Steady Motion of a Slack Belt Drive: Dynamics of a Beam in Frictional Contact With Rotating Pulleys

2020 ◽  
Vol 87 (12) ◽  
Author(s):  
Jakob Scheidl ◽  
Yury Vetyukov
Keyword(s):  
Finite Element ◽  
Steady State ◽  
Boundary Value Problem ◽  
Contact Region ◽  
Boundary Value ◽  
Steady Motion ◽  
Steady State Solution ◽  
Belt Drive ◽  
Creep Theory ◽  
Lagrangian Finite Element

Abstract We seek the steady-state motion of a slack two-pulley belt drive with the belt modeled as an elastic, shear-deformable rod. Dynamic effects and gravity induce significant transverse deflections due to the low pre-tension. In analogy to the belt-creep theory, it is assumed that each contact region between the belt and one of the pulleys consists of a single sticking and a single sliding zone. Based on the governing equations of the rod theory, we for the first time derive the corresponding boundary value problem and integrate it numerically. Furthermore, a novel mixed Eulerian–Lagrangian finite element scheme is developed that iteratively seeks the steady-state solution. Finite element solutions are validated against semi-analytic results obtained by numerical integration of the boundary value problem. Parameter studies are conducted to examine solution dependence on the stiffness coefficients and the belt pre-tension.

scholarly journals Benchmark for the Coupled Magneto-Mechanical Boundary Value Problem in Magneto-Active Elastomers

Materials ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2380
Author(s):  
Philipp Metsch ◽  
Raphael Schiedung ◽  
Ingo Steinbach ◽  
Markus Kästner
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Three Dimensional ◽  
Diffuse Interface ◽  
Two Dimensional ◽  
Benchmark Problem ◽  
Material Models ◽  
Lagrangian Finite Element ◽  
Good Agreement

Within this contribution, a novel benchmark problem for the coupled magneto-mechanical boundary value problem in magneto-active elastomers is presented. Being derived from an experimental analysis of magnetically induced interactions in these materials, the problem under investigation allows us to validate different modeling strategies by means of a simple setup with only a few influencing factors. Here, results of a sharp-interface Lagrangian finite element framework and a diffuse-interface Eulerian approach based on the application of a spectral solver on a fixed grid are compared for the simplified two-dimensional as well as the general three-dimensional case. After influences of different boundary conditions and the sample size are analyzed, the results of both strategies are examined: for the material models under consideration, a good agreement of them is found, while all discrepancies can be ascribed to well-known effects described in the literature. Thus, the benchmark problem can be seen as a basis for future comparisons with both other modeling strategies and more elaborate material models.

The Weighted Error Estimation of Finite Element Method for Two-Point Boundary Value Problem

2012 ◽  
Vol 182-183 ◽  
pp. 1571-1574
Author(s):  
Qi Sheng Wang ◽  
Jia Dao Lai
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Error Estimation ◽  
Boundary Value ◽  
Point Boundary ◽  
Weighted Error ◽  
Element Method

In this paper, the weighed error estimation of finite element method for the two-point boundary value problems are discussed. Respectively, the norm estimation of the H1 and L2 are obtained.

Convergence Analysis for the Nested Refinement of Triangular Finite Element

2011 ◽  
Vol 317-319 ◽  
pp. 1926-1930 ◽  
Author(s):  
Qi Sheng Wang ◽  
Yi Gao Zhao
Keyword(s):  
Finite Element ◽  
Boundary Value Problem ◽  
Convergence Analysis ◽  
Poisson Equation ◽  
Boundary Value ◽  
Triangular Mesh ◽  
Triangular Grid ◽  
Triangular Finite Element ◽  
Convergence Results

In this paper, the method of the nested refinement for triangular mesh and some relevant conclusions are considered. The Κ level triangular grid nested refinement on the plan domain Ω and some related properties are discussed , and the convergence results are obtained for the first boundary value problem of Poisson equation under the nested refinement of triangular finite element.

Stable finite element method for solving the oblique derivative boundary value problems in geodesy

2021 ◽  
Author(s):  
Marek Macák ◽  
Zuzana Minarechová ◽  
Róbert Čunderlík ◽  
Karol Mikula
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
The Finite Element Method ◽  
Gravity Field Modelling ◽  
Field Modelling ◽  
Local Gravity ◽  
Local Gravity Field Modelling ◽  
Element Method

<p><span>We presents local gravity field modelling in a spatial domain using the finite element method (FEM). FEM as a numerical method is applied for solving the geodetic boundary value problem with oblique derivative boundary conditions (BC). We derive a novel FEM numerical scheme which is the second order accurate and more stable than the previous one published in [1]. A main difference is in applying the oblique derivative BC. While in the previous FEM approach it is considered as an average value on the bottom side of finite elements, the novel FEM approach is based on the oblique derivative BC considered in relevant computational nodes. Such an approach should reduce a loss of accuracy due to averaging. Numerical experiments present </span><span>(i) </span><span>a reconstruction of EGM2008 as a harmonic function over the extremely complicated Earth’s topography in the Himalayas and Tibetan Plateau, and (ii) local gravity field modelling in Slovakia with the high-resolution 100 x 100 m while using terrestrial gravimetric data.</span></p><p><span>[1] </span>Macák, Z. Minarechová, R. Čunderlík, K. Mikula, The finite element method as a tool to solve the oblique derivative boundary value problem in geodesy. Tatra Mountains Mathematical Publications. Vol. 75, no. 1, 63-80, (2020)</p>

Analysis of Belt-Driven Mechanics Using a Creep-Rate-Dependent Friction Law

2002 ◽  
Vol 69 (6) ◽  
pp. 763-771 ◽  
Author(s):  
M. J. Leamy ◽  
T. M. Wasfy
Keyword(s):  
Finite Element ◽  
Creep Rate ◽  
Contact Region ◽  
Unit Length ◽  
Velocity Ratio ◽  
Belt Drive ◽  
Friction Law ◽  
Dynamic Finite Element ◽  
Rate Dependent ◽  
Coulomb Law

An analysis of the frictional mechanics of a steadily rotating belt drive is carried out using a physically appropriate creep-rate-dependent friction law. Unlike in belt-drive mechanics analyzed using a Coulomb friction law, the current analysis predicts no adhesion zones in the belt-pulley contact region. Regardless of this finding, for the limiting case of a creep-rate law approaching a Coulomb law, all predicted response quantities (including the extent of belt creep on each pulley) approach those predicted by the Coulomb law analysis. Depending on a slope parameter governing the creep-rate profile, one or two sliding zones exist on each pulley, which together span the belt-pulley contact region. Closed-form expressions are obtained for the tension distribution, the sliding-zone arc magnitudes, and the frictional and normal forces per unit length exerted on the belt. A sample two-pulley belt drive is analyzed further to determine its pulley angular velocity ratio and belt-span tensions. Results from this analysis are compared to a dynamic finite element solution of the same belt drive. Excellent agreement in predicted results is found. Due to the presence of arbitrarily large system rotations and a numerically friendly friction law, the analytical solution presented herein is recommended as a convenient comparison test case for validating friction-enabled dynamic finite element schemes.

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