Chapter 6.P1, Problem 1P

The Undamped Building.

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

$x"+{\omega }_0{}^{2}Kx={\omega }_0{}^{2}f\left(t\right)z$,

Where ${\omega }_0{}^2=\frac{k}{m},x={\left(x_1,x_2,\mathrm{...},x_n\right)}^T,z={\left(1,0,\mathrm{...},0\right)}^T$,

and

$K=\left(\begin{array}{cccccc}2& -1& 0& 0& \dots & 0\\ -1& 2& -1& 0& \dots & 0\\ 0& -1& 2& -1& \dots & 0\\ ⋮& & \ddots & \ddots & \ddots & ⋮\\ 0& \dots & & -1& 2& -1\\ 0& \dots & & 0& -1& 1\end{array}\right)$. (ii)

(b) A real $n\times n$ matrix $\text{A}$ is said to be positive definite if ${\text{x}}^T\text{Ax}>0$ for every real $n$-vector $\text{x}\ne 0$. Show that $\text{K}$ in (ii) satisfies

${\text{x}}^T\text{Kx}=x_1{}^2+{\displaystyle \underset{j=1}{\overset{n-1}{\sum }}{\left(x_j-x_{j+1}\right)}^2}$,

And is therefore positive definite.

(c) Eigenvalues and eigenvectors of real symmetric matrices are (see Appendix $\text{A}.4$). Show that if $\text{K}$ is positive definite and $\lambda$ and $\text{u}$ are an eigenvalue-eigenvector pair for $\text{K}$, then

$\lambda =\frac{{\text{u}}^{\text{T}}\text{Ku}}{{\text{u}}^{\text{T}}\text{u}}>0$.

Thus all eigenvalues of $\text{K}$ in (ii) are real and positive.

(d) For the cases $n=5,10$, and $20$, demonstrate numerically that the eigenvalues of $\text{K}$, ${\lambda }_j={\omega }_j{}^2,j=1,\mathrm{...},n$ can be ordered as follows:

$0<{\omega }_1{}^2<{\omega }_2{}^2<\cdots <{\omega }_n{}^2$.

(e) Since $\text{K}$ is real and symmetric, it possesses a set of $n$ orthogonal eigenvectors, $\left\{{\text{u}}_{\text{1}}{\text{, u}}_{\text{2}}\text{,}\mathrm{...}{\text{, u}}_{\text{n}}\right\}$, that is, ${\text{u}}_i{}^{\text{T}}{\text{u}}_j=0$ if $i\ne j$ ( see Appendix $\text{A}.4$). These eigenvectors can be used to construct a normal mode representation,

$\text{x}=a_1\left(t\right){\text{u}}_{\text{1}}+\cdots +a_n\left(t\right){\text{u}}_{\text{n}}$, (iii)

Of the solution of

$\text{x}"+{\omega }_0{}^2\text{Kx}={\omega }_0{}^{2}f\left(t\right)\text{z}$,

$\text{x}\left(\text{0}\right){\text{=x}}_{\text{0}},x\text{'}\left(0\right)=v_0$, (iv).

Substitute the representation (iii) into the differential equation and initial conditions in Eqs. (iv)and use the fact that $\text{K}{u}_j={\omega }_j{}^2{\text{u}}_j,j=1,\mathrm{...},n$ and the orthogonality of ${\text{u}}_{\text{1}}{\text{, u}}_{\text{2}}\text{,}\mathrm{...}{\text{, u}}_{\text{n}}$ to show that for each $i=1,\mathrm{...},n$, the mode amplitude $a_i\left(t\right)$ satisfies the initial value problem

$a_i{}^{"}+{\omega }_i{}^2{\omega }_0{}^2{a}_i={\omega }_0{}^{2}f\left(t\right)z_i$, $a_i\left(0\right)={\alpha }_i$, $a_i{}^{\text{'}}\left(0\right)={\beta }_i$,

Where

${\text{z}}_i\text{=}\frac{{\text{u}}_i^T\text{z}}{{\text{u}}_i^T{\text{u}}_i}\text{,}{\alpha }_i\text{=}\frac{{\text{u}}_i^T{\text{x}}_{\text{0}}}{{\text{u}}_i^T{\text{u}}_i}\text{,}{\beta }_i\text{=}\frac{{\text{u}}_i^T{\text{v}}_{\text{0}}}{{\text{u}}_i^T{\text{u}}_i}$.

(f) An unforced pure mode of vibration, say, the $j$ thmode, can be realized by solving the initial value problem $\left(iv\right)$ subject to the initial conditions $x\left(0\right)=A_j{u}_j$, where $A_j$ is a mode amplitude factor, in this case, the normal mode solution of the initial value problem $\left(iv\right)$ consists of a single term,

${\text{x}}^{\left(j\right)}\left(t\right)=A_j\cos \left({\omega }_0{\omega }_{j}t\right){\text{u}}_j$.

Thus the natural frequency of the $j$ th mode of vibration is ${\omega }_0{\omega }_j$ and the corresponding period is $2\pi /\left({\omega }_0{\omega }_j\right)$. Assuming that $A_1=1$, ${\omega }_0=41$, and $n=20$, plot a graph of the components (floor displacement versus floor number) of the first mode ${\text{x}}^{\left(1\right)}\left(t\right)$ 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.