Patent Publication Number: US-10788348-B2

Title: Method of determining the left eigenvectors in a flowing Coriolis flowmeter

Description:
FIELD OF THE INVENTION 
     The invention is related to the field of flowmeters, and in particular, to Coriolis flowmeters. 
     BACKGROUND OF THE INVENTION 
     Mass flow rate is measured in a Coriolis flowmeter by vibrating a fluid-carrying tube(s) in a sinusoidal motion and measuring the time delay (or phase angle) between the vibration responses at two or more locations on the tube(s). For practical situations the time delay varies linearly with mass flow rate; however, the time delay is generally not zero at zero mass flow. There is usually a zero-flow delay or offset caused by a number of factors such as non-proportional damping, residual flexibility response, electromagnetic crosstalk, or phase delay in instrument electronics, for example. 
     This zero-flow offset is typically corrected for by measuring the zero-flow offset during a zero-flow condition and subtracting the measured offset from subsequent measurements made during flow. This would be sufficient to correct for the zero-flow offset problem if the zero-flow offset remained constant. Unfortunately, the zero-flow offset can be affected by small changes in the ambient environment (such as temperature) or to changes in the piping system through which the material is flowing. The changes in the zero-flow offset will cause errors in the measured flow rates. During normal operations there may be long periods of time between no-flow conditions, and the flowmeter can only be calibrated by zeroing the meter only during these no-flow conditions. The changes in the zero-offset over time may therefore cause significant errors in the measured flow. 
     The operation of Coriolis flowmeters can be described using mathematical formulas, as more fully described in U.S. Pat. Nos. 7,441,469 and 7,706,987 which are both assigned on their face to Micro Motion, Inc. and are hereby incorporated by reference. The general system of first order differential equations describing the motion of a linear system is: 
     
       
         
           
             
               
                 
                   
                     
                       
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     In Equation (1) M and K are the mass and stiffness matrices of the system and C is a general damping matrix which may have a symmetric component due to damping and a skew symmetric component due to Coriolis force.
 
 Aq+Bq=u   (2)
 
     Equation 1 can be rewritten as Equation 2 where A is equal to the matrix 
               [         C       M           M       0         ]     ,         
B is equal to the matrix
 
               [         K       0           0         -   M           ]               
and u is equal to
 
     
       
         
           
             
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     Insight into the equation of motion can be gained by looking at Equations 1 and 2. The generalized eigenvalue problem associated with Equation (2) may be solved for the right eigenvectors, ϕ (r) , such that:
 
 Bϕ   (r)   =−Aϕ   (r) λ  (3)
 
     For symmetric A and B matrices, the eigenvector can be used to diagonalize, or decouple the equations of motion. Decoupled equations are readily solved. For a non-symmetric system for example, where C includes the Coriolis matrix, the right eigenvectors do not diagonalize the equations of motion, resulting in coupled equations. Coupled equations are more difficult to solve and hinder insight into the solution. Left eigenvectors are required to diagonalize non-symmetric A or B matrixes. The following derivations show the process. The left eigenvectors are obtained by solving the following generalized eigenvalue problem:
 
ϕ (l)     T     B=−ϕ   (l)     T     Aλ 
 
 B   T ϕ (l)   =−A   T ϕ (l) λ  (4)
 
     M and K would generally be symmetric for a Coriolis flowmeter. For no flow, C would also be symmetric, thus, the system matrices, A and B would be symmetric. In this case, Equations (3) and (4) are identical and the left and right eigenvectors are the same. When there is flow, the associated non-symmetry of the C matrix causes the left and right eigenvectors to be different. 
     Consider the j&#39;th right eigenvector:
 
 Bϕ   j   (r)   =−Aϕ   j   (r) λ j   (5)
 
     and the i&#39;th left eigenvector;
 
ϕ (l)     T     B=−ϕ   i   (l)   Aλ   i   (6)
 
     Pre-multiplying Equation (5) by ϕ i   (l)     T   , and post multiplying Equation (6) by ϕ i   (r)     T    and subtracting the two yields:
 
0=−ϕ i   (l)     T     Aϕ   j   (r) (λ j −λ i )⇒ϕ i   (l)     T     Aϕ   j   (r) =0 for  i≠j   (7)
 
     By multiplying Equation (5) by 
             1     λ   j           
and Equation (6) by
 
             1     λ   i           
and going through the same procedure it can be shown:
 
⇒ϕ i   (l)     T     Bϕ   j   (r) =0 for  i≠j   (8)
 
     Equations (7) and (8) show that by pre and post multiplying either of the system matrices, A or B, by the matrix of left eigenvectors, Φ (L) , and the matrix of right eigenvectors, Φ (R) , respectively, the system matrices are diagonalized. 
     
       
         
           
             
               
                 
                   
                     
