Wave propagation in ground-structure systems with line contact

Journal of Sound and Vibration (1985) 101(4), 523-537

WAVE PROPAGATION IN G R O U N D - S T R U C T U R E SYSTEMS WITH LINE C O N T A C T D. TAKAttASrtl

Department of Architectural Engineering, Kyoto University, Yosbida Honmachi Sakyo-ku, Kyoto 606, Japan (Received 28 March 1984, and in revisedform 25 September 1984) The response of a structure in contact with the surface of a viscoelastic half-space (VEHS) is considered. The system is modeled as a ground-structure system. Vibration and radiation of sound into the closed space of the structure, resulting from a harmonic line force applied on the surface of the VEHS, are investigated theoretically. Stress-displacement relationships of the contact area are formulated by using modified forms of Lamb's integral solutions for an elastic half-space as Green functions. A set of integral equations formulated by using this procedure is evaluated and numerically calculated. The structure is modeled a s o n e of thin plates, and the response can be determined in the form of flexurai and quasi-longitudinal waves. Also, acoustic coupling between the flexural modes of vibration and the air is presented as a spatially averaged sound pressure level based on the assumption of a perfectly diffuse sound field.

1. INTRODUCTION Investigations into the problem of vibration and structure-borne noise in buildings due to road traffic, surface and underground railways and so on, should include the following: wave generation and propagation in the ground, wave transmission from the ground to the building, wave propagation in the building structure and sound radiation from the vibrating surface of the structure into the room. Concerning wave generation and propagation, m a n y studies have been performed in earthquake engineering since the pioneering work o f L a m b [1], who studied the response of an elastic half-space excited by various forces and established models for both two- and three-dimensional wave propagation. Wave transmission in the contact area, as reported in m a n y papers and described later, has been studied as a dynamic soil-foundation interaction. Problems in earthquake engineering differ from noise and vibration problems in both the objective frequency range and propagation distance in the ground. This fact requires different modelings of the system. From the vibrational point of view, Gutowski and D y m [2] reviewed the current state of the art regarding wave propagation in the ground and discussed equipment, techniques and measurement data. Dawn and Stanworth [3] discussed the generation and propagation of ground-borne vibration caused by passing trains and the effects on nearby buildings. From the acoustic point of view, Kurzweil [4] presented a very simple method for estimating structure-borne noise due to ground-transmitted vibration in buildings near subways. The purpose of this paper, from the points of view of both vibration and noise, is to establish a model which enables one to estimate the effect of various parameters in the propagation process on the resultant responses. As a first step, a method for analyzing 523 0022-460x/85/160523+ 15 $03.00/0 9 1985 Academic Press Inc. (London) Limited

524

D. TAKAHASHI

the response of a simple two-dimensional structure in contact with a VEHS is presented, together with examples of numerical calculations. In the analysis of wave propagation in ground-structure systems, much effort has been devoted to investigating the contact area of the systems in earthquake engineering. The main objective o f these studies is to determine the relation between the applied force and the displacement of a rigid body on an elastic half-space, which is, in a sense, equivalent to the impedance or adm_~ttance of the elastic half-space. In general, there are three approaches to the analysis o f the contact area; the first approach is to consider a mixed boundary-value problem with a continuity condition at the area and a stress-free condition at the surface away from it, which can be expressed in terms of dual integral equations [5] or an equivalent Fredholm integral equation of the second kind [6-8]. The second approach is a method of assuming an approximate stress distribution throughout the area. Although the analYSiS can be simplified by this assumption, the accuracy depends upon the assumed distribution. These studies, preceded by those of Reissner [9], have been reviewed in reference [10]. The third is the Green function approach which leads to stress-displacement relationships at the contact area. By using this approach, Oien [11] investigated the response of a rigid strip in contact with an elastic half-space due to incident plane waves. In this paper, displacement elements of the contact area o f a two-dimensional VEHSstructure system are related to the corresponding stresses by the Green function approach, regardless of the resultant motion of the structure. Subsequently, the forces and moments acting on the contact area obtained by this approach are applied to the response analysis o f the structure. 2. FORMULATION A system composed of a structure and VEHS in the Cartesian co-ordinates x and z is shown in Figure 1. The response of the structure caused by a harmonic line force qo e -i~" applied on the surface of the VEHS is investigated. Zo

2

-2b~

2bz

Figure 1. Model of a ground-structure system.

