Planar Multibody Systems Research Articles

A recursive algorithm for dynamics of planar multibody systems with frictional unilateral constraints

The contact impact problem in planar multibody systems can be efficiently solved by formulating it as the linear complementarity problem, which requires a complex modeling process. To simplify the process, a recursive algorithm for the dynamics of planar multibody systems with frictional unilateral constraints is proposed based on the reduced multibody system transfer matrix method. Firstly, the contact forces of frictional unilateral constraints are integrated into the recurrence relations of system components using Lagrange multipliers. Subsequently, the relative motion equations of the contact positions are discretized in time, which are then utilized to describe the linear complementarity problem for systems. The dynamics of the system is solved by the Moreau time-stepping method with the recursive method. Finally, the proposed algorithm was validated using the woodpecker toy and used to model a slider-crank mechanism with clearance, which shows its characteristics of facilitating modeling, universal, and highly programmable. This recursive algorithm provides an effective tool for solving non-smooth planar multibody systems while extending the application of the multibody system transfer matrix method.

A new kinematic model for revolute clearance joints with noncircular bushing and pin in planar multibody systems

Due to manufacturing error and wear, the profiles of bushing and pin in revolute clearance joints are usually non-circular, while traditional kinematic models for revolute clearance joints generally assume a circular profile for the pin or bushing, which may not be able to accurately capture the kinematic behavior of real revolute joints in multibody systems. To this end, a curvature center method was proposed for the kinematic modeling of revolute clearance joints with noncircular bushing and pin. In the proposed method, both the noncircular bushing and pin were discretized, and the curvature center and radius corresponding to each discrete point were then obtained to detect the point of contact between the bushing and pin based on their kinematic constraints. The number of discrete points was determined based on both accuracy and efficiency of the proposed curvature center method, which was then applied to model a revolute clearance joint between the slider and the link of a slider-crank mechanism. Via comparison to the traditional geometric center method and discrete point method, it was validated that the proposed method can achieve a comparable accuracy in modeling revolute joints with a circular pin. The unique advantage of the proposed curvature center method is its ability in modeling revolute joints with noncircular bushing and noncircular pin, which was also demonstrated via a series of simulations. Simulation results show a significant influence of noncircular bushing and noncircular pin on the dynamic response of the multibody system. This implies the promising value of the proposed method in studying the effect of manufacturing error or wear of both bushing and pin on the system’s dynamic performance and/or service life.

Study on the vibration characteristics of deep groove ball bearings under time-varying loads

A general methodology for dynamic modeling and analysis of deep groove ball bearing used in the crank-slider mechanism of punching machine is presented in this paper. The time-varying loads applied at the inner ring of this bearing can be obtained by analyzing the planar multibody systems. The bearing joint has been modeled by introducing a nonlinear constraint force system, which takes into account the contact stiffness interaction between the rolling elements and the raceways. The four-degree-of-freedom dynamic equations for the inner and outer rings of the bearings is established by Newton's second law. By numerical calculation, the variations of the load, trajectory, FFT frequency domain response, and x direction phase trajectory and Poincare of the inner ring, and the contact force on each ball element are discussed. The results indicate that the present methodology can not only be used to analyze the overall dynamic behavior of crank-slider mechanism and the deep groove ball bearing used in punching machine, but also to obtain the dynamic loads of the inner ring and ball in bearing. Therefore, the dynamic loads on ball elements can provide a basis for the strength checking, fatigue life calculation and wear analysis of the deep groove ball bearing.

FPGA acceleration of planar multibody dynamics simulations in the Hamiltonian–based divide–and–conquer framework

Multibody system simulations are increasingly complex for various reasons, including structural complexity, the number of bodies and joints, and many phenomena modeled using specialized formulations. In this paper, an effort is pursued toward efficiently implementing the Hamiltonian-based divide-and-conquer algorithm (HDCA), a highly-parallel algorithm for multi-rigid-body dynamics simulations modeled in terms of canonical coordinates. The algorithm is implemented and executed on a system–on–chip platform which integrates a general-purpose CPU and FPGA. The details of the LDUP factorization, which is used in the HDCA approach and accounts for significant computational load, are presented. Simple planar multibody systems with open- and closed-loop topologies are analyzed to show the correctness of the implementation. Hardware implementation details are provided, especially in the context of inherent parallelism in the HDCA algorithm and linear algebra procedures employed for calculations. The computational performance of the implementation is investigated. The final results show that the FPGA–based multibody system simulations may be executed significantly faster than the analogous calculations performed on a general–purpose CPU. This conclusion is a good premise for various model-based applications, including real-time multibody simulation and control.

