Reflection, Transmission and Standing Waves

Chapter 5 – Reflection, Transmission and Standing Waves

Slides to accompany lectures in Vibro-Acoustic Design in Mechanical Systems © 2012 by D. W. Herrin Department of Mechanical Engineering University of Kentucky Lexington, KY 40506-0503 Tel: 859-218-0609 [email protected] Normal Incidence Against a Rigid Barrier

ux = 0

x j(ωt−kx) j(ωt+kx) ux = uie + ure • The amplitude of the reflected wave at x = 0 is equal to that of the incident wave. jωt j(ωt) uie + ure = 0 • The intensity of both waves is equal. jπ ur = −ui = uie

Dept. of Mech. Engineering 2 University of Kentucky ME 510 Vibro-Acoustic Design Normal Incidence Against a Rigid Barrier

ux = 0

j(ωt−kx) j(ωt+kx) ux = uie + ure x j(ωt−kx) j(ωt+kx) First Node (where p(x,t) is zero) ux = uie − uie π jωt − jkx jkx jwt kx0 = − ux = uie (e − e ) = −2 juie sin(kx) 2 2π f 2π u x,t 2u sin kx sin t kx0 = x0 = x0 ( ) = i ( ) (ω ) c λ p x,t 2u ccos kx cos t λ ( ) = iρ0 ( ) (ω ) x0 = − 4

Dept. of Mech. Engineering 3 University of Kentucky ME 510 Vibro-Acoustic Design Standing Waves

Dept. of Mech. Engineering 4 University of Kentucky ME 510 Vibro-Acoustic Design Two Elastic Media

Medium 1 Medium 2

Z1 = ρ1c1 Z2 = ρ2c2 j t k x j(ωt−k1x) (ω − 2 ) pi (x,t) = pie pt (x,t) = pte

j t k x pi j(ωt−k1x) pt (ω − 2 ) ui (x,t) = e ut (x,t) = e ρ1c1 ρ2c2

j(ωt+k1x) pr (x,t) = pre x

pr j(ωt+k1x) ur (x,t) = − e ρ1c1

Dept. of Mech. Engineering 5 University of Kentucky ME 510 Vibro-Acoustic Design Boundary Conditions at x = 0

pi (x = 0,t) + pr (x = 0,t) = pt (x = 0,t)

ui (x = 0,t) + ur (x = 0,t) = ut (x = 0,t)

pi + pr = pt p p p i − r = t ρ1c1 ρ1c1 ρ2c2

pi − pr pi + pr Then = ρ1c1 ρ2c2

Dept. of Mech. Engineering 6 University of Kentucky ME 510 Vibro-Acoustic Design The Reflection Coefficient

p − p p + p i r = i r ρ1c1 ρ2c2 1− R 1+ R p = where R = r ρ1c1 ρ2c2 pi ρ c 1− 1 1 ρ c − ρ c ρ c R = 2 2 1 1 = 2 2 c c ρ1c1 ρ2 2 + ρ1 1 1+ ρ2c2 R is always real for real specific impedances.

Dept. of Mech. Engineering 7 University of Kentucky ME 510 Vibro-Acoustic Design The Transmission Coefficient

p T = t pi

2ρ c 2 T = 2 2 = c c ρ1c1 ρ2 2 + ρ1 1 1+ ρ2c2

T is always real for real specific impedances.

Dept. of Mech. Engineering 8 University of Kentucky ME 510 Vibro-Acoustic Design Three Special Cases

ρ c 1− 1 1 2 ρ2c2 T = R = ρ1c1 ρ1c1 1+ 1+ ρ2c2 ρ2c2

For ρ1c1 < ρ2c2 For ρ1c1 = ρ2c2 For ρ1c1 > ρ2c2 0 < R <1 R = 0 −1< R < 0

pi and pr are T =1 pi and pr are in phase out of phase

Dept. of Mech. Engineering 9 University of Kentucky ME 510 Vibro-Acoustic Design Complex Specific Impedance

Medium 2 typically has losses or is limited in its extent. p p Z =   = u ⋅ n un

Medium 1 Medium 2

Wi Wt

Dept. of Mech. Engineering 10 University of Kentucky ME 510 Vibro-Acoustic Design Complex Specific Impedance

Z − ρ c R = 2 1 1  Z2 + ρ1c1

Medium 1 Medium 2

Wi Wt

Dept. of Mech. Engineering 11 University of Kentucky ME 510 Vibro-Acoustic Design The Absorption Coefficient W W −W W α = t = i r =1− r Wi Wi Wi 2 Ix, r pˆr 2 α =1+ =1− 2 =1− R Ix, i pˆi