                       
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     The fact that the left and right eigenvector matrices diagonalize the system matrices means that both the set of right eigenvectors and the set of left eigenvectors are linearly independent. Either set can be used as a basis of a coordinate system for the response. Recognizing that the difference between the left and right eigenvectors is due to the skew-symmetric Coriolis matrix forms one basis of this invention. 
     In terms of a mathematical model of the flowmeter, the mass, stiffness and damping matrices which model non-Coriolis effects are symmetric. For a no-flow system, the left and right eigenvectors are identical (within an arbitrary scale factor). The Coriolis force associated with flow, however, manifests itself in the mathematical model as a skew symmetric damping matrix (the transpose is the negative of the original matrix). The skew symmetric Coriolis matrix causes the left and right eigenvectors of the system to be different. For a flowing system with no non-proportional damping, the relative phase between different coefficients of the left eigenvectors is equal and opposite to the relative phase between the same coefficients on the right eigenvectors. For a system with non-proportional damping, these phase values are offset equally for both the left and right eigenvectors, however, the difference remains the same. 
     Thus, if the phase characteristics of the left and right eigenvectors can be measured accurately, this characteristic allows the phase attributable to zero-offset from non-proportional damping and the phase attributable to material flow to be distinguished, eliminating associated zero-offset errors. Overall, there is a need for a system and method for accurately calibrating the zero-flow offset during flow. 
     SUMMARY OF THE INVENTION 
     A method is provided comprising placing a material in a flow tube while exciting a vibration mode of the flow tube. Exciting the vibration mode of the flow tube comprises the steps of: periodically driving a first driver with a first signal and periodically driving a second driver with a second signal, wherein the second driver is driven essentially in phase with the first driver, and wherein the first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a minimal amplitude, and the first driver&#39;s drive amplitude modulated signal reaches a minimum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude. Additionally, exciting the vibration mode of the flow tube comprises the steps of measuring the relative phase between a first pickoff and a second pickoff and determining a relative phase of a right eigenvector for the flow tube. 
     A method is provided comprising flowing a material through a flow tube while periodically exciting a vibration mode of the flow tube such that a first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude when a second driver&#39;s drive amplitude modulated signal reaches a minimal amplitude, and the first driver&#39;s drive amplitude modulated signal reaches a minimum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude. The method also comprises the steps of measuring the relative motion of the vibrating flow tube, measuring a relative phase of a right eigenvector while exciting the vibration mode of the flow tube, and determining the material flow through the flow tube using the relative phase of a right eigenvector corrected by a zero offset. A new zero offset is also determined without stopping the material flow through the flow tube using a relative phase of a left eigenvector for the flow tube. Also, the material flow through the flow tube is determined using the relative phase of a right eigenvector corrected by the new zero offset. 
     A vibratory flowmeter is provided, comprising a flowmeter assembly that includes one or more flow tubes and first and second pickoff sensors, as well as first and second drivers configured to vibrate the one or more flow tubes. Meter electronics are coupled to the first and second pickoff sensors and coupled to the first and second drivers, with the meter electronics being configured to provide a first signal to the first driver and a second signal to the second driver, wherein the second driver is driven essentially in phase with the first driver, wherein the first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a minimal amplitude, and the first driver&#39;s drive amplitude modulated signal reaches a minimum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude, and wherein the meter electronics is further configured to measure the relative phase between a first pickoff and a second pickoff and determine a relative phase of a right eigenvector for the flow tube. 
     Aspects 
     An aspect of a method according to an embodiment comprises: placing a material in a flow tube while exciting a vibration mode of the flow tube, wherein exciting the vibration mode of the flow tube comprises the steps of: periodically driving a first driver with a first signal; periodically driving a second driver with a second signal, wherein the second driver is driven essentially in phase with the first driver, wherein the first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a minimal amplitude, and the first driver&#39;s drive amplitude modulated signal reaches a minimum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude; measuring the relative phase between a first pickoff and a second pickoff; and determining a relative phase of a right eigenvector for the flow tube. 
     Preferably, the method further comprises: measuring a frequency shift that occurs between when the first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude and when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude; offsetting the first signal with a phase shift; adjusting the first signal phase offset until the frequency shift is substantially undetectable; and determining the relative phase of the left eigenvector coefficients from the first signal phase offset necessary for rendering the frequency shift substantially undetectable. 
     Preferably, the step of determining an actual flow of the material through the flow tube further comprises: determining an uncorrected flow of the material through the flow tube using the relative phase of the right eigenvector; and determining a zero offset for the flow of the material through the flow tube by comparing the uncorrected flow with the actual flow. 
     Preferably, the method further comprises determining a material flow through the flow tube using the relative phase of the right eigenvector corrected by the zero offset. 
     Preferably, the method further comprises determining the relative phase of a right eigenvector; and determining a zero offset for the flow of the material through the flow tube by weighted averaging the relative phase of the right eigenvector with the relative phase of the left eigenvector. 
     Preferably, the method further comprises estimating a frequency shift caused by the driver cycling; and relating the frequency shift to flow. 
     Preferably, the frequency is estimated with a frequency modulated second-order infinite impulse response adaptive notch filter. 
     Preferably, the method further comprises inputting a notch filter sharpness (a) parameter into the meter electronics; inputting a notch filter modulation frequency (fm) parameter into the meter electronics; determining a notch filter adaptation rate (λ) parameter based on the notch filter modulation frequency (fm) parameter and the sharpness (a) parameter; and inputting a pickoff signal into the adaptive notch filter, wherein the adaptive notch filter has a center frequency adapted to minimize filter output. 
