Transient sound from colliding spheres—Normalized results

Journal of Sound and Vibration (1974) 36(4), 541-553

TRANSIENT SOUND FROM COLLIDING SPHERES-NORMALIZED RESULTS .L.L. Koss Department of Mechanical Engineering, Monash University, Clayton, Victoria 3168, Australia

(Received 30 January 1974, and ht revisedform 29 April 1974) Normalized pressure-time and pressure--frequency equations are derived for predicting the sound radiated due to an elastic collision between two spheres. Important dimensionless groups are defined and their significance interpreted with respect to the experimental variables. The experimental results are then given in their normalized form.

1. INTRODUCTION In a previous paper [1 ] it was shown that the sound radiated by two colliding spheres could be modelled by transient dipole sources. The main objective of that article was to compare experimental results with theoretical predictions based upon the use of the dipole model. This paper is concerned with the normalization or non-dimensionalization of the experimental data obtained from elastic collisions. For the sound radiated by elastic collisions important dimensionless pi groups are defined and the experimental results are given in their normalized form.

2. NORMALIZED EQUATIONS 2.1. REVIEW OF ACOUSTICAL MODEL The physical model used to describe the collision process is shown in Figure 1. A full discussion of this model is given in reference [1]. A sphere called the impactor is given an initial velocity Vo and impacts an initially stationary sphere called the impactee. The impactor is decelerated and the impactee is accelerated to some final velocity determined by the momentum and energy equations and the type of force acting between the spheres. It is during the period of acceleration that the spheres radiate sound. Each sphere is considered to be a _~ z~

Microphone

z,.rz - ' - 7 / / u

Z2

9

XI

Figure 1. Co-ordinate system used to define positions in the radiation field. 54!

542

L . L . KOSS

transient dipole source of sound. The sound pressure phn(rl,Ol, t[) generated by a unit impulsive acceleration of one sphere is given by

t; < o, (1) where i = 1 is for the impactee, i = 2 is for the impactor, p is the density of the air, a t is the radius of sphere i, re is the distance from the centre of sphere i to the microphone, c is the sound speed in the air, It is c[ai, O~is the angle between the microphone and the direction of motion for sphere i, t is time, and = 0,

t/= t

re - - a l C

pim(r. O. t[) is obtained by differentiating Kirchhoff's solution. The sound radiated by one sphere, p(r~,Ot, t[), due to an arbitrary acceleration of that sphere, ac~(t), is given by the convolution of equation (1) with the acceleration time history of the sphere: 0o

p(rl, Ol, t[) = ~ pim (re, Oi, t[ -- x) acl(z) dt,

t[ > O,

0

= 0,

t; < 0,

(2)

where z is a dummy integration variable. For elastic collisions aci(t) is given by Hertzian impact theory [2], and p(r.O.t[) is the sound radiated by each sphere shown in Figure 1. The complete sound pressure time history, p(r, O,t), is given by

p(r,O,t)=p(rl,01,t~)+U(t'~-Td)p(r2,02,

t~),

0<90 ~

(3)

where U(ti - Td) is the unit step function, t~ is re-interpreted as t~ = t; - Td, and Td is the difference in time between the arrival of the sound from the impactee and impactor. Td is approximated for the 0 = 0 ~ direction for equal radii spheres by 2"57al + (r 2 + a2) 1/2 -- rl Td --(4) c 2.2. N O R M A L I Z E D P R E S S U R E TIME E Q U A T I O N S F O R ELASTIC C O L L I S I O N S The sound pressure generated by one sphere undergoing an elastic collision is, from reference [l], for t~ < d, p(r,, 0,, tr)' = Pa'a3c~

r-~l [(2r, _ i . at ,

1) [(813b-41'b3)c~

+812b2sinbt;]-

--4basinbt ' - (813 b + 41, b3) cos bt[ + ( 2rl _ 1/[(4b 3 ! , - 8bl~) cos lt[ \ at

-

(Sb/~ + 4b311) sin/, ( d - • 2/1 ]J

J

/

~ .ao.,~cos 0, sin b,[, 2r 2

(5)

