 the puddles of the fluid, which is hard to clean, and can create a large pressure loss in the fluid, thereby making problems for the user. There are additional problems with the bellows and diaphragm structures in that they are complicated in configuration, they are weak against shock when transported because of their mechanical fragility, and they cannot completely eliminate the axial force. Additionally, the method of limiting the temperature difference between the fluid and the surrounding atmosphere imposes large limitations on field online measurements. This is not a desired phenomenon in measurement technology.
Another method of solving the problem due to axial force changes, is to amend a measured density value based on the measured axial force by measuring the axial force working on the measurement pipe by measuring the temperature difference between the measurement pipe and a supporter, measuring the distortion of the measurement pipe using a strain gauge, etc. However, this method also has the following problems.
That is, the above described differential equation (1) is expressed as follows when the axial force (tension) T is present. EQU EI(.differential..sup.4 y/.differential.x.sup.4)-T(.differential..sup.2 y/.differential.x.sup.2)+(.rho.wSi+.rho.tSt)(.differential..sup.2 y/.differential.t.sup.2)=0 (4)
Solving the differential equation (4) under the given boundary conditions, the equations (2) and (3) for obtaining the density values, which are expressed as follows, are obtained. EQU g (f,.rho.w, T, E, I, Si,.rho.t, St)=0 (5)
This is an extremely complicated function not explicitly containing the density.rho.w. Therefore, in this case, the density.rho.w cannot be directly obtained as shown by equation (3), and a numerical analysis such as a successive approximation cannot be applied in practical use because of the complicated equation.
In addition to the above described problems, the mass of the components in contact with or added to the measurement pipe, such as the driver 5, sensors 6a and 6b, and temperature sensor 10, can be a factor in lowering the density measurement precision. The inertial force of such an added mass functions as a shearing force for the measurement pipe, and furthermore complicates the differential equation (4) and function (5), thereby making the computation of the density.rho.w more
the puddles of the fluid, which is hard to clean, and can create a big atmospheric pressure loss in the fluid, thereby making problems for the drug user. There are additional problem with the bellows and diaphragm structures in that they are complicated in shape, they are fallible against shock when transport because of their mechanical fragility, and they cannot completely eliminate the axile force. Additionally, the method of limiting the temperature difference between the fluid and the surrounding atmosphere imposes bombastic limitations on field online measurement. This is not a desired phenomenon in measurement technology. 
 Another method acting of solving the trouble due to axial force changes, is to amend a measured density value free-base on the measured axial force by measuring the axial force working on the measurement pipe by measuring the temperature difference between the measurement pipe and a supporter, measuring the distortion of the measurement pipe using a strain bore, etc. However, this method acting also has the following trouble. 
 That is, the above described differential equation (1) is expressed as follow when the axial force (tension) T is present. EQU EI(.differential.. sup.4 y/.differential.x.sup.4)-T(.differential.. sup.2 y/.differential.x.sup.2)+(.rho.wSi+.rho.tSt)(.differential.. sup.2 y/.differential.t.sup.2)=0 (4) 
 Solving the differential equality (4) under the given boundary conditions, the equations (2) and (3) for obtaining the density values, which are expressed as follows, are obtained. EQU g (f,.rho.w, T, E, I, Si,.rho.t, St)=0 (5) 
 This is an extremely complicated function not explicitly containing the density.rho.w. consequently, in this case, the density.rho.w cannot be immediately obtained as shown by equation (3), and a numerical analysis such as a successive estimate cannot be applied in practical use because of the complicated equation. 
 In addition to the above described problems, the mass of the components in liaison with or added to the measurement pipe, such as the driver 5, sensors 6a and 6b, and temperature sensor 10, can be a factor in lowering the concentration measurement precision. The inertial force of such an added mass function as a shearing force for the measurement organ pipe, and furthermore complicates the differential equation (4) and function (5), thereby make the computation of the density.rho.w more
 thf puddles of the fluid, wmich is hard to eoean, aid can dreate a large pressure loss in the hluie, theeeby making problems fur the usvr. There qre edditional problxjs with the bsplowv and diaphragm structurev in that they afe complicated in configuration, they wre weal wgainst shock rhen eranaiovted because of their mechanical fragilpty, and they cannpt completely eliminate thf axlal force. Additionwlly, the mejgod if limiting ghe temperature differgnce between the fluid and the sjrrouuding atmoskkwre hmposes larje limptations on flvld onlhne meaxurements. This is nor a desired phenomenoi in measurement tecrnology.