     Preferably, an output of the notch filter is demodulated at the cycling frequency. 
     Preferably, the method further comprises co-locating the first driver with the first pickoff sensor; and co-locating the second driver with the second pickoff sensor. 
     Preferably, the step of periodically driving the first driver with the first signal comprises sinusoidally driving the first driver, and wherein the step of periodically driving the second driver with the first signal comprises sinusoidally driving the second driver. 
     An aspect of a method according to an embodiment comprises: flowing a material through a flow tube while periodically exciting a vibration mode of the flow tube such that a first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude when a second driver&#39;s drive amplitude modulated signal reaches a minimal amplitude, and the first driver&#39;s drive amplitude modulated signal reaches a minimum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude; measuring the relative motion of the vibrating flow tube; measuring a relative phase of a right eigenvector while exciting the vibration mode of the flow tube; determining the material flow through the flow tube using the relative phase of a right eigenvector corrected by a zero offset; determining a new zero offset without stopping the material flow through the flow tube using a relative phase of a left eigenvector for the flow tube; and determining the material flow through the flow tube using the relative phase of a right eigenvector corrected by the new zero offset. 
     Preferably, the method further comprises co-locating the first driver with the first pickoff sensor; and co-locating the second driver with the second pickoff sensor. 
     According to an aspect, a vibratory flowmeter comprises: a flowmeter assembly including one or more flow tubes and first and second pickoff sensors; first and second drivers configured to vibrate the one or more flow tubes; and meter electronics coupled to the first and second pickoff sensors and coupled to the first and second drivers, with the meter electronics being configured to provide a first signal to the first driver and a second signal to the second driver wherein the second driver is driven essentially in phase from the first driver, wherein the first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a minimal amplitude, and the first driver&#39;s drive amplitude modulated signal reaches a minimum amplitude when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude, and wherein the meter electronics is further configured to measure the relative phase between a first pickoff and a second pickoff and determine a relative phase of a right eigenvector for the flow tube. 
     Preferably, the meter electronics are configured to measure a frequency shift that occurs between when the first driver&#39;s drive amplitude modulated signal reaches a maximum amplitude and when the second driver&#39;s drive amplitude modulated signal reaches a maximum amplitude, to offset the first signal with a phase shift, to adjust the first signal offset until the frequency shift is substantially undetectable, and determine the relative phase of the left eigenvector coefficients from the first signal offset necessary for rendering the frequency shift substantially undetectable. 
     Preferably, the meter electronics are further configured to determine an uncorrected flow of the material through the one or more flow tubes using the relative phase of the right eigenvector and to determine a zero offset for the flow of the material through the one or more flow tubes by comparing the uncorrected flow with the actual flow. 
     Preferably, the meter electronics are further configured to determine a material flow through the one or more flow tubes using the relative phase of the right eigenvector corrected by the zero offset. 
     Preferably, the meter electronics are further configured to determine the relative phase of a right eigenvector and to determine a zero offset for the flow of the material through the one or more flow tubes by weighted averaging the relative phase of the right eigenvector with the relative phase of the left eigenvector. 
     Preferably, the meter electronics are further configured to estimate a frequency shift caused by the driver cycling, and relating the frequency shift to flow. 
     Preferably, the vibratory flowmeter comprises a frequency modulated second-order infinite impulse response adaptive notch filter with the meter electronics configured to perform the frequency shift estimate. 
     Preferably, the meter electronics are further configured to receive a notch filter sharpness parameter and a notch filter modulation frequency, wherein the meter electronics is configured to determine a notch filter adaptation rate parameter based on the sharpness parameter and the notch filter modulation frequency; and to receive an a pickoff signal in the adaptive notch filter, wherein the adaptive notch filter has a center frequency adapted to minimize filter output. 
     Preferably, an output of the notch filter is demodulated at the cycling frequency. 
     Preferably, the first driver is co-located with the first pickoff sensor, and the second driver is co-located with the second pickoff sensor. 
     Preferably, the first signal comprises a sinusoid. 
     Preferably, the filter comprises a plurality of adaptable filter coefficients. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a vibratory flowmeter according to an embodiment; 
         FIG. 2A  is a top view of a flow tube in an un-deflected position in an example embodiment of the invention; 
         FIG. 2B  is a top view of a flow tube in a deflected position corresponding to the main bending mode in an example embodiment of the invention; 
         FIG. 2C  is a top view of a flow tube in a deflected position corresponding to a twisting mode induced by Coriolis forces in an example embodiment of the invention; 
         FIG. 3  illustrates an example of transforming a phase into a frequency; 
         FIG. 4  illustrates prior art binary phase switching; 
         FIG. 5A  illustrates drive signals according to an embodiment; 
         FIG. 5B  illustrates periodic phase switching according to an embodiment; 
         FIG. 5C  illustrates an amplitude modulated signal according to an embodiment; 
         FIG. 6  is a flow chart illustrating determining a relative phase of a right eigenvector according to an embodiment; 
         FIG. 7  is a flow chart illustrating determining a relative phase of a left eigenvector according to an embodiment; 
         FIG. 8  is a flow chart illustrating a real time zero offset recalibration of a flowmeter during flow according to an embodiment; 
         FIG. 9  illustrates a left-eigenvector-induced frequency comparison between a Hilbert filter and an adaptive notch filter according to an embodiment; 
         FIG. 10  is a flow chart illustrating an improved frequency modulated adaptive notch filter according to an embodiment; and 
         FIG. 11  is a flow chart illustrating a method to suppress a twist mode according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
       FIGS. 1-11  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. 