543

TRANSIENT SOUND--NORMALIZED RESULTS

and, for t[ > d,

)

p(r,, 0,, t;) = 8(b 4 + 4l~) r~ [\ at

l {[Ob 3 It - 8b 3 it) cos l,(t[ - d) -

- ( 8 b P + 4b 3 it) sin l,(t[ - d)] e -z'cq-d) + [(4b 3 It - 8bP) cos l, t[

_ +,,

+

+,,,,,

e-,,,+ -

+,,

-

(,; -

(6)

where d is the duration of the collision, 3"141593 d

d

and am is the maximum acceleration of the sphere during the collision. The constants am and d can be computed from Hertzian impact theory, given the material properties of the spheres and the impact velocity. Equations (5) and (6) can be normalized or set into non-dimensional form by choosing the following normalizing variables: length--radius of sphere i = a~; time--twice the duration of the collision = d2. Then r* and t* represent the normalized radii and normalized time, respectively; they are given by r! ai

t[ [ t_/.[ g) t* ='~2 \ d , has not been given any specific pi groupin .

(7)

Substituting for rt and t[ in equations (5) and (6) and using equations (7) gives the normalized form of the pressure time history radiated by one sphere. The results of the normalization of equations (5) and (6) are quite easy to visualize as all the terms involving products of the form, e.g., ( 8 1 p b - 4 1 1 b 3) are non-dimensionalized by dividing by (b 4 +41++). The sinusoidal, cosinusoidal, exponential, and cos0t terms are already in non-dimensional form. Thus, just the I # 2 terms must be evaluated openly. Rather than repeat a reproduction of equations (5) and (6) the important non-dimensional ratios, pi groups, will be given as such: rI 7~ 1 ~

--,

at cd2 rc~ = - - , a2

r2

a2

7r 2 ~ - - ~

7~3 ~

at p(r, Ot t[) ~,, = -

-

TC4 ~ - - ,

at

at

Td ~-, = - - r .

,

cd2

--~

(8)

az

Pare at

If the impactor has a different mass than the impactee, then the acceleration of the impactor can be given in terms of the impactee, ac2(t) =----ml acl(t), ]H 2

(9)

544

L.L. KOSS

SOthat another dimensionless group is n8 = m~fin2. Thus, there are eight important dimensionless pi groups. The physical interpretation of groups st, 7r2, rra and rr8 are obvious and deal with lengths and masses. The group n2 is not an independent group if rq,na and the angle 0 are known. The group re6 is a ratio of the acoustic pressure to the force and is in effect the mass of a sphere, pa 3, accelerating at am divided by the area a~: e.g.,

pa 3 am

p(radius,) 3a,. (radius,) 2 ,

a2

(10)

where radius, -- a,. It is interesting to note that this term is independent of the mass ofeither impacting sphere and only depends upon the mass of an equivalent radius sphere. The interpretations of the groupings n.~ and 7rs are not obvious, however; an inspection of equations (5) and (6) indicates that the decay of the transient sound is defined by terms of the following form: exp

(- ca, tilsinCtl. ] a,

(11)

The transient oscillation of a less than critically damped spring mass system is described by the following equation [3]:

X(t)

= e-~'.t sin (~/1 -

~zto.t + ~p),

(12)

where X(t) is displacement, t is time, ~ is the damping ratio, ~o. is the natural frequency of vibration and ~bis a phase angle. If the terms in equations (I 1) and (12) are compared it would seem that c/at is analogous to a damped frequency of vibration and the following relation should apply to equation (12): ~o,, = "~/1 - ~2 o9,,,

(13)

or ~ = 0.707, and thus the analogous damping ratio for the transient sound is 0.707. The natural frequency r 0f the transient sound for sphere i would then be /, to~ = - =

c 1.414-- radians/second, a,

1.414 c c fa, - i = 0"228-- Hertz. 2n al a,

or

(14)

Tile frequency associated with twice the duration of the collision is 1

fY = t--~2'

(15)

so that the groupings n4 and n5 are the ratios of the damped frequency of the transient sound to the frequency associated with twice the duration of collision. As will be shown later, the normalized frequency spectrum peaks atf[f~ = 1, where./is the frequency in Hz, when a,[cd2 equals 0.228, or equivalently whenfl/f~, = I. 2.3. NORMALIZEDPRESSUREFREQUENCYEQUATIONS The Fourier transform of equation (3), the sound pressure time equation, describes the frequency characteristics of the transient sound model. The transform of equation (3) can be taken directly or the transform can be obtained by the product of the transform of the

