Sonic boom analogues for investigating indoor waves and structural response

(1)

SONIC BOOM ANALOGUES FOR INVESTIGATING INDOOR WAVES AND STRUCTURAL RESPONSE

by Sui Lin

(2)

..

.

r

SONIC B00M ANALOGUES FOR INVESTIGATING

lNDOOR

WAVES AND STRUCTURAL RESPONSE

by

Sui Lin

Manuscript received September 1970

(3)

t

ACKNOWLEDGEMENT

The author wishes to express his thanks to Dr. G. N. Patterson for the opportunity to conduct this work at the Institute for Aerospace Studies. I also wish to thank Dr. I. I.Glass for his continued interest and encouragement throughout the course of the project and Dr. H. S. Ribner for many helpful discussions.

Special thanks are due to

Mr.

Albert Perrin for design and construction of the N wave generators and

Mr.

Fred Y. Z. Lee for advice in choosing the

electronic elements and building the e~ectrical analogue.

The help received from

Mr.

W. O. Graf in conducting the computation of experimental data and from

Mr.

D. Appel for reading the manuscript are very much appreciated.

The financial assistance received from the National Research Council of Canada, the Transportation Development Agency and the Civil Aviation Branch of the Department of Transport, and Air Canada is gratefully acknowledged.

(4)

•

,"'

SUMMARY

Experimental results indicate that the maximum amplitude of the indoor pressure wave induced by a sonic boom for the case of a partly.open window is larger than the maximum amplitude of the incident sonic boom. In such a case, the two undesirable effects of the sonic boom are the annoyance it causes people and the effect it has upon structural members are larger indoors than outdoors. The effects of window size, room dimensions, the dimensions and the properties of structural members and the shape of the sonic boom, which influence the

indoor acvustical pressure and the structural dynamic response, are investigated by using an electrical analogue.

The method of design for the electrical analogue is described. The good agreement between the results from the electrical analogue and those of Vaidya (5) shows that the electrical analogue is a suitable device for

investi-gating the room response to sonic booms •

(5)

1. 2.

3.

4.

TABLE OF CONTENTS Notation INTRODUCTION THEORETICAL. BACKGROUND ELECTRICAL ANALOGUE EXPERIMENTAL RESULTS

4.1 Determination of impedance of N-wave generator 1 4.2 Comparison of resu1ts with those of Vaidya

4.3 Effects of dimension1ess paramete~s

CONCLUSION Appendix 1 References Figures 1 to 32 1 2 5

7

8

11 12

(6)

A a

c

c d E F f n I k L L e M m p q R R c S n T T r t t r

v

NOTATION window. area

sound speed in air

capaci tance

damping coefficient; dimensionless parameter defined in Eq.(14)

thickness of glass electrical voltage

force; fundament al dimension of force natural frequency

electrical current spring constant

electrical inductance; fundamental dimension of length effective length defined in Eq. (4)

inertance

inertance of air with vibration of the window 2 (Fig.5) mass

pressure

maximum overpressure of sonic booms electrical charge

resistance

critical resistance switch, n

=

1 to

6

dimensionless time defined in Eq. (14); fundament al dimension of time

dimensionless rise time defined in Eq. (14) time

rise time of sonic booms room volume

(7)

x

z

p T Subscript 1 2 e i st g m volume displacement

linear displacement; dimensionless parameter defined in Eq.

(

1 4)

electrical impedance

critical damping ratio defined in Eq.

(14)

densi ty of air

total duration of sonie booms period ratio defined in Eq.

(14)

window 1; acoustical system

window 2; meehanieal vibrating system eleetrieal system,

gener al subscript, it may be equal to 1 or 2 statie

N-wave generator electrical model

(8)

1. INTRODUCTION

A supersonic aircraft in flight produces a pressure disturbance in its

surroundings, Near the airplane the pressure signature is very complex. At a sufficiently large distance, the characteristic N-wave pattern is built up from the bow and tail shock waves as shown in Fig. 1. The pressure disturbance pro-duced by the supersonic aircraft on the ground is commonly known as a sonic boom.

If this wave was~ sweeping past an observer on the ground, the ear would

respond as shown schematically at the bottom of Fig. 1. The ear is sensitive only

to sudden changes in pressure, and would respond to the steep part of the wave and not to the part which is changing slowly. If the time interval between the two compressions was less than about 0.1 sec. the two booms would not be separately detected, and only one explosive sound would be heard (1).

Typical pressure-time histories of sonic booms measured at ground level are shown in Fig. 2 for small, medium and large aircraft flying at their cruising

altitudes (2). These signals are seen to have the shape of a capital letter N.

Therefore, sonic booms are sometimes termed N-waves.

Figure

3

shows the experimental results (3,4) ~ which were obtained from aircraft flyovers of a specially constructed room-sized cubicle having a window. Tests were conducted with the window closed and also partly opened. The outside

pressure trace is indicated at the top. The inside pressure trace with the

window closed is given in the_'Jb.iddle. The bottom trace represents the condition

for the window partly opened in such a way as to create a Helmholtz resonator.

It can be seen that the pressure fluctuation for the resonator case can be higher

in amplitude than the exciting pressure. It has the appearance of a damped sine

wave, and persists for a longer period of time than the initial excitation wave. Vaidya studied the transmission of sonic boom signals into rooms through open windows

(5).

Rigure

4

shows his experimental results compared with his

theoretical calculations. From this figure, we can see that the result by using the resonator analogy is very close to the experimental data. In the following the same consideration will be taken, that the room which we will investigate will act as a Helmholtz resonator.

