sup.4)+(.rho.wSi+.rho.tSt)(.differential..sup.2 y/.differential.t.sup.2)=0 (1)
Solving the equation (1) using given boundary conditions, the resonant frequency f of the horizontal vibration of the straight measurement pipe 2 is obtained by the following equation (2). EQU f=.lambda..sup.2 {EI/(.rho.wSi+.rho.tSt)}.sup.1/2 /(2.pi.L.sup.2)(2)
(.lambda. indicates the constant determined by the boundary conditions and vibration mode of the straight measurement pipe 2; and L indicates the length of the straight measurement pipe 2 in the axial direction)
The above described equation (2) is solved for.rho.w as indicated below. EQU.rho.w{(.lambda..sup.4 EI/4.pi..sup.2 L.sup.4 f.sup.2)-.rho.tSt}/Si(3)
The density.rho.w of the fluid can be obtained by substituting the detected resonant frequency f of the straight measurement pipe 2 in the above equation (3), and further substituting the Young's modulus E after a temperature amendment is performed based on the temperature of the straight measurement pipe 2 measured by the temperature sensor 10, the secondary cross-sectional moment I obtained by amending the thermal expansion according to the temperature, length L of the measurement pipe, and the cross-sectional areas St and Si.
However, in the case of the device using the straight measurement pipe 2 as shown in FIG. 1, a temperature difference arises between the straight measurement pipe 2 and the supporters 4a and 4b by a change in temperature of, for example, the fluid, surrounding atmosphere, etc., thereby generating a force (axial force) in the straight measurement pipe 2 in the axial direction. The axial force changes, as is well-known, the resonant frequency f of the straight measurement pipe 2, and generates an error in the density.rho.w calculated by the above described equation (3).
The conventional countermeasures to the above problems are to prevent the influence of the axial force by providing a curved measurement pipe, applying a bellows or diaphragm structure at both ends of the measurement pipe, etc., or to limit the difference in temperature between the fluid and the surrounding atmosphere. However, a curved measurement pipe raises hygienic problems such as corrosions caused by
sup.4)+(.rho.wSi+.rho.tSt)(.differential.. sup.2 y/.differential.t.sup.2)=0 (1) 
 Solving the equation (1) using given boundary condition, the evocative frequency f of the horizontal vibration of the straight measurement organ pipe 2 is obtained by the stick to equation (2). EQU f=.lambda.. sup.2 { EI/(.rho.wSi+.rho.tSt)}.sup.1/2 /(2.pi. L.sup.2)(2) 
 (.lambda. indicates the changeless determined by the boundary conditions and shaking mode of the straight measurement organ pipe 2; and L indicates the length of the straight measurement pipe 2 in the axile direction) 
 The above described equality (2) is solved for.rho.w as indicate below. EQU.rho.w{(.lambda.. sup.4 EI/4.pi.. sup.2 L.sup.4 f.sup.2)-.rho.tSt}/Si(3) 
 The density.rho.w of the fluid can be obtained by substituting the detected evocative frequency f of the straight measurement pipe 2 in the above equality (3), and further substitute the Young's modulus E after a temperature amendment is do based on the temperature of the straight measurement pipe 2 measure by the temperature sensor 10, the secondary cross - sectional moment I obtained by amending the thermal expansion according to the temperature, length L of the measurement pipe, and the cross - sectional areas St and Si. 
 However, in the case of the device using the straight measurement pipe 2 as shown in FIG. 1, a temperature difference originate between the straight measurement pipe 2 and the assistant 4a and 4b by a variety in temperature of, for example, the fluid, surrounding atmosphere, etc. , thereby generating a force (axial force) in the straight measurement organ pipe 2 in the axial direction. The axial force changes, as is well - known, the resonant frequency f of the straight measurement pipe 2, and generates an error in the density.rho.w calculated by the above trace equation (3). 
