Modeling of Multibody Flexible Articulated Structures With Mutually Coupled Motions: Part I — General Theory

1992 ◽  
Author(s):  
Jiechi Xu ◽  
Joseph R. Baumgarten
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Local Level ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Moving Frame ◽  
Element Analysis ◽  
Highly Nonlinear ◽  
Geometric Boundary ◽  
System Equations

Abstract In the present paper a general systematic modeling procedure has been conducted in deriving dynamic equations of motion using Lagrange’s approach for a spatial multibody structural system involving rigid bodies and elastic members. Both the rigid body degrees of freedom and the elastic degrees of freedom are considered as unknown generalized coordinates of the entire system in order to reflect the nature of mutually coupled rigid body and elastic motions. The assumption of specified rigid body gross motion is no longer necessary in the equation derivation and the resulting differential equations are highly nonlinear. Finite element analysis (FEA) with direct stiffness method has been employed to model the flexible substructures. Nonlinear coupling terms between the rigid body and elastic motions are fully derived and are explicitly expressed in matrix form. The equations of motion of each individual subsystem are formulated based on a moving frame instead of a traditional inertial frame. These local level equations of motion are assembled to obtain the system equations with the implementation of geometric boundary conditions by means of a compatibility matrix.

Explicit Poincaré equations of motion for general constrained systems. Part II. Applications to multi-body dynamics and nonlinear control

2007 ◽  
Vol 463 (2082) ◽  
pp. 1435-1446 ◽  
Author(s):  
Firdaus E Udwadia ◽  
Phailaung Phohomsiri
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Control ◽  
Rigid Body ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Constrained Systems ◽  
The Third ◽  
Highly Nonlinear ◽  
Body Dynamics ◽  
Multi Body

The power of the new equations of motion developed in part I of this paper is illustrated using three examples from multi-body dynamics. The first two examples deal with the problem of accurately controlling the orientation of a rigid body, while the third example deals with the synchronization of two rigid bodies so that their relative orientations are ‘locked’ through prescribed dynamical relationships. The ease, simplicity and accuracy with which control of such highly nonlinear systems is achieved are demonstrated.

A Hybrid Highly Parallelizable Algorithm for the Dynamics of Rigid Body Chain Systems

1999 ◽  
Author(s):  
Shanzhong Duan ◽  
Kurt S. Anderson
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Constraint Forces ◽  
Dynamics Of Rigid Body ◽  
A Chain ◽  
Explicit Determination ◽  
Order Algorithm

Abstract The paper presents a new hybrid parallelizable low order algorithm for modeling the dynamic behavior of multi-rigid-body chain systems. The method is based on cutting certain system interbody joints so that largely independent multibody subchain systems are formed. These subchains interact with one another through associated unknown constraint forces f¯c at the cut joints. The increased parallelism is obtainable through cutting the joints and the explicit determination of associated constraint loads combined with a sequential O(n) procedure. In other words, sequential O(n) procedures are performed to form and solve equations of motion within subchains and parallel strategies are used to form and solve constraint equations between subchains in parallel. The algorithm can easily accommodate the available number of processors while maintaining high efficiency. An O[(n+m)Np+m(1+γ)Np+mγlog2Np](0<γ<1) performance will be achieved with Np processors for a chain system with n degrees of freedom and m constraints due to cutting of interbody joints.

scholarly journals An Efficient Contact Model for the Simulation of Cargo Airdrop Extraction Phase

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Leiming Ning ◽  
Jichang Chen ◽  
Mingbo Tong
Keyword(s):  
Degrees Of Freedom ◽  
Friction Model ◽  
Rigid Bodies ◽  
Contact Dynamics ◽  
Moving Frame ◽  
Contact Friction ◽  
Practical Implementation ◽  
Contact Formulation ◽  
Efficient Code ◽  
Extraction Phase

A high-fidelity cargo airdrop simulation requires the accurate modeling of the contact dynamics between an aircraft and its cargo. This paper presents a general and efficient contact-friction model for the simulation of aircraft-cargo coupling dynamics during an airdrop extraction phase. The proposed approach has the same essence as the finite element node-to-segment contact formulation, which leads to a flexible, straightforward, and efficient code implementation. The formulation is developed under an arbitrary moving frame with both aircraft and cargo treated as general six degrees-of-freedom rigid bodies, thus eliminating the restrictions of lateral symmetric assumptions in most existing methods. Moreover, the aircraft-cargo coupling algorithm is discussed in detail, and some practical implementation details are presented. The accuracy and capability of the present method are demonstrated through four numerical examples with increasing complexity and fidelity.