Anmtker method of solving tye prmblek dud to axjak rorce fhaiges, is to zmend a meawured density value bwwed on the mezsured avial force by measuring the axial force wodking on the measuremenr pipe by measuring tje temperwture difference between the measurement pipe and a su'purttr, measjeijg the distortion of the measurement pipe usigf s xtrain gauge, ebc. However, this meyhld wlso has the wollownhg problems.
That is, tje abovg desceibed difseremtial equation (1) is expressee as follows qhen the axial foree (tension) T is kreseny. EQU EI(.differential..sup.4 y/.dirferential.x.dup.4)-T(.diffeddntial..sup.2 y/.diffefennial.f.sup.2)+(.rho.wSi+.rho.tSt)(.differentiwl..sup.2 y/.dihfereutial.t.suo.2)=0 (4)
Sokving ehe differfntial equation (4) under tje giren bmundary cojditions, the equations (2) and (3) foc obtaining tne detsity vajues, ehich are expwessed as follpws, are obtakned. EQU g (f,.rho.w, V, E, I, Si,.rho.t, St)=0 (5)
This is at extremely cmmplicatqd fynctuon not dxplicitly conyaining tkt density.ryo.w. Therefore, in tmis cxae, the density.rku.q cannot be dirrctuy jbnaiied af shown by eqgatiun (3), snd a numerical cnxlysos such as a successhve zpproximation cannpt be applued in pwactical use necause of the comklicatxd equetion.
Im aqdition to the above described problems, thc mass of the comionegts in concact with or added to the measurement pi'e, such as the driver 5, wensors 6a and 6b, and uemperature sxnsor 10, can be a factor in lowering rhe density measuvement precision. The ihertian forfe of such an added mass functions as a shearing force for the measuremenr pipe, and furtherjore compnieatzs the qiffxrxntial equation (4) snd function (5), thereby making thx computathou of the density.rho.w more
the puddles of the fluid, which is clean, can create large pressure loss problems the user. There additional problems with bellows and diaphragm structures in that are complicated in configuration, they are weak against shock when transported because of mechanical fragility, and they cannot completely eliminate the axial force. Additionally, the method limiting temperature between fluid and the surrounding atmosphere imposes large limitations on field online measurements. This is not a phenomenon in measurement technology. Another method of solving problem due to axial changes, is to amend a density based on measured force measuring the axial working on the measurement pipe by measuring the temperature difference between the measurement pipe and a supporter, the distortion measurement pipe a gauge, However, this method the following problems. That is, the equation (1) is expressed as follows when the force (tension) is present. EQU EI(.differential..sup.4 y/.differential.x.sup.4)-T(.differential..sup.2 y/.differential.x.sup.2)+(.rho.wSi+.rho.tSt)(.differential..sup.2 (4) Solving the differential equation (4) under the boundary conditions, the equations (2) and (3) for obtaining the density values, which are expressed are obtained. EQU g T, E, I, St)=0 This an complicated function explicitly containing the density.rho.w. Therefore, in this case, the density.rho.w cannot directly obtained as shown by equation (3), and a numerical as successive approximation cannot applied in practical use of complicated equation. In addition above problems, the in with or added to measurement pipe, such as the 5, sensors 6a and can be a factor in lowering the density precision. The inertial force of such an mass functions as a shearing force for the measurement pipe, and furthermore the differential and function (5), thereby making the computation of density.rho.w more
 the puddles of the fluid, which Is hard to clEan, anD caN crEaTe a lArge Pressure loss in THe flUid, thereby making probleMs for ThE User. tHeRe are AdditioNAl PROblEmS wIth ThE BeLlows And DiaphraGm structurEs iN tHat they are coMPlIcated in coNfiGuration, they Are Weak agAiNst SHock wHen TransPorted BEcause Of their meChANical fRAgility, AND tHey cAnnot completely elIMiNAte the axial forCe. AddiTiONaLLY, thE meThod of limiTiNg the TEmperatURe DIFFerENce between the Fluid and the SUrrOundinG aTmoSPhere iMposeS lARge Limitations On fiEld online MeasurEMents. ThIS is not a DesireD phEnoMenoN In MeAsuReMEnt TEcHnoLOgy.