Attention is now restricted to the contact area denoted by x = 0 + b~ and x = Xo+ b 2 o n the x-axis. The motion o f the area may then be considered as the motion o f a rigid plane (denoted by RP.1 or RP.2). As the resultant response of the system, the motion of RP.i ( i = 1, 2) is characterized by the horizontal displacement U~, the vertical displacement V~ of the center and the clockwise angular rotation Ri. When a wave is incident, relative displacements, either with or without the rigid plane, produce new waves and interaction between RP.1 and RP.2. Considering the wave generated by the relative displacements

9

525

WAVES IN G R O U N D - S T R U C T U R E SYSTEMS

of each rigid plane, one obtains the following relations at each area:

u,(x)=[Ute,,+(V,+XRl)e,]-[Uz(X)+U2(X)], UE(X)=[U2ex+(VE+(x-xo)R2)e~]-[Uz(X)+ul(x)],

Ixl
(1) (2)

Here ex and e~ are unit vectors, u,(~) is the surface displacement of the wave influenced by RP.i and ut(x) is the surface displacement of the incident wave. Considering equation (1), one finds that the fight-hand first term is the motion of RP.1 itself and the second term can be considered as a resultant incident wave on RP.1. Consequently u~(x) is the relative displacement which produces horizontal and vertical stresses rl(x) and cry(x). The same can be applied to equation (2), the resultant stresses being represented by r2(x) and cr2(x). The main analysis can be reduced to obtaining the stresses 7j(x) and cri(x) caused by u~(x). In the first step, when RP.1 only is in motion, expressed by the fight-hand first term of equation (1) under the incident wave u t (x), the wave motion for I x - Xol~
Q ' = I ~',(x)dx,

N~= I cr,(x) dx,

M ~ = f o',(x)xdx.

(3)

These quantities contain unknown coefficients U,, V~ and Ri which are determined by the relation to the upper structure at the final stage. A fundamental approach in the formulation process is to formulate the stress-displacement relationship by using the Green function of the VEHS. It is convenient to put x - Xo= X, and introduce the following non-dimensonal quantities:

X=x/b,,

-~r = b,k*,

at. = kL/kr,

2=x/b,,

ft=xt/b,,

fl = b2/b,, 2o=xo/b,,

(4)

5o=zo/b,,

(5)

where k*=kL/~/1-ir/L and k * = k r / d l - i r / r ; kL, kr and r/L, r/r are, respectively, wavenumber and loss factor; subscripts L and T represent longitudinal waves and transverse shear waves in the VEHS respectively. Modified forms of Lamb's integral solutions used as Green functions are

g,(x,~',) =

~_ Gt(k,r

F(k,~',)

eikx

g2(x,~',) = f ~ G2(k'sr~) e i~' dk,

J_~

dk,

G,(k,r = r

G2(k, sr~)= ik(2k 2 - ~'~- 2 vLvr),

(6)

(7)

F(k,r

g3(x,~',) =

I?oo G3(k'sr') e i~ dk, F(k,~',)

G3(k,~,) = r

(8)

where

F(k,~j)=(2k2-K~)2-4k2VLZ, r,

and

~=k2--(aL(,) 2, ~,~-=k2-~'~z, (9,10)

526

D. T A K A H A S H 1

in which ~', represents a wavenumber parameter of transverse shear waves with respect to the contact position: i.e.,

[flk-~r,

~"

i=

"

(I1)

The surface displacements of the VEHS due to a horizontal line force can be expressed by g~ in horizontal motion and g2 in vertical motion, and in the case of a vertical line force, by g2 in horizontal motion and g3 in vertical motion. By using equations (6)-(11), each component o f t h e incident wave u~(x)( = U~(x)ex + v~(x)e~) can be written in the form

ut(x) = (-qo/2rrO*)g2(g-.xb ~,),

or(x) = ( - q o / 2 ~ r G * ) g 3 ( 2 - ~ , ~,),

(12)

where G* is the complex shear modulus of the VEHS. The convolution integral is expressed as JL~ f ( ~ ) g ( x - ~) dE = f ( ~ ) * g ( x - ~) henceforth, for brevity. 2.1. W A V E M O T I O N A T T H E S U R F A C E W I T H O N E R I G I D P L A N E In the surface displacement field of the VEHS, when RP.1 only is in motion, expressed as U, ex+(V~+xR~)e~ under the incidence of the wave u1(x), wave motion U(x) at an arbitrary point can be written as

U(x)=u,(x)+n~

(13)

where u~ is the wave influenced by RP.1, and is the relative displacement, either with or without RP.1. Equating U(x) with the motion of RP.1 for Ixl ~ b~ gives u~

0

0

*

0

= [ U, ex + ( V~+ xRl)ez] - u l (x) = Utu~ + Vlu,2 + bt Rlu 13 "{- (qo/2 lrG )u 14,

with u~ = e~,

u~

e~,

u~ = gez,

U?4 = g 2 ( - ~ - - X l ,

Ixl<~b,,

(14)

~rl)ex+g3(2--2t, ~'t)e=.

(15)

Representing the stresses as r~ in horizontal motion and tr~ in vertical motion caused by the displacement element u~ ( j = 1. . . . . 4), one obtains the stresses r ~ and trlo due to u ~ in the form of a linear summation corresponding to equation (14), where the stressdisplacement relationships between rt~ tr~ and u~ ( = u~ + v~ can be expressed as

goj( r , g,( 2 - r ~,) + 6-~ r * g 2 ( 2 - ~, ~',) = u ~ 2) "[

_~o (6). g2(:~-6, ~',)+,~~162 g3(,~-r ~',)= v~

1,

J

(16)

with ~o = bmrOj27rG,, 6.oj = b~o.Oj2~rG, and j = 1. . . . ,4. The wave u ~ can be obtained by convolution integrals involving r ~ tr ~ and equations (6)-(8). Substituting the expression into equation (13) gives the wave motion U ( = Uxex+ U:ez) at an arbitrary point, which in particular for IXI <~ b2 can be written as [

20 - 21

-qo g = ~ 2 + T Ux(x) = 2-7-~

bi

o

' ~'2)+ 2-Tb-7[r,(D

.

g,(O;+&_r

~r,) IXl<~b2.