In several research papers, for example the paper by Crocker and Hudson

(6),

a structural member was considered to be represented by a mass-spring-damper system. Lowery and Andrews (7) used the mass-spring-damper system with two de-grees of freedom to represent the vibrating system of a ceiling and windows

ex-posed to sonic boom. Only one particular case with an N-wave as the exciting

force was calculated by the authors. Instead of an N-wave they used one cycle of a sine wave as the exciting force and neglected the damping effect. The dy-namic response of different structural configurations under certain conditions were then also numeri~ally calculated.

In this report, a structural member is also considered to be represented by the mass-spring-damper system. This system is combined with an acoustical system. From this combined system, both the indoor acoustical wave and the structural dynamic response induced by sonic booms are studied. The coupling between these two dynamic systems is a~so investigated.

(9)

2. THEORETlCAL BACKGROUND

We consider a room in a large building. lts front wall with an open

window is exposed to sonic booms. On the other side of the room, there is a closed

window (or door) as shown in Fig.

5.

To simplify the problem, we assume that the

rear closed window is the only struc~ural member which will be excited by the

sontc boom. We assume also that the low frequency components of the sonic boom,

which contain most of the acoustical energy, dominate the indorr acoustical wave

and the structural dynamic response, and that their wave lengths are much larger than the dimension of the room. With these assumptions, the room can then be

considered as a Helmholtz resonator.

The mechanical vibrating me mb er , the rear closed window, can be repre-sented by a mass-spring-damper system. The sketch on the right side of Fig.

5

represents the model of the problem, where MI and M

2a are the inertances of the acoustical system, m is the mass of the rear window, k

2 its spring constant and c its damping coefffcent.

2

A

simple acoustical system and a simple mechanical vibrating system can be described by the following differential equations

acoustical

M

mechanical

m

where the symbols are

M:

R:

C:

X:

p: dX +

1:

X

=

p(t) dt

C

d2 dx

2

+ C + kx

=

F(t) dt2 dt inertance [FT2/L

5

J m: resistance[FT/L

5

J c: capacitance[L

5

/FJ k: volume displacement x: [L3 J 2 pressure[F /L J F:

(1)

(2) 2 mass[FT /LJ damping coefficient[FT/LJ spring constant[F/LJ displacement[LJ force[F J

The letters F, Land T in parentheses represent the fundamental dimensions, force, length and time, respectively. These two differential equations are of the same type with different dimensions for the corresponding terms. To describe the

combined system in Fig.

5,

we have to transform the mechanical system into the

acoustical system, or the acoustical system into the mechanical system in order to get the same dimensions. The following transformation relations can be found

[10.,

lIJ:

X

Ax

(10)

~

--- - - - -- ,

-where A is the cross-sectional area of the window 2.in Figure

5.

By using the above relations, the mechanical vibrating system can be transformed into the acoustical system. We have then the combined acoustical system as shown in Fig.

6,

when the symbols are

M_2a: M 2: Cl: C 2:

acoustical inertance of air through the open window 1

acoustical inertance of air with vibration of the window 2 acoustical inertance equivalent of the mass of the window 2 acoustical capacitance of the room

acoustical capacitance equivalent of the mechanical spring constant k

2

acoustical resistance of the open window 1

acoustical resistance equivalent of the mechanical damping coefficient c

2

For the calculation of ~ and M

2a, we consider a Helmholtz resonator. When an

acoustical wave comes into the resonator, there is a certain amount of air moving

as a piston forward and backward. lts effective length is chosen to be [llJ

L

=

0.8

JÄ (4) e

for the case of the open window. For the case of the closed window, the effective

length is assumed equal to half of this. The acoustical capacitance of the room

is then determined by [llJ

(4a)

where V is the volume of the room, p the density and a the sound speed in air. Comparing the following differential equations of an acoustical system and an electrical circuit

acoustical

_d~

_dX 1-M dt2 + R - +

-x

p(t) (5) dt C electrical d2 _dq₊1 ₍₆₎ L - q -,+ R q = E(t) ct.t2 e dt C _e

we find that there is another analogy to the acoustical system. The corresponding

elements in these two systems are

(11)

M:

inertance

L:

inductance

R: acoustical resistance R : electrical resistance e

C: acoustical capacitance C : electrical capaci tance e

X: volume displacement q: electrical charge

dX/dt: volume current dq/dt:= 1: electrical current

p: pressure

E:

voltage

Using this analogy, we can obtain an electrical analogue of the acoustical system as shown in Fig.

7,

where M

2 is neglected, as it.is very ~mall compared to M2• From this circuit, the diffe~ential equations, WhlCh descrlbe the problem, can be easily obtained as follows

and

with the initial

d2Xl . dX_l Ml dt2 + Rl dt d2X 2 M2 dt2 conditions Xl (t X 2 (t dX l (t dX 2 (t for t = 0) = = 0) O)/dt = O)/dt p(t)

o

0 are: 0 0 = 0 0

Suppose the acoustical system is subjected to an N-wave with a finite rise and decay (Fig.8). The pressure history of such an idealized N wave is

p(t) = Po t/t for 0< t < t , r r p(t) = Po T-2t/T-2t for t < t < T-t r' r r -

-p(t) = Po t-T/t for _r T-t < t < T, r - -p(t) = 0 f or - 00 < t < 0 and T <t< 00 , (8) (9) (10) ( 11) (12) time ( 13)

where p is the maximum overpressure, t the rise time and T the total duration

o r

of the N-wave. In this sytem of equations, there are 10 independent variables. To reduce the number of variables, the following dimensionless parameters are introduced:

(12)

X.

=

J. e

=

X.

J. Po C. J. -C 2

C '

₁

X.

J.

_n.