     Residual flexibility, electromagnetic crosstalk and electronic measurement system characteristics all contribute to zero-offset. One interpretation of these effects is that they introduce error in the measurement of the right eigenvector phase. If the drive mode (right eigenvector) could be measured exactly, non-proportional damping would be the only effect causing zero offset and this error would be easily distinguished from flow effects using the left and right eigenvector Δt information. 
       FIG. 1  shows a vibratory flowmeter  5  according to an embodiment. The flowmeter  5  comprises a sensor assembly  10  and meter electronics  20  coupled to the sensor assembly  10 . The sensor assembly  10  responds to mass flow rate and density of a process material. The meter electronics  20  is connected to the sensor assembly  10  via the leads  100  to provide density, mass flow rate, and temperature information over a communication link  26 , as well as other information. A Coriolis flowmeter structure is described, although it is apparent to those skilled in the art that the present invention could also be operated as a vibrating tube densitometer. 
     The sensor assembly  10  includes manifolds  150  and  150 ′, flanges  103  and  103 ′ having flange necks  110  and  110 ′, parallel flow tubes  130  and  130 ′, first and second drivers  180 L and  180 R, and first and second pickoff sensors  170 L and  170 R. The first and second drivers  180 L and  180 R are spaced apart on the one or more flow tubes  130  and  130 ′. In addition, in some embodiments the sensor assembly  10  may include a temperature sensor  190 . The flow tubes  130  and  130 ′ have two essentially straight inlet legs  131  and  131 ′ and outlet legs  134  and  134 ′ which converge towards each other at the flow tube mounting blocks  120  and  120 ′. The flow tubes  130  and  130 ′ bend at two symmetrical locations along their length and are essentially parallel throughout their length. The brace bars  140  and  140 ′ serve to define the axis W and the substantially parallel axis W′ about which each flow tube oscillates. It should be noted that in an embodiment, the first driver  180 L may be collocated with the first pickoff sensor  170 L, and the second driver  180 R may be collocated with the second pickoff sensor  170 R. 
     The side legs  131 ,  131 ′ and  134 ,  134 ′ of the flow tubes  130  and  130 ′ are fixedly attached to flow tube mounting blocks  120  and  120 ′ and these blocks, in turn, are fixedly attached to the manifolds  150  and  150 ′. This provides a continuous closed material path through the sensor assembly  10 . 
     When the flanges  103  and  103 ′, having holes  102  and  102 ′ are connected, via the inlet end  104  and the outlet end  104 ′ into a process line (not shown) which carries the process material that is being measured, material enters the end  104  of the meter through an orifice  101  in the flange  103  and is conducted through the manifold  150  to the flow tube mounting block  120  having a surface  121 . Within the manifold  150  the material is divided and routed through the flow tubes  130  and  130 ′. Upon exiting the flow tubes  130  and  130 ′, the process material is recombined in a single stream within the manifold  150 ′ and is thereafter routed to the exit end  104 ′ connected by the flange  103 ′ having bolt holes  102 ′ to the process line (not shown). 
     The flow tubes  130  and  130 ′ are selected and appropriately mounted to the flow tube mounting blocks  120  and  120 ′ so as to have substantially the same mass distribution, moments of inertia, and Young&#39;s modulus about the bending axes W-W and W′-W′, respectively. These bending axes go through the brace bars  140  and  140 ′. Inasmuch as the Young&#39;s modulus of the flow tubes change with temperature, and this change affects the calculation of flow and density, the resistive temperature detector (RTD)  190  is mounted to the flow tube  130 ′, to continuously measure the temperature of the flow tube. The temperature dependent voltage appearing across the RTD  190  may be used by the meter electronics  20  to compensate for the change in the elastic modulus of the flow tubes  130  and  130 ′ due to any changes in flow tube temperature. The RTD  190  is connected to the meter electronics  20  by the lead  195 . 
       FIGS. 2A-2C  show a top view of a flow tube  130 ,  130 ′ configured to contain a material flowing therethrough.  180 L and  180 R are two drivers (also called actuators) spaced along the flow tube  130 ,  130 ′. In the preferred mode, the two drivers are spaced symmetrically around the axial center of the flow tube. The drivers are configured to impart a force to the flow tube  130 ,  130 ′ to excite a plurality of vibration modes in the flow tube  130 ,  130 ′. The force may be substantially coherent (e.g. confined to a narrow frequency) or may be broadband. The drivers can be such known means as a magnet attached to the flow tube, and a coil attached to a reference, through which an oscillating current is passed, for example, without limitation. 
       170 L and  170 R depict two sensors (also called pickoffs) co-located with drivers  180 L and  180 R. The sensors are configured to produce a plurality of signals representing the location and motion of the flow tube  130 ,  130 ′. The sensors may include a variety of devices, such as coil-type velocity transducers, optical or ultrasonic motion sensors, laser sensors, accelerometers, inertial rate sensors and the like. In this embodiment there are two sensors shown with each sensor co-located with one of the drivers  180 L,  180 R. Other configurations having more than two sensors are also possible. 
       FIG. 2A  shows the flow tube  130 ,  130 ′ in an un-deflected state. By driving the actuators with equal power, the main bending mode of the flow tube can be excited. U.S. Pat. No. 6,092,429, entitled “Driver for Oscillating a Vibrating Flow Tube”, which is assigned on its face to Micro Motion, Inc., and hereby incorporated by reference, discloses drivers  180 L,  180 R configured to excite different modes of vibration in a flow tube.  FIG. 2B  shows the flow tube  130 ,  130 ′ in a deflected state corresponding to the main bending mode of the flow tube. This vibration mode also corresponds to a condition when there is no flow of material through the flow tube. The deflection of the flow tube  130 ,  130 ′ in  FIGS. 2B and 2C  have been magnified for clarity. The actual deflections of flow tube  130 ,  130 ′ would be much smaller. When material is flowing through the vibrating flow tube  130 ,  130 ′, the flowing material causes Coriolis forces to be generated. The Coriolis forces deflect the flow tube  130 ,  130 ′ and excite additional vibration modes.  FIG. 2C  shows the main vibration mode excited by the Coriolis forces. The relative phase difference detected between sensor  170 L and sensor  170 R can be used to determine the material flow through the flow tube  130 ,  130 ′. In no-flow condition (as depicted in  FIG. 2B ), there is no phase difference due to flow detected between  170 L and  170 R. It should be noted, however, that there may be phase differences due to zero-offset conditions. Once material is flowing through the flow tube  130 ,  130 ′, there will be a phase difference between  170 L and  170 R, which is due to flow. The measured phase difference detected between  170 L and  170 R is a measure of the relative phase of the right eigenvector of the system and is proportional to the material flow through the flow tube. Let θ R  equal the relative phase of the right eigenvector, θ 1  be the measured phase of the vibration of the flow tube at sensor  170 L, and θ 2  be the measured phase of the vibration of the flow tube at sensor  170 R, then:
 
