HW 3. CEE 247. Structural Dynamics. UCI. Fall 2006

Nasser Abbasi

Formulas used

Integration by parts is used in many problems below to solve MATH. I derive it once.

Let $u=\tau $ and MATH hence $du=1$ and MATH

Hence

MATH

Hence the integral $I$ becomes

MATH

The above is the form to remember.

or

MATH

For example, when $a=0$, $b=t$, we obtain

MATH

Problem 4.3


prob_4_3.png

Solution

We first assume that the initial absolute state of the girder is MATH, and MATH

This is the force load diagram


force_diagram_prob_4_3.png

The intercept is $F_{0}$ and the slope is MATH hence since the general line equation for $y\left( x\right) $ is MATH, we see that the equation for force loading is

MATH

First we draw the physical model diagram


standard_model.png

Using Duhamel integral, the displacement $u\left( t\right) $ is (using the assumption of no damping)

MATH

Substitute (1) into the above and carry the integration.

MATH

But MATH hence the above becomes

MATH

Now to find the stiffness $k:$

MATH

Hence MATH

Now substitute the above results for $k$ and $\omega_{n}$ in equation (2), and evaluate at $t=0.5$ we obtain

MATH

part(b)

To find maximum displacement MATH we use the response spectrum shown on page 107 of the 5th edition of the text book. First we find the natural period $T.$

MATH

Hence MATH

Hence from the spectrum on page 107, we see that MATH approximately

But MATH

HenceMATH

This is a small program to plot u(t) itself. We see that $u\left( t\right) $ became maximum before $t_{d}$. $u\left( t\right) $ maximum as at about $t=0.25\sec$


prob_4_3_mma_output.png

Problem 4.5


prob_4_5.png

solution

fig P4.5 is


force_diagram_prob_4_5.png

MATH

Hence we need to find $u\left( t\right) $

For $t\leq t_{d}$ and for an undamped simple oscillator, using Duhamel integral, the displacement $u\left( t\right) $ is

MATH

MATH hence the above becomes

MATH

Now we find $DLF$

MATH

Hence

MATH

Now we do the case for $t\geq t_{d}$

MATH

Hence

MATH

But MATH hence the above becomes

MATH

Notice there is a sign difference with the answer on the back of the book. The back of the book gives

MATH

I think the answer in the back of the book is wrong. One way to obtain the book answer from my answer is to replace $t$ by $-t$.

Problem 4.6

Frame shown in problem 4.3 above is subjected to sudden acceleration of 0.5 g applied to the foundation. Determine the maximum shear force in the columns. Neglect damping.

solution

The equation for motion when the system is subjected to ground acceleration can be written as

MATH

Where $u_{r}$ is the relative motion of the girder to the ground, and $\ddot{u}_{g}$ is the ground acceleration (absolute). Hence $-m\ddot{u}_{g}$ is the effective force $F_{e}$

Hence this is the same problem as MATH

which has the solution

MATH

Hence MATH

But from problem 4.3, we calculated $k$ to be MATH lb/in$,$ hence

MATH

Now maximum shear is given by $ku_{\max}$, hence for the left column we have (I will take absolute value of displacement, since we are only interested in maximum value, the sense of shear is not relevant).

MATH

and for the right column MATH

Problem 4.14


prob_4_14.png

solution

$m=100$ lb

$F_{0}=2000\ $lb

$k=1000\ $lb/in

Let $t_{1}=0.2\sec$

$t_{2}=0.4\sec$

Hence for $t\leq t_{1},$

MATH

For MATH

MATH

For an undamped simple oscillator, using Duhamel integral, the displacement $u\left( t\right) $ is

Hence for $0\leq t\leq t_{1}$

MATH

Hence MATH

Note that MATH and MATH

Now for MATH

MATH

But the free vibration response is , using $t_{1}=0.2$

MATH

and the second integral is

MATH

Simplify to

MATH

Hence MATH for MATH is by putting the above result back into (1) we obtain

MATH

But $t_{1}=0.2$, hence

MATH

simplify

MATH

Hence

MATH

Now MATH

Note that at $t_{2}$ we have

MATH

and at $t_{2}=0.4$ we have

MATH

Now for $t\geq t_{2}$ since no force is applied, we use the free vibration solution using the above MATH and MATH as initial conditions

MATH

Now that we have $u\left( t\right) $ for each time segment, we can plot the solution. Here it is for up to $t=0.5$ sec


plot_prob_4_14.png

Here is the solution for up to $t=1.5\sec $


plot_prob_4_14_more.png