Practical calculation of floor impact sound by impedance method

Applied Acoustics 26 (1989) 263-292

Practical Calculation of Floor Impact Sound by Impedance Method Sho K i m u r a & K a t s u o I n o u e College of Science and Technology, Nihon University, 1-8 Kanda-Surugadai, Chiyodaku, Tokyo, Japan (Received 31 August 1988; revised version received 24 October 1988; accepted 10 November 1988)

ABSTRACT This paper describes a practical floor impact sound level calculation method for a heavyweight, soft impact source, such as children jumping and running. The floor impact sound for a heavy, soft impact source is strongly influenced by structural factors such as floor slab stiffness and peripheral support conditions. The manner of analysing these structural factors for performing the sound level calculation is described. A sample of measured values and calculated values from an actual structure is presented to show the accuracy of this method.

1 INTRODUCTION On the calculation of impact sound insulation between dwellings, E. Gerretsen reported the prediction method based on the point admittance and showed its applicability to a typical floor construction.l Previous to this, practical calculation of floor impact sound using an impedance method was presented by the present authors. 2.a According to this calculation method, designers can easily estimate the performance of a floor structure in order to reduce floor impact sound levels while at the design stage. In addition, this method can correspond well to actual values, and it is expected to be an excellent practical method. Under this method, floor impact sound is calculated according to the flowchart indicated in Fig. 1. The flow chart is roughly composed of the following three sections: (1) impulsive force characteristics of an impulsive 263 Applied Acoustics 0003-682X/89/$03-50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

264

Sho Kimura, Katsuo lnoue

source which is input to floor slabs; (2) vibration characteristics of floor slabs; and (3) acoustic characteristics in the room below. The method of calculating floor slab vibration characteristics, which directly contribute to the generation of floor impact sound, is emphasized in this flow. In this paper, the method of handling each factor which composes the flow ~~ -> ~ Impulsive force characteristics (Ff) ,

Composite

Yes

~ Y e s No

I

Equivalent Young's modulus (E) Equivalent slab thickness (h) Equivalent density (p)

Equivalent slab thickness (h}

Equivalent Young's modulus (E) Equivalent slab thickness (h] Equivalent density ( p]

.=

l

.,3

I

-1,

Driving-point impedance of infinite long slab (Zb)

I

Primary vibrational response of each exciting point

I

Full-time response impedance

I

Influence of peripheral fixing (AZb)

1

U.Q

Correction of impedance characteristics at each octave-band (AZ,)

J:

Acoustic effective radiation area (Sat) Acoustic absorption in room below (A) Correction of time constant

J

]

...= g: ca

I Fig. !.

Acoustic radiation coefficient (6R)

Average floor impact sound level (Lf)

.<

Flowchart for floor impact sound calculations by impedance method.

265

Calculation of floor impact sound

of Fig. 1 is explained, and examples are shown where this calculation method is applied to an actual floor structure.

2 EXAMINATION OF EACH FACTOR ON CALCULATION FLOWCHART 2.1 Impulsive force characteristics, F r

JIS A 1418 (Japanese Industrial Standard) stipulates the use of a lightweight, hard impact source (tapping machine) and a heavyweight, soft impact source (see Fig. 2) for measuring the performance of floor impact sound level on site. Heavyweight, soft impact sources are stipulated as a simulation of children running and jumping on a floor. 4 People living in multi-family dwellings in Japan (a country where shoes are not worn in the house) overwhelmingly complain about floor impact sounds caused by children, i.e. heavyweight, soft impact sound sources. Heavyweight, soft impact sources, as shown Fig. 2, show a peak impulsive force of 4000 N and a duration of A

~

4.2

~36

. 2sin [ (t+2) • ~r/241

U

/'

0

"u~ 1.1 et

~N 0

~ i [ t .

~/20]

/ 10

20 22

Impact time t (msec)

Fig. 2. Time characteristics of heavy, soft impulsive force (JIS A 1418).

20ms. The floor impact sound insulation performance is related to the dynamic performance of the building structure. Thus, the JIS is also applicable when checking the rigidity of a floor structure. In this report, the method of calculating the floor impact sounds when the floor slabs are excited by heavy, soft impact sources, as stipulated in the JIS, will be explained. The impulsive force of people running and jumping (the typical heavyweight, soft impact sources generated in daily life), was measured by using the measuring instrument shown in Fig. 3, which uses a force

266

Sho Kirnura, Katsuo Inoue

Exciting force wear plate Impulse (aluminium plate) m / Force transducer ~ / ~ thickness Stationa~TRAHSDUCER// / "~ plate [-----~PCB210B5~/ ' ]/ \ . . \[--~ ~----~/ ~ "--~50em"in diameter .ortar \ l~tl I l lltl" { /~-./..'-'. ,

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CONV.l

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Fig. 3. Measurement and analysis block diagram of impulsive force.

('~'Ju'~ o

PATTERI~-a PEAKt 5 oO04H

IMPULSE=42.THeme

(~)l ~

PATTERN-b PEAK=2,48114

IMPULSE=46.2Hsec

Impulse time (msec)

Impulle time (meet)

(~)1 ~5 IMPULSE=25.0Nsec "

~.5

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........ eI"~-

Impulse time (msec)

2q9.0

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Impulse time (meet)

i) Impulsive Force Characteristics when Child Runs and Jumps

~

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....

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Impulse time (meet)

20

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154

Impulse time (mse¢)

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~--

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} Impulse time (meet)

2

o

o-...

:<~ 7T"::-:": Impulse t:Lee {meec)

2) Impulsive Force Caracteristlcs when Adult Runs and Jumps

Fig. 4. Impulsive force waveform of child and adult (running and jumping).