545

TRANSIENT S O U N D - - - N O R M A L I Z E D RESULTS

impulse pressure solution, equation (1), and the transform of the acceleration time equation. This is valid as convolution in the time domain is equivalent to multiplication in the frequency domain. The Fourier transform of equation (1) can be shown to be pint (r. O. 09) = pa, c [ (a, !, + jr, o9) ((21~-- 09z) _--j209l,) ] cos 0, r~ [ (212 - 092)z + 409212 J

06)

where 09 is angular frequency, j = V " ~ , and the previous definitions apply. The acceleration time function was assumed to be a half cycle of a sinusoid [1] of period d2. The angular frequency of this half sine pulse is equal to b = ~z[d. The Fourier transform ofa halfcycle of a, sinusoid of amplitude a,, and angular frequency b is equal to

arnb e_jwa) ac(09) = b2 _ 092(1 + 9

(17)

The transform of the sound radiated by one sphere is then equal to the product of equations (16) and (17): p(r,, 0,, to) =

a,~b(1 + e -J~) patc [(a~l,+jr~09)((21~-092)-j2091,)] b 2 _ 092 r--f cos 0, [ , (~ ~---~-+-'4-~-~ 2 / ~ ],

(18)

wherep(r,O~,oJ) is the transform of the sound radiated by one sphere. As the sound radiated by the second sphere arrives Td seconds after the sound from the first sphere, the frequency spectrum of the sound from the Second sphere is multiplied by e-J~'ra. Upon expressing the acceleration of the second sphere, ac2(t), as ac2( t ) = _ __ml acl( t ) = - Macl(t )

(19)

1712

then the frequency dependence of the model pressure time history can be shown to be

p(r,O,09)

pcamb (I +e-J~'n)[[ (212-092)-j2091 ] b2 _ m 2 r2r22 , [(212 _092)2 + 409z12] x • [r~alcosOl(all+jr109)-r~Ma2cosO2(a21+jr209)e-J~ra]}.

(20)

Equation (20) is valid for impactor and impactee of equal radii but different masses, so that li = ll = 12 = I and ax = a2. The absolute value squared of p(r,O,09) is obtained by multiplying equation (20) by its complex conjugate. After some algebraic manipulation Ip(r, 0,in)l-" becomes ip(r,O,09)l 2

2(pc)2 n2 2 2 (1 +cos09d) 1 x [r~ a] cos 2 01 (a~ 12 + r ~ 092) + = rlr24---7 d a m (7Z2 _ 092d2)2 9414 + 094 -

-

+ r~ AI 2 a2 cos 202(a~ 12 + r~ o92) - 2r~z r2z al a2 cos Ol cos 02 M [(a I a212 + + rl r2 o92)cos 09Td - (r109a21 - r209al 1) sin 09Td]].

(21)

Equation (21) can be normalized through the use of the following normalizing variables: co'= m x d2,

rl

t

r2

r~ = - - ,

r2 = - - .

a2

a2

(22)

546

L.L. KOSS

co' = co • ,42 has not been given any specific pi grouping. Substituting (22) into equation (21) gives the normalized frequency equation:

n 2 (pca,. d22)2 ( 1 + cos ~1 [ ]P(r'O'co')12-- 2 r P d 4 =2 co'___~ ~ 414d~+co '~ •176 4

[-

+ ri*M cos s 02(12d 2 + r'2s co's) - 2rl s r~s M c o s 01 cos 02 ](12 d 2 + , ,_ +r~rscoZ)c~

,Td ~

,

,Td]]

dslco'(r'x-rz)sinco -~'2]]"

(23)

The important non-dimensional pi groups are, upon assuming a~ = a2, rl

nl=--,

as

r2

~zs=--, as

Td

c d2

=5=1d2 = - ,

ZCT=--

as

d2 ~

ml

ha=--,

1112

=9 =

Ip(r. O, co)l pea,. d~

(24)

If the impactor and impaetee have different radii then Xa = a2/al and ~z4 = cds/al are also needed to describe the normalized frequency spectrum. The non-dimensional groups =1 - =8 are the same as for the time domain. The group ~z9 instead of ~z6 is used for the frequency domain. The term pca"d 2 in =9 can be rearranged as follows:

+o.,,,,,q =

pcamd~ a 3 pa 3 am [ cds \ _ a 2 I\ a+ ] ap

The term cd2/ai can be rearranged as c

cd2

ai

ai

1

n+ or ns,

(26)

d2 where l/d2 can be interpreted as the frequency associated with twice the duration of the collision and is denoted byf.r. Thus, equation (25) can be arranged as follows (upon using n4):

pca,. d 2 = pa3 a2a" (x+) d2.

