Low Order Static Load Distribution Model for Ball Screw Mechanisms Including Effects of Lateral Deformation and Geometric Errors

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Bo Lin ◽  
Chinedum E. Okwudire ◽  
Jason S. Wou
Keyword(s):  
Load Distribution ◽  
Static Load ◽  
High Order ◽  
Contact Model ◽  
Ball Screw ◽  
Distribution Model ◽  
Computational Time ◽  
Geometric Errors ◽  
Lateral Deformation ◽  
Low Order

Accurate modeling of static load distribution of balls is very useful for proper design and sizing of ball screw mechanisms (BSMs); it is also a starting point in modeling the dynamics, e.g., friction behavior, of BSMs. Often, it is preferable to determine load distribution using low order models, as opposed to computationally unwieldy high order finite element (FE) models. However, existing low order static load distribution models for BSMs are inaccurate because they ignore the lateral (bending) deformations of screw/nut and do not adequately consider geometric errors, both of which significantly influence load distribution. This paper presents a low order static load distribution model for BSMs that incorporates lateral deformation and geometric error effects. The ball and groove surfaces of BSMs, including geometric errors, are described mathematically and used to establish a ball-to-groove contact model based on Hertzian contact theory. Effects of axial, torsional, and lateral deformations are incorporated into the contact model by representing the nut as a rigid body and the screw as beam FEs connected by a newly derived ball stiffness matrix which considers geometric errors. Benchmarked against a high order FE model in case studies, the proposed model is shown to be accurate in predicting static load distribution, while requiring much less computational time. Its ease-of-use and versatility for evaluating effects of sundry geometric errors, e.g., pitch errors and ball diameter variation, on static load distribution are also demonstrated. It is thus suitable for parametric studies and optimal design of BSMs.

scholarly journals Numerical characterization of local and global non-uniformities in the load distribution in ball screws

2021 ◽  
Author(s):  
Luca Sangalli ◽  
Aitor Oyanguren ◽  
Jon Larrañaga ◽  
Aitor Arana ◽  
Mikel Izquierdo ◽  
...  
Keyword(s):  
Load Distribution ◽  
High Order ◽  
Ball Screw ◽  
Computational Time ◽  
Ball Size ◽  
Numerical Characterization ◽  
Design Variables ◽  
Characteristic Features ◽  
Load Arrangement

AbstractLoad distribution in ball screws is a representation of the ball contact stress, and it is fundamental to understanding the behavior of these machine elements. This work aims to conduct a multi-variable analysis of the load distribution in ball screws. For this purpose, a numerical tool is developed for the generation and calculation of ball screw finite element (FE) models, which has been validated against the state of the art. This tool is based on the combination of an analytical contact model and the use of high-order FE models for the analysis of the load distribution of ball screws and stands out for its accuracy (less than 1% error against high-order FE models), adaptability, versatility (models are generated with more than 20 design variables and they can be introduced as components in larger models) and efficiency (being the computational time 1.25% of that of a high-order FE models) with respect to other existing models. Many different design variables (number of start threads, pitch, contact angle, ball size, slenderness and load arrangement) are studied in order to obtain a general characterization of the morphology of the load distribution in ball screws. Among them, the most influential variables on the load distribution and therefore on the structural behavior of ball screws are, load arrangement (with ratio r variations of up to 25% on the same ball screw) and slenderness (with ratio variations of up to 13% on ball screws with two turns of difference). The two most characteristic features, the non-uniformity at a local and global level are identified, along with as the possible causes of their appearance and the consequences that they may cause.