267

Calculation of floor impact sound

transducer. Running and jumping characteristics are greatly influenced by the way the knee springs are used. Figure 4 shows typical examples. When each waveform of Fig. 4 is dealt with as a time series datum per second by the FFT, and is calculated as an impulsive force level (20 log10 F, (dB)) of each 1/1 octave band in the range 31.5-500Hz, it becomes as shown in Fig. 5. Figure 5 also shows the analysis results of the impulsive force characteristics of a standard heavyweight impact source, which corresponds to the JIS shown in Fig. 2. Figure 4 indicates the change of impulse time from around 20ms (jumping down heavily without using a knee spring) to 250ms (jumping down lightly with adequate knee spring), and also the change of the peak impulsive force from 200-300 N (500-600 N in the case of an adult), which is equivalent to the weight of a person, to 5000 N (around 7000 N in the case of an adult). The impulsive force characteristics of each octave band in Fig. 5 indicates distribution at a range of around 30dB as a whole. When these impulsive force characteristics at the impact source are compared to the impulsive force characteristics of the JIS, it can be said that the characteristics of the JIS standard are on the whole similar to jumping down frequency characteristics on the whole. Generally, a single impulse (bang) machine (free fall of car tire; Fig. 6) is used as the impact generator which corresponds to these JIS characteristics. Table 1 indicates the impulsive force characteristics of each octave band,

60 50

L

40 30 m t_

Lr~ 0

20 10

c~

0 -10'

A ~ - - A J u m p i n g down (,lOcH,mtural ly) Hdu-,pir~ down ( n a t u r a l l y ) ~ ' ~ ' l t C h i ldren Jumping down (PATTERN-A) ~-~--'AJumping (knee spring) landing section -I~-----OJumping down (knee spring)

i

½

t~

8

I

I

1'6 31'.5 63 125 250 500

0clave-Band Center Frequency (Hz) Fig. 5. Impulsive force frequency characteristics.

-

268

Sho Kimura, Katsuo lnoue

i:i~!? ill!

Fig. 6.

Simple impulse (bang) machine.

which are measured by the device shown in Fig. 3. Therefore, the values in Table 1 are used as the impulsive force, F I, which is to be input to the floor slab for this calculation method.

2.2 Calculation of various floor slab constants (E, p, h) In order to calculate the driving-point impedance and the natural frequency of a test floor slab, the flexural rigidity of the floor slab sectional area has to be calculated. In this case, where the following method is used, the equivalent Young's modulus, E, equivalent thickness, h, and equivalent density, p, of a composite floor sectional area (such as a deck plate floor, a void slab and an additional placing slab) can be obtained. These constants TABLE1 Impulsive Force CharactefisticsofHeavyweightlmpactSource Octave-band center frequency (Hz)

Effective value of impulsive force, Frms (N) Impulsive force level, 201oglo [Frms] (dB)

63

125

250

500

1K

71

13

4

2

1

37

22

12

4

0

Calculation of floor impact sound ......... _ ~,::.,.:::..centei: o f[ Fig. 7.

269

............. fiquz-e.:;:..":~: , h2 ~ ~ 2 ,E2 ,ol '

A d d i t i o n a l placing o f slab.

can be obtained by replacing composite floor materials with a homogeneous

single plate. When an additional slab is placed onto a single plate (see Fig. 7): p = p l h , / ( h x + h2) + pEhE/(hl -1- h2)

(1)

E = ( E l i a + E212)/I x

(2)

h = h a + h2(p2/pl)

(3)

where Ix is the geometrical moment of the total floor sectional area; and I x and 12 are the geometrical moments of sectional area of each member around the neutral axis (x axis). In the case of a void slab and a deck plate slab (see Fig. 8), only the equivalent thickness, h, is obtained by the following equation, as the materials are homogeneous: h = ~

(4)

where Ix is the geometrical moment of the total floor sectional area. When an additional slab is placed onto a void slab and a deck plate slab (see Fig. 9), the equivalent density, p, is calculated using eqn (5), using a sectional area ratio. The equivalent Young's modulus, E, and the equivalent thickness, h, are calculated from the geometrical moment of each member's cross-section against the x-axis, which passes through the total cross-section geometrical moment and the center of the figure, by using eqns (6) and (7) respectively: p = p~Sa/(Sx +

$2) + P282/($1 + $2)

(5)

E = ( E I I I + E212)/Ix

(6)

h= ~

(7)

Here, $1 and $2 are the sectional areas of each member. Center

A

of

I . . . . Fig. 8.

figure

,.

O pE

X p,E

Void slab and deck slab.

270

Sho Kimura, Katsuo Inoue

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~

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Adding onto special slab.

2.3 Calculation of floor slab basic impedance The length of the floor slab is finite. However, when the condition of the surrounding support is excluded from consideration, the driving-point impedance of the floor slab is determined from its cross-sectional characteristics. In other words, the driving-point impedance, Zb, of a floor slab can be calculated as for that of an infinite long slab, using eqn (8): Z b = 8x//-~

- 2.31pl/2E1/2h 2

(8)

where B is the flexural rigidity of the floor slab, m is the surface density of the floor slab, p is the density of the floor slab (equivalent density), E is the Young's modulus of the floor slab (equivalent Young's modulus) and h is the thickness of the floor slab (equivalent thickness). Using this method of calculation, the value obtained in eqn (8) is used as the basic impedance for the calculation process. This impedance value can also be obtained in a finite long slab by using the following measuring method. Figure 10 indicates where the slab surface is excited by a pulse. When the velocity of the flexural wave is Cb, the Impulsive force

Exciting force wave form [ ! Vibration velocity V t ~ wave form of slab excitation-point ~ t ~ ' " Fig. 10.

-~t /'~,f-, ---~%

Response model when floor is excited by impulsive force.

271

Calculation of floor impact sound

~ 10~

1o

Fig. II.

/

.

.

.

Impulsive impedance

.

.

.

.

fbo

.

.

.

.

Frequency (Hz)

.

.

.

