DEPARTMENT OF CIVIL, ENVIRONMENTAL
AND GEOMATICS ENGINEERING
Y4 CE-STRUCTURAL ENGINEERING
CSE 4162-STRUCTURAL DYNAMICS AND
EARTHQUAKE ENGINEERING
II. SDOF
2.1. Viscous Damping
In considering damping forces in the dynamic analysis
of structures, it is usually assumed that these forces are
proportional to the magnitude of the velocity, and
opposite to the direction of motion. This type of
damping is know as viscous damping; it is the type of
damping force that could be developed in a body
restrained in its motion by a surrounding viscous fluid.
II. SDOF
2.2. Equation of Motion
Consider the following structural system modeled as a
simple oscillator:
II. SDOF
2.2. Equation of Motion
Apply Newton’s Law to obtain the differential equation of
motion:
.. .
mu cu ku 0
The equation can be satisfied by an exponential
function of the form
u Ce pt
Substitution of this equation in the equation of
motion leads to the characteristic equation:
mp cp k 0
2
II. SDOF
2.2. Equation of Motion
Which has the following roots:
2
c c k
p1
2m 2m m
And
2
c c k
p2
2m 2m m
The general solution of the equation of the
motion is given by:
ut C1e C 2 e
p1t p2 t
II. SDOF
2.2. Equation of Motion
C1 and C2 are constants of of integration to be
determined from the initial conditions.
The final form of the above general solution
depends on the sign of the expression under the
radical of the roots. Three distinct cases may occur;
the quantity under the radical may either be zero,
positive or negative. The limiting case in which the
quantity under the radical is zero is treated first.
The damping present in this case is called critical
damping.
II. SDOF
2.3. Critically Damped System
For a system oscillating with critical damping:
c cr 2 km
Since the natural frequency of the undamped
system is given by:
the critical damping coefficient given by may also
be expressed in alternative expressions as:
II. SDOF
2.3. Critically Damped System
In a critically damped system the roots of the
characteristic equation are equal:
The general solution would be:
Another independent solution may be found by
using the function:
II. SDOF
2.3. Critically Damped System
Such that the general solution is
II. SDOF
2.4. Over Damped System
In an overdamped system, the damping coefficient
is greater that the value for critical damping. The
roots of the characteristic equations are p1 and p2
and hence the general solution of the equation of
motion is
II. SDOF
2.4. Over Damped System
It should be noted that for the overdamped or the
critically damped system, the resulting motion is not
oscillatory; the magnitude of the oscillations
decays exponentially with time to zero.
II. SDOF
2.5. Under Damped System
When the value of the damping coefficient is less than the
critical value (c < c ), which occurs when the expression
cr
under the radical is negative, the roots of the
characteristic are complex.
Putting the roots in the general solution of the
equation of motion:
II. SDOF
2.5. Under Damped System
With
the damping frequency of the system. Or
The damping ratio of the system is
II. SDOF
2.5. Under Damped System
Graphical record of the response of an
underdamped system with initial displacement u
0
but starting with zero velocity is shown.
II. SDOF
2.5. Under Damped System
It may be seen in this figure that the motion is
oscillatory, but not periodic. The amplitude of
vibration is not constant during the motion but
decreases for successive cycles; nevertheless, the
oscillations occur at equal intervals of time. This
time interval is designated as the damped period
of vibration and is given by:
II. SDOF
2.5. Under Damped System
II. SDOF
2.5. Under Damped System
II. SDOF
2.5. Under Damped System
II. SDOF
2.6. Logarithmic Decrement
A practical method for determining experimentally
the damping coefficient of a system is to initiate
free vibration, obtain a record of the oscillatory
motion, and measure the rate of decay of the
amplitude of motion. The decay may be
conveniently expressed by the logarithmic
decrement δ which is defined as the natural
logarithm of the ratio of any two successive peak
amplitudes, u1 and u2, in the free vibration, that is
II. SDOF
2.6. Logarithmic Decrement
If the decay of motion is very slow, it is desirable
to relate the ratio of two amplitudes several cycle
apart instead of successive amplitudes, to the
damping ratio. Over j cycles the movement
decreases from u1 to uj+1
Such that
II. SDOF
2.6. Logarithmic Decrement
The number of cycles elapsed for a 50% reduction in
displacement amplitude is
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading
We will consider the motion of structures idealized
as single-degree-of-freedom systems excited
harmonically, that is, structures subjected to forces
or displacements whose magnitudes may be
represented by a sine or cosine function of time.
Structures are very often subjected to the dynamic
action of rotating machinery which produces
harmonic excitations due to the unavoidable
presence of mass eccentricities in the rotating parts
of such machinery.
