Dynamic of structures software progress report No 7.

In order to find the response of a structure under an external dynamic force (an earthquake) we solved the eigenvalue problem, whose solutions gives the natural frequencies and modes of a system, u(t)=qn(t)PHIn.

Where the deflected shape PHIn does not vary with time. The time variation of the displacements is described by the simple harmonic function:

qn(t) = AnCosWnt +BnSinWnt

Therefore, u(t)= PHIn(AnCosWnt+BnSinWnt)

Where Wn and PHIn are unkown.

Taking into account the equation of motion: m:u+ku= 0

We could determine: [ -W**2mPHIn+KPHIn]qn(t) =0

Therefore KPHIn=W**2mPHIn

The problem is to determine the scalar W**2 and the vector PHIn.

We obtain the frequency equation: [K-W**2m]PHIn =0

Which has a Non trivial solution: Det[K-W**2n]=0

When the determinant is expanded, a polynomial of order N in Wn is obtained.

The N roots of the frequency equation determine the N natural frequencies Wn (n=1,2,3…) of vibration. When a natural frequency is Known Wn, The equation can be solved for the corresponding vector PHIn to within a multiplicative constant.

There are N independent vectors PHIn, which are also known as natural modes of vibration, or natural mode shapes of vibration.

To commence, we used the elastic bending theory to determine the deformations of a three story structure subjected to unitary load.

We obtained the 6x6 stiffness matrix of the structure:

Then, we used the Goyan method to condense the 6x6 stiffness matrix in order to find the lateral stiffness matrix of the structure.

With the lateral stiffness matrix {Ktt} of the structure, we solved the frequency equation:

Det[Ktt-W**2m]=0

We also solved the polynomial equation to find the four frequencies of vibration Wn of the structure.

Then we normalized the frequency equation to obtain the natural modes of vibration PHIn.

So the latest version of the dynamics of structures software DynRFRv9.0 could calculate the frequencies, periods of vibration and the internal forces of any three story structure.

The next step is to scale up the software in order to solve bigger structures.

The big picture: we would like to use a Maglev Train type of system for buildings, therefore we need to find the size of the magnetic field located at the building foundation in order to reduce the pseudoaceleration effect produced by any earthquake.