Spatial Mechanical Systems Research Articles

The discretization method and multi-contact force model on 3D cylindrical clearance joint for multibody dynamics of spatial mechanical system

The discretization method and multi-contact force model on 3D cylindrical clearance joint for multibody dynamics of spatial mechanical system

Influence of the soil base on the stress-strain state of a large-span building with a cylinder-and-slab roof

The authors consider finite element models of a large-span building with a cylinder-and-slab roof as a large spatial mechanical system with different boundary conditions. The first model represents the superstructure of the building with a fixed-end at the soil base level. In the second model, the superstructure is based on the substructure, which includes a pile foundation and a soil base with different physical and mechanical properties. The purpose of the study is a comparative numerical analysis of the stress-strain state of a large-span building with different boundary conditions. The numerical study revealed the influence of the structural features of the substructure of the building, as well as the physical and mechanical properties of the soil base on the stress-strain state of the long-span roof and the building as a whole. Numerical static analysis of spatial finite element models of a large-span building was carried out in the СAE class Femap NX Nastran software package. The results of the static analysis demonstrated a significant structural influence of the substructure of a large-span building on the characteristics of its stress-strain state. In the next article, it is proposed to conduct a modal analysis for these building models.

Kinematics Analyses of the Spatial Mechanism Using Matlab

The aim of this paper is to present a method of computer formulation and solution of equations of kinematics of spatial mechanical systems. The method of the vector closed loop is generally known way how to built the constraint equations. This method is recommended for planar kinematics. This paper shows the way how to take advantage of the vector cross product, vector magnitude and scalar product of two vectors for getting the constraint equations for spatial mechanism.

Modeling and Analysis of Spatial Flexible Mechanical Systems With a Spherical Clearance Joint Based on the LuGre Friction Model

Abstract A computational methodology for modeling spatial flexible mechanical systems with stick-slip friction in a spherical clearance joint is presented. A modified three-dimensional (3D) absolute nodal coordinate formulation based shear deformable beam element with two nodes is proposed and employed to discretize the flexible components. To avoid locking problems, we employed an enhanced continuum mechanics approach to evaluate the beam element elastic forces. The strain components εyz, εyy, and εzz are approximated using linear interpolation to improve the computational efficiency while the loss of accuracy is acceptable. The contact and friction forces in a spherical clearance joint were evaluated by the hybrid contact and LuGre friction models, respectively. Three numerical examples are presented and discussed. A simple pendulum was utilized to prove the correctness of the modified beam element. A classical slider–crank mechanism was employed to validate the computational methodology. A spatial rigid–flexible slider–crank mechanism with a spherical clearance joint was used to investigate the effect of link flexibility and joint clearance on the dynamic behavior of mechanical systems. Using the LuGre friction model, we reproduced the Stribeck effect as it is expected in real world settings. The components with appropriate stiffness play the role of suspension for spatial mechanical systems with imperfect joints. The vibrations of the flexible components play an active role of intensifying the collision in kinematic joint with clearance.

Probing carbon-based composite coatings toward high vacuum lubrication application

Abstract For challenges in minimizing friction and wear of spatial mechanical systems, synergetic lubrication coatings were prepared by spinning liquid lubricants and hybrid greases on the diamond-like carbon (DLC) films, and were evaluated whether they could achieve a long-term safe and reliable operation under high vacuum. Under high vacuum conditions, liquid lubricants significantly reduce the friction and wear of DLC films. Hybrid greases not only show excellent lubrication performance, but also greatly enhance the tribological behavior of DLC films, especially the grease with optimal proportion. Such excellent tribological performance of DLC-based composite coatings at low applied loads depends on the synergy of DLC and fluid film, in reverse the tribo-chemical reaction film under harsh working conditions.

Designing Self-Balanced Spatial Mechanisms

This paper summarises the results obtained from a study of the conditions required to achieve self-balanced mechanical systems. The design self-balanced systems composed of single-loop spatial mechanisms with a common drive shaft, is reviewed. The paper explains how the problems are solved by selecting the relative angles of the mechanism's cranks, their number and mutual arrangement. The resulting self-balancing criteria enable the construction of spatial mechanical systems with predetermined levels of residual imbalance.

