coriolis-effect in mass flow metering

Archive of Applied Mechanics 61 (1991) 192--214 Archive of

Applied Mechanics �9 Springer-Verlag 1991

Coriolis-effeet in mass flow metering

I I . Raszi l l ier and F. Durst , E r l a n g e n

Summary: The paper aims at a detailed description of the physical background for the so-called Coriolis mass flow meter. I t presents essentially an analysis of the (free) vibration modes of a fluid conveying straight pipe segment. Due to the inertial effects of the flowing fluid, mMnly the Coriolis force, these modes deviate in shape (and in frequency) from those appearing in the absence of fluid motion. The effect of fluid inertia may, therefore, be exploited for the purpose of flow measurement. The analysis is performed under a simplifying approximation: The pipe is considered as a beam, the fluid as a moving string. This approximation leaves the fluid with only one degree of freedom, connected with its mean velocity, and eliminates an infinity of degrees of freedom of the pipe. Yet it keeps the essential features of the phenomenon. The equations deseribing the vibrations are derived variationally, with the constraint of a common vibration amplitude of both fluid and pipe. The Lagrange multiplier associated with the constraint gives the interaction force between pipe and fluid. The modes are determined by a perturbation procedure, wherein the small (perturbation) parameter is related to the fluid velocity. The analysis shows, as mMn result, how the time delay between the vibrations of two appropriately chosen points of the pipe may serve to determine the mass flow rate of the fluid. Other aspects of the problem, like the precise role of the Coriolis force, are considered. The possible improvement of the used approximation is discussed.

Der Coriolis-Effekt in der Dureh~luBmessung

~bersicht: In der Arbeit wird der physikalische tIintergrund des sogenannten Coriolis-DurchfluBmel~geriites beschrieben. Sic stellt im wesentlichen eine Analyse der (freien) Schwingungsmoden eines durehstrSmten Rohres dar. Wegen der Tr~gheitseffekte des strSmenden Fluids, hauptsgchlieh der Coriolis-Kraft, weichen diese Moden in Form (und Frequenz) von denen bei abwesender StrSmung ab. Deshalb kann die Fluidtriig- heir zum Zwecke der DurchfluBmessung genutzt werden. Bei der Ourchffihrung der Analyse wird eine ver- einfaehende Ngherung vorgenommen: Das Rohr wird als dfinner Balken, das Fluid als laufende Suite (oder laufender Faden) betrachtet. Durch diese Ngherung wird ein Tell der Freiheitsgrade des l~ohres eliminiert; dem Fluid bleibt ein einziger Freiheitsgrad, der mit seiner (mittleren) Geschwindigkeit verbunden ist. Die wesentliehenMerkmale des bier betIaehteten Ph~nomens bleiben jedoch davon unbetroffen. Die Schwingungs- gleichungcn werden fiber das Variationsprinzip abgeleitet unter der Zwangsbedingnng einer gemeinsamen Sehwingungsamplitude yon gohr und Fluid. Der mit dem auferlegten Zwang verbundene Lagrange-Multi- plikator ergibt die zwischen l~ohr und Fluid wirkende Kraft. Die Moden werden dutch ein StSrungsver- fahren bestimmt, mit der Fluidgeschwindigkeit als St6rungsparameter. Die Analyse zeigt als Hauptergebnis, in weleher Weise der Zeitunterschied zwisehen den Schwingungen zweier entsprechend gewi~hlter Punkte des Rohres zur Bestimmung des Massendnrchflusses des Fluides benutzt werden kann. Auch andere Aspekte des Problems, wie etwa die genaue golle der Coriolis-Kraft, werden betrachtet. Die mSgliehe Verbesserung der angewandten Ngherung wird besproehen.

1 Introduction

Among the devices employed nowadays for mass flow measurements of fluids a p rominen t p lace is t a k e n b y the in s t rumen t s based on the iner t ia l effects to the flowing f luid in a v ib ra t ing pipe. A n impress ive d ive r s i ty of rea l iza t ion of these mass flow meters wi th var ious degrees of sophist i- ca t ion in de ta i l is avai lable . Al though the common pr inciple under ly ing all these so-called Coriolis mass flow measur ing devices is, qua l i t a t ive ly , a r a the r s imple ma t t e r , i ts qua n t i t a t i ve t r e a t m e n t is, in mos t cases, a t r emendous t a s k which requires a good phys ica l under s t and ing of

I-I. l~aszillier and F. Durst: Coriolis-effeet in mass flow metering 193

the principle of operation and appropriate mathematical tools. I t is therefore not surprising that, in spite of the wide use of this type of instruments, there is so far very little scientific literature available concerning their quantitative description [1].

In the present paper such a description is given in rather detail for the simplest version of the instrument, yet with a number of simplifying approximations. These simplifications allow a clearer look at the essential physical phenomena taking place, without getting lost in a forest of technical complications. With the insight obtained by them one also learns the appropriate way of adding finer details to this description.

The problem investigated in this paper is that of the (free) transverse planar vibrations of a fluid conveying straight pipe segment. The aim is thereby to deduce from the properties of these vibrations the flow rate of the fluid. In order to handle this problem in its full complexity, the mathematical treatment requires from the theory of elasticity the equations of small vibration of a cylindrical shell under external load, and from fluid dynamics the (Euler or Navier- Stokes) equations of fluid flow together with the continuity equation, and to couple them by the appropriate boundary conditions. Before doing this, it is advisable to dedicate oneself to such an approximation of the problem which is able to give not only an important first quantitative insight, but also the necessary hints regarding the steps of further improvements of its theoretical investigation.

The approximation considered here is most conveniently stated in terms of a model to the real problem to be investigated: The (free) transverse vibrations of an Euler beam, coupled (infinitely strong) in its amplitude to a string (or threadline) which moves longitudinally inside the beam without friction. The beam in this model plays the role of the pipe and the string that of an incompressible inviscid fluid. The physical simplification of this model comes from the supression of vibration modes of the (one-dimensional) Euler beams compared to the (two- dimensional) shell and in the total absence of inner degrees of freedom of the (one-dimensional) string compared to the (three-dimensional) fluid. The precise embedding of this model into a systematic approximation procedure to the pipe-fluid problem remains an important mathematical problem to be solved.

This paper is structured as follows: In Sect. 2 the precise mathematical ~ormulation of the model for the fluid conveying vibrating pipe is derived from the principle of stationary action. Then (Sect. 3) the vibration modes of the model system in the absence of flow (string velocity v = 0) are investigated as preparation for the understanding (Sect. 4) of the effect of flow (v # 0) as a perturbation of these modes. A detailed analysis of special modes of the fluid conveying pipe which are of practical relevanee for mass flow metering, is given in Sects. 5 and 6. Finally, in Sect. 7 some additional questions of interpretation of the model and of its improvement are com- menced. Some background material for the understanding of the vibration modes in the presence of flow is given in Appendices A and B.