=

_X

,

_J.

=

s+i for i

T

=

1 T M.C. 21T J. J. 1, 2 f

=

.T, nJ.

S.

J. T r C. R. R. J. J. J.

=

_M.

_"2

=

_R J. ei (14 ) t

/T

r where X

st is the statie volume disp1aeement, fn the natural frequeney and Re the eritieal resistanee. The ealeulation ~nd physieal representation of these para-meters are deseribed in Appendix 1. Equations (7) to (13) ean then be written

as follows: 2

(~f

:?

+

(;~

)2

d x2 1TS₂ _dx~ 2 + (l+e) x 2-xl 0 dT2 .Q2 - - + dT xl (T = 0) 0 x 2 (T 0) 0 dx l (T = O)/dT 0 dx 2 (T O)/dt 0

:cl!2.

= T _{for O<T<T,} Po Tr - - r

:cl!2.

= 1-2T for T < T < l-T , Po 1-2T r r - - r

~=

T-l for l-T < T < 1, Po T r r

-~=O

for - 00 < T < 0 and 1 < T < 00 •

Po

The number of independent variables is now redueed to 7. 30 ELECTRICAL ANALOOUE (8a) (9a) (lOa) (Ua) (12~) (13a)

To solve the system of equations (7a) to (13a), an eleetrieal analogue method is used. The main advantages of using an eleetriea1 analogue to simulate an indoor aeoustiea1 wave and the struetural dynamie response indueed by a son~e

boom are as follows:

1) The eleetrieal model is easy to set up,

2) The variables being investigated ean be easily varied over a wide range,

(13)

3) The space used for the electrical analogue equipment is very small,

4) The voltage and the current of the electrical circuit can be easily measured with simple equipment,

5) The time per test is extremely short, 6) The cost per test is very low.

To build the electrical model, the ranges of the independent variables should be determined first. In this report, the basic data for design of the electrical model are as follows:

volume of the room: area of the windows: thickness of glass:

spring constant of glass(7): sound speed of air (at 80oF): density of air (at atmosphere condition) duration of N-wave:

v

= 12' X 12' x 10' = 1440 ft 3 , A

4'

X

6'

d :;::

1/8

in, k 2 :;:: 21000 lb/ft, a = 1140 f t l sec,

-3

2

1

4

p

=

2.32 x 10 lb-sec f t , T

=

0.1 to 0.4 sec.

By varying the values of these parameters above and below the basic values, the following dimensionless variables cover the ranges

n.

₌

f .T 0.3 to 6,

1 nl

L R./R . :;:: 0.018 to 2 (for i

1 1 Cl 1), 0.0005 to 0.3 (for i

=

2),

C =

c2/cl

= 0.0006 to 2.9.

The electrical model was designed as shown in Fig. 9, where the S , for n

=

1 to 6, indicate the switches at different positions. The points A, B Rnd C are used for measurement. Measuring the voltages at the points A, Band C, we obtain the pressure-time history of the incident N wave signal beside window 1, the acoustical pressure in the room and the compressive stress on the glass of window 2, respectively. By putting switch 8

4 in the cut off position, the mech-anical vibrating system is separated from the acoustical system. Only the in-door acoustical wave is studied in this case. Putting switches Sl and S2 in zero positions and switch S~ in the cut off position, the acoustical system is elimi-nated from this circuit a~d only the mechanical vibrating system of the rear window is studied.

The circle on the left side of Fig. 9 represents an N-wave generator. Two sets of equipment were used to produce N-waves. Set 1 modified the output of a WaveTek Model 116 function generator as shown in Fig.10. The rise time, amplitude and total duration of the N-wave produced by this set can be changed. However, the ratio o~ the rise time to total duration in the high frequency range ~s relatively large. To produce a constant short rise time, a special N-wave

(14)

generator, Set 2, was designed and built as shown in Fig. 11. The rise time is

kept constant at 1 ~sec. This wave generator can also produce the peaked N-wave

shown in Fig. 12.

Figure 33a shows the experimental equipment of set 1, where a is the

WaveTek Model 116 function generator, b the N wave control box, c the electrical

model and d the oscilloscope. Figure 33b shows the experimental equipment of

set 2, where a is N wave generator 2, b the extended duration capacitor box, c

the electrical model, d and e the power supplies, f the amplifier and g the

oscilloscope.

4

•

EXPERIMENTAL RESULTS

4.1 Determination of I~edance of N-Wave Generator 1

Connecting N-wave generator 1 to the electrical model, the electrical

circuit can be represented as follows

where the symbol Z is the impedance. The subscripts g and m indicate N-wave

generator 1 and the electrical model, respectively. The total resistance R,

inductance 1 and capacitance C of this system can be calculated by the following

equations: R ==

R

+

R ,

( 15) g m

1

==

1

+

1 ,

(16 ) g m 1 1 _{+ l -} ₍₁₇₎ C C C g m

Switching the sine wave output of the function generator 116 through the

acous-tical system of the electrical model and varying the values of R ,1 and C ,

the frequency w of the acoustical wave of maximum amplitude canmtemmeasurWd.