Development of Equations of Motion for a Stewart Platform Under Prescribed Mount Motion

2018 ◽  
Author(s):  
Takeyuki Ono ◽  
Ryosuke Eto ◽  
Junya Yamakawa ◽  
Hidenori Murakami
Keyword(s):  
Rigid Body ◽  
Virtual Work ◽  
Equations Of Motion ◽  
Base Plate ◽  
Stewart Platform ◽  
Euler Method ◽  
Moving Frame ◽  
Real Time Control ◽  
Precise Positioning ◽  
Time Control

Analytical equations of motion are critical for real-time control of translating manipulators, which require precise positioning of various tools for their mission. Specifically, when manipulators mounted on moving robots or vehicles perform precise positioning of their tools, it becomes economical to develop a Stewart platform, whose sole task is stabilizing the orientation and crude position of its top table, onto which various precision tools are attached. In this paper, analytical equations of motion are developed for a Stewart platform whose motion of the base plate is prescribed. To describe the kinematics of the platform, the moving frame method, presented by one of authors [1,2], is employed. In the method the coordinates of the origin of a body attached coordinate system and vector basis are expressed by using 4 × 4 frame connection matrices, which form the special Euclidean group, SE(3). The use of SE(3) allows accurate description of kinematics of each rigid body using (relative) joint coordinates. In kinetics, the principle of virtual work is employed, in which system virtual displacements are expressed through B-matrix by essential virtual displacements, reflecting the connection of the rigid body system [2]. The resulting equations for fixed base plate reduce to those for the top plate, obtained by the Newton-Euler method. A main result of the paper is the analytical equations of motion in matrix form for dynamics analyses of a Stewart platform whose base plate moves. The control applications of those equations will be deferred to subsequent publications.

scholarly journals The Dynamical Behavior of a Rigid Body Relative Equilibrium Position

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
T. S. Amer
Keyword(s):  
Rigid Body ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Relative Equilibrium ◽  
Dynamical Behavior ◽  
Vertical Plane ◽  
The Body ◽  
Physical Parameters ◽  
Pendulum Model

In this paper, we will focus on the dynamical behavior of a rigid body suspended on an elastic spring as a pendulum model with three degrees of freedom. It is assumed that the body moves in a rotating vertical plane uniformly with an arbitrary angular velocity. The relative periodic motions of this model are considered. The governing equations of motion are obtained using Lagrange’s equations and represent a nonlinear system of second-order differential equations that can be solved in terms of generalized coordinates. The numerical solutions are investigated using the fourth-order Runge-Kutta algorithms through Matlab packages. These solutions are represented graphically in order to describe and discuss the behavior of the body at any instant for different values of the physical parameters of the body. The obtained results have been discussed and compared with some previous published works. Some concluding remarks have been presented at the end of this work. The importance of this work is due to its numerous applications in life such as the vibrations that occur in buildings and structures.

Contact-space resolution model for a physically consistent view of simultaneous collisions in articulated-body systems: theory and experimental results

2020 ◽  
Vol 39 (10-11) ◽  
pp. 1239-1258
Author(s):  
Shameek Ganguly ◽  
Oussama Khatib
Keyword(s):  
System Dynamics ◽  
Rigid Body ◽  
Degrees Of Freedom ◽  
Null Space ◽  
Rigid Bodies ◽  
Computationally Efficient ◽  
Space Resolution ◽  
Resolution Model ◽  
Space Dynamics ◽  
Contact Space

Multi-surface interactions occur frequently in articulated-rigid-body systems such as robotic manipulators. Real-time prediction of contact-interaction forces is challenging for systems with many degrees of freedom (DOFs) because joint and contact constraints must be enforced simultaneously. While several contact models exist for systems of free rigid bodies, fewer models are available for articulated-body systems. In this paper, we extend the method of Ruspini and Khatib and develop the contact-space resolution (CSR) model by applying the operational space theory of robot manipulation. Through a proper choice of contact-space coordinates, the projected dynamics of the system in the contact space is obtained. We show that the projection into the dynamically consistent null space preserves linear and angular momentum in a subspace of the system dynamics complementary to the joint and contact constraints. Furthermore, we illustrate that a simultaneous collision event between two articulated bodies can be resolved as an equivalent simultaneous collision between two non-articulated rigid bodies through the projected contact-space dynamics. Solving this reduced-dimensional problem is computationally efficient, but determining its accuracy requires physical experimentation. To gain further insights into the theoretical model predictions, we devised an apparatus consisting of colliding 1-, 2-, and 3-DOF articulated bodies where joint motion is recorded with high precision. Results validate that the CSR model accurately predicts the post-collision system state. Moreover, for the first time, we show that the projection of system dynamics into the mutually complementary contact space and null space is a physically verifiable phenomenon in articulated-rigid-body systems.