θ R =θ 1 −θ 2   (10)
 
A time difference, Δt, can be calculated from the phase difference by dividing by the vibration frequency ω.
 
Δ t =(θ 1 −θ 2 )/ω  (11)
 
The time difference Δt is also proportional to the material flow through the flow tube and is a measurement typically used in mass flowmeters. A more accurate determination for the material flow through the flow tube  130 ,  130 ′ can be calculated by correcting the measured material flow with a zero-offset amount, to derive a corrected Δt, Δt C :
 
Δ t   C   =Δt −ZeroOffset  (12)
 
In one example embodiment of the invention, during normal operations, both drivers  180 L,  180 R are used to excite the main bending mode of the flow tube. An example of the material flow through the flow tube is determined by measuring the relative phase of the right eigenvector, converting to a Δt domain, Δt R , and correcting this value with a zero-offset correction amount to determine a corrected Δt R , Δt RC :
 
Δ t   RC   =Δt   R −ZeroOffset  (13)
 
     In an embodiment, the flow tube is excited by periodically driving a first driver and periodically driving the second driver, wherein the first and second drivers are amplitude modulated out of phase from each other. In an embodiment, the drivers  180 L,  180 R are both periodically driving with a frequency corresponding to resonance. The drive signals are amplitude modulated at a much lower frequency. This lower frequency is the cycling frequency at which phase/frequency shift is later demodulated. The drive amplitude modulation signals (at the cycling frequency) are out of phase. The drive signal to each driver at the resonance/drive frequency are essentially identical signals, and are multiplied by the corresponding modulation signal. The periodic sweep may be sinusoidal, square, saw-toothed, etc. Measurements are taken of the phase between the driving signal and the vibration at positions on the flow tube. These measurements are used to determine the relative phase of the left eigenvector of the system. In an embodiment, a frequency shift that occurs between when the first driver&#39;s ( 180 L) drive amplitude modulated signal reaches a maximum amplitude and when a second driver&#39;s ( 180 R) drive amplitude modulated signal reaches a maximum amplitude is measured, and this frequency shift is input into meter electronics  20 . The drive signal driving the first driver  180 L is then offset with a phase shift. The frequency shift which occurs between when the first driver&#39;s ( 180 L) drive amplitude modulated signal reaches a maximum amplitude and when a second driver&#39;s ( 180 R) drive amplitude modulated signal reaches a maximum amplitude is then measured. If the frequency shift is approximately zero, no adjustments need be made. However, if a frequency shift which occurs between when the first driver&#39;s ( 180 L) drive amplitude modulated signal reaches a maximum amplitude and when a second driver&#39;s ( 180 R) drive amplitude modulated signal reaches a maximum amplitude is still detectable, then the first signal is offset again until the frequency shift which occurs between when the first driver&#39;s ( 180 L) drive amplitude modulated signal reaches a maximum amplitude and when a second driver&#39;s ( 180 R) drive amplitude modulated signal reaches a maximum amplitude is substantially undetectable. Once this occurs, the relative phase between the left eigenvector coefficients is calculated from the signal offset necessary for rendering the frequency shift approximately undetectable. 
     Turning to  FIG. 3 , it will be clear that a major difficulty inherent in this approach is that the system&#39;s transformation of phase into frequency is quite sharp, meaning that a very little frequency change occurs for a comparatively large phase change through the meter, which is additionally difficult to invert. As a result, the required frequency accuracy is quite exacting—on the order of nanoHertz. Prior art approaches to this phase switching effectively toggle between drivers  180 L,  180 R in a binary fashion. This is illustrated in  FIG. 4 . This requires a wait of at least several seconds (15-30) for the signal to decay before observing the new frequency from the newly toggled driver. Such a long delay is impractical for a usable system. When such an abrupt drive transition is used to continually cycle the drive back and forth, the transient response of the system to the transitions never actually totally decays, which ends up obscuring the desired drive frequency variation. When the flowmeter  5  is driven with a closed-loop feedback signal, such as when one or more pickoff signals is scaled and fed back to generate the drive signal, this characteristic of phase to frequency conversions is particularly pronounced. Under this feedback drive scheme, something that would otherwise be a phase change in the open-loop characteristics of the system instead manifests as a shift in the drive frequency. This is inherent in the nature of a feedback drive system since phase in the open loop system must, by definition, be made to match the phase in the feedback portion of the system, thus matching these phases requires a change in drive frequency, which automatically falls out of the closed-loop controller. Since, as noted above, a relatively large amount of phase change manifests as a comparatively small frequency shift, observing phase effects in a closed loop drive is prohibitively challenging, as the frequency changes are generally too small to detect. 
     Turning to  FIGS. 5A-5C , an embodiment is illustrated wherein the drive amplitudes are cycled periodically to minimize transient signals. This approach minimizes the slope of the drive amplitude changes in time (given a required cycling frequency), thereby minimizing impulsive shocks to the system. With this approach, the drive is entirely on one driver or the other only momentarily.  FIG. 5A  shows the essentially in-phase nature of the two drive signals.  FIG. 5B  shows the amplitude modulation signals.  FIG. 5C  shows the amplitude modulated drive signals, where the first driver&#39;s amplitude reaches a maximum when the second driver&#39;s amplitude reaches a minimum, and the first driver&#39;s amplitude reaches a minimum when the second driver&#39;s amplitude reaches a maximum. 
       FIG. 6  is a flow chart illustrating the determination of a relative phase of a right eigenvector of the system in an example embodiment. At step  500 , during normal operations, both drivers  180 L,  180 R are used to excite the vibration of the flow tube. The drivers  180 L and  180 R are cycled periodically, wherein when  180 L is at a maximal amplitude,  180 R is driven at a minimal amplitude, and vice versa. At step  502 , driver  180 L is cycled to excite the vibration of the flow tube at a maximum amplitude, while  180 R is cycled to excite the vibration of the flow tube at a minimal amplitude. During this time, the phase between the pickoff sensor  170 L and pickoff sensor  170 R is measured. This will be called phase difference θ 1 . 
     In step  504 , as the drives are cycled periodically, the driver  180 L excites the vibration of the flow tube at a minimum amplitude, while  180 R excites the vibration of the flow tube at a maximal amplitude. The phase between the pickoff sensor  170 L and pickoff sensor  170 R is measured. It should be noted that the phase difference between pickoff sensors  170 R,  170 L may be continually measured, or may be measured at discrete time points. Regardless, this will be called phase difference θ 2 . 
     At step  506 , the relative phase of the right eigenvector is calculated. In an embodiment, system noise and phase due to residual flexibility of higher frequency modes is attenuated by weighted averaging the phase signals, such that the average phase, θ AR , may, in an embodiment, be calculated as: 
                     θ   AR     =       (       θ   1     +     θ   2       )     2             (   14   )               
Other equations, other than equation (14) are contemplated, however.
 
