Systematic evaluation of simplifications used in (ordinary) mobility approach

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Systematic evaluation of simplifications used in (ordinary) mobility

approach

Systematic evaluation of simplifications for (ordinary)

mobility approach for predicting structure borne

power flow from studs to direct attached gypsum

board

Nightingale, T.R.T.

ORAL-203

Slide 1

Systematic evaluation of simplifications for (ordinary)

mobility approach

for predicting structure borne power flow from studs to direct attached gypsum board

T.R.T Nightingale Institute for Research in Construction Katrin Kohler & Jens Rohlfing

Technical University of Stuttgart

Slide 2

Outline

! Wall specimen examined

! List assumptions used in mobility theory

! Resulting power flow expression

! Method for assessing changes in power flow

! Examine assumptions

•Part 1 – “typical” or “ill-defined” fastening points

•Part 2 – “well-defined” fastening points

! Summary

Slide 3

Specimen to be modelled

! 35x85 mm western red cedar studs – no knots

! 16 mm Type X gypsum board – single layer

! Only one stud excited by a single point force

! Source and location & number of fasteners changed

1.22 m 1.22 m 1.22 m 0.406 m

PLATE 3 PLATE 2 PLATE 1

Point Force Excitation Point Force Point Force Excitation 2 .4 3 m 1 .2 2 m 5 0 m m 0.368 m 0.368 m 5 0 m m 4 0 6 m m 3 5 6 m m 3 5 6 m m

Slide 4

What is a force mobility – Y ?

! Paper restricted to force

mobilities, (transmission via moment mobilities to be examined elsewhere)

! Ratio of resulting velocity to

an applied force, both space averaged over the area of the drive point.

! Complex quantity (applied

force and resulting velocity typically not in phase)

! Inverse of impedance

F

v

Y

=

Shaker Accelerometer & Force gauge Mounting stud Element under test Slide 5

Assumptions of mobility models (simplifications)

Previous models by Cremer, as adapted by Craik et al, assumed:

1. Stud velocity constant across stud depth 2. Continuity of velocity at stud/gypsum board

interface

3. Power flow only at fasteners

4. Power flow same for all fastener locations 5. Power flow proportional to number of fasteners 6. Power flow independent of contact area at drive

point (stud to gypsum board)

Power flow – Stud to gypsum board several fasteners, N

[ ]

        + ℜ = 2 2 1 2 2 1 total 12 Y Y Y e v N W , Mobility expression

∑

= = N 1 i i 12 total 12 W W , ,

Total power Flow 1 ➨ 2

2 Gypsum Board mobility Y2 1 Stud mobility Y1 N screws stud velocity V1

[ ]

* v F e W =ℜ General relation

Slide 7

Predicting ideal case – Single fastener and “well-defined” contact

0 5 10 15 20 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] Measured Predicted Point Force D 1 1 7 0 m m 1 1 7 0 m m Slide 8

Assessing power flow from velocity level difference (VLD)

2 2 2 2 12 2 1 1 12 M V M V W = ωη = ωη       =       =         = ω η η η 2 12 1 1 2 12 2 1 2 2 2 2 1 12 W E M M 10 M M 10 V V 10 VLD log log log

SEA Power balance

Velocity Level Difference

! <V12> measured using 18

points on stud face excited by a point force

! <V22> measured using 98

points on the gypsum board

! Change in VLD inversely proportional to change in power flow Point Force <V12> stud space average RMS velocity <V22> gypsum board space average RMS velocity Slide 9

Assumption 1 – Velocity is same on both sides of the stud

! Force applied to the screw by the stud, F1= (<V1>-V1dp)/ Y1

where <V1> is space

average velocity, and V1dpis stud velocity at

the (screw) drive point

! Implies stud does not deform through its cross section Measured velocity level on either side of the stud should be same.

