Damped Free Oscillations Undamped Forced Oscillations Damped Forced Oscillations

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

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:

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:

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

= A = B

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).

A=0 B=2

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

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).

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

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

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

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

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

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:

Undamped Forced OSCILLATION

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

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

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

General solution: General solution = Complimentary + Particular solution 24

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

For any arbitrary time varying force © SB

Driving force: where

Equation of motion x=xr+ixi

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.

Amplitude, Relative Phase © SB

Amplitude and Phase For the case © SB

We have where At resonance [w =wo]

Low Frequency Response Because Stiffness Controlled Regime © SB

High Frequency Response Mass Controlled Regime © SB

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

General solution: 36

Initial conditions: 37

38

(1/2)Sin 2 t Sin

Sin + (1/2)Sin 2

41

42

43

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.

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 3 + A4 Cos 4 + ……. + B1 Sin + B2 Sin 2 + B3 Sin 3 +… 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

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

Examples

(1/2)Sin 2 Sin

Sin + (1/2)Sin 2

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

6 terms of the series

10 terms of the series

20 terms of the series

Sin (1/3)Sin3

Sin + (1/3)Sin3

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

6 terms of the series

10 terms of the series

20 terms of the series

Cos + (1/9)Cos 3

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

8 terms of the series

Damped Forced Oscillations

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

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.

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

Driven Damped Oscillations: Transient and Steady-state behaviours http://comp.uark.edu/~mattclay/Teaching/Spring2013/driven-motion.html