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Example of actual impedance measurement (4.73 x 4.90 x 0.15 m).

displacement of the slab during the excitation, At, proceeds as Cb At for both right and left directions, which is just one wavelength. This width matches the wavelength of flexural waves at the impact frequency [ = 1/(2 At)] of the impact source. If the fixing boundary is not within this displacement, the vibrational response value is determined only by the slab condition. Therefore, if the ratio of the exciting force waveform and the vibration velocity response waveform (the solid line section of the forced response value in Fig. 10) of the floor slab within the exciting force corresponding time is expressed as the impulsive impedance, then this value is the same as the impedance value obtained in eqn (8). Figure 11 shows the actual value of the driving-point impedance at the slab center point, the dimensions of which are 4.73 x 4-90 x 0.15 m. The impulsive impedance Z b is 4.10 x 105 kgs-1 according to the figure. The calculated value based on eqn (8) is: Zb = 2.31 x (2300) 1/2 x (2.6 x 101°) 1/2 x (0.15) 2 = 4.57 x 105 kg s- 1 Thus, both values are almost the same.

2.4 Effect of slab peripheral fixing on impedance characteristics When there are beams or walls within the floor slab--in other words, the excitation point is near to the slab edge--the amount of displacement in the impact time of the impact source becomes smaller, and the impulsive impedance increases due to the effect of the edge fixing. This increasing quantity depends on the fixing condition of the slab edge. The increase in the

272

Sho Kimura, Katsuo Inoue

~

25

~

20

0"~

15 • ,-I

•

Supported by girder

•

10

.,-I

o

5

,--4

0.01

0.10

0.50

LO

xl ,Ib Fig. 12. Impulsive impedance level increasing quantity at slab edge.

impulsive impedance was measured in order to find the actual condition where there are girder supports and beam supports. When these results are adjusted by a ratio of the distance from the slab edge (x) and the flexural wavelength (2b) at the impact frequency [f~ = 1/(2 At), where At is the impact time] of the impact source, they become as shown in Fig. 12. According to these results, when the excitation point comes nearer than 2J2 to the slab edge, the impulsive impedance tends to increase. When the edge is supported by a girder, the increase in the impedance is expected to be + 15 dB, and when it is supported by a beam the increase is around + 10dB. Figure 12 indicates the regression curves for many actual data. Equations (9) and (10) indicate the regression formulae. When supported by a girder, the edge fixing level is high: AL z = 15.37 - 68.86 x ~ + 98-65 x

- 45.36 ×

(9)

When supported by a beam, the edge fixing level is comparatively low: AL= = 10.93 - 55.86 x ~bb+ 92"57 ×

-- 49"72 ×

(10)

AS shown in Fig. 13, the increasing quantity of the impulsive impedance on the slab diagonal line simultaneously receives a fixing effect from both edges (1) and (2), similar to Fig. 12. For this reason, the following two methods to combine the effects of both directions are considered. (i) Calculate the increase in the impedance level, AL~, by using the following equation based on the consideration that the transfer systems from both directions simultaneously affect an optional point diagonally, and an increase of impedance is convolved into the energy:

bLz = 20 logao (al0c2) = bLzl + where ~1 = 10~L"/:° and ~2 = 10AL'2/20"

ALz2

(11)

273

Calculation of floor impact sound

/

Diagonal direction

/ Edge ( 1 ) Fig. 13. Calculation of slab diagonal line impedanceincreasingquantity.

(ii) Calculate the increase in the impedance level, AL,, by using the following equation based on the consideration that the increase in the impedance from both directions is added: A L z = 20 loglo (10AL'*/2° + 10 ALl2/2° - 1)

(12)

In order to verify the applicability of these ideas, a test was conducted using an actual concrete slab, as shown in Fig. 14. Fifteen to twenty excitation points were provided in directions (1) to (3) from the test slab center point P. The impulsive impedance (which is explained in section 2.3) was calculated based on the exciting force and the vibration velocity response near the excitation point by the use of an impulse hammer. These results were expressed by basing the impulsive impedance of the slab center point (the relation of the wavelength (2b) of the flexural wave in the impact frequency of the impact source and the distance (x) from the slab edge at the center point'being x / 2 b > 0.5) and data of other points with B slab I.~ I

3. 490

TM

3 0 0 X 660

A Slab

/

×,

1

3,400

LaDirection (1)

,~1 r

7,150

/ip

540X 660

Fig. 14. Test objectiveslab.

~1 "1

274

Sho Kimura, Katsuo lnoue

A

=~30

~

Calculation

value

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0.01 Fig. 15.

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. . . .

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0.01

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Comparison of impulsive impedance increase at slab diagonal line.

relative levels. The increase in the diagonal impulsive impedance is calculated by eqns (11 ) or (12), based on the above result. Figure 15 indicates the comparison of these increases and measured values. Physically, method (ii) is the general idea. However, when the calculated results are compared with these measured values, there is some difference between them due to the influence of constraints by the column, etc., which exist on diagonal line slabs. As a result, the actual values were situated between method (i) and (ii) calculated values. By considering these test results and simplifying the calculation, it was decided to use method (i) as the method of calculating the impulsive impedance on the slab diagonal line (eqn (11)). 2.5 Driving-point impedance characteristics of a finite floor slab

2.5.1 Calculation of floor slab natural frequency The primary natural frequency of a finite long plate in the peripherally fixed condition (f0, nx) is calculated by eqn (13): = n {2.25 l'4~{E~ */2 f0,~x 4x/~ \ a 2 + - ~ - / / \ p j h

(13)

where a is the length of the short side (m), b is the length of the long side (m), h is the slab thickness (m), E is the Young's modulus (N m-2) and p is the density (kg m - 3). However, the peripheral fixing condition is not completely fixed with a regular floor slab. In many cases, the primary natural frequency is 0-8 to 0-9 times that of the value obtained in eqn (13). Therefore, it is better to use the frequency corrected by eqn (14), as it is expected that this will correspond better with the calculation: fo - 0"8f0,fix

(14)

In many cases, when normal floor slabs are used for a multi-family

Calculation of floor impact sound Span ratio 110" Span ratio

1

100" k

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,0

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0

'

fix

,..,

Solidity rate 0.9 Solidity rate 0.8

31.5Hz band

Fig. 16. Example of calculation for primary natural frequency of floor slab.

dwelling, the slab area is 10-30 m 2 and the span ratio is around 1-2. Thus, in many cases, the primary natural frequency is in the band of 31-5 Hz or 63 Hz, as in an example shown in Fig. 16.

