A Variational Framework for Solution Method Developments in Structural Mechanics

1998 ◽  
Vol 65 (1) ◽  
pp. 242-249 ◽  
Author(s):  
K. C. Park ◽  
C. A. Felippa
Keyword(s):  
Rigid Body ◽  
Virtual Work ◽  
Full Rank ◽  
Structural Mechanics ◽  
Direct Construction ◽  
Parallel Solution ◽  
Variational Functional ◽  
Solution Algorithms ◽  
Variational Framework ◽  
Solution Methods

We present a variational framework for the development of partitioned solution algorithms in structural mechanics. This framework is obtained by decomposing the discrete virtual work of an assembled structure into that of partitioned substructures in terms of partitioned substructural deformations, substructural rigid-body displacements and interface forces on substructural partition boundaries. New aspects of the formulation are: the explicit use of substructural rigid-body mode amplitudes as independent variables and direct construction of rank-sufficient interface compatibility conditions. The resulting discrete variational functional is shown to be variation-ally complete, thus yielding a full-rank solution matrix. Four specializations of the present framework are discussed. Two of them have been successfully applied to parallel solution methods and to system identification. The potential of the two untested specializations is briefly discussed.

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.

The Calculation of Truss Deformation Using ANASYS

2014 ◽  
Vol 1008-1009 ◽  
pp. 1302-1305
Author(s):  
Hui Ding ◽  
Dong Sheng Gu ◽  
Hai Tao Hua ◽  
Tai Quan Zhou
Keyword(s):  
Static Loading ◽  
Virtual Work ◽  
Structural Mechanics ◽  
Truss Structures ◽  
Principle Of Virtual Work ◽  
Test Result

The calculation of truss deformation under static loading is studied in this paper. First, an experiment was conducted to get the truss deformation for calibration. Second, the deformation was calculated by the principle of virtual work of structural mechanics and by Ansys. The calculated results agree well with the test result. Finally, some suggestions are presented to the calculation of more complicated truss structures.

Simulation of Mating Between Non-Analytic Surfaces Using a Mathematical Programming Formulation

2005 ◽  
Author(s):  
Robert Scott Pierce ◽  
David Rosen
Keyword(s):  
Constrained Optimization ◽  
End Milling ◽  
Tolerance Analysis ◽  
Mechanical Assembly ◽  
Solution Algorithms ◽  
Solution Methods ◽  
Gradient Based ◽  
Unconstrained Problem ◽  
Typical Component ◽  
Randomized Search

In this paper we describe a new method for simulating mechanical assembly between components that are composed of surfaces that do not have perfect geometric form. Mating between these imperfect form surfaces is formulated as a constrained optimization problem of the form “minimize the distance from perfect fit, subject to non-interference between components.” We explore the characteristics of this mating problem and investigate the applicability of several potential solution algorithms. The problem can be solved by converting the constrained optimization formulation into an unconstrained problem using a penalty-function approach. We describe the characteristics of this unconstrained formulation and test the use of two different solution methods: a randomized search technique and a gradient-based method. We test the algorithm by simulating mating between component models that exhibit form errors typically generated in end-milling processes. These typical component variants are used as validation problems throughout our work. Results of two different validation problems are presented. Using these results, we evaluate the applicability of the mating algorithm to the problem of mechanical tolerance analysis for assemblies and mechanisms.

On Solution Algorithms for Large Sparse Systems of Normal Equations in Photogrammetry

1981 ◽  
Vol 35 (1) ◽  
pp. 37-51
Author(s):  
Franz Steidler
Keyword(s):  
Bundle Adjustment ◽  
Solution Technique ◽  
Systems Of Equations ◽  
Direct Solution ◽  
Band Matrices ◽  
Normal Equations ◽  
Simple Version ◽  
Solution Algorithms ◽  
Solution Methods ◽  
Sparse Systems

The question of which algorithm is most appropriate for the solution of normal equations with different structures has been investigated. The solution methods are a direct solution for banded and banded-bordered matrices, a special direct solution technique for arbitrary sparse systems of equations and the method of conjugate gradients. The algorithms have first been applied to photogrammetric bundle adjustment with self-calibration, which leads to a banded-bordered matrix of the normal equations, and secondly to calculations of digital height models that are generated by a simple version of the method of finite elements and which lead to band matrices.

