Numerical Algorithms for Simulation of One-Dimensional Mechanical Systems With Clearance-Type Nonlinearities

2019 ◽  
Vol 14 (6) ◽  
Author(s):  
Lina Wramner
Keyword(s):  
Arbitrary Number ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Gear Ratio ◽  
One Dimensional ◽  
Linear Complementary Problem

In many mechanical systems there are nonlinearities of clearance type. This type of nonlinearity often causes problems with convergence and accuracy in simulations, due to the discontinuities at impact. For systems with gap-activated springs connected to ground, it has been proposed in previous work to reformulate the problem as a linear complementary problem (LCP), which can be solved in a very efficient way. In this paper, a generalization of the LCP approach is proposed for systems with gap-activated springs connecting different bodies. The generalizations enable the LCP approach to be used for an arbitrary number of gap-activated springs connecting either different bodies or connecting bodies to ground. The springs can be activated in either compression or expansion or both and a gear ratio can be included between the bodies. The efficiency of the algorithm is demonstrated with an application example of a dual mass flywheel (DMF).

On the Use of the Canonical Equations of Motion for the Dynamic Analysis of Constrained Multibody Systems

1992 ◽  
Author(s):  
E. Bayo ◽  
J. M. Jimenez
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Constrained Multibody Systems ◽  
Constrained Mechanical Systems ◽  
First Order ◽  
Canonical Variables ◽  
Lagrange’S Multipliers

Abstract We investigate in this paper the different approaches that can be derived from the use of the Hamiltonian or canonical equations of motion for constrained mechanical systems with the intention of responding to the question of whether the use of these equations leads to more efficient and stable numerical algorithms than those coming from acceleration based formalisms. In this process, we propose a new penalty based canonical description of the equations of motion of constrained mechanical systems. This technique leads to a reduced set of first order ordinary differential equations in terms of the canonical variables with no Lagrange’s multipliers involved in the equations. This method shows a clear advantage over the previously proposed acceleration based formulation, in terms of numerical efficiency. In addition, we examine the use of the canonical equations based on independent coordinates, and conclude that in this second case the use of the acceleration based formulation is more advantageous than the canonical counterpart.

Approaches to the Numerical Estimates of Grid Convergence of NSE in the Presence of Singularities

2018 ◽  
Vol 19 (3-4) ◽  
pp. 281-287
Author(s):  
Chenguang Zhang ◽  
Krishnaswamy Nandakumar
Keyword(s):  
Numerical Algorithms ◽  
Nonlinear Problems ◽  
Order Of Accuracy ◽  
One Dimensional ◽  
Accuracy Order ◽  
Uniform Grids ◽  
Grid Convergence ◽  
The Common ◽  
Spatial Grid ◽  
Different Levels

AbstractEvaluating the order of accuracy (order) is an integral part of the development and application of numerical algorithms. Apart from theoretical functional analysis to place bounds on error estimates, numerical experiments are often essential for nonlinear problems to validate the estimates in a reliable answer. The common workflow is to apply the algorithm using successively finer temporal/spatial grid resolutions ${\delta _i}$, measure the error ${\isin _i}$ in each solution against the exact solution, the order is then obtained as the slope of the line that fits $(\log {\isin _i}, \log {\delta _i})$. We show that if the problem has singularities like divergence to infinity or discontinuous jump, this common workflow underestimates the order if solution at regions around the singularity is used. Several numerical examples with different levels of complexity are explored. A simple one-dimensional theoretical model shows it is impossible to numerically evaluate the order close to singularity on uniform grids.

Integrated Origami-String System

2019 ◽  
Author(s):  
Ke Liu ◽  
Madelyn Kosednar ◽  
Tomohiro Tachi ◽  
Glaucio H. Paulino
Keyword(s):  
Mechanical Behavior ◽  
Design Space ◽  
Energy Landscape ◽  
Mechanical Systems ◽  
Two Dimensional ◽  
Structural System ◽  
One Dimensional ◽  
Elastic Strings ◽  
Paper Folding

Abstract Origami-inspired mechanical systems are mostly composed of two-dimensional elements, a feature inherited from paper folding. However, do we have to comply with this restriction on our design space? Would it be more approachable to achieve desired performance by integrating elements of different abstract dimensions? In this paper, we propose an integrated structural system consisting of both two-dimensional and one-dimensional elements. We attach elastic strings onto an origami design to modify its mechanical behavior and create new features. We show that, by introducing elastic strings to the recently proposed Morph pattern, we can obtain bistable units with programmable energy landscape. The behavior of this integrated origami-string system can be described by an elegant formulation, which can be used to explore its rich programmability.

