Chapter 6.P1, Problem 1P

The Undamped Building.

(a) Show that Eqs.(1)through(3) can be expressed in matrix notation as

,

Where ,

and

. (ii)

(b) A real matrix is said to be positive definite if for every real -vector . Show that in (ii) satisfies

,

And is therefore positive definite.

(c) Eigenvalues and eigenvectors of real symmetric matrices are (see Appendix ). Show that if is positive definite and and are an eigenvalue-eigenvector pair for , then

.

Thus all eigenvalues of in (ii) are real and positive.

(d) For the cases , and , demonstrate numerically that the eigenvalues of , can be ordered as follows:

.

(e) Since is real and symmetric, it possesses a set of orthogonal eigenvectors, , that is, if ( see Appendix ). These eigenvectors can be used to construct a normal mode representation,

, (iii)

Of the solution of

,

, (iv).

Substitute the representation (iii) into the differential equation and initial conditions in Eqs. (iv)and use the fact that and the orthogonality of to show that for each , the mode amplitude satisfies the initial value problem

, , ,

Where

.

(f) An unforced pure mode of vibration, say, the thmode, can be realized by solving the initial value problem subject to the initial conditions , where is a mode amplitude factor, in this case, the normal mode solution of the initial value problem consists of a single term,

.

Thus the natural frequency of the th mode of vibration is and the corresponding period is . Assuming that , , and , plot a graph of the components (floor displacement versus floor number) of the first mode for several values of over an entire cycle. Then generate analogous graphs for the second and third pure modes of vibration. Describe and compare the modes of vibration and their relative frequencies.