2.5.2 Impedance characteristics of floor slab, Z, A diffused flexural vibration field was assumed on the finite length floor slab, and the impedance characteristics were studied. Basically, it is impossible to have a diffused condition in a natural frequency zone. However, an assumption was made on the premise of simplifying the calculation and to stress the consistency with the value of the actual floor slab. If absorption (transmission) at the peripheral fixing edges is the only loss factor at the floor slab, and the viscous damping in the slab is ignored, the floor slab loss factor, r/, can be expressed by eqn(15), 5 based on twodimensional diffusion theory. Equation (16) indicates the damping time of 60 dB:

l

(15)

Tro - 13"8rcS/Cbhb

(16)

r / - 2.2/(fT6o ) - 32.19 Sx/"f

where Cb is the velocity of the flexural wave (m s- 1), l is the length of the slab circumference (m), ~b is the flexural wave absorption factor around the slab and S' is the slab area (m2).

276

Sho Kimura, Katsuo Inoue

On the basis of eqn (15), the frequency characteristics of the loss factor are proportional to ~b/x/f. If ~b is constant with frequency, the frequency characteristics of the loss factor will indicate a tendency of - 1.5 dB/octave. However, in the actual slab, there is a loss due to floor slab internal viscosity and acoustic radiation, etc., in addition to absorption by slab surroundings. These factors are considered to increase in proportion to the frequency. Thus, it is difficult to determine the frequency characteristics of the loss factor unconditionally. For this reason, it was decided to verify the frequency characteristics of the loss factor by testing an actual slab. Forty-one slab types, the construction methods, thickness, area, etc. of which all differed, were prepared. An excitation was applied to the slab by hitting the slab center with an impulse hammer, and a vibration velocity response waveform was obtained. The loss factor of the resonance frequency was calculated on the basis of this waveform. Figure 17 shows the result. According to this result, the construction method, thickness, etc., does not result in a significant difference. In addition, there is not much change with frequency. Absolute values are between 0.04 and 0"10, with an average of 0.06, and the standard deviation is small (0.01). On the basis of the measured results of actual slab loss factors, it is believed that the absolute value of the slab loss factor to be used for a normal multi-family dwelling is around 0"06, and the frequency characteristics are almost flat. If the loss factor is constant, the 60 dB damping time (T60) of the slab will indicate - 3 dB/octave frequency characteristics according to eqn (15), as shown in Fig. 17. The change in the vibration energy per octave band by resonance is expressed by eqn (17). Here, T~o = T6o/2: A L = 101oglo E o

exp[(-13.8/T;o)t]dt/E o

f f? °

= lOloglo(T;o/Tao)

(17)

According to the above result, L = 10 loglo [T6o/(2T6o)] = - 3 dB. Thus the frequency characteristics indicate + 3 dB/octave at the impedance level. In the frequency domain offo, the primary natural frequency, tOfo/,J2, the impedance can be given as a resonance characteristic of a simple harmonic motion system. When the frequency domain is less thanfo/x/'2, it is directly affected by the. peripheral fixing, and thus it is determined by the flexural stiffness, Kb, of slabs. If the flexural stiffness is given from the maximum displacement of both ends of fixed beams (the spans of which are ! and the thickness of which are h) it becomes constant, regardless of frequency, when

Calculation of floor impact sound

277

1.00. O • • O

.150ram 180ram 200mm 150ram 200mm

thick thick thick t h i c k PC t h i c k composite floor

bd O 0.10 U t~ q~

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0.10

10

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.

o~

tm

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(Hz)

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,.00 o

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1so..

thick

• 180ram t h i c k • 200mm thick - ~. 0 150mm thick PC ~\ zx 2 0 0 m m t h i c k composite floor

.......F r e q u e ni6)0 ' ' '500 cy (Hz)

Observation results of loss factor and 60 dB damping time.

Kb = 16Eh3/l 3 (E indicates Young's modulus). Thus, impedance can be expressed as Zb = jKdco, and characteristics of - 6 dB/octave are indicated. Figure 18 shows schematically the estimated impedance characteristics for all frequency domains. When the impedance reduction quantity in the primary natural vibration zone is calculated from the measured result, ~/, shown in Fig. 17, it is approximately 25 dB. The predictive calculation, however, should be generally conducted for each octave band. It is then convenient to indicate the characteristics of Fig. 18 as an octave band characteristic.

278

Sho Kimura, Katsuo lnoue

u ~3

Impedance value by assuming slab Impedance the infinite Jfn2 /

e-,

loci.

,',~f n6

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r

I

t

t

i

. ---T--.-.~. --a--.+--~.~

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Frequency

Fig. 18.

(Hz)

Estimated impedance characteristics.

For this reason, attention was paid to the impedance reduction quantity at the band which included the primary natural frequency of the slab, and an impedance measurement was conducted at the actual slab center point. When the impedance of the slabs was classified according to their size ( 15 m 2, or greater than and less than 15m2), it becomes as shown in Fig. 19. According to the figure, the slabs which are 15 m 2 and larger are in the 31.5Hz band, which includes the primary natural frequency, and the ~I0~

~lC Less than

15m'

15m ~ or larger

,-4

,-t

,r.t 4J ,-.t

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structure ~

1'8 3f.s Central

frequency

Fig. 19.

(Hz)

lls

Central frequency

Measured impedance value.

