Sound Absorbing Materials

D. W. Herrin, Ph.D., P.E. University of Kentucky Department of Mechanical Engineering Overview Sound Absorption by Porous Materials

D The Basics D Impedance and Absorption D Transfer Matrix Approach D Flow Resistivity

Dept. of Mech. Engineering 2 Noise and Vibration University of Kentucky Short Course Sound Blocking Versus Sound Absorption Sound Absorption by Porous Materials 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 3 Noise and Vibration University of Kentucky Short Course Example Sound Absorption by Porous Materials

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 4 Noise and Vibration University of Kentucky Short Course Sound Absorbing Materials Sound Absorption by Porous Materials

Dept. of Mech. Engineering 5 Noise and Vibration University of Kentucky Short Course NVH Applications Driven by Auto Industry Sound Absorption by Porous Materials

Dept. of Mech. Engineering 6 Noise and Vibration University of Kentucky Short Course Sound Absorbing Materials in Car Sound Absorption by Porous Materials

Dept. of Mech. Engineering 7 Noise and Vibration University of Kentucky Short Course Examples of Sound Absorbing Materials Sound Absorption by Porous 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 8 Noise and Vibration University of Kentucky Short Course Foam Manufacture and Foam Types Sound Absorption by Porous Materials

D Foams are made of various materials including polyurethane, polyethylene, and polypropylene. D 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. D If the cell walls are fractured, the foam is called “open cell” D If the cell walls remain intact, the foam is called “closed cell” D Coverings such vinyl, aluminum, urethane, or aluminized mylar protect the surface, improve appearance, and reduce absorption of liquids, dirt, etc. D Coverings may also act as a barrier, e.g., loaded vinyl

Dept. of Mech. Engineering 9 Noise and Vibration University of Kentucky Short Course Applications of Sound Absorbing Materials Sound Absorption by Porous Materials

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

Dept. of Mech. Engineering 10 Noise and Vibration University of Kentucky Short Course Mechanisms of Sound Absorption Sound Absorption by Porous Materials

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

Two primary mechanisms:

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

Dept. of Mech. Engineering 11 Noise and Vibration University of Kentucky Short Course Vibration of Material Skeleton Sound Absorption by Porous Materials

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 12 Noise and Vibration University of Kentucky Short Course Friction of the Fluid on the Skeleton Sound Absorption by Porous Materials

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

Primary material parameters affecting absorption:

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

Dept. of Mech. Engineering 13 Noise and Vibration University of Kentucky Short Course Overview Sound Absorption by Porous Materials

D The Basics D Impedance and Absorption D Transfer Matrix Approach D Flow Resistivity

Dept. of Mech. Engineering 14 Noise and Vibration University of Kentucky Short Course The Local Reaction Model Sound Absorption by Porous Materials

D The wave component within the material parallel to the surface is attenuated rapidly p D The material is similar to a set of rigid-wall, r parallel capillaries φ

un D The particle velocity un in the material is normal to the surface and only a function of φ ps the local sound pressure p at the surface (u s n p is independent of the form of the incident i wave)

ps z = (independent of pi) un

Dept. of Mech. Engineering 15 Noise and Vibration University of Kentucky Short Course Specific Boundary Impedance Sound Absorption by Porous Materials p z = = r + jx un 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 16 Noise and Vibration University of Kentucky Short Course Specific Boundary Impedance Sound Absorption by Porous Materials x = 0 p A+ B 1+ (B A) p, un zn = = = ρoc un x=0 ⎛ A− B ⎞ 1− (B A) ⎜ ⎟ ⎝ ρoc ⎠

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

Dept. of Mech. Engineering 17 Noise and Vibration University of Kentucky Short Course Absorption Coefficient and Impedance Sound Absorption by Porous Materials

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

An angle of maximum absorption exists: φ = cos−1⎜⎛ (rʹ′)2 + (xʹ′ )2 ⎟⎞ max ⎝ n n ⎠ ʹ′ The maximum absorption here is: 2rn αmax = (znʹ′ + rnʹ′)

