A photo and schematic diagram of the three-story building model used in this lab are shown in Figure 3 below. The model consists of:
in the columns.
The bending of these columns is similar to the extension of a spring;
we can usually assume that the shear force in the columns is constant over the height of the columns and that the shear force 
is proportional
to the column deformation, 
.
In other words, the shear force increases linearly with the column deformation.
The shear stiffness 
of the column is this constant of proportionality, and 
.
The shear forces in the columns act on the mass of the floors, at each floor-column connection.
These forces will contribute to the total acceleration, 
, of the floor masses 
, 
.
|
(a)  (b) |
with respect to the ground, at any point in time, 
),
the forces from the columns below the mass are
acting to the left and
the forces from the columns above the mass are
acting to the right.
| Inertial Forces |
|---|
| We add inertial forces to make the calculations in the reference frame of the floor (all forces add up to zero). While a moving mass does not have a force working against its motion when viewed from stable ground, inertial forces are percieved in the non-inertial reference frame of the floor. |
)
plus the acceleration of the ground (
).
The inertial force will oppose this acceleration
and will act to the left with a magnitude of
.
Now we can write the equations of motion for this first mass.
acting to the left.
From the columns above the second mass, the forces are
acting to the right.
The inertial force on the second mass is
acting to the left.
Summing forces on the second mass gives us
 |
 |
 |
(6) |
 |
 |
|
(7) |
 |
 |
|
(8) |
| Mass and Stiffness Matrices |
|---|
| Note that the mass matrix is diagonal and that both the mass and stiffness matrices are symmetric. In general the mass matrix is not always diagonal, but it is always symmetric. |
is the displacement vector,
is the mass matrix,
is the stiffness matrix,
and 
is a column vector of ones.
We now have a differential equation describing the free vibration of our three-story building model.
However, to more accurately describe the model, we must also include the affects of damping, which we do in the next section.
Then, we will solve the differential equation to find the natural frequencies and mode shapes of the model in order to understand the response of the model in earthquake conditions.
