fig1 - 7 and the following description depict specific examples to teach those skilled in the art how to make and use the best mode of the invention . for the purpose of teaching inventive principles , some conventional aspects have been simplified or omitted . those skilled in the art will appreciate variations from these examples that fall within the scope of the invention . those skilled in the art will appreciate that the features described below can be combined in various ways to form multiple variations of the invention . as a result , the invention is not limited to the specific examples described below , but only by the claims and their equivalents . factors that affect flexural stiffness also affect coriolis flow meter sensitivity ( flow calibration factor ). flexural stiffness is the static spring rate derived from flexing the flow tube with a known force pattern and measuring the flow tube displacement . any force pattern could be used to measure flexural stiffness , as long as it is invariant . as an example , the flexural stiffness for a clamped beam is as follows : k flex = f δ = 192 ⁢ ei l 3 ( 1 ) for a coriolis flow meter , if flexural stiffness changes , then so must calibration factor . flexural stiffness of a coriolis flow meter is defined as : c g — effect of unflexed tube bend geometry on flexural stiffness ; for a straight tube coriolis flow meter with no pre - stress the following expressions show the dependence of calibration factor on ei : m = c ⁡ [ ei l 3 ] ⁢ δ ⁢ ⁢ t ( 3 ) fcf = c ⁡ [ ei l 3 ] ( 4 ) where c is a constant determined by mode shape and pick - off locations . fig1 illustrates a coriolis flow meter 5 capable of directly estimating flexural stiffness by applying a known force to the flow tubes and measuring the resulting deflection . coriolis flow meter 5 comprising a flow meter assembly 10 and meter electronics 20 . meter electronics 20 is connected to meter assembly 10 via leads 100 to provide density , mass flow rate , volume flow rate , totalized mass flow and other information over path 26 . flow meter assembly 10 includes a pair of flanges 101 and 101 ′, manifold 102 and flow tubes 103 a and 103 b . connected to flow tubes 103 a and 103 b are driver 104 , pick - off sensors 105 and 105 ′ and auxiliary position sensor 107 . brace bars 106 and 106 ′ serve to define the axes w and w ′ about which each flow tube 103 a and 103 b oscillates . when flow meter assembly 10 is inserted into a pipeline system ( not shown ) which carries the material being measured , material enters flow meter assembly 10 through flange 101 , passes through manifold 102 where the material is directed to enter flow tubes 103 a and 103 b , flows through flow tubes 103 a and 103 b and back into manifold 102 where it exits meter assembly 10 through flange 101 ′. flow tubes 103 a and 103 b are selected and appropriately mounted to manifold 102 so as to have substantially the same mass distribution , moment of inertia , and elastic modulus about bending axes w - w and w ′- w ′ respectively . the flow tubes extend outwardly from the manifold in an essentially parallel fashion . flow tubes 103 a - b are driven by driver 104 in opposite directions about their respective bending axes w and w ′ and at what is termed the first out of phase bending mode of the flow meter . driver 104 may comprise one of many well known arrangements , such as a magnet mounted to flow tube 103 a and an opposing coil mounted to flow tube 103 b . an alternating current is passed through the opposing coil to cause both tubes to oscillate . a suitable drive signal is applied by meter electronics 20 , via lead 110 to driver 104 . the force pattern generated by driver 104 is fixed over the life of the sensor ( i . e . c p is constant ). driver 104 generates a constant force and auxiliary position sensor 107 determines the tube displacement . as shown in fig1 , the auxiliary sensor would be positioned near the pick - offs 105 or 105 ′ however other suitable locations could be utilized . meter electronics 20 receives the right and left velocity signals appearing on leads 111 and 111 ′, respectively and the tube displacement over lead 112 . meter electronics 20 produces the drive signal on lead 110 which causes driver 104 to oscillate flow tubes 103 a and 103 b . the present invention as described herein can produce multiple drive signals from multiple drivers . meter electronics 20 processes left and right velocity signals and a tube displacement signal