**Strucutral Dynamics /
Inverse Vibration**

**Inverse Vibration Problem for Un-Damped
3-Dimensional Multi-Story Shear Building Models **

Various researchers have
contributed to the identification of the mass and
stiffness matrices of two dimensional (2-D) shear
building structural models for a given set of vibratory
frequencies. The suggested methods are based on the
specific characteristics of the Jacobi matrices, i.e.,
symmetric, tri-diagonal and semi-positive definite
matrices. However, in case of three dimensional (3-D)
structural models, those methods are no longer
applicable, since their stiffness matrices are not
tri-diagonal. In this paper the inverse problem for a
special class of vibratory structural systems, i.e., 3-D
shear building models, is investigated. A
practical algorithm is proposed for solving the inverse
eigenvalue problem for un-damped, 3-D shear buildings.
The problem is addressed in two steps. First, using the
target frequencies, a so-called normalized eigenvector
matrix which is a banded matrix containing the
information related to the frequencies and mode shapes
of the target structural system is determined. In this
regard, similar to the solution of inverse problem for
2-D shear building structural models in which an
auxiliary structure is constructed by adding
constraints (or springs) to the original system,
three auxiliary structures are proposed to solve the
problem for 3-D cases. In the second step, the
normalized eigenvector matrix is utilized to obtain the
normalized stiffness matrix, that in turn this matrix is
decomposed into the stiffness and mass matrices of the
system. Finally, a numerical example is presented to
demonstrate the efficiency of the proposed algorithm in
determining the mass and stiffness matrices of a 3-D
structural model for a given set of target vibrational
frequencies.