Coriolis mass flow/density sensor with a single straight measuring tube

This mass flow/density sensor (10), which can be installed in a pipe and through which a fluid to be measured flows during operation, is to be balanced over a wide density range, so that accurate measurements are possible. A single straight measuring tube (13) having a longitudinal axis (131) extends between its inlet end (11) and the outlet end (12) and is fixed to a support, e.g., a cylindrical tube (14, 14'). The support has a longitudinal centroidal line (141) which is parallel to, but does not coincide with, the longitudinal axis (131) of the measuring tube. A cantilever (15) is fixed to the measuring tube (13) midway between the inlet and outlet ends (11, 12) and in operation causes the measuring tube to vibrate either in a first fundamental flexural mode or in a second fundamental flexural mode having a higher frequency than this first mode. An excitation arrangement (16) disposed midway between the end pieces excites the measuring tube (13) in the second mode. Sensors (17, 18) for the motions of the measuring tube on the inlet and outlet side are positioned between the middle of the tube and the inlet and outlet ends, respectively. The support may be provided with a counterbalance. Because of the torsional vibrations exerted by the cantilever on the measuring tube, the sensor is also well suited for measuring the viscosity of the fluid.

FIELD OF THE INVENTION 
This invention relates to a Coriolis mass flow/density sensor with a single 
straight measuring tube. 
BACKGROUND OF THE INVENTION 
U.S. Pat. No. 5,531,126 describes a Coriolis massflow/density sensor which 
can be installed in a pipe by means of a connecting element at the inlet 
end and a connecting element at the outlet end and through which a fluid 
to be measured flows during operation, comprising: 
a single straight measuring tube having a longitudinal axis and extending 
between and fixed to the connecting elements; 
a straight dummy tube extending parallel to the measuring tube and not 
traversed by the fluid; 
a nodal plate on an inlet side and a nodal plate on an outlet side, 
one of which fixes the inlet-end portion of the measuring tube to the 
corresponding end portion of the dummy tube, and 
the other of which fixes the outlet-end portion of the measuring tube to 
the corresponding end portion of the dummy tube, so that the measuring 
tube and the dummy tube are arranged side by side; 
a support tube having its ends fixed to the respective connecting elements 
and having a longitudinal axis of symmetry parallel to the longitudinal 
axis of the measuring tube; and 
means which act only on the dummy tube to excite the measuring tube into 
flexural vibrations whose frequency is not, however, identical with the 
resonance frequency of the measuring tube, with the measuring tube and the 
dummy tube vibrating in antiphase. 
This prior-art Coriolis mass flow/density sensor is mechanically balanced 
only in a narrow range of density values--approximately .+-.5% of a rated 
density--for a given dimensional design, i.e., only at these density 
values will forces originating from the vibrations of the measuring tube 
be practically not transmitted via the connecting elements to the pipe. 
The above range is extended by the excitation "beside" the resonance 
frequency, but substantially more excitation energy is required than for 
excitation at the resonance frequency. The less balanced the mass 
flow/density sensor is, the more such forces and vibrational energy will 
be transmitted to the pipe; thus, however, vibrational energy is lost and 
measuring inaccuracy increases. 
This unbalance has a disturbing effect not only in case of 
temperature-induced changes in the density of one and the same fluid but 
also particularly during the measurement of different fluids flowing in 
the pipe at different times, for example one after another. 
SUMMARY OF THE INVENTION 
Since Coriolis mass flow/density meters should be suitable for measuring as 
wide a range of very different fluids with different densities as 
possible, it is therefore important to provide Coriolis mass flow/density 
sensors which are balanced in the above sense over a wide density range 
and thus measure accurately. 
To accomplish this, a first variant of the invention provides a Coriolis 
mass flow/density sensor which can be installed in a pipe and through 
which a fluid to be measured flows during operation, comprising: 
a single straight measuring tube having a longitudinal axis, an inlet end, 
and an outlet end; 
a support fixed to the inlet end and the outlet end, 
a longitudinal centroidal line of which is parallel to, but does not 
coincide with, the longitudinal axis of the measuring tube; 
a cantilever 
which is fixed to the measuring tube midway between the inlet end and the 
outlet end, and 
which during operation causes the measuring tube to vibrate either in a 
first fundamental flexural mode or in a second fundamental flexural mode 
having a higher frequency than the first fundamental flexural mode; 
an excitation arrangement for constantly exciting the measuring tube in the 
second fundamental flexural mode 
which is disposed approximately midway between the inlet end and the outlet 
end; and 
a sensor for the motions of the measuring tube on an inlet side and a 
sensor for the motions of the measuring tube on an outlet side which are 
located between the middle of the measuring tube and the inlet end and 
outlet end, respectively, at the same distance therefrom. 
