2. Key Questions Being Answered
1) What are damped Oscillations?
2) What variables are considered in calculating damped
oscillations?
3) What is an underdamped oscillator?
4) What is a critically damped oscillator?
5) What is an over-damped oscillator
6) What is the energy and energy loss in a Damped Harmonic
Oscillator?
7) What questions might be asked on damped oscillation?
3. Question 1: What are damped Oscillations?
Definition: Oscillations In the presence of frictional forces
are called damped oscillations.
The formula F=-bv is used to determine the drag force an
object experiences when moving though a medium.
b = The damping constant or drag constant
V = velocity of the object moving through the medium
The negative sign in the formula indicates that the
direction of the drag is opposite of the velocity of the
object.
4. Question 1: What are damped Oscillations?
(cont.)
It has been shown that two facts are true about damped
oscillations:
1) The amplitude of a damped oscillator decreases
exponentially with time.
2) The oscillation frequency of a damped oscillator is lower
than the oscillation frequency of an undamped oscillator.
The equation used to find the displacement of a damped
oscillator is:
X(t) = Ae^(-bt/2m)cos(W(d)t + Φ)
5. Question 2: What variables are considered in
calculating damped oscillations?
As seen in the last slide, the equation to determine the displacement
of a damped oscillator is:
X(t) = Ae^(-bt/2m)cos(W(d)t + Φ)
The variables considered include:
W(d) = (k/m) – (b^2/4m^2)
K is the spring constant
M is the mass
B is the damping constant
W(d) is known as the oscillation frequency in the presence of drag
force.
W(0) (which is sqrt (k/m)) is called the natural frequency and is when
b = 0.
6. Question 3: What is an underdamped
oscillator?
An Underdamped oscillator is when the natural frequency
(W(0)) is larger than the fraction b/2m.
W(0) > (b/2m)
Underdamped oscillations are when the natural frequency
(W(0)) is close to the oscillation frequency (W(D)).
The main effect of drag force in underdamped oscillations
is to decrease the amplitude exponentially with time.
7. Question 4: What is a critically damped
oscillator?
A critically damped oscillator is when the natural frequency is equal
to b/2m
W(0) = (b/2m)
In this case, the oscillation frequency (W(D)) is ) so there is no back-
and-forth motion in the oscillator.
The question X(t) = (a(1) + a(2)t)e^(w(0)t) is used to find the position
of a critically damped oscillator where:
A(1) and A(2) are determined by the initial conditions X(0) and X(1).
Critically damped oscillators reach its equilibrium position faster than
any other value of the damping constant.
I.e. These oscillators quickly return to their natural position.
8. Question 5: What is an over-damped
oscillator?
An over-damped oscillator is when the value of (b/2m)
surpasses that of the natural frequency.
W(0) < (b/2m)
As the value of b increases, it takes an increasing amount
of time for the oscillator to reach back to its natural
position.
9. Question 6: What is the energy and energy
loss in a Damped Harmonic Oscillator?
The Energy in a damped harmonic oscillator is given by
the equation:
E(t) = (1/2)(kA^2e^(-bt/m))
E(0) = (1/2)kA^2
The fraction energy loss in one oscillation is given by the
equation:
1 – e^(-bT(D)/m)
B is the damping constant
T(D) is the period 2pi/W(D)
10. Question 7: What questions might be asked
on damped oscillation?
Using the example given in the textbook, we are asked to
find the damping constant (b) given the information:
M = 2kg
K = 10N/m
A= 25cm at t=0
The amplitude falls to 75% of its initial value after 4
oscillations.
11. Question 7: What questions might be asked
on damped oscillation?
(cont.)
Step 1: Calculate the total energy
E(0) = (1/2)kA^2 = 0.312J
Step 2: Calculate the period
T = 2pi/(W(0)) = 2pi*Sqrt(m/k) = 2.81s
Step 3: Use the information that the amplitude falls off by
75% in for oscillation.
A(4T) = 3/4A
e^(-2bt/m)=3/4
12. Question 7: What questions might be asked
on damped oscillation?
(cont.)
Step 4: Solve for b.
e^(-2bt/m)=3/4
-2bt/m = ln(3/4)
b = -m/2T*(ln3/4)
b = -(2/2*2.81)*ln(3/4)
b = 0.102kg/2
13. Conclusion
The question that were asked in this power point only
cover the main concepts of the ideas of damped
oscillation.
If you have any questions that could be added to this
power-point, feel free to message me and I will add them
onto this slideshow.
Thank You for Watching
