Download Chapter 2 THE DAMPED HARMONIC OSCILLATOR

Chapter 2
THE DAMPED HARMONIC OSCILLATOR
Reference: George C. Kings, Vibrations and waves, A John Wiley and Sons, Ltd., Publication, 2009.
In this lecture we will concentrate our attention to the damped harmonic oscillator. Before state the
plan of the lecture let us briefly summarize preceeding topic, simple harmonic motion (SHM).
In the previous chapter we have already discussed SHM.
SHM is periodic and neither driven nor damped.
We have stated that;
SHM is that to a good approximation many real oscillating systems behave like simple harmonic
oscillators when they undergo oscillations of small amplitude.
Remember that we have investigated physical nature of the SHM on three fundamental examples:
1. Mass-Spring system
The force acting on the mass is just due to spring. According to
the Hook’s law we can write:
No frictional force (damped force)
No driven force.
Therefore, according to the Newton’s second law
In addition to displacement, velocity, acceleration, force, momentum, energy we define the quantities
Frequency,
(frequency is defined as a number of cycles per unit time.)
Period
(The period is the duration of one complete cycle in a repeating event)
Angular frequency
That describes sSHM.
Note that force is proportional to the
in a SHM then
is defined as
Return (restoring) force per unit displacement, per unit mass.
Solution of the Newton’s equation yields
The constants
can be obtained from initial conditions.
Solution of the equation include information about various physical quantities, such that
Velocity
Acceleration
Force
Momentum
Energy
2. Simple Pendulum
Again we can write the force acting on the mass m. For small oscillations
, we can write the equation of motion:
Its solution is given by
We point out that, for the physical pendulum we consider inertia of the oscillating rod. In that case
3. The last example we have discussed is LC circuit:
Using Kirchhoff’s law
Its solution is again
We also discussed correspondence between physical quantities
of the oscillator
take place of or
take place of
take place of or
IMPORTANT!!
The physical models do not include any damping or driving force! It is not realistic.
We have ignored friction between mass and surface, air resistance, and resistance of the cables etc..
In order to make a more realistic physical model we have to take into consideration, resistive or
frictional forces.
I hope that at the end of this chapter you understand physical characteristic of damping oscillator and
solution of the Newton’s equation including damping forces.
Outline of the Chapter:
2. THE DAMPED HARMONIC OSCILLATOR
2.1 Physical Characteristics of the Damped Harmonic Oscillator
2.2 The Equation of Motion for a Damped Harmonic Oscillator
2.2.1 Light damping (weak damping)
2.2.2 Heavy damping (strong, over damping)
2.2.3 Critical damping
2.3 Rate of Energy Loss in a Damped Harmonic Oscillator
2.3.1 The quality factor Q of a damped harmonic oscillator
2.4 Damped Electrical Oscillations
PROBLEMS
The Damped Harmonic Oscillator
We have described simple harmonic motion that;
a mass i.e mass of the pendulum, swinging back and forth at the end of a string
a mass oscillating up or down or back and forth attached to a sprinq
charges oscillating over capacitor and inductor in a circuit.
Note that this oscillating system is not ideal.
In fact after we set the mass (or charge) in motion, the amplitude of oscillation steadily reduces and the
apple eventually comes to rest.
This is because there are dissipative forces acting and the system steadily loses energy.(remember
energy is proportional to the square of the amplitude of the oscillation!)
For example, the mass will experience a frictional force as it moves through the air.
There exist between mass and surface when the mass
oscillating horizontally attached to a spring.
In a realistic model, there are resistive force acting on the
charge in LC circuit, due to wires and internal resistance of the
devices.
The motion is damped and such damped oscillations are the
subject of this chapter.
We note that all real oscillating systems are subject to damping
forces and will cease to oscillate if energy is not fed back into
them.
How can we model oscillating system with damping?
How can we write the Newton’s equation including damping
force?
How can we relate the damping force with the physical
quantities?
Consider an object moving in a viscous fluids (or gas). Fluid (also gas) experience a force on the
object is known as drug force. (result of interaction between molecules).
Often these damping forces are linearly proportional to velocity. (In fact it is proportional to the nth
power of the velocity but its simplified form for relatively small speeds is proportional to the first
power of velocity).
In an electric circuit this force (potential) is known as resistive force and it is proportional to the
current.
Example: In an electric circuit, power dissipated on a resistor of resistivity 100 Ohm, is 4.0 Watt.
Calculate current through resistor and voltage across the resistor.
Solution: Power of a resistor is defined as follows:
Potential across the resistor is
then power
then
For a mechanical system power can be expressed as
; Here F corresponds potential and
corresponds current .
Fortunately, this linear dependence leads to an equation of motion that can be readily solved to obtain
solutions that describe the motion for various degrees of damping.