In a first preferred embodiment of the first variant of the invention, the 
support is a cylindrical tube having a wall of uniform thickness and a 
longitudinal axis which is parallel to, but does not coincide with, the 
longitudinal axis of the measuring tube. 
In a second preferred embodiment of the first variant of the invention, the 
support is a cylindrical tube having a wall of only partially uniform 
thickness and a longitudinal axis which is parallel to, or coincides with, 
the longitudinal axis of the measuring tube, with the tube wall in the 
region of a first generating line diametrically opposite the cantilever 
being at least partially thicker than the uniform wall thickness and/or 
the tube wall in the region of a first generating line adjacent to the 
cantilever being at least partially thinner than the uniform wall 
thickness in order to form a counterbalance. 
According to a development of the second embodiment of the first variant of 
the invention, a counterweight is attached, partially inserted in, or 
integrally formed on the tube wall diametrically opposite the cantilever. 
In a third preferred embodiment of the first variant of the invention, 
which can be used in the above embodiments and the development of the 
second embodiment, the cantilever has the form of a plate or disk having a 
bore by means of which the plate or disk is slipped over the measuring 
tube. The plate or disk preferably consists of a semicircular ring portion 
and a rectangular portion formed thereon, the semicircular ring portion 
being coaxial with the bore. Advantageously, the plate or disk has a 
thickness equal to approximately half the diameter of the measuring tube. 
According to a development of the first variant of the invention and its 
embodiments, the measuring tube is provided with an annular rib on the 
inlet side and an annular rib on the outlet side which are disposed at the 
locations of the respective sensors. 
In a fourth preferred embodiment of the first variant of the invention, the 
excitation arrangement consists of 
a first portion which acts on the measuring tube in the direction of the 
intersection of a longitudinal axis of symmetry of the cantilever and the 
longitudinal axis of the measuring tube with a first excitation force, and 
a 
second portion, which acts on an end of the cantilever remote from the 
measuring tube with a second excitation force directed opposite to the 
first excitation force. 
A second variant of the invention provides a Coriolis mass flow/density 
sensor which can be installed in a pipe and through which a fluid to be 
measured flows during operation, comprising: 
a single straight measuring tube having an inlet end and an outlet end; 
an inlet plate fixed at the inlet end and surrounding the measuring tube; 
and outlet plate fixed at the outlet end and surrounding the measuring 
tube; 
a first support plate fixed to the inlet plate and the outlet plate and 
extending parallel to a first generating line of the measuring tube; 
a second support plate fixed to the inlet plate and the outlet plate and 
extending parallel to a second generating line of the measuring tube 
diametrically opposite the first generating line; 
a cantilever 
which is fixed to the measuring tube midway between the inlet end and the 
outlet end, and 
which during operation causes the measuring tube to vibrate either in a 
first fundamental flexural mode or in a second fundamental flexural mode 
having a higher frequency than the first fundamental flexural mode; 
a longitudinal bar located opposite the cantilever and fixed to the first 
and second support plates, said longitudinal bar acting as a 
counterbalance; 
an excitation arrangement 
which constantly excites the measuring tube in the second fundamental 
flexural mode, and 
which is disposed approximately midway between the inlet end and the outlet 
end; and 
a sensor for the motions of the measuring tube on an inlet side and a 
sensor for the motions of the measuring tube on an outlet side which are 
located between the middle of the measuring tube and the inlet end and 
outlet end, respectively, at the same distance therefrom. 
In a first preferred embodiment of the second variant of the invention, the 
cantilever is a plate having a front surface, a back surface, an axis of 
torsional vibration parallel to the axis of the measuring tube, and a bore 
by means of which the plate is slipped over the measuring tube, said plate 
consisting of a circular-segment portion, which is coaxial with the bore, 
and a rectangular portion formed thereon, an end surface of which, which 
is cut centrally by a diameter of the measuring tube, is fixed to a 
fastening area of a beam which is longer than the end surface and has a 
first end and a second end which project beyond the end surface and 
comprise respective continuations of the fastening area. 