2 Basic features of the model

The equation describing the amplitude u~(x, t) of small transverse planar vibrations of a straight Euler beam of constant cross section can be derived by the principle of stationary action from the Lagrange function

1 [ (0u~l 2 (~u~t2 /~2u~ 12 ] X~(u~) = - { m~ \ - .~1 -- T, \ ~ x ] - E I \--~x~] ] (2.1)

in which m~ is the beam mass per unit length, I is the (surface) moment of inertia in the direction of vibration, E is the elastic (Young's) modulus of the beam, and ~r is the external tension applied at the ends of the beam. The amplitude us(x, t) of small (transverse) vibrations of a (homogeneous) string moving longitudinally along the x-coordinate with constant velocity v can be derived, similarly, from the Lagrange function

15"

194 Archive of Applied Mechanics 61 (1991)

where md is the (constant) string mass per unit length and Tg is the (constant) string tension. Their coupled transverse vibration with a common amplitude

u~(x, t) = u~(~, t) (2.3)

can also be derived by the principle of stationary action from the Lagrange function

(2.~)

in which the Lagrange multiplier A(x, t) takes care of the constraint (2.3). The equations of vibration and coupling following from the Lagrangian (2.4) are

~S - - = 0 , ( 2 . 5 ) ~A

+ - - 0 , (2.6)

- 0 ( 2 . 7 )

and have due to the Lagrangians (2.1) and (2,2) the explicit form

u~ - u ~ = O, ( 2 . 8 )

m ~ - - -- T ~ - - § E I - A = 0, (2.9) ~t 2 ~x ~ ~x 4

~2U d ~2U d ~2U d m a - - ~ -b 2mav ~-:-:-vt 6x -k (rod v2 -- Ta) ~x-- ~ -Jr ~ = O. (2.10)

Equation (2.8) expresses precisely the constraint (2.3). From (2.9) one can immediately deduce the physical significance of the Lagrange multiplier ,~(x, t) : it represents the (local) force per unit length by which the string acts transversely on the Euler beam, the physical analogue of the transverse force (per unit length) of the (inviscid) fluid on the fluid conveying pipe, and --A(x, t) in (2.10) is the corresponding reaction force on the string. The vibration of the system with the common amplitude u(x, t) = us(x, t) = Ud(X, t) is described by the equation

~ u ~ u ~ u ~ u (m s + rod) - - ~ + 2~ndV--'i-~x + (rod v2 -- T ) - - .-{- E t = 0 (2.11)

~x ~ 3x ~

which results after elimination of A(x, t) from (2.9) and (2.10). This equation does not distinguish, because of

T -- T~ + Td, (2.12)

between the tension T~ in the beam and that in the string, i.e. T d. In order to have a physical predictive power, it has to be supplemented by initial and/or boundary conditions. For the investigation of the free vibration modes they are chosen as

u(O, t) = u(l, t) = 0, (2.13)

~u (o, t) = ~u ~x ~ (1, t) = 0, (2.14)

describing fixed ends of the vibrating segment

0 _~ x ~ l (2.15)

of length l. Since under usual circumstances the longitudinal tension T plays no essential role, the corre-

sponding term in (2.11) will be neglected in the following discussion. The basis of this discussion

I-I. RaszilIier and F. Durst: Coriolis-effect in mass flow metering

will be therefore the equation

@u ~2u ~23 (ms + m~) -~ + 2m~v ~ + m~v 2-~x 2 4- EI

195

~ 4 3 = o. (2.~6)

~ x 4

This equation has been derived previously in various different ways (see e.g. [2]). I ts first three terms have an inertial character: The first one is related to linear acceleration in the transverse direction, the second one is a Coriolis term (O2u/(Ot Ox) is essentially a local angular velocity), and the third one is a centrifugal term (O2w/Ox 2 is the local curvature). I t is the effect of the second term in the vibrations which is looked upon in flow rate measurement, since this te rm contains the product

(~:r = mdv (2.17)

which represents the mass flow rate of the string and stays in this model for the mass flow rate of the fluid through the (vibrating) pipe. This second term - interpreted as Coriolis force -2mavO2u/(~t ~x) when brought to the right hand side of (2.16) -- gives in fact the name to this kind of devices for mass flow rate measurement. In connection herewith one has, however, to remind tha t the Coriolis force is, generally, not the quanti ty which is really measured. The appearance of the string (fluid) mass me also in the other two inertial terms, first of all in (ms + me) ~2u/~t2, asks for a careful analysis of the parameter dependence of the vibration amplitude u(x, t) in various vibration modes. Only with this analysis one can find out, whether and how a direct measurement of the mass flow rate QM is possible through these vibrations.

After the general description of this model for the pipe-fluid system, its small vibrations will be investigated in the next Sections. For the sake of simplicity the terminology will thereby directly adapted to the pipe-fluid system, but one should always remain aware of the fact tha t an approximation is used.

3 Vibration modes without flow

In the absence of flow, (2.16) becomes the usual equation for transverse vibrations of the Euler beam:

~ u ~ u (ms + m~) - ~ + EI ~x 4 = 0 . (3.1)

Its free modes are well known [3], but for convenience they are presented here again, from the perspective needed later.

Equation (3.1) remains unchanged by the substitution

x -+ 1 -- x, (3.2)

i.e. it is invariant under reflection in the (transverse) plane which goes through the midpoint x = !/2 of the pipe segment. Therefore the free vibration modes behave either symmetrically or antisymmetrieal ly under the reflection (3.2). Up to a normalization constant and a constant phase, the symmetric modes are given by

u~)(x, t) = u~l(x) sin (m(,~)t) (n = 1, 2 . . . . ) (3.3) with

The mode constants y(s} are the (positive) solutions of the (transcendental) equation

-- (s) tan y~81 - t a n n y~ , (3.5)

and are approximately given by

y:8): 2.3650, 5.4980, ..., - ~ ( n -- 1 / ~ for n - + e c . (3.6) \ 4 !


The (circular) eigenfrequencies of the modes are

~~ \m~ + me/ (3.7)

Correspondingly, the an t i symmet r i c modes are

u(2)(x, t) = u(na)(x) sin (co~)t) (n = 1, 2 , . . . )

with

= stun 7n sin 7~ ) -- 1 -- sm y~ sinh 7~) -- 1 ,

with the constants ?~) given by the (positive) solutions of

(3.8)

(3.9)

t an y(n a) = t anh 7~ ), (3.10)

app rox ima te ly

~/(a) (7b ~-) : 3.9266, 7.0685, . . . , -+ + ?: for n - - > c ~ , (3.11)

and with the (circular) eigenfrequencies

co n(o) = + m e . (3.12)

The ampl i tudes u(na)(x), u~)(x) of the first two symmet r i c and an t i symmet r ie modes (n = 1, 2) are displayed in Fig. 1.

The modes are o r thogona l to each other, i. e.

1 f ~g)(x) ~)(x) d~ = O, (3.13)

0 l

f ~2(x) ~ ( ~ ) d , = 0 (~, ,~ > 1, ~ =~ ~), (3.14) o

l f u~'(x) ul~)(x) dx = 0 (n, m ~ 1, n 4= m), (3.15)

o

and they are normalized to

]

-2 oosh~ ~, ' ~ - ~ sin (2:,~')

0

+ -2 -- 2y~----- 5 sinh (2y~)) , (3.16)

l y [ 1 ] lu~)(x)/2dx = __l sinh 2y~) 1 + ~ s i n ( 2 y ~ ))

2 0

2 2y~ ) '

and form a complete sys tem of functions. Since the modes of (3.1) are well known, it is convenient to write (2.16) as

~ u ~ u ~2u ~2u (ms + me) - ~ -{- E I = - - 2 m d v - -- mdV2.:

Ox 4 Ot ~x ~x ~

(3.17)

(3.18)

H. Raszillier and F. Durst: Coriolis-effect in mass flow metering 197

u(x)l

1

0

,l I

~ f~,. UI ~ ) 1

'1 X/'~

Fig. 1. Amplitudes U(nS)(x ) and u(na)(x ) for n = l , 2

and to consider it for sufficiently small velocities v as a slightly perturbed form of (3.1). The modes of (2.16) may then be viewed as perturbed modes of (3.1), the perturbation being pro- duced by inertial effects of the flowing fluid, the Coriolis and centrifugal forces -2mav@u/(Ot Ox) - m4@~u/Ox 2. In developing this point of view [4] the perturbat ion procedures, well known e.g. in quantum mechanics, will be taken over without going into the deeper mathematical questions behind them.