+ ~r~162* g2(f12 + 2 ~ ~:' r

-qo Uz(x) = 2--~7

g3~ x , - - ~ - - , ~

{-,2~

-

) 2--~,

D,~ * g2(#2+~o-~, r

(17)

~,VAVES IN GROUND-STRUCTURE SYSTEMS

527

By following the same procedure as above, in the case of RP.2 only being in motion, and expressed as U2e~+(V2+xR2)ez, the wave motion U for Ixl b, can be written as --qo

U~(x) = ~

gz(x -Y,t, ~'1)

"lxl~b,, Uz(x) = 2 ~ *

(18)

g~(~z-~z. ~',)

[2-2o

b2 [ zo(~:) 2~'G*

where r~ and cr~ can be expressed by linear summation of the stress elements r~j and o-~j (j = 1. . . . . 4) corresponding to the relation u~

= [ U2ex + ( 1/2+ xRz)ez] - ul (X + Xo) 0 0 0 = U2u21+ V2u22+b2R2u23+(qo/2~G*)u~

Ixl< b2,

(19)

with U201~ ex,

u~ = ez,

[ "l-x- o- --' x~ ~, ~.2)ex+ g3~,r [ 4" T-%--~t ~2)ez. U04 = g2 ~/~ ,

u~ = ~ez,

(20) The stress elements r~ and cr~ can be determined by solving the following integral equations: ~oj(~), gi(s - ~:, ~2) + ~ o / ~ ) , gz(2 - ~, ~2) = u ~ _< o o ~1;?1- 1, -r~ * g2(,Y- ~, ~'2)+ r * g3()?- ~:, sr2)= v2s(,Y)J

(21)

with .~oj= b2rOj27rG,, 6.o2s= b2orO2j27rG, and u~ = u2je,,+ o o (j = 1 , . . . , 4). vzjez 2.2. STRESSES ACTING ON THE TWO RIGID PLANES Note that the right-hand second terms of equations (1) and (2) can be expressed by equations (18) and (17) respectively. Substituting and rearranging them in terms of the displacement elements gives

ui = UiUil q- Vlui2q- biRiui3-b ( qo/2"n'G*)ui4-t- Ua-iUi5 + V3-~ui6+ b3-iR3-iui7+ (qo/2vrG*)ui8,

(22)

in which the displacement elements uo(i = 1, 2,j = 1, 2 , . . . , 8) can be expressed as follows: for j = l , . . . , 4 : for j = 5 , . . . , 8 :

Uil ~--"e x ,

u i 2 • ez,

Ui3 = :~e=,

ui4 = g2(Xi, ~)ex + g3(Xi, ~'i)ez, (23)

Uij =--[~3~

* gl( Y i - ~, ~3-i)-t-r176 * g2( ~ - ~ : , ~3-1)]ex -o ~.o -I" [T3_i,j_4(bg) * g2( Y / - ~:, ~'3-i) - 3-i,j-4(b~) * g3( Y / - ~:, ~'3-i)]ez,

(24) where

x-xr

"~ J,

Y ' _f(g-2o)/fl'~ -t j

for { i = 121

i=

"

528

D. TAKAHASH

I

Stress elements r u and o'~j corresponding to uo( = uoex + voe~) can be obtained by solving the following integral equations:

-'?,j(~:) * g2(.~-~:, ~',)+ #0(r * g3( : / - ~:, g'r)= v,j(x)J Ixl

(25)

1,

with ?u = b~ru/2r~G* and 6"u = b,o~J2~rG*. Stresses ~'~ and o-~ acting on RP.i can be written in the summation form corresponding to equation (22), and the resultant forces and moment can be written by using equations (3) as

Q, = 2~rG*

;

~(x) dx,

N, = 2~-G*

--1

11

~,(x)

dx,

Ml = 2~rG*b,

6"t(x)x dx,

--1

1

(26) where ?i = biri/2~rG* and ~t = b,cri/21rG*.

3. SOLUTION 3.1. S E R I E S E X P A N S I O N M E T H O D Expressions (16), (21) and (25) each form a pair of coupled Fredholm integral equations of the first kind involving Lamb's solution in integral form, and it is almost impossible to obtain the solution analytically. A series expansion method described in reference [ 11] gives a good approximate solution, which is used here. The desired stress distributions in equations (16) and (21) are written as the following series expansions with unknown coefficients A~ and B~ by using the Tchebychett polynomials T,,(~) and T,(~:):

o(e) =

o

A,jm(1-~

2-,,2 ) Tm(~),

@o(~)= ~ Bo(I_~:2)-,/ZT.(~),

rn=O

I~l~l,

(27)

n=O

and in equations (25) as

~o(~) = ~ m~0

a,jm(l-~2)-'/2rm(~),

6"/~(~:)= ~ BO.(I-~2)-'/2T.(~),

Ir

n=O

(28)