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
The equation of motion is:
The solution of the above equation is in the form of
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
r represents the ratio (frequency ratio) of the
applied forced frequency to the natural frequency
of vibration of the system.
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
Considering initial conditions:
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
As we can see from the above equation, the
response is given by the superposition of two
harmonic terms of different frequencies. The
resulting motion is not harmonic; however, in the
practical case, damping forces will always be
present in the system and will cause the last term,
i.e., the free frequency to eventually vanish.
For this reason, this term is said to represent the transient
response.
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
The forcing frequency term
is referred to as the steady-state response
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Undamped
It can also be seen from the above 2 equations that when
the forcing frequency is equal to natural frequency (r =
1), the amplitude of the motion becomes infinitely large. A
system acted upon by an external excitation of frequency
coinciding with the natural frequency is said to be at
resonance.
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
The equation of motion is
The complete solution of this equation again
consists of the complementary solution uc(t) and the
particular solution up(t).The complementary solution
was given for the underdamped (free vibration).
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
The equation of motion is
The characteristic equation is
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
Substituting u(p) into the above characteristic equation
yields:
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
A more convenient way of solving the equation of motion
is to write it in the following form:
The particular solution of the above equation will
be of the form:
Which gives
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
By using polar coordinates
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
Considering the imaginary part:
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
U is the amplitude of the steady-state motion.
The following equations
can be written as
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
ust = F0/k is seen to be the static deflection of the
spring acted upon by the force F0.
The total response is then obtained by combining
the complementary solution (transient response)
from and the particular solution (steady state
response):
II. SDOF
3. Response of One-Degree-of-Freedom System to
Harmonic Loading: Damped
The ratio of the steady-state amplitude of to the
static deflection defined above is known as the
dynamic magnification factor D.
II. SDOF
4. Response to support Motion
Another very important class of vibration problems
corresponds to the system being excited by the
motion of its base or support. The motion of the
base results in forces being transmitted to the
system and the resulting vibrations can be severe if
the system is not properly designed.
The absolute motion of the mass is considered.
II. SDOF
4. Response to support Motion
II. SDOF
4. Response to support Motion
The differential equation of motion is obtained by
setting equal to zero the sum of the forces
(including the inertial force) in the corresponding
free body diagram.
Substituting in the excitation force gives:
II. SDOF
4. Response to support Motion
The two harmonic terms of frequency in the
right-hand side of this equation may be combined
and rewritten as:
Where
And
II. SDOF
4. Response to support Motion
It is apparent that the above differential is of the
same form as for the exciting harmonic force
considered previously. Consequently, the steady-
state solution is given as before by:
Or by substituting Fo from the previous slide
II. SDOF
4. Response to support Motion
The above equation is the expression for the relative
transmission of the support motion to the system. This is
an important problem in vibration isolation in which
equipment must be protected from harmful vibrations
of the supporting structure. The degree of relative
isolation is known as transmissibility (or transmission
ratio) and is defined as the ratio of the amplitude of
motion U of the system to the amplitude uo, the motion
of the support. It is a measure of the motion that is
transmitted to the mass due to the excitation of the
base.
The transmissibility is then given by
II. SDOF
4. Response to support Motion
We may also find the acceleration transmitted
from the foundation to the mass. The acceleration
transmitted to the mass is given by the second
derivative of ut:
II. SDOF
4. Response to support Motion
The acceleration transmissibility is then given by the
ratio of the amplitudes of the acceleration from
the above two formulas
II. SDOF
4. Response to support Motion
It may be seen that the transmissibility of
acceleration is identical to the transmissibility of
displacements. Hence, the same expression will
give either displacement or acceleration
transmissibility.
II. SDOF
4. Force transmitted to the Foundation
In the preceding section, we determined the
response of the structure to a harmonic motion of
its foundation. In this section we shall consider a
similar problem of vibration isolation; the problem
now, however, is to find the force transmitted to the
foundation.
Consider again the following dynamic system:
II. SDOF
4. Force transmitted to the Foundation
II. SDOF
4. Force transmitted to the Foundation
The equation of motion is
And the steady state solution is :
II. SDOF
4. Force transmitted to the Foundation
The force transmitted to the support through the
spring
. is kuand through the damping element is
cu
Hence the total force transmitted FT is
II. SDOF
4. Force transmitted to the Foundation
The maximum force transmitted to the foundation
is
II. SDOF
4. Force transmitted to the Foundation
The transmissibility is defined as the ratio between the
amplitude of the force transmitted to the foundation
and the amplitude of the applied force.
It is interesting to note that, both, the transmissibility of
motion from the foundation to the structure, and the
transmissibility of the force from the structure to the
foundation are given by exactly the same function.
II. SDOF
4. Force transmitted to the Foundation
The total phase angle is
Or
II. SDOF
EXERCISES
