HW 5. CEE 247. Structural Dynamics. UCI. Fall 2006.

Nasser Abbasi

Problem 7.1

Solution

The idealized physical system is the following

The Lagrangian of the system is

Now apply Euler equation on the Lagrangian to obtain the equation of motion for each degree of freedom. Given the equation of motion for is given by

Hence the equation of motion associated with is given by

And the equation of motion associated with is given by

Hence the equation of motions are

Hence the overall system EQM can be put in a matrix form as follows

Notice that the mass matrix and the stiffness matrix are symmetric. This will always be the case for conservative systems.

Equation (3) can be written as

Now assume the solution is given by

Substitute (5) into (4) we obtain

Since we divide by it and obtain

Factor out

To have a non-trivial solution for the motion the above implies that the determinant of must be zero. Hence we need to solve

Let , and expand the matrices and rewrite we obtain

Now find the numerical values for and plug into the above equation to find

and

Hence eq (6) above becomes

Hence

Hence this is now in standard quadratic format, solve for

Hence

and

Since then

and similarly

Now to find the eigenvectors, since

Then

For the first eigenvalue the above becomes

From first equation we obtain

Hence

Hence we choose the first eigenvector to be

For the second eigenvalue

From first equation we obtain

Hence

Hence we choose the second eigenvector to be

Conclusion

Problem 7.6

Answer

First deterrmine the stiffness and the masses.

I wrote a Mathematica program to solve this. This is the result, and below that I attach step by step run of the program