By using the orthogonality relationship ~r,

I_

(1-~2)-l/2Tm(~)Tn(~) d~:=

re=n=0

7r/2, m = n ~ O

,

(29)

1

0,

otherwise

and the relation

f_ (1 - ~:2)-'/2Tm(~:) e :~k~ d~: = (-i)"'n'Jm ( q: k),

(30)

I

equations (25) yield

aim,Aijm + ~ fli.tBu.= ~,j,, m=O

n=O

- ~ fl,mtaum + ~. T;.,Bu.= ~t'ij,, m=O

n=O

(31)

529

WAVES I N G R O U N D - S T R U C T U R E S Y S T E M S _

.

a,,.,- (-I)

r.+l

co G , ( k", ~';) 2 rr f_~ F ~ , (~- Jm(k)Jt(-k) dk,

-

(

,

~',)

. _ f , , , + t r 2 f ~ Ga(k, ~ . Y;,.,,t- ( ) .I-I F(k, (~) J,,(k)J,(-k) dk,

(32)

_

and qb(/l=

f

t/u(X)(1 --X2)-I/2TI(x)

dx,

~#l =

1

f

vq(x)(l --x2)-l/2Tt(x) dx,

(33)

i

in which Jm( ) denotes the Bessel function of the first kind. Applying the same procedure to equations (16) and (21) yields the same results as in equations (31)-(33) forj = 1 , . . . , 4: i.e.,

A u0 m -__A o , , ,

0 __ BO,,,-Bo,,,,

i = 1,2, j = l , . . . , 4 .

(34)

Considering the even-odd dependence of the integrands on m and ! about the variable k in equations (32) gives a;'~Z=L 0 J'

0;"1= --flitm '

Yi"Z=

"

f~

odd J"

(35)

By substituting equations (23) and (24) into equations (33), q~#l and ~'#t can be determined, and they are given in Appendix A. Solving the simultaneous equations obtained by truncating the series expansions at the same number of terms (insofar as the solution converges) determines the coefficients Ao,, and Born. Subsequently, by substituting the expressions composed of equations (28) into equations (26) and by using relation (29), the desired forces and moment are reduced to the following final results: Q; = 2~'2G*[ U;A~,o + b~RiA~3o+ ( qo/2 rrG*)a;4o + ( U s - i A i $ o + V3-iA;6o+ b s - i R s - ~ A i T o + (qo/2"n'G*)Aiso)],

N; = 2rr2G*[ ViBno+ (dlo/2~G*)Bi4o + ( Us-;B,~o+ Vs-,B,6o+ bs-,Rs-,B,7o+ (qo/2~'G*)B;8o)],

M; = 7r2G* b;[ UiB, t + biRiB;3t + ( qo/ 2 7rG*) B;41 + ( U3-;R;st + V3-;B;6~ + bs-,Rs-;Bm + (qo/2rrG*)B, st)].

(36)

It can be seen that some terms can be omitted because of the even-odd dependence of the stress elements on the displacement elements. If the interaction between RP.1 and RP.2 is neglected, the terms in parentheses in equations (36) are not required, which means that very complicated calculations involving equations (31) with the terms denoted by equations (A3) and (A4) in Appendix A are unnecessary. The results obtained by neglecting the interaction terms also are shown afterwards, which implies that the possibility of extension to multicontact systems can be examined. 3.2. E V A L U A T I O N O F T H E I N T E G R A L S The integrals in the coefficients ai,.t, fli,.t, y~,.t and the terms q~ut, 9"ut of equations (32) and (33) (from equations (A3) and (A4) in Appendix A) cannot be evaluated analytically,

530

D. TAKAHASHl

and direct numerical evaluation is cxceedingly difficult. A detailed discussion on evaluation of the oscillation-type integrals involving Rayleigh's function can be found in references [11] and [12]. The ordinary method is to transform the integral into a more convenient form for numerical evaluation by using Cauchy's residue theorem. However, this method cannot be applied directly to the analysis here because of the loss factor terms, to which particular consideration must be given. Consider the integrals of the form I~)=

2 fo ~ Gj(k,F(k, 5)5)J,,,(k)Jl(-k) dk

(j = 1, 2, 3),

(37)

I~r = I?~ Gj(k, ~5) j , ~, - k~, e ikxdk ( j = 2 , 3 ) , I(aJ)=

I?~oG~(k, ~) j,,,(k)j,(_flk)elkX dk F(k, 5,)

(38)

(j = l, 2, 3):

(39)

Evaluation of the integrals in this form illustrates the problem with this analysis, in which branch points of multivalued functions ~'Land ~'r are not on the real axis, and consequently the branch lines also are not on both the real and imaginary axes because of the complex quantity ~"(or ~'1). In order to avoid this difficulty, the following procedure must be taken prior to the evaluation: changing integration variable k = ~k (or k = and assuming ~/L= ~r, which is acceptable in practice. One then finds that the branch points are on the real axis and the integration path is along the line which runs through the origin, with the slope being about - ~ r / 2 . Consider now the integral I~j~. By using the relation J t ( - k ) = (-1)l[H~l)(k) + H~2)(k)]]2, in which H~~ and H~2)(k) are Hankel functions of the first and second kinds, and by noting that only the case m ~>I is necessary for the evaluation as shown in equations (35), the integration path shown in Figure 2 can be used for the evaluation, in which there are

