Estimating Local Compliance in a Beam From Bending Measurements Part II. Optimal Estimation of Local Compliance
Keywords:Local E, compliance, bending, Kalman, estimation, optimum, ARMA, state-space
We present a method for optimally estimating local elasticity properties along a beam where each estimate is specific to an increment in a subdivision of the beam's length. Previous research indicates that knowledge of localized elasticity values can improve the estimation of strength. Immediate application is expected in the machine stress-rated (MSR) lumber production process. A sequence of bending measurements on overlapping bending spans, as commonly obtained in the MSR process, serves as input to the estimation method.
The sequence of bending measurements is modeled as an autoregressive moving average (ARMA) random process. Autoregression coefficients are estimated from a priori information and refined as additional data are obtained. Moving average weight coefficients come from span functions computed by methods in Part I. A Kalman filter, defined from coefficients of the ARMA process, is applied to the measurements, and local estimates are obtained.
Estimated local elasticity results are presented for both a simulated and a real wood beam. One set of experiments shows that as a modeled correlation coefficient is decreased from an artificially high value, the result evolves from local elasticity estimates that appear much the same as measured elasticity, but without an obvious noise component, to local estimates having more detail. This leads naturally to a suggestion for a practical, non-disruptive introduction of the estimation method to a MSR production line. Grade yield improvement is likely an immediate benefit along with a capability for further research into the estimation method and grading algorithms.
Bechtel, F. K. 1985. Beam stiffness as a function of pointwise E, with application to machine stress rating. Proc. Int'l Symp. on Forest Products Research. CSIR. Pretoria, South Africa.nBechtel, F. K. 2005. Kalman filter derivation per Kalman. Notes available from author.nBechtel, F. K., C. S. Hsu, and T. C. Hanshaw. 2000. High resolution E in lumber from bending measurements. SBIR/USDA Phase I Final Report.nBechtel, F. K., C. S. Hsu, and T. C. Hanshaw. 2006. Method for estimating compliance at points along a beam from bending measurements. U.S. Patent No. 7,047,156.nEubank, R. L. 2006. A Kalman Filter Primer. CRC Press. Boca Raton, FL. 186 pp.nFoschi, R. O. 1987. A procedure for the determination of localized modulus of elasticity. Holz Roh-Werkst.45:257-260.nGuillemin, E. A. 1949. Mathematics of circuit analysis. MIT Press, Cambridge, MA. 590 pp.nHernandez, R., D. A. Bender, B.A. Richburg, and K. S. Kline. 1992. Probabilistic modeling of glued-laminated timber beams. Wood Fiber Sci.23(3):294-306.nKailath, T., A. H. Sayed, and B. Hassibi. 2000. Linear estimation. Prentice-Hall, Upper Saddle River, NJ. 854 pp.nKalman, R. E. 1960. A new approach to linear filtering and prediction problems. Trans. ASME J. Basic Eng. Series 82D. pp 35-45.nKline, D. E., F. E. Woeste, and B. A. Bendtsen. 1986. Stochastic model for modulus of elasticity of lumber. Wood Fiber Sci.18(2):228-238.nLam, F., R. O. Foschi, J.D. Barrett, and Q. Y. He. 1993. Modified algorithm to determine localized modulus of elasticity of lumber. Wood Sci. Technol.27:81-94.nOgata, K. 1987. Discrete time control systems. Prentice-Hall, Englewood Cliffs, NJ. 994 pp.nOppenheim, A. V., and R. W. Schafer. 1989. Discrete signal processing. Prentice-Hall, Englewood Cliffs, NJ. 879 pp.nPapoulis, A. 1991. Probability, random variables, and stochastic processes. McGraw-Hill, New York, NY. 666 pp.nPope, D. J., and F. W. Matthews. 1995. A comparison of deconvolution techniques to improve MOR estimation from stress grading machine output. Wood Sci. Technol.29:431-439.nRichburg, B. A., R. Hernandez, B. J. Hill, and D. A. Bender. 1991. Machine stress grading for determining localized lumber properties. Paper No. 91-4542. International Winter Meeting of the ASAE. Chicago, IL.nRichburg, B. A., and D. A. Bender. 1992. Localized tensile strength and modulus of elasticity of E-rated laminating grades of lumber. Wood Fiber Sci.24(2):225-232.nRosenfeld, A., and A. C. Kak. 1982. Digital picture processing. Academic Press, New York, NY. 435 pp.nSchwarz, R. J., and B. Friedland. 1965. Linear systems. McGraw-Hill, New York, NY. 521 pp.nTaylor, S. E., and D. A. Bender. 1989. A method for simulating multiple correlated lumber properties. Forest Prod. J.39(7/8):71-74.nTaylor, S. E., and D. A. Bender. 1991. Stochastic model for localized tensile strength and modulus of elasticity in lumber. Wood Fiber Sci.23(4):501-519.nTaylor, S. E., D. A. Bender., D. E. Kline, and K. S. Kline. 1992. Comparing length effect models for lumber tensile strength. Forest Prod. J.42(2):23-30.n
The copyright of an article published in Wood and Fiber Science is transferred to the Society of Wood Science and Technology (for U. S. Government employees: to the extent transferable), effective if and when the article is accepted for publication. This transfer grants the Society of Wood Science and Technology permission to republish all or any part of the article in any form, e.g., reprints for sale, microfiche, proceedings, etc. However, the authors reserve the following as set forth in the Copyright Law:
1. All proprietary rights other than copyright, such as patent rights.
2. The right to grant or refuse permission to third parties to republish all or part of the article or translations thereof. In the case of whole articles, such third parties must obtain Society of Wood Science and Technology written permission as well. However, the Society may grant rights with respect to Journal issues as a whole.
3. The right to use all or part of this article in future works of their own, such as lectures, press releases, reviews, text books, or reprint books.