(Hz)

Calculation of floor impact sound 20 ~-~ 10

279

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i

8 16 31.563 125250 5001Z 2K 4K Central frequency (Hz)

Fig. 20. S t a n d a r d i z a t i o n o f i m p e d a n c e characteristics at each band.

impedance reduction quantity is approximately 18 dB less than the value obtained by estimates on the infinite long slab. The primary natural frequency band of slabs which are less than 15 m 2 is at 63 Hz, which indicates a reduction of approximately 16 dB. The estimated impedance which is indicated in Fig. 18 is 18 dB less than the infinite slab impedance at the primary natural frequency band, when the 1/1 octave band is considered. This result can correspond to the result obtained by the actual measurement in Fig. 19.:Figure 19 indicates that the natural frequency of the 63 Hz band can reduce the impedance drop to 2-3 dB compared to when it is at 31.5 Hz. Thus, even when the slab changes, it is considered that the characteristics shown in Fig. 18 can be applied without change. Thus, when it is assumed that the impedance drops 18 dB at 31.5 Hz, the impedance characteristics can be standardized using Fig. 18, even when the slab changes. Figure 20 shows the standardized impedance characteristics of each octave band at the center of slab. As shown in Fig. 17, the loss factor hardly changes, even when the construction method and slab thickness changes. Thus, it is assumed that the standard characteristics of Fig. 20 can be applicable, even when the slab natural frequency changes. Based on this fact, it is also assumed that the value near the actual impedance characteristics can be calculated using the value of characteristics 1, 2 and 3 when the band (which includes the primary natural frequency) is 31.5 Hz, 63 Hz and 125 Hz respectively. The position of relative level, 0 dB, in Fig. 20 indicates the value of the impulsive impedance, which does not receive the influence of the peripheral boundary. This value matches up with the impedance which is obtained by assuming the infinite slab, as explained in section 2.3. When the objective point approaches the peripheral edge of slab, the correction value which is indicated in section 2.4 has to be added to the impulsive impedance.

280

Sho Kimura, Katsuo Inoue

S lab B

Slab A

1 --r],'

I . . . .

l

I

,,

I1

L',iI1

-.

'

':1',

.k4--4---~ II' 11 I lDirec- Ill

is

i1'22o 220350 tlon All I

t

,1' !!1',

t

I

¢,D

D I O T l~ ion A

~1' "'

II0

|

1,8oo Fig. 21.

I

I

1

2,700

Slabs and measuring points used in the test.

2.5.3 Experimental study In the test a 1/2 scale RC floor slab model (A) (the thickness of which was 100 mm), and a real floor slab (B) (the thickness of which was 150 mm) were used, as shown in Fig. 21. Excitation was applied to each excitation point shown in the figure with an impulse hammer, to obtain the vibration velocity response near the excitation point and the slab center point, and the impedance characteristics were then calculated. Figure 22 shows the impedance characteristics of each slab center point as a function of change of excitation point. The values indicated by a horizontal line in the figure are estimated by using the increasing value of impedance with x/2b at the excitation point. Figure 23 shows the impedance characteristics of each octave band at the slab center for various impact points. According to these test results, even when the excitation point is moved to the edge, the impedance characteristics at the slab center correspond well to the total frequency by reflecting the increasing quantity shown in Fig. 12. With the exception ofx/2b = 0.07, which indicates that the excitation point is especially close to the edge, the impedance characteristics are considered to be within 2 dB, which is proportional to the impedance characteristics at the slab center, including the resonance domain. Therefore, when considering the whole slab, a method which adjusts the impedance characteristics obtained in Fig. 20 with the increasing quantity shown in Fig. 12, depending on the position of the excitation point, is appropriate. The validity of the calculation method was examined by estimating the driving-point impedance characteristics of the actual floor slab center point,

Calculation of floor impact sound

281

105 n t :Xl ~ b--0.07 point : X / l b=0.07

105 u

",}/"1"

,, ,~

Impact point :X/A b=0.2

.I,-~"

Impac point :XI 1 b=0.2

v

v

~J ¢O

E

,.:.?,

105

H

H

Impact point :X/A b=0.5

point "XI I b=0.5

Impact point "X/A b=0.818 . . . . . .

10

Fig. 22.

i

.

.

.

.

.

.

.

.

Impact point "X/A b--0.~75 i

100 1000 F r e q u e n c y (Hz) Slab A C e n t e r Point

lo

......

Slab

i6o

......

iiJoo

F r e q u e n c y (Hz) B C e n t e r Point

Results of analysis of slab center impedance according to change of excitation point.

282

She Kimura, Katsuo Inoue

i0-

Slab

10'

A

B

0

>

¢ -i0. >

nter

n X / Z b=0.2

•

-20

$3

C]X / ~, b=O.07

2;0 soo

s

Center

Fig. 23.

-i0

\(~W o x / x b = 0 . s

frequency

"~

-20

(Hz)

Center

frequency

(Hz)

Impedance characteristics of each band for various impact points (slab center).

and then comparing it with the measured results. Two types of slabs were used in the test: (1) RC, 150mm thick and 21.6m 2 large (3.06 × 7.06 m); (2) RC, 150mm thick and 13.5m 2 (3.00 × 4.50m). Figure 24 shows the comparison of the estimated value and the measured value. Good agreement is shown between them, even up to high-frequency regions. 2.6 Effective radiation area of floor slab, S d When complete fixing is assumed at the slab peripheral boundary, and when diffused fiexural vibration is assumed in the slab, the equivalent vibration amplitude is calculated by removing the 1/4 wavelength of the flexural wave in objective frequency from the peripheral boundary (Fig. 25). 107"

1o7

(1)

(2)

1oC%. loct.

® Io6.I~f ./,Jr D~

~ loa

I.

tli

L, \, i

~ ion

~~u loS.

H

N

N

I

-" - - - M e a s u r e d @.----@C a l c u l a t i o n

IO4

10

.....

Fig. 24.

value

,,,i

100

Frequency

(i/!

•

octave-band)

(I/I

value .

octave-band)

, . ,,)el

(Hz)

lk

104 I0

....... ib0 Frequency

....... (Hz)

Comparison of estimated and measured impedance characteristics.

]~

283

Calculation o/floor impact sound Ab

L

{

d X b/4

v

),. b/4

Fig. 25.

Calculation of effective radiation area, Stir.