There is one point of practical importance which is has to be considered before applying these procedures. I t is clear from (3.18) tha t its modes cannot be written, like those of (3.1), as u(x) sin (rot), since the term - 2 m ~ v 3~u/(~t ~x) of the Coriolis force leads to an expression ~-~ cos (rot). One would therefore have to make an ansatz for the modes which consists of one term

sin (rot) and another one --~ cos (cot) with the result that one would arrive at two mode equations, unlike the situation with (3.1) which leads by the ansatz u(x, t) = u ( x ) s i n (rot) to only one equation

E I d4u(x) c@(m~ + md) u(x) = 0 (3.19) dx 4

supplemented by the boundary conditions

u(0) = u(l) = 0, (3.20)

du du d--~" (0) = -~x (1) = 0. (3.21)

The physical reason for the necessity of a second equation in the presence of fluid flow lies in the appearance of nontrivial (local) phases in the vibration modes, besides the local amplitudes. The technically much more convenient way to treat this problem is to consider u(x, t) no more as real but rather as a complex quant i ty u~(x, t) and to make the ansatz

u~(x, t) = u~(x) e ~t (3.22)

with a complex amplitude u~(x) and real ro. In this way one ends up with one complex mode equation instead of the (equivalent) two real equations. The physical mode amplitude u(x, t) can be recovered at the end e.g. as the imaginary par t I m u~(x, t) of Uc(X, t).

4 Perturbation of modes by the flow

The complex equation of vibration modes of the fluid conveying pipe which follows from (2.16) and (3.22), is

E I d%(x) duo(x) @uc(x) § 2imgvw - - 4- m~@ ro2(m~ + rod) Uc(X) = 0 (4.1)

dx 4 dx dx ~


with the bounda ry conditions

u~(O) = u~(1) = 0, (4.2)

du~ du~ d~ (0) = T x (~) = 0 . (4.3)

Flow meter ing as described in this pape r is based on the behaviour of the modes of this equation. Equa t ion (4.1) represents ma thema t i ca l l y a selfadjoint quadrat ic eigcnvalue p rob lem [5].

Under cer tain conditions on its coefficients, which can be in terpre ted as restrictions on the fluid veloci ty v, all the eigenvalues ~ are real, as it was assumed a t the end of Sect. 3. These conditions are ra ther generous f rom the point of view of f low metering. When they are obeyed, all v ibra t ion modes of (2.16) are thus periodic; there is no danger of buckling or f luttering. I n Appendix A such a sufficient condit ion is der ived as

Ivl < c(~ ~) (4 .4)

with

c~) 2r: ( EII1/2 = - - . (4.5)

I t is connected with the buckl ing of the pipe a t v = ~c~ 8) due to the flowing fluid. I t will be assumed, wi th reference to the ma themat i ca l l i terature (e.g. [6]), t h a t the mode

ampl i tudes u~(x) and the mode frequencies ~o are analyt ic functions of the fluid veloci ty v inside a circle of (eventual ly mode dependent) radius c which is a t least equal to c~): c ~ c~ ~). Analy t ic i ty allows convergent power series expansions inside this circle:

~o = ~o(k) (1 v] < c). (4.7) k = 0

Equa t ion (2.16) is no more invar ian t under the reflection (3.2), as (3.1) is, bu t it is invar iant under the subs t i tu t ion

x - + l - x , v - + - v . (4.8)

Therefore the modes of (4.1) behave ei ther symmet r i ca l ly or an t i symmet r ica l ly under the sym- m e t r y opera t ion (4.8), whereas the mode frequencies are symmet r i c functions of v/c. An imme- diate consequence of this are the facts t ha t successive orders in the expansion (4.6) of a mode have opposi te s y m m e t r y proper t ies (parities), i.e.

@ ) ( l - x) @ § - x) = - @ ) ( x ) @ ~ l ) ( x ) , (4.9)

and tha t there are no odd orders in the expansion (4.7) of o , i.e. ~o (2k+1) -- 0 and w is in fact a power series in v2/c2:

~o = _~ ~o(2k') @2 < ca). (4.10) k ' = 0

o r

The t e rm of order zero in the ampl i tude expansion (4.6) of a free mode of (4.1) is real, ei ther

u(~~ = u(,S,)(x), (4.11)

@)(x) = ~ r (4 .12)

therefore in the sequel the index c in u~~ will be omit ted . The t e r m of order zero in the frequency expansion (4.7) is, correspondingly, ei ther

(8) (4.13) (A)(0) ~ ~0 n

H. Raszillier and F. Durst: Coriolis-effeet in mass flow metering 199

o r

~o(~ = co~ ). (4.14)

By substitution of the expansions (4.6), (4.10) into (4.1) one arrives at a system of equations, from which one can compute the higher order terms (k, lc' > 1) of these expansions, which describe the perturbation of the solutions (4.11)--(4.14) by the fluid flow. To the first two orders (/c = 0, 1) these equations ~re

EI d4u(~ ~o(~ + rod)u(~ = 0, (4.15) dx 4

E1 d4u~l)(x) (m~ -~ ?7td) (D(0)2U(1)(X) = -- 2imdcco (~ du(~ (4.16) dx ~ dx

The boundary conditions (4.2), (4.3) imply for the terms of the expansion (4.6)

< k ) ( 0 ) = ~k~(1) = 0 (~ = 0, 1 , . . . ) , (4 .17)

d ~ k) du~ ~) d--~- (0) = - ~ x (1) = 0 (k = 0, 1 , . . . ) . (4.18)

Because of the relations (4.11)-(4.14) one can consider that equation (4.15) is already solved and that the right hand side of (4.16) is known. From (4.16) one can, therefore, compute the lowest order perturbation u~)(x) of u(~ It will be enough for a good approximation of the mode amplitude uc(x), when v 2 is small enough compared to c~:

v 2 ~ c ~. (4.19)

For the search of the solution of (4.16) it proves very helpful that

i m~ ...c (2x -- l) d~u(~ 4 m~ + m a o)(~ dx 2 + (4.20)

obeys this equation, although not the boundary conditions (4.17), (4.18). I~ has only to be com- pleted, therefore, by a solution of the homogeneous equation, of the appropriate symmetry property (p~rity), opposite to tha t of u(~ Such solutions are

du(O)(x) d~u(~ dx ' dx a

therefore, U~1)(32) is of the form

- - ; (4.21)

<~) (~ ) - i md c 4 m s -~ Tit d (D (0)

• [ ( 2 x - l) d2u(~ [ d x 2

+ C1 --du(~ + l~Ca --dau(~ ] (4.22) dx dx ~

with dimensionless constants C1, Ca, which are determined from the boundary conditions (4.17), (4.18) as

CI = - 2 - l dau(~ a [ d~u(~ ~ )-~ ( l ) \ (1)_ , (4.23)

1 d2u(~ /dan(0) ~--1 c 3 - dx---C " ( 4 . 2 4 )