to compute mass flow rate and provide the validation system of the present invention . path 26 provides an input and an output means that allows meter electronics 20 to interface with an operator . fig2 shows a process 200 for validating the flow calibration factor of coriolis flow meter 5 using calculated changes in its flexural stiffness . process 200 begins in step 210 by applying a known force to flow tubes 103 a & amp ; b using driver 104 . in step 220 , auxiliary position sensor 107 determines the tube displacement resulting from the force applied by driver 104 . the flexural stiffness , k new , is measured in step 230 using the force and tube displacement from steps 210 and 220 . a previous flexural stiffness , k old , is retrieved in step 240 . this previous flexural stiffness is determined either at the factory or after the meter is installed . step 250 compares k new with k old to determine if there has been a change in the flexural stiffness of the meter . if no change has been detected , step 260 signals a “ no error ” condition and repeats the process . if a change has been detected , step 270 corrects the flow calibration factor ( fcf ) based on the stiffness change . flow tube flexural stiffness can also be determined by estimating points on a tube frequency response function ( frf ) at given frequencies . these points are then used to fit a single degree of freedom model to the data and finally determine the dc ( e . g . zero crossing ) point on the frf . it is possible to write a general expression for any cross ( not collocated ) mobility frequency response function as a linear combination of sdof responses : v f = h ⁡ ( s ) = ∑ i = 1 n ⁢ ⁢ s m i ⁢ s 2 + c i ⁢ s + k i ( 5 ) v f = h ⁡ ( s ) = ∑ i = 1 n ⁢ ⁢ a i ⁢ s s 2 + 2 ⁢ ζ i ⁢ ω i ⁢ s + ω i 2 ( 6 ) where v is the pickoff velocity , f is the driver force and h ( s ) is the frf of interest . the drive loop frequency response can be approximated well in this manner . in fact , at frequencies near the drive frequency , only one term in the summation is required for good approximation of the frequency response . if more accuracy is required in parameter estimation , then more than one mode can be retained in the model . as an alternative to including more modes , the higher frequency modes could be masked from the drive loop frequency response by selecting the location of the driver and pickoff coils . the second bend mode ( twist mode ) can be masked from the drive loop frequency response because the driver is located on the node of the second mode . if the pick - off &# 39 ; s were located on the nodes of the third mode , then the third mode is masked from the drive loop frequency response as well . with this driver / pickoff pattern , the drive loop frequency response is very close to a sdof system . modeling the drive loop as an sdof system results in the following : v f = h ⁡ ( s ) = s m 1 ⁢ s 2 + c 1 ⁢ s + k 1 ( 7 ) the drive loop frequency response can be converted from mobility ( velocity ) to receptance ( displacement ) by integrating ( multiplying by 1 / s ): x f = h ⁡ ( s ) s = 1 m 1 ⁢ s 2 + c 1 ⁢ s + k 1 ( 9 ) x f = h ⁡ ( s ) s = a 1 s 2 + 2 ⁢ ζ 1 ⁢ ω 1 ⁢ s + ω 1 2 ( 10 ) the stiffness term k 1 is well correlated with flexural stiffness , and thus is a good predictor of changes in flow cal factor caused by changes in flexural stiffness . to estimate the stiffness parameter k 1 a single or multiple frequency estimation can be used . fig3 shows a process 300 for validating a flow calibration factor using single frequency estimation . single frequency estimation 300 begins in step 310 by determining a receptance transfer function ( 9 ). in step 320 , the inverse of the receptance transfer function at iω is determined to get the inverse receptance frequency response : ⅈω h ⁡ ( ⅈω ) = m 1 ⁡ ( ⅈω ) 2 + c 1 ⁡ ( ⅈω ) + k 1 1 = m 1 ⁡ ( ⅈω ) 2 + c 1 ⁢ ⅈω + k 1 ( 11 ) lim ω -& gt ; 0 ⁢ ⅈω h ⁡ ( ⅈω ) = lim ω -& gt ; 0 ⁢ m 1 ⁡ ( ⅈω ) 2 + lim ω -& gt ; 0 ⁢ c 1 ⁢ ⅈω + lim ω -& gt ; 0 ⁢ k 1 ( 12 ) lim ω -& gt ; 0 ⁢ ⅈω h ⁡ ( ⅈω ) = lim ω -& gt ; 0 ⁢ k 1 = k 1 ( 13 ) if m 1 is known or is negligible , then a larger frequency closer to ω 1 can be used to estimate k 1 from : k 1 ≃ ⅈω h ⁡ ( ⅈω ) - c 1 ⁢ ⅈω ( 15 ) if c 1 and m 1 are both known , then any frequency can be used to estimate k 1 from : k 1 ≃ ⅈω h ⁡ ( ⅈω ) ⁢ m 1 ⁡ ( ⅈω ) 2 - c 1 ⁢ ⅈω ( 16 ) in step 340 we calculate k 1 from expressions ( 14 ), ( 15 ), or ( 16 ) with ω in radians / second and correct the flow