According to a development of this first embodiment of the second variant 
of the invention, the excitation arrangement consists of a first 
excitation system, fixed to the continuation of the fastening area of the 
first beam end, and a second excitation system, fixed to the continuation 
of the fastening area of the second beam end, with the first and second 
excitation systems comprising a first coil and a second coil, 
respectively, which in operation are traversed by an exciting current in 
opposite directions. 
According to a second development of the second variant of the invention, 
which can also be used in the first preferred embodiment of the second 
variant, in order to suppress modes of vibration other than the second 
fundamental flexural mode, 
a first part of a first brake assembly based on the eddy-current principle 
is fixed to the front surface of the plate in an area in which the axis 
point through said plate; 
a first part of a second brake assembly based on the eddy-current principle 
is fixed to the back surface of the plate in an area in which the axis of 
torsional vibration has a possible piercing point through said plate; 
the first brake assembly comprises a second part which is attached to a 
first holder fixed at least to the first support plate; and 
the second brake assembly comprises a second portion which is attached to a 
second holder fixed at least to the first support plate. 
In a preferred embodiment of the first development of the second variant of 
the invention, the first parts of the brake assemblies are circular 
cylindrical permanent magnets, and the second parts of the brake 
assemblies are copper disks. 
The two variants of the invention and their embodiments and developments 
may be further improved by extending the measuring tube beyond the inlet 
and outlet ends using respective tube sections of equal length whose 
respective free ends are fixed in a housing. 
According to a further development of the invention, in addition to the 
second fundamental flexural mode, the first fundamental flexural mode is 
excited. 
One advantage of the invention is that the accuracy of the mass flow 
measurement is excellent over a wide density range (0 kg/m.sup.3 to 3000 
kg/m.sup.3 ; 0 kg/m.sup.3 corresponds to a null measurement with a vacuum 
in the measuring tube). Mass flow/density sensors of a preproduction 
series, for example, showed accuracies within 0.1% of the measured value. 
Another important advantage of the invention is that it is also well suited 
for measuring the viscosity of the fluid, which is based on the following 
facts, which are familiar to those skilled in the art: 
The viscosity of a fluid can be measured with a Coriolis mass flow/density 
sensor only if the measuring tube or tubes of the sensor (also) perform a 
torsional vibration, so that shear forces are exerted on the fluid. In the 
case of straight measuring tubes excited in flexural modes of vibration, 
no torsional vibrations, and thus no shear forces, occur at all. 
In the case of bent, particularly U-shaped, measuring tubes, torsional 
vibrations do occur, but their amplitude is so small that a viscosity 
measurement is virtually impossible. Patent documents in which viscosity 
is mentioned in connection with Coriolis mass flow/density sensors are not 
very numerous. 
U.S. Pat. No. 4,938,075, for example, only mentions the Navier-Stokes 
equation, which includes the shear viscosity, which is not measured, 
however. Other patent specifications deal only with the viscosity 
compensation of the measured mass-flow values; see, for example, U.S. Pat. 
Nos. 5,027,662, 4,876,879, and 4,872,351. 
Single-tube Coriolis mass flow/density sensors according to the invention 
perform not only the flexural vibration required and desired for mass-flow 
and density measurements but also, because of the cantilever, a torsional 
vibration in the second fundamental flexural mode around an axis whose 
position is explained below. 
In the invention, the amplitude of this torsional vibration is sufficient 
to permit the viscosity of the fluid to be measured in addition to mass 
flow and density with only little additional electronic circuitry. 
For this viscosity measurement, recourse can be had to methods described in 
the literature which discuss this measurement in connection with the 
measurement of fluid density using vibrating mechanical arrangements, 
particularly tubes. 
According to the journal "IEEE Transactions on Idustrial Electronics and 
Control Instrumentation", August 1980, pages 247 to 253, for example, 
viscosity can be determined if the resonance quality factor of the 
vibrating mechanical arrangement, including the fluid, is measured. This 
can be done, for example, by measuring the electric current with which the 
arrangement is excited.

