Apparatus, methods and computer program products for generating mass flow calibration factors using a normal modal dynamic characterization of a material-containing conduit

[0002] The present invention relates to sensors and related methods and computer program products, and more particularly, to mass flow measurement methods, apparatus, computer program products.

[0003] Coriolis sensors typically operate by detecting motion of a vibrating conduit that contains a flowing material. Properties associated with the material in the conduit, such as mass flow, density and the like, may be determined by processing signals from motion transducers associated with the conduit, as the vibrational modes of the vibrating material-filled system generally are affected by the combined mass, stiffness and damping characteristics of the containing conduit and the material contained therein.

[0004] A typical Coriolis mass flowmeter includes one or more conduits that are connected inline in a pipeline or other transport system and convey material, e.g., fluids, slurries and the like, in the system. Each conduit may be viewed as having a set of natural vibrational modes including, for example, simple bending, torsional, radial and coupled modes. In a typical Coriolis mass flow measurement application, a conduit is excited at resonance in one of its natural vibrational modes as a material flows through the conduit, and motion of the conduit is measured at points spaced along the conduit. Excitation is typically provided by an actuator, e.g., an electromechanical device, such as a voice coil-type driver, that perturbs the conduit in a periodic fashion. Mass flow rate may be determined by measuring time or phase differences between motion at the transducer locations. Exemplary Coriolis mass flowmeters are described in U.S. Pat. No. 4,109,524 to Smith, U.S. Pat. No. 4,491,025 to Smith et al., and U.S. Pat. No. Re. 31,450 to Smith.

[0005] The accuracy of Coriolis mass flowmeters may be compromised by the mountings which constrain vibration of the conduit. The affects of these constraints may be reduced by using flowmeter designs that are balanced to reduce effects attributable to external vibration, and by using frequency domain filters, e.g., bandpass filters designed to filter out components of the motion signals away from the excitation frequency. However, mechanical filtering approaches are often limited by mechanical considerations, e.g., material limitations, mounting constraints, weight limitations, size limitations and the like, and frequency domain filtering may be ineffective at removing unwanted vibrational contributions near the excitation frequency.

[0006] Conventional Coriolis mass flowmeters typically use a “calibration factor” to scale mass flow rate to time difference or phase measurements to generate a mass flow estimate. Typically, in order to generate a calibration factor, a calibration or “proving” procedure is performed in which a material (e.g., water) is passed through the vibrating conduit of the mass flowmeter in a test fixture while signals produced by the motion transducers of the mass flowmeter are processed to determine time or phase differences. This data is then processed to generate an estimated calibration factor for the meter.

[0007] This approach can have several disadvantages. The calibration procedure can be time-consuming, labor-intensive and costly. In addition, because the calibration is typically not performed in the field, it may be subject to inaccuracies due to differences between the test conditions and the conditions of the field, including differences in mounting conditions.

[0008] According to embodiments of the present invention, a calibration factor for a parameter sensor including a conduit configured to contain a material and a plurality of motion transducers operative to generate motion signals representing motion of the conduit is determined from a normal modal dynamic characterization of motion of the conduit. The calibration factor may, for example, relate mass flow rate to a spatio-temporal relationship among motion signals produced by the motion transducers, such as a time difference or phase relationship. The calibration factor may be generated, for example, by generating a solution to a modal domain differential equation of motion, and by generating the calibration factor from this solution.

[0009] According to some embodiments of the present invention, the normal modal dynamic characterization characterizes motion of the conduit as a function of a modal Coriolis term that describes coupling among a plurality of normal modes of the conduit responsive to mass flow in the conduit. Determination of the calibration factor may be preceded by determining the modal Coriolis term from a mode shape function that describes motion of the conduit in a normal mode as a function of location on the conduit. Modal mass and stiffness terms of the normal modal characterization may be generated using, for example, conventional modal analysis techniques.

[0010] For example, in some embodiments, the modal Coriolis term may be determined from a spatial Coriolis characterization that describes motion of a plurality of discrete locations of the conduit in a spatial domain in response to a predetermined mass flow. The modal Coriolis term may be determined from the spatial Coriolis characterization, (e.g., a spatial Coriolis matrix), using a transformation that relates the spatial domain to the plurality of normal modes. The plurality of discrete locations associated with the spatial Coriolis characterization may include locations other than the transducer locations.

[0011] The transformation that relates the spatial domain to the plurality of normal modes may be generated from a mode shape function that describes motion of the conduit in a normal mode as a function of location on the conduit. The mode shape function may be determined from a predetermined eigenvalue and a predetermined boundary condition, for example, from an assumption as to a boundary condition to which the conduit will be constrained.

[0012] According to other embodiments of the present invention, an orthogonality of a mode shape function for a predetermined mass flow in the conduit is determined in order to determine a modal Coriolis term of a normal modal dynamic characterization of conduit motion. Similar orthogonality determinations can be used to generate modal mass and stiffness terms of the normal modal characterization. Such terms may also be generated using conventional modal analysis techniques.

[0013] In other embodiments of the present invention, an estimated spatial response for a plurality of locations of the conduit is generated from a normal modal dynamic characterization, and a calibration factor is generated from the estimated spatial response. For example, a spatio-temporal relationship among movements at a plurality of locations of the conduit, such as a time difference or a phase relationship, may be determined, and the calibration factor may be determined from the spatio-temporal relationship.

[0014] In still other embodiments of the present invention, mass flow of a material in a conduit may be determined by determining a calibration factor from a normal modal dynamic characterization of motion of the conduit, generating a plurality of motion signals representing motion of the conduit at a plurality of locations on the conduit, and processing the motion signals according to the determined calibration factor to produce a mass flow estimate. For example, the calibration factor may be determined from a representation of a modal differential equation of motion, as described above. The mass flow estimate may be generated, for example, by determining a spatio-temporal relationship among the plurality of motion signals and applying the calibration factor to the determined spatio-temporal relationship to generate a mass flow estimate.

[0015] According to aspects of the present invention, the calibration factor may be determined from a mode shape function that describes motion of the conduit in a normal mode as a function of location on the conduit. The mode shape function may be based on an assumed boundary condition for the conduit. Motion of the conduit may be constrained to induce a boundary condition that approximates the assumed boundary condition, for example, by attaching the conduit to structure or by applying a force to the conduit responsive to motion of the conduit to induce a boundary condition that approximates the assumed boundary condition.

[0016] In still other embodiments of the present invention, an apparatus for calibrating a parameter sensor including a conduit configured to contain a material, a plurality of motion transducers operative to generate motion signals representing motion of the conduit and a mass flow estimator circuit operative to estimate mass flow from the motion signals according to a calibration factor is provided. The apparatus includes a calibration factor generator circuit operative to generate a calibration factor from a normal modal dynamic characterization of motion of the conduit, the calibration factor generator circuit including an interface circuit operative to convey the generated calibration factor to the mass flow estimator circuit.