Kinematics and dynamics of planar multibody systems with fully Cartesian coordinates and a generic rigid body

This work describes in detail the theoretical basis of the fully Cartesian coordinates formulation with a generic rigid body and evaluates its performance in the dynamic analysis of planar multibody systems. The topological features of the generic rigid body and the most relevant kinematic constraint equations are presented, aiming its application in the dynamic analysis of complex multibody systems and in advanced teaching.Contrarily to natural coordinates, in which the present formulation lays its foundation, planar rigid bodies are defined with a predetermined kinematic structure that resorts only to the Cartesian coordinates of one point and one unit vector. The introduction of the generic rigid body systematizes the discretization of the multibody system, simplifies the kinematic constraints required for its description, and renders higher physical meaning and sparsity to its mass matrix. However, it also slightly increases the number of generalized coordinates and kinematic constraints, required to model the multibody system, when compared to other global formulations.The present formulation was applied to the analysis of four planar models with different complexity levels, namely the slider-crank, double four-bar linkage, Jansen mechanism and a biomechanical model of the human body, presenting high agreement scores with benchmark and literature data.

Eigenvalue analysis of planar linear multibody system under conservative force based on the transfer matrix method

AbstractThe linear multibody system transfer matrix method (LMSTMM) provides a powerful tool for analyzing the vibration characteristics of a mechanical system. However, the original LMSTMM cannot resolve the eigenvalues of the systems with ideal hinges (i.e., revolute hinge, sliding hinge, spherical hinge, cylindrical hinge, etc.) or bodies under conservative forces due to the lack of the corresponding transfer matrices. This paper enables the LMSTMM to solve the eigenvalues of the planar multibody systems with ideal hinges or rigid bodies under conservative forces. For a rigid body, the transfer matrix can now consider coupling terms between forces and kinematic state perturbations. Also, conservative forces that contribute to the eigenvalues can be considered. Meanwhile, ideal hinges are introduced to LMSTMM, which enables the treatment of eigenvalues of general multibody systems using LMSTMM. Finally, the comparative analysis with ADAMS software and analytical solutions verifies the effectiveness of the proposed approach in this paper.

Research on irregular wear mechanism of planar multi-body mechanical system with multi-clearance joints

An approach for analyzing and quantifying the irregular wear effect of planar multi-body mechanical system with multi-clearances is proposed. The classical planar four-bar linkage is used as a demonstrative example to study the mechanism of irregular wear effect of multi-body system with multi-clearances. The system dynamic equation is established using Lagrange multiplier method, the contact force and friction in the process of contact collision are respectively modeled using the Lankarani-Nikravesh (L-N) nonlinear contact force and LuGre models, and the wear depth prediction model for multi-clearance joints is built based on Archard’s equation. The influence of the crank speed, clearance number, clearance position and clearance size on the wear effect of multi-clearances system is comprehensively studied. Numerical simulation results show that the wear depth joint increases with increase of crank speed, and there are significant differences in the wear depth of different clearance joints. When the clearance size changes, the significant nonlinear wear phenomenon appears at different joints, when the clearance size of different joints meets a certain proportion, the system wear phenomenon is the least. The obtained valuable results provide important theoretical support for the optimal design of multi-clearance joints system.

