will be of the form
.
With the trial function
, the velocities are
, and the accelerations are
.
Substituting this trial solution into the differential equations of motion and rearranging terms, we obtain the matrix equation
(the trivial solution), or if the matrix
is not invertible, i.e., it's determinant is equal to zero.
Because our building's stiffness-mass-damper system has more than one mass, the solution of the
equation (28) requires solving det
= 0
for 
.
Substituting the mass, damping, and stiffness matrices into this determinant equation, we get a sixth degree polynomial.
Unfortunately, unlike the single degree of freedom system, there is no way to solve the determinant equation for in terms of 
,
, and 
.
Luckily, though, if numerical values exist for , , and ,
they can be used to solve the equation instead.
When dealing with the physical model, we do have numerical values.
Because the building is under-damped, be have three complex-conjugate pairs of values for .
), the real parts of the three roots
are all equal to
(derived from equation 25).
Additionally,
the damping ratios are
,
, and
.
Furthermore, since the damping matrix is of the form 
(ie., damping is stiffness and mass proportional), it can be shown that
the natural frequencies of a multi-mass system are independent of damping.
In Figure 6 below, the three natural frequencies,
,
, and
are plotted with respect to the
square root of the ratio of the story shear stiffness, , to the floor mass , 
.
The natural frequencies for this structure are approximately
,
, and
.
 |
| You can use the frequency calculator at right to input different values of mass, stiffness, and damping and find out what the model building's natural frequencies would be in that case. You can see that changing the damping doesn't affect the natural frequencies, just the damped natural frequencies. Also, note the relationships graphed in Figure 6 can be determined from this calculator. What other concepts from this section can be illustrated with this calculator? |
|
With knowledge of natural frequencies and damping ratios, we may now numerically determine the constants
.
This may be done by substituting
and 
into equation (27)
and then substituting into equation (28) and solving for 
.
For every natural frequency
and damping ratio , 
, the vector 
is zero.
The fact that the vectors are real in this problem is another desirable property of proportionally damped systems.
With the simplification that the vectors are real, the solutions become
 |
 |
 |
|
 |
|
 |
are the mode-shape vectors.
Every mode-shape vector has an associated natural frequency and damping ratio.
If we were to apply an initial set of displacements that was proportional to one of the mode-shape vectors, then
the free response of the structure would occur at only that mode's natural frequency.
In general, however, the initial conditions will not be proportional to a particular mode shape, but
may always be represented by a linear combination of mode-shapes, for example,
.
In general, we will assume that the free displacement response of our three-story building
is made up of the sum of three distinct damped oscillations, each with a different frequency 
and damping 
,
.
This trial solution can be written as
,
where the three-by-three modal matrix 
is a column-wise combination of the vectors 
, 
and 
.
If the initial condition 
is only in a single mode, then the free response will remain entirely in that mode.
.