[0017] In some embodiments, the calibration factor generator circuit includes a normal modal dynamic characterizer circuit operative to generate a normal modal dynamic characterization of motion of the conduit and a calibration factor determiner circuit operative to determine the calibration factor from the normal modal characterization. The normal modal characterization may be, for example, a representation of a modal differential equation of motion in a modal domain including a plurality of normal modes of motion of the conduit, and the calibration factor determiner circuit may be operative to generate a solution of the modal differential equation of motion from the representation of the modal differential equation of motion and to generate the calibration factor from the solution of the modal differential equation of motion using, for example, the techniques described above. The calibration factor generator circuit may be integrated with the mass flow estimator circuit in, for example, a mass flow sensor apparatus, or may be separately implemented in, for example, test or process control equipment.

[0018] In yet other embodiments of the present invention, a mass flow sensor includes a conduit, such as a substantially straight tube, that is configured to contain a material. The mass flow sensor further includes an actuator operative to excite the conduit, a plurality of motion transducers operative to generate motion signals representing motion of the conduit and a mass flow estimator circuit configured to receive the motion signals and operative to generate a mass flow estimate therefrom using a calibration factor derived from a normal modal characterization of motion of the conduit that assumes a predetermined boundary condition for the conduit. Means are provided for constraining motion of the conduit to approximate the predetermined boundary condition.

[0019] The means for constraining conduit motion may include, for example, means for attaching the conduit to a structure, such as clamps, welds or other fastening means that attach spaced apart locations of the conduit to a rigid structure. The means for constraining motion of the conduit may also include means for applying a force to the conduit responsive to motion of the conduit to thereby induce a boundary condition that approximates the assumed boundary condition. For example, the means for applying a force to the conduit may include at least two motion transducers operative to generate motion signals representing motion of the conduit, a plurality of actuators operatively associated with the conduit, and a shape control circuit configured to receive motion signals from at least two motion transducers and operative to drive the plurality of actuators responsively thereto.

[0020] According to other embodiments of the present invention, a computer program product for characterizing a parameter sensor including a conduit configured to contain a material and a plurality of motion transducers operative to generate motion signals representing motion of the conduit is provided. The computer program product includes computer-readable program code embodied in a computer-readable storage medium, the computer-readable program code including program code for determining a calibration factor for processing motion signals produced by the plurality of motion transducers from a normal modal dynamic characterization of motion of the conduit. The program code for determining a calibration factor may include program code for generating a solution to a modal differential equation of motion in a modal domain including a plurality of normal modes of motion of the conduit, and program code for generating the calibration factor from the generated solution of the modal differential equation of motion. The normal modal dynamic characterization may characterize motion of the conduit as a function of a modal Coriolis term that describes coupling among a plurality of normal modes of the conduit responsive to mass flow in the conduit. The computer readable program code may include program code for determining the modal Coriolis term from a mode shape function that describes motion of the conduit in a normal mode as a function of location on the conduit.

[0021] FIG. 1 is a schematic diagram of a mass flow sensor apparatus according to embodiments of the present invention.

[0022] FIG. 2 is a schematic diagram of a mass flow sensor apparatus according to other embodiments of the present invention.

[0023] FIG. 3 is a schematic diagram of a calibration factor generator circuit according to embodiments of the present invention.

[0024] FIGS. 4, 5 and 6 are flowcharts illustrating exemplary operations for determining a calibration factor for a mass flow sensor according to embodiments of the present invention.

[0025] FIGS. 7, 8, 9, 10 and 11 are schematic diagrams mass flow sensor apparatus according to various embodiments of the present invention.

[0026] FIG. 12 is a flowchart illustrating exemplary operations for estimating mass flow according to embodiments of the present invention.

[0027] FIG. 13 is a free-body diagram illustrating behavior of a material-containing conduit.

[0028] The present invention now will be described more fully hereinafter with reference to the accompanying drawings, in which embodiments of the invention are shown. This invention may, however, be embodied in many different forms and should not be construed as limited to the embodiments set forth herein; rather, these embodiments are provided so that this disclosure will be thorough and complete, and will fully convey the scope of the invention to those skilled in the art. Like numbers refer to like elements throughout. As will be appreciated by one of skill in the art, the present invention may be embodied as systems (apparatus), methods, or computer program products.

[0029] Embodiments of the present invention described herein relate to Coriolis mass flowmeters that utilize a single, substantially straight conduit that does not require, for example, a mechanically-tuned balance beam. The present invention may be particularly advantageous in such applications. However, those skilled in the art will appreciate that the present invention can be used with other types of flowmeter structures, including conventional curved tube structures and straight tube structures including mechanical balance beams.

[0030] As will be appreciated by one of skill in the art, the present invention may be embodied as an apparatus and/or method and/or computer program product. The present invention may be implemented in hardware or in a combination of hardware and software aspects. Furthermore, the present invention may also take the form of a computer program product including a computer-usable storage medium having computer-usable program code embodied in the medium. Any suitable computer readable medium may be utilized, including semiconductor memory devices (e.g., RAMs, ROMs, EEPROMs, and the like), hard disks, CD-ROMs, optical storage devices, and magnetic storage devices.

[0031] Computer program code for carrying out operations of the present invention may be written in an object oriented programming language, such as Java® or C++, and/or in a procedural programming languages, such as “C.” The program code may execute on a single computer or data processing device, such as a microcontroller, microprocessor, or digital signal processor (DSP), or may be executed on multiple devices, for example, on multiple data processing devices that communicate via serial or parallel data busses within an electronic circuit board, chassis or assembly, or which form part of a data communications network such as a local area network (LAN), wide area network (WAN), or internet.

[0032] The present invention is described below with reference to flowchart illustrations and/or block diagrams of methods, apparatus (systems) and computer program products according to embodiments of the invention. It will be understood that blocks of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program code (instructions). These computer program code may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart and/or block diagram block or blocks. These computer program products also may be embodied in a computer-readable storage medium (e.g., magnetic disk or semiconductor memory, code magnetic memory or the like) that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the computer program stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart and/or block diagram block or blocks. The computer program code may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the code that executes on the computer or other programmable apparatus provides steps for implementing the functions specified in the flowchart and/or block diagram block or blocks.

[0033] According to embodiments of the present invention, a Coriolis mass flow sensor may be calibrated using a normal modal dynamic characterization of the motion of the sensor's material-containing conduit. In some embodiments of the present invention, this normal modal dynamic characterization comprises a modal differential equation of motion that includes a Coriolis term that represents coupling among normal modes of the flow sensor conduit in response to mass flow.