calibration factor based on the calculated k 1 in step 350 . fig4 shows a process 400 for validating a flow calibration factor using multiple frequency estimation . multiple frequency estimation 400 begins in step 410 by identifying constants m 1 , c 1 , k 1 , ζ 1 , ω 1 , a 1 using any time domain or frequency domain system identification method . in step 420 , a curve fitting procedure is used to fit a rational continuous time transfer function model to the complex frequency response vector h at the set of frequencies in vector w ( in radians / second ). the number and location ( in frequency ) of the frf data points does affect the quality of the fit . a good fit is achieved using as few as 2 frequency response data points . the derived model is of the form : the above model is incompatible with the mobility ( velocity ) form of the drive loop frequency response , because there is no way to force b ( n b + 1 )= 0 . in step 430 , we get around this limitation by converting the driver pickoff mobility ( velocity ) frequency response data to the receptance ( displacement ) form . the measured mobility frequency response data h must be multiplied by 1 /( iω ). the measured mobility drive loop frequency response h should be from drive coil current ( proportional to force ) to pickoff voltage ( proportional to velocity ). h ⁡ ( s ) = b ⁡ ( 1 ) a ⁡ ( 1 ) ⁢ s 2 + a ⁡ ( 2 ) ⁢ s + a ⁡ ( 3 ) ( 18 ) where a ( 1 )= 1 . in step 440 , the modal parameters of interest are extracted from the transfer function model as follows : ⁢ ζ 1 = a ⁡ ( 2 ) / 2 / ω 1 ( 19 ) the physical parameters can then be calculated in step 450 using the following equations : once the physical parameters are determined , changes in the flow calibration factor as well as other parameters , including changes in the mass and length of the flow tube , are determined and corrected in step 460 . in addition to correction , warnings and meter shut down can be initiated based on the predetermined criteria . accurately estimating points on the frf can be difficult in real - time . windowing and spectral line estimation often have estimation bias which has a direct impact on the ability to predict fcf changes . in order to more efficiently determine estimating points on the frf and eliminate the need for windowing , a multi - sine input is used . the multi - sine input method applies an efficient spectral line estimation routine using a minimum crest factor multi - sine input excitation signal . the minimum crest factor signal is used to minimize peak drive power and to limit large output signals . large output signals often have non - linear components that cause measurement errors . fig5 shows a process 500 for calculating spectral lines used in validating a flow calibration factor using multiple frequency estimation . process 500 begins with step 510 by determining measurement frequencies of interest . the frequencies of interest are determined from the following : f j + 1 = f j + δf ( j = 1 : n sin ) ( 21 ) next , a multi - sine excitation signal is defined in step 520 as : x = ∑ i = 0 n - 1 ⁢ ∑ j = 0 n ⁢ ⁢ sin - 1 ⁢ sin ⁡ ( 2 ⁢ π ⁢ ⁢ f j ⁢ t i + ϕ j ) ( 22 ) in order to maximize the input energy while minimizing the peak value , a crest factor ( cf ) is minimized with respect to φ j in step 530 . crest factor ( cf ) is defined as max / rms , in order to make efficient use of the multi - sine , total measurement time , step 540 , and total number of averages , step 550 , must be determined . the total measurement time can be calculated using the following equation : equation ( 24 ) yields a measurement time equal to the period of the multi - sine , resulting in a measurement that is minimally affected by parameter changes . the total number of averages required is normally chosen so as to reduce the standard deviation of the estimate due to noise . the signal - to - noise ratio and its direct relationship to the estimated confidence interval , normally determine the total number of averages necessary for a required accuracy . step 560 applies the multi - sine , x , to the input of the system and measures the resulting output , y . step 570 uses x and y to determine the transfer function , stiffness and calibration factor using the techniques described herein . a changing flow calibration factor can also be detected and corrected by utilizing a multiple degree of freedom model . in a multiple degree of freedom model , a change in the flow calibration factor is a function of