Engineering Acoustics -...

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Engineering Acoustics(Mechanic System)

PROF. NING XIANG

GRADUATE PROGRAM IN ARCHITECTURAL ACOUSTICS, SOA

RENSSELAER POLYTECHNIC INSTITUTE, TROY, NEW YORK

WebEx/Xiang, Sept. 8th 2020

Program in Architectural Acoustics2

http://symphony.arch.rpi.edu/~xiangn/XiangTeaching.html

Suggested Solutions

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Program in Architectural Acoustics

Outline Oscillation definition

Categories of oscillations

Basic elements of linear, oscillating,

mechanic systems

Parallel mechanic oscillations

Free / forced oscillations3

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Definition

Oscillation:

An Oscillation is a process with

attributes that are repeated

regularly with time

4


Oscillation: Energy SwingEnergy swinging between:

Kinetic Potential Energy

Electric Magnetic Energy5


Linear Time-Invariant SystemsLinear time-invariant (LTI):

Superposition principle applies

Assuming )()( txFty kk

k

kkk

kk txbFtyb )()(Then:

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Eigen-Function of LTI-Systems

: complex frequency

ttts eAeA j

js

;ˆ jeAA : angular frequency

7

tteeA j tteA t sinjcos


Basic QuantitiesQuantity General Sinusoidal

Velocity

Acceleration

Displacementte jˆ)(t

)(tv tevv jˆ

)(ta teaa jˆ8


Basic Quantities

dt

tdtv

)()(

dttvt )()(

dt

tvdta

)()( dttatv )()(

2

2 )()(

dt

tdta

dtdttat )()(

9

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Sinusoidal Quantities

jv j/v

va j j/av

2a 2/ a10


Basic QuantitiesRelationbetween

Arbitrary time-function Sinusoidal time-function

Displacement

velocity

Velocity

acceleration

Displacement

acceleration

dt

tdtv

)()(

dttvt )()(

dt

tvdta

)()( dttatv )()(

2

2 )()(

dt

tdta

dtdttat )()(

jv v

j

1

va j avj1

2a a2

1

11


Section 2.1

Basic Elements of

Linear, Oscillating

Mechanic Systems

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Basic Assumptions

• Linear relationships of all quantities

• Constant features of elements

• One-dimensional motion (simplified)

13


Basic Quantity: MassamF Newton’s law

)()( tamtF

In sinusoidal cases

mvmamF 2j

(one-port)Mechanic impedance

mZ jmech imaginary

2

2

dt

dm

dt

dvm

Physical unit of a mass: or]kg[ ]/mNs[ 2

14


Basic Quantity: Spring

kFHooke’s law

: Spring constant: Stiffness k: Compliance with a unitkn /1

)(1

tn

F

dtdtan

dtvn

1

1

]m/N[

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Basic Quantity: SpringIn sinusoidal cases:

a

nv

nnF

2

1

j

11

Mechanic impedance:

nZ

j1

mech imaginary

16


Fluid Damper (Dashpot)

ar

rvrF

j

jIn sinusoidal cases:

dtardt

drtvrtF

)()(

Mechanic impedance:

]Ns/m[mech rZ real

17


Force -- ResponsesResponse R Force

----------------Response

Response-----------------

Force

Displacement Dynamic stiffness Dynamic complianceReacceptanceDynamic flexibility

Velocity Mechanic impedance Mobility (mech.admittance)

Acceleration Dynamic mass Inertance,Accelerance

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Parallel Mechanic Oscillators

ndt

dr

dt

dmtF

1)(

2

2

dttvn

tvrdt

dvmtF )(

1)()(

nrm FFFtF )(

19


Section 2.3

Free Oscillations of

Parallel Mechanic

Oscillators20


Free Oscillations of PMO

01

2

2

ndt

dr

dt

dm

0for,0

0for,ˆ)(

t

tFtF

0t

tseTrial: 012 n

srsm

20

22

2

2,1

1

42

mnm

r

m

rs

21

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Quadratic Equation

02 cxbxa

a

c

a

b

a

bx

2

2

2,1 42

22


Free Oscillations of PMO20

22,1 s

mr 2/Damping coefficient:

nm

10 Characteristic angular frequency:

0 : weak damping

0 : strong damping

0 : critical damping

23


Weak Damping22

02,1 j s

2,1s0

: two complex roots

tttjt eeee j

21

For 2/2,121

)cos(2

)(2

2,1

jj

1te

eee t

ttt

(a)