[0034] According to some embodiments of the present invention, the Coriolis term comprises a modal Coriolis matrix that is generated from a spatial (or “physical”) Coriolis matrix that describes mass flow dependent motion of the conduit in a spatial domain (e.g., Cartesion). The modal Coriolis matrix may be derived from the spatial Coriolis matrix using a transformation that is derived from a mode shape function that describes modal behavior of the conduit as a function of location on the conduit. The mode shape function may be generated from motion signals generated by motion transducers operatively associated with the sensor conduit. In particular, discrete modal response data from the transducer locations may be used to generate a continuous mode shape function, which may then be used to calculate a modal-to-spatial transformation Φ_virtualthat includes “virtual” locations of the conduit other than the actual transducer locations, such as locations between the actual transducer locations.

[0035] The spatial Coriolis matrix may represent a combination of local Coriolis matrices that correspond to beamlike segments defined by nodes corresponding to the virtual locations. The local Coriolis matrices represent motion of these segments under a predetermined mass flow condition, (e.g., a predetermined mass flow rate). The transformation Φ_virtualmay be applied to the spatial Coriolis matrix to determine a modal Coriolis term (matrix) of a (vector) modal differential equation of motion that also includes modal mass, modal damping and modal stiffness terms.

[0036] The modal differential equation of motion may be solved to determine a modal response at the predetermined mass flow rate from a known input/excitation. The modal response may be converted to a spatial domain response at the actual transducer locations, and a calibration factor may then be determined based on the estimated spatial domain response and the predetermined mass flow rate. This calibration factor may then be applied to motion signals generated in response to an unknown mass flow to generate an estimate of the unknown mass flow.

[0037] According to other embodiments of the present invention, a more generalized approach to determining a modal Coriolis matrix may be taken. In particular, instead of determining a modal Coriolis matrix from a discrete transformation of a spatial Coriolis matrix, a modal Coriolis matrix may be generated by determining an orthogonality of a mode shape function at a predetermined mass flow rate. A modal Coriolis matrix so determined may be then used to generate a solution to a modal differential equation of motion for the conduit and, from this, a calibration factor may be determined along the lines described above.

[0038] Embodiments of the present invention offer several potential advantages over conventional calibration techniques. In particular, use of calibration techniques according to embodiments of the present invention may eliminate a need to pass material through the mass flow sensor, as is commonly required in conventional proving processes. This can provide, for example, an in situ calibration capability.

[0039] The present invention can also facilitate the practical implementation of straight tube mass flow sensors such as “clamp-on” mass flow sensors, e.g., sensors that include motion transducers, actuators and related components that are configured to operatively engage to a pipeline such that the pipeline itself serves as part of the sensor. A potential difficulty in calibrating such a clamp on sensor is that, unlike a conventional Coriolis mass flow meter, the flow conduit may not be available to the sensor's manufacturer. Although one theoretically could connect a prover to the pipeline and calibrate a clamp-on flow sensor against a master meter, such a procedure may be prohibitively time consuming, costly, or impractical due to difficulty in accessing the sensor, harshness of the pipeline environment, and the like. According to embodiments of the present invention, in situ calibration of such a sensor can be performed by measuring the modal parameters of the installation after the sensor components are installed. This can eliminate the need for a calibration facility or master meter.

[0040] A potential advantage of such a clamp-on flow sensor is that it may be moved to different locations and used, for example, as a temporary sensor for trouble shooting purposes. As each pipeline the sensor components are attached to may differ, a calibration may be needed each time the sensor is relocated. In situ calibration according to embodiments of the invention can provide a practical and cost-effective way to perform such a calibration.

[0041] FIG. 1 illustrates a mass flow sensor apparatus 100 according to embodiments of the present invention. The sensor 100 includes a conduit 103 configured to contain a material 108. An actuator 106 is operative to excite the conduit 103. For example, the actuator 106 may comprise an inertial or relative actuator driven by a drive circuit (not shown) that causes the actuator 106 to vibrate the conduit 103 at a predetermined frequency; however, it will be appreciated that other mechanisms for imparting vibratory motion to the conduit 103 may be used. Motion transducers 105 (e.g., inertial velocity transducers, accelerometers or other motion-sensing devices) are positioned along the conduit 103 and produce motion signals 107 representing motion of the conduit 103 in response to a plurality of forces that may include, for example, a drive force F_Dimparted by the actuator 106, Coriolis forces Fc arising from the flowing material 108, and extraneous forces F_E.

[0042] The sensor interface circuit 100 further includes a mass flow determiner circuit 110 that is operative to receive the motion signals 107 and to generate a mass flow estimate 115 therefrom according to a calibration factor 155. The calibration factor generator circuit 150 determines the calibration factor 155 from a normal modal dynamic characterizer 152, which characterizes the normal mode dynamics of the conduit 103, for example, a modal differential equation of motion as described in detail below. As also described in detail below, the normal modal dynamic characterization may be derived from a mode shape function generated from discrete modal response data, e.g., response data generated by motion signals representing motion of the conduit, such as motion signals generated by the motion transducers 105.

[0043] FIG. 2 illustrates a mass flow sensor apparatus 200 according to other embodiments of the present invention. The sensor 200 includes a conduit 203 configured to contain a material 208. A relative actuator 206 is attached to a structure 204 (which may comprise, for example, a housing that encloses the conduit 203) such that it induces motion of the conduit 203 relative to the structure 204. As shown, relative motion transducers 205 (e.g., velocity transducers, accelerometers or other motion-sensing devices) are positioned along the conduit 203 and measure motion of the conduit 203 relative to the structure 204. The motion transducers 205 produce motion signals 207 representing motion of the conduit 203 in response to forces exerted on the conduit 203. It will be understood that, although the motion transducers 205 are shown as relative motion transducers, absolute or “inertial” motion sensors (e.g., accelerometers), or a combination of relative and absolute motion transducers (e.g., a combination of relative velocity transducers and accelerometers), may be used instead.

[0044] The sensor 200 further includes a mass flow determiner circuit 210 that is operative to receive the motion signals 207 and to generate a mass flow estimate 215 therefrom according to a calibration factor 255. The calibration factor generator circuit 250 determines the calibration factor 255 from a normal modal dynamic characterizer 252, which characterizes the normal mode dynamics of the conduit 103.

[0045] It will be understood that, in general, functions of a calibration factor generator circuit according to embodiments of the present invention, such as the calibration factor generator circuits 150, 250 of FIGS. 1 and 2, may be implemented in analog circuitry, digital circuitry and combinations thereof. This circuitry may be integrated in a single device, such as a microprocessor, microcontroller, digital signal processor (DSP), application-specific integrated circuit or other monolithic device, or may be distributed across multiple devices. It will be further appreciated that a calibration factor generator circuit according to embodiments of the present invention may be integrated within a flow sensor apparatus, e.g., using processing circuitry also used to implement a mass flow determiner circuit such as the mass flow determiner circuits 110, 210 of FIG. 1, or may implemented in separate hardware, such as a piece of test or maintenance equipment or process control computing apparatus. For example, a calibration factor generator circuit could be implemented in test equipment designed to interface with motion transducers of a flow sensor, to determine a calibration factor from motion signals generated by the motion transducers, and to download the determined calibration factor to a mass flow determiner circuit of the flow sensor.