Medium 1 Medium 2

Wi Wt

Dept. of Mech. Engineering 12 University of Kentucky ME 510 Vibro-Acoustic Design Propagation of Plane Waves in 3D Space

ω Wave Fronts  k = n c      k = k ⋅ n = k e + k e + k e x x y y z z λ

 j ωt−kx x−kyy−kzz p(r,t) = pe ( )

2 2 2 k = kx + ky + kz

Dept. of Mech. Engineering 13 University of Kentucky ME 510 Vibro-Acoustic Design Oblique Incidence between Two Fluid Media

Dept. of Mech. Engineering 14 University of Kentucky ME 510 Vibro-Acoustic Design Blackstock, 2000

Oblique Incidence between Two Fluid Media

c1Δt = Δysinθi

c2Δt = Δysinθt c c 1 = 2 sinθi sinθt

The incident, reflected, and transmitted waves have the same periodicity. λ λ λ i = r = t sinθi sinθr sinθt

Dept. of Mech. Engineering 15 University of Kentucky ME 510 Vibro-Acoustic Design Case 1

Case 1 For c1 > c2

c2 sinθt = sinθi c1

θt is redirected to the normal

if θi is 90°

max c2 θt = arcsin c1

Dept. of Mech. Engineering 16 University of Kentucky ME 510 Vibro-Acoustic Design Case 1

Dept. of Mech. Engineering 17 University of Kentucky ME 510 Vibro-Acoustic Design Case 2

Case 2 For c1 < c2

c1 sinθi = sinθt c2

θt > θi

if θt is 90°

cutoff c1 θi = arcsin c2

Dept. of Mech. Engineering 18 University of Kentucky ME 510 Vibro-Acoustic Design Case 2

Dept. of Mech. Engineering 19 University of Kentucky ME 510 Vibro-Acoustic Design Oblique Incidence between Two Fluid Media

The Reflection Coefficient p ρ c cosθ − ρ c cosθ R = r = 2 2 i 1 1 t pi ρ2c2 cosθi + ρ1c1 cosθt

The Transmission Coefficient

p 2ρ c cosθ T = t = 2 2 i pi ρ2c2 cosθi + ρ1c1 cosθt

Dept. of Mech. Engineering 20 University of Kentucky ME 510 Vibro-Acoustic Design Locally Reacting Surface

Every point on the surface is considered to be completely isolated from all other points. p Z = u⊥

Dept. of Mech. Engineering 21 University of Kentucky ME 510 Vibro-Acoustic Design Locally Reacting Surface

Continuity of Sound Pressure

jδr jδt pˆi + pˆre = pˆre

Continuity of Particle Velocity

pˆ pˆ pˆ i − r e jδr = t e jδt  ρ1c1 ρ1c1 Z2

jδt Eliminate pˆte

pˆ j Z cosθ − ρ c R = r e δr = 2 i 1 1  pˆi Z2 cosθi + ρ1c1

Dept. of Mech. Engineering 22 University of Kentucky ME 510 Vibro-Acoustic Design Locally Reacting Surface

The locally reacting assumptions is applicable for: 1. Anisotropic medium (see below) 2. Medium with significant losses like fibers or foams

3. Medium with c2 << c1

Dept. of Mech. Engineering 23 University of Kentucky ME 510 Vibro-Acoustic Design Absorbing Materials

• The Basics • Impedance and Absorption • Measuring Impedance and Absorption • Transfer Matrix Approach • Flow Resistivity

Dept. of Mech. Engineering 24 University of Kentucky ME 510 Vibro-Acoustic Design Sound Blocking Versus Sound Absorption

Relatively massive, stiff barrier having high damping

Some sound is Most sound is transmitted reflected

Less sound is Less sound is transmitted reflected

( These two objectives can be combined)

Dept. of Mech. Engineering 25 University of Kentucky ME 510 Vibro-Acoustic Design Example

Noisy family or playroom Bedroom

Ways to reduce noise level in the bedroom:

Which use sound blocking and which use sound absorption?