Clearly the rate at which the oscillator loses energy will depend on the degree of damping and this is
described by the quality of the oscillator. At first sight, damping in an oscillator may be thought
undesirable (unwanted).
However, there are many examples where a controlled amount of damping is used to quench unwanted
oscillations.
For instance quality of the suspension system of a car depend on damping! If damping is zero then the
car’s bouncing up and down long after it has passed over a bump in the road.
Think! What happens when a bridge does not have a damping? Additional damping was installed on
London’s Millennium Bridge shortly after it opened because it suffered from undesirable oscillations.
2.1 PHYSICAL CHARACTERISTICS OF THE DAMPED HARMONIC OSCILLATOR
A tuning fork is an example of a damped harmonic oscillator. Indeed we hear the note because some
of the energy of oscillation is converted into sound. After it is struck the intensity of the sound, which
is proportional to the energy of the tuning fork, steadily decreases. However, the frequency of the note
does not change!!(Be careful). The ends of the tuning fork make thousands of oscillations before the
sound disappears and so we can reasonably assume that the degree of damping is small. We may
suspect, therefore, that the frequency of oscillation would not be very different if there were no
damping. Thus we infer that the displacement
of an end of the tuning fork is described by a
relationship of the form
where the angular frequency is about but not necessarily the same as would be obtained if there
were no damping.
2.2 THE EQUATION OF MOTION FOR A DAMPED HARMONIC OSCILLATOR
An example of a damped harmonic oscillator is shown in
Figure. It is similar to the simple harmonic oscillator
described in previous section but now the mass is immersed
in a viscous fluid. When an object moves through a viscous
fluid it experiences a frictional force. This force dampens the
motion: the higher the velocity the greater the frictional
force. So as a car travels faster the frictional force increases
thereby reducing the fuel economy, while the velocity of a
falling raindrop reaches a limiting value because of the
frictional force. (Talk about limiting-terminal velocity)
The damping force
acting on the mass in Figure is
proportional to its velocity so long as is not too large (for
large the force is proportional to the higher power of , i.e
for an airplane frictional force is proportional to the square of
the velocity). The resulting equation of motion is
where the minus sign indicates that the force always acts in the opposite direction to the motion. The
constant depends on the shape of the mass and the viscosity of the fluid and has the units of force
per unit velocity.
We introduce the parameters
The equation takes the form:
This is the equation of a damped harmonic oscillator. Now we designate this angular frequency
and describe it as the natural frequency of oscillation, i.e. the oscillation frequency if there were no
damping. This allows the possibility that the damping does change the frequency of oscillation. In the
present example the damping force is linearly proportional to velocity. This linear dependence is
very convenient as it has led to an equation that we can readily solve. As we mentioned before, this
linear dependence is a good approximation for many other oscillating systems when the velocity is
small. The damped oscillator equation can be solved by using various methods. Suppose that the
damping is exponential. The general solution of the equation will be in the form
Substituting
into the equation we obtain:
Then we obtain
Then the general solution takes the form
Depending on the relation between damping constant and natural frequency the degree of damping
involved, corresponding to the cases of (i) light damping, (ii) heavy or over damping and (iii) critical
damping. Light damping is the most important case for us because it involves oscillatory motion
whereas the other two cases do not.
Let us break here. We will continue to explore our analysis next
lecture.
Briefly review previous lecture notes!
Remember, we have obtained a general solution:
Where
, describe type of the oscillation as we
stated before.
2.2.1 Light damping
Consider a mass m immersed in a fluid as in the figure. We
assume that the viscosity of the fluid is low like thin oil or even
just air.
The parameters and are determined solely by the properties
of the oscillator while the constants
are determined by
the initial conditions.
If we let
we obtain, as expected, our previous results for the simple harmonic oscillator.
A
graph
of
is shown in Figure for different values of
Here we are dealing weak damping therefore
for simplicity we set
We see that successive maxima decrease by the same fractional amount. This is logarithmic decrement
2.2.2 Heavy damping(Over damped oscillator, strong damping)
Heavy damping occurs when the degree of damping is sufficiently large that the system returns
sluggishly to its equilibrium position without making any oscillations at all. This corresponds to the
mass being immersed in a fluid of large viscosity like syrup. Consider frequency of the oscillation:
If
is larger than
, then the system is over damped. In this case
becomes imaginary and
Hyperbolic function is not periodic, it is exponential. Therefore the system is over damped. Figure
shows over damped motion of the mass.
  1,  0  12
1

