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By N. Shanmugam, C. Wang

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L. E. (1983), Static, dynamic and stability analysis of structures composed of tapered beams. Computers and Structures, 16(6), 731–748. Kirchhoff, G. (1879), Über die transversalschwingungen eines stabes von veränderlichem querschnitt. Berliner Monatsberichte, 815–828. L. (1952), Vibrations of Tall Buildings, Moscow Press, Moscow. E. E. (1996), Free vibration of beams of bilinearly varying thickness. Ocean Engineering, 23(1), 1–6. S. (2001), Exact solutions for free vibration of shear-type structures with arbitrary distribution of mass or stiffness.

92 The stiffness and mass matrices for each element ( )e may be assembled into the global counterpart matrices K and M using equilibrium and compatibility in the usual way. When the energy function Π remains stationary with respect to the vector of structural or global deformations Q, dΠ/dQ = 0, so that in Eq. 93 The nontrivial solution of Eq. 94 where (K + KR – ω M) is known as the dynamic stiffness matrix of the tapered member. 94 is a standard eigenvalue problem, that can be solved using standard packages to determine the natural frequencies ω.

88) and substitution of Eq. 87) into Eqs. 11) and then into Eq. 92 The stiffness and mass matrices for each element ( )e may be assembled into the global counterpart matrices K and M using equilibrium and compatibility in the usual way. When the energy function Π remains stationary with respect to the vector of structural or global deformations Q, dΠ/dQ = 0, so that in Eq. 93 The nontrivial solution of Eq. 94 where (K + KR – ω M) is known as the dynamic stiffness matrix of the tapered member. 94 is a standard eigenvalue problem, that can be solved using standard packages to determine the natural frequencies ω.

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