Seismic Responses of Oscillatory System on Rigid Half Circular Platform with Lining Outside

Transcription

1 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp Sesors & Trasducers 4 by IFSA Publishi, S. L. Seismic Resposes of Oscillatory System o Riid Half Circular Platform with Lii Outside epu Shi, Yitao Gao, 3 Jua ei,, 3 Key Laboratory of Advaced Maufacturi ad Cotrol Techoloy i Uiversities of Shado, Yatai Uiversity, Yatai, 645, Shado, Chia Tel.: , fax: swp636793@63.com Received: 8 April 4 /Accepted: 3 May 4 /Published: 3 Jue 4 Abstract: The seismic disturbace is ofte complex waves with several differet frequecies, it is very importat for the roud vibratio trasducer to aalyze the disturbace frequecy compoets i order to predict the earthquake disaster ad the occurrece of the earthquake. But due to the mutual effects betwee the roud ad the trasducer, It is a complex elastic wave coupli problem which has ot bee discussed so far. I this paper, the ave fuctio expasio method, Newtoia dyamics equatio ad the vibratio theory are used to study the resposes of the oscillatory system attachi to the half circular riid platform with half circular lii surroudi outside it, the theoretical solutio of the problem is obtaied, the researches show that the icidet wave ale has o effect to the resposes of the trasducer because of the lii ad there is a little differece betwee the displacemet resposes of the roud with or without the existece of the trasducer, the results of the ive examples show the correctess of the coclusios obtaied here. Copyriht 4 IFSA Publishi, S. L. Keywords: Half space, Half circular lii, SH wave, Oscillatory system, ave fuctio expasio method, seismic respose.. Itroductio I a smaller rae, the scatteri problem i the half space is the best theoretical model of studyi the propaatio of the seismic waves, which is oe of the reasos taki it as the ivestiatio obect. For the SH scatteri problem i the half space, may problems ca be solved aalytically by usi the wave fuctio expasio method (Che et al. []; Shi et al. []), if the complex fuctio method, the virtual source method ad the Graf formula are used, the the scatteri problems of several scatters i the half space ca also be solved half-aalytically (Ha et al. [3]; Qi et al. [4]; Lia et al. [5]; Lia et al. [6]; a et al. [7]; Li et al. [8]); Shi et al. [9] employed complex method ad Graf additio formula to ivestiate the scatteri problem of a circular cavity existi i a circular area to i-plae elastic wave, Shi [] used wave fuctio expasio method ad Graf additio formula to propose a half-aalytical method which was employed to obtai the ree fuctio about the scatteri problem of several circular cavities i circular area to SH-waves; Hao [] used the classic wave fuctio expasio method with a ew de-coupli techique to study the diffractio of a two-dimesioal (D) semicircular cavity i a half-space uder icidet SHwaves; Zha et al. [] utilized the separatio of variables method ad the sub-reios divisio techoloy to study the solutio of wave fuctios 34

2 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp for two-dimesioal scatteri ad diffractio of plae SH waves iduced by a symmetrical V-shaped cayo with differet shape ratios; Li, et al. [3] employed complex fuctio method ad movi coordiates ad auxiliary fuctios to ivestiate the ati-plae respose of a scalee triaular hill to icidet SH waves; I the vibratio theory book (Ji et al. [4]), may discussios were doe o the problems about the sesor s priciple, over there, people ofte took the roud disturbace as the kow iput, i practice, the roud disturbaces eerally come from the seismic source vibratio, volcaic eruptio, artificial blasti ad so o, the roud respose is fudametally eeded to be determied, it is the basis ad oal of the seismic sesors, these