Gear mesh excitation and non-uniform Rational B-Splines

Presented in this study is investigation of dynamic behavior of a helical gear reduction by experimental and numerical methods. A closed-loop test rig is designed to measure vibrations of the example system, and the basic principle as well as relevant signal processing method is introduced. A hybrid user-defined element model is established to predict relative vibration acceleration at the gear mesh in a direction normal to contact surfaces. The other two numerical models are also constructed by lumped mass method and contact FEM to compare with the previous model in terms of dynamic responses of the system. First, the experiment data demonstrate that the loaded transmission error calculated by LTCA method is generally acceptable and that the assumption ignoring the tooth backlash is valid under the conditions of large loads. Second, under the common operating conditions, the system vibrations obtained by the experimental and numerical methods primarily occur at the first fourth-order meshing frequencies and that the maximum vibration amplitude, for each method, appears on the fourth-order meshing frequency. Moreover, root-mean-square (RMS) value of the acceleration increases with the increasing loads. Finally, according to the comparison of the simulation results, the variation tendencies of the RMS value along with input rotational speed agree well and that the frequencies where the resonances occur keep coincident generally. With summaries of merit and demerit, application of each numerical method is suggested for dynamic analysis of cylindrical gear system, which aids designers for desirable dynamic behavior of the system and better solutions to engineering problems.

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B-splines on sparse grids for surrogates in uncertainty quantification

Reliability Engineering & System Safety ◽

10.1016/j.ress.2021.107430 ◽

2021 ◽

Vol 209 ◽

pp. 107430

Author(s):

Michael F. Rehme ◽

Fabian Franzelin ◽

Dirk Pflüger

Keyword(s):

Uncertainty Quantification ◽

Sparse Grids ◽

B Splines

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A hybrid analytical-numerical method based on Isogeometric Analysis for determination of time varying gear mesh stiffness

Mechanism and Machine Theory ◽

10.1016/j.mechmachtheory.2021.104291 ◽

2021 ◽

Vol 160 ◽

pp. 104291

Author(s):

Andreas Beinstingel ◽

Michael Keller ◽

Michael Heider ◽

Burkhard Pinnekamp ◽

Steffen Marburg

Keyword(s):

Numerical Method ◽

Isogeometric Analysis ◽

Time Varying ◽

Mesh Stiffness ◽

Gear Mesh ◽

Gear Mesh Stiffness

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Towards B-Spline Atomic Structure Calculations

Atoms ◽

10.3390/atoms9030050 ◽

2021 ◽

Vol 9 (3) ◽

pp. 50

Author(s):

Charlotte Froese Fischer

Keyword(s):

Atomic Structure ◽

Bound States ◽

Virial Theorem ◽

B Spline ◽

B Splines ◽

Self Consistent ◽

History Of ◽

Consistent Method ◽

Atomic Structure Calculations ◽

Structure Calculations

The paper reviews the history of B-spline methods for atomic structure calculations for bound states. It highlights various aspects of the variational method, particularly with regard to the orthogonality requirements, the iterative self-consistent method, the eigenvalue problem, and the related sphf, dbsr-hf, and spmchf programs. B-splines facilitate the mapping of solutions from one grid to another. The following paper describes a two-stage approach where the goal of the first stage is to determine parameters of the problem, such as the range and approximate values of the orbitals, after which the level of accuracy is raised. Once convergence has been achieved the Virial Theorem, which is evaluated as a check for accuracy. For exact solutions, the V/T ratio for a non-relativistic calculation is −2.

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Extended isogeometric analysis using B++ splines for strong discontinuous problems

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2021.113779 ◽

2021 ◽

Vol 381 ◽

pp. 113779

Author(s):

Wenbin Hou ◽

Kai Jiang ◽

Xuefeng Zhu ◽

Yuanxing Shen ◽

Ping Hu

Keyword(s):

Isogeometric Analysis ◽

B Splines ◽

Discontinuous Problems

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Interpolation and Sampling with Exponential Splines of Real Order

Results in Mathematics ◽

10.1007/s00025-021-01404-7 ◽

2021 ◽

Vol 76 (3) ◽

Author(s):

Peter Massopust

Keyword(s):

Positive Real ◽

B Splines

AbstractThe existence of fundamental cardinal exponential B-splines of positive real order $$\sigma $$ σ is established subject to two conditions on $$\sigma $$ σ and their construction is implemented. A sampling result for these fundamental cardinal exponential B-splines is also presented.