(27)

Equation (26) is a ratio of the damped frequency of response of the analogous acoustic system to a frequency given by f : and is dimensionless. The normalizing pressure frequency variable is then equal to the normalizing pressure frequency time variable multiplied by this ratio and by d2.

2.4. EXPERIMENTALRESULTSIN NORMALIZEDFORM--PRESSURETIMEEQUATIONS Experimental pressure time histories obtained from collisions between bronze spheres and steel spheres were normalized. They were obtained by the same techniques as discussed in reference [l]. In Figure 2 are shown normalized pressure time histories obtained for equal

TRANSIENT SOUND--NORMALIZED RESULTS

547

mass and equal radii impactor and impactee. The histories are a function of the parameters

cd2 7C 5

~

- -

=

7T 4 .

a2

As the n4 for the four histories have similar values, from 6.33 to 4-90, the normalized curves virtually coalesce over the normalized time interval 0 < t[d2 < 0.4. For till2 > 0.4 the important parameter is nv = Td/dz and for large values of n7 the normalized histories tend to spread out for t[d2 > 0.4. The amplitudes of the respective three peaks have similar values for the different histories. I

I

I

I

,2t I~- 04[--

~

~

~

-04

I

g E t5 -0.8

-12

-14

qlbo I

I

I

I

I

I

0-2

0.4

06

0.8

I0

1.2

14

D[mens[onless time t / d 2

Figure 2. Normalized pressure time histories for impactor and impactee of equal masses and equal radii,

0 = 0~direction, m/s is metres/second and cm is centimeters. ai

az

r

V

Notation

(em)

(cm)

(m)

(m/s)

Material

9 A D x

1-27 1.27 1.27 1.27

1-27 1-27 1.27 1-27

0.45 0.45 0.45 0.45

0.58 1.20 1-60 2.08

Bronze Bronze Bronze Bronze

7q

n2

n3

n4

n5

n~

n~

al (cm)

rn a-~

r2 a--~"

a2 a'-t"

cdz a-'l-

cd2 a--z-

Td ~

tnl m'-~

0 (degrees)

fl (Hz)

f*l (Hz)

1"27 1"27 1"27 1"27

34-8 34"8 34"8 34"8

36'8 36-8 36'8 36-8

1 I 1 I

6'33 5"46 5"15 4"90

6"33 5"46 5-15 4"90

0"361 0-416 0-441 0'465

1 1 1 1

0 0 0 0

4250 4920 5200 5200

6100 6100 6100 6100

For t/d2 < 0.5 the sphere closest to the microphone is being accelerated, and so the sound generated by this sphere during this interval is normalized by a time variable associated with

548

L.L. KOSS

the d u r a t i o n o f acceleration. This n o r m a l i z a t i o n is also true for the s o u n d generated b y the sphere farthest f r o m the m i c r o p h o n e which arrives at Td/d2 normalized time units later. I f Td/d2 < 0.5 then the s o u n d radiated during the acceleration o f both spheres will be sequentially normalized. H o w e v e r , the sound which is due to the e x p o n e n t i a l decay is n o t normalized with a time variable associated with the decay, e.g., c/a~, so that these times associated with different collisions w o u l d not coalesce over the same n o r m a l i z e d time interval. l

i

I

l

I

I

I

I

I

I 14

I 16

I 18

16

12

_o x o. ~

~

X

X

-04

i:5 -08

-02

-16 I 02

I 04

I 06

I 08

I I0

I 12

Dimensionless

time

20

lid z

Figure 3. Normalized pressure time histories for impaetor and impactee of different masses and different radii, 0 = 180~ direction. al

t/2

r

V

Notation

(cm)

(cm)

(m)

(m/s)

Material

9 [] x

2.54 1.27 0.63

0"63 0.63 0.63

0-21 0.21 0.21

0.75 0.75 0-75

Steel Steel Steel

a,

r_!

r..22

a2

cdz

cdz

Td

m__.!