[0046] The following discussion provides an underlying theoretical description that supports apparatus, methods and computer program products according to embodiments of the present invention. It will be appreciated that the scope of the present invention is not limited to this theoretical explanation. For example, the discussion that follows describes computational operations. Although these operations are presented with a particular sequence and organization to facilitate understanding of the present invention, it will be appreciated that the computational operations may be reordered, combined in a different fashion or otherwise modified within the scope of the present invention. Consequently, it will be further understood that, in general, the present invention includes not only the specific computational operations described herein, but also equivalent operations.

[0047] A modal differential equation of motion for a single straight flowmeter conduit, such as the conduit 103 of FIG. 1, is:

[0048] [mr]{{umlaut over (η)}}+[cr]{{dot over (η)}}+[dr]{{dot over (η)}}+[kr]{η}={N},  (1)

[0049] where {η}, {{dot over (η)}}, {{umlaut over (η)}} represent displacement, velocity and acceleration vectors, respectively, in a normal modal domain comprising a plurality of normal (i.e., orthogonal) modes of vibration of the conduit, [mr] is a modal mass matrix, [cr] is a modal Coriolis matrix that describes coupling between the normal modes of the conduit responsive to mass flow in the conduit, [dr] is a modal damping matrix, [kr] is a modal stiffness matrix, and {N} is a modal excitation applied to the conduit. The modal mass matrix [mr] and the modal stiffness matrix [kr] may be determined using conventional modal analysis techniques. However, determining the modal Coriolis matrix [cr] can be problematic. As described in greater detail below, a flow calibration factor (FCF) may be assumed to be relatively insensitive to damping and strongly correlated to the modal Coriolis matrix [cr].

[0050] A U.S. patent application Ser. No. 09/941,332, entitled “DETERMINING PROPERTIES OF A FLOW TUBE AND OF A FLUID FLOWING THROUGH A FLOW TUBE OF A CORIOLIS FLOWMETER,” filed concurrently herewith and incorporated herein by reference in its entirety, describes how discrete mode shape information for a flowmeter conduit gained from motion transducers that generate motion signals representative of motion of the flowmeter conduit can be used to determine a continuous mode shape function, i.e., a function that describes motion of the conduit in a plurality of normal modes as a function of location on the conduit. From such a continuous mode shape function, “virtual” pickoff responses, i.e., responses for locations other than the transducer locations, may be synthesized.

[0051] The aforementioned patent application Ser. No. 09/941,332 describes continuous mode shape functions of the form: $\begin{array}{cc}{\Phi }_r\left(x\right)=[\begin{array}{cccc}{e}^{\lambda 1_{r}x}& e^{-\lambda 1_{r}x}& e^{j\lambda 2_{r}x}& e^{-j\lambda 2_{r}x}\end{array}]{\left\{\begin{array}{c}{\mathrm{bc}}_1\\ {\mathrm{bc}}_2\\ {\mathrm{bc}}_3\\ {\mathrm{bc}}_4\end{array}\right\}}_r,& \left(2\right)\end{array}$

[0052] wherein Φ_ris a mode shape function for an r-th normal mode, and wherein eigenvalues λ1_r,λ2_rand boundary conditions {bc₁, bc₂, bc₃, bc₄}_rof the mode shape function Φ_rmay be calculated from modal measurements.

[0053] Such a mode shape function can be used to transform a spatial or “physical” Coriolis matrix to a modal Coriolis matrix. In particular, a modal transformation [Φ_virtual] for a plurality of selected locations of a flowmeter conduit may be synthesized using mode shape functions as described above. The number of such “virtual” locations preferably is greater than the number of locations at which motion transducers are present. A modal Coriolis matrix [cr] may then be calculated from a spatial Coriolis matrix [C] for the plurality of virtual locations using [Φ_virtual] according to the following:

[0054] [cr]=[Φ_virtual]^T[C][Φ_virtual]  (3)

[0055] The spatial Coriolis matrix [C] may be generated using conventional techniques. The plurality of virtual locations of the conduit may be viewed as defining a plurality of beamlike segments having first and second nodes that correspond to virtual locations, with a spatial Coriolis matrix for each segment at a predetermined mass flow rate {dot over (m)} being given by: $\begin{array}{cc}\left[C_{\mathrm{local}}\right]=\stackrel{.}{m}\left[\begin{array}{cccc}0& \frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}& 1& -\frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}\\ -\frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}& 0& \frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}& \frac{-l^4}{360\mathrm{gz}+30l^2}\\ -1& -\frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5L^2}& 0& \frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}\\ \frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}& \frac{l^4}{360\mathrm{gz}+30l^2}& -\frac{10\mathrm{gzl}+l^2}{60\mathrm{gz}+5l^2}& 0\end{array}\right]& \left(4\right)\end{array}$

[0056] wherein the first row/column of the “local” Coriolis matrix [C_local]corresponds to the physical displacement of a first node of a segment, the second row and column corresponds to the angular displacement of the first node of the segment, the third row/column corresponds to the physical displacement of a second node of the segment, and the forth row and column corresponds to the angular displacement of the second node of the segment. In equation (4), I is the length of the segment and gz represents a combination material and geometric properties of the conduit, which may be given by: $\begin{array}{cc}\mathrm{gz}=\frac{{\mathrm{EI}}_t}{\hat{k}{\mathrm{GA}}_t}& \left(5\right)\end{array}$

[0057] wherein E is the modulus of elasticity for the conduit material, I_tis the area moment of inertia for the conduit, G is the shear modulus of elasticity for the conduit material, A_tis the cross-sectional area of the conduit, {circumflex over (k)} is a numerical factor dependent on the cross-sectional shape of the conduit. The spatial Coriolis matrix [C] for the conduit may be constructed from the local Coriolis matrices [C_local], using techniques known to those skilled in the art.

[0058] A modal dynamic stiffness matrix [MDS] for a drive frequency excitation, i.e., a frequency at which the conduit is vibrated, can then be created and inverted to determine a modal frequency response function [MDS]⁻¹that can be used to determine a modal response {η} for the predetermined mass flow rate {dot over (m)}:

[0059] {η}=(−ω_drv²[mr]+jω_drv[cr]+jω_drv[dr]+[kr])⁻¹{N}=[MDS]⁻¹{N}.  (6)

[0060] The modal response {η} can be transformed into a spatial domain response vector {y} as follows:

[0061] {y}=[Φ_measured][MDS]⁻¹[Φ_measured]^T{F}.  (7)

[0062] A spatio-temporal relationship between transducer locations, such as a time difference (“delta-t”) or a phase relationship between motion signals corresponding to the transducer locations, for the predetermined mass flow rate {dot over (m)} can be determined from the estimated spatial response {y}. For example, a spatial integration technique for generating a time difference or phase estimate from spatial motion information provided by a plurality of spatially distributed motion transducers may be used. An example of a spatial integration technique is described in U.S. patent application Ser. No. 09/116,845 to Cunningham et al., filed Jul. 16, 1998 and incorporated herein by reference in its entirety [previously filed case on spatial integration].