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Strong / Critical Damping20

22,1 s

2,1s0

: two real roots

tt ee )(

2

)(

1

20

220

2

(b) (strong)

2,1s0 : single real root

te )(21

(c) (critical)

25


Decays of Simple Oscillator( b ) creeping case (aperiodic case)

( a ) oscillating case

( c ) aperiodic limiting case

Fig.2.3

0

0

0

26


Quality / Decay Time

r

mQ 00

2

Q oscillations reduce

to 4% of start value

/9.6T decays 60 dB (reverberation time)

Displacement / velocity decay 1/1000

Power decays 1/100000027

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Applications

28

Poorly damped

Welldamped


Applications

29

Tuning oscillations


Section 2.4

Forced Oscillation of

Parallel Mechanic

Oscillators30

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Forced Oscillation

ndt

dr

dt

dmtF

1)cos(ˆ

2

2

In complex form:

n

rmF1

j2 For velocity:

nrm

v

F

j

1j mechZ

Mechanic admittance:(mobility) mechmech /1 ZY

31


Forced Oscillation

Mechanic impedance:n

rmv

F

j

1j mechZ

Mechanic admittance:(mobility)

mechmech /1 ZY

mechY1

j

1j

nrm

Characteristic angular frequency: nm/10 32


Complex PlaneImpedance Admittance (mobility)

33

4545 Bandwidth

0

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Frequency ResponseDisplacement

12 j

1

r

nm

F

22

21

1

rmn

F

34


Frequency Response

0Q

Velocity1

j

1j

nrm

F

v

22

1

1

rn

mF

v

35

4545 Bandwidth


Complex PlaneImpedance

0

36

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Complex PlaneAdmittance (mobility)

37


Frequency Response (Velocity)

0Q

2

0Q

238

Magnitude

Phase


Section 2.5

Energies and

Dissipation Losses

39

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Energy / Dissipation Losses

dttvn

tvrdt

dvmtF )(

1)()( )(tv )(tv

2)(tv

Instantaneous power: d

dttv )(

1

1

0

,0 )(t d

t vdttFW

111

00

2

0

1t dtt

dtvn

dtvrdtdt

dvvm

1

0

)(

dtF

d

dttv )(

40


Energy / Dissipation LossesEnergy (work) :

21

0

221

0

1

2

1

2

1)(

11

ndtvrvmdtF

t

Kinetic energy Potential energy

Friction loss

Lossless: )(2

1)(

2

1)( 22 t

ntvmtW

41


Lossless Energies

)(2

1)(

2

1)( 22 t

ntvmtW

)(2

1)(

2

1 2

0

2

0t

nWtvmW

v

00v

At instant all energy is kinetic

At instant all energy is potential42

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Frictional Losses0r

1

0

2 )(t

r dttvrW

averaging over one T

2

0

22 ˆ2

1)(cosˆ

1vrdttv

TrP

T

r

Power has to be supplied:

1

0

2 )(t

r dttvdt

drP

rmsrms2/ˆˆ vFvF 43

For single‐tones:


Section 2.6

Basic Elements of

Linear, Oscillating

Acoustics Systems

44


Acoustic Elements: MassVolume velocity

)(tvAdt

dA

dt

dVq

21 ppp

Pressure difference

dt

qdmtp a

)(

aa mq

pZ j

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Acoustic Elements

dtqn

tpa

1)(

an

qp

j

qrp a aa r

q

pZ

aa nq

pZ

j1

46


Section 2.7

Helmholtz

Resonator

1821 - 189447


Helmholtz Resonator

aaaa n

rmq

pZ

j

1j

aaa nrmppp

p

48

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Helmholtz Resonator

49

lV

Ac

lAma 0

220 Ac

Vna

aa nm/10 3

0

0

2

c

Vr

mQ

a

aa A

l

: Cross-section area

: Neck length

V : Cavity volume

A

l


Assignment #2 Problem 2.1 -2.7

Problem 2.2 -- review Sec 2.3

Problem 2.3 -- review Sec 2.4

Problem 2.5 - review Appendix 15.2

Problem 2.6 - electrical circuit in series

Due on Wed. Sept. 16th 2020

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