Effect of novel continuous friction model on nonlinear dynamics of the mechanisms with clearance joint

A general methodology for the dynamic modelling and analysis of planar multi-body systems with a continuous friction model in joint clearance is presented. Joint clearance is the critical factor that influences the dynamic response and the performance of mechanisms for high-speed application. In light of recent developments in the joint clearance studies, the number of contact force models has been introduced with ignoring friction continuity. The selection of an appropriate continuous friction model is still challenging and essential, which requires further development. Therefore, a perfect continuous friction model, including the Stribeck effect, static, dynamic and viscous friction terms, is proposed and validated. Investigating the dynamic modelling and analysis of double rocker four-bar linkage mechanisms with frictional revolute clearance joints is presented to investigate friction models' effect when surfaces collide with a non-zero tangential velocity. Unlike the smooth crank input mechanism, a double rocker four-bar linkage mechanism is analysed as a challenging problem in the impact mode. Resolving this concern, the novel friction model avoids discontinuity at zero velocity considering the accurate static friction zone. The results reveal that the novel friction model, compared with the piecewise friction model, is more effective in reflecting the mechanical systems' dynamic behaviour. In order to grasp the nonlinear characteristics of the high-speed four-bar linkage mechanism with our model in joint clearance, the Poincaré portrait, and Fast Fourier transformation plot are employed. It is proved that chaos exists in the dynamic response with the influence of the restitution coefficients and kinetic coefficient of friction.

Study on Transfer Matrix Method for the Planar Multibody System With Closed-Loops

Abstract The conventional transfer matrix method for multibody systems (MSTMM) with closed-loops (CLs) has superiority of avoiding the global dynamics equations. However, it requires a transfer equation to link multiple-input single-output (MISO) rigid body with multihinge subset and supplement equations caused CLs. In order to simplify the deduction processing and improve the numerical stability, the Riccati transformation is introduced and the Riccati transfer matrix method for multibody systems (RMSTMM) with CLs is proposed. In a system with CLs, each CL is cut off at the connection point, and the new unknowns generated at the cutoff point are introduced into the Riccati recurrence relation. The numerical results of the conventional MSTMM and the RMSTMM are compared, and the reliability of the RMSTMM is verified. Meanwhile, the constrained Jacobian matrix is used to eliminate the constraint violation of the system. The variations of the constraint violation error are compared to validate necessarily of constraints.

Study on dynamic behavior of planar multibody system with multiple lubrication clearance joints

With the wide application of multi-link mechanism, it is necessary to establish an accurate and reliable dynamic model to analyze dynamic response of multi-link mechanism. In the practical application of multi-link mechanism, lubricating oil has been widely used in kinematic pair to reduce influence of clearance on accuracy and stability of mechanism. Research on dynamic response of multi-link mechanism with clearance mainly focuses on effect of dry friction clearance, and effect of lubrication clearance on mechanism mainly focuses on simple mechanism. There are few studies on dynamic response and nonlinear characteristics of multi-link complex mechanisms with multiple lubrication clearance joints. It is rare to study multiple DOFs complex mechanism test-bed with multiple lubrication clearances. Lubrication force of lubrication clearance is established by modified Pinkus-Sternlicht model. A transition force model considering both lubrication and dry friction is established. Dynamic model of 2 DOFs 9-bar mechanism considering multiple lubrication clearances is established by Lagrange multiplier method. Effects of dry friction clearance and lubrication clearance on dynamic behavior of mechanism are compared and analyzed. Effects of dynamic viscosity of lubricating oil, crank speed and clearance value on dynamic response of mechanism with lubrication clearances are analyzed. Test platform with lubrication clearance is built, influences of clearance value and driving speed on test platform of mechanism considering lubrication clearance are studied, and correctness of theoretical model is verified by experimental data.

Sensitivity analysis of deployable flexible space structures with a large number of design parameters

The reliability and dynamic performance of deployable flexible space structures significantly depend on their key design parameters. Sensitivity analysis of these design parameters in the frame of multibody dynamics can serve as a powerful tool to evaluate and improve the dynamic performances of deployable space structures. Nevertheless, previous studies on the sensitivity analysis are mainly confined to planar multibody systems with a few design parameters. In this study, an efficient computational methodology is proposed to perform the sensitivity analysis of the complex deployable space structures with a large number of design parameters. Firstly, the analytical sensitivity analysis formulations of objective functions with the parameter-dependent integration bounds are deduced via the direct differentiation method and adjoint variable method. A checkpointing scheme is further introduced to assist the backward integration of the high-dimensional differential algebraic equations of the adjoint variables. The flexible beams in the deployable flexible space structure are described by the locking-free three-node spatial beam elements of absolute nodal coordinate formulation. Furthermore, a parallelized automatic differentiation algorithm is proposed to efficiently evaluate the complex partial derivatives in the sensitivity analysis formulations. Finally, four numerical examples are provided to validate the accuracy and efficiency of the proposed computational methodology.

