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This article deals with Hadamard inequalities for strongly s,m-convex functions using generalized Riemann–Liouville fractional integrals. Several generalized fractional versions of the Hadamard inequality are presented; we also provide refinements of many known results which have been published in recent years.

This paper proposes an efficient and automated scheme to predict and identify the position and motion errors of rotary axes on a non-orthogonal five-axis machining centre using the double ball bar (DBB) system. Based on the Denavit-Hartenberg theory, a motion deviations model for the tilting rotary axis B and rotary C of a non-orthogonal five-axis NC machine tool is established, which considers tilting rotary axis B and rotary C static deviations and dynamic deviations that total 24. After analysing the mathematical expression of the motion deviations model, the QC20 double ball bar (DBB) from the Renishaw Company is used to measure and identify the motion errors of rotary axes B and C, and a measurement scheme is designed. With the measured results, the 24 geometric deviations of rotary axes B and C can be identified intuitively and efficiently. This method provides a reference for the error identification of the non-orthogonal five-axis NC machine tool.

In order to diagnose the unbalance of aerostatic guideways, displacement and acceleration sensors were used to measure the vibration signals of the unbalance of the aerostatic guideways caused by the gas fluctuation. The frequency characteristics for the unbalance of aerostatic guideways caused by gas fluctuation has been extracted from the measured signals by power spectral density, and the basic frequencies of the guideway system have been diagnosed according to spectral characteristics, in agreement with the results calculated by the corresponding motion equations and the finite element method.

The Navier velocity-slip boundary conditions are adopted to fluid continuous flow equation. Modified Reynolds equation in micro-scale is obtained. Based on the conservation of oil film flow and the assumption of linearization distribution of oil film pressure, the load capacity is gained in the consideration of slip. Through comparison and analysis, the performance of hydrostatic bearing under the micro-scale is different from tradition. With the increase of slip length, load capacity is reduced. The oil film thickness of the maximum load capacity decreases. Slip phenomenon is more serious, the effect is more obvious. So the design of bearing must consider the influence of the micro-scale.