MathematicalPhysics

MP205 Vibrations and Waves

Lecturer:              Dr.  Jiří  Vala  Office:                Room  1.9,  Mathema<cal  Physics,    

               Science  Building,  North  Campus  Phone:              (1)  708  3553  E-­‐Mail:              jiri.vala@nuim.ie  

Lecture  9  -­‐  10  

Example:

A (nearly) undamped pendulum of length 38.8 cm, i.e. of a natural frequency ω0 = 8π/5 rad.s-1, with the forcing done by moving the suspension point back and forth 2 millimetres each way, but with different frequencies:

ω = ω0/2 ω = ω0 ω = 2ω0

resonance

Complex representation of the sinusoidal driving force and displacement vector in the forced oscillations:

Barton’s pendulum

in which several pendula of differing lengths are all driven by an oscillation of the same frequency:

Time: t t + Δt

Barton’s pendulum:

The driving frequency is ω = 6π/5 rad.s-1, the damping constant is the same, i.e. Q = 1.

The ten blue pendula which would have the above behaviour in real life are of lengths from 6.9 cm to 110.3 cm in increments of 13.8 cm. The red pendulum is at resonance, i.e. its natural frequency is identical to the driving frequency, and would be 69.0 cm long.

Note that all pendula shorter than this red one lead it, while the longer ones lag behind.

Example: the effect of damping on kicked pendula:

pendula are kicked by a half-second long oscillation of amplitude 3.9 cm, with the damping constants such that

Note that, in both the under- and overdamped cases, the pendulum slighly overshoots its initial position, whereas in the critically damped case, it gradually approaches the vertical without overshooting. This is why the indicator needles in instruments like ammeters and voltmeters are critically damped, so that they quickly approach a final reading rather than wobble about it.

Q = 2 underdamped

Q = 1/2 critically damped

Q = 2/5 overdamped

Sharpness of tuning of a resonant system with Q: example: Barton’s pendulum

Q1 < Q2 < Q3

transient (both natural & driving vibrations)

t = 0: the driving force is turned on

steady-state (no natural vibrations)

time

Response to a periodic driving force:

Undamped harmonic oscillator – the beat pattern continues indefinitely

Transient behavior of damped harmonic oscillator driven off-resonance

Transient behavior of damped harmonic oscillator driven at resonance

Mean power absorbed by a forced oscillator

ω/ω0

P(ω)

_

in units of F0

2/2mω0

Pmax= QF02/2mω0

_

Q = 1

Q = 3

Q = 10

Q = 30

Sharpness of resonance curve determined in terms of power curve

ω0 ω0 – γ/2 ω0 + γ/2

Pmax

Pmax/2 Width of power resonance curve at half-height = γ or ω0/Q very nearly