Analysis of load distribution and contact stiffness of the single-nut ball screw based on the whole rolling elements model

2019 ◽  
Vol 43 (3) ◽  
pp. 344-365 ◽  
Author(s):  
Ye Chen ◽  
Chun-yu Zhao ◽  
Si-yu Zhang ◽  
Xian-li Meng
Keyword(s):  
Load Distribution ◽  
Contact Stiffness ◽  
Structural Parameters ◽  
Contact Angles ◽  
Ball Screw ◽  
Distribution Model ◽  
Axial Stiffness ◽  
Feed System ◽  
Normal Contact ◽  
Rolling Elements

This paper aims to investigate the load distribution and contact stiffness characteristics of the single-nut ball screw pair (SNBSP). First, the transformed relationship of coordinate systems is established. Then, the whole rolling elements load distribution model of the SNBSP is presented. Based on this, the whole rolling elements contact stiffness model is obtained. Applying the Newton–Raphson iterative method to solve the model, the normal force of rolling elements and the contact angles between balls and raceway surface are determined. The calculation results are reasonably consistent with those of the half pitch model. Then, the local contact stiffness and global contact stiffness are obtained. Furthermore, the effects of axial load and structural parameters of the SNBSP on the normal contact force, contact angle, and local and global contact stiffness are discussed using numeric analysis. Finally, a dynamic model of the z-axis feed system with time-varying axial stiffness is established, and the accuracy of the model is verified by experiments.

scholarly journals A Stiffness Formulation for Spline Joints

2016 ◽  
Vol 138 (4) ◽  
Author(s):  
J. Hong ◽  
D. Talbot ◽  
A. Kahraman
Keyword(s):  
Stiffness Matrix ◽  
Load Distribution ◽  
Dynamic Models ◽  
Torsional Stiffness ◽  
Distribution Model ◽  
Computational Time ◽  
Inversion Method ◽  
Time Required ◽  
Dynamic Phenomena ◽  
Diagonal Coupling

Due to the lack of knowledge in terms of their flexibility and deformation, spline joints are typically assumed to be rigid in dynamic models of gearboxes, transmissions, and drivetrains. As various dynamic phenomena are associated with the stiffness of a spline joint, any high-fidelity dynamic model of drivetrains must properly capture the stiffness of spline joints. In this study, a general analytical stiffness formulation for spline joints is proposed based on a semi-analytical spline load distribution model. This formulation defines a fully populated stiffness matrix of a spline joint including radial, tilting, and torsional stiffness values as well as off-diagonal coupling terms. A blockwise inversion method is proposed and implemented with this analytical formulation to reduce computational time required. At the end, a detailed parametric study is presented to demonstrate the sensitivity of the spline stiffness matrix to torque level, tooth modifications, misalignments, and tooth indexing errors.

Partial Meshing of Synchronous Belt Teeth

2007 ◽  
Author(s):  
Tomas Johannesson
Keyword(s):  
Load Distribution ◽  
Static Load ◽  
Power Transmission ◽  
Transmission Error ◽  
Distribution Model ◽  
Velocity Difference ◽  
Line Segments ◽  
Tooth Stiffness ◽  
Overlap Area ◽  
Difference Effect

Synchronous belts have been used in power transmissions where synchronization is also needed since the 1940’s. In the 1960’s overhead camshaft engines were introduced and synchronous belts were used as cam belts. This made way for a new standard for belts: improvements were made in materials and profile geometry. These new belts had lower noise emissions and, at the same time, greater durability. Often, both wear and noise are generated when a belt tooth seats or unseats a pulley. A tooth is considered to be fully meshed when the whole belt pitch forms a circular arc. This is not the case for teeth in partial mesh, which occurs in seating and unseating zones. In these zones force peaks are often present. These peaks are believed to arise mainly as a result of two phenomena: one is the overlap effect due to the belt geometry not fitting the pulley, and other is the velocity difference effect. The latter is speed-dependent while the former depends on the belt and pulley profile geometries and the belt teeth positions relative to the pulley. Although force peaks of high magnitude occur, they are present at a such small part of the engagement that their contribution to power transmission can be neglected. This indicates that the positions of the belt pitches relative to the pulley pitches can be established by the load distribution from fully meshed conditions. Although the characteristics of partial mesh teeth have been improved by the introduction of new profiles and materials, problems of durability, noise and transmission error, arising from partially meshed teeth, are still present. Therefore it is important to study belt mechanics in seating and unseating zones. This paper describes a method to calculate force peakson seating and unseating. An overlap area (geometrical interference) is formed by giving belt teeth profiles displacement and checking for interference with the pulley profile. Since it is assumed that the seating and unseating force peaks do not influence the load distribution, the positions of the first and last teeth are superimposed on belt teeth profiles using the results from a quasi-static load distribution model covering fully meshed conditions. The superimposed first and last belt teeth profiles are modelled by line segments. A pulley profile is also modelled by line segments and the profiles are checked for interference. Where interference occur an overlap area is formed. The overlap is translated to a force value via correlation with belt tooth force measurements. Results from the model show good agreement with measurements when force peaks are small. This is due to the fact that the quasi-static load distribution model produces correct belt displacements for these cases. For measured force peaks of higher amplitude the seating and unseating effects are under estimated by the method. The semicircular belt geometry in combination with the hyperelastic nature of the elastomer is probably the reason. A solution is to implement a non-linear force-overlap relation. Another effect not included is the velocity difference effect. The results are sensitive to belt tooth height and radial tooth stiffness.

