition of gear teeth. Finally, two numerical examples of application are presented to demonstrate and validate the proposed methodology.
In the current study, the behavior of a crank–slider mechanism with single and multiple clearance joints is analyzed. For this purpose nonlinear dynamics of the system are discussed, using Poincare maps and bifurcation diagrams. Subsequently, the effects of joint friction on dynamical behavior and nonlinear response of the mechanism are investigated. Afterward, a control scheme providing continuous contact in joints, with the aim of maintaining a more stable behavior, is proposed. The control scheme is easy to apply and, since continuous contact between bodies in joints is established, several undesired effects normally caused because of clearance joints are prevented or reduced. Also the limitation of the proposed control method is investigated.
It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.
The main purpose of this paper is to present and discuss a methodology for a dynamic modeling and analysis of rigid multibody systems with translational clearance joints. The methodology is based on the non-smooth dynamics approach, in which the interaction of the elements that constitute a translational clearance joint is modeled with multiple frictional unilateral constraints. In the following, the most fundamental issues of the non-smooth dynamics theory are revised. The dynamics of rigid multibody systems are stated as an equality of measures, which are formulated at the velocity-impulse level. The equations of motion are complemented with constitutive laws for the normal and tangential directions. In this work, the unilateral constraints are described by a set-valued force law of the type of Signorini’s condition, while the frictional contacts are characterized by a set-valued force law of the type of Coulomb’s law for dry friction. The resulting contact-impact problem is formulated and solved as a linear complementarity problem, which is embedded in the Moreau time-stepping method. Finally, the classical slider-crank mechanism is considered as a demonstrative application example and numerical results are presented. The results obtained show that the existence of clearance joints in the modeling of multibody systems influences their dynamics response.
This paper aims to study the dynamic behavior of a slider–crank mechanism with flexible components and a multijoint clearance. A numerical investigation was developed for this objective. The mechanism model used for the simulation tests has been performed under MSC ADAMS software using the contact force under the “Impact-function” library. The obtained results illustrate that the mechanism performance is more significantly influenced with multiple joints with clearance. Three contact modes are involved: (i) a continuous contact motion, (ii) a free motion, and (iii) an impact motion. Numerical outcomes prove that the clearance dimension and location have a determinant effect on the slider responses precision. The mechanism reliability decay and a random overall behavior govern the dynamic response along the free flight mode.
In this paper, the dynamic response of a planar rigid multi-body system with stick–slip friction in revolute clearance joints is studied. LuGre friction law is proposed to model the stick–slip friction at the revolute clearance joints. This is because using this law, one can capture the variation of the friction force with slip velocity, thus making it suitable for studies involving stick–slip motions. The effective coefficient of friction is represented as a function of the relative tangential velocity of the contacting bodies, that is, the journal and the bearing, and an internal state. In LuGre friction model, the internal state is considered to be the average bristle deflection of the contacting bodies. By applying the LuGre friction law on a typical slider–crank mechanism, the friction force in the revolute joint having clearance is seen not to have a discontinuity at zero slip velocity throughout the simulation unlike in static friction models. In addition, LuGre model was observed to capture the Stribeck effect which is a phenomenon associated directly with stick–slip friction. The friction forces are seen to increase with increase in input speed. The effect of stick–slip friction on the overall dynamic behavior of a mechanical system at different speeds was seen to vary from one clearance joint to another.
This article is devoted to the analysis of the influence of the joint clearances in a mechanism of a circuit breaker, which is a 42 degree-of-freedom mechanism made of seven links, seven revolute joints, and four unilateral contacts with friction. Spatial (3D) revolute joints are modeled with both radial and axial clearances taking into account contact with flanges. Unilateral contact, Coulomb’s friction and Newton impact laws are modeled within the framework of nonsmooth mechanics without resorting to some regularizations or compliance/damping at contact. The nonsmooth contact dynamics method based on an event-capturing time-stepping scheme with a second order cone complementarity solver is used to perform the numerical integration. Furthermore, the stabilization of the constraints at the position level is made thanks to the stabilized combined projected Moreau–Jean scheme. The nonsmooth modeling approach together with an event–capturing time-stepping scheme allows us to simulate, in an efficient and robust way, the contact and impacts phenomena that occur in joints with clearances. In particular, comparing with the event-detecting time-stepping schemes, the event-capturing scheme enables us to perform the time-integration with a large number of events (impacts, sliding/sticking transitions, changes in the direction of sliding) and possibly with finite-time accumulations with a reasonable time-step length. Comparing with compliant contact models, we avoid stiff problems related with high stiffnesses at contact which generate some issues in contact stabilization and spurious oscillations during persistent contact periods. In the studied mechanisms of the circuit breakers, the numerical method deals with more than 70 contact points without any problems. Furthermore, the number of contact parameters is small—one coefficient of restitution and one coefficient of friction. Though they are sometimes difficult to measure accurately, the sensitivity of the simulation result with respect to contact parameters is low in the mechanism of the circuit breaker. It is demonstrated that this method, thanks to its robustness and efficiency, allows us to perform a sensitivity analysis using a Monte Carlo method. The numerical results are also validated by careful comparisons with experimental data, showing a very good correlation.