       FIG. 7  is a flow chart illustrating the determination of a relative phase of a left eigenvector in an example embodiment. At step  600 , during normal operations, both drivers are used to excite the vibration of the flow tube. The drivers  180 L and  180 R are cycled periodically, wherein when  180 L is at a maximal amplitude,  180 R is driven at a minimal amplitude, and vice versa. At step  602 , a frequency shift between the driving signal used by driver  180 L and the driver  180 R is measured. 
     At step  604 , the first signal is offset with a phase shift as the drives are cycled periodically. At step  606 , the first signal is offset until the frequency shift is substantially undetectable. At step  608 , the relative phase of the left eigenvector coefficients is determined from the first signal offset necessary for rendering the frequency shift approximately undetectable. 
       FIG. 8  is a flow chart illustrating a real time zero offset recalibration of a flowmeter during flow in one example embodiment. In step  700 , during normal operations, both drivers  180 L,  180 R are used to periodically excite the vibration of the flow tube. An uncorrected relative Δt for the right eigenvector is determined. The uncorrected relative Δt of the right eigenvector is then corrected by using a zero offset. The flow through the meter is determined using the corrected relative Δt of the right eigenvector. Periodically, in step  702 , the relative Δt of the left eigenvector is determined. The relative Δt of the left eigenvector may be corrected for residual flexibility and electronic crosstalk effects. In step  704 , the relative Δt of the left eigenvector and the uncorrected Δt of the right eigenvector are used to determine a new zero offset. The new zero offset is substituted for the old zero offset and the process resumes at step  700 . By calculating and substituting the new zero offset into the meter, the meter has been recalibrated for the zero flow condition during material flow through the meter. 
     In one example embodiment, the determination for when the re-calibration should occur may be done by using a fixed time interval between calibrations. In another example embodiment, a re-calibration may be done when changes in the environment or the piping system are detected. For example, when a change in temperature is greater than a threshold amount, a re-calibration may be performed. The determination for when re-calibration occurs may be a combination of a periodic timer and detecting changes in environment. The time period between recalibrations may be shorter for systems that require higher accuracy than for systems that have less stringent accuracy requirements. Switching between drivers  180 L and  180 R in order to measure the relative phase of the left eigenvector does not imply that the normal operation of the flowmeter has to be interrupted (i.e. measuring flow using Δt of the right eigenvector). In yet another example, a flowmeter is simply monitored for changes to the zero, so an embodiment for a zero verification tool is contemplated. This is useful in a number of applications, such as custody transfer, for example without limitation, wherein the zero is not permitted to be changed, except under particular circumstances. 
     As noted, a problem encountered in prior art approaches, however, relates to isolating the phase effects due to switching drivers  180 L,  180 R from the other transient effects from the controller. In particular, when drivers  180 L,  180 R are switched abruptly, the system may react with unwanted transient responses. The drive controller tends to reinforce these step changes in the system. When this abrupt drive transition is used to continually cycle the drive back and forth, the transient response of the system to the transitions tends to never decay, so the desired drive frequency variation is obscured. Another issue is the sensitivity of the frequency measurement. If one simply examines the raw frequency signal when cycling drivers  180 L,  180 R, the changes of interest are obfuscated by noise effects. Signal processing methods and electronics are described in U.S. Pat. No. 5,734,112, which is assigned on its face to Micro Motion, Inc. and is hereby incorporated by reference. 
     For the various Δt calculations to be made, signal processors rely on frequency estimations of flow tube bending mode vibrations. A Hilbert filter-based approach for estimating frequency is known from the prior art, but proves too noisy for the accuracy desired. This sinusoidal cycling approach may, in an embodiment, be augmented with an improved frequency estimator filter. In particular, a modified adaptive notch frequency estimator is employed. The adaptive notch frequency estimator provides a far more accurate frequency estimate than a Hilbert filter.  FIG. 9  illustrates a left-eigenvector-induced frequency variation using the two frequency estimators—a Hilbert filter (H) and adaptive notch filter (AN). The adaptive notch filter is much cleaner, which is particularly clear where the phase of the left eigenvector has been tuned out. The noise present when using the Hilbert filter has a standard deviation over 500 times greater than that of the adaptive notch filter. 
     In an embodiment, the adaptive notch filter comprises a frequency modulated second-order infinite impulse response filter that is applied to an incoming signal, with its center frequency adapted to minimize the output. For a purely tonal signal, this will be when the filter&#39;s frequency matches that of the input signal. For multi-tonal signals, the filter locks onto the largest tone present. Beyond the adapted center frequency, the filter&#39;s behavior is controlled by two design parameters: the sharpness of the filter, α, and the adaptation rate of the frequency, λ. In an embodiment, sharpness (α) is kept as a static design parameter, and typically defined during the manufacturing stage. The adaptation rate of the frequency (λ) parameter influences sensitivity to a frequency modulated signal. In an example embodiment, the adaptive notch filter&#39;s modulation frequency is defined according to the following expression: 
     
       
         
           
             
               
                 
                   
                     f 
                     m 
                   
                   = 
                   
                     
                       0.155 
                       dt 
                     
                     ⁢ 
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             a 
                           
                           ) 
                         
                         ⁢ 
                         
                           ( 
                           
                             1 
                             - 
                             λ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Where:
         α=sharpness of filter   λ=adaptation rate of the frequency   dt=sample rate of the filter   fm=modulation frequency       

     Thus, given a sample rate, a value for a, and a modulation frequency fin, the appropriate value for the adaptation rate of the frequency (λ) may be found as: 
     
       
         
           
             
               
                 
                   λ 
                   = 
                   
                     1 
                     - 
                     
                       
                         
                           ( 
                           
                             6.45 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               fm 
                               · 
                               dt 
                             
                           
                           ) 
                         
                         2 
                       
                       
                         1 
                         - 
                         α 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     In this manner, frequency modulation is tracked at a known modulation frequency several orders of magnitude more accurately than with a generic, prior art, adaptive notch filter. 
       FIG. 10  is a flow chart that illustrates the use of an improved frequency modulated adaptive notch filter according to an embodiment. From the relative phase of the left eigenvector, a frequency estimate is ultimately calculated. In step  900 , the relative phase of the left eigenvector is calculated. In step  902 , the sharpness parameter, a, is input into the filter, which corresponds to a response time of the filter. In an embodiment, this parameter is determined at the time of flowmeter assembly and verification. In step  904 , the notch filter modulation frequency (fm) is input into the filter. In step  906 , the adaptation rate parameter (λ) is calculated based on the sharpness (a) and notch filter modulation frequency (fm) parameters. In an example embodiment, expression (15) is employed for this calculation, but other expressions are contemplated, and nothing limits embodiments to expression (15). Step  908  corresponds to inputting the frequency shift that occurs between when the first driver&#39;s ( 180 L) drive amplitude modulated signal reaches a maximum amplitude and when a second driver&#39;s ( 180 R) drive amplitude modulated signal reaches a maximum amplitude, and offsetting the first drive signal with a phase shift offset. This entails adjusting the first signal offset until substantially no frequency shift is detectable. In step  910 , the relative phase of the left eigenvector coefficients is determined from the first signal offset necessary for rendering the frequency shift substantially undetectable. 
     In another related embodiment, instead of utilizing a pure frequency estimator, as described above, a frequency-modulated signal is extracted at a known modulation frequency. Consider a frequency-modulated signal, with a known, fixed modulation at the frequency ω c :
 
 y ( t )=cos((ω 0 +Δω cos(ω c   t )) t )  (17)
 
In an embodiment, a goal is to measure Δω, and one method for such a measurement comprises comparing signal y to a time-delayed version of itself, and computing the phase difference, φ:
 