According to (3.4), (3.9) their values are

C --2(1 + y(n s) tanh v(s)~ = [ n / ,

for u(~ = u~)(x): 1 1 eoth y~) (4.25) Ca 2 7~ )


and

/C1 - 2 ( 1 + y(~) eoth (~)~ = Yu ),

= (4.26) for u(~ u(~)(x): 1 1 - (a) C3 2 7(~ ~) tann ~n �9

I t is to be noticed that the constants C1, C3 are real. Now it is appropriate to come back to the real amplitude u(x, t) according to

u(x, t) = Im [u~(x, t) e i~t]

= Re u~(x) sin (o)t) -4- Im u~(x) cos (cot). (4.27)

To the first two orders discussed this gives

t~e u~(x) = u(~ (4.2s)

with

Im u~(x) -- v m ~ u(1)(x), (4.29) l~o(~ m s q- m d

l [ d2u(~ du(~ dSu(0)(x) ] u(1)(x) = -~- (2x - l) dx----- 7 - + C1 d-----~ + 12C~ - - d x a . (4.30)

I t is important to mention here, that the functions u(~ u(1)(x) as given by (4.11), (4.12) and (4.30), resp., are dimensionless and independent of physical parameters of the pipe-fluid system. The parameter dependence of

v m-----A--d uO)(x) cos (oJ(~ + 0 --~ (4.31) u(x, t) = u(~ sin (~0(~ leo(~ m~ + m e

appears only through the frequency m(0) and the number vme/[l~o(~ -? me) ]. As it was expected, the critical velocity c does not appear explicitly in the formula (4.31)

of the amplitude u(x, t). Since this velocity describes the radius of convergence of the series expansions, it cannot be extracted from the first few (two) terms of the series only [6]. The important fact for the purpose of flow metering is, that c ~ c[ ~). I t allows by

< ~ (4.32)

to give a quantitative estimate of the ratio v/c from the values of v and c~ s). Thereby one gets a quantitative formulation of the statement ([vl ~ c) that the fluid velocity v is small enough, that the first order perturbation already reflects all the essential features of the vibration modes in the presence of the fluid flow.

In Appendix B the perturbed vibration modes (4.31) are viewed as superpositions of the vibration modes in the absence of flow. Thereby it turns out that the perturbation of a mode, caused by the flow, consists mainly in its mixture with (small amounts of) its spectral neighbours. This view shows a close connection between the (unperturbed) vibration spectrum of the pipe and the sensitivity of Coriolis-type flow metering instruments.

5 FIow metering with fundamental mode

The expression of the fundamental mode of the fluid conveying pipe is given for small fluid velocities v, Iv] ~ {c~S)l, by the perturbation formula (4.31) with u(~ = u~S)(x), ~o(~ = rg~ sl and 7(10) = 2.3650 from Sect. 3. The behaviour of its amplitude over a period is illustrated in Fig. 2; the flow is from left to right (v > 0). For comparison, the behaviour of the amplitude in the absence of flow (v = 0) is displayed in Fig. 3. One dearly notices that due to the flow there

H. Raszillier ~nd F. Durst: Coriolis-effeet in mass flow metering 201

ulx, t)i

1

~s)t/~; = 0.5

// .." i - - \< o.87s

1

5

i

Fig. 2. Behaviour of the amplitude u(x, t) over one period for u(~ = U(lS)(x), co(~ = o~ s) (s) and vmd/[l~o 1 (m s -~ rod) ] 0.2

Full lines: w~s)t = 0.5r~ (symmetric), 0.75% 0.875rc, ,~ (antisymmetric), 1.125u, 1.25~ (down- w~rds). Broken lines: eg(ls~t = 1.5~ (symmetric), 1.75,% t.875u, 2,-: (~ntisymmetric), 2.125rr 2.25u (upwards)

u(x,t) olivet/== 0.5

1 ~ ' 0 . 7 5 , 2.25

/ / ~ ~ 0.875, 2.125

0

' ' 5 , 1 . 8 7 5

_1L , , ," 1.5

Fig. 8. Behaviour of the amplitude u(x, t) over one period for u(~ = u~S)(x), o)(~ = ~o~ s) in the absence of flow (v = 0) The values of co(ls)t are the same as in Fig. 2. The last five broken lines coincide here with the first five full lines. All curves are symmetric (U(lS)(l -- x) = u~S)(x))

is a phase lag of the in le t p a r t of the p ipe compared to the out le t p a r t which can be expla ined qua l i t a t i ve ly b y the ac t ion of a Coriolis force 2 m e v • ~t, I~[ = @u/(~t Ox). The qua n t i t a t i ve analys is of the behav iou r of th is mode, for the p ropose of precise flow meter ing, concent ra tes on the pase behav iou r of the mode, in pa r t i cu l a r on the compar ison of the t imes v~S)(x) of zero crossings a t var ious po in ts x of the p ipe :

u(x , = 0 .

.According to (4.31) the ins tances z~s) are de t e rmined b y

(5.1)

t a n [~Is)~i ')(x)] - v md u(1)(x) (5.2) loJ~ s) ms 4- md u~S)(x) "


The behaviour of the (antisymmetrie) ratio (s~ uO)(x)/u I (x) is shown in Fig. 4. Its limit values at the ends are determined by

lira u(1)(x) _ _ _ y~s) t a n h y ~ s) (y~s) tanh y~81 _ 1) (5.3)

to 3.075 6 at x = 0 and -3 .075 6 at x = I. This ratio is almost a linear function of x/l. Figure 5 gives a representation the phase ~)(x) = r of the zero crossings in its dependence on x with the period 7: of the tangens function: For a sufficiently small value (8) of vmd[1091 (ms + md)] the branches of the function ?~S)(x) are also linear. Then one may approximate

~S)(x) = tan ~?)(x) (rood ~) (5.4)

and arrive thereby at

~8)(x ) _ v m~ ua)(x) (rood ~). (5.5) l G ~) ~8 + m~ u~)(x)

The error implied by the approximation (5.4) is smaller than

1 ( v m__~ 12 [ua)(x)12 3 + m d " (5.6)

Now it is appropriate to m~ke the important observation for flow metering, that the phase ~)(x) is proportional to the mass flow rate QM = m~v, but one has to add immediately, that it depends besides that also on other physical parameters of the pipe-fluid system:

~S)(x ) _ QM 1 u(1)(x~) (rood ~). (5.7) l (s) u~'(x) 091 i s "~- ~ffbd

All these parameters may be considered as system constants except the fluid mass md which has to be considered as a variable quantity. Because of the fluid mass dependence of the frequency 09~s), i.e.

091s) = + - - , (5.s)

one can write ~S)(x) as

q~s)(x) = QM" 09(s) ( 2--~-ls) ) . . ~ . E118 u~S)(x)UO)(x) (mod u) (5.9)

which allows the conclusion that QM may be determined from two measurements: One of the phase ~S)(x) (at a given point x) and one of the frequency 09~s). Only if

m~ ~ ms, (5.10)

U(I](X)/U~S){~t 1- L o , ,

-5 Fig. 4. I{epresentation of "atD(X)/u[S)ix) in the range 0.04 < xfl < 0.96

H Raszillier and F. Durst: Coriolis-effeet in mass flow metering 203

~S~{x)/2g

I , , I [ ~ 1 1

Fig. 5. Phase ~ S ) ( x ) = w(ls)v(S)(x) of the zero crossings u(x, T~S)(x))= 0 for u(~ u~S)(x) and vm~/[/~)(m~ + md)] = 0.2

A~o ~(I-xL, xL)~ l

q I

2

1

0 0.2 v/c

-1

-2

Fig. 6. Representation of A~(ls)(l -- xf,, xL) as ~ function of v/c = wnd/[lw(~s)(ms +. rod) ] for xL/l = 0.25

when the frequency (o~ ~) is insensitive to the fluid mass me, one can consider also ~o~ ~) as a charac- teristic constant of the system. I n this limit si tuation a phase measurement alone determines, under knowledge of the constant caracteristics of the system, the mass flow rate.