~ik),

LI

r, Lz

Figure 2. Integration paths in the complex/~-plane.

no singularities except Rayleigh poles. Choice of the contour YI+ F I + L~ for integrals involving the Hankel function of the first kind, and y2+F2+L2 for the second kind, ensures that the boundedness and outgoing wave condition are satisfied. Therefore the integral is evaluated in the form 2zri ~ ReS--SL~ --SL2"

WAVES IN GROUND-STRUCTURE SYSTEMS

531

Considering the same conditions as described above with respect to e ~kx in order to evaluate integrals ltj ~and It~~, one must choose the closed contour of the upper half-plane about the same line. The integrals can then be evaluated by the residue terms and branch line integrals. The final results are given in Appendix B. 4. RESPONSE OF STRUCTURES By using the forces Q~, N~ and the moment M~ acting on the contact area, the response of a structure with an arbitrary upper construction in contact with the VEHS can be obtained. Here, for example, consider-a two-dimensional model o f the structure as shown in Figure 1. Analyses of both the vibration response and sound radiation into the closed space of the structure are given. 4.1. VIBRATION OF THE STRUCTURE Thin plate theory is assumed, being the most simple model, and so each member of the structure, to which a number is given as shown in Figure 1, responds in the form of flexural and quasi-longitudinal waves. Each member is denoted by MEM.i (i = 1. . . . ,4) henceforth. The displacements of MEM.i with respect to each wave are denoted by w~ and wL~,and the corresponding wavenumbers are kF~ and kL~ respectively. The equations of motion are, for flexural waves, 4

4

4

d w r d d X i - knnFi = 0,

(40)

d 2 w u / d X ~+ k~iwt_l = O,

(41)

and for longitudinal waves,

where

i i=3,4J" Let the shear force, axial force and moment of MEM.i be Qij, Nu and M U at the junction of MEM.i and MEM.j. The boundary conditions at each junction are then as follows: at the junction of V E H S - M E M . 4 - M E M . i (i = 1, 2), w ~ = wt~ = ~ ,

wLi = we4 = V~,

Q , a - e,N4i = Qi,

dwri/dz = -dWF4/dx = -Ri,

N i a - e,Q4, = Ni,

Mi4 + e,M4, = Mi ;

(42)

at the junction of M E M . 3 - M E M . i (i = 1, 2), w ~ = WL3, Qi3 + eiN31 = O,

WLI = V.'r3,

d w F J dZ = - d w F a / dx,

Ni3 + e~Q3i = O,

Mi3 - eiMai = O;

(43)

where

,{::

i:1},

,44,

General solutions for equations (40) and (41) with unknown coefficients C o (i = 1. . . . ,4, j = 1, 2 , . . . , 6) can be written in the form WFi = Cil e ik~xJ + CI2 e -ik~x~ + Ci3 e kF'xl + Ci4 e -ktlXj,

WLi = Cis e iku'x'j + Ci6 e - k u x i .

(45, 46)

532

D. T A K A H A S H I

Substitution of equations (45) and (46) into equations (42) and (43) yields simultaneous equations in 30 unknowns with respect to U~, V~, Ri and C~j. 4.2. S O U N D R A D I A T I O N I N T H E C L O S E D S P A C E In problems o f structure-borne noise caused by ground vibration, the choice of the acoustic index is also a problem. In view o f the main purpose o f this paper, it is desirable to use an index which properly reflects the effects of various parameters concerned in the process of wave propagation. Here, acoustic power supplied to the closed space can be selected, and the spatially averaged sound pressure level (SPL), which is familiar in noise problems, may be substituted for the power. By tentatively assuming a perfectly diffuse sound field within the space, the SPL can be obtained easily from the mean-square sound pressure Pr.m.s. 2 as

SPL = 10 log,o(p 2.. . . . /p2) = 10 log,o{(pocowP/La)/p~},

(47)

where P denotes the total radiated sound power in W / m , L the perimeter, ff the average absorption coefficient, Po the reference value o f the sound pressure ( = 2 • 10 -s N / m s) and poCothe characteristic impedance of the air. In the strict sense, the power P must be determined from the particle velocity and the corresponding sound pressure on the boundary surface, obtained by solving the Helmholtz equation with the continuity conditio n at the surface. However, exact analysis is considered to be unnecessary for a problem of this nature. An approximate method is applied here, in which the power is considered to be supplied independently from the surface o f each member o f the structure. This method implies that the total power can be calculated by summation of each power radiated from a vibrating surface in a rigid baffle. Now consider the vibrating surface in the region x~ ~
pCx, z) = pow f x2 r

H(ol)Ckor) dx',

(48)

2 Jx,

where r = [(x-x')2+z2]~/2and ko= o)/Co. The far field expression p(r, O) can be obtained by using the relation H ( o l ) ( k o r ) - - - ~ e iko('-x'~i"~ as r ~ o o , which is

p(r, O) = poCo

e i(k~

, ~:(x') e -ik:'~i" o dx'.

