tailieunhanh - Reduced impedance of branch component with hyperstatic interface

This paper has its genesis in an early study [3] in which reduced impedance of a statically determinate branch component was found by using component mqdes. It is intended to apply the method to the case of hyperstatic interface. | Journal of Mechanics, NCNST of Vietnam T. XIX, 1997, No 1 (42- 47) REDUCED IMPEDANCE OF BRANCH COMPONENT WITH HYPERSTATIC INTERFACE NGUYEN THAC Si Hanoi University of Mining and Geology 1. Introduction An analytical technique proposed by Berman [1] allows an exact representation of a branch component within the model of the main structure by considering the impedance matrix of the component and its inverse. a This paper has its genesis in an early study [3] in which reduced impedance of statically determinate branch component was found by using component mqdes. It is intended to apply the method to the case of hyperstatic interface. 2. Definition Consider a structural system composed of the main component "M" and a branch component "k?' as illustrated in . "k" f It is convenient to rearrange and partition the elements of the two impedance matrices in the following way: z;,] zk , ee where f refers to interface coordinates and i. to non-interface coordinates. The impedance of the system, then, may be forined by superimposing these matrices in the form: z7,]. z;, If a valid model of component "k" could be formed using only the interface coordinates, the impedance of the system could be written as Z, = [Zee z,, z,,z,,+ z,, l , "k 42 where ZJ1 is called reduced impedance of component "k". 3. Reduced Impedance Consider a branch component the interface of which is assumed hyperstatic. Its ·displacement may be expressed by the vector q(t) = [ :~ J U:J () = in which qr are rigid-body displacement on the connection interface and qc the remainder of displacements on the connection interface·. The equation of motion of the component is Mij+Kq=F, () where the mass, stiffness and force matrices are F=[~]· The displacement of any point is found by superimposing the motion excited by the main component through the interface and the elastic motion relative to the latter, so that ()