model parameters ( i . e . mass , stiffness , mode shape , complex pole , etc . ), fluid properties ( density , pressure , etc .) and environmental conditions ( temperature , vibration etc ). in order to determine parameters that affect the flow calibration factor , a response model must first be determined . response models are valuable in that they are direct measurements of the actual structure . the structure &# 39 ; s response from a known excitation is usually in the form of an impulse response function ( irf ) or frequency response function ( frf ). after the response model is determined , the response model is converted into a modal model , and then into a spatial model . for example , in a modal model , the flow calibration factor is a function of the mode shape and modal frequencies as evidenced by the following expressions : δ % mfcf = f ( ρ fluid , k )= c f ρ fluid + c k k by periodically measuring modal parameters , fluid properties and other environmental parameters , a change in the flow calibration factor can be detected and corrected . fig6 shows a process 600 for validating the flow calibration factor of a coriolis meter using a multi - degree of freedom model . process 600 begins in step 610 by creating a response model of a structure . step 620 converts the response model from step 610 into a modal model . the modal model is then converted into a spatial model in step 630 . in step 640 , physical parameters , such as flexural stiffness , can be determined from the spatial model . step 650 compares the current physical parameter data with previously determined physical parameter data , to establish if a change has occurred . if no change has occurred , measurement can continue to take place at a predefined interval at the request of a user . if a change has occurred , an error signal is generated in step 660 . after the error signal in generated , step 670 corrects the calibration factor . correction can either be done automatically or only at the request of the user . correction of the flow calibration factor can be accomplished using various methods including coefficient estimation , multi - fluid calibration and trending . coefficient estimation uses equations , like equation 25 above , with an estimation of c f and c k . for instance , if the material properties of a meter are affected by temperature , values for c f and c k can be estimated by finite element analysis and determination of a nominal material property change with temperature . multi - fluid calibration involves the calibration of a meter using at least two fluids at two different stiffnesses . coefficients c f and c k can be determined by using the following least squares fit : trending techniques can also be used to correct the flow calibration factor . initial model parameters , fluid properties , and temperature data is collected in the transmitter and the coefficients for a polynomial , n - dimensional “ operating surface ” f ( ρ fluid , k , t , σ , p , . . . ) would be determined . because there is generally a linear relationship between flow calibration factor , fluid properties , model parameters , and temperature , a change would be identified when newer data showed coefficients changing with respect to the initial coefficients . as an example , assume the fluid density has no effect on flow calibration factor , or c f is zero . the measured stiffness will change with temperature due to changes in the flow tube &# 39 ; s material properties . furthermore the measured stiffness will change due to temperature affecting a transducer &# 39 ; s material properties and sensitivity . therefore the rate of change of stiffness due to temperature is unknown . nevertheless , the stiffness and temperature can be measured at a specific time , i . e . time of installation . the data can be used to establish the relationship between stiffness and temperature . most material properties change linearly with temperature over a wide range as shown by : k ( t ) = f ( t )= c k c trans1 δt + c trans2 ( 27 ) initial data taken at the time of installation or at the factory over a range of flow tube temperatures allows you to calculate c k , c trans1 and c trans2 . the stiffness measured at any temperature will fall in a trend line . when enough stiffness measurements are made over a reasonable temperature range , the slope , or the rate of change of the stiffness with respect to temperature can be estimated using a least squares fit . this is the common change