DETAILED DESCRIPTION OF THE DRAWINGS 
The partially sectioned longitudinal view of FIG. 1 shows a Coriolis mass 
flow/density sensor 10 according to the first variant of the invention, 
and FIG. 2 shows a section taken along line II--II of FIG. 1. When in use, 
the mass flow/density sensor 10 is installed in a pipe of a given diameter 
through which flows a fluid to be measured, the pipe being not shown for 
the sake of clearness. The sensor is connected to the pipe fluid-tight. 
FIGS. 1 and 2 show flanges 111 and 121 for this purpose, which are 
connected via short tube sections 112 and 122 to end pieces 113 and 123, 
respectively, in which terminate and are fixed an inlet-end portion 11 and 
an outlet-end portion 12, respectively, of a single measuring tube 13; the 
measuring tube is straight and has a longitudinal axis 131. Mass 
flow/density sensor 10 may also be installed in the pipe via conventional 
fixing means other than flanges 111, 121. 
Inlet end 11 and outlet end 12 of measuring tube 13 are fixed to a support 
in the form of, e.g., an open or closed frame or a cylindrical tube 14. 
The closed frame or the cylindrical tube enclose the measuring tube 13 in 
the manner of an encasement. Measuring tube 13 and end pieces 113, 123 as 
well as the latter and the support are preferably welded together. 
The support has a longitudinal centroidal axis 141 which is parallel to, 
but does not coincide with, the longitudinal axis 131 of measuring tube 
13. This noncongruence is apparent from FIGS. 3 and 4, which show sections 
taken along line III--III of FIG. 1 and line IV--IV of FIG. 2, 
respectively. 
Each of FIGS. 1 to 4 shows a circular cylindrical tube 14 of uniform wall 
thickness. The longitudinal centroidal line of tube 14 is therefore 
identical with the longitudinal axis of the tube, and measuring tube 13 
and tube 14, because of the above-mentioned parallelism of their axes, are 
not concentric, i.e., not coaxial. 
Fixed to measuring tube 13 midway between end pieces 113, 123 is a 
cantilever 15, which may be a plate or disk with a bore by means of which 
the plate or disk is slipped over measuring tube 13. In FIGS. 3 and 4 it 
can be seen that in the embodiments shown therein, the plate consists of a 
semicircular ring portion 151, which is coaxial with the bore, and a 
rectangular portion 152 integrally formed thereon. 
FIG. 5, a sectional view similar to FIGS. 3 and 4, shows that the 
cantilever 15 may also be a circular disk 153 which is slipped over 
measuring tube 13 by means of an eccentric bore and is fixed on this tube. 
Circular disk 153 and measuring tube 13 are thus not concentric. 
The plate or disk serving as cantilever 15 in FIGS. 2 to 5 preferably has a 
thickness equal to approximately half the diameter of measuring tube 13. 
In FIGS. 2 to 4, an excitation arrangement 16 can be seen which is disposed 
approximately midway between end pieces 113, 123. It is, for example, an 
electromagnetic shaker comprising, for example, a coil assembly 161, 
mounted on the support or tube 14, and a permanent magnet 162, attached to 
cantilever 15. 
The excitation arrangement may be any of the various types of excitation 
arrangements described for this purpose in the prior art relating to 
Coriolis mass flow/density sensors and Coriolis mass flow meters. 
The excitation arrangement may also consist of a first portion, which acts 
on the measuring tube in the direction of the intersection of a 
longitudinal axis of symmetry of the cantilever and the longitudinal axis 
of the measuring tube with a first excitation force, and a second portion, 
which acts on an end of the cantilever remote from the measuring tube with 
a second excitation force directed opposite to the first excitation force 
(not shown for the sake of simplicity). 
In operation, excitation arrangement 16 excites measuring tube 13 in 
flexural modes of vibration whose frequency is equal to the mechanical 
resonance frequency of the measuring tube. This resonance frequency, as 
has been known for decades also in connection with Coriolis mass flow 
sensors, is a measure of the density of the fluid to be measured, cf., for 
example, U.S. Pat. No. 4,187,721. Further details of the excitation of 
vibrations are given below in connection with the explanation of FIGS. 12 
and 13. 
FIGS. 2 also shows schematically a sensor 17 for the motions of measuring 
tube 13 on the inlet side, which can also be seen in FIGS. 3 and 4, and a 
sensor 18 for the motions of measuring tube 13 on the outlet side. The 
sensors 17 and 18 are located between the middle of the measuring tube and 
inlet-end piece 113 and outlet-end piece 123, respectively, at the same 
distance therefrom. Preferably, annular ribs 132, 133 are provided on 
measuring tube 13 at the locations of the respective sensors. 
