Aeroelasticity by Raymond L. Bisplinghoff, Holt Ashley, Robert L.Halfman

By Raymond L. Bisplinghoff, Holt Ashley, Robert L.Halfman

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For most engineering problems, [A] and [B] will be symmetric matrices of order n. À is a scalar (called the eigenvalue) and λ is a column vector with n components (called the eigenvector). If the physical problem is the free vibration analysis of a structure, [A] will be the stiffness matrix, [B] will be the mass matrix, λ is the square of natural frequency and X is the mode shape of the vibrating structure. The eigenvalue problem given by Eq. e. γι\ - Xbyii a12 Û22 - A&12 ··· CL\n - Xbin — AL?

792 ... 3729 ωχ = ΓΕΪ L2 Rayleigh-Ritz method If we want many natural frequencies, we have to substitute a solution, made up of a series of admissible functions which satisfy the forced boundary conditions, in Eq. 42). For example, if we want n frequencies, we take W(x) = Cifx(x) + C2f2(x) + ... 45) where Cl9 C2, ... , fn are admissible functions. If we substitute Eq. , Cn. » Cn such that they make ώ a minimum. For this, we set iiSil = 1(ωϋ= ... i @ Ü . , Cn, and hence we will be able to solve them.

Area A3 -IX/L) area = A (x) = A . e (b) Fig. 7 A one-dimensional tube of varying cross section Assume the area of cross section of the tube as A(x) = A0*e~^D' · Solution Note: In this case the terminology of solid mechanics like solid body, displacement, stiffness matrix, load vector and equilibrium equations has to be replaced by the terms continuum, potential function, characteristic matrix, characteristic vector and governing equations. (i) Idealization Divide the continuum into two finite elements as shown in Fig.

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