                   φ   =     ∠   ⁢       y   ⁡     (     t   +     Δ   ⁢           ⁢   t       )         y   ⁡     (   t   )                   (   18   )               
Since phase is defined as the integral of frequency, in an example embodiment, without limitation, phase may be computed as:
 
                     φ   ⁡     (   t   )       =         ω   0     ⁢   Δ   ⁢           ⁢   t     +         Δ   ⁢           ⁢   ω       ω   c       ⁢     (       (       sin   ⁡     (       ω   c     ⁢   t     )       ⁢     cos   ⁡     (       ω   c     ⁢   Δ   ⁢           ⁢   t     )         )     +       cos   ⁡     (       ω   c     ⁢   t     )       ⁢     sin   ⁡     (       ω   c     ⁢   Δ   ⁢           ⁢   t     )         -     sin   ⁡     (       ω   c     ⁢   t     )         )                 (   19   )               
Since the modulation frequency, ω c , is known, the time delay, Δt, is selected as a quarter period of the modulation frequency, so:
 
                       ω   c     ⁢   Δ   ⁢           ⁢   t     =     π   2             (   20   )               
Expression (19) simplifies to:
 
                     φ   ⁡     (   t   )       =           ω   0     ⁢   Δ   ⁢           ⁢   t     +         Δ   ⁢           ⁢   ω       ω   c       ⁢     (       cos   ⁡     (       ω   c     ⁢   t     )       -     sin   ⁡     (       ω   c     ⁢   t     )         )         =         ω   0     ⁢   Δ   ⁢           ⁢   t     +       2     ⁢       Δ   ⁢           ⁢   ω       ω   c       ⁢     cos   ⁡     (         ω   c     ⁢   t     +     π   4       )                     (   21   )               
This represents a DC value plus a sinusoid at the modulation frequency, the amplitude of which is proportional to Δω. Therefore, the amplitude of the frequency modulation at the known modulation frequency may be extracted by comparing the phase of the signal with an appropriately delayed version of itself and calculating the amplitude of the resulting sinusoidal frequency modulation. It will therefore be obvious that this technique is sensitive to modulation at the expected frequency, yet provide rejection of the majority of out-of-band noise.
 
     In the embodiments above, both drivers  180 L,  180 R are utilized to determine the relative Δt of the left eigenvector. In yet another embodiment, only a single driver may be used at a time. For example, driving a single driver,  180 L, measuring the phase at a pickoff sensor,  170 L, and then driving the other driver,  180 R, and subsequently measuring the phase at the same pickoff,  170 L, can be used to allow the unique contributions of each driver to be discerned. An inherent drawback on a lightly-damped flowmeter stems from the fact that it takes up to, and even potentially longer than, five minutes for the response from the first driver to decay, which is necessary before the second driver can be activated. In many real-world situations, this is a virtually useless approach due to the long time lag, as environmental conditions may change fast enough to render any measurements obsolete. Thus, a solution to this time lag problem is provided in an embodiment. 
     In an embodiment, driver forces are calibrated so to improve system accuracy. In traditional single-driver meters, the sensor is designed to be balanced, with the drive coil located at a node of the twist mode&#39;s modeshape. By doing so, it is extremely difficult for the driver to directly excite the twist mode, and thus direct twist mode excitation does not significantly influence the drive phase measured at the pickoff sensors  170 L,  170 R, thereby removing one potential source of zero error in the flow measurement. With a dual driver approach, however, the drivers  180 L,  180 R are not located at twist mode nodes, and thus there exists the risk of directly exciting the twist mode. Direct twist mode excitation is therefore a significant problem. The relative twist mode response (as compared to drive mode response) may change as a function of density and temperature. Thus, if the drive mode is relatively close to the twist mode, the relative response of the twist mode will increase. This is not a pure zero effect, but rather a flow error that is dependent, at least partially, on drive frequency. If the twist mode is not excited to begin with though, such as due to driver placement on single driver meters, this error source is eliminated. 
     Therefore, in an embodiment, a driver force projection vector (Ψ dr ) and a pickoff modal filter vector (Ψ po ) are selected that correct for any system gains and allow driving of the drive mode while suppressing the twist mode—the primary goal being the suppression of the twist mode at both the drivers  180 L,  180 R and the pickoff sensors  170 L,  170 R, as illustrated in  FIG. 11 . This is an improvement over the standard single-driver meter which inherently allowed only for suppression at the driver, but did nothing to prevent the pickoff sensors  170 L,  170 R from measuring the twist mode response. 
     In order to excite the drive mode without exciting the twist mode, the twist mode modeshapes (i.e. eigenvectors) from the resulting residue matrix are extracted in step  1100 . With the twist mode left and right eigenvectors (i.e. pickoff and driver modeshapes) in hand, drive mode vectors are designed that are orthogonal, as shown in step  1102 . In an embodiment, this is accomplished using the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       φ 
                       t 
                       
                         ( 
                         l 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   k 
                                   1 
                                 
                               
                             
                             
                               
                                 
                                   - 
                                   
                                     k 
                                     2 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⇒ 
                         
                           Ψ 
                           dr 
                         
                       
                       = 
                       
                         [ 
                         
                           
                             
                               
                                 1 
                                 / 
                                 
                                   k 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 1 
                                 / 
                                 
                                   k 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     φ 
                     t 
                     
                       ( 
                       r 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 g 
                                 1 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   g 
                                   2 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⇒ 
                       
                         Ψ 
                         po 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               1 
                               / 
                               
                                 g 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               1 
                               / 
                               
                                 g 
                                 2 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     By driving a dual driver meter via the vector Ψ dr  and combining the pickoff sensors  170 L,  170 R for feedback via the vector Ψ po , as in step  1104 , the desired suppression of the twist mode is realized, since:
 
ψ T   dr φ t   (l) =1−1=0
 
ψ T   dr φ t   (r) =1−1=0  (23)
 
     While it may be ideal to measure twist mode modeshapes, in an alternate embodiment, only the drive mode data is utilized. The approach is similar to that described above, but with an additional assumption that the meter being driven is balanced so the drive and twist mode shapes are generally: 
     
       
         
           
             
               
                 
                   
                     φ 
                     d 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               1 
                             
                           
                           