Criterion and Test:

2 Gypsum Board mobility Y2 1 Stud mobility Y1 Screw V2dp V1dp Stud velocity V1

Slide 10

Velocity level across a stud

-7 -5 -3 -1 1 3 5 7 9 11 13 15 63 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [d B] Exciter 2.43m 1.21m 0.40m NOT TO SCALE A1 B1 B2 A2 Point 1 near exciter (A1-B1) Point 2 away from exciter (A2-B2)

Slide 11

Velocity level across a stud

!Assumption of same velocity is not valid at all

frequencies

•Low – acceptable approximation

•Mid – some variation but trends are correct

•High – systematic difference causing

underestimation of predicted VLD (stud to gypsum board)

!Model could be extended by allowing for

deformation of stud through the cross section

Assumption 2 – Continuity of velocity at interface

! Velocity of the stud is the same as the gypsum board at the interface, V1dp= V2dp ! Implies fastener applies

sufficient force to ensure no relative motion, hence minimum screw torque

Evaluate VLD’s for screw torques 1.2 and 2.0 Nm, where 1.8 Nm is required to drive the screw head just into the sheet.

Criterion and Test:

2 Gypsum Board mobility Y2 1 Stud mobility Y1 Screw V2dp V1dp Stud velocity <V1>

Slide 13

VLD as a function of screw torque Effect of Screw Torque

-5 0 5 10 15 20 63 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] 2.0 Nm torque 1.2 Nm torque

Screws loosened so heads 2 mm above surface of gypsum board

Slide 14

Assumption 3 – Only significant power flow at fasteners

! Expression only considers power flow at fasteners

! Implies power flow at ill-defined point contacts due to irregularities in the stud and gypsum board is insignificant Mobility expression

When there are no fasteners, the power flow should be zero, and the VLD infinite, or at least much greater than with fasteners

Criterion and Test:

[ ]

        + ℜ = 2 2 1 2 2 1 total 12 Y Y Y e v N W , Slide 15

VLD without any fasteners Point force applied immediately

opposite to Screw -5 0 5 10 15 20 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [d B] Range for No Screws

Upper & Lower limits for a single screw Point Force Excitation 2 .4 3 m 4 0 6 m m 4 0 6 m m A B C D E F G 3 5 6 m m 4 0 6 m m 4 0 6 m m 3 5 6 m m

Slide 16

Summary – Part 1 Assumptions relating to “ill-defined” fastening points

! Velocity across the stud depth is significant

•Violates a fundamental assumption

•Predicted VLD will be underestimated, especially at high frequencies

! Screw torque need not be modeled explicitly

! There is significant power flow, away from fastener(s) at “ill-defined” contact points

•Effect should be included in models with few fasteners

Slide 17

Part 2 – Experiments with “well-defined” fastening points

!Assumptions relating to the “well-defined”

fastening points

•Number of fasteners

•Location of fasteners

•Contact area

Creating “well-defined” contact points

!Goal to assess power flow at

fastening points

•Need to remove power flow

at “ill-defined points”

!Place thin (2 mm) spacers

between stud and gypsum board to create a “well-defined point” of known area

2 mm thick disc 15 mm dia. stud gypsum board

Slide 19

Assumption 4 – Power flow proportional to number fasteners

! Expression indicates power flow is directly proportional to number of fasteners

! Implies motion at each fastener is incoherent and that fastener spacing defines applicable frequency range Mobility expression

Using measured VLD’s for individual points, check that the sum is the same as measured when there is a fastener at each point

Criterion and Test:

[ ]

        + ℜ = 2 2 1 2 2 1 total 12 Y Y Y e v N W , Slide 20

Measured and estimated VLD for 3 fasteners (with defined contact)

0 5 10 15 20 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] Measured 3 screws located at poitions A, D, & G Estimated 3 screws

from individual VLD's for single screws at

A, D, G PointForce A D G 1 1 7 0 m m 1 1 7 0 m m Slide 21

Measured and estimated VLD for 7 fasteners (with defined contact)