For the sensors 17, 18, the various types of sensors described for this 
purpose in the prior art relating to Coriolis mass flow/density sensors 
and Coriolis mass flow meters can be used, such as displacement, velocity, 
or acceleration sensors which operate electrodynamically or optically, for 
example. 
FIGS. 6 to 11, in sectional views corresponding to FIG.1, show different 
implementations of a second embodiment of the first variant of the 
invention using a tube 14' which is again circular cylindrical in terms of 
its inside diameter, but which, unlike the first embodiment, shown in 
FIGS. 1 to 5, has only a partially uniform wall thickness, so that its 
longitudinal centroidal line does not coincide with the longitudinal axis 
of tube 14'. 
On the other hand, this longitudinal axis of tube 14' coincides with the 
longitudinal axis of measuring tube 13. The two tubes in FIGS. 6 to 11 are 
thus coaxial although tube 14' does not have a uniform wall thickness. 
This coincidence, however, is not mandatory: The two longitudinal axes may 
also be parallel to each other. 
In the implementations of FIGS. 6 to 11, the mandatory parallelism of the 
longitudinal axis of measuring tube 13 and the longitudinal centroidal 
line of the support, particularly of tube 14', follows from the nonuniform 
wall thickness of tube 14'. As a result of this nonuniform wall thickness, 
the wall of tube 14' acts as a counterbalance to cantilever 15, i.e., the 
unbalance caused by the cantilever is offset. 
In the first implementation of the counterbalance, shown in FIG. 6, the 
wall of tube 14' along a first generating line diametrically opposite 
cantilever 15 is thicker than the uniform wall thickness of the remainder 
of this tube over the entire length. This can be achieved, for example, by 
providing a longitudinal rib 142 along the first generating line, 
particularly by welding or soldering it on. The width and height of the 
longitudinal rib and its material must be selected taking into account the 
mass of cantilever 15 and the uniform wall thickness of tube 14'. 
In the second implementation of the counterbalance, shown in FIG. 7, the 
wall of tube 14' along the first generating line diametrically opposite 
cantilever 15 is thickened only in a region opposite cantilever 15 by a 
counterweight 143. 
Counterweight 143 may again be fixed on tube 14' by being welded or 
soldered on, for example, or it may be inserted into and fixed in a bore 
or a blind hole made in the wall of the tube, or it may be formed 
integrally with the tube as shown. The width, height, and length of 
counterweight 143 and its material must be selected taking into account 
the mass of cantilever 15 and the uniform wall thickness of tube 14'. 
In the third implementation of the counterbalance, shown in FIG. 8, the 
wall of tube 14' along the first generating line diametrically opposite 
cantilever 15 is thicker than the uniform wall thickness of the remainder 
of tube 14 only over two sections 144, 145 of the entire length. Sections 
144, 145 extend from the respective ends of tube 14' toward the middle of 
the tube, thus forming a central section 146 which has the uniform wall 
thickness of tube 14'. 
The thickening in sections 144, 145 may again be achieved, for example, by 
welding or soldering respective longitudinal ribs to tube 14' or forming 
such ribs integrally with the tube. The width, height, length, and 
material of sections 144, 145 must be selected taking account of the mass 
of cantilever 15 and the uniform wall thickness of tube 14'. 
In the fourth implementation of the counterbalance, shown in FIG. 9, the 
wall of tube 14' along a second generating line adjacent to cantilever 15 
was made thinner than the uniform wall thickness of the remainder of the 
tube over its entire length. 
In the fifth implementation of the counterbalance, shown in FIG. 10, the 
wall of tube 14' along the second generating line adjacent to cantilever 
15 was made thinner only in a central section 147 opposite cantilever 15 
by removing material from the wall; the remainder of tube 14 has the 
uniform wall thickness. 
In the sixth implementation of the counterbalance, shown in FIG. 11, the 
wall of tube 14' along the second generating line adjacent to cantilever 
15 was made thinner than the uniform wall thickness of the remainder of 
the tube over two sections 148, 149 of its entire length. Sections 148, 
149 extend from the respective ends of tube 14' toward the middle of the 
latter, thus forming a central section 150 which has the uniform wall 
thickness of tube 14'. 