0

ff

fol

Az

(cm)

al

ai

a--~

al

a2

"~z

m2

(degrees)

(Hz)

(Hz)

(Hz)

0.63 0"63 0-63

37-9 35.8 32"8

32.8 32.8 32.8

0-25 0.50 1"0

1"35 2.70 4.52

5"35 5"35 4-52

0.42 0-42 0'50

64 8 1

180 180 180

I0 000 10 000 12 000

3050 6100 12 200

12 200 12 200 12 200

N o r m a l i z e d pressure time histories for i m p a c t o r a n d impactee ofdifferent sizes are s h o w n in F i g u r e 3, the results being for the 0 = 0, = 02 = 180 ~ direction. F o r these experiments the i m p a c t o r was a 1.27 centimeter (cm) d i a m e t e r sphere and the impactee was either a 1-27 cm, 2.54 cm, o r 5.08 cm d i a m e t e r sphere. T h e n o r m a l i z i n g length variable was the radius o f the i m p a c t o r : e.g., a~ = a2 = 0.635 cm. F o r t/d2 < 0.4 the pressure time histories coalesce due to the similar values o f rt5 = cd2/a2. F o r t/d2 > 0.5 the plots tend to diverge a n d the third p e a k for the case zt3 = I, zr4 = ~5, and rcs = 1 is greater t h a n for the o t h e r two cases, a n d is due to the difference in the pi groupings. I f the radius o f the impactee, a~, h a d been used to n o r m a l i z e the

TRANSIENT SOUND--NORMALIZED RESULTS

549

pressure time histories the normalized results for the 0 = 180 ~ direction would not have collapsed onto the same curve for t/d2 < 0-5 due to the very different values of rr4. In the next paragraph the normalized results for the 0 = 0 ~ direction will be given for the different size impactor and impactee. ---r

1

T

T

r

1

1

08

i

06

o Sound from small sphere

i

0 4

I9 9~ Io~Je . sphere

II

[

t

Q.!~ 0 2

",

B. '~ -02

.?~

c ~ -04

i!

-06

\J

-08 _-_._.1

04

.t

"v. L _ _

08

12

. t 16

- _ _ ~ 20 - 24

28

32

Dimensionless time t / d e

Figure 4. Normalized pressure time histories for impactor and impactee of different masses and different radii, 0 = 0 ~ direction. al

az

r

Ip

Notation

(cm)

(cm)

(m)

(m/s)

Material

x 9

2.54 1.27

0.63 0-63

0.21

0-75

0-21

0.75

Steel Steel

a~ (cm)

rl a--s

r2 a~

az a--~

ode a-q-

cdz a--S-

Td ~'-~2

ml m'-~

0 (degrees)

fs (Hz)

f~l (Hz)

f~, (Hz)

0.63 0"63

29"8 31-8

34"8 34"8

0.25 0-50

1-35 2"70

5"35 5"35

!-70 0.85

64 8

0 0

I0 000 10 000

3050 6100

12 100 12 100

The normalized pressure time histories for the 0 = 0 ~ direction are shown in Figure 4; the normalizing length variable used was a2, the same as for the 0 = 180 ~ direction. The histories do not collapse onto the same curve due to the greatly differing values o f rr4 = cd2[a2 and rt7 = Td[d2. It is interesting to point out that as rrs = m~/m2 increases, for spheres o f the same material, the radiation becomes more impulsive in character. That is, the radiation defined by equations (5) and (6) is dominated by the decay terms. The group rr4 can be rearranged as dz[(a,[e) which is a ratio o f a forcing duration to a decay time constant. For the same value of d2 the smaller the value of Tr4the more important the decay terms become; or, the forcing time is small in comparison to the natural response of the system. This implies a shift towards an impulsive acceleration of the sphere. For a true impulsive acceleration, the first compression