3

2
2
2
  1,  0  6
1

3

2
2
2
  4,  0  1
1

2

3
2
2
  6,  0  1
1


2
3
2
2
2.2.3 Critical damping
An interesting situation occurs when
equation becomes
. In this case the solution of the damping oscillator
This is the case of critical damping. Here the mass returns to its equilibrium position in the shortest
possible time without oscillating. Critical damping has many important practical applications. For
example, a spring may be fitted to a door to return it to its closed position after it has been opened. In
practice, however, critical damping is applied to the spring mechanism so that the door returns quickly
to its closed position without oscillating. Similarly, critical damping is applied to analogue meters for
electrical measurements. This ensures that the needle of the meter moves smoothly to its final position
without oscillating or overshooting so that a rapid reading can be taken. Springs are used in motor cars
to provide a smooth ride. However damping is also applied in the form of shock absorbers
as illustrated schematically in Figure. Without these the car would continue to bounce up and down
long after it went over a bump in the road.
Schematic diagram of a car
suspension system showing the
spring and shock absorber.
Shock absorber consists essentially
of a piston that moves in a cylinder
containing a viscous fluid. Holes in
the piston allow it to move up and
down in a damped manner and the
damping constant is adjusted so that
the suspension system is close to the
condition of critical damping. You
can see the effect of a shock absorber
by pushing down on the front of a
car, just above a wheel. The car
quickly returns to equilibrium with
little or no oscillation. You may also
notice that the resistance is greater
when you push down quickly than
when you push down slowly. This
reflects the dependence of the
damping force on velocity.
  2,  0  1
1


2
3
2
2
To appreciate the physical origin of these different types of motion, we recall that is the damping
term while
is proportional to the spring constant .
When the damping term is small compared with
the motion is governed by the restoring force of
the spring and we have damped oscillatory motion. Conversely, when the damping term is large
compared with
the damping force dominates and there is no oscillation. At the point of critical
damping the two forces balance. We finally note that the relative size of
compared with
also determines the response of the oscillator to an applied periodic driving force, as we shall see in
next Chapter.
Worked example (I will add an example)
2.3 RATE OF ENERGY LOSS IN A DAMPED HARMONIC OSCILLATOR
The mechanical energy of a damped harmonic oscillator is eventually dissipated as heat to its
surroundings. We can deduce the rate at which energy is lost by considering how the total energy of
the oscillator changes with time. The total energy E is given by
For undamped oscillator the energy is given by
For damped oscillation the displacement can be written as:
We can write
In this case energy takes the form:
where
is the total energy of the oscillator at t = 0. We have the important and simple result that the
energy of the oscillator decays exponentially with time as shown in Figure.
We also have an additional physical
meaning for . The reciprocal of is the
time taken for the energy of the oscillator to
reduce by a factor of . Defining
,
we obtain
where has the dimensions of time and is
called the decay time or time constant of
the system. There are many examples of
both classical and quantum-mechanical
systems that give rise to exponential decay
of their energy with time as described here
and for some of these
is called the
lifetime.
The energy of an oscillator is dissipated because it does work against the damping force at the rate
(damping force × velocity). We can see this by differentiating . Power dissipated by resistive or
damped or drag force is
2.3.1 The quality factor Q of a damped harmonic oscillator
From the foregoing analysis, it is clear that the damped oscillator is characterized by two parameters,
. The constant
is the angular frequency of undamped oscillations and is the
reciprocal of the time required for the energy to decrease to
of its initial value. Thus
are
quantities of the same dimensions. For convenience in applying our results to diverse kinds of physical
systems, we define a parameter called the value ( for quality) of the oscillatory system, given by
the ratio of these two quantities:
Note that
meaning of the
have the same dimensions therefore
then we define -value
is a pure number. Remember physical
Angular frequency can be expressed as:
It is quite evident that the higher the
value, the greater the number of oscillations.
We can define the quality factor in a different way by considering the rate at which the energy of the
oscillator is dissipated. If we consider the energy of a very lightly damped oscillator one period apart
we have
Giving
The series expansion of
Thus, when If
is
,
to a good approximation. And therefore
where we have substituted
. The fractional change in energy per cycle is equal to
so the fractional change in energy per radian is equal to
.
2.4 DAMPED ELECTRICAL OSCILLATIONS
In our mechanical example of a mass moving through a fluid we
saw that the fluid offered a resistance that damped the motion. In
the case of an electrical oscillator it is the resistance in the circuit
that impedes the flow of current. An electrical oscillator is shown
in Figure. It consists of an inductor and capacitor as before but
now there is also the resistor
We charge the capacitor to voltage
switch. Kirchoff’s law gives
and
and then close the
This has the identical form to Equation,
and we recognise the analogous quantities we met before: is equivalent to , to and to
.
However, we see that
is analogous to the mechanical damping constant and so
is the
equivalent of
. Since the above differential equations have identical forms, their solutions
also have identical forms.
The importance of this is that we can use our results for the mechanical oscillator to immediately write
down the corresponding results for the electrical case. Thus solution of the equation is
Where
, describe type of the oscillation as we stated before. Here
is initial charge
on the capacitor and
Similarly we find that the quality factor
of the circuit is given by
For example, with
, which is a typical value for
an electrical oscillator. Again we emphasize the exact correspondence between the equations and
solutions that describe the mechanical and electrical systems, so that mechanical systems can be
simulated by electrical circuits. Such analogue computers can greatly facilitate the design and
development of mechanical systems.
Now, we have completed chapter 2. Next lecture I will solve sample problems.