In the case of the mass flow/density sensors of FIGS. 9 to 11, the 
reduction in the wall thickness of tube 14' is achieved by removing, e.g., 
planing or milling, material from the wall of the tube along the second 
generating line. The respective mass of the material to be removed must be 
determined taking into account the mass of cantilever 15 and the uniform 
wall thickness of tube 14'. 
As an introduction to the explanation of the FIGS. 12 and 13, it should be 
recalled that a cantilever-free tube fixed at two points and set into 
flexural vibration between these points has a single flexural mode of 
vibration. It vibrates in this mode like a string does at its fundamental 
resonance frequency, which, in the theory of vibrations, is also referred 
to as the fundamental tone or first harmonic, and which is numerically the 
lowest possible resonance frequency. In the steady state, the string, and 
thus the tube, has a single antinode, and therefore no vibration node, 
between the two fixing points; vibration nodes are located only at the 
fixing points. 
Unlike such a tube without a cantilever, according to the invention, 
measuring tube 13, provided with cantilever 15, vibrates in a first and a 
second flexural mode, as will now be explained with reference to FIGS. 12 
and 13. These figures show schematic central cross sections of the 
vibrator consisting of measuring tube 13 and cantilever 15, with the rest 
position of the vibrator shown in the middle of each of the figures and 
the positions of the vibrator at maximum deflection shown at the left and 
right, cf. the double-headed arrows indicating the direction of vibration. 
FIG. 12 shows the conditions for the first fundamental flexural mode of 
vibration, in which cantilever 15 moves around the axis of measuring tube 
13 toward the left when the measuring tube moves to the left, and toward 
the right when the measuring tube moves to the right. As the cantilever 
rotates slightly outward, the measuring tube performs a pure flexural 
vibration. 
This first fundamental flexural mode has a--"first"--resonance frequency 
corresponding to the above-mentioned resonance frequency which is 
numerically the lowest possible frequency; for a given measuring tube of a 
predetermined diameter, predetermined length, and predetermined wall 
thickness and a cantilever with a predetermined mass and predetermined 
dimensions, this frequency is 400 Hz, for example. 
FIG. 13 shows the conditions for the above-mentioned second fundamental 
flexural mode of vibration, in which cantilever 15 moves about the axis of 
measuring tube 13 toward the right when the measuring tube moves to the 
left, and toward the left when the measuring tube moves to the right, 
i.e., the cantilever rotates inward. Thus, a torsional vibration is 
superimposed on the flexural vibration which the measuring tube performs 
like in the first fundamental flexural mode. 
The axis of this torsional vibration is obviously not the same as the axis 
of the measuring tube but is parallel to this axis. The axis of torsional 
vibration is identical with the centroidal line of all mechanical masses 
which contribute to the vibration in the second fundamental flexural mode. 
These are the masses of the measuring tube, including the mass of the 
fluid, the mass of the cantilever, the mass of the parts of the excitation 
arrangement fixed to and vibrating with the cantilever, and, if present, 
the first parts of the brake assembly explained below. 
In FIG. 13, a possible piercing point of the axis of torsional vibration 
through the plane of the paper is denoted by X; it is located, as shown, 
on the vertical centerline of the cantilever if the mass of the excitation 
arrangement is symmetrical with respect to this centerline. The piercing 
point X moves back and forth slightly as a function of the density of 
fluid. 
The second fundamental flexural mode of vibration has 
a--"second"--resonance frequency which is higher than the 
above-defined--"first"--resonance frequency of the first fundamental 
flexural mode; for the above-mentioned given measuring tube, it is 900 Hz. 
According to the invention, measuring tube 13 is constantly excited in the 
second fundamental flexural mode, in which cantilever 15 moves toward the 
measuring tube when the latter moves outward, as is shown in FIG. 13 and 
as was explained above. 
As the resonance frequency of the second fundamental flexural mode is, or 
can be made, twice as high as the resonance frequency of the first 
fundamental flexural mode, an excitation circuit in the form of a 
phase-locked loop required to energize the excitation arrangement 16, cf., 
for example, U.S. Pat. No. 4,801,897, can be readily designed to excite 
only the second fundamental flexural mode. 
