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Further development of Ko displacement theory for deformed shape predictions of nonuniform aerospace structures
U.S. Government
Paperback. Books LLC, Reference Series 2011-09-28.
ISBN 9781234059644
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Förlagets beskrivning
Original publisher: Edwards, Calif. : National Aeronautics and Space Administration, Dryden Flight Research Center, [2009] OCLC Number: (OCoLC)672029577 Excerpt: ... case expressions for uniform beams, ( ). The modified slope and deflection equations for slightly c = c i ? 1 i nonuniform cantilever beams are derived in the subsequent sections. SLIGHTLY NONUNIFORM CANTILEVER BEAMS When the cantilever beam is slightly tapered ( that is, ), the logarithmic terms in the slope ( c c ) ? 1 i i ? 1 and deflection equations ( eqs. ( 14 ) and ( 16 ) ) will approach zero ( that is, ). Therefore, the log ( c c ) ? 0 e i i ? 1 perturbation method must be used to expand the logarithmic term, , in the neighborhood of log ( c c ) e i i ? 1 to obtain nonzero mathematical expressions, so that the slope and deflection equations can ( c c ) ? 1 i i ? 1 degenerate into those for the uniform cantilever beam case, . ( c c ) = 1 i i ? 1 Slope Equations For a slightly tapered beam, the logarithmic term, , in slope equation ( 11 ) can be log ( c c ) e i i ? 1 expanded in the neighborhood of ( that is, ). Carrying out the expansion up to ( c c ) ? 1 log ( c c ) ? 0 i i ? 1 e i i ? 1 2 the second-order terms in causes the logarithmic term, , to take on the following log ( c c ) ( c ? c ) e i i ? 1 i ? 1 i form as approaches [ ] ( see Appendix B for details of mathematical expansions; c ( c c ) ? 1 c i ? 1 i i ? 1 i refs. 2, 6 ): c c c ? c i i i ? 1 i ? 1; log ? ( c ? 3c ) e i i ? 1 ( 17 ) 2 c c 2c i ? 1 i ? 1 i ? 1 Substitution of equation ( 17 ) into slope equation ( 11 ) yields the following slope equation for the slightly nonuniform cantilever beam ( ref. 2 ), with the ( ) factor in the denominators eliminated: c ? c i ? 1 i ? ? ? ? l c ? i; c ? c ? ? ? n? tan = 2 ? + + ta ( 18 ) ? ? ? i ? 1 i i i ? 1 i i ? 1 ? ? ? 2c c ? i ? 1 i ? 1 ? Applying the descending recursion relationship causes slope equation ( 18 ) to become i ? ? ? ? ? c ?l 1 j ? tan? = 2 ? ? + ? + tan?; c ? c ( 19 ) ? ? ? ?
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Further development of Ko displacement theory for deformed shape predictions of nonuniform aerospace structures
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