In this work, we propose a game theoretic framework to analyze the behavior of cognitive radios for distributed adaptive channel allocation. We define two different objective functions for the spectrum sharing games, which capture the utility of selfish users and cooperative users, respectively. Based on the utility definition for cooperative users, we show that the channel allocation problem can be formulated as a potential game, and thus converges to a deterministic channel allocation Nash equilibrium point. Alternatively, a no-regret learning implementation is proposed for both scenarios and it is shown to have similar performance with the potential game when cooperation is enforced, but with a higher variability across users. The no-regret learning formulation is particularly useful to accommodate selfish users. Non-cooperative learning games have the advantage of a very low overhead for information exchange in the network. We show that cooperation based spectrum sharing etiquette improves the overall network performance at the expense of an increased overhead required for information exchange.
A novel computational approach for the dynamic analysis of a large scale rigid–flexible multibody system composed of composite laminated plates is proposed. The rigid parts in the system are described through the Natural Coordinate Formulation (NCF) and the flexible bodies in the system are modeled via the finite elements of Absolute Nodal Coordinate Formulation (ANCF), which can lead to a constant mass matrix for the derived system equation of motion. For modeling composite laminated plates accurately, a new composite laminated plate element of ANCF is proposed and the corresponding efficient formulations for evaluating both the elastic force and its Jacobian of the element are derived from the first Piola–Kirchhoff stress tensor. To improve computational efficiency, the sparse matrix technology and graph theory are used to solve the huge set of linear algebraic equations in the process of integrating the equations of motion by using the generalized-a method, and an OpenMP based parallel scheme is also introduced. Finally, the effectiveness of the proposed approach is validated through two numerical examples. One is the static simulation of a single composite laminated plate under gravity and the other is the dynamic simulations of unfolding process of a satellite system with a pair of complicated antennas.
Capturing the non-cooperative space debris has gained increasing attention in the past decades. As an alternative of the rigid robot, the flexible tether-net space robot systems (TNSRS), like ROGER, are proposed by many research institutes, which can significantly reduce the risk of capturing process. However, their poor manoeuvrability and the lack of abilities to keep the net shape may lead to the failure of the capture process. Thus, a manoeuvrable tether-net space robot system (MTNSRS), as a potential approach to improve TNSRS, is proposed. In order to simplify the dynamics, we introduce the assumption that the Young’s modulus of the cord in the net is infinite when in tension while it is zero when being slack. Then, the contact dynamics of rigid robots is employed to solve the unilateral constraints within the above assumption, and the T3 element is introduced to approximate the shape of the net. Furthermore, the coordinated controller for MTNSRS is designed by transferring the inverse dynamics to be a double-level optimization problem. Finally, the simulation results show that without active control, the net will gradually close in the approaching phase, and this process will be significantly accelerated even by a small dragging force in the connecting tether. It is also shown that our controller can ensure MTNSRS to successfully capture the target and can resist the effects of initial state errors, measurement noise and kinetic parameter errors.
Clearance is inevitable for assembly and mobility in the kinematic joints of mechanisms. Excessive value of this clearance leads to poor operational characteristics, and these result in losses in kinematic and dynamic performances of mechanism. In this study, effects of joint clearances on vibration and noise characteristics of mechanism are investigated. An experimental test rig has been set up, and a planar slider-crank mechanism having two joints with clearance has been used as a model mechanism. Joint clearance is modeled as a massless virtual link and continuous contact mode between journal and bearing in joint connection is considered in theoretical analyses. Three accelerometers and two microphones have been located at different points to measure the vibrations and noises on system during the mechanism motion. The results obtained for the cases with and without joint clearance are evaluated for vibration and noise characteristics of mechanism.