FIG. 14, a representation similar to FIG. 2, shows a development of the 
first variant of the invention which can be used in all the embodiments 
explained so far and in the embodiments shown in FIGS. 15 and 16. 
Reference characters of FIG. 14 corresponding to reference characters used 
so far have the same numeral but are provided with an asterisk. 
In the mass flow/density sensor 10' of FIG. 14, the vibrating measuring 
tube 13' was lengthened beyond the inlet and outlet ends by means of 
equally long, straight tube sections 13" and 13*, respectively, which are 
in alignment with measuring tube 13'. The respective free ends of tube 
sections 13", 13* are fixed in a housing 19. Housing 19 comprises, for 
example, flanges 111', 121' corresponding to flanges 111, 121 of FIG. 1. 
The perspective top view of FIG. 15 and the perspective bottom view of FIG. 
16 show a second variant of a Coriolis mass flow/density sensor 10" 
according to the invention. This sensor has a single straight measuring 
tube 13" with an inlet end 11" and an outlet end 12". An inlet plate 213 
and an outlet plate 223 which completely surround the measuring tube 13" 
are fixed at the inlet end and the outlet end, respectively. 
Fixed to inlet plate 213 and outlet plate 223 is a first support plate 24, 
which extends parallel to measuring tube 13". Also fixed to the inlet 
plate and the outlet plate is a second support plate 34, which extends 
parallel to the measuring tube diametrically opposite to the first support 
plate 24. Thus, side surfaces of the two support plates 24, 34 facing 
toward each other are also parallel to each other. 
Fixed on measuring tube 13" approximately midway between inlet end 11" and 
outlet end 12" is a cantilever 15" which in operation causes measuring 
tube 13" to vibrate either in a first fundamental flexural mode or in a 
second fundamental flexural mode having a higher frequency than the first 
fundamental flexural mode. 
The member acting as counterbalance in the second variant is a longitudinal 
bar 25 which is located opposite to cantilever 15" and is fixed to the 
first and second support plates 24, 34. In FIGS. 15 and 16, longitudinal 
bar 25 extends substantially parallel to the entire vibratory length of 
measuring tube 13"; this is not mandatory, however; it is so in this 
embodiment only. 
The system consisting of the two support plates 24, 34, inlet plate 213, 
outlet plate 223, and longitudinal bar 25 has a longitudinal centroidal 
line parallel to the axis of measuring tube 13". With respect to this 
property, the arrangement of FIGS. 15 and 16 is thus comparable with the 
arrangement of FIGS. 1 to 5. 
Measuring tube 13" is excited in the second fundamental flexural mode of 
vibration by an excitation arrangement 16" which acts on cantilever 15" 
and is thus again disposed approximately midway between the inlet end and 
the outlet end. Special details of the excitation arrangement will be 
explained below in connection with FIG. 17. 
The motions of measuring tube 13" on the inlet and outlet sides are sensed 
with sensors 17" and 18", respectively, which are located between the 
middle of the measuring tube and the inlet and outlet ends, respectively, 
at the same distance therefrom. 
In FIGS. 15 and 16, it is indicated by the heads of the screws shown that 
the above-mentioned fixing of support plates 24, 34 to end plates 213, 223 
and to longitudinal bar 25 may be done by screwing. This is not mandatory, 
however; it is also possible to use other suitable forms of fastening 
familiar to those skilled in the art. 
The second variant, shown in FIGS. 15 and 16, is similar in the design of 
its mechanical vibrating system to the development of the first variant 
shown in FIG. 14. In the second variant, too, measuring tube 13" was 
lengthened beyond the inlet and outlet ends by means of equally long, 
straight tube sections 13.sup.# and 13.sup.+, respectively, which are 
aligned with measuring tube 13". 
The respective free ends of tube sections 13.sup.#, 13.sup.+ are fixed in 
a housing, of which only housing caps 191, 192 are shown, while a 
remaining, tube like part which interconnects and hermetically seals the 
housing caps 191, 192 is not illustrated so as not to cover the internal 
parts shown. 
FIGS. 15 and 16 also show that housing caps 191 and 192 have respective 
connecting portions 193 and 194 formed thereon via which the Coriolis mass 
flow/density sensor 10" can be installed in the above-mentioned pipe 
fluid-tight. 