Dynamic Analysis of Planar Multibody Systems Considering Contact Characteristics of Ball Bearing Joint

A comparative study of dynamic analysis for planar multibody systems with ball bearing joints is conducted in this study. The transmission mechanism is used as the exemplar case for illustrating the effect of ball bearing joints on the dynamic behavior of multibody systems. To reflect the energy loss, the models of continuous contact force and modified Coulomb’s friction are considered in the kinematic equations for the multibody system with ball bearing joint. With this, the dynamic characteristics of the mechanism are studied. Meanwhile, an experimental platform is built to generate the test data for demonstrating the effectiveness and correctness of the proposed method. Moreover, the effects of driving speed and clearance size on the dynamic behavior of the multibody system are investigated. The numerical results indicate that the dynamic behavior of the mechanical system is sensitive to the variation of the design parameters and the selection of parameters can affect greatly the accuracy of the mechanism with clearance joints.

Multibody Modeling Method for UHV Porcelain Arresters Equipped with Lead Alloy Isolation Device

This paper presents a multibody modeling method for seismic analysis of UHV porcelain surge arresters equipped with a kind of seismic isolation device. An UHV arrester is modeled as a planar multibody system whose number of DOF is equal to the number of the arrester units. Joint coordinate method is adopted to construct the governing equations of motion. The seismic isolation device utilizing a number of lead alloy dampers as its core energy dissipation components is also investigated. An analytical model of this device is given by modeling each lead alloy damper as a hysteretic spring and reducing the entire device to a planar system consisting of a range of hysteretic springs. Its mechanical characteristic is derived theoretically, and the obtained moment-angle relationship is expressed as a system of differential algebraic equations. The initial rotational stiffness of the device is formulated in terms of the structural and mechanical parameters of the device. This analytic expression is used in estimating the fundamental frequency of the isolated equipment. By this modeling method, it is easy to construct the governing equations of motion for the isolated system. An UHV arrester specimen is analyzed by this proposed method. The effectiveness of the isolation device in terms of reducing the internal base moment is significant and the influence of system parameters on the effectiveness is also discussed. The proposed method shows its potential usefulness in optimal design of the isolation device.

Dynamic computation of 2D segment-to-segment frictional contact for a flexible multibody system subject to large deformations

A new formulation is presented to simulate the segment-to-segment frictional contact dynamics of planar multibody systems that are subjected to large deformations and large motions. The contact bodies are meshed via planar elements of the absolute nodal coordinate formulation, which offers a C1-continuous surface representation for contact regions. To maintain the optimal convergence, the mortar method is used to discretize the contact constraints, which are established in an integral form on the nonconforming contact surface. By using Coulomb's friction law, the switch between stick and slip status can be realized. The penalty approach and return mapping algorithm are adopted to enforce contact constraints. The rescale and reconstruction strategy proposed in the previous study of authors is improved to recover the gap functions and the tangential friction force. In the mortar space, the magnitudes of the normal contact force and tangential friction force are discretized via the third-order Hermite interpolation, while their directions are along the normal vector and tangential vector of the contact surface, respectively. Several numerical examples are presented to demonstrate the effectiveness of the formulation.

Robust Optimization of Planar Constrained Mechanical System with Varying Joint Clearance Size Based on Sensitivity Analysis

This paper is devoted to a general methodological study on sensitivity analysis and robust optimization for a planar crank-slider mechanism in presence of joint clearances and random parameters and investigate the effects of parameter uncertainty on optimization results when joint clearance sizes are constantly changing due to wear. The first-order sensitivity analysis based on the response surface proxy model is performed. Then, a multiobjective robust optimization algorithm based on sensitivity analysis is carried out to reduce the undesirable effects of joint clearances and random parameters. In the algorithm, a multiobjective robust optimization model derived from the mean and variance of the objective function is constructed. Here, the objective function is defined based on the consideration of reducing the contact force generated at all clearance joints. Additionally, in order to balance computational accuracy and efficiency in the multiobjective robust optimization process, high-precision Kriging agent models are established. The optimum values of design variables are determined by combining Monte Carlo sampling and multiobjective particle swarm optimization method. By combining the Baumgarte approach with Lankarani–Nikravesh contact force model and Coulomb friction model, the dynamic equations of the planar multibody system with clearance joints are established. The uniform probability distribution is applied for characterizing random parameters. Simulation results show that the influence of design variable variations on the objective function changes in relation to the joint clearance size, but their relative influence degree on the objective function will not vary with the size of joint clearances. Moreover, the optimal solution selected on the Pareto front will affect the average levels and peak fluctuations of the dynamic responses in multibody systems.