I n practice one does of course not determine the phase ~S)(x) in a single point x, bu t ra ther the phase difference

AV~)(xR, XL) = q~IS)(XL) -- q~S)(XR) (5.11)

between two points xn and XL; this difference is given by

I t is mos t convenient to choose the two points XL, XR symmetr ical ly (XL = 1 -- XR, XR > l/2), because then

U(1)(XL~)) -~ -- U(1)(XR~------~)" (5.13) ~?)(x~) q~)(x.) '

the phase delay of the left point XL with respect to its (right) symmetr ic image XR = 1 -- XL is

( r 13 u(1)(xL) �9 ~o (~) ( m o d r:). ( 5 . 1 4 ) A ~ ) ( / -- XL, XL) = QM 1 8 E I U(I~)(XL)


Figure 6 shows A~v~s}(/ -- XL, XL) as a function of the dimensionless quanti ty vmd/[le)~)(m, + md)] = Q ~ , ( 2 r l ~)) ~ ~/(EI) for x~/~ = 0.25.

The mass flow rate QM can thus be determined by a measurement of the phase delay A~) ( I - XL, XL) and by one of the frequency ~o~ ~). Since the fluid mass md is given by the product of the fluid density ~ and the inner cross-section S of the fluid conveying pipe, i.e.

me = OjS, (5.15)

the measurement of (o[ s) is in fact a measurement of the fluid density. Therefore the measurement of the phase delay A~s~)(/- XL, XL) alone does generally, when (5.10) is not obeyed, not provide a measurement of the mass flow rate of the fluid; an additional density measurement is necessary.

Things look differently, if the t ime delay

1 Ar~)(xR, XL) = ~ AV~)(xR, XL) (5.16)

is measm'ed, because it does not contain other variable quantities than the mass flow rate QM:

1)~ l~ [u(~)(xL)

Measurement of t ime delay

8EI U~)(XL) rood ~o~-- 5 (5.18) is therefore really a direct measurement of mass flow rate.

For purpose of practical illustration a realistic example will now be considered [7]: it refers to a circular straight pipe segment of length l = 0.24 m, outer diameter 2Ro = 12.7 mm, wall thickness Ro - Ri = 0.66 mm, Young's modulus E = 110 GPa, and pipe density ~s = 4.5 g/cm 3. From

m s = ~(R~ -- R~) ~ , (5.19)

2 (5.20) m d ~ ~ R i Q d ,

I = ~ - (R~o - R~) (5 .21) 4

one gets, for water as fluid (~a = 0.998 g/cma), the numbers

m~ = 1.1234 g/cm, (5.22)

m d = 1.0151 g/cm, (5.23)

and

m._..~_e = 0.903 6, (5.24) m s

- - -- 944.4 Hz, (5.25) 2re

c~ ~) = 580.5 m/s, (5.26)

(. 1 ~ 1-1 ~- 3.000" 10am/s. (5.27) loJ~ ~ m~ + rod/

These numbers clearly show that condition (5.10) is not obeyed. They also show that the ratio

vmd/[l~o~8)(m~ + md)] is here indeed small, as long as v is of the order of several m/s. For this order of magnitude v is also seen to be very small compared to the critical velocity c~ ~).

H. Raszillier ~nd F. Durst: Coriolis-effect in mass flow metering 205

6 Pecul iar i t ies wi th other modes

I t is, of course, not compulsory to s t ick to the f u n d a m e n t a l mode of the v ib ra t ing pipe-f lu id sys tem for flow ra te measurement . This mode is only especia l ly convenient , since the a m p l i t u d e u~81(x) has no o ther nodes t h a n the ends, and therefore the ra t io u(1 ) ( x ) /u~ ) ( x ) never gets s ingular in the in t e rva l 0 < x < l. F o r the o the r modes th is is no more t rue and so add i t i ona l care has to be t a k e n if t h e y are employed for flow ra te measurement . The pecul ia r i t ies encountered with these modes will be i l lus t r a t ed here on the example of the p e r t u r b e d lowest a n t i symme t r i c mode for which u(~ = u ~ ( x ) and 09) = c~ ~). The behav iou r of the amp l i t ude u ( x , t) of this mode, according to (4.31), over a per iod is shown in Fig. 7; for comparison, the behav iour for v = 0 is shown in Fig. 8. Like in the f u n d a m e n t a l mode, the amp l i t ude crosses zero a t a po in t x a t

ulx, t},

1 i \ I

O-

, , ../wc~)t/ar = 2.25

-1

Fig. 7. Behaviour of the amplitude u(x, t) over one period for u(~ = u~a)(x), a~(~ = c ~ a~ and vm~/[ l~a l (ms § rod) ] = 0.075 Full lines: ~ a ) t = 0.5= (antisymmetric), 0.75rc, 0.875=, = (symmetric), 1.125=, 1.25= (on the right side: downwards). Broken lines: ~o~a)t = 1.5~ (antisymmetrie), ].75=, 1.875=, 2~ (symmetric), 2.i25~, 2.25~ (on the right side: upwards)

uCx, t)[ / \

/ / \ \

/ ~

/

(~)t/~ = 0.5

.75, 2.25

0.875, 2.125

1,2

'~ ,'r

", /'%'1.5 x _ /

Fig. 8. Behaviour of the amplitude u(x , l) over one period for u(~ = u(la)(x ), w(o) = w(la) in the absense of flow (v = 0) The values of eg~a)t are the same as in Fig. 2. The last five broken lines coincide here with the first five full lines. All curves are antisymmetrie (u~a)(1 - - x) = --u(la)(x))


u ( l l (x) /u% ~ (x) l

10

-10

I , p l I

1 x ~ �84

Fig. 9. Bepresentation of u(1)(x)/ u(la)(x ) in the r~nge 0.02 < x/l

0.98

~~

1

0 q p ~ ~ , ~ ~ , I

Fig. 10~ Phase ~'~a)(x) = o)~al~a)(x) of the zero crossings u(x, T~a)(x))= 0 for u(C)(x) = u~a)(x) and vmd/[ICo(la)(~?, s -~ rod) ] : 0.075

i n s t ances v~a)(x) d e t e r m i n e d b y

(a) (a) x v md u(1)(x) (6.1) t a n [5o I T 1 ( ) ] - lo~i ~ ms + m~ u (p (x ) '

b u t t he b e h a v i o u r of u(1)(x)/u~)(x) is n o w d i f fe ren t (Fig. 9). A t x = l/2 i t b lows up, because

u~a)(x) has the re a node u~a)(l/2) = 0. I t s b e h a v i o u r a t t he ends is d e t e r m i n e d b y

lira u(1)(x) - 7~ a) c~ co th y~a) __ 1) (6.2) ~_~ ~~

which leads w i th y~a) = 3 .9266 to t h e va lues - 1 1 . 5 1 2 5 a n d 11.5125 of the ra t io a t x = 1 and x = 0, respec t ive ly . Th e m i n i m a of lu(1)(x)/u~a)(x)l lie a t t he ( symmet r ic ) po in t s x = 0.337l a n d x ---- 0.663/. Th i s b e h a v i o u r is ref lected in t h a t of t he phase ~a)(x) __ ~'(a)~(a)C~ ~ , shown in Fig. 10 for a va lue 0.075 of t h e p a r a m e t e r vmd/[lco~)/(ms + rod) ]. This va lue cor responds a lmos t to t he 0.2 in Fig. 5, as c an be seen if one t akes i n to a c c o u n t

