Influence of human operator on stability of haptic rendering: a closed-form equation

This paper considers leakage in mechanical seals under hydrostatic operating condition. A contact model based on the Greenwood and Williamson contact of rough surfaces is developed for treating problems involving mechanical seals in which both the micron scale roughness of the seal face and its macro scale profile are used to obtain either a closed-form equation or a nonlinear equation relating mean plane separation to the mass flow rate. The equations involve the micron scale geometry of the rough surfaces and physical parameter of the seal and carriage. Under hydrostatic condition, it is shown that there is an approximate closed-form solution in which mass flow rate in terms of the mean plane separation, or alternatively, the mean plane separation in terms of the leakage mass flow rate is found. Equations pertaining to leakage in nominally flat seal macro profile is considered and closed form equation relating to leakage flow rate to pressure difference is obtained that contain macro and micron geometries of the seal.

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Comparison of Static and Dynamic Stiffness for Beams Undergoing Flexural and Torsional Loading With an Intermediate Mass

10.1115/imece2000-1635 ◽

2000 ◽

Author(s):

Arnoldo Garcia ◽

Arnold Lumsdaine ◽

Ying X. Yao

Keyword(s):

Closed Form ◽

Natural Frequency ◽

Cross Sections ◽

Dynamic Stiffness ◽

Closed Form Solution ◽

Frequency Equation ◽

Form Solution ◽

Torsional Loading ◽

Intermediate Mass ◽

Form Equation

Abstract Many studies have been performed to analyze the natural frequency of beams undergoing both flexural and torsional loading. For example, Adam (1999) analyzed a beam with open cross-sections under forced vibration. Although the exact natural frequency equation is available in literature (Lumsdaine et al), to the authors’ knowledge, a beam with an intermediate mass and support has not been considered. The models are then compared with an approximate closed form solution for the natural frequency. The closed form equation is developed using energy methods. Results show that the closed form equation is within 2% percent when compared to the transcendental natural frequency equation.

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Determination of critical loads for global buckling of axially loaded pultruded fiber-reinforced polymer members with doubly symmetric cross sections

Advances in Structural Engineering ◽

10.1177/1369433218759572 ◽

2018 ◽

Vol 21 (12) ◽

pp. 1911-1922

Author(s):

Yang Zhan ◽

Gang Wu

Keyword(s):

Closed Form ◽

Cross Sections ◽

Reduction Factor ◽

Fiber Reinforced Polymer ◽

Design Stage ◽

Fiber Reinforced ◽

Original Solution ◽

Form Equation ◽

Reinforced Polymer ◽

Global Buckling

This article proposes a new closed-form equation to determine the reduction factor for global buckling of concentrically loaded pultruded fiber-reinforced polymer struts based on the Ayrton–Perry formula and observed initial out-of-straightness of pultruded fiber-reinforced polymer members measured by other researchers, which makes the original solution recommended by Eurocode 3 easy to be used to predict the global buckling loads of doubly symmetric pultruded fiber-reinforced polymer members subjected to axial compression. The influence of the geometric imperfections of pultruded fiber-reinforced polymer profiles is considered in this new closed-form equation. Validation of the solution including the parameter of the reduction factor for global buckling of pultruded fiber-reinforced polymer columns is performed by comparison with published experimental evidence. In addition, compared with the five closed-form solutions available in the literature, this solution exhibits higher accuracy in predicting the global buckling capacity of concentrically loaded pultruded fiber-reinforced polymer struts with doubly symmetric cross sections. The solution implemented into the new reduction factor equation for global buckling of pultruded fiber-reinforced polymer members can be conveniently used by structural engineers at the preliminary engineering design stage for accurately assessing the reliability and safety of composite structures under concentric compressive loading.

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