Bistatic RCS of Electrically-Large Objects Calculations with High-Order Parabolic Equation Method

2013 ◽  
Vol 765-767 ◽  
pp. 557-561 ◽  
Author(s):  
Meng Kong ◽  
Ming Sheng Chen ◽  
Jin Hua Hu ◽  
Xian Liang Wu ◽  
Yuan Cheng
Keyword(s):  
Parabolic Equation ◽  
High Order ◽  
Numerical Results ◽  
Equation Method ◽  
Computational Time ◽  
Two Dimensional ◽  
Parabolic Equation Method ◽  
Order Parabolic Equation ◽  
Low Order ◽  
Crank Nicolson

A high-order parabolic equation (PE) method is applied to calculate the Bistatic RCS of two-dimensional (2-D) Electrically-large Objects. According to the shapes of objexts, Crank-Nicolson and Pade (1,0) schemes are introduced to solve the high-order PE. The numerical results demonstrate that the method not only extends the accurate angle range but also decreases the rotation times and computational time in comparison with the traditional low-order parabolic equation method.

scholarly journals A Multivariate High-Order Markov Model for the Income Estimation of a Wind Farm

Energies ◽  
2021 ◽  
Vol 14 (2) ◽  
pp. 388
Author(s):  
Riccardo De Blasis ◽  
Giovanni Batista Masala ◽  
Filippo Petroni
Keyword(s):  
Markov Model ◽  
Electricity Market ◽  
Wind Farm ◽  
Central Italy ◽  
High Order ◽  
Distribution Model ◽  
Spot Price ◽  
Model Parameters ◽  
Price Series ◽  
Cross Correlations

The energy produced by a wind farm in a given location and its associated income depends both on the wind characteristics in that location—i.e., speed and direction—and the dynamics of the electricity spot price. Because of the evidence of cross-correlations between wind speed, direction and price series and their lagged series, we aim to assess the income of a hypothetical wind farm located in central Italy when all interactions are considered. To model these cross and auto-correlations efficiently, we apply a high-order multivariate Markov model which includes dependencies from each time series and from a certain level of past values. Besides this, we used the Raftery Mixture Transition Distribution model (MTD) to reduce the number of parameters to get a more parsimonious model. Using data from the MERRA-2 project and from the electricity market in Italy, we estimate the model parameters and validate them through a Monte Carlo simulation. The results show that the simulated income faithfully reproduces the empirical income and that the multivariate model also closely reproduces the cross-correlations between the variables. Therefore, the model can be used to predict the income generated by a wind farm.

Polynomial Moving Fitting Method for Edge Identification

2011 ◽  
Vol 308-310 ◽  
pp. 2560-2564 ◽  
Author(s):  
Xiang Rong Yuan
Keyword(s):  
Edge Detection ◽  
Curve Fitting ◽  
Polynomial Function ◽  
High Order ◽  
Polynomial Fitting ◽  
Fitting Method ◽  
Order Polynomial ◽  
Better Than ◽  
Low Order

A moving fitting method for edge detection is proposed in this work. Polynomial function is used for the curve fitting of the column of pixels near the edge. Proposed method is compared with polynomial fitting method without sub-segment. The comparison shows that even with low order polynomial, the effects of moving fitting are significantly better than that with high order polynomial fitting without sub-segment.

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