Dept. of Mech. Engineering 26 University of Kentucky ME 510 Vibro-Acoustic Design Sound Absorbing Materials

Dept. of Mech. Engineering 27 University of Kentucky ME 510 Vibro-Acoustic Design NVH Applications Driven by Auto Industry

Dept. of Mech. Engineering 28 University of Kentucky ME 510 Vibro-Acoustic Design Sound Absorbing Materials in Car

Dept. of Mech. Engineering 29 University of Kentucky ME 510 Vibro-Acoustic Design Examples of Sound Absorbing Materials

a: fully reticulated plastic foam (x14) b: partially reticulated plastic foam (x14) c: glass fiber (x14) d: mineral (rock) wool (x14)

Dept. of Mech. Engineering 30 University of Kentucky ME 510 Vibro-Acoustic Design Foam Manufacture and Foam Types

• Foams are made of various materials including polyurethane, polyethylene, and polypropylene. • Foams are created by pouring premixed liquid products onto a conveyor and allowing the chemical process to create cells (voids) of various size as the foam cures and hardens. Similar to bread rising. • If the cell walls are fractured, the foam is called “open cell” • If the cell walls remain intact, the foam is called “closed cell” • Coverings such vinyl, aluminum, urethane, or aluminized mylar protect the surface, improve appearance, and reduce absorption of liquids, dirt, etc. • Coverings may also act as a barrier, e.g., loaded vinyl

Dept. of Mech. Engineering 31 University of Kentucky ME 510 Vibro-Acoustic Design Applications of Sound Absorbing Materials

• Suspended baffles in gymnasiums or factories • Under hood applications for engine noise • Vehicle interiors • Inside building walls – improves transmission loss • Inside office and computer equipment – reduces reverberant buildup of sound • Ceiling tiles and carpeting • HVAC applications – duct liner

Dept. of Mech. Engineering 32 University of Kentucky ME 510 Vibro-Acoustic Design Mechanisms of Sound Absorption

Sound is “absorbed” by converting sound energy to heat within the material, resulting in a reduction of the sound pressure.

Two primary mechanisms:

• vibration of the material skeleton - damping • friction of the fluid on the skeleton - viscosity

Dept. of Mech. Engineering 33 University of Kentucky ME 510 Vibro-Acoustic Design Vibration of Material Skeleton

Vibration of the material matrix is caused by sound pressure and velocity fluctuations within the material.

Damping of the material converts sound to heat.

Important for light materials at low frequencies.

Difficult to model and measure (ignored here)

Dept. of Mech. Engineering 34 University of Kentucky ME 510 Vibro-Acoustic Design Friction of the Fluid on the Skeleton

The oscillating fluid particles within the material rub against the matrix and create heat by friction (viscosity).

Primary material parameters affecting absorption:

• porosity (fraction of air volume in the material) • structure factor (orientation of fibers, tortuosity) • flow resistance

Dept. of Mech. Engineering 35 University of Kentucky ME 510 Vibro-Acoustic Design Absorbing Materials

• The Basics • Impedance and Absorption • Measuring Impedance and Absorption • Transfer Matrix Approach • Flow Resistivity

Dept. of Mech. Engineering 36 University of Kentucky ME 510 Vibro-Acoustic Design The Local Reaction Model

• The wave component within the material parallel to the surface is attenuated rapidly

• The material is similar to a set of rigid-wall, pr parallel capillaries φ

un • The particle velocity un in the material is normal to the surface and only a function of φ ps

the local sound pressure ps at the surface (un is independent of the form of the incident pi wave)

ps z = (independent of pi) un

Dept. of Mech. Engineering 37 University of Kentucky ME 510 Vibro-Acoustic Design Specific Boundary Impedance

p z = = r + jx u n surface resistance reactance

Pa −2 −1 Units: = kg m s ≡ rayl (named in honor of Lord Rayleigh) ms−1

Absorption coefficient: Sound Energy Absorbed α(φ) = Sound Energy Incident

Dept. of Mech. Engineering 38 University of Kentucky ME 510 Vibro-Acoustic Design Specific Boundary Impedance

x = 0 p A+ B 1+ (B A) p, un z = = = ρ c n u o 1 B A n x=0 ⎛ A− B ⎞ − ( ) ⎜ ⎟ ⎝ ρoc ⎠

− jkx u Ae zn 1+ R n = Be+ jkx ρoc 1− R B R = A R is called the pressure reflection coefficient

Dept. of Mech. Engineering 39 University of Kentucky ME 510 Vibro-Acoustic Design Absorption Coefficient and Impedance

While the specific boundary impedance is independent of angle of incidence, the absorption coefficient is not:

4rnʹ cosφ rn xn α(φ)= 2 2 where rnʹ = xnʹ = (1+ rnʹ cosφ) + (xnʹ cosφ) ρoc ρoc # & 1 An angle of maximum absorption exists: ϕ = cos−1 % ( max % 2 2 ( $ (rn") +(x"n ) ' The maximum absorption here is: 2rnʹ αmax = (znʹ + rnʹ)