The relation be~ween this frequency and the values of the electrical elements

is given by

w ==

m (18)

Theoretically, we can determine the three unknown values of R , 1 and C by

g g g

using three results of w measured

1 and C. In order to ~educe the

m m

with three sets of the known values of

R ,

error of the measurement,

9

values of w m

m

were measured. The results are:

R

g(average)

11 /1 1

_g _m <

4% ,

45 ohms,

(15)

1 C g / 1 C m

I

< 0.1%

These results show that Land C can be neglected.

g g

The N-wave generator 2 did not have the ability to produce a sine wave

hence the impedance of this generator wasscrrm determined. By estimation, the

im-pedance of N-wave generator 2 would be of the same order as that of N-wave generator 1. The resistance of this generator is assumed to be equal to 45 ohms for any

calculation •

4.2 Cornwarison of Results with Those of Vaidya

Vaidya (5) used a room with inner dimensions 11 ft height, 14.25-ft width and 15.50 ft depth to study the transmission of sonic boom signals into rooms

through open windows. The room had two lightweight doors. The wallof the room

was 4.5" thick and made of brick. The window sizes used for the tests were

4' x 3', l' X 3' and l' x 1'. The N waves were simulated with explosive charges. In his paper, the pressure signatures inside and outside the test room with N

waves of duration T

=

200, 100 and 40 msec are reported. The values of

n

for

the three windows wi th T= 100 msec can be calcula ted as follows: 1

0.412

However, the values of the resistance for these tests were unknown. To compare

the experimental results from the electrical model with those of Vaidya, the same

values of

n

_l were used. For the resistance, on~y the two lowest values of the

electrical model were taken. The experimental data used are: M

l

=

1.0 mH, Cl = 8.2 nF,

for the window 4'x3' for the window l'x3' for the window l'xl'

T

=

13.8

Il s ec _{9.7 Il s ec} 7.4 Il s ec.

Rl

=

45 ohms 45 ohms 96 ohms

sl 0.0642 0.0642 0.137

The incident waves and the acoustical pressure signatures measured from the elec-trical model are shown in Fig. 13, 14 and 15 for window sizes, 4' x 3' ,1' X 3' and l' x 1', respectively. Figure 16 shows these results compared to the incident waves and the pressure signatures measured at the middle of the room by Vaidya.

Even though the critical damping ratios Sl were chosen arbitrarily in the

electri-cal model and the incident waves were not exactly the same, the agreement between the results from the electrical model and those of Vaidya is very good.

4.3 Effects of Dimensionless Parameters

a) Effect of the rise time ratio TI

=

t~

To study the effect of the rise time ratio T , one cycle of a

triangu-r

lar wave (T

=

1/4) and an N wave with t

=

111sec were used. The maximum

(16)

amplitude of these indicent waves was kept constant, equal to 0.2 volt. These inci-dent waves with a duration of 15 ~sec are shown in Fig. 17. Figure 18 and 19 show the acoustical and mechanical vibration waves induced by these indicent waves. The maximum amplitudes of the acoustical wave in Fig. l8b and the mechanical vibration

wave in Fig. 19b induced by the triangular wave are larger than the waves in Fig. 18a and 19a induced by the N-wave. Extending .the tot al duration of the incident

waves from 15 ~sec to 25 ~sec (Fig. 20), the maximum amplitude of the acoustical

wave induced by the triangular wave (Fig. 21b) is still larger than th at induced by the N-wave (Fig. 21a). However, the maximum amplitude of the mechanical

vibra-tion wave induced by the N-wave (Fig. 22a) is in this case much larger than that

induced by the triangular wave (Fig. 22b). These results show that the maximum amplitude of waves induced by N-waves with small rise time ratio is not always larger than that induced by N-waves with large rise time ratio.

b) Effect of the critical damping ratio Sl

=

R./R .

--lL:::cl-Figures 23 and 24 show the acoustical and mechanical waves induced by the triangular wave shown in Fig. 17b. The acoustical waves in Fig. 23a and 23b were measured with Sl

=

0.0588 for Rl

=

45 ohms and sl

=

0.189 for R

l

=

145 ohms, respectively. For both cases, the parameter Ol

=

0.915. The mechanical waves

in Fig. 24a and 24b were measured with S2

=

0.0045 for R

2

=

45 ohms and s2

=

0.0245 for R

2

=

245 ohms, respectively. The parameter O2 was equal to' ~.~92 for these

cases. From these figures, we can see that the amplitude of the induced waves decreases as the resistance increases. However, the acoustical wave is much more

sensitive to change in resistance than the mechanical wave.

c) Effect of the capacitance ratio c

The dimensionless parameter c contains the capacitance Cl of the

acou-stical system and the capacitance C

2 of the mechanical system. It is the only

parameter which controls the direct coupling between these two systems. By

changing the value of c by means of changing the value of C., the other patameters

1

O. and S. are also changed, as they contain the factor C .• To study the effect of

1 1 1

c, the parameters O. and S. have to be kept constant by adjusting the values of

1 1

M. or R .• The followi~g values were used for studying the effect of c:

]. 1 = 2.0 mH 0.5 mH 195 ohms 45 ohms 1.68nF 6.8 nF = 10 mH 40 mH = 51 ohms 200 ohms 0.4 nF 0.1 nF c

=

0.2352 c

=

0.0147

Figures 26a and 26b show the acoustical waves induced by the incidertt I~-wave shown in Fig.25 for c

=

0.0147 and 0.2352, respectively. Figures 27a and 27b show the mechanical vibration waves induced by the same incident waves for

(17)

c

=

0.0147

and

0.2352,

respectively. From these figures, we can see that the amplitudes of the acoustical waves change only a little, while the mechanical waves change significantly.

d) Effect of the Period Ratio D.

=

f . T

---l----nl---The ratio of the maximum amplitude of the vibration wave induced by the sonic boom tothe maximum amplitude of the incident N-wave,

PmlP ,

is chosen as a characteristic parameter for comparing different vibration wave~. In the follow-ing, this ratio will be called the amplitude ratio. Figure

28

shows the amplitude ratio of the acoustical waves as a function of Dl with ~l as parameter for

T

=

0.15.