550

L.L. KOSS

p e a k w o u l d be located at an infinite distance a l o n g the n o r m a l i z e d time axis. T h e n o r m a l i z i n g time variable for an impulsive acceleration should then be a,/c rather t h a n d2. 2.5. EXPERIMENTALRESULTS IN NORMALIZEDFORM--PRESSURE FREQUENCY EQUATIONS The experimental pressure time histories were digitized a n d t r a n s f o r m e d into the frequency d o m a i n . T h e spectra were normalized as described in section 2.3: the frequency was n o r m a l ized w i t h f l a n d the t r a n s f o r m e d pressure with pcamd 2. N o r m a l i z e d spectra o b t a i n e d from collisions between steel spheres, a n d bronze spheres, are p l o t t e d in Figure 5 as functions o f the p a r a m e t e r s ~1 = rdax a n d zr4 = cd2/a,. F o r f f e f a , the spectra peak to the right o f f / f : = 1, forf:"f,~ the s p e c t r a p e a k at f / f : = 1, a n d for f : > f ~ the spectrum p e a k s to the left-of

E

o E #

o

0 02 04 06 08 I0

12

14 16

18 2 0 2 2 2 4 26

f/ff

Dimensionless frequency

Figure 5. Normalized pressure frequency spectra with zq and rr~ as parameters. r

V

(cm)

(m)

(m/s)

Material

9 ,x [] o O

1.27 1-27 1.27 2-54 1.27

1"27 1-27 1.27 2"54 1.27

0.45 0'45 0.45 0-45 0-45

0-28 1-20 3.05 0-74 2.60

Bronze Bronze Bronze Steel Steel

7TI

~'2

,n'3

~4

7T5

7[7

7l"8

rz at

a_2

cdz

cdz

__Td

m__L

0

(em)

rt "fit

al

at

az

d2

m2

(degrees)

h (Hz)

(Itz)

1-27 1"27 1"27 2"54 1-27

34"8 34"8 34"8 16"7 34"8

36"8 36-8 36-8 18"8 36'8

I I 1 1 1

0 0 0 0 0

3700 4920 5910 2950 7300

6100 6100 6100 3050 6100

al

9 A [] o O

al (cm)

a2

Notation

7.48 5"26 4.50 4-48 3.50

7-48 5.26 4.50 4'48 3-50

0-313 0"417 0.503 0-507 0.654

1 1 1 1 1

fa!

TRANSIENT SOUND----NORMALIZED RESULTS

551

f/ff = 1. As mentioned in section 2.2,fo, (which is 0.228 c/a1) is the natural frequency o f the transient sound. The smaller the value off/f.r at which the sound spectrum is a m a x i m u m the more the transient decay terms play an important role in determining the sound wave profile. The effect o f the variation o f 7zl = rl[a~ on the spectrum is quite obvious as can be seen f r o m Figure 5. I

I

l

I

I

I

I

I

z I0

t 12

T 14

I

I

I

l

I

10-3

10.4 -.

/ ~o-~

0

t t t r 02 04 06 08

t t i i ~I~ T 16 18 20 2 2 2 4 26

Dimensio~ess frequency flft

Figure 6. Normalized pressure frequency spectra for 0 = 0~ and 0 = 180~ for various n groups.

at (cm)

al

a2

r.

V

Notation

(cm)

(cm)

(m)

(m/s)

Material

o zx 9 []

1-27 2.54 1-27 2.54

0.63 0.63 0.63 0.63

0.21 0.21 0-21 0.21

0.75 0.75 0-75 0.75

Steel Steel Steel Steel

~1

~2

7%3

/'g4

KS

;%7

~8

rl a-~

r2 a--7

a, a"]"

cd, a"~"

cdz a'-]-"

Td d2

mi m2

0 (degrees)

.Is (Hz)

f*l (Hz)

fa~ (Hz)

o

0.63

35.8

32-8

0.50

2-70

5-35

0.42

8

180

10 000

6100

12 200

A 9 []

