Geometric Covariance Modeling for Surface Variation of Compliant Parts Based on Hybrid Polynomial Approximation and Spectrum Analysis

Part variation characterization is essential to analyze the variation propagation in flexible assemblies. Aiming at two governing types of surface variation,warping and waviness,a comprehensive approach of geometric covariance modeling based on hybrid polynomial approximation and spectrum analysis is proposed,which can formulate the level and the correlation of surface variations accurately. Firstly,the form error data of compliant part is acquired by CMM. Thereafter,a Fourier-Legendre polynomial decomposition is conducted and the error data are approximated by a Legendre polynomial series. The weighting coefficient of each component is decided by least square method for extracting the warping from the surface variation. Consequently,a geometrical covariance expression for warping deformation is established. Secondly,a Fourier-sinusoidal decomposition is utilized to approximate the waviness from the residual error data. The spectrum is analyzed is to identify the frequency and the amplitude of error data. Thus,a geometrical covariance expression for the waviness is deduced. Thirdly,a comprehensive geometric covariance model for surface variation is developed by the combination the Legendre polynomials with the sinusoidal polynomials. Finally,a group of L-shape sheet metals is measured along a specific contour,and the covariance of the profile errors is modeled by the proposed method. Thereafter,the result is compared with the covariance from two other methods and the real data. The result shows that the proposed covariance model can match the real surface error effectively and represents a tighter approximation error compared with the referred methods.

Part variation characterization is essential to analyze the variation propagation in flexible assemblies. AL ming at two governing types of surface variation, warping and waviness, a comprehensive approach of geometric covariance modeling based on hybrid polynomial approximation and spectrum analysis is proposed, which can for- mulate the level and the correlation of surface variations accurately. Firstly, the form error data of compliant part is acquired by CMM. Thereafter, a Fourier-Legendre polynomial decomposition is conducted and the error data are approximated by a Legendre polynomial series. The weighting coefficient of each component is decided by least square method for extracting the warping from the surface variation. Consequently, a geometrical covariance ex- pression for warping deformation is established. Secondly, a Fourier-sinusoidal decomposition is utilized to approx- imate the waviness from the residual error data. The spectrum is analyzed is to identi^y the ~requency and the am- plitude of error data. Thus, a geometrical covariance expression for the waviness is deduced. Thirdly, a compre- hensive geometric covariance model for surface variation is developed by the combination the Legendre polynomials with the sinusoidal polynomials. Finally, a group of L-shape sheet metals is measured along a specific contour, and the covariance of the profile errors is modeled by the proposed method. Thereafter, the result is compared with the covariance from two other methods and the real data. The result shows that the proposed covariance model can match the real surface error effectively and represents a tighter approximation error compared with the referred methods.