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Eigenvalue Sensitivities of a Linear Structure Carrying Lumped Attachments

2011; American Institute of Aeronautics and Astronautics; Volume: 49; Issue: 11 Linguagem: Inglês

10.2514/1.j050808

1533-385X

D. Philip, Andrew B. Sabater,

Structural mechanics and materials

a = length of a plate along the x direction B = p p characteristic matrix b = length of a plate along the y direction bij = i; j th element of B c = viscous damping coefficient D = flexural rigidity of a plate E = Young’s modulus g = user-defined function that depends on the characteristic equation of the combined system h = thickness of a plate I = area moment of inertia of the cross section of a beam I = identity matrix j = imaginary unit Ki = ith generalized stiffness of a host linear structure Kpq = generalized stiffnesses of a host plate K = N N diagonal stiffness matrix whose elements are Ki k = translational spring constant kt = torsional spring constant L = length of a beam Mi = ith generalized mass of a host linear structure Mpq = generalized masses of a host plate M = N N diagonal mass matrix whose elements are Mi m = lumped mass parameter N = number of generalized coordinates p = number of lumped attachments q = number of attachment locations s = number of parameters that characterizes a lumped attachment t = time variable u xa = u1 xa u2 xa uN xa T ui = u xi ui x = a function that depends on the ith eigenfunction of a linear structure evaluated at x w x; t = lateral deflection of a linear structure evaluated at x and t x = spatial coordinate along a linear structure xa = attachment location in the x direction xi = ith attachment location ya = attachment location in the y direction i = ith physical parameter of a lumped attachment = vector of the perturbations of the system parameters and attachment locations r = change in the rth lumped system parameter from its nominal value xs = change in the sth attachment location from its nominal value ji = Kronecker delta i t = ith generalized coordinate evaluated at t = column vector of the perturbed eigenvalues = diagonal matrix consisting of the exact eigenvalues of a linear structure 0 = column vector of the unperturbed eigenvalues = eigenvalue of the combined system I = imaginary part of i = ith eigenvalue of the combined system or the ith eigenvalue of the perturbed system i0 = ith eigenvalue of the unperturbed system R = real part of = Poisson’s ratio = mass per unit length of a beam or mass per unit area of a plate = a parameter that describes the lumped attachment i = parameter that describes the ith lumped attachment i x = ith eigenfunction of a host linear structure evaluated at x i x = derivative of i x with respect to x ! = natural frequency of a combined system !hi = ith natural frequency of a host linear structure r = matrix of the eigenvalue gradients