of the stiffness with respect to temperature . when the stiffness and temperature are measured and the data point does not fall on the curve , then a change in the sensor has occurred and the mfcf has also changed . the fact that the stiffness estimate is changing with temperature indicates that the flow tube material and transducer material properties are in slight error . a correction can be made in the temperature correction coefficients to make the measured stiffness vs . temperature slope zero . the correction may be made by an optimization process to minimize the error . subsequent to this correction , a stiffness change will indicate the change in the mfcf . the concept of a multiple degree of freedom model can be applied to other parameters , including flow calibration factor changes due to fluid density , pressure , external vibration , etc . the described solutions to a multiple - degree of freedom model are ideal when the structure does not change over time . however , in some applications a structure changes over time . in the case of a flow sensor , the density of a process fluid conveyed by the sensor may change over time resulting in a change in mass of the sensor . furthermore , ambient temperature and process fluid temperature may change over time , altering the material properties of the sensor . in order to collect meaningful data at varying conditions , a normalization process needs be implemented . for the case of a flow sensor , there are three main types of normalization . the first is to normalize that data with respect to a resonant frequency . various process fluid properties , such as fluid density , flow , and pressure , cause the resonant frequency to shift . the second is to normalize that data with respect to a reference temperature . the third is to normalize the data with respect to the displacement , or some other response variable . for example , a typical coriolis flow meter operates at a specific resonance for a specific material density . at one fluid density , the resonant frequency is at 152 hz . if the fluid density increases , the resonant frequency shifts down to 121 hz . the position of the resonant frequency relative to the test frequencies causes significant errors in the estimates of the model parameters . therefore , the data needs to be normalized in order to produce meaningful results . fig7 shows a process 700 for normalizing data in order to correct the flow calibration factor of a coriolis meter using a multi - degree of freedom model . process 700 begins in step 710 by receiving data from the structure . once the data is received it is normalized with respect to the resonant frequency . this allows the model parameters to be calculated from data measured for any fluid density . tracking the resonant frequency allows the test frequency to be set as a scalar of the resonant frequency . next in step 730 , we normalize the data with respect to a reference temperature . this corrects errors resulting from data that is measured at two different temperatures . for example , two important material properties change with temperature causing a discrepancy in the measured data . the modulus of elasticity of the tube material changes by approximately − 4 . 2 %/ 100 ° c . since most coriolis flow meters use voice coil transducer to measure flow and drive , changes in temperature will change the strength of the magnetic field and change the length of the wire in the coil . for a typical alnico magnet , the magnetic flux density changes by − 2 . 1 %/ 100 ° c . for both the driver and pickoff . also , a coriolis flow meter &# 39 ; s modulus will change with temperature . coriolis flow meters typically utilize a temperature sensor in order to report the temperature of the fluid in the sensor . using this temperature measurement , the frf measurements can be corrected back to some reference temperature . in step 740 , the data is normalized with respect to displacement or some other a response variable . this is done by integrating from mobility ( velocity / force ) to receptance ( displacement / force ). the following is an example of the equation used to normalize the measured data : h normalized ⁡ ( ω test_freq , t ref ) = [ 1 = c g ⁡ ( t - t ref ) ] h . ⁡ ( ω test_freq , t ) [ 1 + c drive_mag ⁡ ( t - t ref ) ] [ 1 + c pickoff_mag ⁡ ( t - t ref ) ] ( jω test_freq ) ( 28 ) once the data has been normalized , step 750 corrects the flow calibration factor using the methods described herein .