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Engineering Acoustics (Mechanic System)
Transcript of Engineering Acoustics (Mechanic System)
9/10/2021
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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
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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
its attributes that are repeated
regularly with time.
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Oscillation: Energy SwingEnergy swinging between:
Kinetic Potential Energy
Electric Magnetic Energy5
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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
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ttA jee ttA t sinjcose
ttA je
Program in Architectural Acoustics
Basic QuantitiesQuantity General Sinusoidal
Velocity
Acceleration
Displacementt jeˆ)(t
)(tv tvv jeˆ
)(ta taa jeˆ8
Program in Architectural Acoustics
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
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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
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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
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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 )(
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Program in Architectural Acoustics
Section 2.3
Free Oscillations of
Parallel Mechanic
Oscillators20
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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
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Quadratic Equation
02 cxbxa
a
c
a
b
a
bx
2
2
2,1 42
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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
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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
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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
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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)
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4545 Bandwidth
0
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Frequency ResponseDisplacement
12 j
1
r
nm
F
22
21
1
rmn
F
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Frequency Response
0Q
Velocity1
j
1j
nrm
F
v
22
1
1
rn
mF
v
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4545 Bandwidth
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Complex PlaneImpedance
0
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Complex PlaneAdmittance (mobility)
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Frequency Response (Velocity)
0Q
2
0Q
2
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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)(
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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
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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:
Program in Architectural Acoustics
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
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Acoustic Elements
tqn
tp d1
)(a aj n
qp
qrp a arq
pZ a
aa j
1
nq
pZ
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Section 2.7Helmholtz Resonator
1821 - 189447
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Helmholtz Resonator
aaaa j
1j
nrm
q
pZ
aaa nrmppp
p
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Helmholtz Resonator
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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
Program in Architectural Acoustics
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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