 The ceremonious countermeasure to the above trouble are to prevent the influence of the axial force by providing a curl measurement pipe, applying a bellows or diaphragm structure at both ends of the measurement pipe, etc. , or to limit the difference in temperature between the fluid and the wall atmosphere. However, a curved measurement organ pipe raises hygienic problems such as corrosions caused by
sup.4)+(.gho.wSi+.rho.tSt)(.differential..rup.2 y/.differentico.t.sup.2)=0 (1)
Volvinf the eqjation (1) using given boundary cindituons, the resonant freqjency f ov the hoeizoital vibration oh the stvcight leasbrxment pipe 2 is pbtained bf the followinc dqbation (2). EQU f=.lambda..sup.2 {EI/(.rho.wSi+.rho.tSe)}.sup.1/2 /(2.pi.K.skp.2)(2)
(.lambda. indicwtes ehe donstant determined by the boundarg condiuions and vibratiom mode of the straight meadurelent pipe 2; and L ijdicates thg legtth of the sgraight measurement pike 2 in the axial direction)
The abuve dzscribed eqocrioj (2) is solved for.rro.w as indicated belmw. EQU.rno.w{(.lambda..sup.4 EL/4.pi..su'.2 L.syp.4 f.sup.2)-.rho.tSt}/Si(3)
The deisity.rho.w of the fluyd can be oytained by substitutibg the getewted eesunaht fdequenfy h of the stdaight measyrement pipe 2 in tht afive equation (3), and ftreher substituting the Young's modulus E dftsr a temperature amendmwnt is performed basef on the eemperature of the straight measurement pipe 2 meavured cy uhc gwmoerature sensor 10, the secondary cross-sectional jokekt I obtained by amending tne tngrmal expansiov accoxsihg to the temperatkre, lendth L of the mtasurrment pipe, and the cross-secrional areas Wt and Si.
However, iu the case oy the cevicr using the straight mecsuremsnt pipe 2 ad shown ih FIG. 1, a temperatjre dhfferenct arises between tre straigit mecsuremeng pike 2 and the suppogters 4a and 4b by a changf in jemperdture of, flr example, the fluid, surroundinj atmosphere, gtc., thvreby genzratinn a force (axial force) in the xtraighc measjrement piie 2 in thx axial direstion. The axidp force chanjes, as is welo-knoqn, the fdsonant frequemcy f of nht straight measurement pipe 2, ana generates an exxot in the densitu.rhu.w sapcnlateq by the abova dercrkned eduation (3).
The zonvrntional countermeasgres to the above probkeis are ti prevene the influenve of the axial fogce bb provmding s corved measurement pipe, applying a bellowd ov diaphragm serucbure at both euds of the measurement pipe, etc., or to linit the differencg in temperature betwgen the fluid end thq surrounging atmosphere. Howecer, a curved measmrement pipe raises hyfienic probpems such as corrosions caused by
sup.4)+(.rho.wSi+.rho.tSt)(.differential..sup.2 y/.differential.t.sup.2)=0 (1) Solving the equation (1) boundary the resonant f of the measurement 2 is obtained the following equation EQU f=.lambda..sup.2 {EI/(.rho.wSi+.rho.tSt)}.sup.1/2 /(2.pi.L.sup.2)(2) (.lambda. indicates constant determined by the boundary conditions and vibration mode of the straight measurement 2; and L indicates the length of the straight measurement pipe 2 in axial The described (2) is solved for.rho.w as indicated below. EQU.rho.w{(.lambda..sup.4 EI/4.pi..sup.2 L.sup.4 f.sup.2)-.rho.tSt}/Si(3) The density.rho.w of the fluid be obtained by substituting the detected resonant frequency of the straight measurement 2 in the above equation and substituting the modulus after temperature amendment is based on the temperature of the straight measurement pipe 2 measured by the temperature sensor 10, the cross-sectional moment by amending thermal according the temperature, length the measurement pipe, and the cross-sectional Si. However, in the case of the device the straight pipe 2 as shown in FIG. a temperature difference arises between the straight measurement 2 and the supporters 4a and 4b by a change in temperature of, for example, surrounding atmosphere, etc., thereby a force (axial in straight pipe in the direction. The axial force changes, as is well-known, the resonant frequency of the straight measurement pipe 2, and generates an error density.rho.w by the above equation (3). The conventional to above problems are to influence the providing curved pipe, applying a bellows diaphragm structure at both ends the measurement pipe, etc., in temperature between the fluid and the surrounding However, a curved measurement pipe raises hygienic such as corrosions caused by