~he value of the amplitude ratio increases rapidly in the range of

r

the period ratio D

<

1. The largest value of pip is located at a point for which D ~ 1. For large values of D, the amplQtWdePratio approaches a limit.

Figure

29

shows the amplitude ratio of the acoustical waves with and without coupling with the mechanical vibrating system. Curve (a) is the same as the curve with ~l

=

0.141

in Fig.

28

without coupling. The curves band c indicate the cases with coupling. The values us~d for the measurement are:

0.14,

~2 =

0.0225,

T =

0.15,

r

c

=

1/39

for curve band

5/39

for curve c.

Figure

30

shows the amplitude ratio of the mechanical system with (curves band c) and without (curve a) coupling with the acoustical system. The values used for the measurement are:

~2

=

0.225,

T =

0.15,

r

c

5/39

for curve band

5/3.9

for curve c.

From Figures

29

and

30

we can see that the amplitude ratio for the case with coupling can be larger or smaller than that for the case without coupling. For further information about the coupling effect, systematic measurements are needed.

5.

CONCLUSIONS

The good agreement between the results of acoustical response from the electrical analogue and those of Vaidya shows that the electrical analogue is a suitable device for investigating the room response to sonic booms. Based on thè ,.asslJ1Il.Ptidm tMiat the2lbw-Jitèquetlcy componenpsuGf the.;solÏlic Jboom.dominate the indoor acoustical wave, this analogue method is expected to be more accurate for booms of larger duration or for rooms with smaller dimensions.

For predicting subjective response, the electrical analogue needs further investigation.

(18)

1. Parker, Miss M. A. 2. Maglieri, D. J. 3. Hubbard, H. H. Mayes, W. H. 4. Hubbard~ H. H. Maglieri, D. J. Mayes~ W. H. 5. Vaidya, P. G. 6. Crocker,

M.

J. Hudson, R. R. 7. Lowery, R. L. Andrews, D. K. 8. Cheng, D. H. Benveniste, J. E. 9. Cheng, D. H. Benveniste, J. E. 10. Sutherland, R. L. 11. KinsIer, L. E. Frey, A. R. REFERENCES

The Sonic Boom Problem. Aircraft Engineering, August 1968, p.30.

Sonic Boom Flight Research---Some Effects of Airplane Opera ti ons and the Atmosphere on Sonic Boom. NASA AP-147 (Ed. A. R. Seebass), 1967.

Sonic Boom Effects on People and Structures. NASA AP-147 (Ed. A. R. Seebass), 1967.

Results of Recent NASA Research Pertinent to Aircraft Noise and Sonic-Boom Alleviation. A general lecture at the Sixth Congress of ICAS, 1968.

The Transmission of Sonic Boom Signals into Rooms Through Open Windows, Part 3: Experimental Work and General Discussion.Prepared for NASA CR Report. Structural Response to Sonic Booms. J. Sound Vib.

9 (3), p.454, 1969.

Acoustical and Vibrational Studies Relating to an Occurrence of Sonic Boom Induced Daroage to a Window Glass in a Store Front. NASA CR-66170, 1966.

Dynamic Response of Structural Elements Exposed to

Sonic Booms. NASA CR-1281, 1969.

Sonic Boom Effects on Structures~--A Simplified Approach. Trans. N.Y. Acad. Sci. Ser.II, 30, p.457,

1968.

Engineering Systems Analysis. Addison-Wesley Publishing Company, 1958.

(19)

APPENDIX 1: THE CALCTJLATION ~"'D PHYSleAL REPRESENTATION OF THE DlMENSIONLESS

PARAMETERS

In the follow~ng, the dimensionless parameters are indieated in parentheses in order to distinguish them from others. For example, (x ) represents the

dimen-sionless parameter in Eq. (14), and x

2 represents the posi€ion eoordinate in Fig.5o By using Eqs.

(3),

(4) and (4a), the dimensionless parameters in Eq. (14) cau be described as follows:

(xl) Xl _.-poel X (x 2) 2 =

P

_o

c

₂ (U 1) 1 :;-~Cl (Q2) .. _M1. 2C2 R

(

~1) ,- 1 2 R 2 (~2)

_.

₂ C 2 Cc) = - = el (T_r )

=

t

/T,

r CT) t/T

.

Xl _{dynamic volume displacement}

=

Xs+_l statie A 2x2 x2

=

7

:= poA2 2

_-ç

Po

K2

T 1 = 27T _pAlLel A2 1 T f n2T, 27T

--el Rl

_v

-- :::

2 ~-

2 _pa C 2 2 c 2 A2

-

=

y{-

_2A2 2 K 2 2 A2 A2 V ₂

- / - =

_K 2 pa 2 V volume displacement

,

x₂ := - - = X_s+₂ V

2

_pa o.8p

JÄ

_l m 2 A2 ₂ 2 pa K 2 dynamic statie T

--27T d i sp lac emer:~ di splae ement T 27T

,

c2 1

=

damping coeffieient 2 m 2K2 critical damping ,

(20)

....

FLiGHT PATH

II

P

I

X

__ "'\:

"~

NEAR FIELD

GROUND LEVEL

l1p I

~

PRESSURE DlsrRIBUTION

Î

~

FAR FIELD

EAR

R~roNSE~~~~~~~~~~

· ~B_O_~_S~h_~_r_d~._

FIG. I

SHOCK

~TTERN

a

GROUND PRESSURE DISTRIBUTION FOR AN

AIRCRAFT

FLYING AT

SUPERSONIC SPEED

( Ref. I )

(21)

F-I04

8-58

PEAKED

~

_-~

NORMAL

~

ROUNDED

X8-70

L

...