Kinematic Analysis of Spatial Mechanical Systems Using a New Systematic Formulation Method for Lower and Higher Kinematic Pairs

This paper presents a new systematic formulation method to describe all kinds of lower and higher kinematic pairs and to express the pertaining kinematic relationships. The method can be applied to any mechanical system, which may be a mechanism or a manipulator, but it may especially be convenient for a system with multijoint links and multiaxis joints, such as a parallel manipulator. The method is based on the kinematic elements of the joints. In the first stage of the method, a joint frame is attached to every kinematic element hosted by the links of the system. It is attached in such a way that its relative position with respect to the link frame of the hosting link is described by a minimal number of essential parameters that are necessary and sufficient to represent all the characteristic features of the kinematic element. To systematize the attachment of the joint frames, the kinematic elements are classified into six types according to their geometric complexity. The link frames may also be attached judiciously to further minimize the total number of parameters required by the whole system. In the second stage of the method, the necessary equations are written to express the relative position between the mating kinematic elements of each joint. In the paper, such equations are written for a set of typical lower and higher kinematic pairs including samples ranging from a revolute joint up to a spatial cam joint. The application of the method is demonstrated on two mechanisms. One of them is a two-loop spatial mechanism with five different joints and the other one is a single-loop spatial cam mechanism with ellipsoidal and cylindrical cams. For each mechanism, the loop closure equations are first written and then simplified to prepare for solution to determine the unspecified joint variables. Afterward, the semi-analytical solutions of the loop closure equations are described and discussed.

Study Concerning the Effect of the Bushings' Deformability on the Static Behavior of the Rear Axle Guiding Linkages

In this paper, we attempt to study the influence of the bushings (compliant joints) on the static behavior of the guiding system used for the rear axle of the vehicles. In fact, we are interested to determine the difference in behavior between the compliant model (with bushings) and the rigid model (bushings modeled as spherical joints) of the axle guiding mechanism. The static model, in which the car body is attached to ground, is a constrained, multi-body spatial mechanical system, in which the bodies are connected through geometric constraints, compliant joints, and force elements. The external loading is made by vertical forces applied to the wheels, in stationary regime. The study is made for the general groups of guiding mechanisms, with M=1 and M=2 degrees of mobility, by using the MBS (Multi-Body Systems) environment ADAMS of MSC Software.

Active stabilization of an antenna system

A spatial mechanical system including a tethered balloon with a platform attached to it is considered. Suspended from the platform by a pendulum system is a radar. The system has six degrees of freedom. In order to perform a simplified analysis and develop a control algorithm, the case of plane vibrations is considered. The system is divided into two subsystems. Using the dynamic programming method, an algorithm for designing an optimal (in terms of the integral quadratic criterion) feedback control law is developed for the antisymmetric subsystem.

Kinematic Analysis of Spatial Mechanical Systems Using Exponential Rotation Matrices

The mathematical properties associated with the exponential rotation matrices are reviewed. By means of two typical mechanism examples, it is demonstrated that these properties constitute a versatile analytical tool, which can be used effectively in kinematic studies on spatial mechanical systems involving position, velocity, acceleration, and singularity analyses.

A recursive algorithm for generating the equations of motion of spatial mechanical systems with application to the five-point suspension

In this paper, a recursive formulation for generating the equations of motion of spatial mechanical systems is presented. The rigid bodies are replaced by a dynamically equivalent constrained system of particles which avoids introducing any rotational coordinates. For the open-chain system, the equations of motion are generated recursively along the serial chains using the concepts of linear and angular momenta. Closed-chain systems are transformed to open-chain systems by cutting suitable kinematic joints and introducing cut-joint constraints. The formulation is used to carry out the dynamic analysis of multi-link five-point suspension. The results of the simulation demonstrate the generality and simplicity of the proposed dynamic formulation.

Static equilibrium analysis of compliant mechanical systems using relative coordinates and loop closure equations

This paper deals with the static equilibrium formulation of mechanical systems, including rigid and compliant elements, and the determination of their final equilibrium configuration under known loads. Statically determinate as well as statically indeterminate systems are analysed. Relative coordinates are used as generalised coordinates. The system is modeled using homogeneous transformation matrices and loop closure equations. In the first phase of analysis (assembly), initial correct configuration of the system is obtained, from approximate input provided by the user. For the solution of the equilibrium problem, a potential function is used. Its minimisation, subject to rigid constraints yields the equilibrium configuration. The approach is implemented in a program and tested on a number of planar and spatial mechanical systems. The software program is intended to be used as a design aid for engineers.