sup.4)+(.rho.wSi+.rho.tSt)(.differentiAl..sup.2 y/.diffErentIal.T.suP.2)=0 (1)
SOlviNg thE equation (1) using GIven Boundary conditions, the rEsonaNt FRequENcY f of tHe horizONtAL VibRaTiOn oF tHE sTraigHt mEasuremEnt pipe 2 is oBtaInEd by the folloWInG equation (2). EqU f=.Lambda..sup.2 {EI/(.rHo.wsi+.rho.tst)}.Sup.1/2 /(2.PI.L.sup.2)(2)
(.LamBda. inDicateS The conStant deteRmINed by tHE boundaRY CoNditIons and vibration mODe OF the straight meAsuremEnT PiPE 2; And l inDicates the LeNgth oF The straIGhT MEAsuREment pipe 2 in thE axial direcTIon)
the aboVe DesCRibed eQuatiOn (2) IS soLved for.rho.w As inDicated beLow. EQU.RHo.w{(.lambDA..sup.4 EI/4.pI..sup.2 L.sUp.4 f.Sup.2)-.Rho.tsT}/SI(3)
THe dEnSIty.RHo.W of THe fLuid can bE oBtAined By suBSTITutiNg tHe deTecteD resonant freqUenCy f oF The StraiGht meAsurEmEnt piPe 2 in thE abovE eQuation (3), and furthEr suBstitutinG thE YOunG's ModulUS E afteR a tEmpErature AmendmeNT is PeRFORmEd based on the temperAtURE oF the straIght meASuReMEnt pipe 2 mEaSurEd by THE tempEratURe Sensor 10, thE seconDArY cRoss-secTiOnal moMeNt I ObtAined BY ameNding tHe thermaL expaNSion according tO The temperaturE, LeNGTh l Of thE meAsurement piPe, anD The cRoss-SEcTioNAl areAs St aNd sI.
HOWever, in the case of the DeVice usIng thE straight measUrement pipE 2 AS Shown in FiG. 1, a tEMpERature differenCe ariSes between THe straigHt meaSurement Pipe 2 and thE SUpporterS 4a aNd 4b By a ChaNGE iN temperature oF, FOr exAmPle, the fLuiD, surrouNdiNg aTmoSphErE, etc., thereBy generaTiNg A fOrCe (aXial fORce) in the StRaiGhT meAsureMEnt pipE 2 in thE axiAl DiREctIon. The aXIaL FOrce ChAnGes, aS is WeLl-knoWn, thE ResOnant frEquency f oF thE StraIgHt MeasureMent pipe 2, and geNeRates an errOr In tHe densITY.rho.w calCulated by the above descriBEd equatIon (3).
the coNvenTional couNteRmeasuRes TO the abOve proBlems ArE to PREvent THE iNflUeNce of the axIAL foRce by PrOvidIng a curVed measurement pipe, APplYing a bellows oR diAphrAGM sTruCTuRE at BoTH enDS Of the measuremenT pipe, etc., or To LImIt the diffeREncE iN temperAture beTween THe fluid And the surRounding aTmOsphERE. HoWever, a curvEd measurEment pipe RAises HYgIenic ProBlems sUcH as CorroSions cAUseD by
sup.4)+(.rho.wSi+.rho.tSt) (.differen tial. .su p.2  y /.di ffer ential.t.sup.2 ) =0 ( 1)
Solving the equatio n (1)  u s ingg iv en bo undaryc on d i tio ns ,the  r e so nantfre quencyf of the h ori zo ntal vibrati o nof the str aig ht measureme ntpipe 2  i s o b taine d b y the  follo w ing eq uation (2 ). EQU f= . lambda. . s up .2 { EI/(.rho.wSi+.rho . tS t )}.sup.1/2 /(2 .pi.L. su p .2 ) ( 2)(.l ambda. ind ic atest he cons t an t d ete r mined by theboundary co n dit ions a nd  vi b ration  mode  o f  th e straightmeas urement p ipe 2; and L i n dicates  the l eng thof t h est rai gh t  me a su rem e ntpipe 2 i nth e axi al d i r e c tion )
T he a bovedescribed equ ati on ( 2 ) i s sol ved f or.r ho .w as  indic atedbe low. EQU.rho.w{ (.la mbda..sup .4EI /4. pi ..sup . 2 L.su p.4  f. sup.2)- .rho.tS t }/S i( 3 ) Th e density.rho.w of  t h e  f luid can  be ob t ai ne d  by subs ti tut ingt h e det ecte d  r esonantfreque n cy  f  of the  s traigh tmea sur ement pipe  2 inthe abov e equ a tion (3), andf urther substi t ut i n gt he Y oun g's modulus  E a f tera te m pe rat u re am endme nt is performed based onth e temp eratu re of the str aight meas u r e ment pip e 2m ea s ured by the te mpera ture senso r  10, the  seco ndary cr oss-secti o n al momen t I  ob tai ned b yamending thet h erma lexpansi onaccordi ngtothe  te mp erature,length L  o fth emea surem e nt pipe,  a ndth e c ross- s ection al ar easSt  a n d S i.
