engineering acoustics (mechanic system)

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Engineering Acoustics(Mechanic System)

PROF. NING XIANG

GRADUATE PROGRAM IN ARCHITECTURAL ACOUSTICS, SOA

RENSSELAER POLYTECHNIC INSTITUTE, TROY, NEW YORK

Greene 204, Sept. 8th 2021

Program in Architectural Acoustics2

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

Course Materials

Click on here Teaching

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

its 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 )()( txty kk

k

kkk

kk txbtyb )()(Then:

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Eigen-Function of LTI-Systems

: complex frequency

tsAe

js

;eˆ jAA : angular frequency

7

ttA jee ttA t sinjcose

ttA je


Basic QuantitiesQuantity General Sinusoidal

Velocity

Acceleration

Displacementt jeˆ)(t

)(tv tvv jeˆ

)(ta taa jeˆ8


Basic Quantities

t

ttv

d

)(d)(

ttvt d)()(

t

tvta

d

)(d)( ttatv d)()(

2

2

d

)(d)(

t

tta

tttat dd)()(

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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

t

ttv

d

)(d)(

ttvt d)()(

t

tvta

d

)(d)( ttatv d)()(

2

2

d

)(d)(

t

tta

tttat dd)()(

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)

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Basic Quantity: MassamF Newton’s law

)()( tamtF

In sinusoidal cases

mvmamF 2j

(one-port)Mechanic impedance

mZ jmech imaginary

2

2

d

d

d

d

tm

t

vm

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

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Basic Quantity: Spring

kFHooke’s law

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

)(1

tn

F

ttan

tvn

dd1

d1

]m/N[

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Basic Quantity: SpringIn sinusoidal cases:

a

nv

nnF

2

1

j

11

Mechanic impedance:

nZ

j1

mech imaginary

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Fluid Damper (Dashpot)

ar

rvrF

j

jIn sinusoidal cases:

tart

rtvrtF dd

d)()(

Mechanic impedance:

]Ns/m[mech rZ real

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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

nt

rt

mtF1

d

d

d

d)(

2

2

ttvn

tvrt

vmtF d)(

1)(

d

d)(

nrm FFFtF )(

19


Section 2.3

Free Oscillations of

Parallel Mechanic

Oscillators20


Free Oscillations of PMO

01

d

d

d

d2

2

nt

rt

m

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 j

21eeee

For 2/2,121

)cos(e2

)ee(e2

2,1

jj

1tt

ttt

(a)

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Strong / Critical Damping20

22,1 s

2,1s0

: two real roots

tt )(

2

)(

1

20

220

2

ee

(b) (strong)

2,1s0 : single real root

t e)(21

(c) (critical)

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

( a ) oscillating case

( c ) aperiodic limiting case

Fig.2.3

0

0

0

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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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Section 2.4

Forced Oscillation of

Parallel Mechanic

Oscillators30


Forced Oscillation

nt

rt

mtF1

d

d

d

d)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

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Complex PlaneImpedance Admittance (mobility)

33

4545 Bandwidth

0


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

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Complex PlaneImpedance

0

36


Complex PlaneAdmittance (mobility)

37


Frequency Response (Velocity)

0Q

2

0Q

2

38

Magnitude

Phase

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Section 2.5

Energies and

Dissipation Losses

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Energy / Dissipation Losses

ttvn

tvrt

vmtF d)(

1)(

d

d)( )(tv )(tv

2)(tv

Instantaneous power: d

ttv d)(

1

1

0

,0 d)(t d

t tvtFW

111

00

2

0

d1

ddd

dt dtt

tvn

tvrtt

vvm

1

0

d)(

tF

d

ttv d)(

40


Energy / Dissipation LossesEnergy (work) :

21

0

221

0

1

2

1d

2

1d)(

11

ntvrvmtF

t

Kinetic energy Potential energy

Friction loss

Lossless: )(2

1)(

2

1)( 22 t

ntvmtW

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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


Frictional Losses0r

1

0

2 d)(t

r ttvrW

averaging over one T

2

0

22 ˆ2

1d)(cosˆ

1vrttv

TrP

T

r

Power has to be supplied:

1

0

2 d)(t

r ttvdt

drP

rmsrms2/ˆˆ vFvF 43

For single‐tones:


Section 2.6

Basic Elements of

Linear, Oscillating

Acoustics Systems

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Acoustic Elements: MassVolume velocity

)(d

d

d

dtvA

tA

t

Vq

21 ppp

Pressure difference

t

qmtp

d

d)( a

aa j mq

pZ

45


Acoustic Elements

tqn

tp d1

)(a aj n

qp

qrp a arq

pZ a

aa j

1

nq

pZ

46


Section 2.7Helmholtz Resonator

1821 - 189447

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Helmholtz Resonator

aaaa j

1j

nrm

q

pZ

aaa nrmppp

p

48


Helmholtz Resonator

49

lV

Ac

lAm 0a

220

a c A

Vn

aa0 /1 nm

3

0a

0a c

2

Vr

mQ a A

l

: Cross-section area

: Neck length

V : Cavity volume

A

l


Assignment #2 Problem 2.1 -2.7

Problem 2.3 -- review Sec 2.3

Problem 2.4 -- review Sec 2.4

Problem 2.5 - review Appendix 16.2

Problem 2.6 - electrical circuit in series

Due on Wed. Sept. 15th 2021

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engineering acoustics (mechanic system)

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