A vibration test was conducted to verify the validity of this process using actual slabs. The test procedure was as follows. When exciting the center of an RC structured floor slab (3940 x 6250 mm at a thickness of 150 mm, all four sides being supported by girders) the vibration velocity response of 1/4 slab intervals was finely detected. The vibration velocity response was indicated as the relative level of each frequency band based on the velocity level near the excitation point (center of the slab). These results are shown in Fig. 26. The sine-wave-like reduction of the response level is recognized as corresponding to the flexural wave of the frequency when approaching the edge of the slab. The distance equivalent to 1/2 wavelength

~

Exciting-point (center of slab) 2X b

Exciting-point (center of slab) 3! .51,1z

63Hz

30

{0 20 30

Exciting-point

Exciting-point

center of slab)

center of slab)

250Hz

|25Hz

b

dB0 lO

Fig. 26.

Distribution of vibration velocity response.

Sho Kimura, Katsuo lnoue

284

~Partltlon wall

LB

p.

c

o

°1

•

350L

5,000 Fig. 27.

Plan of floor slab to be measured.

of each test frequency from the long side edge in the figure was marked with arrows. The position of the arrows corresponds comparatively well to the measured values. Thus, it is clear that indication of an equivalent amplitude by the method shown in Fig. 25 is basically valid. When the sound receiving room under the test slab is partitioned with wood or a light steel flame instead of beams and concrete walls, the restraining ability of the partition affects the effective radiation area. However, it is very difficult to estimate the restraint. Therefore, an experimental study using an actual partition was carried out. Figure 27 indicates the concrete floor slab used in the test. A wooden and light steel frame partition was provided, connecting two opposite long sides of the slabs, and passing through point B, as shown in Fig. 27. Excitation was applied to the floor slab from the slab edge point A to the slab center point C using an impulse hammer. The exciting force and the vibration velocity response near the excitation points were measured, and then the impulsive impedance was calculated. Figure 28 shows the result. If the restraint of the floor slabs is increased due to wooden and light steel frame partitions, the impulsive impedance should !

A

1

IP------mWithout partition walls

~20 I = L

A-------4Llght steel-frame partition walls • .... • W o o d e n partition walls

~15 ~ . ~

A

~IO >

+ I

•

•

B

~

c

Partition ~Slab center "~O

I

=0

0.05 Fig. 28.

0.1

0.5

1.0

X/A b

Increase of impulsive impedance at edge and influence of partition walls.

Calculation of floor impact sound

x/Xb=O.lO 0.13m away and

from edge of beam 1.28m away from partition

, ~

285

wall

I m~---IWithout partition wall A--.-4&Light wall A--.-4&Light steel-frame s eel-framq partition par O---OWooden partition wall

~

10°

........

u m

vvv --

10a

i

E

H

•

10

. . ,

....

,

.

100 Frequency

,

. . . . . . .

1K

(Hz)

Center

frequency

(Hz)

X / I b= 1.06 1.41m away from edge of beam a n d o n t o p of p a r t i t i o n

10e'

~-

0)

. . . . ,~

°[ dE

-

---

. . . . . . . . . .

~ 10°

' c__

..................

I0

100 Frequency

I~

1631.563 125250500 IK C e n t e r f r e q u e n c y (Hz)

(Hz)

X / A b = 1.81 2 . 4 1 m a w a y f r o m e d g e of b e a m and 1.99m away from partition

0.J

wall

~

10e

105

c~ E

t-I

10

100 Frequency

Fig. 29.

IK (Hz)

16 31.563 125 250500 lk Center frequency (Hz)

Full-time response impedance before and after providing partition walls.

286

Sho Kimura, Katsuo lnoue

increase, especially around B in the figure. However, Fig. 28 indicates that the increase of the impulsive impedance around B is similar to that of the slabs without partitions. Thus, no dynamic restraint can be recognized. Figure 29 shows the comparison of impedance at driving points A, B and C with and without partitions. According to this result, there is a slight increase of impedance at the wooden partition (B) in the primary natural vibration region (31.5 Hz band) of the slab. However, other points did not show any significant differences. In particular, the impedance was almost the same between slabs with and without partitions at the test floor impact sound of 63 Hz or higher. According to these tests, the influence of wooden and light steel frame partitions on the vibration condition of the floor slab could be ignored. Thus, it is appropriate to count the distance up to the partition wall without removing 2b/4for the calculation of an effective radiation area downstairs, as in the case of beam and concrete wall supports.

2.7 Acoustic radiation coefficient of floor slab, k There are various suggestions for acoustic radiation characteristics from a finite length slab. However, in this paper eqn (18), indicated by Lyon, 6'v was used and the acoustic characteristics were verified experimentally:

f=f~ />>f~

k=l~.45/~

(18)

where k is the acoustic radiation coefficient, S is the radiation area (m2), l is the slab circumference (m), and 2c is the wavelength at the coincidence threshold frequency, fc(m). The experimental floor slab shown in Fig. 30 was used, and excitation was

,

,e

Ill

: o 6 o 4 o2 i:l I

! °I

J

l|

Plan

I

3,~

"t

Sectional view

Fig. 30. Outline of slab used for experiment and measuring points.

Calculation of floor impact sound

I

L Exciting

i I

I

point

P

I 4 ..... I

z ....

287

I J

t

', f+-..

,,

! t

! !

!

Fig. 31. Calculation method of radiation area.

applied at points 1-7 using an impulse hammer, and a measured acoustic radiation coefficient was then obtained from the vibration velocity response at the b o t t o m face of the slab under the excitation points, as was the sound pressure response 10 cm under the slab. Calculation values used in eqn (18) are as follows. The circular portion Fc

10t *~-

~

~,

^18

I Center

frequency

(Hz)

16 31 5 63 125 250 500 1000

l,,.or.,io.1 ,e~..:

2-101

Heasured

value

i

~ l A c t u a l slab center point -~v

(3.40X 5.4Ore)

_ 1o[

FC

125[.A center frequency (nz)

eo

8 1~. 31.~.~.~ .z~t~,,,!ooo

0 ~-10

I

-2C

Actual

slab

~I I0 f

FiCcenterfrequency (Hz)

°[8 0

/

center point (3.85X 4.48m)

16 31.s ~ L ~ ° ~ 1 0 0 0 _

_

I ~[Experimental -zu floor

-/4"

slab

_-

:

P'I

' " P'S ~. ." p-~ - - P-7

Fig. 32. Result of analysis of acoustic radiation characteristics.

