
The Undamped Building.
(a) Show that Eqs.(1)through(3) can be expressed in matrix notation as
x"+ω02Kx=ω02f(t)z,
Where ω02=km ,x=(x1,x2,...,xn)T, z=(1,0,...,0)T,
and
K=(2−100…0−12−10…00−12−1…0⋮⋱⋱⋱⋮0…−12−10…0−11). (ii)
(b) A real
n×n matrix
A is said to be positive definite if
xTAx>0 for every real
n-
xTKx=x12+n−1∑j=1(xj−xj+1)2,
And is therefore positive definite.
(c) Eigenvalues and eigenvectors of real
λ=uTKuuTu>0.
Thus all eigenvalues of K in (ii) are real and positive.
(d) For the cases n=5,10, and 20, demonstrate numerically that the eigenvalues of K, λj=ωj2,j=1,...,n can be ordered as follows:
0<ω12<ω22<⋯<ωn2.
(e) Since K is real and symmetric, it possesses a set of n orthogonal eigenvectors, {u1, u2,..., un}, that is, uiTuj=0 if i≠j ( see Appendix A.4). These eigenvectors can be used to construct a normal mode representation,
x=a1(t)u1+⋯+an(t)un, (iii)
Of the solution of
x"+ω02Kx=ω02f(t)z,
x(0)=x0, x'(0)=v0, (iv).
Substitute the representation (iii) into the differential equation and initial conditions in Eqs. (iv)and use the fact that Kuj=ωj2uj, j=1,...,n and the orthogonality of u1, u2,..., un to show that for each i=1,...,n, the mode amplitude ai(t) satisfies the initial value problem
ai"+ωi2ω02ai=ω02f(t)zi, ai(0)=αi, ai'(0)=βi,
Where
zi=uTizuTiui, αi=uTix0uTiui, βi=uTiv0uTiui.
(f) An unforced pure mode of vibration, say, the j thmode, can be realized by solving the initial value problem (iv) subject to the initial conditions x(0)=Ajuj, where Aj is a mode amplitude factor, in this case, the normal mode solution of the initial value problem (iv) consists of a single term,
x(j)(t)=Ajcos(ω0ωjt)uj.
Thus the natural frequency of the j th mode of vibration is ω0ωj and the corresponding period is 2π/(ω0ωj). Assuming that A1=1, ω0=41, and n=20, plot a graph of the components (floor displacement versus floor number) of the first mode x(1)(t) for several values of t 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.

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