kid of problems ca be solved by usi elastic wave theory ad vibratio theory. But from the literatures, the related researches about these problems are little to see. I this paper, the wave fuctio theory ad vibratio theory are utilized to study the seismic resposes of several oscillatory systems o the half circular riid platform with lii surroudi outside it, ad the theoretical solutios are obtaied, the umerical results of the ive example show the correctess of the coclusios obtaied here.. Problem Model ad Derivatios As show i Fi., the sheari modulus ad the mass body desity of the medium i the half space are μ, ρ respectively, ad the sheari modulus ad the mass body desity of the lii medium are μρ, respectively, the sheari modulus ad the mass body desity of the half riid body medium are μ, ρ ( μ μ( μ ), ρ ρ( ρ )) respectively; the iside radius ad outside radius of the lii are RR, respectively, ad the boudaries of the lii are deoted by, respectively. Suppose the oscillatory system is fixed o the half circular riid body, ad it is also fixed o the lii. (I) Half circular riid body α o y α μρ, Half circular lii Oscillati system Fi.. Aalytical model. x (r) Geerally speaki, for the steady scatteri problem of the out-of-plae waves, the displacemet of the medium is U( r )exp( i ωt), it satisfies with the followi Helmholtz equatio U + K U, () where U () r is the displacemet amplitude, r (, r θ ) is the locatio vector of the poits i the medium, K ω / V is the wave umber; V is the wave velocity, it is defied as the square of the ratio of the sheari modulus ad the mass body desity; ω is the disturbi circular frequecy. Suppose the icidet disturbi displacemet wave is expressed as: () I w exp[ik rcos( θ α)], () where is the amplitude of the icidet wave; α is the icidet ale; i is the imaiary uit, defied as i ; K ω / Vis the wave umber of the half space medium, with defiitio K ω/ V ω/ μ / ρ. I the complete half space, because of the reflectio of the free boudary, the reflectio wave (r) amplitude w is defied as: ( r) w exp[ik rcos( θ + α)], (3) Cosideri of the existece of the lii, the scatteri wave of the lii to the icidet wave ad the reflectio wave, deoted as w which ca satisfy with the boudary free stress coditio of the half space boudary ca be ive as: ( s ) () (s) w A H ( K r)cos( θ ), (4) where A (,,..) are the ukow coefficiets, () H () is the first kid of -th Hakelfuctio. At the same time, iside the lii, there is a refractio wave propaati iside comi from the ( f ) outside boudary of the lii, deoted as w, which also satisfies with the free stress coditio of the horizotal boudary, ad ca be expressed as: ( f ) () w B H ( Kr)cos( θ ), (5) where B (,,..) are the ukow coefficiets. There is the scatteri waves propaati outside comi from the half riid body, deoted as w ( s), which also satisfies with the free stress coditio o the horizotal boudary, ad ca be expressed as: 35

3 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp ( s ) () w C H ( Kr)cos( θ ), (6) where C (,,..) are the ukow coefficiets. Besides, the respose displacemet half riid body is defied as: ( ) w () of the w exp( i ω t), (7) where is the out-of-plae displacemet amplitude of the half riid body, which eeds to be determied. Based o the above aalyses, the total displacemet amplitude i the half space medium, ca be ive usi the superpositio priciple: w + w + w, (8) (I) (r) (s) The total displacemet amplitude which is the total wave fuctio i the lii, ca also be obtaied by superpositio priciple as followi: w + w, (9) (s) (f) Accordi to the boudary coditios of the problem show i Fi., the followi equatios ca be ive: μ r r μ, () Besides, the half circular riid body also satisfies the followi dyamic equatio: μ Rd F θ + r, () R ω ρ where F is the acti force amplitude of the oscillatory