1. Oscillations
Different Situations

2. Oscillations Free Oscillations Forced Oscillations Under NO damping
(Undamped Oscilations)
Under damping
(Damped Oscillations)

3. x(t)=Real part of z(t)= x+iy
Undamped Free Oscillations
x(t)=Real part of z(t)= x+iy

4. Damped Free Oscillations
Resistive force is proportional to velocity

5. Undamped Forced Oscillations

6. Damped Free Oscillations

7. Linear Differential Equations
and it Solutions

8. homogeneous inhomogeneous
Linear differential equation of order n=2
homogeneous or
inhomogeneous

9. Complementary Solution
Take trial solution : x = emx, m is constant
m1, m2,……….will be the roots
If all roots are real and distinct, then the general solution: x =c1em1t+c2em2t+…………….
If some roots are complex, if a+ib is one then a-ib will be the other, solution: x = eat(c1 cos(bt) +c2 sin(bx)) +……
If some roots are repeated, say m1 repeated k times, then
the solution: x = (c1 + c2t+ …..cktk-1)em1t

10. General solution = Complimentary + Particular solution
Particular Solution & General Solution
General solution = Complimentary + Particular solution
For the inhomogeneous one, the complementary solution is obtained in the same way ,i.e., by making f(t)=0
Then the particular solution is added: the solution to be assumed depending on the form of f(t)

11. 2nd Order Linear Homogeneous Diff. Eqn
2nd order linear homogeneous differential equation
with constant coefficients
General solution :
x1 (t) and x2 (t) are linearly independent,
i.e x1 (t) NOT proportional to x2 (t)

12. 2nd Order Linear Inhomogeneous Diff. Eqn
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

13. Free Oscillations: SHO
A simple harmonic oscillator is an oscillating system which satisfies the following properties
1. Motion is about an equilibrium position at which point no net force acts on the system
2. The restoring force is proportional to and oppositely directed to the displacement
3. Motion is periodic
This is the first slide

14. Elements of an Oscillator
need inertia, or its equivalent
mass, for linear motion
moment of inertia, for rotational motion
inductance, e.g., for electrical circuit
need a displacement, or its equivalent
amplitude (position, voltage, pressure, etc.)
need a negative feedback to counter inertia
displacement-dependent restoring force: spring, gravity, etc.
electrical potential restoring charges

15. Harmonic Oscillator Potential

16. Model System
Consider a mass m attached to one end of a light (massless) helical spring whose other end is fixed.
We choose the origin x=0 where spring is unstretched.
The mass is in stable equilibrium at this position and it will continue to remain there if left at rest
We are interested in a situation where the mass is disturbed from equillibrium
The mass experiences a restoring force from the spring if it is either stretched or compressed.

17. Simple Pendulum

18. Torsional Oscillation

19. Compound Pendulum

20. Electrical Oscillations

21. Solution
A=Amplitude,
=Phase

22. we see the equation of motion is
Solution: Form-1
We wish to find x(t) :
we see the equation of motion is
the obvious solution is of the form of sine or cosine function
x(t) is a function describing the oscillation
what function gives itself back after twice differentiated, with negative constant?
cos(at), sin(at) both do work
exp(at) looks like it ought to work...

23. Uniform Circular Motion vs SHO

24. So the general solutions is
Solution: Form-1
So the general solutions is
where A1 and A2 are determined by the values of x and dx/dt at a specified time
So the general solution is given by the addition or superposition of both values of x so we have …

25. Solution: Form-2
If we rewrite the constants as
Where  is a constant angle, then
So that
And finally

26. Exponential Solution: Form-3
A=Complex amplitude
Real and imaginary parts of z(t) satisfy
simple harmonic equation of motion
x(t)=Re z(t)

27. Solution in Three Forms

28. Another Example

29. The Problem Cube of mass M, side 2a
Cylinder of radius r, fixed along a horizontal axis
Cube rocks on the cylinder for small angles, does not slip
Find time period

30. Torque about Q
= MgQM
= Mg(PS-PR)
PS=rθcosθ
PR=asinθ

31. I parallel axis = IG + Md2 Parallel Axis Theorem
For Cube of sides 2a : IG= ⅔ Ma2
d → QG = [a2 + (rθ)2]½ ≈ a (for small angle)
IQ = ⅔Ma2 + Ma2 = 5/3Ma2
@AR

32. Equation of Motion
Torque about Q:
Moment of inertia of the cube about a horizontal axis passing through Q is I
I =
Using small angle approximation,

33. Energy of SHO

34. K.E. & P.E. also vibrates

35. Phase space