FIG. 17 is a scaled-up perspective top view of the cantilever 15" and the 
excitation arrangement 16" of FIG. 15 and of further parts preferably 
associated with the excitation arrangement. In a preferred embodiment of 
the second variant of the invention, cantilever 15" is implemented as a 
plate. It has a front surface 154 and a parallel back surface, which is 
not visible in FIGS. 15 and 17. Cantilever 15" has a bore by means of 
which the plate is slipped over and fixed on measuring tube 13". 
The plate--similarly as in the arrangement of FIGS. 1 to 4--consists of a 
circular-segment portion 151", which is coaxial with the bore, and a 
rectangular portion 152" integrally formed thereon. An end surface 155 of 
rectangular portion 152" is cut centrally by a diameter of measuring tube 
13"; only a small portion of this end surface can be seen in FIG. 17, 
namely in bores of a beam 163. 
Beam 163 is fixed in a fastening area (not visible) to end surface 155. 
Screws used for this purpose can be seen in FIG. 15. Beam 163 is longer 
than end surface 155 and has a first end 163' and a second end 163" with 
respective continuations of the fastening area, which ends project beyond 
side surface 155. 
The excitation arrangement 16" consists of a first excitation system 26, 
fixed to the continuation of the fastening area of the first beam end 
163', and a second excitation system 36, fixed to the continuation of the 
fastening area of the second beam end 163". The first excitation system 26 
contains a first coil fixed to support plate 24, and the second excitation 
system 36 contains a second coil fixed to support plate 34. In operation, 
the first and second coils are traversed by an exciting current in 
opposite directions. The coils cooperate with first and second permanent 
magnets fixed to beam ends 163' and 163", respectively. 
According to a further development of the invention, in order to suppress 
modes of vibration other than the second fundamental flexural mode, 
particularly in order to suppress the first fundamental flexural mode and 
its harmonics, a first part 27 of a first brake assembly and a first part 
37 of a second brake assembly based on the eddy-current principle are 
provided. 
The first part 27 of a first brake assembly is fixed to front surface 154 
of the plate in an area where the above-explained axis of the mechanical 
torsional vibration system has a possible piercing point through front 
surface 154. 
In similar fashion, the first part 37 of the second brake assembly is fixed 
to the back surface of the plate in an area where the above-mentioned axis 
of torsional vibration has a possible piercing point through the back 
surface of the plate. 
A second part 28 of the first brake assembly, which end surface is only 
visible in FIG. 15, is secured to a first holder 29 fixed to the first and 
second support plates 24, 34. This holder is only shown in FIG. 15; in 
FIG. 17 it has been omitted in order not to obstruct the view of 
cantilever 15" and other elements. 
A second part 38 of the second brake assembly is secured to a second holder 
39 fixed to the first and second support plates 24, 34, namely by means of 
angled "foots", as shown. The two holders are preferably made of soft 
magnetic material. 
The two first parts 27, 37 of the brake assemblies comprise circular 
cylindrical permanent magnets, of which only permanent magnet 271 can be 
seen in FIG. 17 together with an associated holder 272. The two second 
parts of the brake assemblies are copper disks. 
By means of the brake assemblies, the instantaneous position of the 
above-mentioned axis of torsional vibration is stabilized. This prevents 
the building up of another vibration mode, particularly the first 
fundamental flexural mode and its harmonics, and/or of those harmonics of 
the second fundamental flexural mode which have an axis of torsional 
vibration other than the latter. 
The buildup of such a mode or of such harmonics would be equivalent to a 
considerable back-and-forth motion of the axis of torsional vibration, 
cf., for example, FIG. 12. This stabilization takes place as long as the 
axis of torsional vibration is in the area of the second parts 28, 38 of 
the brake assemblies. 
A further increase in measurement accuracy is possible in both variants of 
the invention if in addition to the second fundamental flexural mode, the 
first fundamental flexural mode is excited; then, the brake assemblies of 
FIGS. 15 and 17 must, of course, be omitted. 
The excitation of the first fundamental flexural mode can be achieved by 
using a further phase-locked loop which operates on the first resonance 
frequency. In that case, the signals provided by the sensors will contain 
both a Coriolis-effect-induced phase-shift component of the vibrations of 
the second resonance frequency and a Coriolis-effect-induced phase-shift 
component of the vibrations of the first resonance frequency. Since these 
two resonance frequencies differ by a factor of about 2, the two 
phase-shift components can be readily separated by electronic means.