system to the half circular riid body, which is to be determied. The time factor exp( i ωt) is all omitted i the above equatios (8~). Furthermore, for several lumped parameter vibratio systems (Temporarily ot thiki of the dampi ad frictio i the vibratio system), F is also the force of the half circular riid body acti o all oscillatory systems, so the equatios of which the vibratio systems satisfy with (time factor exp( i ωt) omitted) are kζ mω ( + ζ ) Q, () F kζ where k, m(,,..., Q) are the stiffess coefficiet ad the mass of the th oscillatory system; ζ is the displacemet amplitude of the th vibrator relative to the half circular riid body. Followi equatio (), the followi equatio roup ca be obtaied: ω ζ F m,,,..., Q k mω Q kmω k mω, (3) Cosideri of the equatios (9,,, 3), the followi equatio ca be deduced: d θ β, (4) r Q km Rρ where β [ + ] ω. μrk ( mω ) μ Equatio (4) ca be simplified as: B χ ( ) + C χ ( ), (5) B C '() () KH + βh, where χb ( ), () β H, '() () KH + βh, χc ( ). () β H, From equatio (5), the followi equatio ca be obtaied: where () '() KH B () '() KH KR C β β BH cos( θ ) + +, (6) CH cos( θ) + ( ) () () '() H H H, () () '() H H H. 36

4 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp Cosideri of the first coditio of the boudary coditio (), the followi equatio ca be obtaied: AH + BH, (7) + CH i Jcos( α) () () () ε, where ε, J () is the first kid of 4, th Bessel fuctio,,,.... Cosideri of the secod coditio of the boudary coditio (), the followi equatio ca be obtaied: AH + ψ BH, (8) + ψ CH ε i Jcos( α) where '() '() '() ' H H () H (), () () '() H () H+ () H () ; ' J () J+ () J () ; ψ μk / μ K μv /( μ V) () () '() + () μρ /( μ ρ ),,,... Solve the equatios (6~8), the ukow Coefficiets A, B, C(,,,...) ca be obtaied. For the seismic sesor, the ratio modulus γ (,,..., Q) of the vibrator displacemet to the half circular riid body is a importat parameter, which ca be expressed as: γ ζ / Kmω [ B H C H ] '() '() + ( k mω ) β Cosideri of the above equatio, there are oly the coefficiets B, Ceeded to be determied. So the equatios about B, Cca be obtaied: where qb + qc b, (9) qb + qc b '() () q χ () ( ) ( ) B KH KR + βh KR, '() () q χ () ( ) ( ) C KH KR + βh KR, '() () q ψ H ( KR ) H ( KR ) () '() H ( KR ) H ( K R ), b ; q ψ H ( KR ) H ( K R ) '() () () '() H H, ' () b ε [ J H '() J H ] ; Solve the equatio (), B, Cca be obtaied as: bq B q q bq C q q bq q q bq q q, () I order to aalyze the displacemet of the half space medium ad the lii ad riid platform, the followi equatio s roups eed to be solved: BχB() + CχC(), () () AH + BH () + CH ε J( KR ), '() '() AH + ψ BH '() ' + ψch εj( KR ) BχB( ) + CχC( ), () () AH ( KR ) + BH () + CH εi J( KR )cos( α), '() '() AH ( KR ) + ψ BH '() ' + ψ CH εi J( KR )cos( α) () () The displacemet amplitude of the riid platform ca be deduced as: K[ B H + C H ]/ β, (3) '() '() If the oscillatory system is ot cosidered here, the displacemet amplitudeu of the riid platform ca be expressed as: U K[ B H + C H ]/ β, (4) '() '() where β.5 Rρω / μ. The differece modulusζ is defied as: ζ U, (5) 3. Numerical Results of Example ad Discussios I this paper, oly two oscillatory systems are cosidered, the umerical results about γ, γ ad ζ are ive as Fis. ~7. 37

