E12 Assignment 09

Description: 

Problems: 
1. Oscillations in LC Circuit (Double Pendulum Analog)

For the circuit shown L1=1,  L2=2, C=1.

a) Determine the circuit equation in the form:

b) Find frequencies of vibrations, and mode shapes (do by hand, you can check with Matlab).

c) Draw the force-current analog of the system.

2. E12 Vibration Code

In class I used code (code from class) to calculate modes and frequencies of vibration, and displayed mode shapes and mass trajectories. Add comments to the code between the line starting with %% Begin Comments here and ending at the line starting with %% Stop Comments here. The comments should be written for yourself, yesterday. In other words, make the comments such that if you had read them yesterday you could understand how the code is used to solve the problem and display the results.

If you are using an older version of MATLAB that generates an error, you can comment out the lines that say set(gca,'ColorOrderIndex',1);

3. Oscillation of Four Story Building (four masses with springs)

The system shown is a very crude model of a building with no damping. The masses can only move in a horizontal direction.

a) draw free body diagrams

b) write equations of the form

c) If m1=m2=m3=m4=1, and k=1, find mode shapes and associated frequencies (you can use the code from class).  Note: these mode shapes are similar to those of the air pressure in wind instruments.

d) Examine the mode shapes and associated frequencies. Is the lowest frequency associated with the expected mode shape? Why or why not?

Note: you have just developed the solution to an 8th order problem.

4. Undamped Building with Ground Shift

The system shown is a very crude model of a building with no damping. The masses can only move in a horizontal direction. The ground under the building has moved suddenly to the left at t=0, stretching the bottom spring. If we call the ground position "zero" this corresponds to the initial conditions:
x1(0)= x2(0)= x3(0)= x4(0)=1.

a) Which mode do you think will have the largest magnitude? Why?

b) Find the mode weights (i.e., the γ's) and verify your answer.

c) simulate the system and plot the trajectories of the masses.  (you can use the code from class)

5. Four Story Building, excitation of lowest frequency mode

For the previous problem

a) find initial conditions such that only the lowest frequency mode is excited.  

b) simulate the system and plot the trajectories of the masses.  (you can use the code from class)

6. Oscillations, 3 Masses with Springs, Middle mass is heavier

For the system shown

As you are doing this problem, think about the how the system would behave as the middle mass gets to be much heavier than the other two.   This should allow you to figure out in advance approximately what the mode shapes and frequencies should be.

a)   If m1=m3=1, m2=10, and k=1, find mode shapes and associated frequencies (you can use the code from class).   Try to figure out approximately what the mode shapes and frequencies will be before solving.  Recall that for a mass-spring system the natural frequency is (k/m)½.

b) Find the weights  (i.e., the "γ" terms) associated with each eigenvector and plot trajectories for the initial conditions .