Workspace Analysis of Multibody Mechanical Systems Using Continuation Methods

1988 ◽  
Author(s):  
D.-Y. Jo ◽  
E. J. Haug
Keyword(s):  
Mechanical Systems ◽  
Point Of View ◽  
Continuation Methods ◽  
Dimensional Manifold ◽  
One Dimensional ◽  
Generalized Coordinates ◽  
Workspace Analysis ◽  
Manifold Mapping ◽  
Manifold Theory

Abstract A new approach to numerical analysis of workspaces of multibody mechanical systems is developed. Numerical techniques that are based on manifold theory and utilize continuation methods are presented and applied to a variety of mechanical systems, including closed-loop mechanisms. Generalized coordinates that define kinematics of a system are classified and interpreted from an input-output point of view. Boundaries of workspaces, which depend on the classification of generalized coordinates, are defined as sets of singular points of Jacobians of the kinematic equations. Numerical methods for tracing one dimensional trajectories on a workspace boundary are outlined and example are analyzed using one dimensional manifold mapping computer programs, such as PITCON and AUTO.

Numerical Algorithms for Mapping Boundaries of Manipulator Workspaces

1996 ◽  
Vol 118 (2) ◽  
pp. 228-234 ◽  
Author(s):  
E. J. Haug ◽  
Chi-Mei Luh ◽  
F. A. Adkins ◽  
Jia-Yi Wang
Keyword(s):  
Continuation Method ◽  
Initial Point ◽  
Numerical Algorithms ◽  
Companion Paper ◽  
Dimensional Solution ◽  
One Dimensional ◽  
Bifurcation Points ◽  
Input And Output ◽  
Taylor Expansions ◽  
Tangent Calculation

Numerical algorithms for mapping boundaries of manipulator workspaces are developed and illustrated. Analytical criteria for boundaries of workspaces for both manipulators having the same number of input and output coordinates and redundantly controlled manipulators with a larger number of inputs than outputs are well known, but reliable numerical methods for mapping them have not been presented. In this paper, a numerical method is first developed for finding an initial point on the boundary. From this point, a continuation method that accounts for simple and multiple bifurcation of one-dimensional solution curves is developed. Second order Taylor expansions are derived for finding tangents to solution curves at simple bifurcation points of continuation equations and for characterizing barriers to control of manipulators. A recently developed method for tangent calculation at multiple bifurcation points is employed. A planar redundantly controlled serial manipulator is analyzed, determining both the exterior boundary of the accessible output set and interior curves that represent local impediments to motion control. Using these methods, more complex planar and spatial Stewart platform manipulators are analyzed in a companion paper.

A Sparsity-Oriented Approach to the Dynamic Analysis and Design of Mechanical Systems—Part 1

1977 ◽  
Vol 99 (3) ◽  
pp. 773-779 ◽  
Author(s):  
N. Orlandea ◽  
M. A. Chace ◽  
D. A. Calahan
Keyword(s):  
Computer Simulation ◽  
Dynamic Analysis ◽  
Numerical Solutions ◽  
Sparse Matrix ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Mechanical Systems ◽  
Numerical Algorithms ◽  
Front Suspension ◽  
Analysis And Design

The work described herein is an extension of sparse matrix and stiff integrated numerical algorithms used for the simulation of electrical circuits and three-dimensional mechanical dynamic systems. By applying these algorithms big sets of sparse linear equations can be solved efficiently, and the numerical instability associated with widely split eigenvalues can be avoided. The new numerical methods affect even the initial formulation for these problems. In this paper, the equations of motion and constraints (Part 1) and the force function of springs and dampers (Part 2) are set up, and the numerical solutions for static, transient, and linearized types of analysis as well as the modal optimization algorithms are implemented in the ADAMS (automatic dynamic analysis of mechanical systems) computer program for simulation of three-dimensional mechanical systems (Part 2). The paper concludes with two examples: computer simulation of the front suspension of a 1973 Chevrolet Malibu and computer simulation of the landing gear of a Boeing 747 airplane. The efficiency of simulation and comparison with experimental results are given in tabular form.

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