(49)

The radiated power can be obtained by integrating the radial intensity Ip(r, 0)12/2poCo over a hemicylindrical surface of radius r. By taking into account the flexural mode of MEM.i concerned in the acoustic coupling, and by using equation (45), the radiated power P~ can be expressed as P,

f=/z Ip,(r, 0)12 2poCo

= a-,~/2

r dO,

(50)

in which pi(r, O) is expressed by equation (49) where ~ =-itowvl. Substituting the total 4 P~ into equation (47) gives the SPL, which is here written in the form power p =~i=1

SPL = 10 log1 o([p I

/I aol

4(Xo-Zo)ff ,=l a-=z2

s=,

- 6.02 + VALo, KS

(51) dO,

(52)

~,VAVES

533

IN G R O U N D - S T R U C T U R E SYSTEMS

K j = I - i ( k o s i n O--ejkF,), t--ikosin O+ej_2k~,

j = 1,2"[ j=3,4J'

(Zo, ' = 1,2"[ X i = LXo, i = 3 , 4 J '

(53)

in which ej is given by equation (44), and ao is the calculated reference value of the vertical acceleration at the reference point on the ground surface. By letting the reference point be near the source, the value of the incident wave may be taken as the value of ao. This expression means that the SPL may be predicted by measurements of the vertical vibratory acceleration level VALo (dB re 10-5 m/s 2) at the reference point. 5. NUMERICAL RESULTS AND DISCUSSION A non-dimensional wavenumber/~r = 2~fbl/cr of transverse shear waves with speed cr was introduced as a frequency parameter. Subsequently, numerical calculations were carried out for the frequency range noted for problems of both vibration and structureborne noise. There remained a problem in determining the values of cr, 7/r and aL which characterize the VEHS. Here, the property of either soil or soil-asphalt, which is intermediate between soil and asphalt, was assumed and in accord with references [13, 14], the following values were used: soil: c.r=160m/s, 7/T=0"5; soil-asphalt: c r = 4 5 0 m/s, ~r = 0.1. In both cases Poisson's ratio was assumed to be ] (aL = 0"5). For the structure 2O

i

IO

(Q)

(c)

(b)

(d)

I

i

i

i

I

I

I

I

I

I

I

I

I

I 0-,5

I I

I 2

I 4

8

I 16

I

/

0 -I0 -20 -30 -40 v -M

-,50 -60 20 I0 0 -I0 -20

--4300

f

-50 -60

I 1

2.5

2 1

5

4 I

IO

B I

20

16 I

40

I 32

I 64

I I

I I 128 0.25 I

80 IO0 200 Frequency

I~r

400

I

I

2-5 .5 (Hz) for bj = 0.1 m

t

IO

I

20

1

40

II

I 32 (,10 "z) I

80 I00 200

1

400

Figure 3. Vibratory acceleration level of the model floor (a) in horizontal motion on a soil ground, (b) in vertical motion on a soil ground, (c) in horizontal motion on a soil-asphalt ground, and (d) in vertical motion on a soil-asphalt ground. , Exact; ...... , approximate.

534

D. T A K A H A S H I

model, constructed of four members of dense concrete with the same thickness, the following normalized values were taken: distance from the source gl = - 1 5 0 , distance between two points of contact Xo= 80, height of the structure E'o= -40. The frequency response curves presented by showing the vibratory acceleration level (VAL), averaged across the width of the model floor, are shown in Figure 3. The level is normalized by the vertical acceleration of the incident wave at 10 b~ from the source point. In Figure 3, the results of an approximate solution obtained by neglecting the interaction between two contact points are also shown. From these results, it can be considered that the effect of the interaction is only limited and within a restricted frequency range. In the case of a soil ground the effect is so small that it is not shown. 60 /

I

50 I

(o)

I

I

1

1

I

I

I

I

32

64

128

I

i

I

i

I

i

i

i

I

I 0-5

I I

I 2

I 4

I 8

I 16

I 52

I I 80100

I 200

40

30

E "o

20 I0 0 -I0 -20 -30 -40

2

4

8

16

I

I

l

I

I

2-5

5

I0

20

40

I

F i g u r e 4. S o u n d , Exact; ......

I I

l

I O'25

l/or

I

l

80100 200 400 Z.5 5 Frequency (Hz) for b t = 0"1 rn

p r e s s u r e l e v e l i n t h e m o d e l r o o m ( a ) o n a soil g r o u n d , , approximate.

I I0

I 20

I 40

and (b) on a soil-asphalt

~I0-2)

400

ground.