α(0°)→1 if rnʹ →1 and xnʹ → 0

Dept. of Mech. Engineering 40 University of Kentucky ME 510 Vibro-Acoustic Design Example Impedance of Foam

8 6 Resistance r' Reactance x' 4 2 0 Impedance -2 0 1000 2000 3000 4000 -4 Dimensionless Boundary Dimensionless -6 Frequency (Hz)

Dept. of Mech. Engineering 41 University of Kentucky ME 510 Vibro-Acoustic Design Example Sound Absorption Coefficient

0.8

0.6

0.4 phi = 0 degrees phi = 30 degrees

Absorption Coefficient 0.2 phi = 60 degrees

0 0 1000 2000 3000 4000 Frequency (Hz)

Dept. of Mech. Engineering 42 University of Kentucky ME 510 Vibro-Acoustic Design Common Glass Fiber

0.8

0.6

0.4 Glass Fiber 0.2 Absorption Coefficient 0 0 1000 2000 3000 4000 Frequency (Hz)

Dept. of Mech. Engineering 43 University of Kentucky ME 510 Vibro-Acoustic Design Closed Cell vs. Open Cell Foam

1 Closed Cell Foam 0.8 Open Cell Foam

0.6

0.4 Absorption CoefficientAbsorption 0.2

0 0 1000 2000 3000 4000 5000 Frequency (Hz)

Dept. of Mech. Engineering 44 University of Kentucky ME 510 Vibro-Acoustic Design Foam Absorbers must be Open Cell

Closed-cell Open-cell

Dept. of Mech. Engineering 45 University of Kentucky ME 510 Vibro-Acoustic Design Effect of Thickness

0.8

0.6 glass fiber 0.4 Absorption Coefficient Absorption 0.2 2 in. thick 1 in. thick 0 0 1000 2000 3000 4000 5000 Frequency (Hz)

Dept. of Mech. Engineering 46 University of Kentucky ME 510 Vibro-Acoustic Design Effect of Covering an Absorber

0.8

0.6

0.4 Absorption CoefficientAbsorption cover 0.2 Foam without Cover Foam with Cover 0 0 500 1000 1500 2000 Frequency (Hz)

Dept. of Mech. Engineering 47 University of Kentucky ME 510 Vibro-Acoustic Design Layering of Materials

0.8

2” 0.6 2” 1.4 ” 0.4 Absorption Coefficient Absorption 0.2 air gap

0 0 500 1000 1500 2000

Frequency (Hz)

Dept. of Mech. Engineering 48 University of Kentucky ME 510 Vibro-Acoustic Design Lining of Partial Enclosures

100

Sound Power Level (dB) No Absorption (108.2 dBA) 50 With Absorption (97.6 dBA) 40 0 1000 2000 3000 4000 5000 Frequency (Hz)

vibrating surface

Dept. of Mech. Engineering 49 University of Kentucky ME 510 Vibro-Acoustic Design Full Enclosure

100

95 Top and Bottom Front and Rear 90 Sides Empty 85

75 SPL (dB) SPL Plexiglass loudspeaker 70 enclosure 65

50 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz)

Placement of material is often flexible

Dept. of Mech. Engineering 50 University of Kentucky ME 510 Vibro-Acoustic Design Absorbing Materials

• The Basics • Impedance and Absorption • Measuring Impedance and Absorption • Transfer Matrix Approach • Flow Resistivity

Dept. of Mech. Engineering 51 University of Kentucky ME 510 Vibro-Acoustic Design Measurement of Sound Impedance

ASTM E1050-95 Test Method

microphones Transfer function

driver (loudspeaker) sample

Dept. of Mech. Engineering 52 University of Kentucky ME 510 Vibro-Acoustic Design Coordinate System and Microphone Locations

Material sample Sound source

Dept. of Mech. Engineering 53 University of Kentucky ME 510 Vibro-Acoustic Design Plane Wave Theory

Total sound pressure at any point in the tube: P(x)= Ae− jkx + Be jkx

+x traveling wave -x traveling wave

The transfer function between points 1 and 2:

− jkx2 jkx2 − jkx2 jkx2 P(x2 ) Ae + Be e + Re H = = = 12 − jkx1 jkx1 − jkx1 jkx1 P(x1 ) Ae + Be e + Re B R = is the pressure reflection coefficient of the material A

Dept. of Mech. Engineering 54 University of Kentucky ME 510 Vibro-Acoustic Design Solving for Material Properties

e− jkx2 − H e− jkx1 Solving for R: R = 12 jkx1 jkx2 H12e − e

Sound absorption coefficient (normal incidence) of the material:

Sound Energy Absorbed 2 α(φ) = =1− R Sound Energy Incident Normalized specific boundary impedance: z 1+ R n = ρoc 1− R

Dept. of Mech. Engineering 55 University of Kentucky ME 510 Vibro-Acoustic Design Research Foundation

1. Seybert, A.F. and Ross, D.F., Experimental Determination of Acoustic Properties Using a Two-microphone Random Excitation Technique, J. Acoust. Soc. Am., 61, 1362-1370 (1977) 2. Chung, J.Y. and Blaser, D.A., Transfer Function Method of Measuring In-duct Acoustic Properties, II: Experiment, J. Acoust. Soc. Am, 68, 914-921 (1980) 3. Seybert, A.F., Two-sensor Methods for the Measurement of Sound Intensity and Acoustic Properties in Ducts, J. Acoust. Soc. Am, 83, 2233-2239 (1988) 4. Tao, Z. and Seybert, A. F., A Review of Current Techniques for Measuring Muffler Transmission Loss, Proceedings, SAE Noise and Vibration Conference, Traverse City, paper 2003-01-1653 (2003)

Dept. of Mech. Engineering 56 University of Kentucky ME 510 Vibro-Acoustic Design Two-Microphone Standards

1. ISO 10534-2, Acoustics-Determination of sound absorption coefficient and impedance in impedance tubes - Part 2: Transfer-function method

2. ASTM E1050-98, Standard Test Method for Impedance and Absorption of Acoustical Material Using a Tube, Two Microphones and a Digital Frequency Analysis System

Dept. of Mech. Engineering 57 University of Kentucky ME 510 Vibro-Acoustic Design Sound Absorption Measurement

Sample holder with rigid piston

Dept. of Mech. Engineering 58 University of Kentucky ME 510 Vibro-Acoustic Design Absorbing Materials

• The Basics • Impedance and Absorption • Measuring Impedance and Absorption • Transfer Matrix Approach • Flow Resistivity

Dept. of Mech. Engineering 59 University of Kentucky ME 510 Vibro-Acoustic Design Sound Propagation – Porous Material

Plane waves in a porous material: • amplitude decreases with distance • p and u are not in phase

• characteristic impedance z’c is a complex number • wave number k’ is a complex number − jkʹx p(x) = Poe k'= β − jγ x p(x) = zʹ u(x) c Attenuation constant (responsible for wave attenuation) complex

Dept. of Mech. Engineering 60 University of Kentucky ME 510 Vibro-Acoustic Design The Basic Idea

The sound pressure p and the particle velocity v are the acoustic state variables

1 For any passive, linear component:

p1 = Ap2 + BS2u2

any acoustic 2 S1u1 = Cp2 + DS2u2 component or p1, u1 ! % ! % # p1 # ( A B +# p2 # " & = * -" & S u C D S u $# 1 1 '# ) ,$# 2 2 '# p2, u2

Transfer, transmission, or four-pole matrix (A, B, C, and D depend on the component)

Dept. of Mech. Engineering 61 University of Kentucky ME 510 Vibro-Acoustic Design The Straight Tube

jkx jkx −1 dp L p(x) = Ae− + Be+ u(x) = jkρ c dx A o p(0) = p1 = A + B B S A − B u 0 = u = p , u p ,u ( ) 1 1 1 2 2 ρoc (x = L) − jkL + jkL (x = 0) p(L) = p2 = Ae + Be − jkL + jkL must have plane waves Ae − Be u(L) = u2 = ρoc

Solve for A, B p1 = p2 cos(kL) + u2 ( jρoc)sin(kL)

in terms of p1, u1 u1 = p2 ( j ρoc)sin(kL) + u2 cos(kL) then put into ) , equations for p , u . jρoc 2 2 " & + cos(kL) sin(kL) ." & $ p1 $ + S2 .$ p2 $ # ' = # ' S u + jS S . S u %$ 1 1 ($ + 1 sin(kL) 1 cos(kL) .%$ 2 2 ($ c S *+ ρo 2 -. (note that the determinant A1D1-B1C1 = 1)

Dept. of Mech. Engineering 62 University of Kentucky ME 510 Vibro-Acoustic Design Straight Tube with Absorptive Material

k’,zc (complex wave number and complex characteristic impedance)

( + jzc ! % * cos(k'L) sin(k'L) -! % # p1 # * S2 -# p2 # " & = " & S u * jS S - S u $# 1 1 '# * 1 sin(k'L) 1 cos(k'L) -$# 2 2 '# z S )* c 2 ,-