A Unified Framework for Dynamics and Lyapunov Stability of Holonomically Constrained Rigid Bodies

2005 ◽  
Vol 9 (4) ◽  
pp. 387-394 ◽  
Author(s):  
Khoder Melhem ◽  
◽  
Zhaoheng Liu ◽  
Antonio Loría ◽  
◽  
...  
Keyword(s):  
System Dynamics ◽  
Lyapunov Stability ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Minimal Realization ◽  
State Variables ◽  
Unified Framework ◽  
Constrained System ◽  
And Control

A new dynamic model for interconnected rigid bodies is proposed here. The model formulation makes it possible to treat any physical system with finite number of degrees of freedom in a unified framework. This new model is a nonminimal realization of the system dynamics since it contains more state variables than is needed. A useful discussion shows how the dimension of the state of this model can be reduced by eliminating the redundancy in the equations of motion, thus obtaining the minimal realization of the system dynamics. With this formulation, we can for the first time explicitly determine the equations of the constraints between the elements of the mechanical system corresponding to the interconnected rigid bodies in question. One of the advantages coming with this model is that we can use it to demonstrate that Lyapunov stability and control structure for the constrained system can be deducted by projection in the submanifold of movement from appropriate Lyapunov stability and stabilizing control of the corresponding unconstrained system. This procedure is tested by some simulations using the model of two-link planar robot.

Dynamic Response Analysis of Large Latticed Space Structures

1989 ◽  
Vol 4 (1) ◽  
pp. 25-42 ◽  
Author(s):  
A.R. Kukreti ◽  
N.D. Uchil
Keyword(s):  
Dynamic Response ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Time ◽  
Space Structures ◽  
Response Analysis ◽  
Actual System ◽  
Large Space ◽  
Element Analysis ◽  
Dynamic Response Analysis

In this paper an alternative method for dynamic response analysis of large space structures is presented, for which conventional finite element analysis would require excessive computer storage and computational time. Latticed structures in which the height is very small in comparison to its overall length and width are considered. The method is based on the assumption that the structure can be embedded in its continuum, in which any fiber can translate and rotate without deforming. An appropriate kinematically admissable series function is constructed to descrbe the deformation of the middle plane of this continuum. The unknown coefficients in this function are called the degree-of-freedom of the continuum, which is given the name “super element.” Transformation matrices are developed to express the equations of motion of the actual systems in terms of the degrees-of-freedom of the super element. Thus, by changing the number of terms in the assumed function, the degrees-of-freedom of the super element can be increased or decreased. The super element response results are transformed back to obtain the desired response results of the actual system. The method is demonstrated for a structure woven in the shape of an Archimedian spiral.

Three-Dimensional Creep Analysis of Solder Joints in Surface Mount Devices

1995 ◽  
Vol 117 (1) ◽  
pp. 20-25 ◽  
Author(s):  
Pardeep K. Bhatti ◽  
Klaus Gschwend ◽  
Abel Y. Kwang ◽  
Ahmer R. Syed
Keyword(s):  
High Performance ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Nonlinear Material ◽  
Computational Time ◽  
Nonlinear Creep ◽  
Element Analysis ◽  
Surface Mount ◽  
Creep Analysis ◽  
Highly Nonlinear

Three-dimensional finite element analysis has been applied for determining time-dependent solder joint response of leaded surface mount components under thermal cycling. Two main challenges are the geometric complexity in mesh development and computationally intensive analysis because of the highly nonlinear material properties. Advanced techniques have been applied, including multi-point constraints for mesh transition, which reduces the number of degrees of freedom in the model, and substructuring, which effectively reduces computational time in the iterative analysis. The result is a generic approach for nonlinear creep analysis using commercial FEA software on a high performance workstation. Illustrations are provided for J and gullwing leaded packages.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

1999 ◽  
Vol 66 (4) ◽  
pp. 986-996 ◽  
Author(s):  
S. K. Saha
Keyword(s):  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Kinematic Chain ◽  
Rigid Bodies ◽  
Dynamic Equations ◽  
Orthogonal Complement ◽  
Forward Dynamics ◽  
Velocity Constraints

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n—n being the degrees-of-freedom of the system at hand—inverse dynamics and order n3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the system’s accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

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