An approach for dynamic analysis of planar multibody systems with revolute clearance joints

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.

Dynamics of luffing motion of a flexible knuckle boom crane actuated by hydraulic cylinders

In this paper we present a modeling procedure for a flexible knuckle boom crane, which is actuated by hydraulic cylinders and is modeled as a planar multibody system. We propose a convenient framework where both rigid body velocities and velocities caused by flexible behavior are represented as twists. Such formulation allows for using screw transformations, which leads to systematic derivations. Dynamics of a crane and mass balance of hydraulic cylinders are coupled using the bond graph method. In addition, we present a procedure for the determination of reaction forces in passive joints, which is conveniently given as an extension of the dynamic modeling procedure. Both procedures are presented in a general and systematic manner such that they can be applied for a group of planar flexible manipulators.We study the dynamics of luffing motion of a crane by numerical simulation and provide the simulation results, as well as determine the reaction forces in passive joints. The simulation results are validated by the ANSYS finite element analysis. The derived model provides a basis for design of luffing cylinders and can potentially be used for studying performance of a crane control system.

A methodology for modeling and simulating frictional translational clearance joint in multibody systems including a flexible slider part

The objective of this paper is to present a simple and effective methodology for modeling and simulating frictional translational clearance joint in planar multibody systems including a flexible slider part. By using the finite element method (FEM), the slider is divided into finite elements, and the distributed body forces and boundary stresses (contact forces) on the slider are equivalent to the nodal forces. The normal contact forces are described by a nonlinear viscoelastic contact force model, while the frictional forces are evaluated by the Coulomb dry friction model. By lumping the mass matrix and presenting it in diagonal form, the equations of motion are then inertially decoupled, and the frictional forces are solved independently via trial-and-error algorithm. A penalty method for the revolute joint connecting the slider and other bodies is applied, and equations of motion are integrated numerically. Finally, two numerical examples are given to test the feasibility and effectiveness of the methodology in this paper.

Dynamic computation of 2D segment-to-segment frictionless contact for a flexible multibody system subject to large deformation

A new formulation of segment-to-segment frictionless contact dynamics is proposed for the planar multibody systems subject to large deformations, based on the absolute nodal coordinate formulation, which is not only able to describe both large deformations and overall motions, but also to offer the C1-continuous surface representation for contact problems. The mortar method is used to discretize the contact constraints, established in an integral form on the nonconforming contact surfaces. A rescale and reconstruction strategy is proposed to recover the gap function as following. The gap function is revised via a scale factor first, and then the slope of the gap function is reconstructed via weighted average method. To satisfy the frictionless contact condition, the magnitude of the normal contact force is discretized via the third-order Hermite interpolation, while the direction of the force is along the normal vector of the contact surface. The penalty method is used to enforce the contact constraints. The generalized forms and Jacobians of the contact forces are derived. The generalized-alpha method is adapted to solve the dynamic equations. Finally, four numerical examples are given to verify the accuracy, convergence, and robustness of the proposed algorithm.

Effects of passive vibration absorbers on the mechanisms having clearance joints

In the current study, the dynamic behavior of two planar mechanisms with revolute joints, in the presence of clearances is investigated. Subsequently, a control scheme with the aim of restraining the clearance and maintaining a more stable behavior is proposed. This approach is based on using tuned mass damper (TMD) in order to reduce the effects of clearances in mechanisms for passive control purpose. The applied absorber’s mass is insignificant and they are little in size. By attaching two perpendicular absorbers in order to control the clearance, the oscillations are notably reduced. The proposed methodology is applied to planar multibody mechanical systems with revolute clearance joints with a view to demonstrating its features. The absorber performance is evaluated for changes around the designed state and the results show that the designed absorbers have a good robustness with variations to changes in different parameters.