Dept. of Mech. Engineering 63 University of Kentucky ME 510 Vibro-Acoustic Design Transfer Matrix Approach

For each layer: ! p $ ! A B $! p $ # i & = # i i & # i+1 & # Su & # C D &# Su & air air " i % " i i %" i+1 % Overall:

⎡AT BT ⎤ [Ttotal ]= [T1][T2 ][T3 ]...[Tn ]= ⎢ ⎥ ⎣CT DT ⎦

p SA Layer 1 Layer 2 Perforate Layer n Z = 1 = T u1 CT

Plane Wave Assumption

Dept. of Mech. Engineering 64 University of Kentucky ME 510 Vibro-Acoustic Design Example - Layered Materials

0.8

0.6 Units: mm 51 51 36 0.4

Absorption Coefficient Absorption 0.2 Predicted Measured 0 0 500 1000 1500 Frequency (Hz)

Dept. of Mech. Engineering 65 University of Kentucky ME 510 Vibro-Acoustic Design Absorbing Materials

• The Basics • Impedance and Absorption • Measuring Impedance and Absorption • Transfer Matrix Approach • Flow Resistivity

Dept. of Mech. Engineering 66 University of Kentucky ME 510 Vibro-Acoustic Design Designing the Absorber from Scratch

Bulk Surface Layered

k’ and z’c z and α z and α

????

Dept. of Mech. Engineering 67 University of Kentucky ME 510 Vibro-Acoustic Design Absorption Coefficient vs. Frequency

αo Different densities and thicknesses

Frequency (Hz) (Mechel, 1988)

Dept. of Mech. Engineering 68 University of Kentucky ME 510 Vibro-Acoustic Design Flow Resistivity and Absorption

αo

ρof/σ

Dept. of Mech. Engineering 69 University of Kentucky ME 510 Vibro-Acoustic Design Flow Resistance and Flow Resistivity σ

Flow resistance: Sample ΔP (thickness t ) ΔP r = s u

u (velocity) Flow resistivity:

rs Vacuum source σ = t

Dept. of Mech. Engineering 70 University of Kentucky ME 510 Vibro-Acoustic Design ASTM C522-03 Flow Resistance Measurement

Manometer Pipe Vacuum pump Flow Meter

Specimen Valve Valve Pump Manometers Flow Meters

Specimen Holder

Specimen

Dept. of Mech. Engineering 71 University of Kentucky ME 510 Vibro-Acoustic Design Absorption Coefficient

φ sound energy absorbed α (φ ) = sound energy incident φ

Dept. of Mech. Engineering 72 University of Kentucky ME 510 Vibro-Acoustic Design Absorption Coefficient and Flow Resistivity

σt ≈ 2 ρc

AbsorptionCoefficient

Flow Resistivity

fluid solid

Dept. of Mech. Engineering 73 University of Kentucky ME 510 Vibro-Acoustic Design Three Empirical Models

X = ρ f σ o Wu, 1988 – 17 plastic foam materials; −0.554 −0.592 kʹ k = (1+ 0.188X ) − j0.163X f = 200-2000 Hz; 2900 ≤ σ ≤ 24300 −0.548 −0.607 rayls; 0.01 < X < 0.83 zcʹ zo = (1+ 0.209X ) − j0.105X

kʹ k = (1+ 0.0978X −0.700 ) − j0.189X −0.595 Delaney and Bazly, 1970 – fibrous materials; f = 250-4000 Hz; σ = ?; −0.754 −0.732 zcʹ zo = (1+ 0.0571X ) − j0.087X 0.012 < X < 1.21 for X < 0.025: kʹ k = (1+ 0.136X −0.641) − j0.322X −0.502 −0.699 −0.556 zcʹ zo = (1+ 0.081X ) − j0.191X Mechel (after Fahy) – fibrous materials; for X > 0.025: f = ?; σ = ?; 0.002 < X < 0.5 kʹ k = (1+ 0.103X −0.716 ) − j0.179X −0.663 −0.725 −0.655 zcʹ zo = (1+ 0.0563X ) − j0.127X

Dept. of Mech. Engineering 74 University of Kentucky ME 510 Vibro-Acoustic Design VAC Toolbox Approach

Simón et al. A Fitting Method to Estimate the Air Flow Resistivity of Porous Materials (ICSV 13, 2006)

Measure α

Calculate α Compare in a Assume Using Selected σ Least Squares Sense Empirical Model

Minimize the error to find optimal σ

Dept. of Mech. Engineering 75 University of Kentucky ME 510 Vibro-Acoustic Design ESI FOAM-X Software

AutoSEA2

ASTM E1050 Test System FOAM-X

RAYON

SYSNOISE VIOLINS

Other acoustic Programs

Dept. of Mech. Engineering 76 University of Kentucky ME 510 Vibro-Acoustic Design ESI FOAM-X Software

Frequency dependent fitting algorithm

Dept. of Mech. Engineering 77 University of Kentucky ME 510 Vibro-Acoustic Design FOAM-X Users Guide Biot Properties Determined by FOAM-X

Flow Resistivity (Rayls/m or N⋅s/m4) The static airflow resistivity (σ) expresses the frictional retardation to a quasi-static airflow through the pores.