α(0°)→1 if rnʹ′ →1 and xnʹ′ → 0

Dept. of Mech. Engineering 18 Noise and Vibration University of Kentucky Short Course Example Impedance of Foam Sound Absorption by Porous Materials

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

Dept. of Mech. Engineering 19 Noise and Vibration University of Kentucky Short Course Example Sound Absorption Coefficient Sound Absorption by Porous Materials

0.8

0.6

0.4 phi = 0 degrees phi = 30 degrees

Absorption Coefficient Absorption 0.2 phi = 60 degrees

0 0 1000 2000 3000 4000 Frequency (Hz)

Dept. of Mech. Engineering 20 Noise and Vibration University of Kentucky Short Course Common Glass Fiber Sound Absorption by Porous Materials

0.8

0.6

0.4 Glass Fiber 0.2 Absorption Coefficient

0 0 1000 2000 3000 4000 Frequency (Hz)

Dept. of Mech. Engineering 21 Noise and Vibration University of Kentucky Short Course Closed Cell vs. Open Cell Foam Sound Absorption by Porous Materials 1 Closed Cell Foam 0.8 Open Cell Foam

0.6

0.4 Absorption Coefficient Absorption 0.2

0 0 1000 2000 3000 4000 5000 Frequency (Hz)

Dept. of Mech. Engineering 22 Noise and Vibration University of Kentucky Short Course Foam Absorbers must be Open Cell Sound Absorption by Porous Materials Closed-cell Open-cell

Dept. of Mech. Engineering 23 Noise and Vibration University of Kentucky Short Course Effect of Thickness Sound Absorption by Porous Materials 1

0.8

glass fiber 0.6

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 24 Noise and Vibration University of Kentucky Short Course Effect of Covering an Absorber Sound Absorption by Porous Materials

0.8

0.6

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

Dept. of Mech. Engineering 25 Noise and Vibration University of Kentucky Short Course Layering of Materials Sound Absorption by Porous Materials 1

0.8

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

0 0 500 1000 1500 2000

Frequency (Hz)

Dept. of Mech. Engineering 26 Noise and Vibration University of Kentucky Short Course Lining of Partial Enclosures Sound Absorption by Porous Materials

100

Sound Power Level (dB) Level Power Sound 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 27 Noise and Vibration University of Kentucky Short Course Full Enclosure Sound Absorption by Porous Materials

100

95 Top and Bottom Front and Rear 90 Sides Empty 85

Plexiglass SPL (dB) 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 28 Noise and Vibration University of Kentucky Short Course Overview Sound Absorption by Porous Materials

D The Basics D Impedance and Absorption D Transfer Matrix Approach D Flow Resistivity

Dept. of Mech. Engineering 29 Noise and Vibration University of Kentucky Short Course Sound Propagation – Porous Material Sound Absorption by Porous Materials

Plane waves in a porous material: D amplitude decreases with distance D p and u are not in phase D characteristic impedance z’c is a complex number D wave number k’ is a complex number − jkʹ′x p(x) = Poe k'= β − jγ x p(x) = zʹ′ u(x) c Attenuation constant (responsible for wave attenuation) complex

Dept. of Mech. Engineering 30 Noise and Vibration University of Kentucky Short Course The Basic Idea Sound Absorption by Porous Materials

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 p1, u1 or ! % ! % # p # ( +# p # " 1 & = * A B -" 2 & $# S1u1 '# ) C D ,$# S2u2 '# p2, u2