Modeling and analytical methods for a mounting system with double stage isolation

2021 ◽  
pp. 095440702110632
Author(s):  
Yawei Zheng ◽  
Wen-Bin Shangguan ◽  
Yingzi Kang
Keyword(s):  
Free Vibration ◽  
Rigid Body ◽  
Motion Control ◽  
Three Dimensional ◽  
Vibration Equation ◽  
Proposed Model ◽  
Solution Methods ◽  
Rigid Body Modes ◽  
Simulation Results ◽  
Solutions Of Equations

A calculation method for obtaining the displacements and rigid body modes of a Powertrain Mounting System (PMS) with double stage isolation is proposed in this paper. Firstly, the PMS with double stage isolation is modeled as a 12 Degree of Freedoms (DOFs) model, which includes six DOFs for the powertrain and the subframe respectively. The mounts are simplified as a three-dimensional spring along each axis of its Local Mount Coordinate System (LMCS), which takes the non-linear relation of the force versus the displacement of each spring into account. Secondly, the quasi-static equilibrium equation and the free vibration equation as well as the forced vibration equation of the proposed model are derived and the solutions of equations are presented. Then, the calculation and solution methods are validated by the simulation results. The differences of rigid body modes and displacements of the powertrain between single and double stage isolation are estimated, which demonstrates that the proposed model is more accurate, especially when powertrain mounts are stiff. Also, the effect of locations for powertrain mounts on car body is investigated, which shows that is beneficial for motion control of powertrain.

The Development of Force-Deflection Relationships for Compliant Mechanisms

1994 ◽  
Author(s):  
Larry L. Howell ◽  
Ashok Midha
Keyword(s):  
Rigid Body ◽  
Compliant Mechanisms ◽  
Virtual Work ◽  
Compliant Mechanism ◽  
Flexible Link ◽  
Model Concept ◽  
Body Model ◽  
Design Flexibility ◽  
Rigid Body Model ◽  
Manufacturing Time

Abstract The advantages of compliant or flexible link mechanisms include increased design flexibility and reduction in manufacturing time and cost. The analysis of such mechanisms may be difficult and time consuming due to the nonlinearities introduced by large deflections. Also, unlike rigid-body mechanisms, the type and form of motion of a compliant mechanism is dependent on the location and magnitude of applied loads. The pseudo-rigid-body model concept has been developed to simplify the analysis of compliant mechanisms by allowing them to be modeled as rigid-link mechanisms with springs. This work uses the principle of virtual work and the pseudo-rigid-body model concept to develop force-deflection relationships for compliant mechanisms. Several examples are presented, and general design equations are derived for pseudo-rigid-body four-bar and slider-crank mechanisms.

Bilevel Integer Programs with Stochastic Right-Hand Sides

2021 ◽  
Author(s):  
Junlong Zhang ◽  
Osman Y. Özaltın
Keyword(s):  
Structural Properties ◽  
Large Scale ◽  
Value Function ◽  
Integer Program ◽  
Value Functions ◽  
Integer Programs ◽  
Solution Algorithms ◽  
Right Hand ◽  
Solution Methods ◽  
The Value Function

We develop an exact value function-based approach to solve a class of bilevel integer programs with stochastic right-hand sides. We first study structural properties and design two methods to efficiently construct the value function of a bilevel integer program. Most notably, we generalize the integer complementary slackness theorem to bilevel integer programs. We also show that the value function of a bilevel integer program can be characterized by its values on a set of so-called bilevel minimal vectors. We then solve the value function reformulation of the original bilevel integer program with stochastic right-hand sides using a branch-and-bound algorithm. We demonstrate the performance of our solution methods on a set of randomly generated instances. We also apply the proposed approach to a bilevel facility interdiction problem. Our computational experiments show that the proposed solution methods can efficiently optimize large-scale instances. The performance of our value function-based approach is relatively insensitive to the number of scenarios, but it is sensitive to the number of constraints with stochastic right-hand sides. Summary of Contribution: Bilevel integer programs arise in many different application areas of operations research including supply chain, energy, defense, and revenue management. This paper derives structural properties of the value functions of bilevel integer programs. Furthermore, it proposes exact solution algorithms for a class of bilevel integer programs with stochastic right-hand sides. These algorithms extend the applicability of bilevel integer programs to a larger set of decision-making problems under uncertainty.

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