Open Porosity The open porosity (φ) is defined as the fraction of volume that is occupied by the air in the interconnected porous network.

Viscous Characteristic Length The viscous characteristic length (Λ) is an average macroscopic dimension of the cells related to viscous losses.

Thermal Characteristic Length The thermal characteristic length (Λ′) is an average macroscopic dimension of the cells related to thermal losses. It may be seen as an average radius of the larger pores of a porous aggregate.

Geometrical Tortuosity The geometrical tortuosity (α∞), or identically the structure factor (ks), is a geometrical measurement of the deviation of the actual path followed by an acoustical wave from a direct path.

Dept. of Mech. Engineering 78 University of Kentucky ME 510 Vibro-Acoustic Design Sound Absorption (Thickness: 1 inch)

1.0

0.8

0.6 Absorption Absorption

0.4

0.2 Measurement Simón et al., R=4930 rayls/m

FOAM-X, R=5178 rayls/m 0.0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Frequency (Hz)

Dept. of Mech. Engineering 79 University of Kentucky ME 510 Vibro-Acoustic Design Surface Impedance (Thickness: 1 inch)

3.0

1.0 Real

-1.0

-3.0

SurfaceImpedance Imaginary -5.0

Measurement -7.0 Simón et al., R=4930 rayls/m

FOAM-X, R=5178 rayls/m -9.0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Frequency (Hz)

Dept. of Mech. Engineering 80 University of Kentucky ME 510 Vibro-Acoustic Design Absorption of Sample (Thickness: 3 inch)

1.0

0.8

0.6 Absorption Absorption 0.4

0.2 Measurement Simón et al., R=4930 rayls/m

FOAM-X, R=5178 rayls/m

0.0 0 500 1000 1500 2000 2500 3000 3500 4000 4500

Frequency (Hz)

Dept. of Mech. Engineering 81 University of Kentucky ME 510 Vibro-Acoustic Design Absorption of Layered Materials

1.0

0.8

0.6 1.125” 1” Absorption Absorption

0.4

0.2 Measurement

Simón, et al.

0.0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Frequency (Hz)

Dept. of Mech. Engineering 82 University of Kentucky ME 510 Vibro-Acoustic Design Eigen Frequencies and Modes in 3D

For a parallelepiped:

 j ωt±kx x±kyy±kzz p(r,t) = pˆe ( )

All possible combinations of propagation directions can occur:

" pˆ e− j(kx x+kyy+kzz) pˆ e j(kx x+kyy+kzz) % $ 1 + 2 +' $ ˆ − j(kx x+kyy−kzz) ˆ j(kx x+kyy−kzz) '  p3e + p4e + ω 2 2 2 p r,t = $ ' k = = kx + ky + kz ( ) − j k x−k y+k z j k x−k y+k z c $ pˆ e ( x y z ) + pˆ e ( x y z ) +' $ 5 6 ' $ − j(kx x−kyy−kzz) j(kx x−kyy−kzz) ' # pˆ7e + pˆ8e &

Dept. of Mech. Engineering 83 University of Kentucky ME 510 Vibro-Acoustic Design Eigen Frequencies and Modes in 3D

ux = 0 at x = 0 and x = lx Boundary Conditions: uy = 0 at y = 0 and y = ly

uz = 0 at z = 0 and z = lz

 ˆ jωt p (r,t) = pcos(kx x)cos(ky y)cos(kzz)e  k ˆ x jωt ux (r,t) = p sin(kx x)cos(ky y)cos(kzz)e Solution: jωρ0  k ˆ y jωt uy (r,t) = p sin(ky x)cos(kx y)cos(kzz)e jωρ0  k ˆ z jωt uz (r,t) = p sin(kz x)cos(kx y)cos(kyz)e jωρ0

