Associated Legendre Polynomial Approximations

1951 ◽  
Vol 22 (1) ◽  
pp. 87-89 ◽  
Author(s):  
R. Landauer
Keyword(s):  
Legendre Polynomial ◽  
Polynomial Approximations

Nonparametric nonlinear restoring force and excitation identification with Legendre polynomial model and data fusion

2021 ◽  
pp. 147592172199474
Author(s):  
Bin Xu ◽  
Ye Zhao ◽  
Baichuan Deng ◽  
Yibang Du ◽  
Chen Wang ◽  
...  
Keyword(s):  
Data Fusion ◽  
Legendre Polynomial ◽  
Structural Parameters ◽  
Polynomial Model ◽  
Magnetorheological Damper ◽  
Dynamic Responses ◽  
Identification Accuracy ◽  
Restoring Force ◽  
Nonlinear Restoring Force ◽  
Dynamic Loadings

Identification of nonlinear restoring force and dynamic loadings provides critical information for post-event damage diagnosis of structures. Due to high complexity and individuality of structural nonlinearities, it is difficult to provide an exact parametric mathematical model in advance to describe the nonlinear behavior of a structural member or a substructure under strong dynamic loadings in practice. Moreover, external dynamic loading applied to an engineering structure is usually unknown and only acceleration responses at limited degrees of freedom of the structure are available for identification. In this study, a nonparametric nonlinear restoring force and excitation identification approach combining the Legendre polynomial model and extended Kalman filter with unknown input is proposed using limited acceleration measurements fused with limited displacement measurements. Then, the performance of the proposed approach is first illustrated via numerical simulation with multi-degree-of-freedom frame structures equipped with magnetorheological dampers mimicking nonlinearity under direct dynamic excitation or base excitation using noise-polluted measurements. Finally, a dynamic experimental study on a four-story steel frame model equipped with a magnetorheological damper is carried out and dynamic response measurement is employed to validate the effectiveness of the proposed method by comparing the identified dynamic responses, nonlinear restoring force, and excitation force with the test measurements. The convergence and the effect of initial estimation errors of structural parameters on the final identification results are investigated. The effect of data fusion on improving the identification accuracy is also investigated.

scholarly journals Boundary Fitting Problems Associated with Coronal Magnetic Models

1974 ◽  
Vol 57 ◽  
pp. 89-91 ◽  
Author(s):  
Kenneth H. Schatten
Keyword(s):  
Magnetic Fields ◽  
Legendre Polynomial ◽  
Magnetic Monopole ◽  
Polynomial Fit ◽  
Coronal Magnetic Fields ◽  
Boundary Fitting

The calculation of coronal magnetic fields was first suggested by Gold (1958). Altschuler and Newkirk (1969) and Newkirk et al. (1968) used a Legendre polynomial fit to the photospheric observations of magnetic fields whereas Schatten (1968) with Wilcox and Ness (Schatten et al., 1969) use a magnetic monopole fit, first incorporated by Schmidt (1964).

Verifying Tolerant Systems Using Polynomial Approximations

2009 ◽  
Author(s):  
Pavithra Prabhakar ◽  
Vladimeros Vladimerou ◽  
Mahesh Viswanathan ◽  
Geir E. Dullerud
Keyword(s):  
Polynomial Approximations

scholarly journals Smooth Wavelet Approximations of Truncated Legendre Polynomials via the Jacobi Theta Function

2014 ◽  
Vol 2014 ◽  
pp. 1-24 ◽  
Author(s):  
David W. Pravica ◽  
Njinasoa Randriampiry ◽  
Michael J. Spurr
Keyword(s):  
Fourier Transform ◽  
Theta Function ◽  
Legendre Polynomial ◽  
Legendre Polynomials ◽  
Inverse Fourier Transform ◽  
Recursion Relations ◽  
Jacobi Theta Function ◽  
Convergence Results ◽  
The Family ◽  
Reciprocal Symmetry

The family ofnth orderq-Legendre polynomials are introduced. They are shown to be obtainable from the Jacobi theta function and to satisfy recursion relations and multiplicatively advanced differential equations (MADEs) that are analogues of the recursion relations and ODEs satisfied by thenth degree Legendre polynomials. Thenth orderq-Legendre polynomials are shown to have vanishingkth moments for0≤k<n, as does thenth degree truncated Legendre polynomial. Convergence results are obtained, approximations are given, a reciprocal symmetry is shown, and nearly orthonormal frames are constructed. Conditions are given under which a MADE remains a MADE under inverse Fourier transform. This is used to construct new wavelets as solutions of MADEs.

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