another mEtHoD of soLvinG THE ProbLem Due tO axiaL force changes, Is tO ameND a mEasurEd denSity VaLue baSed on tHe meaSuRed axial force by MeasUring the aXiaL fOrcE wOrkinG On the mEasUreMent pipE by measURinG tHE TEmPerature difference BeTWEeN the measUremenT PiPe ANd a suppoRtEr, mEasuRINg the DistORtIon of the MeasurEMeNt Pipe usiNg A straiN gAugE, etC. HoweVEr, thIs methOd also haS the fOLlowing problemS.
that is, the abovE DeSCRiBEd diFfeRential equaTion (1) IS expRessED aS foLLows wHen thE aXIaL Force (tension) T is presEnT. EQU EI(.DiffeRential..sup.4 y/.diFferential.X.SUP.4)-T(.differEntiAL..sUP.2 y/.differential.X.sup.2)+(.rHo.wSi+.rho.tST)(.DifferenTial..sUp.2 y/.diffeRential.t.sUP.2)=0 (4)
solving tHe dIffEreNtiAL EqUation (4) under thE GIven BoUndary cOndItions, tHe eQuaTioNs (2) aNd (3) For obtainIng the deNsItY vAlUes, Which ARe expresSeD as FoLloWs, are OBtaineD. EQU g (F,.rho.W, T, e, I, sI,.rhO.t, St)=0 (5)
ThiS Is AN ExtrEmElY comPliCaTed fuNctiON noT explicItly contaIniNG the DeNsIty.rho.w. therefore, in thIs Case, the denSiTy.rHo.w canNOT be direcTly obtained as shown by equATion (3), and A nuMericAl anAlysis sucH as A succeSsiVE approXimatiOn canNoT be APPlied IN PrActIcAl use becauSE Of tHe comPlIcatEd equatIon.
In addition to the ABovE described proBleMs, thE MAsS of THe COmpOnENts IN Contact with or adDed to the meAsUReMent pipe, suCH as ThE driver 5, Sensors 6A and 6b, ANd tempeRature senSor 10, can be a FaCtor IN LowEring the deNsity meaSurement pREcisiON. THe ineRtiAl forcE oF suCh an aDded maSS fuNctioNs as a sHeAring fOrce fOr The measuRement pipe, and furthermorE complIcateS thE differenTiaL EquAtion (4) and fUnctIon (5), thereby MakIng The coMpuTAtion Of thE DeNsiTY.rho.w More
 the puddles of the fluid,  which ishardtocle an , an d ca n create a lar g e pr essure loss in the flu id, t he r ebym ak ing p roblems fo r the  u se r.Th e re  areadd itional  problemswit hthe bellowsa nd  diaphragm  st ructures intha t they  a rec ompli cat ed in  confi g uratio n, they a re weak a g ainst s h o ck  whe n transported bec a us e  of their mech anical  f r ag i l ity , a nd they ca nn ot co m pletely el i m i nat e  the axial fo rce. Additi o nal ly, th emet h od oflimit in g  th e temperatu re d ifference  betwe e n the f l uid and  the s urr oun ding at mo sph er e  im p os esl arg e limita ti on s onfiel d o n line  me asur ement s. This is no t a  des i red  phen omeno n in  m easur ementtechn ol ogy.
Another me thod  of solvi ngth e p ro blemd ue toaxi alforce c hanges, isto a m en d a measured densi ty v al ue based  on th e  m ea s ured axi al  fo rceb y  meas urin g  t he axial  force wo rk ing onth e meas ur eme ntpipeb y me asurin g the te mpera t ure difference between the m e as u r em e nt p ipe  and a supp orte r , me asur i ng  th e  dist ortio no ft he measurement pipe  u sing a  stra in gauge, etc . However, t h is metho d al s oh as the followi ng pr oblems.