[0063] The time difference or phase relationship, which may be referred to generally as Δ, and the predetermined mass flow rate {dot over (m)} can be used to determine a flow calibration factor FCF in the appropriate units (e.g., mass/sec/sec or mass/sec/degrees): $\begin{array}{cc}\mathrm{FCF}=\frac{\stackrel{.}{m}}{\Delta }.& \left(8\right)\end{array}$

[0064] A time difference or phase relationship measurement (generally, Δ_measured) generated responsive to an unknown mass flow rate {dot over (m)}_unknownmay be multiplied by the calibration factor FCF to generate an estimate of the unknown mass flow rate {dot over (m)}_unknown:

[0065] {dot over (m)}_unknow=FCF×(Δ_measured−Δ₀),  (9)

[0066] where Δ₀is an offset (e.g., time or phase) between transducer signals at substantially zero flow (“zero offset”).

[0067] Experimentation indicates that, in many practical applications, mass flow rate is nearly linearly proportional to time-difference over a range of mass flow rates, such that FCF may be assumed to be constant. However, if FCF is not sufficiently constant with flow, it could be estimated over a range of assumed flow rates to develop, for example, a lookup table or functional representation thereof. FCF could also be modified based on other parameters, such as temperature, e.g., one could compensate for a change in FCF due to changes in modulus (E) with temperature.

[0068] If it is assumed that the ends of a flow conduit of a mass flow sensor are constrained such that forced response of the motion of the conduit ends is about zero, and if it is further assumed that the case or other structure supporting the motion transducers is stiff and massive enough such that it acts like a rigid body, the motion signals generated by the sensor's motion transducers can be treated as approximating absolute velocity signals. Under such conditions, the flow conduit can be treated as having approximately fixed boundary conditions at the frequency of excitation of the conduit. The boundary conditions and eigenvalues may, therefore, be determined, at least for Euler beam assumptions.

[0069] For a flow sensor conduit 1310 as illustrated in the free-body diagram of FIG. 13:

[0070] V′−ρ_tA_t−dÿ+Fc+τ{overscore (c)}(θ+α)+F(x,t)=0

[0071] M′−Sy′+V−ρ_tI_t−Mc=0  (10)

[0072] where V is the shear force exerted on the conduit 1310, ρ_tis the density of the material of the conduit 1310, A_tis the cross-sectional area of the conduit 1310, d is a damping coefficient, ÿ is the acceleration of the conduit 1310, Fc is force coupling between a material 1320 contained in the conduit 1310, τ is viscose shear force between the material 1320 in the conduit and the inner wall of the conduit 1310, θ is the angular displacement of the conduit 1310, α is the shear distortion of the conduit 1310, F(x,t) is the drive force applied to the conduit 1310, M is the bending moment of the conduit 1310, S is the tension acting on the conduit 1310, ρ_tI_tis the inertia per unit length of the conduit 1310, and {overscore (c)} is circumference of the inner surface of the conduit 1310, and wherein dots indicate time derivatives and primes indicate spatial derivatives with respect to length. For the material 1320 contained within the conduit 1310: $\begin{array}{cc}-{\rho }_f{A_f\left(\frac{\partial }{\partial t}+v_f\frac{\partial }{\partial x}\right)}^{2}y-\mathrm{Fc}-\tau \stackrel{\_}{c}\left(\theta +\alpha \right)=00-{\rho }_f{I}_f\ddot{\theta }+Mc+{\mathrm{PA}}_f{y}^{\prime }=0& \left(11\right)\end{array}$

[0073] where ρ_fA_fis the mass per unit length of the material 1320 in the conduit 1310, y is the displacement of the conduit 1310, v_fis the velocity of the material 1320 in the conduit 1310, ρ_fI_fis the inertia per unit length of the material 1320 in the conduit 1310, {umlaut over (θ)} is the angular acceleration of the conduit 1310, P is the pressure of the material 1320 in the conduit 1310, and A_fis the cross-sectional area of the material 1320 in the conduit 1310.

[0074] Eliminating coupling forces (Fc), moments (Mc) and fluid shear ( ) on the wall of the conduit “beam”: $\begin{array}{cc}{V}^{\prime }-{\rho }_t{A}_t-d\ddot{y}+{\rho }_f{A_f\left(\frac{\partial }{\partial t}+v_f\frac{\partial }{\partial x}\right)}^{2}y+F\left(x,t\right)=0M^{\prime }-{\mathrm{Sy}}^{\prime }+V-\left({\rho }_t{I}_t+{\rho }_f{I}_f\right)\ddot{\theta }+{\mathrm{PA}}_f{y}^{\prime }=0& \left(12\right)\end{array}$

[0075] From basic elasticity principles:

[0076] y′(x,t)=θ(x,t)+α(x,t)V(x,t)={circumflex over (k)}GAα(x,t)M(x,t)=EI_tθ′(x,t)  (13)

[0077] This can be reduced to two equations in terms of y(x,t) and x,t. $\begin{array}{cc}\hat{k}{\mathrm{GA}}_t\left(y^{″}-{\theta }^{\prime }\right)-{\rho }_t{A}_t\ddot{y}-d\stackrel{.}{y}-{\rho }_f{A_f\left(\frac{\partial }{\partial t}+v_f\frac{\partial }{\partial x}\right)}^{2}y+f\left(x,t\right)=0{\mathrm{EI}}_t{\theta }^{″}-{\mathrm{Sy}}^{\prime }+\hat{k}{\mathrm{GA}}_t\left(y^{\prime }-\theta \right)-\left({\rho }_t{I}_t+{\rho }_f{I}_f\right)\ddot{\theta }+{\mathrm{PA}}_f{y}^{\prime }=0& \left(14\right)\end{array}$