....

/

FIG.

2 VARIATION

OF

MEASURED SONIC

BOOM

PRESSURE

SIGNATURES AT

GROUND

LEVEL

FOR

SMA~.L t

MEDIUM

er

LARGE

AIRCRAFT

IN

STEADY

LEVEL FUGHT

(22)

...

l::t.

P

pst

~

~O.Jsec.

~

Internol ---.,...

measure

EXTERNAL

--.L

INTERNAL - WINDOW OPEN

FIG. 3

INTERNAL ROOM

PRESSURE

TIME

HISTORIES

DUE TO

SONIC

BOOMS

FOR

BOTH

WINDOW-CLOSED

a

WINDOW-OPEN

CONDITIONS

(23)

WINDOW SIZE 3

1

x

4

1

_{- - - OBSERVED RESPONSE}

- - - -RESPONSE CALCULATED

BY APPROXI MATE

NORMAL MODE FORMULAE

... RESPONSE CALCULATED

BY USING THE

RESONATOR

ANALOGY

WINDOW SIZE

3

1

X

1

WINDOW SIZE 1

1 X

1

FIG. 4

A

COMPARISON

BETWEEN THE

EXPERIMENTAL

a

THEORETICAL RESULTS

FROM REF. 5

(24)

2

"7 > > > 7 7 7 7 7 7

t

àp

I

X;mr

~-

I ...•

-I

I

X4

L __

~

M,

~_J

FIG.

5:

Physical model of the sonic boom problem

'\.

Mechanical

System

Acoustical

System

(25)

FIG.

7:

FIG.

6:

Combined acoustical system

Rf

dJC, "\

- dt

\

Electrical analoguel of the combined acoustical system shown in Fig.

6

(~a is neglected)

ptt)

~---~~---,---~t

t----~-J

~---~-T---~

FIG.

8:

Normalized N wave

(26)

\.

o

mH

o. f

I

~

I

o.S

/

rnm'. LO '2.0 '2,s"

M,

o

ohm

St

too

t50

'200

00 Rf

~l

vI

~l

Ni

~

o

rrI

H

I

:itj

S4 2.~

o

nf

/

.

0.0'24 Sr:

--f

I

5".0

I

0.1

--I

fO.o

~j

0-4

--'11,."

- f

-'1-0.0

t.~ ~' 40.0 . <D.4

--f

CUT OFF' _LO

•

• ~

Mz

C2.

~l ~l

_ql

~l ~l ~.

ro T t--

_T

vT

(J\ T

"

- <:-& C'O ro ('f\

Cf

::l \J

FIG.

9:

Electrical model

o

ohm

I

51 - I

_fOo

I

tSo

'200 '2.5"0 00

• 1<'2.

(27)

J L

5<f,uc:u"~

wa""

(r~Qr of

Wavc\ek

14~ )

J L

front

r4!!ar

of Wave.Tek

1\(0 .

~,uFI

(outward

CDnt'l4!ctor ') ph~

selector

~

I

U J , ~

oscillator

,K

,--

_I

470 •

fund: ion

vC<:a

Tris,er

~--1--IJ

phase..

FIG. 10: N wave generator 1

'-I

model

...IL

I

osci

lloscope..

i

o

~IOI<

2.'2K

I

~

'27

I

5t

SL

10 vc~ 'NQve"'~1<. f1iD ~v

I

---

T

~K 2'" 4S1S'!

~UJT

To -t'S~nc.. OSc:.illo~ Qnd Trij' (in)

Wave..Te.k 11"

(28)

\. Initia) condiliorr +21Z.; v Inte~rotor 10K pu be ~,..er(ltor (Re.petiti\l~ wave) (automc:l'tie.. ) /OIJUF'

I

100 /OO~ -I " ... ,.. , "

"OH

f"

T

I

_{':' "'}_.₇ K 680 Q'l 3.~ \<. 2.7 K Exte..rnal outfut .A\""\~\ifiers c:.arQc.itor In r',c.ofarads otheI"IoI;S" Marked

~

5'c1C> 100 K ' ; ' \ow.r \ev~l Rs-base cdj ... t ... t S;;;é!~""" li ... set { io -biCl1!o

(-,.,.

... )

trian~u\ar pip -share"" 'Diff~ré!.l\tiato,.. lo",~r tevc.l det~t.,...

'SGh",.td Tri~,~r reut \ +3." V pip relative heijht odjust '-80 74 ~ 10 K ~ K 2.2~ 10 K~ K Al' np" transistors '2N 41'2.4 ~_ Mc.724P

ELECTRONlc:,.

0.02 )AF 10K pips out SHOGI(. 'WAVE

~I.g

l;:,.ternal trjcw~1" SIMULATOR FIG. 11: N wave generator 2 lOOK 1001<. pulse.. -= '3enerator (()n~ shot) (_) s~,.c-. oUt

O/p

(29)

'"

:"11 ~

~

I

_~_;Ij I1 11:

_~

_I_,

~:"II

, J lIi:'!

,iI

,

I I I i ₁ I I D' n' , I I 111 ( a) 11\

E

IC

LI

1::::1 _J ;:00 I: lIIi:

~

I: I ~;!

..

_I

_~

~

I: ~ lIi:!

_

I

I 1 11 I lIi I I ~ (b)

FIG. 12: (a) Normal N wave and (b) peaked N wave produced by the

N wave vertical: 0.1 vOlt/div.

(30)

( a)

(b)

FIG.13: Acoustical wave (b) induced by the incident wave (a) used in comparison with those of Vaidya for window size

4'x3'

and

(31)

(a)

(b)

FIG.