0"63 0"63 0'63

37"9 35"8 37"9

32"8 32"8 32"8

0"25 0"50 0"25

1"35 2"70 1"35

5"35 5"35 5"35

0'42 0"85 1"70

64 8 64

180 0 0

10 000 I0 000 I0 000

3050 6100 3050

12 200 12 200 12 200

Normalized sound frequency spectra for different size impacting spheres are shown in Figure 6. These results were obtained for 1.27 cm diameter steel impactors, impacting 2.54 cm and 5-08 cm diameter impactees. F o r the 0 = 180 ~direction the sound spectra peak atf/fs ~ 1.1 as f~ < f ~ c F o r the 0 = 0 ~ direction the normalized sound spectra have more than one important peak. F o r the 0 = 180 ~ direction the sound pressure is dominated by the sound radiated by the impactor (see Figure 3) and only one spectral peak would be expected. F o r the 0 = 0 ~ direction the sound from the impactee and impactor are important (see Figure 4) and more than one important spectral peak can be anticipated. The first peak for the 0 = 0 ~ direction is associated with the analogous natural frequency o f the larger sphere as can be seen from the ratio offa2[fl. This further indicates that the transient decay is important relative to the sound radiated during the acceleration o f the spheres.

552

L . L . KOSS

3. DISCUSSION The frequency analysis of the experimentally obtained pressure time histories [1] showed that the relationship between the frequency at which the spectrum is a maximum, f~,ak and the radius of the impacting spheres (for equal size impactor and impactee) is fp,ak =

76.1 al

Hz,

(28)

where at is measured in meters. Ifthe speed of sound c = 340 meters/second is substituted into equation (14) the analogous natural frequency of the air system is f.E =

77-5 Ol

Hz,

(29)

where ai is measured in meters. Equation (28) obtained from experiments and equation (29) obtained from theory agree quite well. These results show that the analogous natural response of the radiated sound is important in determining the sound wave profile. The dimensionless pressure term p(r,,Oi,t;)/(pama~)can be considered to be a ratio of an acoustic pressure to an inertial pressure. For constant values of the pi groupings, except for zr, and gs, the peak amplitude of the dimensionless pressure time terms show a constant value (see Figure 2). This, however, is not true for the peak amplitudes of the dimensionless pressure frequency results (see Figure 5). The reason for this is that the term pama,contains am, which normalizes the peak pressure time amplitude. The term pcamd~ contains amd~, where both am and d] normalize the peak pressure frequency amplitude. The pressure frequency normalizing variable,

pca,.d2,=pa'am(a-~y ) ,/2, a~

(30)

can be con.sidered to be the product of a pressure, (pa~a,.[a~)(c[alf.r),and a time, dz. This is equivalent to an impulse of momentum per unit area. Equation (30) describes both the peak acceleration amplitude and how the sphere is accelerated in time. The dimensionless pressure in the frequency domain can be considered to be the ratio of the pressure transform to an impulse of momentum per unit area.

4. CONCLUSIONS The pressure time and pressure frequency equations which describe the transient dipole sound model have been normalized. Nine dimensionless pi groups were identified. The dimensionless pressure in the time domain was shown to be the ratio of the measured pressure to the force required to accelerate a sphere ofmass equal to pa~ at a rate equal to the maximum acceleration am, divided by a~, where p is the density of air and a, is the radius of one of the impacting spheres. The normalized time is given by t/d2where d, is twice the collision duration and t is time. In the frequency domain the normalized pressure is equal to the ratio of the pressure transform to an impulse of momentum per unit area. The normalized frequency is given byf/f: wherefis frequency and f : equals lid2. If the pi groupings for one collision are the same as those for another collision then both of the pressure time histories will coalesce into one curve in dimensionless pressure time domain and the pressure transforms will coalesce into one curve in the dimensionless pressure frequency domain.

TRANSIENTSOUND---NORMALIZEDRESULTS

553

It has been shown that the analogous natural frequency of the collision sound radiation is equal to 0-228 c[ai, where c is the sound speed in air, and corresponds to the peak in the sound frequency spectrum for equal size impacting spheres. The normalized pressure frequency spectrum is a maximum atf/fs equal to unity, whenfs = 0.228 c[ai. The analogous damping ratio of this system has been shown to be equal to 0.707.

REFERENCES 1. L. L. Koss and R. J. ALFREDSON1973 Journal of Sonndand Vibration 27, 59-75. Transient sound radiated by spheres undergoing an elastic collision. 2. W. GOLDSMITH1960 Impact. London: Edward Arnold. See Chapter 4. 3. W. THOMPSON1965 Vibration Theory and Applications. Plainfield, New Jersey. Prentice-Hall. See Chapter 2.