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Non-uniform subdivision for B-splines of arbitrary degree

Computer Aided Geometric Design ◽

10.1016/j.cagd.2007.12.005 ◽

2009 ◽

Vol 26 (1) ◽

pp. 75-81 ◽

Cited By ~ 26

Author(s):

S. Schaefer ◽

R. Goldman

Keyword(s):

Arbitrary Degree ◽

B Splines

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Three-Dimensional Computerized Anthropometry of the Nose: Landmark Representation Compared to Surface Analysis

The Cleft Palate-Craniofacial Journal ◽

10.1597/06-021 ◽

2007 ◽

Vol 44 (3) ◽

pp. 278-285 ◽

Cited By ~ 14

Author(s):

Virgilio F. Ferrario ◽

Fabrizio Mian ◽

Redento Peretta ◽

Riccardo Rosati ◽

Chiarella Sforza

Keyword(s):

Three Dimensional ◽

Digital Data ◽

Limits Of Agreement ◽

B Splines ◽

Surface Areas ◽

Using Data ◽

Difference Volume ◽

T Values ◽

Plaster Models ◽

Plaster Casts

Objective: To compare three-dimensional nasal measurements directly made on subjects to those made on plaster casts, and nasal dimensions obtained with a surface-based approach to values obtained with a landmark representation. Methods: Soft-tissue nasal landmarks were directly digitized on 20 healthy adults. Stone casts of their noses were digitized and mathematically reconstructed using nonuniform rational B-splines (NURBS) curves. Linear distances, angles, volumes and surface areas were computed using facial landmarks and NURBS-reconstructed models (surface-based approach). Results: Measurements on the stone casts were somewhat smaller than values obtained directly from subjects (differences between −0.05 and −1.58 mm). Dahlberg's statistic ranged between 0.73 and 1.47 mm. Significant (p < .05) t values were found for 4 of 15 measurements. The surface-based approach gave values 3.5 (volumes) and 2.1 (surface area) times larger than those computed with the landmark-based method. The two values were significantly related (volume, r = 0.881; surface, r = 0.924; p < .001), the resulting equations estimated actual values well (mean difference, volume −0.01 mm3, SD 1.47, area 0.05 cm2, SD 1.44); limits of agreement between −2.89 and 2.87 mm3 (volume); −2.88 and 2.78 cm2 (area). Conclusions: Considering the characteristics of the two methods, and for practical purposes, nasal distances and angles obtained on plaster models were comparable to digital data obtained directly from subjects. Surface areas and volumes were best obtained using a surface-based approach, but could be estimated using data provided by the landmark representation.

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Vibration and sound radiation analysis of the final drive assembly considering the gear-shaft coupling dynamics

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406216632021 ◽

2016 ◽

Vol 230 (7-8) ◽

pp. 1258-1275 ◽

Cited By ~ 9

Author(s):

Yawen Wang ◽

Junyi Yang ◽

Dong Guo ◽

Teik C Lim

Keyword(s):

Dynamic Models ◽

Acoustic Analysis ◽

Bending Moment ◽

Moment Of Inertia ◽

System Dynamic ◽

Dynamic Responses ◽

Propeller Shaft ◽

Hypoid Gear ◽

Gyroscopic Effect ◽

Gear Mesh

A generalized dynamic model of driveline system is formulated that includes the coupling effect and gyroscopic moments of the propeller shaft and hypoid gear rotor assembly. Firstly, the dynamic models with only gear-shaft coupling, with only gyroscopic effect, and with both gear-shaft coupling and gyroscopic effect are analyzed and compared. The results show that the combined effects of the gear-shaft interaction and gyroscopic behavior have considerable influence on the system dynamic responses surrounding gear bending resonances, especially for the bearing responses. However, the gear out-of-phase torsional modes still dominate the gear mesh frequency response. Secondly, the influence of pinion bending moment of inertia, propeller shaft stiffness and bearing stiffness on the system dynamic responses are examined. The system responses are then applied to perform further vibration and acoustic analysis for an axle housing structure. Computational results reveal that NVH (noise, vibration, and harshness) refinement can be achieved by tuning the pinion bearing rotational stiffness and pinion bending moment of inertia for the example considered. This study provides an understanding of the interaction between hypoid gear pair and propeller shaft, and can be employed to enhance driveline system design.

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