MP205 Lecture 5-8 2 · 2013. 2. 27. · Barton’s pendulum: The driving frequency is ω = 6π/5...
Transcript of MP205 Lecture 5-8 2 · 2013. 2. 27. · Barton’s pendulum: The driving frequency is ω = 6π/5...
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: [email protected]
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