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

Dept. of Mech. Engineering 31 Noise and Vibration University of Kentucky Short Course The Straight Tube Sound Absorption by Porous Materials − 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 Ae− jkL − Be+ jkL must have plane waves u(L) = u2 = ρoc = + ρ Solve for A, B p1 p2 cos(kL) u2 ( j oc)sin(kL)

in terms of p1, u1 u1 = p2 ( j ρoc)sin(kL) + u2 cos(kL) then put into ) ρ , equations for p , u . j oc 2 2 " & + cos(kL) sin(kL) ." & $ p1 $ + S2 .$ p2 $ # ' = + .# ' $ S1u1 $ jS1 S1 $ S2u2 $ % ( + sin(kL) cos(kL) .% ( ρ *+ oc S2 -. (note that the determinant A1D1-B1C1 = 1)

Dept. of Mech. Engineering 32 Noise and Vibration University of Kentucky Short Course Straight Tube with Absorptive Material Sound Absorption by Porous Materials

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

( + jzc ! % * cos(k'L) sin(k'L) -! % # p1 # * S2 -# p2 # " & = * -" & # S1u1 # jS1 S1 # S2u2 # $ ' * sin(k'L) cos(k'L) -$ ' )* zc S2 ,-

Dept. of Mech. Engineering 33 Noise and Vibration University of Kentucky Short Course Transfer Matrix Approach Sound Absorption by Porous Materials

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 34 Noise and Vibration University of Kentucky Short Course Example - Layered Materials Sound Absorption by Porous Materials 1

0.8

0.6 Units: mm 5 1 51 3 6 0.4

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

Dept. of Mech. Engineering 35 Noise and Vibration University of Kentucky Short Course Overview Sound Absorption by Porous Materials

D The Basics D Impedance and Absorption D Transfer Matrix Approach D Flow Resistivity

Dept. of Mech. Engineering 36 Noise and Vibration University of Kentucky Short Course Designing the Absorber from Scratch Sound Absorption by Porous Materials

Bulk Surface Layered ’ ’ k and z c z and α z and α

????

Dept. of Mech. Engineering 37 Noise and Vibration University of Kentucky Short Course Absorption Coefficient vs. Frequency Sound Absorption by Porous Materials

αo Different densities and thicknesses

Frequency (Hz) (Mechel, 1988)

Dept. of Mech. Engineering 38 Noise and Vibration University of Kentucky Short Course Flow Resistivity and Absorption Sound Absorption by Porous Materials

αo

ρof/σ

Dept. of Mech. Engineering 39 Noise and Vibration University of Kentucky Short Course Flow Resistance and Flow Resistivity σ Sound Absorption by Porous Materials

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

rs Vacuum source σ = t

Dept. of Mech. Engineering 40 Noise and Vibration University of Kentucky Short Course ASTM C522-03 Flow Resistance Measurement Sound Absorption by Porous Materials Manometer Pipe Vacuum pump Flow Meter

Specimen Valve Valve Pump Manometers Flow Meters

Specimen Holder

Specimen

Dept. of Mech. Engineering 41 Noise and Vibration University of Kentucky Short Course Absorption Coefficient Sound Absorption by Porous Materials

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

Dept. of Mech. Engineering 42 Noise and Vibration University of Kentucky Short Course Absorption Coefficient and Flow Resistivity Sound Absorption by Porous Materials σt ≈ 2 ρc

Absorption Coefficient

Flow Resistivity

fluid solid

Dept. of Mech. Engineering 43 Noise and Vibration University of Kentucky Short Course Three Empirical Models Sound Absorption by Porous Materials X = ρo f σ Wu, 1988 – 17 plastic foam materials; kʹ′ k = (1+ 0.188X −0.554 ) − j0.163X −0.592 f = 200-2000 Hz; 2900 ≤ σ ≤ 24300 −0.548 −0.607 zcʹ′ zo = (1+ 0.209X ) − j0.105X rayls; 0.01 < X < 0.83

−0.700 −0.595 kʹ′ k = (1+ 0.0978X ) − j0.189X Delaney and Bazly, 1970 – fibrous −0.754 −0.732 materials; f = 250-4000 Hz; σ = ?; 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 ʹ′ = + −0.699 − −0.556 zc 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 44 Noise and Vibration University of Kentucky Short Course