5 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp Ratio modulus γ ad γ ave umber of icidet wave K γ, γ alo Fi.. Variatios of the ratio modulus with the dimesioless wave umber K. Ratio modulus γ ad γ 4 x ave umber of icidet wave K γ, γ alo Fi. 3. Variatios of the ratio modulus with the dimesioless wave umber K. 8 x -3 x -9 Ratio modulus γ ad γ 6 4 Differece amplitude ζ ave umber of icidet wave K ave umber of icidet wave K γ, γ alo Fi. 4. Variatios of the ratio modulus with the dimesioless wave umber K. Fi. 5. Variatios of the differece amplitudeζ alo with the dimesioless wave umber K. 3 x - 3 x - Differece amplitude ζ Differece amplitude ζ ave umber of icidet wave K ave umber of icidet wave K Fi. 6. Variatios of the differece amplitudeζ alo with the dimesioless wave umber K. Fi. 7. Variatios of the differece amplitudeζ alo with the dimesioless wave umber K. The kow coditios of every example are listed as followi. Example : The kow parameters are listed as followi: ρ / ρ. /.7, μ / μ /3, 5 ρ, m, m 5, k 5, k, R / R 5/4,, K. ~.5 ;the computatio results are show i Fi. ad Fi. 5. Example : The kow parameters are listed as followi: ρ / ρ. /.7, μ / μ /3, ρ, m, m 5, k 5, k 5, R / R 5/4,, K. ~.5,the computatio results are show i Fi. 3 ad Fi. 6. Example 3: The kow parameters are listed as followi: ρ / ρ. /.7, μ / μ /3, ρ, m, m 5, k, k, R / R 5/4,, K. ~.5, the computatio results are show i Fi. 4 ad Fi

6 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp From the above computatio results, whe the atural frequecy of the oscillatory system is i close proximity to the disturbace frequecy, the system resposes of the oscillatory systems are very stro (show i Fi., Fi. 3, Fi. 4), differet frequecy of the vibrati system will have differet reactios (show i Fi. ad Fi. 4); he two systems have similar atural frequecies, they will have similar resposes (show i Fi. 3). Because the seismic disturbace wave is ofte a complex wave with several frequecies, by desii a vibrati system roup with differet atural frequecy, they will have differet stro resposes to the icidet wave, to some extet, the frequecy compoets of the disturbace ca be aalyzed by usi their differet resposes, this is very importat for further studies o the characteristics ad the effects of the earthquake. Of course, the aalyses ad the coclusios i this paper ca also be used for further calculatio of displacemet field ad stress wave field i the medium. From the above theoretical aalyses, because of the existece of the lii, the icidet ale of icidet wave has o obvious effect o the displacemet respose of the vibratio systems. I additio, i eeral books about the vibratios, the ive examples about the roud trasducer are ofte discussed assumi that the roud motio is kow, but i practice, the roud motio is ot easy to kow, because the roud trasducer is attached to the roud, it has iertia, whe the roud is movi due to the icidet wave disturbace, the trasducer has a resistat force to the roud motio, there are some differeces betwee ad U, that is to say, ζ is ot zero, but from the umerical results show i Fi. 5, Fi. 6 ad Fi. 7, ζ is approximate to zero with hih frequecy, ζ is bier with low frequecy, such results is due to the mutual effects betwee the trasducer ad the roud, so the roud motio is ofte ukow beforehad, it should be eerally solved toether with the resposes of the oscillatory systems, that is to say, this kid of problem is ofte a coupled problem of the wave ad vibratio, which ca be easily foud from the equatio (), the results of the ive examples show the correctess of this coclusios. From the above discussios, if the oscillatory systems are ot istalled o a riid body, it is directly istalled o a elastic medium (such as the roud), the ivolved elastic wave problem would be too difficult to be solved, which ca oly be umerically solved, because the reasoable aalytical model ca ot be ive, however, whe the oscillatory system is istalled o a riid body, the problem will be a simpler oe which ca be solved with hih precisio. This idea ca be used for referece i the desi of the seismic trasducer. 