Revolute joints in applications always show clearance between pin and bushing due to manufacturing tolerances, the need of relative motion or progressing wear. Many researchers developed and investigated methodologies to calculate the dynamic behavior of mechanisms with such imperfect joints. Very often they use a simple slider-crank mechanism to test or demonstrate the capability of their approaches. In this paper, a methodology for simulating a slider-crank mechanism with an imperfect revolute joint in RecurDyn, a commercial multibody simulation tool, is presented. Therefore, a thorough investigation of existing contact, damping and friction force models as well as different ways of modeling revolute joints in RecurDyn was conducted. For the investigation of the damping models, a special program for calculating the model parameters for a given coefficient of restitution was developed. Only one damping model was capable of reproducing the experimental results, which were found in literature. Some characteristic results of the slider-crank mechanism are presented in a way that they can be compared to results in other papers. Thereby. a good correlation was achieved, demonstrating the capabilities of the methodology.
An effective reduction technique is presented for flexible multibody systems, for which the elastic deflection could not be considered small. We consider here the planar beam systems undergoing large elastic rotations, in the floating frame description. The proposed method enriches the classical linear reduction basis with modal derivatives stemming from the derivative of the eigenvalue problem. Furthermore, the Craig–Bampton method is applied to couple the different reduced components. Based on the linear projection, the configuration-dependent internal force can be expressed as cubic polynomials in the reduced coordinates. Coefficients of these polynomials can be precomputed for efficient runtime evaluation. The numerical results show that the modal derivatives are essential for the correct approximation of the nonlinear elastic deflection with respect to the body reference. The proposed reduction method constitutes a natural and effective extension of the classical linear modal reduction in the floating frame.
This paper describes the fractional modeling and control of an industrial selective compliant assembly robot arm (SCARA); the fractional model was obtained by using the Euler–Lagrange and Hamilton formalisms. Each joint of the robot manipulator was driven by an induction motor. In this work, the fractional model of each induction motor was formulated, and the matching of the induction motors with the SCARA robot is shown. For comparison purposes, the SCARA robot control was formulated by conventional PI and PD and by fractional PI ς and PD δ controllers. So each induction motor was controlled by using PI and fractional PI ς controllers, and for trajectory tracking control, PD and fractional PD δ controllers were designed. For tuning the PI, PI ς , PD, and PD δ controllers, the PSO algorithm was used; the same restrictions were used for the PI and PD classical controllers, and ITAE index was used as a cost function to be minimized. For computing the fractional derivatives and to obtain the numerical solution of the system, the Riemann–Liouville and Grünwald–Letnikov approaches were used. The numerical simulations have shown the effectiveness of the use of fractional PI ς and PD δ controllers.
A variant of the generalized-α scheme is proposed for constrained mechanical systems represented by index-3 DAEs. Based on the analogy with linear multistep methods, an elegant convergence analysis is developed for this algorithm. Second-order convergence is demonstrated both for the generalized coordinates and the Lagrange multipliers, and those theoretical results are illustrated by numerical tests.
The aim of the paper is to develop a fully 3D simulation technique for rockfall dynamics taking rock shape into account and using the state-of-the-art methods of multibody dynamics and nonsmooth contact dynamics. The rockfall simulation technique is based on the nonsmooth contact dynamics method with hard contact laws. The rock is modeled as an arbitrary convex polyhedron and the terrain model is based on a high resolution digital elevation model. A specialized friction law for rockfall is proposed which allows for the description of scarring behavior (i.e., rocks tend to slide over the terrain before lift-off). The influence of rock geometry on rockfall dynamics is studied through two well-chosen numerical simulations.
Virtually all machines and mechanisms use mechanical joints that are not perfect from the kinematic point of view and for which tolerances, in the fitting of their components, are specified. Together with such controlled clearances, mechanical joints may require the use of bushing elements, such as those used in vehicle suspensions. Furthermore, in many situations the joints exhibit limits (stops) in their translational or rotational motion that have to be taken into account when modeling them. The dynamic response of the mechanical systems that use such realistic mechanical joints is largely dependent on their characteristic dimensions and material properties of the compliant elements, implying that correct models of these systems must include realistic models of the bushing/clearance joints and of the joint stops. Several works addressed the modeling of imperfect joints to account for the existence of clearances and bushings, generally independently of the formulation of the perfect kinematic joints. This work proposes a formulation in which both perfect and clearance/bushing joints share the same kinematic information making their modeling data similar and enabling their easy permutation in the context of multibody systems modeling. The proposed methodology is suitable for the most common mechanical joints and easily extended to many other joint types benefiting the exploration of a wide number of modeling applications, including the representation of cut-joints required for some formulations in multibody dynamics. The formulation presented in this work is applied to several demonstrative examples of spatial mechanisms to show the need to consider the type of imperfect joints and/or joints with stops modeling in practical applications.
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.