14: Acoustical wave (b) induced by the incident wave (a) used in comparison with those of Vaidya for window size l'x3' and

(32)

. f

FIGo 15:

( a)

Acoustical wave (b) induced by the incident wave (a) used in comparison with t hose of Vaidya for window size 1 IX 1 I and

(33)

(a) (b) (c)

p

....--Po

t

T

p

-

_Po

t 0

T

r

-

_Po

f

T

P

-

Po

t

₃

~

T

4 --P

Po

_,

T

-1.

Po

,

4 0

₃

T

FIG. 16: Comparison between results from the present work shown in

Fig.13, 14, and 15 (---)and those of Vaidya[5] (---) for T

=

100 msec.

(a) window size 4'x3' ~b) window size l'x3'

(34)

~--~-~----~---~

---FIG. 17:

(a)

(b)

Incident waves for

T .

testing the effect of the rise time ratio

r

(a) N wave with T

=

1/15, (b) Triangular wave with T

r r

=

1/4.

(35)

(a)

(b)

FIG. 18: (a) and (b) acoustical waves induced by the N wave and the

triangular wave in Fig. 17, respectively, for 0l = 0.915

and ~l

=

0.0588 (~

=

1.0

mH,

Cl

=

6.8

nF,

Rl = 45

ohms).

(36)

(a)

(b)

FIGo 19: (a) and (b) mechanical vibration waves induced by the N wave

and the triangular wave in Fig. 17, respectively, for 02

=

1.192

and ~2

=

0.0045 (M₂

=

10

mH,

C₂

=

0.4 nF, _R2

=

45 o~). Scale: (a) vertical: 0.1 volt/div., horizontal:

5

~sec/div.

(37)

FIGo 20: Incident waves for testing the effect of the rise time T • (a) N wave with T = 1/25, (b) triangular wave with T

~

1/4.

r r

(38)

.

-(a)

(b)

FIG.

21: (a) and (b) acoustical waves induced by the N wave and the

triangular wave in Fig. 20, respectively, for U

l = 1.525 and

~l = 0.0588 (~

=

1.0 mH, Cl

=

6.8 nF, Rl

=

45

ohms).

(39)

(a)

(b)

FIG. 22: (a) and (b) mechanical vibration waves induced by the N wave and the triangular wave in Fig. 20, respectively, for

n

2= 1.99 and ~2

=

0.0045

(M2=

10

mH,

c

2

=

0.4

nF,

R2

=

45 ohms). Scale: vertical: 0.1 volt/div., horizontal:

5

~sec/div.

(40)

(a)

FIG. 23: Acoustical waves induced by the triangular wave in Fig. 17b for

n

l = 0.915 (~= 1.0

mH,

Cl= 6.8

nF).

(a) ~l= 0.0588 (R

l= 45 ohms), (b)~l= 0.189 (Rl

=

145

ohms).

Scale: vertical: 0.1 volt/div., horizontal:

5

~sec/div.

(41)

(a)

FIGo 24: Mechanical vibration waves induced by the triangular wave in

Fig.lTb for Q2= 1.192 (~= 10 mH, C

2= 0.4

nF).

(a) ~2= 0.0045 (R

2= 45 ohms), (b) ~2= 0.0245(R2= 245 ohms). Scale: vertical: 0.1 volt/div., horizontal:

5

~sec/div.

(42)

FIG. 25: Incident N wave with T

=

6/50 for testing the effect of the capacitance ratio c. r

(43)

( a)

FIG. 26: Acoustical waves induced by the N wave in Fig. 25.

(a) c = 0.0147, (b) c = 0.2352.

(44)

(a)

",

FIG. 27: Mechanical vibrat"ion waves induced by the N wave in Fig. 25. (a) c

=

0.0147, (b) c

=

0.2352

(45)

t

Pm

~

2.0

0.8

0.4

r

o

2

3

4

FIG. 28: Amplitude ratio of the acoustical waves as a function of Ul with Sl as parameter for T

=

0.15.

5

6

(46)

f

Pm

-~

2. a:

A

b:

G

c:

0 -1.21-

Ii#

"

e/

~

"

R

dT·1r=--~R~~~--~

o.

0.4 o

o---P--0-

-cr----O-FIG. 29: Amplitude ratio of the acoustical waves as ·a function of Dl for Tr

=

0.15 and ~l

=

0.141.

(a) without coupling with the mechanical system

(b) with coupling for c 1/39 and ~2'

=

0.0225 (c) with coupling for c

=

5/39 and ~2

=

0.0225

2

3

4

5

6

(47)

t4

..!in.

Q,

3

2

3 a:

a

b:

A

c:

0 -4

FIG. 30: Amplitude ratio of the mechanical vibration waves as a function of 02 for T_r

=

0.15 and ~2

=

0.0225.

(a) without coupling with the acoustical system (b) with coupling for c

=

5/39 and ~l

=

0.141

(c) with coupling for c

=

5/3.9 and ~

=

0.141

6

5

6

(48)

. . , . - - - -- - - - -- - - ---

-(a)

FIG. 31: (a) and (b) acoustical waves induced by the N wave and the

peaked N wave in Fig. 12, respectively.

(~

=

1.0 mH, Cl

=

6 .

8 nF,

Rl

=

45

o~)

(49)

(a)

~)

FIG. 32: (a) and (b) mechanical vibration waves induced by the N wave and the peaked N wave in Fig. 12, respectively.

(~

=

10 mH, C2

=

0.4 nF, R

2

=

45 ohms)

(50)

( a)

FIG.