288

Sho Kimura, Katsuo Inoue

(S = n2~/4), the diameter of which is the slab flexural wavelength of the exciting impact frequency, was used as the acoustic radiation area, S. The circumference of the same circle (n2b) was used as the circumference, l. When the span, a, of the floor slab was shorter than )-b, it was approximated as 2 b = a (Fig. 31). Figure 32 shows the comparison of the above calculated result and the measured values. Acoustic radiation characteristics of other actual slabs, which were also obtained using the same method, are also shown in the same figure. It is clear from the figure that the calculated values and the measured values correspond very well. Based on these results, it can be said that the method of applying eqn (18) to the floor slab acoustic radiation coefficient is highly valid.

2.8 Sound-absorbing power of room below The average living room of a multi-family dwelling has a floor area of 10-30m 2 and, in most cases, is rectangular in shape. These rooms do not present a uniform sound pressure distribution, due to the great influence of modes in low-frequency regions. However, a diffuse sound field is assumed in all frequency regions, in order to calculate the average sound pressure level in the room, and to simplify the calculation as much as possible for practical use. The average sound absorption coefficients were obtained from the measured reverberation time in many regular living rooms with a floor space of 10-30m 2. According to these results, the average sound absorption coefficient is dispersed within a range of around 0.1 to 0.2, and this value does not change much, regardless of the size of the room. There is a tendency for the sound absorbing power to be reduced slightly in the 63 Hz band, although the frequency characteristics are basically flat.

2.9 Correction of sound level meter dynamic characteristics, A C JIS A 1418 stipulates the use of the peak indicated value of the 125 ms time constant (FAST characteristics) for measuring floor impact sound levels, while a series of data analyses shown in this paper are all indicated by the energy integral value per second. Therefore, in order to examine the agreement of these calculated values with the measured FAST peak value, or to obtain the sound insulation grade, L, which is the rating number of the JIS floor impact sound level from the calculated value, the agreement of both values has to be examined and the correction factor must be provided. This correction factor is expected to change, depending on the time characteristics of the sound pressure response. F o r this reason, the floor impact sound of an actual living room was measured, and the difference

Calculationoffloor impactsound

289

between the output value of the time constant circuit and the digital processed value of the response per second was then obtained. According to this result, the difference between the FAST peak value and the energy integral level is 5-10dB on the whole, and there is no extreme difference against the floor structure. There is a tendency for the difference to gradually decrease when the frequency increases; thus it cannot be dealt with as for flat frequency characteristics. Table 2 shows the result of digitally processed half-sine waves by the FFT method as a response wave, and also the difference from the peak value recorded in the level recorder by time weighting of FAST through the band-pass filter as the correction value. This correction should be added to the integral value per second to approximate the output of the time constant circuit FAST peak. TABLE 2

Dynamic Characteristics Correction Value of Sound Level Meter Octave-band center frequency(Hz) Correction value (dB)

63 9.8

125 8.3

250 6'5

500 5.6

lk 4.9

2k 4.9

4k 4.9

3 C A L C U L A T I O N OF FLOOR IMPACT SOUND LEVEL The floor impact sound level inside the sound receiving room can be calculated by eqn (19), by using the results derived in each part of section 2:

[F2ms 4"X L:=lOlogto~-~-bPoCoSeefk~)+ 120+ AC

(19)

where L: is the floor impact sound level per octave band (dB); poCo is the acoustic resistance of air (kg m-2 s-1); Seeris the effective radiation area (m 2) (Fig. 25); A is the sound absorbing power of the sound receiving room (m2); AC is the dynamic characteristics correction value of the sound level meter (dB) (Table 2); Frms is the impulsive force effective value of the heavyweight impact source (N) (Table 1); Zb is the impedance value of each excitation point, which is obtained by adjusting the full-time response standard impedance with the slab peripheral fixing (kg s- 1) (Figs 12 and 20); and k is the acoustic radiation coefficient (eqn (18)). 4 CALCULATED E X A M P L E OF FLOOR IMPACT SOUND LEVEL The floor impact sound level was calculated by considering an actual floor slab and by using eqn (19), and then comparing with the measured value: Fig. 33 shows these results.

290

Sho Kimura, Katsuo Inoue

i

8o 7O ~a0

40

Span ratio: 1.35T % . "

"

g

~

~

~

I~:~,:l . . . . . . . | 1

T

" ' , i "~

•~

"~ ~ "

-kX"+._ + ~4:"--~-~+

~0C~atr p e t

. ",

,:,,

"

~s

............

Llght-weight , ~ [ ~ c o n c r e t e 'value . t-52

t~- :--k~~

-

~o 3

"l-Sectlonalvlew'"• Calculated 0 ~ ~ 125 250 500 l k 2k 4k Center frequency (Hz)

|0

Sectional vlew 63

63

125 250 500

lk

2k

4k

Center frequency (Hz)

8C

A

7O L-~O

6C o t0

ca

40 C a r p e t " ~ Felt -' " Add I t l o n a I qNJ c o n c r e t e p l PC b o a r d

I

o

:-

k

-

Actual ---l-~-~_ *. v a l u e L'50 ~ • Calculated a c J n g ~ "'-.

50

L. ,L & c t u a l

9.n r a t

4C 30 o

2O l0

~ect lonal 63

'vle~

'

lO

125 Z30 500 l k 2k ~k Center frequency ( H z )

63

|000 5ectlona 1 view 125 250 500 l k 2k 4k Center frequency (Hz)

9O

80

•-,, , ,,

\

/ \\

//

70

~=80

60

030

o30 span ra¢io:1.28 ~-~-" ~" A r e a : 8 . 0 0 # "~ "-,] ~ . . . . . ~ . . . . . L-45 ' I I ",T~--T~C"~"-J f'

30

Ethylene thick hott .....

