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Lecture 1 - Background from 1A. Revision: Resonance and Superposition Aims: Continue our review of driven oscillators: Velocity resonance; Displacement resonance; Power absorption Impedance matching (electrical circuits). Superposition of oscillations: Same frequency; - PowerPoint PPT Presentation
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1 Waves 2
Lecture 1 - Background from 1ALecture 1 - Background from 1A
Revision: Resonance and SuperpositionRevision: Resonance and Superposition
Aims:Aims: Continue our review of driven oscillators:
Velocity resonance; Displacement resonance; Power absorption Impedance matching (electrical circuits).
Superposition of oscillations: Same frequency; Different frequency (beats).
Transient response of a driven oscillator
2 Waves 2
Impedance.Impedance.
Mechanical impedance Mechanical impedance (Section 1.3.1)(Section 1.3.1) Last lecture we had: Z = force applied / velocity response
Magnitude: Minimum value is b, when m = s/.
Phase: = 0: phase = - / 2
= o: phase = 0
: phase = + / 2
bsmZ
smbZ
ismibZ
1
2122
tan:Phase
:Magnitude
Q =2Q = 5Q = 15
3 Waves 2
Displacement resonanceDisplacement resonance
Velocity resonanceVelocity resonance Occurs when = o.
The lower the damping the greater the “response”. (the lower the damping, the greater the amplitude of the velocity response).
Displacement resonanceDisplacement resonancealgebra is a little more complicated: solution of eq.[1.3] (last lecture) gave:
Maximum when magnitude of denominator is smallest i.e.
Resonance frequency is always less than o.(But usually only by a small amount)
bims
F
i
mFA
o
222 2
02222
bms
dd
22
2
2
2
11
2 Qm
bms
o2resonance
4 Waves 2
Velocity resonanceVelocity resonance
Magnitude and phase vs frequencyMagnitude and phase vs frequency Curves for Q=2; Q=5; Q=15.
Note: maximum velocity response at =o.
5 Waves 2
Displacement responseDisplacement response
Magnitude and phase vs frequencyMagnitude and phase vs frequency Curves are for Q=2; Q=5; and Q=15.
Note: max displacement response at < o.Phase curves shifted by -/2 but otherwise the same as for velocity resonance.
6 Waves 2
ViolinViolin
Violin bridgeViolin bridge real-life mechanical system:
Ref: “The physics of the violin”, L Cremer, MIT Press, (1983).
Impe
danc
e
7 Waves 2
Power absorptionPower absorption
cos21
21
21 22
Z
Fe
Z
F
Z
ff i
fv21 Mean power absorbed: Mean power absorbed: (sect. 1.1.3)(sect. 1.1.3)
from fig.
Notes:Notes: Power absorption -> 0 as -> 0, and as -> ,
(since Z -> ). Power absorption is maximum when = o.
The max value is
]4.1[21
21
cos
22
2
bvbZ
F
Zb
o
power mean
bF 22
8 Waves 2
Impedance matching, IImpedance matching, I
Power transmission from source to load:Power transmission from source to load: Electrical circuit:
Source impedance Zs
Load impedance Zl
Power dissipated
lsls
ooo XXiRR
V
Z
VI
tio
tio
ll
tio
llout
out
eIeVZ
iXR
eVZ
iXRV
IV
21
.
,21
Power
where
9 Waves 2
Impedance matching, IIImpedance matching, II
Notes:Notes: Rs and Rl are always positive, Xs and Xl may be
positive or negative. Maximum power transmitted when:
Impedance of the load must be equal to the complex conjugate of the impedance of the source.i.e. when there is an impedance match.
22
2
2
2
21
21
21
lsls
o
ooo
ll
XXRR
RV
RZ
V
Z
VV
Z
iXR
0
ls
ls
XX
RR
10 Waves 2
1.4 Superposition of oscillators1.4 Superposition of oscillators
Linearity:Linearity: Our equations are linear in z. Thus solutions
can be superposed.
Vibrations with equal frequency:Vibrations with equal frequency: Two forcing terms, with different amplitude and
phase.
Coherent excitation: const. interference
Incoherent excitation:Energy is simply the sum of energies of the two excitations.
otio
iiti
titi
eAeAeAe
eAeAzzz
21
21
21
2121
122122
21
2 cos2 AAAAAo
Interference termInterference termA2 energyA2 energy
0cos 12
12
11 Waves 2
Superposition cont…...Superposition cont…...
Vibrations of different frequencyVibrations of different frequency
(for simplicity) take
Beats:Beats: When there are many
rotations of Ao before the length changes significantly.
Time between successive maxima in amplitude is 2/(1-2) .
The beat frequency is the frequency difference.
221121
titi eAeAz
.;0 2121 AAA
ti
tititi
etA
eeAez
221
222
21
212121
2cos2
2121
12 Waves 2
TransientsTransients
Full solution for the forced oscillator.Full solution for the forced oscillator.Sum of two parts: Particular integral: i.e. solution of
Complementary function: i.e. solution of
(decays with time, and oscillates for a lightly damped system).
tiFeczzbzm
0 czzbzm