Node Sampling for Nonlinear Vibration Analysis of Structures with Intermittent Contact

A = assembly operator B = set of degree-of-freedom indices corresponding to boundary nodes Ccp = set of degree-of-freedom indices corresponding the contact pairs D = set of degree-of-freedom indices corresponding to deleted degrees of freedom F̂ = virtual impulse associated with the amount of penetration f = contact force vector G = set of degrees of freedom corresponding to all generalized internal degrees of freedom H = coordinate transformation matrix of the Hintz–Herting component mode synthesis method I = identity matrix I = set of degree-of-freedom indices corresponding to internal nodes ke = equivalent spring constant per unit length on crack faces L = set of degree-of-freedom indices corresponding to degrees of freedom not exposed to nonlinearity M = set of degree-of-freedom indices corresponding to modal coordinates M, K = finite-element mass and stiffness matrices M, K = finite-element mass and stiffness matrices projected onto span HPR MH ,KH = finite-element mass and stiffness matrices projected onto span H N = set of degree-of-freedom indices corresponding to degrees of freedom exposed to nonlinearity n = size of the finite-element mass, stiffness, and damping matrices nm = number of free-interface normal modes nj1, n j 2, n j 3 = normal vectors at the jth contact pair O = set of degree-of-freedom indices corresponding to the nodes directly used in the structural analysis P, P = coordinate transformation matrix associated with the normal vector at jth contact pair and the assembled form of all these matrices q = modal coordinates corresponding to (HPR) R = set of degrees of freedom corresponding to all the relative degrees of freedom R, R = coordinate transformation matrix associated with the relative displacement at the jth contact pair and the assembled form T = period of vibration u = x1 component of the nodal displacement u, v = relative displacements normal to the surface, corresponding to the jth contact pair up = amount of penetration along the surface normals on crack surface X , Y, Z = set of degree-of-freedom indices corresponding to x1, x2, and x3 x = finite-element nodal displacement vector x1, x2, x3 = perpendicular axes of Cartesian coordinate system A;B = boundaries involving intermittent contact Presented as Paper 2009-2493 at the 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, 4–7 May 2009; received 14 July 2009; revision received 12 May 2010; accepted for publication 18 May 2010. Copyright © 2010 by Akira Saito, Bogdan I. Epureanu, Matthew P. Castanier, and Christophe Pierre. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/10 and $10.00 in correspondence with the CCC. Graduate Student Research Assistant, Department of Mechanical Engineering; currently Assistant Researcher, Toyota Central R&DLabs, Inc., Vehicle Mechanism Laboratory, Vehicle and Biomechanics Systems Division, Nagakute, Aichi 480-1192, Japan. Student Member AIAA. Associate Professor, Department of Mechanical Engineering. Member AIAA. Mechanical Engineer, Research Business Group. §Dean, Faculty of Engineering. Senior Member AIAA. AIAA JOURNAL Vol. 48, No. 9, September 2010