- = 0 3 6 2 s ( 6 . 3 ) OJi a) \ ~21a} ]

a n d 0 .3628 �9 0.2 = 0.07256. Since la) u(1)(x)/ui (x) is u n b o u n d e d , t h e ve loc i ty range w i t h i n which the a p p r o x i m a t i o n

t a n cf~a)(x) = cf~a)(x) ( m o d 7:) (6.4)

H. Raszillier and F. Durst: Coriolis-effeet in mass flow metering 207

- 0 2 / 0 . - . . . . . . . 0.2

-2

Fig. 11. l~epresentation of AT~a)(I - - xL, XL) as a function of v / c ~ vmd/ [ lo )<a) (ms + rod) ] for c%fl = 0.337 ( ) ~nd x~/1 ~ 0.475 ( . . . . . . . )

r,(~l(x) 0.5

/

/ O

Fig. 12. Representation of r~S)(x) ~ uO>(x) (ldu~S)(x)/dx) - i

is accurate enough depends very sensitively on the chosen point x; the opt imal range is certainly achieved with Ix - 1/21 = 0.163/. Figure 11 shows

A ~ ~ - xL, x ~ ) = ~ ) ( x L ) - ~(a)(~ _ xL) ( 6 . 5 )

as a funct ion of vmd/[loo~)(m~ + rod) ] = Qi~o~a)(2?~a))-413/(EI), for the opt imal coordinate XL/l = 0.337, and, for comparison, also for XL/l ---- 0.475.

Otherwise, there are no basic obstacles in applying this mode - as well as others - for the purpose of flow metering, according to

or

w , - xL, x~) = Q M ~ 7 ) l ~ i o ) / S E ~ ~i~ (mod ~) (6.6)

8 E I U~(XL) ~o~dl~ i . (6.7)

7 Comments

I t has been shown tha t by measuring the zero crossings of the ampli tude of a fluid conveying vibrat ing pipe one can determine the mass flow rate of the fluid. The quant i ta t ive investigation of the underlying physical problem has been performed on the basis of an approximat ion which ignores pa r t of the degrees of freedom of the pipe and all the internal degrees of freedom of the fluid. I n this approximat ion the pipe behaves like a thin beam and the fluid like a running string.

16


The mass flow rate of the fluid follows either from the phase delay AV between the zero crossings of the vibration amplitude at two different points xL and XR and from the vibration frequency, in a certain chosen mode, or alone from the direct measurement of the time delay between the zero crossings in these points. In principle the chosen mode is not of importance, but the fundamental mode has particularly convenient features.

I t is not essential that one detects delays precisely in the zero crossings of the amplitude; equally suitable is the detection of delay in the maxima (or minima) of the amplitude. Whereas the (periodic) instances ~o(~ of zero crossing are given according to (5.2) by

tan [~o(~176 -- v md ua)(x) (7.1) le)(~ ms § m d u(O)(x)

(in the notation of Sect. 4), the instances T~~ of the amplitude extrema du(x, ~~ = 0 are given by

co~ [w(~176 - v m~ ua)(x) (7.2) l~o(~ m~ + m d u(~ "

There is thus a constant delay between ~0(~ and r(e~ i.e.

(mod =), (7.3)

and equality between the phase delay of zero crossing, i.e.

~%(~ ~L) = v(o~ - ~(o~ (rood ~), (7.4)

and those of the amplitude extrema, i.e.

Aq~~ XL) = q~O)(xL) -- Cf~O)(XR) (rood =), (7.5)

in the chosen mode, with V(0~ = o)l%0(~ ~~ = (o(~176

A~(00)(XR, XL) --~ A~p~O)(zR, XL) (rood ,~). (7.6)

A detector may possibly distinguish between the two directions of zero crossings or (algebraic) maxima or minima of the vibration amplitude, If this is the case, then in the corresponding equations for ~(~ and A?(~ XL) the (mod ~:)-statement has to be replaced by a (mod 2~)-statement. This has as a consequence that in the Figs. 5, 6 and 11, 12 every second branch of the curve disappears which may possibly increase the range of attainable flow rates. I t is, however, generally to be expected that for small fluid velocities v ~ c where the performed approximations are valid this (mod 7:)-problem of the phases will play no practical role.

I t is useful to take a look on the force 2(x, t) by which the fluid acts on the (unit length of the) pipe. Once the amplitude u(x, t) is known, this force can be computed, e.g. from (2.10) (with Ta = O, as it was assumed). Because of

~2u_2QM ~ u (v~) ~(~, t) = - ~ - ~ ~ + o ~ , (7.7)

one gets from the form (4.31) of the amplitude u(x, t) the result

),(~ t) = mdco(~176 sin (~o(~

~ o ( ~ 1 7 6 ) 1 md u(l)(x)) cos ((D(~ . (7.8) - 2 z d - - - - ~ + 2 ~n~+~-------~

I t consists of a sum of two terms which have opposite parities. The one has the pari ty of the unperturbed mode u(~ and reflects the reaction to the linear (transverse) acceleration of the fluid by the pipe. One contribution to the other term is the Coriolis force which acts on the pipe in the limit situation when the vibration mode of the pipe is not perturbed by the fluid flow. The other contribution to this term is the (first, order) effect of this perturbation. In the fundamental

H. l~aszillier and F. Durst: Coriolis-effect in mass flow metering 209

mode u(~ = u~l(x) the Coriolis t e rm of the force is antisymmetric. In principle, it can be measured by the difference

2(*)(xl , , t)~ = 2~)(x, t) -- ,~)(l -- x, t) (7.9)

of the force acting on two pipe elements situated symmetrically with respect to x = 1/2:

o)~ ~) ( du~8'(x) l md ) 2~S)(x, t)~ = --4 �9 OM l - - + - - u(1)(x) COS (~o[s)t). (7.10)

1 dx 2 m~ § m~

Should it turn out tha t the ratio

1 m d uO)(x) 1 me _ _ _ , . t ) ( x )

2 m, + m d ldu~)(x) 2 ms + r o d

dx

(7.11)

is very small, then the measurement of ,~S)(x, t)a and tha t of the frequency ~o~ s) may serve as a measurement procedure for the mass flow rate QM. The ratio r~)(x) is represented in Fig. 12; it decreases from the middle x = 1/2 of the pipe segment toward the ends x = '0, x = 1 with the values

2sinhy~ ~) eos~y~8 ) ~ o s ~ ~ 7_ e~sh--~i ] ( = 0.4106) (7.12)

and r~S)(0) = r(2)(l) = 0. In the correction te rm (7.11) of the Coriolis force the factor r~)(x) dependends on the detection point and the chosen mode, whereas ma/[2(m~ + me)] depends on the ratio of the pipe and fluid mass, i.e. also on the nature of the fluid. For an estimate one may take md/[m~ + ms] = 0.4747 from (5.22), (5.23) and the detection points XL/l = 0.25, XR/l = 0.75. Then the correction to the Coriolis force turns out to be