Inverse dynamic analysis of mechanical systems in joint coordinate space

An inverse dynamic analysis algorithm for spatial flexible mechanical systems with closed loops is developed in the relative joint coordinate space. System equations of motion and constraint acceleration equations are derived using the velocity transformation technique. An inverse velocity transformation operator, which transforms the Cartesian velocities to the relative velocities, is derived systematically, corresponding to the types of kinematic joint connecting the bodies. Using the resulting matrix, the joint reaction forces and moments are analysed in the Cartesian coordinate space. The joint coordinates and the deformation modal coordinates are used as the generalized coordinates of a flexible mechanical system. The algorithm is verified by means of two numerical examples.

Dynamics of Rigid Multibody Systems with Clearances in the Joints*

ABSTRACT An approach for calculation of reaction and friction forces between pairs of bodies in spatial mechanical systems, as well as contact points in joints with clearances is presented. Coulomb friction is taken into consideration. Problems are solved iteratively, to satisfy both kinematic and dynamic equations. Matrix methods are used to derive the nonlinear kinematic constraint equations. Newton-Euler dynamic equations are applied for solution of direct and inverse problems of dynamics. External and inertia forces for each configuration of the kinematic chain the contact points and corresponding normal forces are calculated. Relative positions of the coordinate systems of the links are computed and, their new positions relative to the inertia reference frame are estimated. An iterative algorithm is suggested for calculation of forces and computing generalized coordinate increments for achieving the prescribed accuracy of system end-effectors. Kinematic and dynamic equivalents of rotational, translational, and prismatic pairs with clearances are suggested. A model of joints with clearances is applied for solution of the direct and inverse problems of dynamics, taking into account presence of unilateral reactions, viscous friction, contact friction, and impact in bearings. An example of a triple pendulum with rotational pairs, a driving force on the end-effector, and large clearances is presented.

Generalized Constraint and Joint Reaction Forces in the Inverse Dynamics of Spatial Flexible Mechanical Systems

In both the augmented and recursive formulations of the dynamic equations of flexible mechanical systems, the inerita, constraints, and applied forces must be properly defined. The inverse dynamics is a commonly used approach for the force analysis of mechanical systems. In this approach, the system is kinematically driven using specified motion trajectories, and the objective is to determine the driving forces and torques. In flexible body dynamics, however, a force that acts at a point on the deformable body is equipollent to a system, defined at another point, that consists of the same force, a moment that depends on the relative deformation between the two points, and a set of generalized forces associated with the elastic coordinates. Furthermore, a moment in flexible body dynamics is no longer a free vector. It is defined by the location of its line of action as well as its magnitude and direction. The joint reaction and generalized constraint forces represent equipollent systems of forces. Both systems in flexible body dynamics are function of the deformation. In this investigation, a procedure is developed for the determination of the joint reaction forces in spatial flexible mechanical systems. The mathematical formulation of some mechanical joints that are often encountered in the analysis of constrained flexible mechanical systems is discussed. Expressions for the generalized reaction forces in terms of the constraint Jacobian matrices of the joints are presented. The effect of the elastic deformation on the reaction forces is also examined numerically using the spatial flexible multibody RSSR mechanism that consists of a set of interconnected rigid and elastic bodies. The procedure described in this investigation can also be used to determine the joint torques and actuator forces in kinematically driven spatial elastic mechanism and manipulator systems.

Application of Composite Plate Theory and the Finite Element Method to the Dynamics and Stress Analysis of Spatial Flexible Mechanical Systems

This investigation concerns itself with the dynamic and stress analysis of thin, laminated composite plates consisting of layers of orthotropic laminae. It is assumed that the bonds between the laminae are infinitesimally thin and shear nondeformable. The finite element formulation presented is sufficiently general to accept an arbitrary number of layers and an arbitrary number of orthotropic material property sets. In the dynamic formulation presented, the laminae is assumed to undergo large arbitrary rigid body displacements and small elastic deformations. The nodal shape functions of the laminae are assumed to have rigid body modes that need to describe only large rigid body translations. Using the expressions for the kinetic and strain energies, the lamina mass and stiffness matrices are identified. The nonlinear mass matrix of the lamina is expressed in terms of a set of invariants that depend on the assumed displacement field. By summing the laminae kinetic and strain energies, the body mass and stiffness matrices are identified. It is shown that the body invariants can be expressed explicitly in terms of the invariants of its laminae. Numerical examples of a spatial RSSR mechanism are presented in order to demonstrate the use of the present formulation.

