1. Damped Free Oscillations
Undamped Forced Oscillations
Damped Forced Oscillations

2. Damped Free OSCILLATION
Resistive force is proportional to velocity
Where,
Or sometimes given in the form...
Where, g =r/m and

3. Solution
The equation is a second order linear homogeneous equation with constant coefficients.
Solution can be found which has the form: x = Cept where C has the dimensions of x, and p has the dimensions of T-1.
Trivial solution
Solving the quadratic equations gives us the two roots:
The general solution takes the form:

4. Case I: Overdamped (Heavy damping)
The square root term is +ve: The damping resistance term dominates the stiffness term.
Let:
Now, if:
Then displacement is:

5. Non-oscillatory behavior can be observed.
But, the actual displacement will depend upon the boundary conditions

6. = A
= B

7. CaseII: Critical damping
The damping resistance term and the stiffness terms are balanced.
When r reaches a critical value, the system will not oscillate and quickly comes back to equilibrium.
The quadratic equation in p has equal roots, which, in a differential equation solution demands that C must be written as (A+Bt).

8. A=0 B=2

9. Case III: Underdamped 4m2
The square root term is -ve: The stiffness term dominates the damping resistance term.
The system is lightly damped and gives oscillatory damped simple harmonic motion.
4m2

10. Features of underdamped motion
The underdamped motion has two features:
Its frequency is reduced:‘<0 – which means that the time period is increased and
2) Its amplitude decays exponentially (as seen in the next graph).

11. Underdamped oscillations
Note that the logarithmic decrement is defined as the natural logarithm of the ratio of successive amplitudes:

12. Logarithmic decrement
Q: How is the energy (PE) changing??

13. Relaxation time
Relaxation time is the time taken for the amplitude to decay to 1/e of its original value.
Note: 1/e = 0.368
When t = relaxation time

17. Undamped free oscillation
Envelope function
Undamped free oscillation
Damped oscillation
Energy decay

18. Quality factor (Q-value) of an oscillator
Q value measures the rate at which the energy decays
Since amplitude decays as:
The decay of energy is proportional to:
Now,
(Energy value at t=0)
where
The time taken for energy to decay to
is t = m/r
During this time the oscillator will have vibrated through
radians.
Now, we define the Quality factor:
It is the number of radians through which the damped system oscillates as its energy decays to

19. High Q low damping Quality factor:
If r is very small, then Q is very large and
becomes
which is a constant for the damped system
And to a very close approximation:
High Q low damping
Hence,
As Q is a constant, the following ratio is also a constant:
This gives the number of cycles through which the system moves in decaying to
It can be shown that:

20. Undamped Forced OSCILLATION

21. Equations of Motion Linear differential equation of order n=2
inhomogeneous 21

22. 2nd order linear inhomogeneous differential equation
with constant coefficients
General solution :
Complementary function
Particular integral: obtained by special methods, solves the equation
with f(t)0; without any additional parameters
A & B : obtained from initial conditions

23. Complementary solution:
C(t)
Particular solution:
P(t)

24. General solution:
General solution = Complimentary + Particular solution 24

25. External Forcing SHO with an additional external force
Why this particular type of force ?
© SB

26. For any arbitrary time varying force
© SB

27. Driving force:
where

28. Equation of motion
x=xr+ixi

29. Obtaining the particular integral
Note: As the complementary solution has been discussed extensively earlier, we shall ignore this term here.
Obtaining the particular integral
Trial solution:
Note: Here w is the angular frequency of the external driving force.

30. Amplitude, Relative Phase
© SB

31. Amplitude and Phase
For the case
© SB

32. We have
where
At resonance [w =wo]

33. Low Frequency Response
Because
Stiffness Controlled Regime
© SB

34. High Frequency Response
Mass Controlled Regime
© SB

35. Summary Undamped forced oscillation
Stiffness controlled regime (w<w0)
Resonance (w=w0)
Mass controlled regime (w>w0)

36. General solution: 36

37. Initial conditions: 37

38. 38

39. (1/2)Sin 2 t
Sin

40. Sin +
(1/2)Sin 2

41. 41

42. 42

43. 43

44. Fourier Series
A Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

45. Fourier Series: f(t), t < < T f(t) = A0 + A1 Cos + A2 Cos 2
<
< T
f(t) = A0 + A1 Cos A2 Cos 2
+ A3 Cos A4 Cos …….
+ B1 Sin B2 Sin B3 Sin …
For a periodic function f(t) that is integrable on [−π, π] or [0,T], the numbers An and Bn are called the Fourier coefficients of f.
A0 =
f(t)
An / 2 =
f(t)
Cos n
Bm / 2 =
f(t)
Sin m

46. Cos n Sin m = 0 Cos m Sin n are different n and m The infinite sum
is called the Fourier series of f.

47. Examples

48. (1/2)Sin 2
Sin

49. Sin +
(1/2)Sin 2

50. Sin
+ (1/2)Sin 2
+ (1/3)Sin 3
+ (1/4)Sin 4

51. 6 terms of the series

52. 10 terms of the series

53. 20 terms of the series

54. Sin
(1/3)Sin3

55. Sin +
(1/3)Sin3

56. Sin
+(1/3)Sin3
+(1/5)Sin5
+(1/7)Sin7

57. 6 terms of the series

58. 10 terms of the series

59. 20 terms of the series

60. Cos
+ (1/9)Cos 3

61. Cos
+ (1/9)Cos 3
+ (1/25)Cos 5
+ (1/49)Cos 7

62. 8 terms of the series

63. Damped Forced Oscillations

64. The solution for x in the equation of motion of a damped simple harmonic oscillator driven by an external force consists of two terms:
a transient term (‘=temporary’)
and
a steady-state term

65. Transient Term Steady-state Term
The transient term dies away with time and is the solution to the equation discussed earlier:
This contributes the term: x = Cept
which decays with time as e-βt
Steady-state Term
The steady state term describes the behaviour of the oscillator after the transient term has died away.

66. Solutions Complementary Functions are transients
Both terms contribute to the solution initially, but the ultimate behaviour of the oscillator is described by the ‘steady-state term’.
Solutions
Complementary Functions are transients
It always dies out if there is damping. As a practical matter, it often suffices to know the particular solution.
Steady State behaviour is decided by the Particular Integral

67. Driven Damped Oscillations:
Transient and Steady-state behaviours