""-.

concrete

" " ~ . . . . . L-35~

Calculated

o

o~

20

2C

v a l u e L'59 63 125 250 500 Ik Center frequency

Fig. 33.

2k

v a l u e L-61 4k (Hz)

Sectional

10

63 125 250 500 ]k Center frequency

view 2k 4k (HZ)

Example of comparing the floor impact sound level.

Calculation of floor impact sound

291

According to these results, the calculated value and the measured value correspond comparatively well, regardless of differences in the floor structure. However, when a hollow slab (such as a void slab) is used for the floor structure, the air in the hollow layer acts as an elastic body and generates secondary sound in middle and high-frequency regions. Thus, the calculated value does not correspond to the measured value at 500 Hz and above. The method for dealing with secondary generated sound is a subject for future study. The figure also indicates the evaluation curve (L curve) of the floor impact sound by the JIS standard, where the correspondence of the L curve rating value is within one rank of the sound insulation grade.

5 CONCLUSION An estimation method for heavyweight floor impact sounds by use of the impedance method was proposed. More than 200 actual slabs, all with differing structures and manufacturing methods, were calculated using this estimation method. When these calculation values were compared with the measured values, approximately 90% of all the values fell within one rank of the sound insulation grade. Thus, the validity of the calculation method is very good, although some bold assumptions and manipulations were used when making calculations in the practical application of the method.

REFERENCES 1. Gerretsen, E., Calculation of airborne and impact sound insulation between dwellings. Appl. Acoust., 19 (1986) 245-64. 2. Kimura, S. & Inoue, K., Floor impact sound and vibration characteristics of the floor slabs. Trans. Archit. Inst. Japan, No. 332 (Oct. 1983), 83-93 (in Japanese). 3. Kimura, S., Inoue, K. & Arai, A., Estimation for driving point impedance characteristics of concrete slab in case of floor impact. J. A rchit., Piann. Environ. Engng (Trans. AIJ), No. 363 (May 1986) 1-8 (in Japanese). 4. Kimura, S., Method for field measurement of floor impact sound level using heavy impact source. Proceedings of the Tenth International Congress on Acoustics, Sydney, 1980. 5. Yasuoka, M., A calculation method for floor impact sound. Archit. Acoust. Noise Control, 6(4) (1977) 1-27 (in Japanese). 6. Lyon, R. H. & Maidanik, G., Power flow between linearly coupled oscillations. J. Acoust. Soc. Am., 34 (1962) 623-39. 7. Maidanik, G., Response of ribbed panels to reverberant acoustic fields. J. Acoust. Soc. Am., 34 (1962) 809-26.

292

Sho Kimura, Katsuo lnoue

APPENDIX: SPECIFICATION OF HEAVYWEIGHT FLOOR I M P A C T S O U N D G E N E R A T O R 1'2 (section 2.2.2, JIS A 1418) (1) (2)

(3)

(4) (5)

(6) (7)

(8)

The h e a v y w e i g h t impact s o u n d g e n e r a t o r shall be provided with one heavy i m p a c t source, which can h a m m e r the floor surface vertically. Except d u r i n g h a m m e r i n g , the h e a v y impact source shall be kept apart f r o m the floor surface a n d shall hit the floor surface o n l y once at one fall. The surface o f the h e a v y impact source to c o m e into c o n t a c t with the floor shall be convex, with its radius o f c u r v a t u r e 0"1 to 0.3 m and the m a x i m u m area 0.015 to 0.025 m 2. W h e n the heavy impact source is r e g a r d e d as a single-mass system, its equivalent mass shall be 7.3 _ 0.4 kg. The d y n a m i c equivalent stiffness, 3 d e t e r m i n e d f r o m the total d u r a t i o n o f impulsive force, on the a s s u m p t i o n t h a t the system is linear, o f the heavy i m p a c t source seen f r o m its surface in c o n t a c t with the floor shall be (1-6 + 0.1) x 105 N m - t a n d the non-linearity 4 within its d e f o r m a t i o n range shall be less t h a n + 20%. The coefficient o f restitution o f the h e a v y i m p a c t source shall be 0-8 + 0-1. The speed o f the heavy i m p a c t source at the m o m e n t o f its collision against the floor surface shall be equivalent to t h a t realized w h e n the source's impact surface comes into c o n t a c t with the floor after its free fall f r o m a height o f 0.9 + 0.1 m above the floor. The h e a v y w e i g h t floor i m p a c t s o u n d g e n e r a t o r shall w o r k stably a n d neither generate a n y mechanical noise, vibration, etc., which can disturb the m e a s u r e m e n t with it n o r affect the floor's vibration characteristics.

1The heavy impact source can be such that it does not always comply strictly with the requirement given in items (4), (5) and (7), as far as the impulsive force-time characteristics obtained from its fall on a smooth and rigid surface with sufficient effectivemass comes within the range shown in Fig. 2. 2 A 5"20-10-4Pr. tire for an automobile, specified in JIS D 4202 'Principal particulars of tire for automobile' approximately satisfies this specification at its pressure (1"5 + 0.1) x 105 Pa. It is possible to adjust the impact time with mass and stiffness, and the impulsive force with mass and height of fall. 3A value of the dynamic equivalent stiffness k (N/m-1), calculated using the following formula: k = n 2 M / T 2 as a single resonance system in which the wave form of impulsive force is assumed to be half-sine and the loss is neglected, from the force's duration T(s) and the equivalent mass M(kg) between the moments when the heavy impact source comes into and out of contact With the floor surface shall be used. The non-linearity shall be expressed with a deviation of the dynamic equivalent stiffness determined for at least four stages of height of fall, that is, 0-03, 0.1, 0.3 and 0.9 m, from its value for the height of 0"9m.