33:

(a) Experimental equipment of set 1.

a: Wavetek model 116 function generator, b: N wave control box

c: electrical model, d: oscilloscope

(b) Experimental equipment of set 2

a: N wave generator 2, b: extended duration capacitor box, c: electrical model, d and e: power supplies,

(51)

(52)

UTIAS TECHNICAL NOTE NO. 158

10,"'''',

10, As,",p". Studi", Uoi,.nity of T o,oolo ~

"

.

.,

Sonie Boom Analogues for Invest1gatlng Indoor Waves and Structural 'Response

Lin, Su! ₁₂_pages 33 ~igures

1. Sonie Boom 2. Acoustlc-Mechanlcal-Electrlcal Analogues 3. Bonie Boom Stru-ctural Response. 4. gonie Boom Indoor Response

1. Lin. Sui II. UTIAS Teohnioal Note No.158

Experimental results indioate that the maximum amplitude of the indoor pressure

wave indueed by a Bonie boom for the case ot a partly open windov ls larger than

the maximum amplitude of the incident Bonie boom. In Buch a case, the two

unde-sirable etfects ot the Bonie boom are the annoyance i t causes peop~e and the effect it has upon structural members are larger indoors than outdoors. The

effects ot window aizet room dimensions, the dimensions and the properties or

struetural members and th~ shape or the sonie boom, whieh influenee the indoor aeoustieal pressure and the struetural dynamie response, are investigated by using an eleetrieal analogue. The method or design for the eleetrieal analogue

is described. The good agreement between the results trom the electrieal analogue

and those of Vaidya (5) shows th at the e1eotrioa1 analogue is a suitab1e device

tor investigating the sonie boom problem.

Available copies of this report are limited. Return this card to UTIAS, if you require a copy.

Institul:e for Aerospace Studies, University of T oronl:o

~

Sonie Boom Analogues for Investigating Indoor Waves and Struetural Response

Lin, Sui _{12 pages} 33 figures

1. Sonie Boom 2. Aeoustie-Meehanieal-E1eetrieal Analogues 3. Sonie Boom

Stru-etural ~espoDse. 4. Sonie Boom Indoor Response

I. Lin, Sui 11. UTIAS Teohnica1 Bote No.158

Experimental results indieate that the maximum amplitude or the indoor pressure

wave indueed by a Bonie boom tor the case of a partly open window is 1arger than

the maximum amplitude of the incident sonie boom. In sueh a case, the two

unde-sirable ettects of the sonie boom are the annoyanee i t eauses peop~e and the

effect i t has upon struetural members are larger indoors than outdoors. The effects of window size, room dimensions, the dimensions and the properties of

structural members and th~ shape of the sonie boom, whieh int1uence the indoor

aeoustical pressure and the struetural dynamie response, are investigated by

using an eleetriea1 analogue. The methad or design tor the eleetrieal analogue

is deseribed. The gaod agreement between the resu1ts trom the eleetrieal ana10gue and those ot Vaidya (5) shovs that the eleetr1cal analogue 1s a suitable device tor investigat1ng the aonie boom problem.

Available copies of this report: are limil:ed: Return I:his card to UTIAS, if you require a copy.

Institute for Aerospace Studies, University of T oronto

Sonie Boom Analogues tor Investigating Indoor Waves and Structural Response

~

Lin, Sui 12 'pages 33 tigures

1. Sonie Boom 2. Aeoustie-Mechanieal-Eleetrieal Analogues 3. Sonie Boom

Stru-etural Response. 4. Sonie Boom Indoor Response

I. Lin. Sui 11. UTIAS Teohnioal Note No.158

Experimental results lndicate that the maximum amplitude or the indoor pressure

wave indueed by a sonie boom for the case ot a partly open windov is larger than the maximum amplitude of the incident sonie boom. In such a case, the two

unde-sirable efteets of the sonie boom are the annoyanee i t causes people and the

effect i t has upon struetural members are larger indoors than outdoors. The efteets of window size, room dimensions, the dimensions and the properties ot

struetural members and tha shape ot the sonie boom, whieh influenee the indoor aeoustieal pressure and the structural dynamie response, are investigated by using an eleetrieal analogue. The method of design for the eleetrieal analogue is described. The good agreement between the results trom the electrical analogue

and those of Vaidya (5) shows th at the e1eotrioa1 ana10gue is a suitab1e devioe tor investigating the Bonie boom problem.

Available copies of I:his reporl: are limit:ed. Rel:urn I:his card 1:0 UTIAS, if you require a copy.

Inst:il:ul:e for Aerospace Studies, Universil:y of T oronto

Sonie Boom Ana10gues far Investigating Indoor Waves and Structural Response

~

Lin, Sui ₁₂_p_ages 33 figures

1. Sonie Boom 2. Aeoustic-Meehanieal-Eleetrlcal Analogues 3. Sonie Boom

Stru-etural Response. 4. Sonie Boom Indoor Response

I. Lin, Sui 11. UTIAS Technica1 Note No.158

Experimenta1 results indieate that the maximum amplitude of the indoor pressure

wave indueed by a sonie boom for the case of a part1y open windaw is larger than

the maximum amplitude of the incident sonie boom. In 6uch a case, the two

unde-sirable effects of the sonie boom are the annoyanee i t causes people and the

effect it has upon struetural members are larger indoors than outdoors. The effects of window size, room dirnensians, the dimensions and the properties of

structural members and tha shape of the sonie boom, whieh influenee the indoor acoustical pressure and the structural dynamie response, are investigated by using an eleetrical analogue. The method of design for the eleetrieal analogue

ls described. The good agreement between the results trom the electrieal analogue

and those of Vaidya (5) shows tb at the electrieal analogue 1s a 6uitable device for 1nvestigating the sonie boom problem.