The spatially averaged sound pressure level in the model room is shown in Figure 4, in which the results are shown by omitting both the scaling length b, and absorption coefficients c / o f the room. Adding the values of both 10 log,o(b2/6) and VALo gives the actual SPL. The approximate results obtained by neglecting the interaction between RP.1 and RP.2 also are shown. Characteristics similar to the results for the vibrations can be seen. 6. CONCLUSIONS Wave propagation in a simple two-dimensional model of a ground-structure system has been investigated theoretically. A method for analyzing the vibrational response and sound radiation into the closed space of the structure under a harmonic line force applied on the ground surface has been presented, with numerical examples. Although the model of the system is simple in comparison with the actual system, an outline of the response characteristics can be obtained. In general, the response of structures in contact with the ground depends upon the considerable number of parameters concerned in the overall process of wave propagation. These are, for example, the properties of the ground, thecondition of the contact between the ground and structure, construction and structural materials. For problems of vibration

535

WAVES IN GROUND-STRUCTURE SYSTEMS and structure-borne these p a r a m e t e r s on m o d e l o f the system such e s t i m a t e s to be

noise c o n t r o l in b u i l d i n g s , it is i m p o r t a n t to e s t i m a t e the effect o f the r e s u l t a n t r e s p o n s e s b o t h q u a l i t a t i v e l y a n d quantitatively. T h e a n d t h e analysis p r e s e n t e d in this p a p e r a r e c o n s i d e r e d to e n a b l e a c h i e v e d to s o m e degree.

REFERENCES 1. H. LAMB 1904 Philosophical Transactions of the Ro)'al Society of London Series A 203, 1-42. On the propagation of tremors over the surface of an elastic solid. 2. T. G. GUTOWSKI and C. L. DYM 1976 Journal of Sound and Vibration 49, 179-193. Propagation of ground vibration: a review. 3. T. M. DAWN and C. G. STANWORTH 1979 Journal of Sound and Vibration 66, 355-362. Ground vibrations from passing trains. 4. L. G. KURZWELL 1979 Journal of Sound and Vibration 66, 363-370. Ground-borne noise and vibration from underground rail systems. 5. A. O. ABOJOt3I and P. GROOTENHUIS 1965 Proceedings of the Royal Society of London Series A 287, 27-63. Vibration of rigid bodies on semi-infinite elastic media. 6. P. KARASUDHI, L. M. KEER and S. L. LEE 1968 Journal of Applied Mechanics 35, 697-705. Vibratory motion of a body on an elastic half plane. 7. I. A. ROBERTSON 1966 Proceedings of the Cambridge Philosophical Society 62, 547-553. Forced vertical vibration of a rigid circular disc on a semi-infinite elastic solid. 8. G. M. L. GLADWELL 1968 International Journal of Engineering Science 6, 591-607. Forced tangential and rotatory vibration of a rigid circular disc on a semi-infinite solid. 9. E. REISSNER 1936 Ingenieur-Archiv 7, 381-396. Station[ire, axialsymmetrishe, durch eine schiittelnde Masse erregte Schwingungen eines homogenen elastischen Halbraumes. 10. F.E. RICIIART, JR., J. R. HALL, JR. and R. D. WOODS 1970 Vibrations of Soils and Foundations. Englewood Cliffs, New Jersey: Prentice-Hall. See chapter 7. 11. M. A. OIEN 1971 Journal of Applied Afechanics 38, 328-334. Steady motion of a rigid strip bonded to an elastic half space. 12. W. M. EWING, W. S. JARDETZKY and F. PRESS 1957 Elastic Waves in Layered Media. New York: McGraw-Hill. See chapter 2. 13. E. E. UNGAR and E. K. BENDER 1975 Inter-Noise 75, 491-498. Vibrations produced in buildings by passage of subway trains; parameter estimation for preliminary design. 14. L. CREMER, M. HECKL and E. E. UNGAR 1973 Structure-Borne Sound. Berlin: Springer-Verlag. See page 216.

APPENDIX A T h e t e r m s t/5~1 a n d ~o~ c a n be e x p r e s s e d as follows: for i = 1, 2,

~qs=~

0,

j=2

,

O,

j = 3J

w h e r e 8 is the K r o n e c k e r d e l t a ; for i = 1, 2, qb,4' = (_i),,.r f ~ G2(k, ~',) J , ( - k ) eikx, dk, 3 - oo F(k, ~l)

g'ol=

8ojrr,

j=

j = 1, 2, 3:

,

(A1)

81zTr/2, j = j=4"

~i4t = (-i)lcr

f ~ -G3(k,~',) ~ F (-k , ~i) J l ( - k ) eikx' dk, (A2)

where

ix',=

(Xo-X,)l,6,

i

;

536

D. TAKAHASHI

for i = 1, 2, j = 5 , . . . , 8 : --o

tPO! = '

....

+, 2~'~ G , ( / q ~ ' l ) S "t k ~J t ak~eik~o ' "

dk

+ ~ no . . . . +' 2f~ G2(k'r .=o IJ='s-4""t-U rr j_~ F(k, ~,)

i=1 (A3)

--o

....

-..=o'a"s-'"t-u

+t 2( ~ G,(k,~_~L)j,.(k)j,(_flk)ei~od k ~r j_| F(k, ~,)

I i~.+'r ~',) J , ( k ) J , ( - f l k ) ei~o dk, - z., B o~.s-4.,~s !I"~ O~(k, ~.--g-7~.7"-; n=0

J-oo

Ao

....