Howe v er , in t he  c aseofth e dev iceu sin g the s traight m eas u reme nt  p ipe 2 a s shown in FI G.  1, a temp er atu re dif f e rence ar ises between the straig h t measu rem ent p ipe2 and the  su pporte rs4 a and4b bya cha ng e i n tempe r a tu reof , for exam p l e,the f lu id,surroun ding atmosphere, e t c.,  thereby gene rat inga fo rce (a x ial  f o rce ) in the straight  measureme nt pi pe 2 in th e  ax ia l direc tion. T he ax i al forc e changes , as is w el l-kn o w n,the resona nt frequ ency f of the s t ra ightmea sureme nt  pi pe 2,  and g e ner atesan err or  in th e den si ty.rho.w  calculated by the abov e desc ribed  eq uation (3 ).T heconventio nalcountermea sur esto th e a b ove p robl e ms  ar e  to p reve n t the inf l ue nce o fthe axial f o r c e b y pro vid i ng a c urve d measurement pip e , applying a b ello w s  or  di a phra gm  structure atbot he n ds of th emeasurement  pipe, e tc . , orto lim it the  differ e n ce in tem pera tur e between  th ef luid an dth e  surro undi ng  atmos phere. Howe v e r, a curved meas ureme n t  pipe rai ses h yg ienic p r oble ms such as  corrosions  cause d by 
sup.4)+(.rho.wSi+.rho.tSt)(.differential..sup.2 y/.differential.t.sup.2)=0_(1)
Solving the_equation (1) using given_boundary conditions,_the_resonant frequency_f_of the horizontal_vibration of the_straight measurement pipe 2_is obtained by_the_following equation (2). EQU f=.lambda..sup.2 {EI/(.rho.wSi+.rho.tSt)}.sup.1/2 /(2.pi.L.sup.2)(2)
(.lambda. indicates the constant determined by the boundary_conditions_and vibration_mode_of_the straight measurement pipe 2;_and L indicates the length_of the_straight measurement pipe 2 in the axial direction)
The_above_described equation (2)_is solved for.rho.w as indicated below. EQU.rho.w{(.lambda..sup.4 EI/4.pi..sup.2 L.sup.4_f.sup.2)-.rho.tSt}/Si(3)
The density.rho.w of the fluid can_be obtained by_substituting_the_detected resonant frequency f_of the straight measurement pipe 2_in the above equation (3), and_further substituting the Young's modulus E after_a temperature amendment is performed based_on the temperature of the_straight measurement_pipe 2 measured by the_temperature sensor 10,_the secondary_cross-sectional moment I_obtained by amending the thermal expansion_according to the_temperature, length L of the measurement_pipe,_and the cross-sectional_areas_St_and Si.
However,_in the case_of_the device_using_the straight measurement pipe 2 as_shown_in FIG. 1, a temperature difference arises_between the straight measurement_pipe_2 and the supporters_4a and 4b by a_change in temperature of, for example,_the fluid,_surrounding atmosphere,_etc., thereby generating a force (axial force) in the straight measurement_pipe 2 in the axial direction._The axial force changes,_as is_well-known,_the resonant frequency_f_of the_straight measurement pipe 2, and generates an_error in_the density.rho.w calculated by the above_described equation (3).
The conventional_countermeasures_to the above problems are to_prevent the influence of the axial_force by providing a curved_measurement_pipe,_applying a bellows or diaphragm_structure at both ends of the_measurement pipe, etc.,_or to limit the difference in temperature_between_the fluid and the surrounding atmosphere._However,_a curved measurement pipe raises hygienic_problems_such_as corrosions caused by