Dept. of Mech. Engineering 84 University of Kentucky ME 510 Vibro-Acoustic Design Eigen Frequencies and Modes in 3D

sin(kxlx ) = 0 In order to satisfy boundary conditions: sin(kyly ) = 0

sin(kzlz ) = 0

Thus: nxπ kx = where nx = 0, 1, 2, … lx

nyπ ω 2 2 2 ky = where ny = 0, 1, 2, … k = = kx + ky + kz ly c

nzπ kz = where nz = 0, 1, 2, … lz

Dept. of Mech. Engineering 85 University of Kentucky ME 510 Vibro-Acoustic Design Modal Behavior of Rooms

Natural frequencies of a rigid-wall enclosure or room:

2 2 2 flmn (Hz) = (c 2π ) (nxπ lx ) +(nyπ ly ) +(nzπ lz )

nx ny nz flmn nx ny nz flmn 1 0 0 85.8 0 2 1 285.8 0 1 0 114.3 2 2 0 285.8 Natural 1 1 0 142.9 1 2 1 298.4 frequencies: 0 0 1 171.5 2 2 1 333.3 2 0 0 171.5 0 0 2 343.0 lx = 2 m 1 0 1 191.7 1 0 2 353.6 ly = 1.5 m 0 1 1 206.1 0 1 2 361.6 2 1 0 206.1 1 1 2 371.6 lz = 1 m 1 1 1 223.2 2 0 2 383.5 c = 343 m/s 0 2 0 228.7 2 1 2 400.2 2 0 1 242.5 0 2 2 412.2 1 2 0 244.2 1 2 2 421.1 2 1 1 268.1 2 2 2 446.5

Dept. of Mech. Engineering 86 University of Kentucky ME 510 Vibro-Acoustic Design Modeling Interior Acoustics

Dept. of Mech. Engineering 87 University of Kentucky ME 510 Vibro-Acoustic Design Experimental Setup

Measuring MIC Transfer Function

Measurement Grid

Reference MIC

Or Electrical Signal Identical to Experimental Modal Analysis for Structures

Dept. of Mech. Engineering 88 University of Kentucky ME 510 Vibro-Acoustic Design Measuring Acoustic Modes

Microphones

Loudspeaker

Dept. of Mech. Engineering 89 University of Kentucky ME 510 Vibro-Acoustic Design Solid and FEM Models

Dept. of Mech. Engineering 90 University of Kentucky ME 510 Vibro-Acoustic Design Mode 1

99.5 Hz 107.1 Hz

Dept. of Mech. Engineering 91 University of Kentucky ME 510 Vibro-Acoustic Design Mode 2

123.7 Hz 131.8 Hz

Dept. of Mech. Engineering 92 University of Kentucky ME 510 Vibro-Acoustic Design Mode 4

191.7 Hz 195.6Hz

Dept. of Mech. Engineering 93 University of Kentucky ME 510 Vibro-Acoustic Design Mode 11

262.8 Hz 267.0 Hz

Dept. of Mech. Engineering 94 University of Kentucky ME 510 Vibro-Acoustic Design Mode 18

347.9 Hz 350.6 Hz

Dept. of Mech. Engineering 95 University of Kentucky ME 510 Vibro-Acoustic Design Modal Table

Mode No. FEM (Hz) Experiment (Hz) Error (%) 1 99.5 107.1 -7.1 2 123.7 131.8 -6.1 3 157.7 165.6 -4.8 4 191.7 195.6 -2.0 5 200.1 206.9 -3.3 6 219.9 223.5 -1.6 7 229.9 232.4 -1.1 8 239.1 241.8 -1.1 9 245.6 250.7 -2.0 10 257.2 259.4 -0.8 11 262.8 267.0 -1.6 12 286.6 13 293.3 293.2 0.0 14 304.0 305.6 -0.5 15 313.1 315.1 -0.6 16 319.2 325.6 -2.0 17 333.4 331.9 0.5 18 347.9 350.6 -0.8 19 351.4 351.9 -0.1 20 354.6 356.2 -0.4

Dept. of Mech. Engineering 96 University of Kentucky ME 510 Vibro-Acoustic Design Mode 12

286.6 Hz

Dept. of Mech. Engineering 97 University of Kentucky ME 510 Vibro-Acoustic Design Measuring Interior Noise

Foam Lining

Dept. of Mech. Engineering 98 University of Kentucky ME 510 Vibro-Acoustic Design Model Validation

50 SPL (dB)

40 Measured (97.7 dBA) 30 SYSNOISE (IBEM, 99.7 dBA) SYSNOISE (FEM, 99.6 dBA) 20 0 500 1000 1500 2000 Frequency (Hz)

Dept. of Mech. Engineering 99 University of Kentucky ME 510 Vibro-Acoustic Design