                             
                               1 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       and 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         φ 
                         t 
                       
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1 
                           
                         
                         
                           
                             
                               - 
                               1 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     These are then corrupted only by the transducer gains, such that 
                         φ   d     (   l   )       =     [           k   1               k   2           ]       ,       φ   t     (   l   )       =     [           k   1               -     k   2             ]         ⁢     
     ⁢         φ   d     (   r   )       =     [           g   1               g   2           ]       ,       φ   t     (   r   )       =     [           g   1               -     g   2             ]                 (   25   )               
Thus, for a symmetric flowmeter, left and right eigenvectors are measured, as described above, so an estimate of the twist mode for left and right eigenvectors may also be estimated. From that point, the same approach as outlined above is utilized to arrive at a driver projection vector and a pickoff modal filter vector that suppress the twist mode. Either approach is applicable to meter calibration, either initial factory calibration or field calibrations. So to avoid directly exciting the twist mode, driver and/or pickoff gains are adjusted to obtain more stable flow estimates. Though these methods are well suited to dual driver systems, they are applicable to systems having greater than two drivers  180 L,  180 R and/or greater than two pickoff sensors  170 L,  170 R.
 
     For a dual drive flowmeter, unwanted electronic zero drift due to drift between the front end circuitry of the left and right pickoff channels may be an issue that interferes with a flowmeter&#39;s accuracy. Of course, the zero drift may be compensated for by the methods described above, for example without limitation, but these methods alone do not address another source of inaccuracy known to affect flow measurements that stems from phase and amplitude drift on the analog-to-digital (A/D) converters and the input circuitry and digital-to-analog (D/A) converters and the corresponding output circuitry. In an embodiment, phase and amplitude drift associated with A/D and D/A converters is compensated for. 
     A/D converters typically receive an analog signal from their respective flow tube sensor output signals. A processor applies control signals to the A/D converters, and receives digitized sample values from the A/D converters. For example, without limitation, a processor may determine a Δt value from the phase difference between sampled channels. A D/A converter then converts the digital signal value into an analog signal proportional to a mass flow rate. One of ordinary skill in the art will readily recognize that clocking signals required by the various components may be generated by any well-known clock generation techniques such as crystal controlled oscillators, or any of several commercially available clock generation integrated circuits. 
     The A/D converters are typically embodied within a single integrated circuit with multiple converters and a single communication bus connection to a DSP processor. This helps assure that the phase relationship between the sampled signals is due to the Coriolis effects of the vibrating flow tubes rather than effects of signal trace routing on a printed circuit board to physically separate A/D converter circuits. Many such stereo A/D converter chips are known to those skilled in the art. One example of such a chip is the Crystal Semiconductor CS5329, a 2-channel stereo A/D converter device. These signal processing methods and electronics, as noted above, are more fully described in U.S. Pat. No. 5,734,112. 
     Input phase offset is simply an electronic zero, which is well-known in the art. A constant offset in phase between the two input channels can be simply subtracted out before converting a Δt signal to flow. Historically, input phase drift has largely been addressed by precise matching of the input circuitry components. Input amplitude offset, on the other hand, which is not a very large concern for a standard single drive flowmeter, is more important to address for dual drive meters. It is desirable to ensure that modal filter vectors derived from the pickoff signals are optimized. Even though pickoff locations—i.e. the eigenvector coefficients at the pickoff sensors  170 L,  170 R—are symmetric, the input circuitry and A/D converters for the pickoff sensors  170 L,  170 R could have a bias or offset between each other. 
     Similarly, input amplitude offset is not a very large concern for a standard single drive flowmeter and is usually neglected. Pickoff amplitude is used for drive control with the largest pickoff signal used as the amplitude feedback signal. Smart Meter Verification typically measures a fairly large difference between the right and left pickoff stiffnesses. This offset is addressed by normalizing the stiffnesses to a factory baseline stiffness for each pickoff, ignoring any amplitude offset. 
     It is desirable to ensure that the modal filter vectors as derived from the pickoff signals are optimized. This assumes that the pickoff locations—i.e. the eigenvector coefficients at the pickoff sensors  170 L,  170 R—are symmetric. However, the input circuitry and A/D converters for the pickoff sensors  170 L,  170 R could have a bias or offset between each other. A preliminary calibration of the input circuitry could be effectuated, and during this calibration step one pickoff may be fed into both channels simultaneously. A software gain can therefore be adjusted to ensure that the front end circuitry reads the same voltage. This calibration step is typically done before calculation of the modal filter vectors, and can be easily accomplished in a production environment. With the precise matching of the components necessary for minimizing phase drift, it can reasonably be expected that the amplitude drift would be negligible. In an embodiment, the current through the dual drivers is measured as an input, and the input circuitry and A/D converters for the current measurement may be calibrated as for pickoff input channels. Calibrating the current measurement may, in some embodiments, comprise additional hardware on the embedded electronics such as relays or externally accessible jumpering, for example without limitation. 
     An amplitude offset in the two output channels for the drivers  180 L,  180 R would excite the residual flexibility of the twist mode, resulting in an interfering signal. Calibrating the output circuitry is accomplished in an embodiment by sending the same signal into both drive channels. The measured currents could then be compared and a gain on one of the output channels may be adjusted until the measured currents are equal. This would ensure that the output currents for the two drivers  180 L,  180 R are equal, after which the force projection vectors as discussed above could be accurately calculated. Therefore, with proper matching of the electronics components/signals the amplitude drift would is negligible. 
     The detailed descriptions of the above embodiments are not exhaustive descriptions of all embodiments contemplated by the inventors to be within the scope of the invention. Indeed, persons skilled in the art will recognize that certain elements of the above-described embodiments may variously be combined or eliminated to create further embodiments, and such further embodiments fall within the scope and teachings of the invention. It will also be apparent to those of ordinary skill in the art that the above-described embodiments may be combined in whole or in part to create additional embodiments within the scope and teachings of the invention. Thus, although specific embodiments of, and examples for, the invention are described herein for illustrative purposes, various equivalent modifications are possible within the scope of the invention, as those skilled in the relevant art will recognize. The teachings provided herein can be applied to other vibrating systems, and not just to the embodiments described above and shown in the accompanying figures. Accordingly, the scope of the invention should be determined from the following claims.