[0078] Non-dimensionalizing the basic independent variables in equation (14), where the under-bar indicates a dimensionless variable of space or time and the pointed brackets indicate the U.S. Customary units for the variables: $\begin{array}{cc}x=L\underset{\_}{x}dx=Ld\underset{\_}{x}t=\frac{1}{{\omega }_1}\underset{\_}{t}dt=\frac{1}{{\omega }_{1·}}d\underset{\_}{t}\underset{\_}{{\omega }_r}=\frac{{\omega }_r}{{\omega }_1}\underset{\_}{\omega }=\frac{\omega }{{\omega }_1}{\eta }_r\left(t\right)=e^{j\omega t}={\eta }_r\left(\underset{\_}{t}\right)=e^{j\frac{\omega }{{\omega }_1}\underset{\_}{t}}=e^{j\underset{\_}{\omega }t}⟨\mathrm{in}⟩\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)=\frac{1}{L}{\eta }_r\left(\underset{\_}{t}\right)⟨0⟩{\Phi }_r\left(x\right)={\Phi }_r\left(\underset{\_}{x}\right)=\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)⟨0⟩{\Phi }_r\left(x\right)=[\begin{array}{cccc}{e}^{\mathrm{\lambda 1},x}& e^{-\mathrm{\lambda 1},x}& e^{j\mathrm{\lambda 2},x}& e^{-\mathrm{j\lambda 2},x}\end{array}]\left\{{\mathrm{BC}}_r\right\}⟨0⟩{\Phi }_r\left(\underset{\_}{x}\right)=[\begin{array}{cccc}{e}^{\mathrm{\lambda 1},L\underset{\_}{x}}& e^{-\mathrm{\lambda 1},\underset{\_}{L}x}& e^{j\mathrm{\lambda 2},L\underset{\_}{x}}& e^{-\mathrm{j\lambda 2},L\underset{\_}{x}}\end{array}]\left\{{\mathrm{BC}}_r\right\}⟨0⟩\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)=[\begin{array}{cccc}{e}^{\mathrm{\lambda 1},L\underset{\_}{x}}& e^{-\mathrm{\lambda 1},L\underset{\_}{x}}& e^{j\mathrm{\lambda 2},L\underset{\_}{x}}& e^{-\mathrm{j\lambda 2},L\underset{\_}{x}}\end{array}]\left\{{\mathrm{BC}}_r\right\}⟨0⟩{\Psi }_r\left(x\right)={\Psi }_r\left(\underset{\_}{x}\right)=\frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)⟨1/\mathrm{in}⟩{\Psi }_r\left(x\right)=[\begin{array}{cccc}{e}^{\mathrm{\lambda 1},x}& e^{-\mathrm{\lambda 1},x}& e^{j\mathrm{\lambda 2},x}& e^{-\mathrm{j\lambda 2},x}\end{array}]\left[u_r\right]\left\{{\mathrm{BC}}_r\right\}⟨1/\mathrm{in}⟩{\Psi }_r\left(\underset{\_}{x}\right)=[\begin{array}{cccc}{e}^{\mathrm{\lambda 1},L\underset{\_}{x}}& e^{-\mathrm{\lambda 1},L\underset{\_}{x}}& e^{j\mathrm{\lambda 2},L\underset{\_}{x}}& e^{-\mathrm{j\lambda 2},L\underset{\_}{x}}\end{array}]\left[u_r\right]\left\{{\mathrm{BC}}_r\right\}⟨1/\mathrm{in}⟩\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)=[\begin{array}{cccc}{e}^{\mathrm{\lambda 1},L\underset{\_}{x}}& e^{-\mathrm{\lambda 1},L\underset{\_}{x}}& e^{j\mathrm{\lambda 2},L\underset{\_}{x}}& e^{-\mathrm{j\lambda 2},L\underset{\_}{x}}\end{array}]\left[\underset{\_}{u_r}\right]\left\{{\mathrm{BC}}_r\right\}⟨0⟩\left[\underset{\_}{u_r}\right]=L\left[u_r\right]y\left(x,t\right)=\sum _r{\Phi }_r\left(x\right){\eta }_r\left(t\right)=\sum _r{\Phi }_r\left(\underset{\_}{x}\right){\eta }_r\left(\underset{\_}{t}\right)=y\left(\underset{\_}{x},\underset{\_}{t}\right)=⟨\mathrm{in}⟩\underset{\_}{y}\left(\underset{\_}{x},\underset{\_}{t}\right)=\frac{1}{L}\sum _r{\Phi }_r\left(\underset{\_}{x}\right){\eta }_r\left(\underset{\_}{t}\right)=\sum _r{\Phi }_r\left(\underset{\_}{x}\right)\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)⟨0⟩\theta \left(x,t\right)=\sum _r{\Psi }_r\left(x\right){\eta }_r\left(t\right)=\sum _r{\Psi }_r\left(\underset{\_}{x}\right){\eta }_r\left(\underset{\_}{t}\right)=\theta \left(\underset{\_}{x},\underset{\_}{t}\right)=⟨0⟩\underset{\_}{\theta }\left(\underset{\_}{x},\underset{\_}{t}\right)=\sum _r\frac{1}{L}\sum _r\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)L\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)=\sum _r\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)⟨0⟩& \left(15\right)\end{array}$

[0079] Differentiating and integrating for the non-dimensionalized mode shapes Φ_r,Ψ_r: $\begin{array}{cc}\frac{\partial y\left(x,t\right)}{\partial x}=\frac{1}{L}\frac{\partial {\Phi }_r\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}{\eta }_r\left(\underset{\_}{t}\right)=\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)⟨0⟩& \left(16\right)\\ \frac{{\partial }^{2}y\left(x,t\right)}{\partial x^2}=\frac{1}{L^2}\frac{{\partial }^2{\Phi }_r\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}{\eta }_r\left(\underset{\_}{t}\right)=\frac{1}{L}\frac{{\partial }^2\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)⟨\frac{1}{\mathrm{in}}⟩& \\ \frac{\partial y\left(x,t\right)}{\partial t}={\omega }_1{\Phi }_r\left(\underset{\_}{x}\right)\frac{\partial {\eta }_r\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}={\omega }_{1}L{\Phi }_r\left(\underset{\_}{x}\right)\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}⟨\frac{\mathrm{in}}{\sec }⟩& \\ \frac{{\partial }^{2}y\left(x,t\right)}{\partial t^2}={\omega }_1^2{\Phi }_r\left(\underset{\_}{x}\right)\frac{{\partial }^2{\eta }_r\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}={\omega }_1^{2}L{\Phi }_r\left(\underset{\_}{x}\right)\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}⟨\frac{\mathrm{in}}{{\sec }^2}⟩& \\ \frac{{\partial }^{2}y\left(x,t\right)}{\partial x\partial t}=\frac{{\omega }_1}{L}\frac{\partial {\Phi }_r\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\frac{\partial {\eta }_r\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}={\omega }_1\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}⟨\frac{1}{\sec }⟩& \\ \frac{\partial \theta \left(x,t\right)}{\partial x}=\frac{1}{L}\frac{\partial {\Psi }_r\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}{\eta }_r\left(\underset{\_}{t}\right)=\frac{1}{L}\frac{\partial \underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)⟨\frac{1}{\mathrm{in}}⟩& \\ \frac{{\partial }^2\theta \left(x,t\right)}{\partial x^2}=\frac{1}{L^2}\frac{{\partial }^2{\Psi }_r\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}{\eta }_r\left(\underset{\_}{t}\right)=\frac{1}{L^2}\frac{{\partial }^2\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)⟨\frac{1}{{\mathrm{in}}^2}⟩& \\ \frac{\partial \theta \left(x,t\right)}{\partial t}={\omega }_1{\Psi }_r\left(\underset{\_}{x}\right)\frac{\partial {\eta }_r\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}={\omega }_1\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)=\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}⟨\frac{1}{\sec }⟩& \\ \frac{{\partial }^2\theta \left(x,t\right)}{\partial t^2}={\omega }_1^2{\Psi }_r\left(\underset{\_}{x}\right)\frac{{\partial }^2{\eta }_r\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}={\omega }_1^2\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)=\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}⟨\frac{1}{\sec }⟩& \\ \frac{\partial {\Phi }_r\left(x\right)}{\partial x}=\frac{1}{L}\frac{\partial {\Phi }_r\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}=⟨1\text{/}\mathrm{in}⟩& \\ \frac{\partial {\Psi }_r\left(x\right)}{\partial x}=\frac{1}{L}\frac{\partial {\Psi }_r\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}=⟨1\text{/}{\mathrm{in}}^2⟩& \\ {\int }_{x_L}^{x_R}{\Phi }_r\left(x\right)dx=L{\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\Phi }_r\left(\underset{\_}{x}\right)d\underset{\_}{x}=⟨\mathrm{in}⟩& \left(17\right)\\ {\int }_{x_L}^{x_R}{\Phi }_m\left(x\right){\Phi }_n\left(x\right)dx=L{\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\Phi }_m\left(\underset{\_}{x}\right){\Phi }_n\left(\underset{\_}{x}\right)d\underset{\_}{x}=⟨\mathrm{in}⟩& \end{array}$