-10 -5 0 5 10 15 20 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] Measured 7 screws Estimated 7 screws from individual VLD's for single screws at positions A through G Point Force 4 0 6 m m 4 0 6 m m A B C D E F G 3 5 6 m m 4 0 6 m m 4 0 6 m m 3 5 6 m m Spacing Fastener 2 B< λStud Gyspum board Spacing Fastener 2 B< λ

Slide 22

Assumption 5 – Power flow is the same at all fastening points

! Expression indicates power flow at a fastening point is independent of location

! Implies, for a single fastener VLD will be independent of fastener location

Mobility expression

Compare measured VLD’s for different fastening points

Criterion and Test:

[ ]

        + ℜ = 2 2 1 2 2 1 total 12 Y Y Y e v N W , Slide 23 VLD as a function of location Source located opposite to fastener

(using average stud levels)

0 5 10 15 20 25 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] A B C D E F G Assumption 6 – Mobility is independent of contact area between stud and gypsum board

!Ordinary mobilities are used almost exclusively in prediction models

!Assumptions in typical models:

•“Effective Contact Area” – where the stud

and gypsum board have the same velocity – is “infinitely small”

•Power flow is independent of area for

Slide 25

Effect of contact area – stud to gypsum board for single screw

Effect of Contact Area between the stud and gypsum board (single screw and drive point at D)

0 5 10 15 20 25 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] 9 dia. = 64 mm^2 15 dia. = 177 mm^2 16x35 = 560 mm^2 35x35 = 1225 mm^2 70x35 = 2450 mm^2 140x35 = 4900 mm^2 280x35 = 9800 mm^2 15 mm dia. spacer not changed

spacer dimensions being systematically changed Increasing contact area Slide 26

Summary – Implications for models

➻ ➻ ➼

Power flow independent of contact area at fastener

➼ ➼ ➻

Power flow proportional to number of fasteners

➼ ➼ ➻

Power flow same at all fasteners

➻ ➻ ➻

Power flow only at fasteners

➼ ➼ ➼

Screw torque is not important

➻ ➻ ➼

Velocity constant across stud depth

High Mid Low Assumption Slide 27 Questions

Slide 28

What contact area gives power flow like direct attachment?

-5 0 5 10 15 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B] Direct contact No Spacer, one screw Single spacer 289x35 mm (9800 mm^2), one screw Slide 29

Effect of contact area – screw head to gypsum board

Effect of Contact Area between screw and gypsum board (single screw and drive point at D)

0 5 10 15 20 25 125 250 500 1k 2k 4k Frequency, [Hz] V L D, [ d B]

15mm dia. plexi disc 35x35x2mm plexi plate 38x38x2mm steel plate 15 mm dia.spacer not changed spacer dimensions being systematically changed

Mobility as function of location – Gypsum board 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 63 125 250 500 1k 2k 4k Frequency, Hz Real p a rt o f M o b ilit y , m/Ns Theoretical Values Center-Located Point Edge-Located Point 9.5 mm 19 mm 1200 mm plate center 50 mm

Slide 31

Practical implications for increasing sound insulation

!Use studs that contribute higher velocity level

difference across the stud

• wood studs with low shear modulus

(western red cedar was used – very low G)

•steel studs with shape that is compliant to

normal forces

!Minimise number of contact points (fasteners)

!Reduce contact area between stud and

gypsum board

Slide 32

Selecting expressions for mobilities Allowed Not possible Not possible Deformation Arbitrary size Arbitrary size Infinitely small point Contact area Volumetric Near Field [3], [4] Interface [2] Ordinary [1] Drive point

1: Structure borne sound, Cremer Heckl and Ungar 2: Petersson, JSV(1997), 202(4), pp. 511-537 3: Petersson & Heckl, JSV (1996), 196(3), pp. 295-296 4: Petersson, JSV(1999), 224(2), pp. 243-266

Slide 33

Summary – Implications for models

! Power flow is reasonably independent of fastener location

! Total power flow is approximated by sum of powers at individual points

•(tends to fail when points are in phase – i.e., there is a minimum separation)

! “ill-defined” contact points contribute significantly when there are few fasteners

! Contact area between stud and gypsum board is very important