4. Coclusio I this paper, the wave fuctio expasio method ad the Newtoia dyamics equatio are used to study the seismic resposes of several oscillatory systems attached to the half circular riid body with lii surroudi outside o the surface of the half space to the out-of-plae seismic disturbace, the followi coclusios ca be obtaied: ) The accurate aalytical solutios ca be obtaied; ) differet vibrati systems with differet atural frequecies have differet resoat resposes to the seismic disturbace, they ca feel the differet disturbace frequecy compoets respectively. 3) To some extet, the existece of the lii has shieldi effects which ca make the respose of the vibratio sesor ot obvious to the ale of the icidet wave. 4) A vibrati system with differet atural frequecies ca be desied o a platform, which ca sese the icidet disturbace with differet frequecies, this is very importat for aalyzi the frequecy compoets of the complex icidet waves. 5) The trasducer attached to the roud has effects o the seismic resposes of the roud, but the effects oly bri a little differece which ca ofte be omitted for better approximatios, but i some occasios, the differeces may be bi which caot be omitted. 6) The method proposed here ca also be used i solvi the similar coupled problems with several cavities or flaws i the half space with the trasducer attached o the roud. Ackowledemets The research was supported by Shado Provice Sciece ad Techoloy Research Proect (G3) ad Shado Provicial Natural Sciece Foudatio (ZRAM), Chia. Refereces []. Z. G. Che, D. K. Liu, Gree fuctio i elastic half space impacted by out plae harmoic lie source loadi at horizotal surface, Joural of Harbi Eieeri Uiversity, Vol. 3, Issue 4,, pp. -5. [].. P. Shi, S. J. Fa, C. P. Zha, et al., Fuzzy respose of the scatteri of the half circular cayo o the boudary of the half space to SH waves, Joural of Mechaical Streth, Vol. 33, Issue 3,, pp [3]. F. Ha, G. Z. a, H. Che, Research o scatteri of SH-waves o multiple hills ad cayos, Applied Mathematics ad Mechaics, Vol. 34, Issue 4, 3, pp [4]. H. Qi, G. C. Zha, D. N. Che, et al., Scatteri of SH-wave by the circular lii with a iterface crack i a bi-material half-space, Explosio ad Shock aves, Vol. 3, Issue 5,, pp

7 Sesors & Trasducers, Vol. 73, Issue 6, Jue 4, pp [5]. J.. Lia, J. Q. Che, Z. N. Ba, 3D scatteri of obliquely icidet SH waves by a cylidrical cavity i layered elastic half-space, Methodoloy ad verificatio, Acta Seismoloica Siica, Vol. 34, Issue 6,, pp [6]. J.. Lia, X. Y. Mei, Z. N. Ba. Three-dimesioal scatteri by a alluvial valley i a layered half-space for obliquely icidet plae SH waves, Earthquake Research i Chia, Vol. 8, Issue 4,, pp [7]. H.. a, D. K. Liu, F. Q. Qiu, et al. Dyamic aalysis of multiple semi-cylidrical alluvial valleys with multiple subsurface circular cavities uder SH waves, Joural of Natural Disasters, Vol. 5, Issue 3, 6, pp [8]. M. Li,. Be, Y. T. Fe, Dyamic stress aalysis for homocetric cavities uder a rae of semicylidrical homocetric hills impacted by SH-wave, orld Earthquake Eieeri, Vol. 4, Issue, 8, pp [9].. P. Shi, J. ei, Studies o the scatteri problem of circular cavity i circular area to i-plae wave, Joural of Applied Scieces, Vol. 3, Issue, 3, pp [].. P. Shi. Gree fuctio solutio of scatteri problem of several circular cavities i circular area to SH-waves. Joural of Applied Scieces, Vol. 3, Issue, 3, pp []. H. Luo, V.. Lee ad J.. Lia, Ati-plae (SH) waves diffractio by a uderroud semi-circular cavity: aalytical solutio, Earthquake Eieeri ad Eieeri Vibratio, Vol. 9, Issue 3,, pp []. N. Zha, Y. F. Gao, D. Y. Li, et al., Scatteri of SH waves iduced by a symmetrical V-shaped cayo: a uified aalytical solutio, Earthquake Eieeri ad Eieeri Vibratio, Vol., Issue 4,, pp [3]. S. Z. Li, F. Q. Qiu, D. K. Liu, Scatteri of SH waves by a scalee triaular hill, Earthquake Eieeri ad Eieeri Vibratio, Vol. 9, Issue,, pp [4].. M Ji, T. Fa, S. Q Che, Mechaical Vibratio, Beii, Sciece Press, Copyriht, Iteratioal Frequecy Sesor Associatio (IFSA) Publishi, S. L. All rihts reserved. ( 4