[0080] In equation (17), the limits of integration x_R/L and x_L/L are typically 0 to 1 or −½ to +½ depending on where the coordinate system has been placed. Equation (18) is the same as equation (14), except all the non-dimensional relationships have been included: $\begin{array}{cc}\frac{\hat{k}{\mathrm{GA}}_t}{{\mathrm{EI}}_t}\left(\frac{1}{L}\frac{{\partial }^2\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}-\frac{1}{L}\frac{\partial \underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\right)\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)-\frac{\rho A{\omega }_1^{2}L}{{\mathrm{EI}}_t}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}-\frac{d{\omega }_{1}L}{{\mathrm{EI}}_t}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}-\dots \frac{2{\rho }_f{A}_f{v}_f{\omega }_1}{{\mathrm{EI}}_t}\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}-\frac{{\rho }_f{A}_f{v}_f^2}{{\mathrm{EI}}_t}\frac{1}{L}\frac{{\partial }^2\underset{\_}{{\Phi }_r}\left(x\right)}{\partial {\underset{\_}{x}}^2}{\eta }_r\left(\underset{\_}{t}\right)+\frac{1}{{\mathrm{EI}}_t}F\left(\underset{\_}{x_{\mathrm{drv}}},\underset{\_}{t}\right)=0\frac{1}{L^2}\frac{{\partial }^2\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)+\frac{\left({\mathrm{PA}}_f-S\right)}{{\mathrm{EI}}_t}\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)+\frac{\hat{k}{\mathrm{GA}}_t}{{\mathrm{EI}}_t}\left(\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}-\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\right)\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)-\dots \frac{\rho I{\omega }_1^2}{{\mathrm{EI}}_t}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}=0& \text{(18)}\end{array}$

[0081] Multiplying through equation (18) by L²and L, respectively: $\begin{array}{cc}\frac{\hat{k}{\mathrm{GA}}_t{L}^2}{{\mathrm{EI}}_t}\left(\frac{1}{L}\frac{{\partial }^2\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}-\frac{1}{L}\frac{\partial \underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\right)\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)-\frac{\rho A{\omega }_1^2{L}^4}{{\mathrm{EI}}_t}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\frac{{\partial }^2\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}-\frac{d{\omega }_1{L}^4}{{\mathrm{EI}}_t}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}-\dots \frac{2{\rho }_f{A}_f{v}_f{\omega }_1{L}^3}{{\mathrm{EI}}_t}\frac{1}{L}\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}-\frac{{\rho }_f{A}_f{v}_f^2{L}^2}{{\mathrm{EI}}_t}\frac{1}{L}\frac{{\partial }^2\underset{\_}{{\Phi }_r}\left(x\right)}{\partial {\underset{\_}{x}}^2}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)+\frac{L^2}{{\mathrm{EI}}_t}F\left(\underset{\_}{x},\underset{\_}{t}\right)=0\frac{1}{L}\frac{{\partial }^2\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)}{\partial {\underset{\_}{x}}^2}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)+\frac{\left({\mathrm{PA}}_f-S\right)L^2}{{\mathrm{EI}}_t}\frac{1}{L}\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)+\frac{\hat{k}{\mathrm{GA}}_t{L}^2}{{\mathrm{EI}}_t}\left(\frac{1}{L}\frac{\partial \underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)}{\partial \underset{\_}{x}}-\frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\right)\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)-\dots \frac{\rho I{\omega }_1^2{L}^2}{{\mathrm{EI}}_t}\frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\frac{{\partial }^2\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}=0& \text{(19)}\end{array}$

[0082] In equation (19), S is positive for tension, therefore S is negative when the tension is positive: $\begin{array}{cc}\begin{array}{c}\underset{\_}{\rho A}=\frac{\rho A{\omega }_1^2{L}^4}{{\mathrm{EI}}_t}\underset{\_}{S}=\frac{\left({\mathrm{PA}}_f-S\right)L^2}{{\mathrm{EI}}_t}\underset{\_}{\rho I}=\frac{\rho I{\omega }_1^2{L}^2}{{\mathrm{EI}}_t}\\ \underset{\_}{A}=\frac{{\rho }_f{A}_f{v}_f^2{L}^2}{{\mathrm{EI}}_t}\underset{\_}{B}=\frac{\hat{k}{\mathrm{GAL}}^2}{{\mathrm{EI}}_t}\underset{\_}{\stackrel{.}{m}}=\frac{{\rho }_f{A}_f{v}_f^2}{{\mathrm{EI}}_t}\\ \underset{\_}{f}=\frac{L^3}{{\mathrm{EI}}_t}f\left(x,t\right)\underset{\_}{F}=\frac{L^2}{{\mathrm{EI}}_t}F\left(x_{\mathrm{drv}},t\right)\underset{\_}{d_r}=\frac{2{\zeta }_r{\omega }_r\rho A{\omega }_1{L}^4}{{\mathrm{EI}}_t}\end{array}& \text{(20)}\end{array}$

[0083] Putting equation (19) in compact matrix form: $\begin{array}{cc}[\begin{array}{cc}\underset{\_}{\rho A}& 0\\ 0& \underset{\_}{\rho I}\end{array}]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}\frac{{\partial }^2\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}+[\begin{array}{cc}\underset{\_}{d}+2\underset{\_}{\stackrel{.}{m}}\frac{\partial •}{\partial \underset{\_}{x}}& 0\\ 0& 0\end{array}]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}+\dots \left[\begin{array}{cc}\left(\underset{\_}{A}-\underset{\_}{B}\right)\frac{{\partial }^2•}{\partial {\underset{\_}{x}}^2}& \underset{\_}{B}\frac{\partial •}{\partial \underset{\_}{x}}\\ -\left(\underset{\_}{S}+\underset{\_}{B}\right)\frac{\partial •}{\partial \underset{\_}{x}}& \underset{\_}{B}•-\frac{{\partial }^2•}{\partial \underset{\_}{x}}\end{array}\right]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)=\left\{\begin{array}{c}\underset{\_}{F}\\ 0\end{array}\right\}& \left(21\right)\end{array}$

[0084] It may be noted that the mode shape functions Φ_r(x) and Ψ_r(x) are non-dimensional. The 1/L term is carried along to non-dimensionalize the integration when calculating orthogonality. The “box” symbol is a placeholder for the differential operator. The first term of equation (21) accounts for all the mass and inertia of the beam. The second term accounts for proportional damping and the Coriolis effects. The third term accounts for the stiffness, centrifugal forces, pressure, and tension in the beam. The stiffness matrix is mainly diagonal with some slightly skew symmetric terms.