1 md r~)(xL.~) ~ 0.06. (7.13) 2 2ras + m d

The main contribution to 27)(x, t)~ is thus really given by the unperturbed Coriolis force, but the effect of the correction te rm in

�9 dq~)(x) ( Xt'(x, t)o = - 4 . t ) Q ~ ~ 1 + 1 .ze r~)(x)t cos o~)t (7.14)

2 2ms + m d ]

may become quite large�9 A few comments should be made here, also some regarding the approximation used in this

paper. The pipe has been treated here as a beam and the fluid as a running string. Thereby the parameters of the system (the masses m~ and me, the moment of inertia I , the Young modulus E, and the velocity v) have all been taken as constants. The beam is thus geometrically uniform and physically homogeneous, the flow is stat ionary: dQM/dt = 0. With an appropriate non- uniform mass distribution in the pipe (ms -+ ms(x), I --~ I(x)) one could influence the frequency spectrum of the vibrating pipe and thereby the sensitivity of the measuring device for the mass flow rate. A variable velocity ( v -~ v(t)) lets one expect new physical phenomena not unlike those related to the Hill equation. The equations describing the system with variable parameters can be derived from the suitably modified Lagrange function (2.4).

Within the approximation chosen all internal fluid effects are ignored. There is no room left for a quant i ta t ive investigation of the effect of vibration induced secondary fluid flow, in parti- cular of the role played in detail by the fluid viscosity. For the approach of this kind of questions one has possibly to s tar t from the shell equations instead tha t of the beam for the pipe, but certainly from the fundamental equations of fluid dynamics instead the string equation. Their coupling becomes then a more difficult problem than in the approximation used here. So far only modest steps have been done in this direction [2, 8]. Therefore a careful analysis of this

16"


improved frame, possibly with the construction of a suitable perturbation scheme, is highly desirable.

Finally, it has to be stated that the behaviour of the pipe-fluid system under external exci- tation which is of special practical importance has not been considered here. Yet all elements for its spectral approach, similar to that taken in Appendix B, are available in the paper. Its proper presentation, however, requires the extent of a separate publication.

Appendix A: Spectrum of the fluid conveying pipe

In Sect. 4 of the paper a condition is given which is sufficient for the existence of only real eigenvalues in the problem (4 .1) - (4.3). In order to arrive simply at a condition of this kind one may observe that from (4.1) one derives the following equation in co with real coefficients:

l l ;__ ; .lxl E I d4u(x) u*(x) dx + mev ~ - ~ u*(x) dx

~ d x 4 ,~

o o

l I

(du(x)dx f + 2im~vo j u*(x) dx -- ~o2(m, + rod) [u(x)] 3 dx = 0. (A.1)

0 0

This equation has real roots for any eigenfunction u(x) if

I m~v3 [']du(x) u*(x) dx => - (m~ 4- m~) lu(x)[ 2 dx

$ 0

( / ; ) • E I d~u(x) u*(x) dx + may 2 d3u(x) u*(x) dx (A.2) dx 4 , dx 3

0

is valid for it. Since the left hand side of (A.2) is positive, it is sufficient for this that the right hand side is negative, i.e.

I I (d3u(.) E I dtu(x) u*(x) dx + m4v 2 u*(x) dx ~ O. (A.3)

,~ dx 4 , , I dx ~ -- 0 0

The eigenfunctions of (4.1)--(4.3) satisfy the (Poincar~) inequality,

I l s du(x) 2 13 / " d~u(x) 3 1 dx < - j dx, (A.4)

3 1 d* I = s 0 0

which implies l l

8 E I [" d3u(x) E I ( dau(x) u*(x) dx -4- - - , u*(x) dx > O. (A.5) . ] dx 4 l 3 J dx 3 =

o o

From this inequality it follows, that at least for

v ~' < c~,, c~ - ( a . 6 ) l \rod 1

the inequality (A.2) is obeyed and, consequently, the spectrum of (4.1)-(4.3) is real. Condition (A.6) already assures the existence of a real spectrum (i.e. of only periodic vibra-

tion modes) in a very wide range of velocities which goes much beyond that attained in flow

H. Raszillier and F. Durst: Coriolis-effect in mass flow metering 211

metering. Yet this range is an underestimate which can be improved by using other means for the analysis of (A.2) instead of the Poincard inequality. The inequality (A.2) says that for a real spectrmn of (4.1)-(4.3) it is sufficient that the spectrum of the (selfadjoint) operator defined by

d ~ A : = E 1 ( t4 + mev 2 - (A.7) dx ~ dx 2

and by the boundary conditions (4.2), (4.3) is nonnegative, i.e. if u~(x) is an eigenfunction of A with eigenvMue #n, i.e.

then

_ _ . d%~(x) E I d4u~(x) + m.v 2 - - - #,~u~(x), (A.8) dx 4 dx ~

#, > 0 (n = L2, ...)

should be fulfilled; because of

#I < ~ < ' " < # ~ < " "

this is equivalent with

# 1 ~ 0 .

If one differentiates the equality l l

f A~(z) ~*(x) dx = ~ f I~(x)i ~ dx 0 0

with respect to @, one arrives at l l

d ~ f lu~(x)]2 dx = - m e f du~(x) ' dx d ~ ! J l d~ I

0 0

which impIies

d/~ - - < 0 . dv 2

(A.9)

(A.m)

(A.11)

(A.12)

(A.13)

(A.14)

One is led, therefore, to ask for the smallest value of the velocity v, for which the operator A has an eigenvMue zero. In other words, one has to look for the solutions of the equation

E I d%(x) d2u(x) + mdv ~ - - - 0 (A.15) dx 4 dx 2

with the boundary conditions (4.2), (4.3), essentially an eigenvalue problem for v 2, and pick up the smallest eigenvalue @. Equation (A.15) describes the buckling of the Euler beam under the influence of the compressively acting centrifugal force; it is the same as the well known equation for the determination of the compressive critical load Tc = -[Tel ( < 0), one has only to make the substitution ITc] = rod@ with the critical velocity v = c [3]. The compressive action of the centrifugal force can be inferred directly from the opposite signs of the centrifugal and external tension terms in (2.11).