:.s-,.~t-,

,.:o

i=2

F ~ a . ~ ~,11

+t ", foo G~(k, ~) j,(k)jm(_flk) eik~~ dk ~ J ~ F(k, ~,-~-S

:~ ~o , i,.%.2 f ~(k,r .-.=o 2.s-,,..,-, J-oo-f(k,,r J~(k)J"(-/~k)e'~~

i=1 (A4)

~o G

o 2(/q r m~_oA,j_4,m( - i ) m+~ "a"2 f-oo "-~,;~-~ Jm(k)J'(-flk)eik~~

.=o

0 9 n+l Bx.S_4.,,(-i ) ~r2

=

- -

F(k, ~ , ) J,,(k)J:(-flk)

e iu~ dk,

i= 2

APPENDIX B

The results of the evaluation of integrals (37)-(39) are as follows: Gi(K , 1) I ~ ' ) = 2 ~ ' i ( - 1 ) t - F'(K) J"(~'~:)Hlt)(~rK)-2i(-1)l I f L ~2(k)Jm(srk)H, " (') (sr-) k dk _2i(_l)l

ch3(k)J,,(~k)H~(I) (~k)dk,

(B1)

L

I~2)= 2rri(-1)'

G2(K, 1) F'(K) J,~(srK) U~u(srK)+ 7r im+~

I;

~p3(r)I,.(~'r)K,(~'r) d r

qSs(k)Jm(~k)H~')([k) dk,

4-4(-1)'

(B2)

L

I~3)=2rti(-1)'G~;)J..(r162162162 -8i(-1)'

q~.(k)J.,(~k) H~u(~rk) dk,

(B3)

L

IP) = 2 r r i ( ' l ) ' G2(K, F'(n) 1) Jl(~'K) e i ~ x + 4 ( - - 1 ) l i(23)= z~rl(_l)^ .t G3(K,_fiT~K)I) Jl(~'K)"" e i c " x - 2 ( - i ) I -2i(-1) I

Io"~

,_45(k)Jz(~'k) e i~kx dk,

(B4)

ioo

~bt(r)Ii(g'r) e -c~x d r

t~l(k) Jl(~'k) e i~kx d k - 8 i ( - 1 ) t

Io 4,(k)J,(~k) L

e ~k'~ dk,

(BS)

537

WAVES IN GROUND-STRUCTURE SYSTEMS

9

-i

ICa')=2~r'(-1)

G~(K, 1)

F---~Ki Jm(~r'K)J'(fl~"K) e ~ ' ' x

-2ira§

ioo

t

- 2i(-1) t

02(r)I,.(~',r) Ii(fl~:') e - q ' x d r

4~(k) Jm(~,k) J z ( ~ , k ) e ~ , ~ dk

Io"

- 2i(-1) a

4 3 ( k ) Jm(~,k) J t ( ~ , k )

(B6)

e ~'~ d~

L

1(32)= 2.~. '.1. .( - - 1 ) t G2(K, .... F--~K)1) Jm~'(~lKS J,(fl~,K) e ic''x

+4(-1) a

Io 4~(k) Jm(~',k) J ~ ( ~ k ) e ~ , ~ dk,

(B7)

L

1 ei~'l~x 1(33)= 2~'i(- 1)' G3(K, F , ( K ) ) Jm(~',K) Jl(~g'~K)

-2im+'(-~) ~

I; 0,(~-) I,.(~:-) I~(~:-) e -~:~ d~-

-2i(-1) ~

4~(k) Jm(~k) J ~ ( ~ k ) e ~ , ~ dk

I:

-8i(-1) ~

~4(k) Jm(~'~k) J~(fl~'~k) e i~,kx dk,

(BS)

L

4~-

k~

t~,(k) - (2k 2 - 1 ) 2 + 4 k 2 ~

lx/]-~_ k2,

(2k 2 - 1)2,~-_ k 2 q53(k) - (2k 2_ 1)4+ 16k4(k 2 - t~2)(1 - k2) '

l~-k 2 ~2(k) = (2k2 - 1)2 + 4 k 2 ~

lx/]--~_ k2,

k2( k2 - a 2 ),f-~- k 2

~b4(k) - (2k2 - 1)4+ 16k4(k2_ a2)( 1 _ k2),

k(2k 2 - 1 ) ~ l~-k 2 ~bs(k) - (2k2 _ 1)4+ 16k4(k 2 _ a2)( 1 _ k2 ), r

=

~+: (2~.2 + 1)2 _ 4 z 2 ~

(a9)

141414141414141414?~: l,,/]--~r2,

02(~') = (2r2 + 1)2_ 4 7 . 2 ~

9 (2z2+ 1 - 2 ~ l~f~ z 2) 4'3('0 - (2r:+ 1)5 - 4 r 2 ~ 4 1 - - + - - ~ 2'

1,~-~z2, (BIO)

in which F ' ( K ) = [ d F ( k , 1)/dk]k=,,, K is the Rayleigh pole and Ira(), Kin( ) denote the modified Bessel functions of the first and second kinds.