[0085] To calculate the orthogonality, or “self-adjointedness” of any two modes m and r, each term of the differential equation may be integrated over the entire length of the beam: $\begin{array}{cc}{\int }_{x_L}^{x_R}{\left\{\begin{array}{c}{\Phi }_m\left(x\right)\\ {\Psi }_m\left(x\right)\end{array}\right\}}^T[\begin{array}{cc}\underset{\_}{\rho A}& 0\\ 0& \underset{\_}{\rho I}\end{array}]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}dx\frac{{\partial }^2\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}+{\int }_{x_L}^{x_R}{\left\{\begin{array}{c}{\Phi }_m\left(x\right)\\ {\Psi }_m\left(x\right)\end{array}\right\}}^T[\begin{array}{cc}\underset{\_}{d}+2\underset{\_}{\stackrel{.}{m}}\frac{\partial •}{\partial \underset{\_}{x}}& 0\\ 0& 0\end{array}]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}dx\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}+\dots {\int }_{x_L}^{x_R}{\left\{\begin{array}{c}{\Phi }_m\left(x\right)\\ {\Psi }_m\left(x\right)\end{array}\right\}}^T\left[\begin{array}{cc}\left(\underset{\_}{A}-\underset{\_}{B}\right)\frac{{\partial }^2•}{\partial {\underset{\_}{x}}^2}& \underset{\_}{B}\frac{\partial •}{\partial \underset{\_}{x}}\\ -\left(\underset{\_}{S}+\underset{\_}{B}\right)\frac{\partial •}{\partial \underset{\_}{x}}& \underset{\_}{B}•-\frac{{\partial }^2•}{\partial {\underset{\_}{x}}^2}\end{array}\right]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}dx\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)={\left\{\begin{array}{c}{\Phi }_m\left(x_{\mathrm{drv}}\right)\\ {\Psi }_m\left(x_{\mathrm{drv}}\right)\end{array}\right\}}^T\left\{\begin{array}{c}\underset{\_}{F}\\ 0\end{array}\right\}& \text{(22)}\end{array}$

[0086] Modifying the integration such that the integration is non-dimensionalized over the length L: $\begin{array}{cc}L{\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T[\begin{array}{cc}\underset{\_}{\rho A}& 0\\ 0& \underset{\_}{\rho I}\end{array}]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}d\underset{\_}{x}\frac{{\partial }^2\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial {\underset{\_}{t}}^2}+L{\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T[\begin{array}{cc}\underset{\_}{d}+2\underset{\_}{\stackrel{.}{m}}\frac{\partial •}{\partial \underset{\_}{x}}& 0\\ 0& 0\end{array}]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}dx\frac{\partial \underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)}{\partial \underset{\_}{t}}+\dots L{\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T\left[\begin{array}{cc}\left(\underset{\_}{A}-\underset{\_}{B}\right)\frac{{\partial }^2•}{\partial {\underset{\_}{x}}^2}& \underset{\_}{B}\frac{\partial •}{\partial \underset{\_}{x}}\\ -\left(\underset{\_}{S}+\underset{\_}{B}\right)\frac{\partial •}{\partial \underset{\_}{x}}& \underset{\_}{B}•-\frac{{\partial }^2•}{\partial \underset{\_}{x}}\end{array}\right]\left\{\begin{array}{c}\frac{1}{L}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \frac{1}{L}\underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}d\underset{\_}{x}\underset{\_}{{\eta }_r}\left(\underset{\_}{t}\right)={\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(x_{\mathrm{drv}}\right)\\ \underset{\_}{{\Psi }_m}\left(x_{\mathrm{drv}}\right)\end{array}\right\}}^T\left\{\begin{array}{c}\underset{\_}{F}\\ 0\end{array}\right\}& \text{(23)}\end{array}$

[0087] The length L cancels out, yielding non-dimensional modal mass, damping, Coriolis, and stiffness matrices mr, dr, cr, kr: $\begin{array}{cc}{\underset{\_}{\mathrm{mr}}}_{m,n}={\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T[\begin{array}{cc}\underset{\_}{\rho A}& 0\\ 0& \underset{\_}{\rho I}\end{array}]\left\{\begin{array}{c}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}d\underset{\_}{x}& \text{(24)}\\ {\underset{\_}{\mathrm{kr}}}_{m,n}={\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T[\begin{array}{cc}\left(\underset{\_}{A}-\underset{\_}{B}\right)\frac{{\partial }^2•}{\partial {\underset{\_}{x}}^2}& \underset{\_}{B}\frac{\partial •}{\partial \underset{\_}{x}}\\ -\left(\underset{\_}{S}+\underset{\_}{B}\right)\frac{\partial •}{\partial \underset{\_}{x}}& \underset{\_}{B}•-\frac{{\partial }^2•}{\partial {\underset{\_}{x}}^2}\end{array}]\left\{\begin{array}{c}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}d\underset{\_}{x}& \text{(25)}\\ {\underset{\_}{\mathrm{dr}}}_{m,n}+{\underset{\_}{\mathrm{cr}}}_{m,n}={\int }_{\frac{x_L}{L}}^{\frac{x_R}{L}}{\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T\left[\begin{array}{cc}\underset{\_}{d}+2\underset{\_}{\stackrel{.}{m}}\frac{\partial •}{\partial \underset{\_}{x}}& 0\\ 0& 0\end{array}\right]\left\{\begin{array}{c}\underset{\_}{{\Phi }_r}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_r}\left(\underset{\_}{x}\right)\end{array}\right\}d\underset{\_}{x}& \text{(26)}\\ N_m={\left\{\begin{array}{c}\underset{\_}{{\Phi }_m}\left(\underset{\_}{x}\right)\\ \underset{\_}{{\Psi }_m}\left(\underset{\_}{x}\right)\end{array}\right\}}^T\left\{\begin{array}{c}\underset{\_}{F}\\ 0\end{array}\right\}& \text{(27)}\end{array}$