Equation (A. 15) which can be brought by two quadratures to

E I d2u- = [ ( - -- 1) + K ~ ] (A.16) dx ~ + m~c2u me@ K~ 2 xl

with two undetermined constants K~, K~ has the solutions

u~(x) = D a s i n [ 2 ( 2 1 - - 1 ) ] + K a ( 2 x - l 1) (antisymmetrie) (A.18)

212

where

Archive of Applied Mechanics 61 (1991)

l {mdC2~ 1/2 A = -~- --~--] . (A.19)

The boundary conditions (4.2), (4.3) lead to the eigenvalue equations

2 sin ~ = 0 (A > 0) (A.20)

for the symmetric solutions, and

cos ~ - sin A = 0 ()~ ~ 0), (A.21)

for the antisymmetric ones. The eigenvalues , , ~ (s) of (A. 20) are

2(~)" z~, 27, nT:, (A.22)

those (~)) of (A.21)

A(a)" 4.4930, 7.7291, ..., --> (n + l ) r: for n - + o o . (A.23) n �9

By (A. 19) the eigenvalues a n~(s), --n~(a) imply the critical velocities

_(~) --n __El (A.24) % = 1 \ m~ l

_(~) _ --n _ _ (A.25) % l \ m~ I "

By invoking physical plausibility arguments the solutions u~')(x), u~)(x) of the buckling equation (A. 15) which correspond to ~n~(~>, "'n~(a) can be considered as limit forms of the modes U(nS)(x, t), u(a)(x, t) of (3.16), determined in Sect. 4 (equ. (4.31)) for small fluid velocities. Along this line of argumentation the critical velocity c in the expansions (4.6), (4.7) can be interpreted as

c(8) for u(~ ul~)(x) (A.26) n a n d as

c(~) for u(~ U(n~)(x). (A.27) n

The condition (4.19) that the fluid velocity v should be small has thus to be interpreted as

22(~ ) ( EII ' /2 (for u(~ -~ u:)(x)) (A.28)

or as 22(a) -{ Ex-/1; (for = (A.20) Iv] - 7 - - \

The smallest eigenvalue is

C~8) 27I: ( EY/1/2 (A.30)

= 7 \mall as given by formula (4.5). I t assures the reality of the spectrum of (4.1)-(4.3) in a velocity range

which is by a factor = / ] /2 -= 2.22 larger than the value cp following from the simple appli- cation of the Poincar6 inequality.

Appendix B: Spectral properties of modes

The modes (4.31) display clearly a periodic tumbling between symmetric and antisymmetric shapes of their amplitudes. At the moments to, defined by ~o(~ = (n + 1/2) 7:, n c Z, i.e. twice during one vibration period the amplitude takes a form with the parity, the symmetry behaviour under the reflection (3.2), of u(~ :

u(x, to) = ( - 1)" u(~ (B.1)

H. t~aszillier and F. Durst: Coriolis-effect in mass flow metering 213

At the m o m e n t s tl , defined b y co(~ = nrc, n r Z, again twice during a v ibra t ion period, i t takes a fo rm with the (opposite) pa r i t y of u0)(x) :

u(x, tl) = - ( - - 1 ) ~ v m~ ua)(x). (B.2) l r m s - J -T t$ d

As one expects f rom the discussion related to the s y m m e t r y (4.8) of the mode equat ion (4.1), the definite pa r i t y of the modes is b roken b y the fluid flow.

The compac t form (4.31) of the modes is par t icu lar ly useful in m a n y respects. I t is however not very helpful if one t r ies to unders tand these modes as a superposi t ion of the unper tu rbed v ibra t ion modes of equat ion (3.1). The in te rp lay of the unper tu rbed modes in (4.31) can be un- covered if one is able to express the funct ion uO)(x) in t e rms of the sys tem {u(m -)(x)}m= 0 - - ~u(a)Ix u oo t m \ ) l m = O

or Llu(~)(xU~m t m~=0 for u(~ = uT)(x) or u(~ = uT)(x), respect ively:

u(1)(x) = E <;-)u(g)(~). (B.3) / t t ~ 0

The weight p(,~-) of a mode u(m-)(x) in u(1)(x) is propor t ional to the square la(~-)i 2 of the expansion coefficient a(,~) :

l

f I<;)(.)E~ dx p ~ - ) = i.~7)1 ~ o ( B . 4 )

l

f / u ( ~ ) ( x ) l 2 d x 0

One does, however, not arr ive a t a very t r ansparen t result if one computes a~ -) direct ly f rom its definit ion

l

a( - ) = o (B .5 ) l

f I,~(~-'(~)j ~ d~ 0

Ins tead , one m a y s ta r t f rom the equat ion

E I d~uO)(x) co(~ = 21~o( ~ du(~ (B.6) m~ + mj dx 4 dx

~ ( - ) oo which it obeys according to (4.16) and (4.29). I f one expands du(~ into { ~, (X)}m=0, i.e.

du(~ 1 - E b(,7)u(~-)(~), (B.7)

dx 1

then (B.6) leads with the expansion (B.3) to the sys tem of equat ions

(o~I;)2 - ~ ( j )~ )~ ( , ; ) = 2 ~ ( 0 ) % 7) ( .~ = 1, 2 . . . . ) (B.S)

for the coefficients a(m -) wherein co(m -~ is the e igenfrequency of u(,-[)(x), ~,n (a) or co (s)m, according to the case

E 1 d4ul~)(x) (m~ + ma)'~m(-)2~ ~ ) = 0. (B.9) dx a

The coefficient ]a~-)l 2 in the weight p(~-) is thus given by

F r o m the denomina to r (co(,~ -) - co(~ 2 one expects t ha t the pe r tu rba t ion of the mode u(~ is ma in ly a mixing with the modes of the neighbouring frequencies. For the fundamen ta l mode u(~ = u~S)(x) there is only one neighbour, the lowest an t i symmet r i c mode u(la)(x). For all o ther modes u(~ there will be two neighbours, an upper and a lower one.


For the quant i ta t ive comparison of the weights p(~-) one needs the values of the coefficients

l

f du(~ (x) Zd ~ ~-)(~)d~ b~2~ = o (B.11)

" l

f lu~-)(x)[ ~ dx 0

Their denominators are given by the expressions (3.16), (3.17), the numerators are

I

f du(n~)(x) (a)[ x

o

= 4y~)2 ( __1 1 .) y : ) (8) sin y : ) s i n k 7 : ) y~)2 y~)2 y~,2 + y(a)2 cos cosh y,,

+

l

f aun~ ~)(.) dx

o

cosn y, cos sink y~) + cos y,, sink y~s) sin ~(m a) cosh y - ~ = ~Td) z (sin y~) - (8,

(s) (a) 2 y n r m h /8) y(~) y(d) (8) y~) (~)~ cos ~ n + cos y(m a) sink rm J, y~)2 + yl~)~- (sin y?) sin eosh cos Y, sinh

(B.~2) l

f du~S')(x) u~)(x) dx. (]].13) d x = - - j -~x

0

References

1. Sultan, G. ; Hemp, J. : Modelling of the Coriolis mass flowmeter. J. Sound. Vibration 132 (1989) 473--489 2. Chen, S.-S.: Flow-induced vibration of circular cylindrical structures. Washington: Hemisphere 1987 3. Landau, L. D.; Lifshitz, E. M. : Theory of elasticity. Oxford: Pergamon 1959 4. Handelman, G. H.: A note on the transverse vibration of a tube containing flowing fluid. Quart. Appl.

Math. 13 (1956) 326--330 5. Markus, A. S.: Introduction to spectra] theory of operator pencils (in Russian). Kishinev: Stiintsa 1986

(Engl. Transl.: AMS Translations of Mathematical Monographs, vol. 71, 1988) 6. Baumggrtel, H.: Analytic perturbation theory for matrices and operators. Basel: Birkh~iuser 1985 7. Keita, N. M.: Contribution to the understanding of zero shift effects in Coriolis mass flowmeters. Flow

Measurement and Instrumentation 1 (1989) 39--43 8. Niordson, F. I. N. : Vibrations of a cylindrical tube containing a flowing fluid. Trans. Royal Inst. Techn.

No. 73 (1953) 1--27

Received February 20, 1990

Dr. H. Raszillier Prof. Dr. F. Durst Lehrstuhl fiir Str6mungsmechanik Universit~t Erlangen-- Nfirnberg Cauerstr. 4 W-8520 Erlangen (FRG) Federal Republic of Germany

coriolis-effect in mass flow metering

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