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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

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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

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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

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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.

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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.

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ViolinViolin

Violin bridgeViolin bridge real-life mechanical system:

Ref: “The physics of the violin”, L Cremer, MIT Press, (1983).

Impe

danc

e

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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

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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

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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

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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

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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

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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