Dynamics of Initially Curved Plates in the Analysis of Spatial Flexible Mechanical Systems

In this paper, the effect of the initial curvature on the nonlinear dynamics of plates in constrained flexible multibody systems is examined. A set of nonlinear differential equations that describe the motion of the elastic bodies which undergo large reference displacements is derived using the principle of virtual work in dynamics. The resulting nonlinear mass matrix is written as the sum of two nonlinear matrices. These are the conventional mass matrix that arises when the effect of initial curvature is neglected and the curvature mass matrix that represents the effect of the initial curvature. It is shown that all the components of the generalized centrifugal forces depend on the initial curvature. In particular, the generalized centrifugal forces associated with the large rigid body translation and the elastic deformation are linear functions of the initial curvature, while the centrifugal forces associated with the finite rotation are quadratic functions of the initial curvature. It is also shown that the generalized Coriolis forces associated with the finite rotation are linear functions of the initial curvature, while other Coriolis components do not depend on the initial curvature. It is demonstrated numerically that the use of the normal modes of vibration of flat plates in the analysis of the initially curved elastic components leads to a significant error in the dynamic response. Numerical results presented in this investigation are obtained using the spatial flexible multibody RSSR mechanism.

Automatic generation and numerical integration of differential-algebraic equations of multibody dynamics

Abstract An algorithm for computing the matrix and vector functions in the numerical integration method of constrained equations of motion in multibody mechanical system dynamics is presented. Cartesian coordinates and Euler parameters have been used as the state variables to construct the mathematical model for a spatial multibody mechanical system. Four basic constraints, which describe the orthogonality and the parallelism of pairs of vectors, have also been written and will serve as building blocks of the kinematic joint library. Derivatives of these basic constraints are then derived by applying differential operators to the multivariable and vector-valued constraint functions. Verification is done by comparing the numerical solution with results computed by a symbolic computation software. In multibody dynamics, the acceleration of constrained equations of motion is a function of the state variables. Using the differentials of joint library functions and generalized force functions, the Jacobian of the acceleration with respect to the state variables is obtained. The efficiency and accuracy of computing this analytic Jacobian matrix are demonstrated by comparing execution time and numerical results with those obtained by applying a first order linearization method. The proposed computational technique is also suitable for various formulations of the equations of motion, for instance, the Euler angles instead of the Euler parameters can be used for the orientation variables. The final goal is to construct a standard library of joint constraints and force expressions for automatic generation of constrained equations of motion and to enhance the numerical integration methods for multibody dynamics.

Analysis and Optimal Design of Spatial Mechanical Systems

This paper presents a new approach to optimal design of large multibody spatial mechanical systems which takes advantage of both numerical analysis and symbolic computing. Identification of system topology is carried out using graph theory. The equations of motion are formulated in terms of relative joint coordinates through the use of a velocity transformation matrix. Design sensitivity analysis is carried out using the direct differentiation method applied to the relative joint coordinate formulation for spatial systems. The symbolic manipulation program MACSYMA is used to automatically generate the necessary equations for both dynamic and design sensitivity analyses for any spatial system. The symbolic equations are written as FORTRAN statements that are linked to a general purpose computer program which performs dynamic analysis, design sensitivity analysis, and optimization, using numerical techniques. Examples are presented to demonstrate reliability and efficiency of this approach.

Dynamics of Flexible Mechanical Systems Using Vibration and Static Correction Modes

A finite-element-based method is developed and applied for geometrically nonlinear dynamic analysis of spatial mechanical systems. Vibration and static correction modes are used to account for linear elastic deformation of components. Boundary conditions for vibration and static correction mode analysis are defined by kinematic constraints between components of a system. Constraint equations between flexible bodies are derived and a Lagrange multiplier formulation is used to generate the coupled large displacement-small deformation equations of motion. A standard, lumped mass finite-element structural analysis code is used to generate deformation modes and deformable body mass and stiffness information. An intermediate-processor is used to calculate time-independent terms in the equations of motion and to generate input data for a large-scale dynamic analysis code that includes coupled effects of geometric nonlinearity and elastic deformation. Two examples are presented and the effects of deformation mode selection on dynamic prediction are analyzed.