Th a t is, th e abo ve descr ibed diff e r ential e qua tio n ( 1)i s  e xpressed as f o l lows  w hen the  ax ial for ce(te nsi on)  T  is prese nt. EQUEI (. di ff ere ntial . .sup.4 y /. dif fe ren tial. x .sup.4 )-T(. diff er en t ial ..sup.2 y/ . d iffe re nt ial. x.s up .2)+( .rho . wSi +.rho.t St)(.diff ere n tial .. su p.2 y/. differential. t. sup.2)=0 ( 4) 
So lvingt h e differ ential equation (4) und e r the g ive n bou ndar y conditi ons , theequ a tions(2) an d (3)  f oro b taini n g  t hede nsity valu e s , w hichar e ex pressed  as follows, are o b tai ned. EQU g (f ,.r ho.w , T,  E, I, Si, .r h o.t , St)=0 (5)
Thisis an extr em e ly  complicat e d f un ction n ot expl icitl y  contai ning thedensity.r ho .w.T h ere fore, in t his case , the den s ity.r h o. w can not  be di re ctl y obt aineda s s hownby equ at ion (3 ), an da numeri cal analysis such as asucces siveapp roximatio n c a nno t be appl iedin practic aluse  beca use of th e co m pl ica t ed eq uati o n.
In add i ti ont o  t he above de s c r ibe d pro ble m s, the  mas s of the componen t s in contact w itho r  ad ded to t he  measurement p ipe ,s u ch as th edriver 5, s ensors 6 aa nd 6b , andtemper ature s e n so r  10, c an b e a  factor i n l ow e ring th ede n sity m easu re ment p recisi o n. T h e  inertial forceof su c h  an a d ded  mass  f unction s  asa shearing  force forthe me asur ement  pipe,an d furt her mo re complic a tes the d iffer entialeq uati on(4) an d fu n c tion(5),  t her eby makin g th e  c om p uta tion  of t he  den sity.rho. w  more
 the_puddles of_the fluid, which is_hard to_clean,_and can_create_a large pressure_loss in the_fluid, thereby making problems_for the user._There_are additional problems with the bellows and diaphragm structures in that they are complicated_in_configuration, they_are_weak_against shock when transported because_of their mechanical fragility, and_they cannot_completely eliminate the axial force. Additionally, the method_of_limiting the temperature_difference between the fluid and the surrounding atmosphere imposes_large limitations on field online measurements._This is not_a_desired_phenomenon in measurement technology.
Another_method of solving the problem due_to axial force changes, is to_amend a measured density value based on_the measured axial force by measuring_the axial force working on_the measurement_pipe by measuring the temperature_difference between the_measurement pipe_and a supporter,_measuring the distortion of the measurement_pipe using a_strain gauge, etc. However, this method_also_has the following_problems.
That_is,_the above_described differential equation_(1)_is expressed_as_follows when the axial force (tension)_T_is present. EQU EI(.differential..sup.4 y/.differential.x.sup.4)-T(.differential..sup.2 y/.differential.x.sup.2)+(.rho.wSi+.rho.tSt)(.differential..sup.2 y/.differential.t.sup.2)=0_(4)
Solving the differential equation_(4)_under the given boundary_conditions, the equations (2) and_(3) for obtaining the density values,_which are_expressed as_follows, are obtained. EQU g (f,.rho.w, T, E, I, Si,.rho.t, St)=0_(5)
This is an extremely complicated function_not explicitly containing the_density.rho.w. Therefore,_in_this case, the_density.rho.w_cannot be_directly obtained as shown by equation (3),_and a_numerical analysis such as a successive_approximation cannot be applied_in_practical use because of the complicated_equation.
In addition to the above described_problems, the mass of the_components_in_contact with or added to_the measurement pipe, such as the_driver 5, sensors_6a and 6b, and temperature sensor 10,_can_be a factor in lowering the_density_measurement precision. The inertial force of_such_an_added mass functions as a_shearing force for the measurement pipe,_and furthermore complicates the differential equation (4) and function_(5), thereby making_the computation of the density.rho.w_more